Technical Report: 026511-2-T MODELING MICROWAVE BACKSCATTER FROM DISCONTINUOUS TREE CANOPIES Kyle C. JJ1cDonald Fawwaz T. Ulaby JPL Contract C-958437 (NASA Prime) Principal Investigator: Fawwaz T. Ulaby June, 1991

ABSTRACT Forest ecosystems represent a significant portion of Earth's vegetation cover. While playing an important role in the global carbon cycle, the areal extent of forests, the rate of global deforestation and the amount of forest biomass remain key unknown parameters in understanding atmospheric carbon dioxide flux from these biomes. Spaceborne microwave imaging systems have been proposed as a means of assessing biophysical parameters of vegetation canopies. Such systems can monitor regions of the globe where environmental conditions render optical techniques ineffective. Scattering models that accurately simulate forest canopy backscatter allow measured radar data to be coupled to canopy parameters and significantly aid in applying remotely sensed data to understanding canopy physiological state. The goal of this work is to develop a robust microwave scattering model for forested areas. A tree canopy is characterized as an inhomogeneous medium comprised of discrete scatterers that represent the trunks, branches, stems, needles and leaves. Radiative transfer theory is applied to derive a first-order fullypolarimetric solution for backscatter. The fundamental contribution of this thesis is the development of a model that accounts for backscatter from forest canopies that have discontinuous crown layer geometries. By treating parameters describing the size, shape and location of individual tree crowns as random variables, a statistical approach is taken that defines the expected value of canopy backscatter. Application of the radiative transfer equations to the discontinuous canopy geometry is reviewed. The application of random variables defining the crown geometry and the incorporation of these variables into the radiative transfer solution is discussed. The resulting model is valid for microwave frequencies over a wide range of radar incidence angles. Model simulations are compared to results derived with the ii

continuous canopy model. The effect of the open crown geometry is found to be most significant at shallow incidence angles and at high frequencies for trees with well-developed crowns. The model successfully couples canopy biophysical parameters to radar backscatter measurements. When compared to measured radar data, variations in backscatter that occur because of changing environmental conditions that cause changes in canopy water status are accurately predicted. iii

TABLE OF CONTENTS ABSTRACT....................................................................................ii LIST O F FIG U R ES.............................................................................vii LIST OF TABLES..............................................................................xxi LIST O F APPENDICES.....................................................................xxiv CHAPTER I. IN TR O D U C TIO N..................................................................1 II. B A C K G R O U N D.......................................................... 2.1 Tree Canopy Param eters......................................................5 2.2 Radiative Transfer Theory....................................................9 2.3 Radar Polarim etry............................................................ 15 III. RADAR BACKSCATTER MODEL FOR A CLOSED-CROWN TR EE C A N O PY....................................................................20 3.1 MIMICS I Solution for the Radiative Transfer Equations...............21 3.2 Modeling Ground Surface Cover..........................................40 3.2.1 Ground Surface Covered by a Foliar Understory.............41 3.2.2 Ground Surface Covered by a Snow Layer.....................44 3.3 Applicability of MIMICS I................................................47 IV. RADAR BACKSCATTER MODEL FOR AN OPEN-CROWN CANOPY - MIMICS II.........................................................50 4.1 Radiative Transfer Solution for an Open-Crown Canopy.............. 53 4.2 Application of Canopy-level Statistical Parameters......................57 4.2.1 Crown Layer Transmissivity....................................57 iv

4.2.2 Crown Layer Phase Matrix............. 61 4.2.3 Effective Scatterer Number Density........ 62 4.3 Polarimetric Solution...................... 63 4.4 Scalar Solution.......................... 74 V. SHAPE STATISTICS FOR TREE CROWNS......... 76 5.1 Calculation of Within-Crown Propagation Length PDF for Different Crown Shapes..................... 76 5.1.1 Spheroid.....................77 5.1.2 Square Column...................81 5.1.3 Cone........................ 86 5.1.4 Partial Crown Shapes............... 88 5.1.5 General Shapes..................92 5.2 Within-Crown Propagation Length PDF for Crowns Distributed in Size and Location....................... 94 5.3 Calculation of Crown Cross-Sectional Area.......... 102 5.3.1 Cross-Sectional Area of Selected Crown Shapes with Specified Size and Location............ 103 5.3.2 Cross-Sectional Area for Crowns Distributed in Height 104 5.3.3 Cross-Sectional Area for Crowns Distributed in Size 108 5.3.4 Cross-Sectional Area for Crowns Distributed in Height and Size....................... 112 5.4 Summary of Crown Shape Statistics............. 115 VI. MODELING ANALYSES AND APPLICATIONS...... 116 6.1 MIMICS I Corn Canopy Modeling Using SIR-B Data..... 116 6.2 Eos Synergism Study...................... 126 6.2.1 Orchard Canopy Characteristics...........126 6.2.2 Modeling Analysis................... 134 6.3 ERS-1 Alaskan Boreal Forest Study............ 142 6.3.1 Test Site Description and Canopy Properties.... 143 6.3.2 Boreal Forest Transmissivity Analysis...... 148 6.3.3 Boreal Forest Backscatter Analysis.......... 154 6.3.4 Boreal Forest Multi-Season Simulation....... 167 6.4 MIMICS II Simulations of Open-Crown Canopies....... 170 6.4.1 Black Spruce Simulation............... 171 6.4.2 Coniferous Canopy Simulation............ 186 6.4.3 Deciduous Canopy Simulation............ 195 6.4.4 Summary of MIMICS II Results.......... 203 VII. CONCLUSIONS AND RECOMMENDATIONS........ 205 7.1 Sum m ary.......................... 205 v

7.2 Recommendations for Future Work............... 207 APPENDICES..................................209 BIBLIOGRAPHY................................ 344 vi

LIST OF FIGURES Figure 2.1 Geometry of a tree canopy with (a) a continuous (closed) crown layer and (b) a discontinuous (open) crown layer............... 6 2.2 Polarization ellipse illustrating the polarization state of an electromagnetic wave traveling in the k direction (out of the page). The angles, and b are measured clockwise................. 16 2.3 Polarization response of a short, thin conducting cylinder oriented vertically................................ 19 3.1 Forest Canopy Model........................... 21 3.2 Problem Geometry showing the positive- and negative-going intensities in the crown and trunk layers................... 23 3.3 First-order contributions to bistatic scatter.............. 32 3.4 First-order contributions to canopy backscatter............ 35 3.5 Contributions to canopy backscatter from the canopy understory.. 41 3.6 Contributions to canopy backscatter from an underlying snow layer. 44 4.1 First-order backscatter terms for a discontinuous canopy....... 51 4.2 Model geometry for a canopy with a discontinuous crown layer... 54 4.3 Illustration showing one individual tree crown, the within-crown propagation length s and the projected shadow area on the ground A, for a view angle 0......................... 58 4.4 Canopy illumination geometry showing ovelapping crowns in th crown layer and the corresponding overlapping shadows on the ground. The number density of scatterers in the region where the two crowns overlap is twice that within one individual crown volume......... 59 vii

4.5 Cross-sectional area Ac at depth z of a crown centered at zi.... 63 5.1 Geometry of a sphere of radius R showing a within-crown path length s at a distance r from the center of the sphere............. 78 5.2 Transformation of a sphere to a prolate spheroid........... 79 5.3 Illustration of the within-crown propagation length for a crown with a rectangular shape factor. Case I occurs for a radar look angle 0 > Oc. Case II occurs for a radar look angle 0 < 0c. The critical angle 0c is determined from tan 0c = a/c. Sm represents the maximum value of s for a given crown. The solids have depth b into the paper.... 82 5.4 Geometry used for deriving p(sll) for a square column....... 83 5.5 p(sll) for a square column with a = b = c = 1. Values are shown for three incidence angles.......................... 86 5.6 Geometry of a conical crown with height c, basal diameter a and apex angle 'c............................... 87 5.7 p(sll) for several incidence angles for a right circular cone with a = b = 0.5 and c = 1............................. 87 5.8 Geometry of spheroid crowns with propagation depth Izl < c.. 88 5.9 p(sll) of a spheroid with a = b = 2 and c = 10 for several penetration depths. The incidence angle 0 = 30.................. 89 5.10 p(sll) for several incidence angles for an spheroid with propagation depth z = -c/2. The ellipsoid has a = b = 2 and c = 10....... 90 5.11 Geometry for computing p(sll) of a frustum of a right circular cone. 91 5.12 p(sll) of a cone frustum for several heights at an incidence angle 0 = 30~. The cone has a= b = 0.5 and c = 1.............. 92 5.13 Geometry of a mixed spheroid. The upper spheroid has a = b = 2 and c = 10. The lower spheroid has a = b = 2 and c = 4....... 93 5.14 p(sll) of a mixed spheroid shape at several incidence angles. 93 viii * * Vlll

5.15 Illustration of ellipsoidal crowns in a crown layer of thickness d. the crowns are distributed in both height c and center location zi. z' represents the depth in the layer at which the value of the propagating intensity is to be estimated....................... 94 5.16 p(s) at various depths z' in the crown layer for a spherical crowns with centers located at zi = -4 meters and size uniformly distributed between 4 < c < 8 with a/c= 1 and 0 = 30~.............. 95 5.17 p(s) at various depths z' in the crown layer for conical crowns with centers located at zi = -4 meters and size uniformly distributed between 4 < c < 8 with a/c = 0.5 and 0 = 30~............. 96 5.18 p(s) at various depths z' in the crown layer for a spherical crown with size c = 6 meters, a/c = 1, center location uniformly distributed between -7 < zi < -3 and 0 = 30~................... 97 5.19 p(s) at various depths z' in the crown layer for a conical crown with size c = 6 meters, a/c = 0.5, center location uniformly distributed between -7 < z, < -3 and 0 = 30~.................. 97 5.20 p(s) at various depths z' in the crown layer for a spherical crown with crown center location uniformly distributed between -8 < zi < -3 and size varying linearly with zi between 1 < c < 6 and a/c = 1 such that the smaller crowns are low in the canopy and the larger crowns are high in the canopy. The incidence angle 0 = 30~.... 98 5.21 p(s) at various depths z' in the crown layer for a conical crown with crown center location uniformly distributed between -8 < zi < -3 and size varying linearly with zi between 1 < c < 6 and a/c = 0.5 such that the smaller crowns are low in the canopy and the larger crowns are high in the canopy. The incidence angle 0 = 30~.... 99 5.22 PDF of crown center height p(zi) for lognormal and uniform distributions.................................. 100 5.23 p(s) at z' = -10 meters in the crown layer for a spherical crown with center location having three different distributions between -8 < zi < -3 and size varying linearly with zi between 1 < c < 6 and a = c such that the smaller crowns are low in the canopy and the larger crowns are high in the canopy. The incidence angle 0 = 30~.. 101 ix

5.24 p(s) at z' = -10 meters in the crown layer for a conical crown with center location having three different distributions between -8 < zi < -3 and size varying linearly with zi between 1 < c < 6 and a = 0.5c such that the smaller crowns are low in the canopy and the larger crowns are high in the canopy. The incidence angle 0 = 30~.. 101 5.25 Cross-sectional area Ac at depth z of a crown centered at zi..... 102 5.26 Expected value of crown cross-sectional area at depth z in a crown layer for a single square column crown with center uniformly distributed over z1 < zi _< 2 with a = c 2.............. 107 5.27 Expected value of crown cross-sectional area at depth z in a crown layer for a single spheroidal crown with center uniformly distributed over z< Zi _< 2 with a = c = 2.................... 107 5.28 Expected value of crown cross-sectional area at depth z in a crown layer for a single conical crown with center uniformly distributed over z1 < zi < z2 with a =c = 2.................... 108 5.29 Expected value of crown cross-sectional area at depth z in a crown layer for a single square column crown with c uniformly distributed between cl and c2, a = c and zi = -2.................. 110 5.30 Expected value of crown cross-sectional area at depth z in a crown layer for a single spherical crown with c uniformly distributed between cl and c2, a = c and zi =-2................... 111 5.31 Expected value of crown cross-sectional area at depth z in a crown layer for a single conical crown with c uniformly distributed between cl and c2, a = c and zi = -2....................... 111 5.32 Expected value of crown cross-sectional area at depth z in a crown layer for a single square column crown with center uniformly distributed between z1 and z2 and c uniformly distributed between 1 and 3 meters with a = c........................ 113 5.33 Expected value of crown cross-sectional area at depth z in a crown layer for a single spherical crown with center uniformly distributed between z1 and z2 and c uniformly distributed between 1 and 3 meters with a = c............................. 113 x

5.34 Expected value of crown cross-sectional area at depth z in a crown layer for a single conical crown with center uniformly distributed between z1 and Z2 and c uniformly distributed between 1 and 3 meters with a = c................................ 114 6.1 Comparison of the polarization phase difference calculated by MIMICS to that extracted from the aircraft SAR imagery......... 119 6.2 L-band c~ HH-polarized backscatter response to changes in volumetric soil moisture. Data measured by the SIR-B SAR are compared with MIMICS simulations for canopies with (a) dry stalks with gravimetric moisture = 0.125, (b) wet stalks with gravimetric moisture = 0.9, (c) stalks with gravimetric moisture coupled to the soil moisture via mg = 1.S mv- 0.12........................ 120 6.3 Co-polarized L-band response of a corn canopy at 0 = 30~ for (a) wet conditions with soil moisture = 0.3 and stalk moisture = 0.6 and (b) dry conditions with soil moisture = 0.08 and stalk moisture = 0.35................................... 123 6.4 Corn canopy backscatter response to changes in volumetric soil moisture and gravimetric stalk moisture for an incidence angle 0 = 20~. (a) HH-polarized response, (b) VV-polarized response, (c) polarization phase difference response..................... 124 6.5 Solution set for estimation of canopy moisture conditions at 0 = 20~ for (a) wet conditions with soil moisture = 0.3 and stalk moisture = 0.6 and (b) dry conditions with soil moisture = 0.08 and stalk m oisture = 0.35............................. 125 6.6 Illustration of a walnut tree showing the four branch classes and the leaves............................ 127 6.7 Comparison of a periodic piecewise fit to measured L-band trunk dielectric constant data for real and imaginary parts......... 130 6.8 Behavior of the soil dielectric constant showing the estimated behavior of the L- and X-band dielectric constant. The symbol (i) indicates the beginning of a, 2.5 hour irrigation period......... 131 6.9 Dielectric constants of woody constituents for (a) L-band and (b) X -band.................................. 133 xi

6.10 Comparison of MIMICS results with measured L- and X-band multiangle data. (a) compares L-band modeled total canopy backscatter to the scatterometer measurements for like- and cross-polarized configurations (HH, VV, HV). (b) compares X-band modeled direct crown backscatter to the scatterometer measurements for these same polarizations.............................. 135 6.11 Comparison of MIMICS results with measured backscatter recorded during the three day diurnal experiment for (a) HH polarized L-band backscatter, (b) VV polarized L-band backscatter, (c) HV polarized L-band backscatter, (d) HH polarized X-band backscatter, (e) VV polarized X-band backscatter and (f) HV polarized X-band backscatter. The X-band HV MIMICS data has been offset 8 dB to account for multiple scatter............................ 139 6.12 Walnut orchard backscatter response to changes in canopy biomass for (a) VV-polarization, (b) VH-polarization. The incidence angle 0 = 30~.................................. 141 6.13 Transmission loss for one-way propagation through the alder canopy. Measurements are shown for four trihedral targets at (a) C-band and (b) X -band................................ 150 6.14 Transmission loss for one-way propagation through the mixed balsam poplar/alder canopy. Measurements are shown for seven trihedral targets at (a) C-band and (b) X-band................. 151 6.15 Transmission loss for one-way propagation through the mixed white spruce/balsam poplar/alder canopy. Measurements are shown for nine trihedral targets at (a) C-band and (b) X-band......... 152 6.16 Comparison of MIMICS simulated and measured transmission loss for one-way propagation through the alder, balsam poplar and white spruce canopies at C-band. The best-fit straight lines are shown for each canopy, together with their respective correlation coefficients p. 153 6.17 Comparison of measured canopy backscatter to MIMICS simulated backscatter for (a) HH-polarization and (b) VV-polarization.... 156 6.18 MIMICS-simulated L-band polarization response of frozen white spruce stand W S-5................................ 158 6.19 MIMICS-simulated L-band polarization response of thawed white spruce stand WS-5............................ 159 xii

6.20 Measured L-band polarization response of frozen white spruce stand WS-5................................... 160 6.21 Measured L-band polarization response of thawed white spruce stand WS-5.................................. 161 6.22 MIMICS-simulated L-band linear polarization response of thawed white spruce stand WS-5........................ 162 6.23 Comparison of measured canopy backscatter to MIMICS simulated backscatter for (a) C-band and (b) X-band. The data have been normalized to the backscatter from white spruce stand WS-1 for each SAR pass....................... 163 6.24 Total canopy backscatter for black spruce stand (BS-1) at L-band under frozen canopy conditions for (a) ground layer consisting of a snow half-space and (b) ground layer consisting of a snow layer on top of a soil half-space..................... 165 6.25 Total canopy backscatter for black spruce stand (BS-1) at L-band under thawed canopy conditions for (a) ground layer consisting of a snow half-space and (b) ground layer consisting of a snow layer on top of a soil half-space....................... 166 6.26 MIMICS simulated canopy backscatter response to environmental state for ERS-1 parameters (C-band, VV-polarization, 0 = 23~)... 169 6.27 Canopy geometry used in MIMICS II simulation of black spruce stand BS-1............................... 172 6.28 Comparison of net canopy backscatter from a black spruce canopy modeled with a continuous crown layer (MIMICS I) and a discontinuous crown layer (MIMICS II) at L- C- and X-bands for (a) VV, (b) HH, and (c) HV polarizations................... 175 6.29 Comparison of crown layer transmissivity through a black spruce canopy modeled with a continuous crown layer (MIMICS I) and a discontinuous crown layer (MIMICS II) at L- C- and X-bands for (a) V-polarization and (b) H-polarization.................. 176 6.30 Comparison of L-band VV-polarized backscatter from a black spruce canopy modeled with (a) a continuous crown layer (MIMICS I) and (b) a discontinuous crown layer (MIMICS II)............. 178 xiii

6.31 Comparison of L-band HV-polarized backscatter from a black spruce canopy modeled with (a) a continuous crown layer (MIMICS I) and (b) a discontinuous crown layer (MIMICS II)............. 179 6.32 Comparison of net canopy backscatter from a black spruce canopy modeled with a continuous crown layer (MIMICS I) and a discontinuous crown layer (MIMICS II) at VV, HH and HV polarizations for incidence angles of (a) 20~ and (b) 60~ as a function of the density multiplication factor........................... 180 6.33 Comparison of one-way transmissivity through a black spruce canopy modeled with a continuous crown layer (MIMICS I) and a discontinuous crown layer (MIMICS II) for V and H polarizations for incidence angles of (a) 20~ and (b) 60~ as a function of the density multiplication factor........................... 181 6.34 VV-polarized canopy backscatter as a function of crown side length with total crown layer biomass held constant. Results are for L, C and X bands at an incidence angle of 0 = 20~. Results for the equivalent closed crown canopy (MIMICS I) are also shown..... 183 6.35 L-band backscatter from a black spruce canopy for VV, HH and HV polarizations as a function of canopy cover fraction.......... 185 6.36 L-band VV -polarized backscatter from a black spruce canopy at an incidence angle of 0 = 20~ as a function of canopy cover fraction... 185 6.37 One-way V-polarized transmissivity through a white spruce crown layer modeled as continuous (MIMICS I) and discontinuous (MIMICS II) for (a) thawed conditions and (b) frozen conditions..... 189 6.38 Backscatter from a thawed white spruce canopy modeled with a closed crown layer (MIMICS I) and with an open crown layer (MIMICS II) at (a) VV, (b) HH and (c) HV-polarization.......... 190 6.39 Backscatter from a frozen white spruce canopy modeled with a closed crown layer (MIMICS I) and with an open crown layer (MIMICS II) at (a) VV, (b) HH and (c) HV-polarization.............. 191 6.40 Contributions to net canopy backscatter at X-band from a thawed white spruce canopy for (a) a closed-crown canopy (MIMICS I) and (b) an open-crown canopy (MIMICS II)............... 192 xiv

6.41 Contributions to net canopy backscatter at X-band from a frozen white spruce canopy for (a) a closed-crown canopy (MIMICS I) and (b) an open-crown canopy (MIMICS II)................ 193 6.42 Response of L-band like-polarized canopy backscatter to changes in volumetric soil moisture for a closed-crown canopy (MIMICS I) and an open-crown canopy (MIMICS II) for a canopy with (a) dry trunks and (b) wet trunks. Simulations are for an incidence angle of 0 = 20~.194 6.43 Comparison of MIMICS I and MIMICS II simulations of one-way transmissivity through a deciduous canopy for (a) L-band and (b) X-band........................... 198 6.44 Comparison of MIMICS I and MIMICS II simulations of like-polarized X-band backscatter from a deciduous canopy............. 199 6.45 Comparison of contributions to net backscatter for (a) MIMICS I and (b) MIMICS II simulations of VV-polarized X-band backscatter from a deciduous canopy......................... 200 6.46 Comparison of contributions to net backscatter for (a) MIMICS I and (b) MIMICS II simulations of HH-polarized X-band backscatter from a deciduous canopy......................... 201 6.47 MIMICS II like-polarized X-band backscatter sensitivity to changes in crown diameter. Simulations are shown for incidence angles of 20~ and 60~. The MIMICS I closed-crown canopy simulation is also shown................................ 202 E.1 Terms contributing to direct crown backscatter in the second-order solution............................... 239 G.1 Side view (a) and top view (b) of the walnut orchard showing the scatterometer measurement geometry. For a scatterometer beamwidth /, the sensing volume V is defined at a given slant range R by the pulse width and the scanning arc angle a.............. 251 G.2 Uncalibrated L- and X-band canopy backscatter versus slant range at 0 = 55~. The foliage fraction was computed for the estimated X-band sensing volume and scaled to fit on the dB axis....... 252 G.3 Illustration of a walnut tree showing the four branch classes and the leaves.................................. 255 G.4 Branch orientation probability distribution functions (PDFs).... 256 XV

G.5 Geometry used to model an a x a leaf folded along its midrib. The folding angle X shown in (a) defines the distance s between the opposite edges of the leaf. Chord length s and arc length a define the sector of a circle with radius p shown in (b).............. 258 G.6 Backscatter from a curved leaf, ac, normalized to af, the backscatter from a flat leaf of equal area. Backscatter is shown for a leaf curved to fit a cylindrical surface with radius of curvature pi = 7.7 cm and for a leaf curved to fit an ellipsoidal surface with pi = 7.7 cm and P2 = 10 cm........................... 260 G.7 Comparison of a periodic piecewise fit to measured L-band trunk dielectric constant data for (a) two insertion depths and (b) real and imaginary parts............................. 264 G.8 Behavior of the soil dielectric constant showing (a) the fits to the measured L-band dielectric constants of the irrigated and non-irrigated areas and (b) the estimated behavior of the L- and X-band dielectric constant for the combination of irrigated and non-irrigated areas. (i) indicates the beginning of a 2.5 hour irrigation period........ 266 G.9 Comparison of measured leaf water potential to piecewise fit.... 269 G.10 Dielectric constants of woody constituents for (a) L-band and (b) X-band................................ 271 G.11 Comparison of MIMICS results with measured L- and X-band multiangle data. (a) compares L-band modeled total canopy backscatter to the scatterometer measurements for like- and cross-polarized configurations (HH, VV, HV). (b) compares X-band modeled direct crown backscatter to the scatterometer measurements for these same polarizations.;.............................. 274 G.12 Components of canopy backscatter for HH, VV and HV polarizations.277 G.13 Comparison of MIMICS results with measured backscatter recorded during the three day diurnal experiment for (a) HH polarized L-band backscatter, (b) VV polarized L-band backscatter, (c) HV polarized L-band backscatter, (d) HH polarized X-band backscatter, (e) VV polarized X-band backscatter and (f) HV polarized X-band backscatter. The X-band HV MIMICS data has been offset 8 dB to account for multiple scatter............................ 279 xvi

G.14 L-band MIMICS response to changes in trunk and primary branch dielectric constant............................ 281 G.15 L-band MIMICS response to changes in soil and trunk dielectric constant.................................. 282 G.16 X-band HH-polarized direct crown backscatter response to changes in (a) leaf gravimetric moisture and leaf area index, (b) primary and secondary branch gravimetric moisture and (c) primary and secondary branch gravimetric moisture with primary branches assigned a sin 0 orientation function....................... 285 G.17 Walnut orchard backscatter response to changes in canopy biomass for (a) VV-polarization, (b) VH-polarization. The incidence angle 0 = 30~.................................. 287 H.1 Transmission loss for one-way propagation through the alder canopy. Measurements are shown for four trihedral targets at (a) C-band and (b) X-band............................ 303 H.2 Transmission loss for one-way propagation through the mixed balsam poplar-alder canopy. Measurements are shown for seven trihedral targets at (a) C-band and (b) X-band.................. 304 H.3 Transmission loss for one-way propagation through the mixed white spruce-balsam poplar-alder canopy. Measurements are shown for nine trihedral targets at (a) C-band and (b) X-band......... 305 H.4 Comparison of MIMICS simulated and measured transmission loss for one-way propagation through the alder, balsam poplar and white spruce canopies at C-band. The best-fit straight lines are shown for each canopy, together with their respective correlation coefficients p. 307 H.5 Comparison of MIMICS simulated one-way canopy transmissivity for a mature white spruce stand (WS-2) at C-band to the average measured transmissivity of the mixed-species white spruce stands. Error bars are based on the mean value ~ one standard deviation.. 308 H.6 Comparison of MIMICS simulated one-way canopy transmissivity for a mature white spruce stand (WS-2) at X-band to the average measured transmissivity of the mixed-species white spruce stands. Error bars are based on the mean value ~ one standard deviation.. 309 xvii

H.7 MIMICS simulated one-way canopy transmissivity for a mature white spruce stand (WS-2) at L-, C- and X-bands for (a) frozen canopy conditions and (b) thawed canopy conditions............. 311 H.8 MIMICS simulated one-way canopy transmissivity for a black spruce stand (BS-1) at L-, C- and X-bands for (a) frozen canopy conditions and (b) thawed canopy conditions................... 312 H.9 Comparison of measured canopy backscatter to MIMICS simulated backscatter for (a) HH-polarization and (b) VV-polarization..... 315 H.10 MIMICS-simulated L-band polarization response of frozen white spruce stand W S-5................................ 318 H.1 1 MIMICS-simulated L-band polarization response of thawed white spruce stand WS-5............................ 319 H.12 Measured L-band polarization response of frozen white spruce stand WS-5................................... 320 H.13 Measured L-band polarization response of thawed white spruce stand WS-5.................................. 321 H.14 MIMICS-simulated L-band linear polarization response of thawed white spruce stand WS-5......................... 322 H.15 Comparison of measured canopy backscatter to MIMICS simulated ba.ckscatter for (a) C-band and (b) X-band. The data have been normalized to the backscatter from white spruce stand WS-1 for each SAR pass...............................323 H.16 MIMICS simulated canopy backscatter for a white spruce stand (WS-5) at L-, C- and X-bands under frozen canopy conditions for (a) HH-polarization, (b) VV-polarization and (c) VH-polarization.. 325 H.17 MIMICS simulated canopy backscatter for a white spruce stand (WS-5) at L-, C- and X-bands under thawed canopy conditions for (a) HH-polarization, (b) VV-polarization and (c) VH-polarization.. 326 H.18 Canopy backscatter components for white spruce stand (WS-5) at L-band under frozen canopy conditions for (a) HH-polarization, (b) VV-polarization and (c) VH-polarization................ 328 xviii

H.19 Canopy backscatter components for white spruce stand (WS-5) at L-band under thawed canopy conditions for (a) HH-polarization, (b) VV-polarization and (c) VH-polarization................ 329 H.20 Canopy backscatter components for white spruce stand (WS-5) at C-band under frozen canopy conditions for (a) HH-polarization, (b) VV-polarization and (c) VH-polarization................ 330 H.21 Canopy backscatter components for white spruce stand (WS-5) at C-band under thawed canopy conditions for (a) HH-polarization, (b) VV-polarization and (c) VH-polarization................ 331 H.22 Total canopy backscatter for frozen and thawed white spruce stands at L-band for (a) HH-polarization, (b) VV-polarization and (c) VHpolarization................................. 332 H.23 Total canopy backscatter for frozen and thawed white spruce stands at C-band for (a) HH-polarization, (b) VV-polarization and (c) VHpolarization................................. 333 H.24 MIMICS simulated canopy backscatter for a white spruce stand (WS-5) at L-, C- and X-bands under frozen canopy conditions for (a) HH-polarization, (b) VV-polarization and (c) VH-polarization.. 335 H.25 MIMICS simulated canopy backscatter for a black spruce stand (BS1) at L-, C- and X-bands under thawed canopy conditions for (a) HH-polarization, (b) VV-polarization and (c) VH-polarization.... 336 H.26 Canopy backscatter components for black spruce stand (BS-1) at L-band under frozen canopy conditions for (a) HH-polarization, (b) VV-polarization and (c) VH-polarization............... 337 H.27 Canopy backscatter components for black spruce stand (BS-1) at Lband under thawed canopy conditions for (a) HH-polarization, (b) VV-polarization and (c) VH-polarization............... 338 H.28 Canopy backscatter components for black spruce stand (WS-5) at C-band under frozen canopy conditions for (a) HH-polarization, (b) VV-polarization and (c) VH-polarization................ 339 H.29 Canopy backscatter components for white spruce stand (WS-5) at C-band under thawed canopy conditions for (a) HH-polarization, (b) VV-polarization and (c) VH-polarization................ 340 xix

H.30 Total canopy backscatter for black spruce stand (BS-1) at L-band under frozen canopy conditions for (a) ground layer consisting of a snow half-space and (b) ground layer consisting of a snow layer on top of a soil half-space.......................... 342 H.31 Total canopy backscatter for black spruce stand (BS-1) at L-band under thawed canopy conditions for (a) ground layer consisting of a snow half-space and (b) ground layer consisting of a snow layer on top of a soil half-space.......................... 343 XX

LIST OF TABLES Table 2.1 Canopy dielectric parameters....................... 7 2.2 Canopy geometric parameters..................... 7 2.3 Values of Shape Factor Constants for Simple Crown Shapes.... 9 3.1 Terms contributing to canopy backscatter............... 36 5.1 Integration limits for computing (Ac (z, t)).............. 105 5.2 Cross sectional area for crowns with centers uniformly distributed between zl and Z2............................ 106 5.3 Integration limits for computing (Ac (z, z)).............. 109 5.4 Cross sectional area for crowns with height uniformly distributed such that cl < c < C2 and a = c..................... 110 6.1 Corn canopy parameters for fields observed by the aircraft SAR. 118 6.2 Corn canopy parameters for fields observed by the SIR-B SAR.... 119 6.3 Canopy Branch Classes......................... 128 6.4 Leaf Characteristics........................... 128 6.5 Effects of Leaf Curvature at L- and X-Bands............. 129 6.6 Canopy Dielectric Characteristics................... 134 6.7 Modeled Dielectric Characteristics of Snow for Frozen and Thawed Conditions............................... 143 6.8 Summary of Stand Biophysical Parameters.............. 144 xxi

6.9 Summary of Mean DBH, Height and Basal Area for All Stands.. 145 6.10 Equations Defining Height-to-DBH Relationship......... 145 6.11 Geometry of Crown Layer Constituents................ 6.12 Number Density of Canopy Constituents........ 6.13 Relative Dielectric Constant for Tree Constituents.......... 6.14 Comparison of MIMICS Estimates to Measured L-band SAR Data (dB ).................. 6.15 Specified Environmental and Phenologic Conditions for MIMICS Simulation.......................... 6.16 Ground Surface Roughness Parameters for the MIMICS II Simulations. 146 147 147 155 168 171 6.17 Tree level parameters for black spruce stand BS-1... * o 6.18 6.19 6.20 6.21 6.22 G.1 G.2 G.3 G.4 Black Spruce Canopy Dielectric Characteristics..... Tree level parameters for'the coniferous stand...... Coniferous Canopy Dielectric Characteristics....... Tree-level parameters for the open-crown deciduous stand. Canopy Dielectric Characteristics....... Canopy Branch Classes.................. Leaf Characteristics.................. Effects of Leaf Curvature at L- and X-Bands....... Canopy Dielectric Characteristics................. 173...... 173...... 186...... 187..... 195...... 196...... 257...... 258...... 261...... 272 H.1 Modeled Dielectric Characteristics of Snow for Frozen and Thawed Conditions............................... H.2 Summary of Stand Biophysical Parameters.............. H.3 Summary of Mean DBH, Height and Basal Area for All Stands... 292 294 295 xxii

H.4 Equations Defining Height-to DBH Relationship........... 295 H.5 Dry Biomass Fractions of Canopy Components as Percent of Total. 296 H.6 Geometry of Crown Layer Constituents................ 297 H.7 Number Density of Canopy Constituents................ 297 H.8 Relative Dielectric Constant for Tree Constituents.......... 298 H.9 Stand Characteristics in the Neighborhood of the Trihedral Reflectors.300 H.10 Comparison of MIMICS Estimates to Measured L-band SAR Data (dB).................................... 314 H.11 Comparison of Measured and Modeled Polarization Ratios at L-band.316 H.12 Comparison of the Effects of Freeze/Thaw State on L-band Backscatter.................................... 317 xxiii

LIST OF APPENDICES Appendix A. DIELECTRIC MODELS FOR CANOPY CONSTITUENTS..... 210 A.1 Dielectric Behavior of Vegetation......... A.1.1 Model in Terms of Volumetric Moisture A.1.2 Model for Leaves........... A.2 Dielectric Behavior of the Ground Surface.... A.2.1 Soil................ A.2.2 Standing Water............. A.2.3 Snow Layer...................... 210....... 210....... 212....... 212....... 212....... 214....... 215 B. SCATTERING MODELS FOR ROUGH SURFACES.... B.1 Geometrical Optics Model............. B.2 Physical Optics Model.................. B.3 Small Perturbation Model................ C. SCATTERING MODELS FOR TRUNKS AND BRANCHES..... 217... 218.... 219.... 221.... 224 C.1 Scattering Matrix for Prolate Spheroids........... 224 C.2 Scattering Matrix for Long Thin Cylinders.......... 226 C.3 Scattering Matrix for Large Cylinders............. 227 D. SCATTERING MODELS FOR LEAVES................ 230 D.1 Scattering Matrix for Oblate Spheroids........... 230 D.2 Physical Optics Model...................... 232 E. SECOND ORDER SCATTERING IN THE CROWN LAYER.... 235 F. CONNECTING MODELS FOR CANOPY BIOPHYSICAL PARAMETERS....................................241 F.1 Elemental Volume Fractions................. 241 F.2 Leaf Area Index......................... 242 xxiv

F.3 Biomass Parameters....................... F.3.1 General Definitions.................. F.3.2 Constituent Biomasses and Water Contents..... G. MODELING MULTIANGLE AND MULTITEMPORAL BACKSCATTER FROM A WALNUT ORCHARD - THE EOS SYNERGISM ST U D Y................................... 243 243 244 246 G.1 Study Objectives and Test Site Description.......... 247 G.2 Scatterometer Measurement Procedure............. 248 G.2.1 System Description and Operation.......... 248 G.2.2 Calibration....................... 253 G.3 Orchard Canopy Characteristics................ 254 G.3.1 Canopy Architecture................. 254 G.3.2 Canopy Dielectric Characteristics.......... 262 G.4 Modeling Analysis........................ 272 H. MODELING MICROWAVE ATTENUATION AND BACKSCATTER FROM ALASKAN BOREAL FOREST CANOPIES. 289 H.l Test Site Description and Canopy Properties... H.I.1 Stand Selection................ H.1.2 Temperature Conditions........... H.1.3 Ground Surface Characteristics........ H.1.4 Stand Geometry................ H.1.5 Stand Dielectric Characteristics....... H.2 Boreal Forest Transmissivity Analysis......... H.3 Boreal Forest Backscatter Analysis........... H.3.1 Comparison with Measured Data...... H.3.2 White Spruce Simulations.......... H.3.3 Black Spruce Simulations............... 290.... 291.... 291.... 292.... 293.... 298.... 298.... 310.... 314.... 324.... 334 XXV

CHAPTER I INTRODUCTION Microwave radar has been used in remote sensing applications since the early 1960s and is becoming a major tool for observing Earth's biosphere (Ulaby et al.,[70]; Ulaby and Elachi, [62]). By exploiting their ability to provide an independent source of illumination and their capability to penetrate clouds, space-borne microwave imaging systems can provide long and short term terrain monitoring in areas of the globe where environmental conditions render optical techniques ineffective. Microwave remote sensing has proven useful in studying properties of the ocean surface, polar ice, land cover, and in some cases subsurface terrain features. Another advantageous property of microwaves is their ability to penetrate more deeply into vegetation canopies than optical waves. The extent of this penetration depends on several factors, including canopy moisture content, radar frequency and radar incidence angle. In general, many canopy characteristics strongly influence the radar signature. Along with the moisture content of the vegetation itself, these include such properties as soil moisture content, total vegetation biomass, physical location and structure of the vegetation constituents, ground surface state (bare soil, snow cover, vegetation cover, flooded) and canopy phenological and biophysical state. Forest ecosystems represent a significant portion of Earth's vegetation cover. 1

2 While playing an important role in the global carbon cycle, the areal extent of forests, the rate of global deforestation and the amount of biomass in existing forests remain key unknown parameters in understanding atmospheric carbon dioxide flux from these biomes. Because of their aforementioned capabilities, microwave imaging systems have been proposed as a means of assessing these parameters. The development of scattering models that accurately simulate backscatter from such canopies allows measured data to be mathematically coupled to canopy parameters and significantly aids in understanding the physiological state of the canopy. The goal of this work is to develop a robust microwave scattering model for forested areas. A tree canopy may be characterized as an inhomogeneous medium comprised of discrete scatterers that represent the trunks, branches, stems, needles and leaves. Most models for radar scattering from vegetation treat the canopy as a uniform layer of some specified height containing a. random distribution of scatterers (Attema and Ulaby,[2]; Eom and Fung,[19]; Fung and Ulaby, [22]; Karam and Fung, [31]; Lang and Sidhu, [35]; Richards et al., [48]; Tsang and Kong, [59]). Models based on the field approach (Fung and Ulaby,[22]; Tsang and Kong,[59]) account for the inhomogeneity of the medium through the correlation function characterizing the fluctuating component of the dielectric constant of the medium whereas models based on the radiative transfer intensity approach (Durden et al.,[17]; Eom and Fung,[19]; Tsang et al.,[60]; Ulaby et al.,[70]) account for the inhomogeneity by averaging the Stokes matrix over the statistical distributions characterizing the sizes, shapes, and orientations of the canopy elements. In general, the field approach is appropriate for weakly scattering media in which the ratio of the fluctuating component of the dielectric constant to the mean value for the medium is small (Lee and Kong, [36]; Ulaby et al.,[70]). For a medium such as vegetation in which the individual scatterers

3 have discrete configurations and have dielectric constants that are much larger than that of the background (air), the radiative transfer approach is more appropriate. A first-order raditive transfer model for simulating backscatter from tree canopies has been under development at The University of Michigan Radiation Laboratory for some time (Ulaby et al., [67], [68],[69]). This model, known as the Michigan Microwave Canopy Scattering (MIMICS) model, is fully polarimetric and is designed to function for a frequency range extending from 0.5 to 10 GHz and over a wide range of incidence angles. Several variations of MIMICS have been under development. The first-generation model, MIMICS I, models tree canopies that have continuous or closed crown layer geometries. The second generation model, MIMICS II, models canopies with discontinuous or open crown layer geometries. The fundamental contribution of this thesis is the introduction of MIMICS II. To this end, Chapter II begins with a presentation of the definitions used in describing tree canopy parameters. Sizes and orientations of the canopy elements are defined in terms of random variables with specified probability density functions (PDFs). Brief reviews of the fundamentals of radiative transfer theory and radar polarimetry are also presented. Chapter III presents the development of the closed-crown canopy model. The radiative transfer equations are applied to the continuous canopy geometry and an iterative technique is used to find for the first-order solution. A technique is then introduced by which ground vegetation or snow cover may be modeled. Chapter IV presents the development of the open-crown canopy model. In developing MIMICS II, the same radiative transfer approach that was applied in developing MIMICS I is used except that now the scattering and extinction properties of the crown layer are specified in a statistical fashion, in terms of random variables

4 that depend on the location, size and shape of individual tree crowns. The resulting expression is then in terms of the expected value of the Canopy Backscattering Transformation matrix. Statistical parameters that define the structure of the crown layer are introduced and polarimetric and scalar expressions for canopy backscatter are derived. Chapter V presents the derivation of the crown layer shape statistics for several types of canopy architectures. Parameters are considered on the individual crown level as well as for collections of crowns. It is necessary to introduce these statistics into the MIMICS II radiative transfer solution in order to compute backscatter for a particular canopy geometry. Chapter VI then presents several modeling analyses and applications of both the closed-crown and open-crown canopy models. These analyses include sets of measured radar data obtained with truck-mounted scatterometer systems and with aircraft-mounted synthetic aperture radars (SAR).

CHAPTER II BACKGROUND 2.1 Tree Canopy Parameters For modeling purposes, a tree canopy may be characterized by two layers of vegetation distributed over a ground surface. Figure 2.1(a) shows the geometry of a tree canopy with a continuous or closed crown layer, and Figure 2.1(b) shows the geometry of a canopy with a discontinuous or open crown layer. In both cases, the canopy vegetation is distributed between two regions. The upper region has vertical extent d and contains the foliage that comprises the tree crowns. The lower region contains the tree trunks and has a height equivalent to the average trunk height Ht. The closed canopy geometry is characterized by a crown layer whose elements are distributed within a layer of uniform thickness d whereas the open canopy geometry has a crown layer of total vertical extent d whose constituents are contained within individual tree crown volumes. In general, the crown and trunk layers may overlap such that the total canopy height is H < Ht + d. Two classes of parameters are relevant for model development. These are: (1) the dielectric parameters that specify the electrical properties of canopy constituents and (2) the geometric parameters that specify the shapes, sizes and spatial distributions of the canopy constituents. The dielectric parameters, which are summarized 5

6 Ht (a) Continuous Crown Layer d Ht (b) Discontinuous Crown Layer Figure 2.1: Geometry of a tree canopy with (a) a continuous (closed) crown layer and (b) a discontinuous (open) crown layer. in Table 2.1, may be used as direct inputs to the model or may be inferred from the appropriate constituent parameters listed in the table. These parameters include moisture content, constituent temperature, vegetation bulk density, and soil textural composition. Models that link these parameters to the dielectric constants are reviewed in Appendix A. Table 2.2 summarizes the canopy geometric parameters. Two types of random variables are used to define the statistical properties of the canopy geometry. The tree-level random variables define geometrical parameters within individual trees

7 Table 2.1: Canopy dielectric parameters. Constituent Dielectric Relevant Constituent Parameters Vegetation Dielectrics: Trunk Er Moisture content Branch ~e Temperature (freeze/thaw state) Needle Er Bulk density p Leaf e Surface Dielectric Surface composition (soil,snow,standing water) Moisture content Temperature (freeze/thaw state) Soil textural components (sand, silt, clay) Table 2.2: Canopy geometric parameters. Tree-level Random Variables Canopy-level Random Variables Branches and Needles: Canopy height parameters: Cylinder PDF: fc (lc, d; c, c) Trunk layer height: Ht Number Density: N, Crown layer height: d (cylinders per unit volume) Crown parameters: Leaves: Crown shape factor Disk PDF: fd (dd, td; Od, Od) Crown size PDF: fcr (-') Number Density: Nd Area density: Nt (disks per unit volume) (trees per unit area) Trunks: Ground Surface Roughness: Cylinder PDF: ft (It, d; Ot, It) Is, m and allow for determination of the microwave scattering characteristics of individual canopy components. Canopy-level random variables, on the other hand, define parameters on the scale of the total canopy and thus describe the gross canopy structure. Both sets of random variables are assumed to be statistically homogeneous over the region of interest. In addition, the tree-level random variables are assumed to have identical distributions throughout all tree crown volumes and are therefore independent of the canopy-level random variables. This property will be referred to as local statistical homogeneity. The size and orientation of each class of vegetation constituent is defined in terms of a probability density function (PDF). In general, the crown layer consists of K

8 constituent classes such as primary and secondary branches, leaves, and needles, each with a corresponding number density of Nk particles per unit volume. The joint probability density function fk(sk; Ok, Ok) describes the distribution in size and orientation of constituent class k. The vector sk consists of the size parameters that are relevant to class k. Branches and needles are described in terms of the length 1, and diameter d, of an equivalent cylinder so that (1,, dc) E sc. Leaves are described in terms of the diameter dd and thickness td of an equivalent circular disk so that (dd, td) E Sd. The parameters (Ok, Ok) describe the orientation of class k in inclination and azimuth. Trunk size and orientation are defined in terms of similar random variables (It, dt; Ot, qt). Each of these classes may be divided further into sub-classes as appropriate for the canopy being considered. A given canopy may, for example, include two or three classes of branches representing primary, secondary and tertiary branching structures. Canopy-level random variables include the crown shape factor and PDF describing the size distribution of crowns. These shape parameters are relevant only for canopies with open crown layer geometries. Equations (2.1) and (2.2) conveniently describe a wide variety of crown shape factors in Cartesian coordinates with the z-axis directed in the zenith direction: + 13 - Y 1 () a ( b ) c )(2.1) ( +(I -( -) =0. (2.2) a ) b ) c) Equation (2.1) describes crowns with ellipsoidal shapes while (2.2) describes those with conical shapes. By choosing appropriate values for the shape factor constants a, b, c, a, /b and -, it is possible to simulate many different crown shapes. Table 2.3 lists values of the constants for a few of the simple shapes. The constants a or b

9 and the constant c represent the maximum diameter and height of the crown. More complex shapes may be simulated by allowing values for these constants to vary with quadrant or by combining the two equations (Horn, [26]). The crown shape PDF, fcr (a, b, c; a, /, -y), is defined over the domain of the shape factor constants, allowing for a convenient statistical description of the shape and size of tree crowns within a canopy. Table 2.3: Values of Shape Factor Constants for Simple Crown Shapes. Crown Shape Shape Factor Constants Equation 2.1: Spherical a = b = c; a = 3 = a = 2 Oblate Spheroid a = b, c < a; a = = = 2 Prolate Spheroid a = b, c > a; a = = = 2 Square a = b = c; a = p = 7 oo Oblate Square Column a = b,c < a; a = 3 = y oo Prolate Square Column a = b,c > a; a = 3 = 7 - oo Equation 2.2: Circular Cone a = b; a = / = 2 2.2 Radiative Transfer Theory Radiative transfer theory deals with the transport of energy through a medium containing particles that absorb, emit and scatter radiation (Chandrasekhar [4]; Ishimaru [27]; Tsang et al., [60]; Ulaby and Elachi, [62]; Ulaby et al., [70]). At microwave frequencies, the intensity that propagates through a vegetation medium will in general exhibit some degree of polarization. Therefore, this discussion defines quantities that are applicable to the solution of the vector radiative transfer equation, which accounts for the polarization state of the intensity. The vector radiative transfer equation is formulated in terms of the vector specific intensity I(r, p) where r is a position vector and p denotes the direction of prop

10 agation. The electric field of an elliptically polarized monochromatic plane wave propagating in the direction k in a medium with intrinsic impedance r7 may be described as E= (Ev + Eh hI) eir, (2.3) where v and h are the unit vertical and horizontal polarization vectors. The corresponding vector specific intensity I is defined in terms of the modified Stokes vector Fm by the relation: Iv I Ev 12 Ih I Eh 12 I - Fm /1 = / Eh I ~ (2.4) U 2 Re(Ev E*) 17 2 Im(EE E*) The specific intensity Is scattered by a single particle is related to the incident specific intensity Ii by is (Os, Os) = 2- Cm (Os;,; Oi, /i; Oj, j) P (2o, i), '(2.5) where (Os, 4s) defines the scattered direction in inclination and azimuth, (O., Xi) defines the incident direction, (Oj, qij) defines the particle orientation, and is$,12 ISvhi2 Re(ShSVV) -Im(S Sv,) ISh 2 Shl Il2 Re(Sh Shv) -Im(ShvShh) (2.6) 2R.e(SvvSv,) 2R.e(S hvSh5) Re(SvS^ h + SvhSv) -Im(SvvSh- ShSv) 2Im(,SSo) 2Im(ShSbh) Im(SVS^ h + SvhSLv) Re(SvSh - SvhS*) is the modified Mueller matrix, defined in terms of the elements of the particle scattering matrix S that relate the incident and scattered electric fields E? and Es as follows: E e]ik Sv, Suh For a distrited target, the scattered specific intensity must be defined in terms of the net electric field ta, epresenting the d secifitor sum of all fields scattered in terms of the net electric field Es, representing the vector sum of all fields scattered from

11 an illuminated area A subtending a solid angle dQ = A cos 8s/r2 at the range r from the surface. The angle Os is the angle between the outward normal to A and the scattering direction. In this case, IV v Iv I Es 12 Is dQ = Ih d /. (2.8) us 2Re(E Eh*) V s ' 2Im(EsES*) The integral form of the vector radiative transfer equation for a medium in a free space background is (Chandrasekhar [4]): I(yt, 0, z) = e-6IzqI(it, 0, 0) + j e-t(Z-z')/F(it, 0, z)dz', (2.9) where K is the 4 x 4 extinction matrix of the medium, F is the source function, # = cos 0 and z > 0. The first term accounts for extinction of the intensity as it propagates through the distance z/ji in the direction (it, (). The source function F(ji,, z') = Jf (s0, 0s; i, q) I (t,,) dr, (2.10) with dQi = dtuid-i = sin OidOidqi accounts for the scattering of energy propagating in the direction (Oi, qij) into the direction (Os, 0s) via the phase matrix P (Os, &; i, 1 i). The effect of self emission has been ignored here because in radar remote sensing its contribution is negligibly small in comparison to the other terms in the equation. One of the fundamental assumptions of radiative transfer theory is that in a medium containing a random distribution of particles, the waves scattered from the particles are random in phase, thereby allowing the addition of the waves to be performed incoherently. That is, the Stokes parameters of the mixture are the sum of the respective Stokes parameters of the separate waves (Tsang et al., [60]; p. 127).

12 If the medium contains Nk particles per unit volume that are distributed over the size and orientation parameters Sk and (k1;, k), then 'Pk (Os, s; Oi, i) = Nk ( Lk (Os, O; Oi, 0i; Sk; k, Ok)) = Nk JJ fk(sk;Ok, k) *Ck (Os, Os.; Oi, i'; Sk; ~k, O k) dSkdOkdo k (2.11) where fk (Sk; 0k, qk) is the joint probability density function over the shape, size, and orientation parameters (sk; Ok, Ok) of the particles. If the medium contains K classes of particles, the total phase matrix is K -P (Os, q.s; 0i, <kz) = Z Pk (Os, Os; s i, k) k=1 (2.12) where the summation over k represents an addition over all of the K classes. For a medium containing arbitrary particles, the extinction matrix (Ishimaru and Cheung [28]): constituent is given by -2Re( -Re() -Re(M) 0 -2e(i, -Re()) -Re( ) -2Re(M/,lh) -2Re(M/h) -Re(MA,, + Mhh) 2Im(Aihv) -2I1m(Mh) -Inm(A/Iv - Ahh) -Im(Mvh) Im(Mh,) Im(M, - Mhh) -Re(Mvv + M1hh) (2.13) where AdMmn = 1 k (S11,k(Oi, 04; 0i, Oi; Oj, 5j-))k; 7n, n = v, h. (2.14) k=l o The summation over k represents an addition over the K particle classes, Nk is the number density per unit volume of ea.ch class, ( )k represents statistical averages over the size and orientation distributions of class Ic, and Smnk is the scattering amplitude for forward scattering corresponding to polarization mn for class k.

13 The quantity e-z/l1 is defined by etZ/~l = ~E(yt, )D)(/, 0; _-z/)S-1 (~, 0), (2.15) with the 4 x 4 eigenmatrix E(y, 4) given by 1 bl b2 I b2 12 1 b 12 bi b 1 ~(, ) = (2.16) 2Re(bl) 1 + blb* 1 + btb2 2Re(b2) -2Im(bi) -i(l- blb) i( l- bb2) 2Im(b2) where b, = -A Ih + (2.17) A.v,, - A'Ihh + 7' b2 = 2A h (2.18) -Afvv + 4hh - r r = { (MAv - 'lhh)2 + 4AMhvMIh }1/2. (2.19) ZD(i, O; -z/li) is a diagonal matrix with elements: [VD(/t,;-z/Jt)]ii = e-'i(")/z (2.20) where 0,Q(y, >) represents the ith eigenvalue of o(i, <). The eigenvalues are given by 3l(0, q$) -r — r/32(O, ) 1 -r+ r* P(S) = -Re(A/lv + MAhh) + (2.21) /33(0, ) - r-* /4(0,) ) r+ r2IrmK iKW - il, =,i~s'2* - i~~s(2.22) 2iK - iK2 2ImK2

14 where K1 = ko - [Mvv + Mhh + r] (2.23) K2 = ko- [Aiv + Mhh-r] (2.24) For particles that do not depolarize, M/Ah = -1Mvh = 0 and E becomes 1 000 O O 0 1 E~= - (2.25) 0 1 1 0 0110 0 -i i 0 In solving (2.9) in a vegetation canopy, a.n expression is sought that relates the vector specific intensity Ii incident on the canopy to that which is scattered from the canopy IP. As will be seen in Chapter III, this is achieved through a 4 x 4 transformation matrix T such that Is = T Ii. T may be used to compute the linearly polarized canopy backscattering coefficients: ~V = 47rcos o[T]1, (2.26) ahh = 47rcos Oo[T]22 (2.27), = 47r cos o [T]21 (2.28) ach = 47rcos o []12 (2.29) or, by applying wave synthesis techniques, the matrix may be used to compute the backscattering coefficient for any transmitting and receiving polarization combination. The techniques involved in doing this are discussed in Section 2.3.

15 2.3 Radar Polarimetry The specific intensity scattered by a medium may be related to the intensity incident upon the medium by Is = T I (2.30) where T is a 4 x 4 transformation matrix. Given T, the received power may be computed for any possible combination of transmitting and receiving polarizations. The process is called polarization synthesis (van Zyl et al.,[75]; Zebker et al., [85]; Ulaby and Elachi [62]). This section reviews some of the principles of radar polarimetry as they apply to polarization synthesis and to the radiative transfer problem. The polarization state of an electromagnetic wave propagating in the k direction may be defined in terms of the ellipse shown in Figure 2.2. The electric field for such a wave is described by E= (Ev + Eh h) ekk (2.31) where the electric field components E, and Eh are complex quantities, v and h are the polarization basis vectors and the wave travels in the k direction (out of the page). The unit vectors v and h have been defined such that they are oriented in a direction consistent with scattering in a standard spherical coordinate system. The polarization ellipse represents the locus of points that the tip of the electric field vector traces out as a function of time. In the general case, the ellipse axes < and ( are rotated with respect to the polarization basis vectors through a rotation angle V measured clockwise from the v axis. The degree of ellipticity is defined by the ellipticity angle X measured clockwise from the < axis. The limits of ' are -90~ > 4 < 900 and the limits of x are -45~ > X < 45~. The sense of rotation of the electric field vector around the ellipse corresponds to the handedness of the wave.

16 Major Axis h / \ Polarization Minor EllipseAxis v Figure 2.2: Polarization ellipse illustrating the polarization state of an electromagnetic wave traveling in the k direction (out of the page). The angles X and 4, are measured clockwise. The wave is left-handed if X > 0 and right-handed if X < 0. The wave is linearly polarized for y = 0 and circularly polarized for X = 45~. The polarization state of the wave may be expressed in terms of the modified Stokes vector Iv I Ev2 E( Ih Eh 12 Fm U 2 Re(Ev Ell) V 2 Im(E, E*) where Io = IEI12 + IEhI2 is proportional to the completely polarized wave, the elements of the I I 1 + cos 2 cos2,X) 1 - cos 2' cos 2X) o (2.32) sin 2Vb cos 2X sin 2x total intensity of the wave. For a Stokes vector are related by I2 =

17 (Iv - Ih)2 + U2 + V2 Given the polarization angles (Ott, Xt) of a particular transmitted wave and (Ir, Xr) for a received wave, the scattering cross section of the target is given by art(Or,, Xr,,t, Xt) = 47r A, (br, Xr) * MmAt (t, Xt) (2.33) where At ((t,Xt) = Ft (t, Xt)/Io and Ar(Or,Xr) = Fr(Or,Xr)/Io are the normalized modified Stokes vectors for the transmit and receive waves and AMm is the modified Stokes scattering operator of the target. For a point target.MMn = QLm (2.34) where Cm is the modified Mueller matrix given by Equation (2.6) and 1 0 0 0 0 1 0 0 Q =. (2.35) 0010 0 0 0 - 0 0 0 -1 The modified Stokes scattering operator MAm for a distributed medium illuminated over an area A is.MmA = A cos 0,QT (2.36) where T is the transformation matrix of the medium and 0, is the angle between the surface normal and the scattered direction. For a distributed target, the scattering cross section is at = 47r cos0,Ar -Q T A. (2.37) The polarization signature or polarization response (Agrawal, A. P., and W. M. Boerner, [1]; Ulaby and Elachi, [62]; van Zyl,[74]; van Zyl et al., [75]; Zebker et al., [85]) of a point or distributed target is a convenient graphical representation of

S1 (2.33) or (2.37) that consists of a plot of the synthesized scattering cross section as a function of the polarization angles (Xt, xt) of the transmitted wave. Three types of responses will be presented in this study. The co-polarized response represents the scattering cross section synthesized for receive and transmit antennas having identical polarization (i,. = it, Xr = Xt). The cross-polarized response corresponds to the case in which the receive antenna is polarized orthogonal to the transmit antenna (r = -bt + 90~, X, = -Xt). The linear-polarized response corresponds to the synthesized scattering cross section for all combinations of linear polarization. This response is synthesized by setting X, = Xt = 0 and kr = -t + 6 where 0 < 6 < 90~ is defined as the linear polarization angle. The condition 6 = 0 represents likepolarized cross section and 6 = 900 represents cross-polarized cross section. All polarization responses presented here for a given target will be normalized to the maximum scattering cross section synthesized for that target. As an example, the modified Stokes scattering operator of a short, thin conducting cylinder of radius a and length 1 oriented parallel to the v axis is (Ulaby and Elachi,[62]): 1 0 0 0 *44 - k4l6 0 0 0 0 (2.38) M.li = ] (2 3S) 9 [ln(41/a) - 1]2 0 0 0 0 0 0 0 0 The polarization response may be synthesized from (2.33) and is presented in Figure 2.3. The co-polarized response shows a maximum value at a,, ({ = 0, X = 0) and a minimum at ahh ('0 = ~90~, = 0). The cross-polarized response shows minima at a, and chh. The linear-polarized response illustrates the behavior of a,t as a function of the linear polarization angle 6.

19 S i 4,45. 'g f.lireidt Atogl % (a) Co-polarized response. z (b) Cross-polarized response. 0. 0riea o. Iti% 45 45. (c) Linear-polarized response. Figure 2.3: Polarization response of a short, thin conducting cylinder oriented vertically.

CHAPTER III RADAR BACKSCATTER MODEL FOR A CLOSED-CROWN TREE CANOPY A first-order solution for the radiative transfer equation useful in simulating microwave backscatter from tree canopies with continuous (closed) crown layer geometries has been developed at The University of Michigan and is presented by Ulaby et al., [67],[68], [69], Ulaby and Elachi, [62] and Sarabandi [50]. This model, which represents the first version of the Michigan Microwave Canopy Scattering model (MIMICS I) has been developed specifically for modeling microwave backscatter from tree canopies but is also useful for a variety of other applications in studying remote sensing of vegetation. MIMICS I is a fully polarimetric model and is valid over a wide range of incidence angles. The first section of this chapter presents the derivation of MIMICS I. Then, Section 3.2 introduces a technique that is useful in accounting for various types of ground surface states, including the case of a snowcovered ground surface or a foliar canopy understory. Section 3.3 briefly discusses some applications of this model. Detailed modeling analysis and applications will be presented in Chapter V1. 20

21 3.1 MIMICS I Solution for the Radiative Transfer Equations For purposes of solving the radiative transfer equations, a forest canopy is modeled as shown in Figure 3.1. The canopy consists of two distinct horizontal vegetation layers over a dielectric ground surface. The top (crown) layer consists of the tree crowns and is comprised of an appropriate combination of leaves, needles, and branches. This layer has height d, is statistically homogeneous, and is continuous in the. horizontal direction. The bottom (trunk) layer has height Ht and consists of the tree trunks. Io (-Ito, ho) I (/fo, o0 + 7r) Tt (\yo, o + 7r) Io(-/o, 0o) z=0 Crown Layer z=-d Trunk Layer z=-(d+H,) Ground Surface Figure 3.1: Forest Canopy Model. The total backscattered intensity I(yuo, qo + 7) may be related to the intensity

22 Io incident upon the canopy through the transformation matrix Tt(uo, fo + 7r) by I (go, oo + 7) r= t(ao, qo + r)Io(-[o, ko). (3.1) The 4 x 4 matrix Tt(go, qo + wr) will be called the total canopy backscattering transformation matrix. To solve for "t(Io, qo + 7r), the problem is divided into two parts. First, a solution is found with the ground surface treated as a specular dielectric interface. This assumption is reasonable as long as the intensity scattered by this surface is dominated by its coherent component in the specular direction. If the surface is very rough or its mean slope is not zero relative to the vertical direction, this assumption will not hold. The scattered intensity for a vegetation layer over a specular surface is I*(to, qo + 7r) = -;(2o, qo + 7r)Io(-Po, qo), (3.2) where T,(co, 0o) is the canopy transformation matrix relating the incident and scattered intensities in the absence of direct backscatter from the ground surface. In the second part of the problem, direct backscatter from the ground surface is accounted for by expressing the backscattered intensity as I(,o, > o + 7) = (,to, <o + r)Io(-~o, Oo), (3.3) where T9(1o,I O + 7) is the transformation matrix that accounts for propagation through the canopy down to the ground surface, backscatter by the ground surface, and propagation again through the canopy back to the radar. Combining (3.2) and (3.3) yields the total backscattered intensity i terms of the total canopy backscattering transformation matrix Tt: It(Io, >o + 7r) = [T(flo, ho + w) + g(io, ~o + r)] Io(-Po, 4o) (3.4)

23 = Tt(o, o0 + w)Io(-o, 0o). (3.5) Consider first the problem of two homogeneous layers over a specular surface. To solve for Tc(yo, 0o), the integral form of the radiative transfer equation is set up separately in the crown and trunk layers. Boundary conditions are applied at the layer interfaces and the equations are solved iteratively to obtain the zero- and firstorder solutions. This technique is applicable for weakly scattering media in which the scattering albedo is small. The geometry of this problem is illustrated in Figure 3.2. The specific intensity in each layer is separated into upward-going I+(,,,z) and downward-going I-(-j/, X, z) components, noting that u = cos 0 and 0 varies between 0 and r/2. The radiative transfer equation will then be expressed as a set of coupled equations in each layer. I(- glo') 0(00o) ( z=0 Diffuse Boundary (-,O>) Crown LayerI+( z=-d Diffuse Boundary \ I-'1) Trunk Layer 1 ) z=-(d+H,) Specular Surface Figure 3.2: Problem Geometry showing the positive- and negative-going intensities in the crown and trunk layers.

24 In the crown layer, the coupled radiative transfer equations are I+(t, q, z) =e- I -d) +f e-K+ (z)/ +,, z') dz' (3.6) I-(-,, z) = e'z/I-(-[, q, 0)+ J eKc-(z-z')/ (-(-, z')dz' (3.7) where the subscript c has been used to denote quantities specific to the crown layer. Similarly, the equations: I+(tL, Z) = e-K+[Z+(d+Ht)]/I+,, -(d + Ht)] + J e-(-)/+ /IY(ei, -, z') dz' (3.8) J-(d+Ht) I7-(-L, q, z^ ) = eK(z+d)/I(j (-_1, -d) + 7 e -(z -F-')/. ' (-l,, z') dz' (3.9) apply in the trunk layer. In the trunk layer, for cylindrical trunks oriented near-vertical with length greater than the propagating wavelength (Ht > A), only two types of scattering will be considered. These are forward scattering from the trunks that gives rise to extinction of the forward propagating field and specular scattering from the trunks that gives rise to a ground-trunk interaction contribution to the field backscattered from the canopy. This means that the phase matrix in the trunk layer is significant only in the forward and specular scatter directions and that direct backscatter form the trunk layer may be neglected. The source functions in the crown and trunk layers are then F+(/l+, z)=1 [o j P1(,; i ', s q')I+(t', z) dQ (i; it z) d Q' (3.10) Jo Jo 1, J5- I —i )d

25 F-(-/t,5,z)= 1 2 PC(-^I -' 2,, I)I r27r 1 /o oC(-,;- )I (-L,, z) dQ Jo ~(/~ Jo- ~ ' )I(z)d (3.11) and F+t(t,, = / P, z); P( ', jo I; (/)u', ', Z)k - ) (t ) dQ (3.12) F (-t,;, z)=- /t Pt(-', q; -I', ')I( ', '(, Z)6 t (/1- ') dQ' (3.13) where 6k (t - I') is the KIronecker delta function defined by f1; i = -' 5k (it - it) = (3.14) 0; otherwise. The air-crown interface (z = 0) and the crown-trunk interface (z = -d) may be treated as diffuse boundaries, in which case the intensities across these interfaces become continuous. This assumption yields the boundary conditions: I ' (- i,,0) It (-A, J( -d) rs(/^, ) = Io (-Po, o) 6 (1 - o) 6 ( - oo) =I+ (IL, 0,-d) = IC (-i,,-d) = I+ (,, oO). (3.15) (3.16) (3.17) (3.18) At the trunk-ground interface where z = - (d + Ht), we have I+' [pi ol - (d + Ht)] = IZ (/,) I- [-pl 0, - (d + Ht)] (3.19) where 1? (,u) is the reflectivity matrix of the specular surface: I rv2 0 0 0 0 Irh 12 0 0 R(p1) = 0 0 Re(rwr*) -Im(r),r) 0 0 Im(rr*) Re(rr*) (3.20)

26 where r, and rh are the Fresnel reflectivity coefficients for the specular surface at vertical and horizontal polarizations, respectively. To solve for the specific intensities in each layer, these boundary conditions are applied to (3.6) through (3.9) yielding: I (-", q, z) It-(-,, Z) It+ (p, 4, Z) eKc Z/Io (-Po, qo) S (It - ao) S ($ - ~o) + eK (-z')/.''7 (-IL, q, z') dz' /Z (3.21) z Lz e t (+d)/, -d) + 1et7(z-z')/I,'7(-,, q, z') dz' = er-(+d)/e- d/hIo ( —/o, o) (/ - /o) S (q - qo) + et(z+d)I / eJ (-d-')l/ (-, z ' ) dz' J-d + eKt (z-z')/- (-,,z')dz' (3. Jz 22) =e-t [z+(d+Ht)]/pR (~) I- [-g,,- (d + Ht)] + Lr + e-H, + (-zz')/ Ft+ (,,, z') dz' J (d+Ht) e-+t [z+(d+Ht)]/tX7 (-) e- Ht -'/e- d/'.Io (-Lo, o) 6 (it - o) 6 (q - ko) + e- [z+(d+Ht)]/R7 (it) e- Ht/l' |_ e - -(_ -, z')dz' + e-c +[z+(d+Ht)]/,R ] (i) t( - (d+H )-t) t (P, ') dz' +e- +(-z').(/, e, z') dz' (d+Ht) (3.23)

27 I+(,, z) = e- (z+ d)/It+ (,, -d) + e-; (Z-z')/I.c+( (,, z') dz' =e- e-(+d)/,e-kt+,/,7 (i) e-n Ht/ e-I~Cd/ *Io (-gto, qo)S(( -,o) (q - o)o) + e-K (z+d)/ie-Kt Ht/l () e-tHt/lt -d _- e'q,(-(d+"')-z')l/"7(_-,;, z') dz' -(d+Ht) ^+ eK (z+d)/LL eKe K' 1Z(, /U) + e-S (+)// e+ (a+ ')/t+ z) dz' (d+Ht) + Je-1(z' (,, 4z, z') dz' (3.24) In the bistatic case, the intensity scattered from the canopy is: IsI (,) -I+ (g, 4,O) (3.25) = e-;+d/e-K+ Ht/HZl (p) e-tHt/e- d// ~Io ( —o, o) ( (4 - -o) + e- CKd/e't-+H,/ (it) e- t H'/C. e;c(-d-z')/p.I (_, (, z')dz' + e-+ de-C+Ht/, (i ) -d | IdH eKT(-(d+Ht)-z')/ti-(-, z') dz' -(d+Ht) + e-cd/+ / _ e+t (d+'')/.T+ (p, q, Z') dz' J-(d+Ht) + J e' )'/. +(p, 4) z') dz' (3.26) Solving (3.21) through (3.24) with Ff = -F = 0 gives the zero-order solution at any depth z: I!~)+(,,, z) = e~+(z+d)/ e-tHt/ ( ) e-H'/t-/Kc-d' e a(~~~~t

28 Io (-/,o, qo) S(i - 1io)6( - qo) (3.27) I(~)-(-/,, z) = eKcZ/"IIo (-/Lo, qo) 6(/ -,o)6( - o) (3.28) I()+(, z) = e-t(z+(d+Ht))/ () e-;Ht'/-e- d Io (-Pto, ko) 6(/ - uo)W(q - qo) (3.29) I)-( —, ~, z) = eKt7(z+d)/'e-K'd/Io ( —/to, o) 6( - to)6(O -- o). (3.30) These zero-order intensities are now substituted into (3.10) through (3.13), yielding the zero-order source functions: F( )+(G), ) = ]- \ |(/[, r; t, )I)+, z) d' + j27r |j p(4p,; p-1, )I()- (-, ', z) dQ] = [Pc(t,; /o, o)e-~ (z+d)/~oHe-tHt/loRZ (/Lo) e-Ht/-/o F~)-(-1, z) = 1 [j '|Pc(-/', & [', )IJO)+(t q,,z) dQ + j2r j 'Pc(-1, 6;-t', ')I?)-(-t', ', z) dQ'] = 1 [Pc(-tt ~; to, ~o)e-S;C+(~+a)/"~e-tCH/"~T~o (to) e-^t F'/~ 21 *e-;-d/l(o + 7Pc(-, 4;-tO, ~o)et;Z/'~] Io ( ---to, 40) (3.32) - 1 Pt(p,; po,o o)e-St+()+( )/I~ (to) e-tIoH'/~ *e- C d//Io (-;-o, o) 65 ((1 - /Zo) (3.33) F!O$-(-1, ) = -r j Pt(-[, 4;-/', ')I ( —t, ', z)k ( -o) d *e- c /~oIo (-io0, o) k (1 - Po) (3.34) where c+/l- c i= t(~t, q)/lt and i!/p1o = (i(~- o, o)/po.

29 Substituting these source functions into (3.21) through (3.24) and separating terms gives the first-order solution for the specific intensities: l()-(_y,^) = exc/Zl, (, - o) (P - ~o) + 1 |[f ecc (Z')/,Pc/(-,; /o, )o)e- (z+ d.e-IC+ -R (po) e- tCe I+op edl(zz')/AlPcV11, k —/10o, 4bO)eIZO dz'l.Io (-1o, I'o) (3.35) {eKr(z+d)/lie-c d/6b (1_ - o) 6 (4) - )o) + e edM + e7t (z+d)/t,. [deC ( -d-pz)/''p,( —, <; to, o)e '+d/.e- t Ht/~O-R (A/o) e-~T Ht/~'e-K;Cd/ ~ + eKr(z+d)/l 1 \ ~ e-(-d-z')/pt(-[t, -o )e;z'd dz' 1 I d-d- (Z-Z)/t-pt(-P) };-o ) (z+d)/o dz-o, o) + I -o ( -p.e - rd/.ok (/ - Po))Io (-/o, 0o) (3.36) (?)+ ( /,, z) = {e-"+^ [z+(d+Ht)]/HIZ (e) e — Htl/ e-t- d/t (1 - o) 6 (4 - )o0) Ie t^i~i2) ~ \ + e,+[z+(d+Ht)]/"~ (/.) e- T I'/ ' +I_ e (-f 4)/[Pc-o, )e C (z +d)/,o0 dz'.e-n Ht/"~oR. (Lo) e-t Ht/AO~e-'/ + e-nt tz+(d+Ht)l/'R (p1) e-n' Ht/' + e t ~ ~ ~ ~ /o

I(, +( = Z) = 30 1[f0 e (-dz')/p7:(, p,c /; -po, q/o)eKc Z'/IO dz'] /I [J-d J + e-Kt +(+Ht)]/7p(<) 1 [I_- et(-(d+Ht)z')I/u7p:( —, V); — Po, o) t ( d+ JL [J-(d+Ht).et (z'+d)/o dz'] e- cd/ok (, - Po) + d+) e-K(z -')/Pt(/p,; /to, o)e-K '+(d+H ( )) dz'] It f/-(d+Ht) J *7. (Po) e-Kt Ht/~oe- Cd/ o6k (I - Po)}Io (-/lo, >o) (3.37) e-~+ (z+d)/~e-cH t () e-t H /e- /e- c - (I, - Ho) (q> - 0o) + e- C +(z+d)/e-t 'Ht/7 (I) e-It7Hnt/ + e-tc+(+d)/,e-Kt+H,/7 ( t)t e-;t H'/ I l[/det7-(-d-z')/,pc(_-/,4; -/_o, o)et z'/ dz] p J-d + 'et+ H/ (#o) e-t Ht/ e-t d/~ ~ e.t (z'+d)/o dz'] e- c-/~~ (p - po) + e-t(z+d)/~, 1 [ /- e '+(d+z')/'pt(l; o, o) # c z(d+n,)

31.e-t (z'+(d+Ht))/4lo dz'] *. (go) e- t /~e- c d/~k (P - [0) + 1 e-c(z-z')/^c(,; -go, qo)eKz'/Io dz'] } Io (-Po, ~'o) (3.38) The first-order solution for the bistaticalscattered intensity emerging from the canopy is i (PI 4) = I?)+ (pO, o) = Tc (,I, ) Io (-I0o, o) (3.39) (3.40) where c(pi) = e- +d/pe-K+/I-RI (z) e-/H-/l-e —a-c sd/g' ( - o) 6 ( - ) o) + -e c- ed/e-t/HII (it ) e-.THt/IAi (p,; po, qo) 1 I e-g+ u'/u'~ (po) e- H/Htllo e-7 d/,o + -e ic IH /I? e-e i-H / " LA2 (,) e-; go0,,o) p + - A3 (P, 4; Po, o) e-t+nHt/,,o ('o) e-t. /oe- d/o + -e -c +d/e-I ()) (,4 (p, /; Po), Ko) e-;d/l~Sk ( - Io) It + ie- dAs5 (,; o, 0o) R. (po) e-itH' /~e-Icd/lo k ( - Po) + (3.41) + - (,; o, o) (3.41) with Ai (p, q; Io, qo) = e-c (d+2')'P (-,; Io, o)e-I (Z'+d) dz' J-d A2 (IL O; to, o, o) = e-c I(d+z')/ c(_; o, O; )e-;oz', L dz' (; o,-o) = ez',; o, o)e-(z+d)/o dz' A3 (,O, 0; PO, 00) = eK+ Z/Ipc (it, 0; /io, Oo)e- c //AO dz' J-d (3.42) (3.43) (3.44)

32 A4(t,, 0; Po, 0o) = -d -K.-(d+Ht+z')/I'7: (_'; _^ (z'+ d) _-d t-Pe ' -po, Co)e" /140 dz' (3.45) A5(/,;/ o,q~o) = L+ et (d+'+z )/t(p(, q; Po, o)e- '(z'+d+t)/lo dz' (3.46) -(d+H + t) A6 (P, 0; po, Co) - e +/' Z7'"P:(t, q; -po, oo)e^~;'/~0 dz'. (3.47) J-d Tc (p, q) is the canopy bistatic transformation matrix and relates the incident and scattered intensities for the bistatic scattering case. Each of the seven terms in (3.41) corresponds to one of the scattering processes illustrated in Figure 3.3. These mechanisms are now examined in order of their appearance in (3.41): ss 1 2a 2b 3a 3b 4 Figure 3.3: First-order contributions to bistatic scatter. Term ss This term represents propagation of the incident intensity down through the crown and trunk layers to the ground surface, specular scatter by the ground surface at the angle 00, and propagation back up through the trunk and crown layers. This terms exists only in the specular direction (0, () = (00, o0).

33 Term 1 This term represents propagation of the incident intensity down through the crown and trunk layers to the ground surface, specular scatter by the ground surface at an angle 00, propagation up through the trunk layer, scatter by the crown layer back down to the ground in a direction (r - 0, >), specular scatter again by the ground layer at an angle 0 and propagation back up through the trunk and crown layers in a direction (0, q). Term 2a This term represents propagation of the incident intensity down into the crown layer, scatter by the crown layer into the direction (7r-0, q) such that, after specular scatter by the ground surface at an angle 0, the scattered intensity propagates in a direction (0, q) up through the trunk and crown layers. Term 2b This term in the compliment of term 2a. It represents propagation down through the crown and trunk layers in a direction (rt - 00, 4o), specular scatter by the ground surface at an angle 00, propagation in a direction (00, 0o) up into the crown layer and scatter by the crown layer into the direction (0, q). Term 3a This is a trunk-ground interaction mechanism that represents propagation of the incident intensity into the trunk layer, scatter by the trunk layer into the direction (rt - 00, ), specular scatter by the ground layer at the angle 00 followed by propagation up through the trunk and crown layers in the direction (00, 0). Term 3b This is the complement of term 3a. It represents propagation down through the crown and trunk layers in a direction (7r - 00, (o), specular scatter by the ground surface at an angle 00, scatter by the trunk layer into the direction (0, q), followed by propagation crown layer in the direction (0, q). Term 4 This term represents direct scatter by the crown layer of the incident

34 intensity into the direction (0, q). Note that the only contribution to Tc (I, q) from direct ground scatter comes from a term that is specularly scattered by the ground surface (term ss). Taking Lt = fo and q = oQ + 7r in Tc (u, d) yields the canopy backscatter in the absence of direct ground backscatter. To obtain the contribution from direct ground backscatter, the second part of the canopy scattering problem is now considered. Direct backscatter from the ground is described by propagation down through the crown and trunk layers in a direction (-'o, qo), backscatter by the ground surface, and propagation back up through the trunk and crown layers in the direction (Lo, qbo + r). Then in (3.3), T(lo, qo + r) = e- d/lo e-+I Ht/Io~ (0o) e-t H'/oe-";d/lo (3.48) where g (0o) is the ground backscattering matrix that accounts for direct backscatter by a rough surface at an angle 00. The form chosen for this matrix depends on the roughness parameters of the ground surface. Techniques used to model Q (00) for different surface roughness states are presented in Appendix B. The total canopy backscattering transformation matrix is obtained by adding 7(go, qo + r) to TC ([o, qo + 7r): Tt (xo, Oo + eo) e-= eo Ale () o, e +!e; /o /o) Yo *e -t H'/ o~ (so) e-I-Ht /llo e-Icd/Mto +d/,o -,+-e110 + -A3 (Io, Ho + t; Ko, (o) e-1E).H/ (Io) e-t;H/~Koe-c; d/o Ito + e-cde- H/7 (#o) A4 (P0o, o + 7r; Pot, 0o) e-t7/ gPo

35 + — e As (P(o, + 7r;,o, qo) IZ (0o) e -t /"o~e-cdIlo 'Po + -6 (o, o + 7r; 0o, o0) Yo + e -~;d/o~e~'+Ht/~O (0o) e-~t Ht/Oe-~Kcd/o. (3.49) Figure 3.4 shows the seven contributions to the backscattered specific intensity. Combining term 2a with its complement 2b and term 3a with 3b yields an effec 1 2b 2a 4 3a 3b 5 Figure 3.4: First-order contributions to canopy backscatter. tive number of 5 contributions. These terms will be referred to by the mechanism names listed in Table 3.1. The integrals defined by (3.42) through (3.47) may be computed by applying (2.15). In the backscatter case: Al (Po, o0 + 7T;o, qo) =

36 Table 3.1: Terms contributing to canopy backscatter. Mechanism Mechanism Relationship to Number Name Listed Term 1 Ground-Crown-Ground multiple bounce Term 1 2 Crown-Ground interaction Terms 2a + 2b 3 Trunk-Ground interaction Terms 3a + 3b 4 Direct Crown backscatter Term 4 5 Direct Ground backscatter Term 5 JO d Ec(-,to, qo + r)Zc(-Ipo, qo + r; -(d + z')/,o)E;'(-,o, qo + rT) J-d "Pc( —o, Io + 7r; JoI, /o))Ec(Lo, Io)Dc(Io, o; -(d + z')/yzo)E-l((o, Oo) dz' (3.50) 4A2 (o, o + 7r;,o, o) = E ec(-,O, qO + 7r)Vc(-Lo, bo + r; -(d + z')/,o)F-l(-~,o, qo + r) -d 'Pc(-tlo, ko + r.; -Lo,,o)Ec(-to,; /o)~,(-po, So; z'/lo)E'l(-1o, o) dz' (3.51) A3 (J0, o0 + 7; Po, 4o) = r0 J ~C(jo, qo + 7r),c(io, Oo + 7r; z'/Io)c l(po, qo + w) d- d -'c(,o, Oo + r; to, 0o)Ec(o0, Oo)EDc(o0, Oo; -(z' + d)l/o)~l (,o, Oo) dz' (3.52) A4(j0o, o + 7; Po, qo) = -d -(d+Ht) et(-uo, q0 + T)Dt(-po, qo + 7r; -(d + H + z')/io)E~t (-1o, 0o + r) 'Pt(-1o0, 4o + r; — o, qo)Et(-1-o, qo)VDt(-uo, o; (z' + d)/io)~1-(-po, qo) dz' (3.53) As5(uO,q0O + 7r; o, o0- ) = (-d - ~ t(po, XO + 7r)Dt(t, >; (d + z')/lo)E'(p,\ (o + +r) J-\d+Ht)

37 -,t(/o, 0o + r; o,, o)Et(,uo, ~o)Vzt(lo, oo; -(z' + d + Ht)/1o)~t l(o, o o) dz' (3.54) A6 (/o, qo + T; Po, 7o) = J ~E,(o, qo + 7r)D>c(o, o + T; z'/1o)EZ'(Qo, ko + 7r) -d *Pc(1Po, 'o + 7; -g/o, o)ec(-Po, 0o)ZVc(-& o, 1o; z'/p1o)e-1 (-po, ~o) dz' (3.55) For a canopy symmetric in ^, I+ = c-, K+ = K7t and the 4 x 4 A, matrices are given by A1 ([o, ~o + Tr; [o, 0o) = Ec( - -o, <o + 7r)B11(0o, ~o + T; Po, q0o)E (Io, Io) [B1 (to, o + 7r; [/o, o)]ij = 1 -exp[-(/ic(-Lto, qo + 7r)/Po +,i3(fo, q o)/[to)d] i3c(-o, qo + 7r)//o + fjc(lo, qo)/[o *[~E1 (-/to, 0 + 7r)'( — o, o0 + T; Io0, 0o)Ec(go, 0o)]ij (3.56) A2 (/to, 1o + +r; to, ko) = ec(-t0o, 'o + )52([to, o + 7r; o10, qo)E6' (-~o,o o) [B2(/o, 0 + r; Uo, )o)]ij = de-(/(p~)d/~t [f~ '(-uo, 0o + 1r) ~PC(-[o, 0o + 7r; -/o0, (o)Ec(-/o., 4o)] J exp[-jf(-[to, q o + 7r)d/po] - exp[- ij(-/[o, o)d/po] -#i(-/to, o + + lr)/,o + /j(-to, qo)/tPo i j ~[e-1(-.o, 1o + w7)'PC(-/o, 'o + T7; -/o, 1o)Ec(-o0, 1o)]ij (3.57) Aa3 (/o, 4o + 7r; P/o, qo) = ~c(Po, 0o + 7r)33(/o, 0o + 7r; P0, 0o)ecl' (t, 0o)

38 [33(jio, 0o + 7r; 9,o, 0o)]ij = deI-((Io)d/o. [e-1(o0, o0 + 7r) -'c(P(o, 0o + iX; xo, o)~cQ(-o, 4oo)I ) exp[ —c(/o, )o)d/to] - exp[-ic(,1o, Oo + r)d/u] /ic(,o, 0o + 7r)/Pto - PC(,po, Oo)/,o ~ *[EC-'(o, 4o + r)'Pc(to, o + 7r; to, (o)ec(tIo, 4o)] i = j isj IJ (3.58) -A4(o, 4o + 7r; [o, 40) = ~t(-Lo, 4)o + 7r)t34(0,o, o + 7r; [o, oo)Et- (-_o, qo) [34(Io, 0o + 7; -0o, o)]ij = Hte-,(o~o)H/'o. [t1(-L-0, 1 +0 - ) I=j *Pt(-Io, 0o + 7r; -Io, oo)Et(-lo, o)],, exp[-fit(-tpo, qo + 7r)Ht/to] - exp[-fi(-t-o, qo)Ht/[to] -fi[(-Fo, 4o + 7r)/o + fji(-to, qo)/o i j [E-l(-Pto0 - 7+)t(-Ito, 40 + 77; -)t-, + o, o)t(-[to, 0O)]ij (3.59) A5( o, + 7; to, qo) = et(P(o, qo + 7r) s5 ( o, Q + -r; to, o0)E -l(/o, ~o) [t5([o, co + +r; [o, oo)]-3 = Hte- (o)Hz/o. [1(Po, o + ) =j -P't(Po,?o + 7r; Po, )o)Et([o., o)]i J exp[-flit(to, qo)Ht/po] - exp[-fi(to, Oo + r)Ht/o] i([ ( o, o + lr)/tto - foJ( to, o )o)/o i o *[E'l(t, l)o0 + 7rPt(tto, 00o + 7) t(to, + o)et((o, )o)]ij (3.60) A6 (o1, )o + 7r; to, )) = (, o) =c(o o + 7r)36([o, qo + r; [o, Oo)Ec1 (-_to, 4o)

39 [B36(go, o0 + r; go0, =o)]ij 1 - exp[-(ficf(o, qo + 7r)/#o +,/j(-/zo, qo)/Lo)d] p/i (Ho, qo + 7)/yo + Cjo(- o, ko)/,o *[ec' (go, o + r)'Pc(ulo, 0o + 7r; -go, Oo)~c(-io, qo)]ij (3.61) where fl/ and /c are the nth eigenvalues of ri and iP, respectively. The phase and extinction matrices, P and rc, may be computed by applying (2.12) and (2.13) together with appropriate scattering models for the individual vegetation constituents. The choice of which scattering models to apply for a given canopy depends on the shapes of the scattering constituents present in the canopy, as well as on their sizes relative to the radar wavelength. Constituent scattering models that are applicable to a wide variety of canopy architectures are presented in Appendices C and D. Tt (11o, o0 + r) represents a first-order solution for canopy backscatter. This firstorder approximation is reasonable at lower frequencies in cases where the scattering albedo of the medium is small. At higher frequencies, the effects of multiple scatter become more important and this approximation may break down (Ulaby et al.,[65]; McDonald et al. [43],[44j). In such cases, it may be appropriate to examine a secondor higher-order solution for backscatter. A second-order MIMICS solution may be derived by using the first-order intensities as new source functions in the radiative transfer equations and continuing with the iterative technique. Appendix E presents such a solution for direct crown backscatter. Although straightforward to derive, in the general case determination of the second-order backscatter contribution becomes computationally prohibitive and will not be examined in this study.

40 3.2 Modeling Ground Surface Cover The model discussed in Section 3.1 works well for modeling many types of closed crown canopies situated on level ground if the ground surface may be accurately modeled by a specular surface for purposes of describing its forward scatter, and if the direct ground backscatter may be accurately modeled by applying an appropriate rough surface scattering model for G(00). As seen in some MIMICS I modeling analyses (Dobson et al.,[13]), ground surface cover can have a substantial effect on the canopy backscatter. If the ground is covered with snow or with a dense foliar understory, the accuracy of the specular forward scatter approximation and of the ground surface backscattering matrix comes into question. This section discusses techniques in which underlying ground cover may be accounted for in MIMICS. Two basic approaches may be considered for modifying MIMICS I to account for ground cover. In one approach, an additional layer representing the ground cover may be added to the forest geometry and the radiative transfer equations solved for a three-layer forest medium over a dielectric half-space. In the other approach, the ground backscattering matrix Q(8) and the ground reflectivity matrix IR(P) may be modified to account for differences in the backscattered and specular scattered intensities from the ground surface. In this section, the latter approach is employed. This technique allows for a straightforward modification to the solution derived in Section 3.1 that still accounts for the first-order interaction mechanisms that occur between the canopy overstory and the underlying surface. Section 3.2.1 presents an approach for modeling a ground surface covered by a vegetation understory and Section 3.2.2 discusses an approach for modeling a ground surface covered by a snow layer.

41 3.2.1 Ground Surface Covered by a Foliar Understory Canopy understory may be modeled as a single layer of vegetation over a ground surface. Figure 3.5 shows the scattering processes applicable to this problem. The understory layer has height H, a diffuse upper boundary, and is azimuthally symmetric. Figure 3.5(a) illustrates the scattering processes that modify the canopy direct ground backscatter. The specific intensity backscattered by the understory, Ii(-os)) I +b (goo+T) b 2ab 2bb 3b 4b u u U U U U U t (a) Terms affecting the direct ground backscatter. lIS(go1o) l I(go,~o+K) Iu(-go,~o+/) u\1kg)god U - ~~ uP (-40900) U\/. (b) Terms affecting ground specular scatter. Figure 3.5: Contributions to canopy backscatter from the canopy understory. Ib (-Ho, qo + 7r), is related to the intensity incident on the understory, I (-Ao,' o),

42 through an effective ground backscattering matrix Qu(Oo) by the relationship I' (-uo, ^o + T) - G (0o) Ir (-ao, qo) (3.62) where the subscript u has been used to denote ground backscatter in the presence of a canopy understory. Qu (o0) is found by solving the radiative transfer problem for a single vegetation layer of height Hu over a ground surface (Ulaby and Elachi, [62]). This may be accomplished by employing (3.49) with no trunk layer (Ht = 0) and with the crown layer height d set equal to Hu' g6 (Oo) = ' neI Hu/o7R (to) A1 (/o, o + r;,o, o) Z (1,o) e-Hu/o tto /lo i -e+H /o + — e cHu (>lo) R 4 (Po, 2 o + =; Po, 0o) Ito +. A3 (/o, qko + 7r; Po, qko) R7Z (lo) e-KU"HU/1o 11o + — A6 (Qo, qo + 7T; /o, 0o) so + e-+HUU/o0 (0o)e e-KHu/Io (3.63) where +i = -, is the understory extinction matrix and the A, values are computed by applying (3.56) through (3.61) to the understory layer. The five terms in (3.63) correspond to terms 1', 2ab, 2bb, 3b and 4, respectively, in Figure 3.5(a). The direct ground backscatter transformation matrix becomes 7(11o, 4o + 7r) = e-~ d/~oe-tH/O~u (o)e- Htd/e o (3.64) where 5(0o) has been replaced by,u (00) in (3.49). Figure 3.5(b) illustrates the effect of the canopy understory on the ground specular scatter. This effect modifies the trunk-ground and crown-ground interaction scattering mechanisms, along with the ground-crown-ground multiple bounce term.

43 The intensity scattered by the understory IP (To, qo0) is related to the incident intensity I (-/Lo, qo) through the two specular scattering mechanisms shown as part of term 1'. Similarly, 1s (Xto,,bo + ir) is related to I1 (-a0o, qo + r) through the mechanisms shown as part of term 2. Terms 1, and 2' are identical for an azimuthally symmetric understory. Each of the two terms consists of two components. These represent an intensity scattered by the ground surface attenuated by two-way propagation through the understory layer and an intensity scattered by the vegetation itself. These components comprise the effective ground reflectivity matrix 7u(yo) which relates the incident and scattered intensities as Is ( —o, 0o) = Ru (Po) I, (-to, o) (3.65) and Is (-o,qo + 7r) = RU (go) I (-Lo, o0 + 7r) (3.66) where the subscript u denotes reflection by the ground layer in the presence of a canopy understory. To solve for Ru (ylo), the radiative transfer equations are applied to a single vegetation layer over a specular ground surface. The solution is found by applying the canopy bistatic transformation matrix in (3.41) in the specular scatter direction (1 = /to and q = 0o) with Ht = 0 and d = Hu. In addition, it is assumed that the component that is reflected from the ground is dominated by the zero-order intensity I(O)+, thereby neglecting terms 1 - 3 in Figure 3.3. Under these assumptions, 7Ru (go) becomes, in terms of (3.41), u (ILo) = e-u+ Hu/og7 (o)e-t/Hu/'o + -.A6 (lo, o; Ito, 0) (3.67) /o where K+ = e, is the understory extinction matrix and.A6 is computed by applying

44 (3.47) to the understory layer. The effect on the ground specular scatter is then accounted for by setting 7R (go) = 7R (go) in (3.49). 3.2.2 Ground Surface Covered by a Snow Layer I'sn(*-o1,o) I sn(go,Wo+g) (a) Terms affecting the direct ground backscatter. sn(-golo) is (gO3'00) is ~g~+n (b) Terms affecting ground specular scatter. Figure 3.6: Contributions to canopy backscatter from an underlying snow layer. At low frequencies, a snow layer may be modeled as an attenuating layer over a dielectric half space as shown in Figure 3.6. The snow layer has height H, and relative dielectric Esn. The ground half-space has relative dielectric ET > c,. Under this assumption, most scattering will occur at the snow-ground interface, thus allowing

45 any scattering from the snow surface and any volume scatter inside the snow layer to be neglected. This assumption works well at low frequencies (P- and L-bands) but may be inappropriate at higher frequencies where the contribution of volume scatter in the snow becomes more important. Figure 3.6(a) illustrates the effect of the snow layer on direct ground backscatter. The intensity backscattered from the snow I, (puo, qo +?r) is related to the incident intensity I, (-Hio, ko) through the effective ground backscattering matrix Qs,(Oo) as I, (-;0o, qo + 7r) = Qsn (0o) Isn (- 0o, o) (3.68) where the subscript sn in used to denote ground backscatter in the presence of a snow layer. The angle of refraction in the snow layer 0' is related to the angle of incidence on the snow surface 00 through Snell's law. For a lossless layer in which fsn is a purely real number, sin 0' sin 00. (3.69) X/Csn This relationship changes, however, for a layer in which Er,s is complex although Snell's law still holds in a purely formal way (Stratton, [56] pp. 500-505; Ulaby et al., [70] pp. 76-78). In such a layer ( 1 /(p2 ) + q2)/ where p = 2ca3 (3.71) q = 32-a2 - k2 sin2 0 (3.72) a = ko iIm /sn (3.73) = lkoRe/sn (3.74)

46 O' is equivalent to the local angle of incidence on the ground surface. The snow layer therefore acts as a lens, focusing the incident intensity onto the ground surface. The effective ground backscattering matrix is given by Gsn(0o) = e- ^nHs//l 'Qsg (')61e-snHs1A' (3.75) where Ksn is the snow extinction matrix, s,,,g(O') is the ground backscattering matrix evaluated at 0' for the snow-ground interface and y' = cos 0'. The extinction coefficient of the snow layer is related to the relative dielectric of the snow by Ksn = 2ko IImf/iT| (3.76) Since Kisn is the same for both horizontal and vertical polarizations, the extinction matrix may be written in diagonal form: Ksn 0 0 0 0 Ksn 0 0 Ksn = (3.77) 0 0 Ksn 0 O O 0 0 sn so that the transmissivity of the snow layer is Ys, (n') = e-"CnH"ls (3.78) 1000 0 100 =e-2snHs/ 0 (3.79) 0010 0001 Eiquation (o. to) may then be written as,sn(0o) = e-4'"KnHs/'G9S..9(0). (3.80)

47 Figure 3.6(b) illustrates the effect of the snow layer on the ground specular scatter. The intensity scattered from the snow In (Cio, <o) is related to the incident intensity In (-plo, qo) through the effective ground reflectivity matrix ZRsn(lo). The scattered and incident intensities In (uo, qo + 7r) and 'In (-/0o, qo + 7r) are similarly related. 7sn(jio) is given by R",n(ILo) = e- "H//7Z,,Z (Ie')e-KsnH.s/ (3.81) = e-4,nH^./ s-_g (ttI) (3.82) where lRg,, (1u') is the reflectivity matrix of the specular surface at the snow-ground interface, evaluated at the local angle of incidence 0'. Note that both s,,,g (,u') and Qs-_g (0') must be evaluated for scatter from an interface with a relative dielectric given by the ratio of the snow and ground dielectrics. The effect of the snow layer on the ground specular scatter is now accounted for by setting 7t (do) = tZsn (Io) in (3.49). 3.3 Applicability of MIMICS I This chapter has presented the development of the MIMICS I model. MIMICS I represents the first in a planned series of radar backscatter models for use in modeling microwave backscatter from tree canopies. This model has been developed for application to azimuthally symmetric tree canopies on flat ground and with closed crown layer geometries. In this development, the canopy layers have been assumed to be statistically homogeneous. Thus, it cannot account for the effects of geometries such as row structure, as would be seen in many agricultural canopies. MIMICS I has been found to function very well in a number of modeling studies. McDonald et al. [41], [42], [43], [44] and Dobson et al. [13] have used MIMICS I to model multi-angle and multi-temporal scatterometer measurements of a walnut

48 orchard. These analyses were performed as part of the Eos Synergism Study at the Kearney Agricultural Center in Fresno County, California during the summer of 1987 (Cimino et al. [9], Dobson et al. [15]). Here, MIMICS I has been shown to account for variations in canopy backscatter caused by changes in canopy water status as observed over a period of several days. Dobson et al. [12], [13] and Kasischke et al. [32] have used MIMICS I to study multi-frequency, multi-polarization backscatter and extinction properties of several types of tree canopies in the Alaskan Boreal forest. In these analyses, the model has been shown to predict the behavior of canopy backscatter over changes in environmental conditions that caused the canopy to cycle between frozen and thawed states. This application is being extended by Way et al. [76], [78], [79], [80], for monitoring seasonal environmental and phenologic state of Alaskan forests. The applicability of MIMICS I for predicting the sensitivity of microwave backscatter to changes in canopy biomass for Black Spruce stands has been examined by Skelly [53] and Skelly et al. [54], [55]. Data simulated in this study, although not compared directly with backscatter measurements, demonstrate how MIMICS may be applied to provide greater understanding of the use of SAR for estimating forest biomass. Although developed primarily for application to tree canopies, MIMICS I may also be applied to model backscatter from many other types of vegetation canopies. For example, Ulaby and Elachi [62] p. 184 applied MIMICS in modeling like-polarized phase difference of backscatter from corn canopies (Ulaby et al., [66]). These results have been extended in Chapter VI to illustrate a technique useful for monitoring soil moisture in corn canopies. A major limitation of the MIMICS I model is that it was developed specifically for tree canopies with continuous crown layers. In order to more fully understand the

49 effect that discontinuous or open crown layer geometries have on canopy backscatter, a new version of MIMICS must be developed. To this end, the MIMICS II model is proposed in Chapter IV. Details of the statistics required for this model are examined in Chapter V. Chapter VI will then present specific modeling analyses and applications for both MIMICS I and II.

CHAPTER IV RADAR BACKSCATTER MODEL FOR AN OPEN-CROWN CANOPY - MIMICS II Although the closed-crown tree canopy model presented in Chapter III has been used successfully in many modeling applications, it does not account for the effects that discontinuities in the crown layer have on canopy backscatter. The purpose of this chapter is to introduce a radiative transfer-based model for tree canopies with discontinuous, or open, crown layer geometries, which shall be referred to as MIMICS II. In this case, backscatter mechanisms similar to those found for a closed crown canopy must be accounted for while allowing for opens areas, or gaps, in the crown layer. Figure 4.1 illustrates the types of backscattering mechanisms that must be considered. Terms 1 through 5 illustrate mechanisms which interact with the crown layer similar to those presented in Figure 3.4 while terms 6 through 10 illustrate the effects crown layer gaps have on these mechanisms. A variety of work has been performed in modeling the interception of radiation with vegetation canopies that have discontinuous crown layers (Charles-Edward and Thorpe, [5]; Ferguson, [21]; Jackson and Palmer, [29]; Li, [37]; Li and Strahler, [38],[39]). Much of this work has been carried out for application to models describing processes at optical wavelengths. Typically, these analyses consider one of two 50

51 7 8 \\\^^^^s~~~~~:~~~~:::''f~~~''l;~::~ wr~~~~~~~~~~~~~~~~ii~~:'::"~: 1 4 2a 2b "I'll I ~f: +z;, ft:'.:,tf' ~ -. I. P. u:~I::~~~:::':t '~:~:~~'~;:;~'~;~.:f 5~5~~5~1 --- —--- IrbkdwFswrwsggswsw _ 5 6 Is J ^. ~~~:!~~~~:::'::~~Z~~: 4 -;-~.~.=~.5~.~.;~.=,,Ss~5.~.~;..... ~'"ff~~f:''stzzzt~ ~~ ~~z..lttt.''+I.s-;.-t~f~f:~~'''2'':::::''5"'.'~55' ~~S-'-.;.;.t;fS.:.;....... ';:-: ~2~5iS'tl 1.55:S:::::ff:;5.3==: '' ~;~.5~;~;~-~:~'~-~.....~:.s:::;;;fr: ss...;~2;~ j ~~ — ~PI~S~RgRP 1 m ISEE I I I Figure 4.1: First-order backscatter terms for a discontinuous canopy. general cases. These are the deterministic case, in which the shapes, sizes and locations of the tree crowns are precisely specified, and the statistical case in which these parameters are not precisely known but rather are specified in terms of probability distribution functions (PDFs). In general, deterministic modeling techniques may be applied to modeling radiation interception by canopies whose crown geometries and locations are well-specified, e.g. orchards, whereas statistical techniques may be applied when considering canopies whose crown geometries are not as well or

52 dered. When considering natural forest stands, the latter approach is generally more appropriate. Although several models have been proposed for continuous vegetation canopies (Attema and Ulaby, [2]; Durden et al., [17]; Eom and Fung, [19]; Fung and Ulaby, [22]; Karam and Fung, [31]; Lang and Sidhu, [35]; Richards et al., [48]; Tsang and Kong, [59]; Ulaby et al., [69]), very little work has been done in developing models for canopies with discontinuous geometries at microwave frequencies. To date, Sun et al. [57] have proposed one such model for tree canopies. In this model, canopylevel statistics are used in the development of probability factors that describe the interception of radiation by the individual tree crowns. The MIMICS I solution is multiplied by these factors in order to account for the discontinuous nature of the crown layer. Essentially, this solution modifies the MIMICS I solution by a series of weighting factors. This chapter proposes an approach to this problem that is based directly on the radiative transfer solution. This model represents the second version of the Michigan Microwave Canopy Scattering model, MIMICS II. Development of MIMICS II begins with the MIMICS I radiative transfer solution and proceeds by applying the canopylevel random variables to characterize an additional averaging process over and above that required in accounting for size and orientation on the level of the individual constituents. First, Section 4.1 presents the form of the radiative transfer equations and transformation matrix for a canopy with a discontinuous crown layer. Given the form of these equations, a basic understanding of the statistics of open crown geometries must be developed. Section 4.2 develops these ideas by discussing the adaptation of canopy-level statistical parameters for application to the radiative transfer solution.

53 Two forms of the total canopy transformation matrix are then presented. Section 4.3 describes the fully polarimetric form and section 4.4 describes the scalar form. Detailed modeling examples will be presented in Chapter VI. 4.1 Radiative Transfer Solution for an Open-Crown Canopy To determine the solution of the radiative transfer equations for a canopy with a discontinuous crown layer, a statistical method may be employed in which the gross crown layer geometry is described by a set of random variables with specified distribution functions. Once the radiative transfer solution is found for such a canopy, these canopy-level statistics may be introduced to estimate the canopy backscatter. To this end, a radiative transfer approach similar to that used in MIMICS I is applied with appropriate canopy-level statistical parameters introduced where necessary. Again, the tree canopy is considered to have a three-layer structure as shown in Figure 4.2 with the crown and trunk layers each occupying distinct layers. The height of the trunk layer is defined by the average height of the trunks Ht and the height of the crown layer is defined by the maximum vertical distance d through which individual tree crowns are distributed. The total equivalent canopy height is d' = Ht + d. As with MIMICS I, the problem is solved in two parts. First, the problem of backscatter from a two-layer canopy over a specular ground surface is addressed. Then, an appropriate term is added to account for backscatter directly from the ground surface. For the closed-crown geometry, the incident and scattered specific intensities are related through the total canopy backscattering transformation matrix Tt (Qo, qo + 7r), which is a function of the phase and extinction matrices of the crown and trunk layers. These quantities depend directly on an averaging process

54 Io (-Po, qo) I (go, o + r) = (T \ (o, 0o + 7r))Io (-ho,l o) z=0 Crown Layer z=-d Trunk Layer z=-(d+Ht) Ground Surface Figure 4.2: Model geometry for a canopy with a discontinuous crown layer. performed over distribution functions that describe tree-level random variables defining constituent size and orientation. For the open crown layer case, (It(Co, o0 + 7)), = (Tt(go, qo + 7)), Io(-go, qo) (4.1) where crown layer discontinuities have been accounted for through an additional averaging process over the canopy-level random variables, as indicated by the notation ( {.)c. This specifies the backscatter solution in terms of an extended set of parameters that includes the canopy-level random variables. The function (Tt(ro, qo + 7)),, which represents the expected value of Tt, is itself the sum of the expected values of the canopy and ground backscattering transformation matrices: (Tt(po, I$0 + 7r))L = (T(o0, O0 + 7r)) + (T9(po, qo + T))c - (4.2)

55 where (T(po, q0o + r)), is the canopy backscattering transformation matrix relating incident and scattered intensities for the canopy over a specular ground surface: (I(/,o, qo + rT))c = (T(o, ko + 7r)), Io(-~o, ko) (4.3) and (Tg (po, qo + 7r))y is the ground backscattering transformation matrix that accounts for the contribution of direct backscatter from the ground: (Ig(o, 0o + 7r)) = (Tg(1po, o + 7r)), Io(-jo, Iqo) (4.4) The effects of the open crown layer on the upward- and downward-propagating intensities in the crown and trunk layers may also be examined: (I, z ))^ = (e-+(z+d)i/Ij+(i, -d)), + ( e- (z-z')/lF+(, q, z')dz' (4.5) (i (-,,, z)) (e,,i:(-,,, o)) =+ (J0 ec (z-)/'F'(_-#,,z')dz' (4.6) (I+(,, z)) e-+d'))/ (I+(,,-d')) -+ K(12 e (z-z')/F+(p l, z')dz') (4.7) (It(-i, k, z)) et e (z+d)/h (It(-I, $, -d)) + (j" eKt (z-2')/ F-(-L,, z')dz' (4.8) where (. )c denotes the expected value of the enclosed expression, indicating their dependence on the averaging process at the canopy level. Similarly, the source functions that account for coupling between the upward and downward propagating intensities are (F+(,,z))c = K( f Pc(pf,; [', )I+(', q', z)dQ') ~,,I / ~ Y/ le \-L $ /, )CLJ

56 (F (-1,<$z)) = (F+(it Z))C (Ft -(-/,Z))C - KF-(- p,0, z))= + (1 c(; )I (-', z)dQ') (4.10) + ( I r (; )I,;-$', z')Id -'I, I z)d ') (4.11) \ fJ[ Jo,o - c (^ r [1 ),( ', z)da' + 1 r 1gI(/t (4.10) 'Pt(p,;#', )+(li',4',z) d',k (I- /)) (4.11) (1 j|r j Pt(-i,;-', q')I7(-', ', z)dQ'k( - ')). (4.12) (4.12) The iterative approach identical to that discussed in Chapter III is used to find the solution to (4.5) through (4.8), yielding (Tt (0o, q0 + 7I))c 1 (e-+ d/,O e- t+ Ht/o OR (Lo) e-t "H'/Io A (1lo, qO + 7r; fto, qo) /to.e-+:l-Ht/l/o7R (1o0) e-to HtI/loe-K d/lo ) c + -(e kC ~d10e1 Ht/OZ (,o) e —Ht/oA2 (o, q +; ~o )) $+ — L(A3 (/.o, 0 + T7; lo, ~0) e-;t H/oSR (0o) e-)c'/^e-^/) + — {e-t +d/o~e-+ H 7'/tZo (yo) A44 (tto, 0o + 7r; Lo, Oo) e- ' /)o /to + -(e- /A45 (Lo, + 7r; gxo, Oo) -C (yo) e-;-Lt u'/"e-c od/l )c PUo + -(A6 (11o, Io + 7r; to, 0o))c lto + ~(e-fd/0-A H(110 4o +r1-(1Ht 10)-Ke^d/ 1 oe /1o)c + (e-Id/1'~Oe- tI+ll~ (0o)e-t/"~e-t~d/11~.)e (4.13) where the A, matrices are given by (3.56) through (3.61). Quantities describing trunk layer and ground surface scattering remain unchanged from the closed-crown layer case. Quantities in (4.13) that depend on the PDFs describing the canopy-level random variables include the crown layer phase matrices, c(.....) and the crown layer transmissivity matrix e- CZ.

57 4.2 Application of Canopy-level Statistical Parameters In order to perform the averaging process over the canopy-level random variables, expressions relating the crown layer architecture to the parameters used in the radiative transfer solution must be developed. This section defines the attributes of the canopy-level statistical parameters necessary to describe crown-layer transmissivity and scattering properties. 4.2.1 Crown Layer Transmissivity An important parameter in modeling the amount of radiation intercepted by a vegetation canopy is the probability of gap, or gap probability, PGAP of the canopy (Li and Strahler,[39]). Historically, PGAP has been used to describe transmissivity through canopies at optical frequencies. In these applications, the individual canopy constituents have infinite optical thickness and thus the gap probability represents the fraction of radiation that passes through a gap in the canopy. That is, PGAP corresponds to the fraction of incident radiation not intercepted by any canopy constituent. In general, PGAP may be defined as the probability that a portion of incident radiation will pass through a canopy unintercepted, i.e. the fraction of radiation that is unattenuated by the canopy. At microwave frequencies, this is equivalent to the canopy transmissivity. For a continuous crown layer (Li and Strahler,[39]), PGAP = e- (4.14) where r is the extinction per unit length of the crown layer and s, the withincrown propagation distance, is the distance of propagation through the crown layer. Similarly, at microwave frequencies a continuous crown layer is characterized by an

58 extinction matrix nc and the transmissivity at a depth z in the crown layer is TY (Z) = -e'tz/ (4.15) where At = cos 0, z/i is the total propagation distance in the crown layer and z < 0 represents the path length in the layer. To find the transmissivity of a discontinuous crown layer, the geometry shown in Figure 4.3 is considered. In general, the distance s, which is a random variable As Figure 4.3: Illustration showing one individual tree crown, the within-crown propagation length s and the projected shadow area on the ground As for a view angle 0. representing the total within-crown propagation distance for radiation incident at an angle 0, is a function of both 0 and location (x, y) in the horizontal plane. If a PDF p (s) describing s (x, y, 0) in the entire crown layer is known, then PGAP may be expressed as an expected value: PGAP (0) = p(s) e-Tds. (4.16) Similarly, the expected value of transmissivity of the crown layer is (C (0))C= jp(s)e-Kcsds. (4.17)

59 More generally, the expected value of the transmissivity for propagation I times through the same location in the crown layer is (T (0))c= p(s)e-l'csds. (4.18) The functional form of p (s) depends on the shape and size of the tree crowns. Li and Strahler, [39], have previously derived p(s) for an entire crown layer and their approach is applied here. Consider a point on the ground (x, y). As illustrated in Figure 4.4, this point may be covered by no crown shadows, one shadow or any number n of shadows, where a shadow is defined from a radar-backscatter perspective. The probability that a ray passes through a given number of individual crowns Crown Layer Overlapping Crowns Overlapping Shadows Figure 4.4: Canopy illumination geometry showing ovelapping crowns in th crown layer and the corresponding overlapping shadows on the ground. The number density of scatterers in the region where the two crowns overlap is twice that within one individual crown volume. is equivalent to the probability of that number of shadows overlapping a point (x, y). For trees that are randomly spaced such that the crown shadows fall with equal likelihood anywhere on the ground, this probability is characterized by a Poisson PDF (Gedis and Jackson,[23]): e-(ANt) P (n) = (A,Nt,)n (4.19) n!

60 where Nt specifies the average number of trees per unit area in the canopy and As is the mean shadow size of the individual crowns. This distribution has a mean of = AsNt. It is now assumed that the number density of scatterers within the overlapping portion of the tree crowns increases proportionally with the number of intersecting crowns. That is, the number of scatterers per unit volume is doubled, tripled, etc. corresponding to the number of crown volumes that overlap. If the propagation distance through the ith crown is si and if n individual crowns are intersected, then the total within-crown pathlength is n s =- si. (4.20) i=1 The distribution of this within-crown pathlength given that the propagation is occuring through n crowns is p(sln) =p(sl) * p (s21)*... *p(Sn1) (4.21) where * represents convolution. Then the PDF of s, taking into account all possibilities of multiple shadowing, and including the no-shadow case, is given by 00 p (s) = P (n) p (sin) (4.22) n=O where P(0) represesnts the no-shadow case. From (4.18), the average crown transmissivity becomes (Tc(0)) = P(0)+ P(n)j p(sln) e-tKcsds (4.23) 00 n 00 -=P (0) + 1 P (n) T) (s ) 6cds (4.24) n=1 = O+pn)l PWsf-i^F'sds (4.24) where e-lcs has been treated as -a Fourier kernel. If p (sll) is the same for all of the crowns, then (c(0)) = P () + 3P (n) p (sIll) e-ltcds] (4.25) n=l

61 where I = 1 for one-way transmissivity and I = 2 for the two-way case. The probability p(sll) represents the distribution of s for propagation through a single crown and may be derived analytically or numerically for a wide variety of tree crowns. Derivation of p (sll) is discussed in Chapter V for various crown shapes. The transmissivity at any depth z in the crown layer may also be found from (4.25) by computing p (s I) for the partial crown shapes that result from slicing the crown layer at the depth z. 4.2.2 Crown Layer Phase Matrix The crown layer phase matrix Pc is a random variable that depends on depth z in the crown layer. The expected value of 'Pc for an intensity incident in the (Oi, 0i) direction and scattered in the (Os, qs) direction is (TIC (OS( Is; 0, Oi; z)), K -= (p'k (Os, s; Oi, Xi; Z))c (4.26) k=l K -= J(Nk (z) I ifk (sk; Ok, k) k (o0,,s; Oi, i; Sk; 0k, k) dskdOkdk) (4.27) k=l IC K = (Nk (z))c Lk (O,, qs; Oi, Xi; Sk; Sk, Ok) (4.28) k=l where the summation over k represents an addition over the K constituent classes within the crown layer (branches, leaves, needles, etc.), Nk (z) is the number density per unit volume of each class, Lk (Os, s; Oi, Xi; Sk; 0k, Ok) is the Mueller matrix for class k with size and orientation specified by Sk and (Ok, qk), respectively, and fk (Sk; Ok, Ok) is a distribution function over the size and orientation parameters. In general, for discontinuous crown layers, the number density of each constituent class is a random variable that depends on location in the crown layer. Thus, Nk(z) at a particular depth z is an effective density that depends on the shapes and locations of

62 the individual crowns. In going from (4.27) to (4.28), the location of the scatterers in the crown layer is assumed to be independent of their scattering characteristics, thus imposing the condition of local statistical homogeneity. 4.2.3 Effective Scatterer Number Density The expected value of the constituent number density (Nk (z))c is now considered. If number density increases proportionally with the number of intersecting crowns, (Nk (z)), = Nk (n(z))c (4.29) where Nk is the number density of scatterers in class k within one individual crown volume and (n(z))c is the expected number of crowns overlapping at point (x, y) at depth z in the layer. For a canopy with density Nt trees per square meters that are randomly placed, the crown overlapping statistics are described by the Poisson distribution such that 00 (n(z))c = nP(n) (4.30) 0 = (Ac(z))c Nt (4.31) where (Ac(z)), is the expected value of cross sectional area of a single crown at depth z. Equation (4.31) is simply the expected value of the Poisson distribution P(n). As illustrated in Figure 4.5, the cross sectional area Ac of an individual crown is not only a function of z but also depends on random variables describing crown size and center location zi. The expected value of the crown cross section is (Ac (z)) = A (z, zi, t)pz, (zi) pt (t) dzidt (4.32) where Pz, (zi) and pt (t) are the PDFs for the crown center location zi and crown size parameters t, and Ac (z, zi, t) is a function describing crown cross sectional area at

63 O --- —------------------------------------ -d Figure 4.5: Cross-sectional area Ac at depth z of a crown centered at zi. a depth z for a crown characterized by size parameter t and whose center is at point Zi. The quantity (A, (z)), may be computed for a variety of combinations of crown shapes and center location distributions. Derivation of (A (z))c for a variety of crown shapes and crown layer geometries is presented in Chapter V. Given these expressions, the effective scatterer number density is (Nk (z)) = NkNt (Ac (z))c. (4.33) 4.3 Polarimetric Solution When seeking a polarimetric solution for canopy backscatter, all of the boldface terms in (4.13) represent 4x4 matrices. Thus, since order of matrix multiplication is important, the trunk and crown layer transmissivities and phase matrices cannot be easily separated. To derive a solution, the matrix multiplications may be expanded and the canopy-level random variables applied to the individual elements. For example, to determine the elements of the crown-layer transmissivity matrix,

64 consider (T(0o)) = (e- cs) (4.34) = op(s)e-Kcsds (4.35) ro00 = p(s)Ec (0o, (o) Vc (glo, o; s) E'1 (/o0, 0o) ds (4.36) where e-Ai (,o,o)s 0 0 0 0 e-~2(L~OO)S 0 0 VD (#o, 4o; s)= (4.37) 0 0 e- 0 34(A~')5 O ~ O O e-4 (Ao,00o)S and the elements of E, which are defined in (2.16), are constant throughout the crown volumes. Expanding the matrix multiplications gives 4 ([YC (0o)](ij) = [~c (o, qo)](i,/) [E-1 (ho, qo)] (I j, p(s)e-Alds (4.38) where p(s) is the PDF of within-crown propagation lengths. Each of the (i,j) elements of (Yr (0o))c is seen to depend on the four eigenvalues defined by A1. Note that j p(s)e-A'ds = P(0) + P(n) [p(sIll) e-lds (4.39) n=1 and for complex Al: p(s)e1 - ds P(0) + E P(n) p(sIl) eRe(A cos[Im(A)s] ds n=1 -i j p(sll)e Re(A')ssin [Im(A)s] ds} (4.40) This matrix expansion technique may be applied to each of the scattering contributions in (4.13):

65 * Ground-Crown-Ground Multiple Bounce (Tgcg ( o, o + T)) = e ( -,K.Oe-Kt/~o (go) e-":/,o L/o 0 * [f e-;( )/ P(- /o, o + 7r; 1o, /o) e-;( ')/" dz'] *e- H't/IwR (go) e-H. e- d/~) (4.41) The paths of the upward and downward intensities, as they propagate completely through the crown layer, are assumed to be independent of the path of the intensity that is reflected upward by the ground and then backscattered by the crown down to the ground. First, let A49c9 = Ji e-C (d+z')/ooPC (-#o, C$o + r; o10, Co) e- cK(z'+d)ltodz/ -d = / [~c Vc [-(d+z') /lo] -~ 1] *Pc (-0o,o + t; 0o, 0o) * [ec * Dc [- (z' + d) /o] - e-1] dz' (4.42) so that for element (m, n) of M4gc, 4 4 [.;gc-q](mn) = E Zec (m, k) -1 (1, n) 1=1 k=l 4 4 E E c 'Tc(J, i)c ('l t) c;1 (k,j) j=l i=1 -f e-(d+ k)ldz/ (4.43) -d 4 4 = Z, ~c (m, k) 1 (1, n) 1=1 k=l *,Uo~ (j, i) l (, j) [1 - e-d(At+Ak)/]. j=i- i=1 (1 + 1k (4.44) It then follows that 4 4 K([9gcg](m^n))c = > ~E (m, k) C~-1 (1, n) 1=1 k=1

66 * (ioP (j, i) ) (i, l) cl (k j) j=l i=1 \cAl + ^/k [1 - o p(s)e-('+k)ds]. (4.45) NFow define 7Z' ([to) = e-K+Ht/o/lZ (to) e-etIHtI/o (4.46) SCD (7gcg ( Lo, o + 7))c 1 (-[c * )c * -1] [R' (0) * Mgcg,t (0)]A [tO *[ V *c * Ec1] (4.47) = - ([c * c * E 1 ], 2cg) [c' * )c E 1] ) (4.48) wvhere M(2) = Z' ([o). Mgcg * R7 (o). (4.49) FPinally, 4 4 ([Tp] _1 ~= (m, k) E-_ (1,n) [co 1=1 k=1 4 4 * ~.&(~(ji)~ c(i, l)~ -1 (k,j) e-*(x+xk) = -ZZ ^^(m,k)ez-(l n) [ j0p(s)e-S(t\+Ak)ds]. (4.51) T[his gives the expression for the (m, n) element of (Tgcg9).

67 * Crown-Ground Interaction Here, the paths of the upward- and downward-propagating intensities are assumed to be independent, so (Tcg( uo, o + 7r)) = -(e- d/e/o e-,t H,/oJo (o,) e-,Ht//o * 1 e (d+z)/LoP (-Ih, qo + T; -o, qo) eKczh/"O~dz) (4.52) = - (e-Ird//o) g' (/10) JPo * Ki e-c(d+z')/oPc (-Lo, o + r; -/o, qo) 6e'Iodz' ) (4.53) =,(Y (Oo))c 1R (dlo) lao * (0 e-cc (d')/^pPC (-/^, 0 + T; -o, Co) e8 /""dz') (4.54) Acg = J - e-c(d+z' )/oPc (-f0o, 0o +; — to, 4o) eKc z'/~odz' (4.55) d c = Jd [ec * [-(d + z') /lo] E1] Pc (-izo, o + r;-o, o) * [Ec - C[(z') /o,-] *E' ] dz' (4.56) Expanding gives [9c](mn) = 1E Z Ec(m, k) C1 (, n) ~d 1=1 =l =(* (Oo ' (,)()1(o) j=1 i=1.e-(d+z')0A/do+z'AL/odz'l (4.57) d 4 =- ( I, o (m, k) ''d (, 55n) =1 k =l 0 4 4 []n* E - 'c ()E, k),-1 (, n) j=1 i=l 1 -/ [edko_,) - dAo(i /] (4) 1 k58) A/I-lo~~ ~ -tLO dAa L (4.57) Alt- Xk

68 4 4 - EZ Ze(n, k) - ( n) 1=1 k=l 4 4 ~ ~e (i,l)~-' (k,j)Q(lk) (4.59) j=1 i=1 where odPc (j, i) e-dAk/l~ I - k Q(l. k) - AO -=k(4.60).oLPc(ji) [e-dAk/.o _ e-dAx/1o] I y k so 4 4 ([-cMg](m,))) = Z > Ec (mn,.k) ~, 1 (1, n) 1=1 k=l 4 4 E E ~~ (i,;1) ~E- (k, )(Q(, k))c (4.61) j=l i=1 with |uod (Pc (j, i)) fo p(s)e-sAkds 1 = k (4.62) e; Rr. t (4-62) o ( fi) rp(s) [e "*k -e - sA] ds I 5 k The entire term is then given by the product of three matrices: (Tcg( oo,.o+ ))c 1 = (rY (0o))C R' (Yo)(A4,). (4.63) Io * Ground-Crown Interaction Derivation of this term is very similar to that of the crown-ground interaction: (Tgc( Io, o + 7))c = -L([j| eKtC'Iopc (,/o, 0o + 7r; Po, o0) e-i (z'+d)/-/dz'] 110 -d.e- tnH'/"~ (yo) e —n Ht/ oe- C;d/,o ) (4.64) = — ([/ et Z/~pc (o, o0 + r; lo, rO) e-;'+( )/~dz'] ) to d c ~ Z' (1o) (Tc(o))c (4.65)

69 Letting gc = etz'lTPc (o, o +; o, o0) e-o+(z'+d)/odz' (4.66) -d = J. E- [ z~o * D ] - 1 P (Lo, o0 + 7; 0o, 0o) d [~E. V [-(z' + d)/po] *;1] dz' (4.67) where 0 4 4 [A9dc]s(mn k) d ~c (m, k) ~ (1, n) \^E](m,) ^J-d 1k 1=1 k=1 4 4 ~ E E-Pc (j, i) ~c (i, 1) ~:-1 (k,j) ez'X\klo-(z'+d),l/~Odz' (4.68) 4 4 -- ~ sE, ec(m, k)C-1 (ln) 1=1 k=1 4 4 * E E c (i, 1) E-1 (k,j) Q(l k) (4.69) j=1 i=1 with p odTc (j, i)e- dxAO l = k Q(,1 k)= (4.70) L oPc.(ii) [e-dAl/o _ -dWk/ o I k, we have 4 4 <([ c](^m,))c = — E ZEc (m, k)~-1 (1, n) 1=1 k=l 4 4 ~ ~, ~~ (i, 1) E~'1 (k,j) (Q(/, k))c (4.71) j=1 i=1 where od (Tc (j, i)) fo p(s)e-sAkds = k (Q(I, k)) - (4.72) /to Ak-l rO p([e-\ -- ] ds 1 k. Finally, (Tc ( o,o + 7r))c - (M9c)c R' (io) (:Fc(Oo)), (4.73) Po

70 yields the ground-crown interaction mechamism. * Trunk-Ground Interaction This mechanism is independent of the crown layer statistics except as they apply to crown layer transmissivity. The propagation paths in the crown layer for positive- and negative-going intensities are assumed to be independent. Hence, (Ttg( Ito + +T)),1 (e-f dlo/e-n+ Ht/uo (io) [| e- (d'+ z)/op t (-lo, ko + r; -/to, qo) e' ]z+)" ~odz] - eI-d/go}) (4.74) Noting that.Mtg = e-" Ht/~ozR (o) [ j. e-(d'+z')/opt (-/.o, +,; -/o, o) e(+d)/Ldz] J-d' (4.75) is identical to the MIMICS I solution for trunk-ground scattering in the absence of a crown layer, it follows that (Ttg) = -(e d/. Mtg e-c p) (4.76) /to c /to = -o [Ec Dc* E s'] * t, tg [e T.c * D.] ) (4.77) = ([EC.DC.E-1]).A Mt, * ([6E-c *E~-;1]) (4.78) with elements (m, n) given by [Ttg](m,n,,) ) Ek- (1, n),n) 1 E E E~, Eln) PO 1I=1 k;=1

71 4 4 * E E /t (j, i) Ec (i, 1) E-1 (k, j) (e-s') (e-sk)(4.79) j=l i=l E E c (m, k) -1 (, n) PO 1=1 k=l 4 4 * E i.' (j, i) ~C (i, l)13 1 (k, j) j=l i=1 * [ p (s) e-S'ds] [ p (s) e- ds]. (4.80) * Ground-Trunk Interaction Expansion of this mechanism follows in the same way: (Tgt( [o,o+7))c= -+d [I e K+(d+z')/~Opt ([to, qO + 7r,;o, 0) e (d+)/dz godi * 1?,(pio) e -tHCIe-c4d/"o) (4.81) 1 (e-(-d/L [Agt] e-cc d/Io) (4.82) 1o where = [I K e+(d+z')'Ptt~t o-,0 + -,; to, qo) e-'i+(d'+z')/odz'] J-d' ~*R (,o) e-KtHt'/lO (4.83) is identical to the MIMICS I case. Then (Tgt) = - l(e -d/.o M. gt e-IC-d/o) (4.84) = 1 K([ec Dc l] A4,t [ C DC. l1]) (4.85) ito = JLo ([EC D, c e]) Agt ([Ec * c * ])c (4.86) and expanding gives: 4 4 (K[TPgt](n))_ = 1 E > ~c, (m, k) E-1 (1, n) mO 1=1 kl=

72 j=1 i=1 4 4 1 -c (ml k) C-1 (1, n) l0o 1=1 k=l 4 4 *. A4gt (j, i) ~ (i, 1) ~-1 (k,j) j=l i=1 * [jo p (s) e- 'ds] [J p (s) e-Ak ds] (4.88) * Direct Crown Backscatter Here, (Tc( o o 7o+ ))c = -(l (J/ I/"~p: c (Pou, q0 + 7r,; -~o, o) eoc z'/"L~dz') (4.89) = (o [Cf, * DC (Z') (-C] * 'Pc (0, o 0 + r,; -O, i0o) ~ [~. 3Dc (z') ~;'] dz') (4.90) Expanding gives -- I 4 4 [Tc](n)) = (I c ( k) 1 (1, n) o \J'o dl k — * E E PC (j, i) ~ (i, 1) ~ 1 (k,j) ez'(x'+xk)/ ~dz') (4.91) j=l i=1 c 1 - ]E ~ (m, k) ~c (I, n) tO l=1 k=l = i=1 Pc (j, i) c -1 ( k j) Ak+ \A-k+/, n [1 -j p(s)e-S(X +XAk)ds (4.92) * Direct Ground Backscatter In this case, the paths of the upward- and downward-propagating intensities in the crown layer are identical. Hence, (Tg ( [o, o + 7))c=

73 Defining e- ed/-o e+Ht/lo~ (0o) e-K Ht/oe-c d/"o) AM9 = e fCtHt/og (Oo)e-KtHIt/o (4.93) (4.94) so that (Tg)c = (e-. /me-...d) =- ([Ec.Dc. E ] e g * [EC - c ]) fto ~ c (4.95) (4.96) gives 4 4 [Tg](m.n))c = S E- (-m,;) Ec (1, n) =o 1 k=1 4 4 110 1=1 k=l 'EEg(jni6g (icl)~ E (k,) j=l i=1 [ p( s) e-X +Ak)ds] (4.98) -- o -- These terms may now be combined to obtain an estimate of the total canopy backscatter. The canopy-level statistics come into play here in computing the average crown layer phase matrix (P)c and in integrating over exponentials that are dependent on the eigenvalues of the crown layer extinction matrix: J~ p(s)e-sAtds. Jo Quantities of the form /P(i,j)\ A~1 Ak/c

74 represent ratios of phase matrix elements to eigenvalues of the extinction matrix. These quantities are constant throughout the crown volumes and thus do not depend on the canopy-level random variables. 4.4 Scalar Solution When considering only HH and VV polarization configurations, the solution may be simplified considerably by considering the scalar solution for which the boldface terms in (4.13) become scalar quantities. This allows those variables that depend on trunk layer parameters to be factored out of terms that include the crown layer statistics. If the constituents in the crown and trunk layers are distributed uniformly in azimuth, then i~ = ct. Making these simplifications to (4.13) leads to: c c 1T(,, PC ~o + / (-Po, ~o+r; )A 1 /0o/>~) (T, (uo, o + 0 r))c = 2 (KT(o))K( -Pc( —o-o - -; o, 'o) ) ( -_ (T (o))).e-4"tHt/.lo [G(0o)]2 d 0 + - (TC (~O))C (PC (-HIo, ~o + i; -Hio, +~) e-2KtHt/Io G(O) + - (Tc (0o))C (PC (-o, o0 + 7r; -o, o)) [to.e-2^tHt/Lo G(0o) 1 + - (Tc (0o)) e-2^'H' R (o) HtPt (- o, ~o + r; -to, bo) /~o 1 2 -2H + - (Tc (0o))c e2^'HL/ R (o) HtPt (Ao, /o + rT; o, do) Ho 2 Pc (uo, 0 + 7r; o,o)\ ( T(0)) O K; ic - c + (T2 (Oo)) e-2^tHt/oG (0o) (4.99) where crown layer transmissivities have been separated into one- and two-way transmissivities, corresponding to propagation through differing or identical propagation

75 paths. The quantity Pc{(st,<s,;i, ~i) /ic represents the ratio of scattering to extinction for intensity incident in the (pi, qi) direction and scattered in the (iy, q,) direction. This quantity, is constant throughout the crown volumes and therefore is not affected by the canopy-level random variables.

CHAPTER V SHAPE STATISTICS FOR TREE CROWNS This chapter presents the derivations of the statistics associated with specific crown shapes. These statistics are necessary for applying the radiative transfer solution derived in Chapter IV to particular canopy geometries. Section 5.1 presents the derivation of the probability density function p(sjl) of the within-crown propagation length for various shapes of tree crowns. The derivation considers several classes of crown shapes for crowns that have a specified size. Section 5.2 then generalizes these results to account for crowns distributed in size and location throughout the crown layer. The derivation of the crown cross-sectional area is presented in Section 5.3. Again, several classes of crown shapes are considered and results are generalized for crowns distributed in size and location. 5.1 Calculation of Within-Crown Propagation Length PDF for Different Crown Shapes The distribution of within-crown propagation pathlength for single tree crowns, p (sIl), may be computed numerically for arbitrarily shaped crowns. However, some simple shapes allow for straightforward derivation of analytical expressions for p (sl l). In either case, p(sll) is most easily derived for shapes that can be easily defined mathematically. Equations (5.1) and (5.2) allow for simple mathematical definition 76

77 of a wide variety of crown shapes in Cartesian coordinates (Horn [26]). (a + + = 1, (5.1) (2a)a+ (2Y ) -(1 —) = 0. (5.2) Equation (5.1) describes crowns with ellipsoidal shapes while (5.2) describes those with conical shapes. The specific shapes considered in this section are (1) spheroid, (2) square column and (3) cone. In addition, examples are shown for a mixed spheroid for which the shape factor constants in (5.1) assume different values in the upper and lower half-spaces. 5.1.1 Spheroid The spheroid is a natural choice for modeling the crown shape for many types of trees. This section presents the derivation of p(sll) for a sphere as well as for prolate and oblate spheroids. The derivation for a sphere is considered first and then the result is generalized to prolate and oblate spheroids. Sphere An analytical expression for p(sll) for a sphere has been derived by Li and Strahler, [39], and is reproduced here for completeness. Letting a = b = c and a = / = a = 2 in (2.1) yields the equation of a sphere. Figure 5.1 diagrams a sphere and the within-crown path s at a distance r from the center of the sphere. For a given r, the sphere will contain a cylinder of length s along the illumination direction, as indicated by the shaded region in the figure. As r increases to r + dr, s decreases to s' = s - ds. The quantity p(sll)ds represents the proportional rate of change of s at a distance r. This is the proportion of the area of the circle represented by the

78 Figure 5.1: Geometry of a sphere of radius R showing a within-crown path length s at a distance r from the center of the sphere. shaded annulus in the figure. The infinitesimal area in which s and r are constant is 2rrdr. Thus the proportion by which s changes at distance r is -2rrdr/7R2 so -2irr p(sil)ds = - -dr. (5.3) irR2 Furthermore, r R2 1 -( )2 (5.4) so dr _-s ds 4R- ()2.~~= —.^~ ~(5.6) 4r It follows that 2irr dr p(s1l) -R2 d (5.7) 2R=~~~~2 wt(5.8) 2R2

79 For a sphere with diameter a = 2R, 2s p(sll) 2 a2 (5.9) where 0 < s < a. Equation (5.9) represents a ramp function rising from 0 to 2/a as s goes from 0 to a. Generalization to prolate and oblate spheroids Letting a = b and a = # = 7 = 2 in (2.1) yields the general equation of a spheroid. The spheroid is prolate when c > a and oblate when c < a. As illustrated in Figure 5.2, p(sll) for a spheroid is easily derived from (5.9) through a linear z al2 z c/2 a/2 Figure 5.2: Transformation of a sphere to a prolate spheroi transformation of the z axis. For example, a sphere of diameter a is transformed to a prolate spheroid with major axis c and minor axis a through a stretching of its z-axis. If the original axes are x' and z' and the transformed axes are x and z, then the points in the sphere (x zl) and (x2, z2) are transformed to the points (x1, z1)

80 and (X2, Z2) in the prolate spheroid. If f = c/a then z1 = f * and Z2 = * Zf while X1 = x' and x2 = x. The difference values are defined as Ax' = x2 -xI (5.10) Az' = Z-Z4 (5.11) AX = X2 - 1- f Ax' (5.12) Az = Z2-Z1 =f Az' (5.13) Furthermore, Ax' tanX'= =f tanX (5.14) Az' and the path lengths are ' = /(Ax')2 + (Az)2 = Az/1 + f2 tan2 x (5.15) = /(A)2 + (Az)2 = Az1 + tan2 (5.16) It follows that _ Az/1+tan2X A /~z'/l ^ ~+ftan2^x (5.17) ' A^z'l + f2 tan2 X so I = a lAz +tan (5.1 Az' 1+f2 tan2 ( \1 + tan2 2 = I'f 1 +tanx (5.19) l + f2tan2 = 'A(x) (5.20) is the new path length where a 1+ ()2 tan X a a() 2(.1

81 For a radar incidence angle 0, the sphere diameter a in (5.9) must be scaled by A(0), giving p(sl) = )] (5.22) [aA(O)]2 2 a 2 1 + tan2 2 = [1+( )2tan2l c4S (5.23) where 0 < s < aA(O). Since this process is a simple linear transformation, (5.23) represents a ramp function rising from 0 to 2/[aA(O)] as s goes from 0 to aA(O). The validity of (5.23) is easily verified by considering two special cases. First, let 0 = 0 so the spheroid is viewed along the z-axis. Here, A(O) = c/a and p(sll) = 2 (5.24) 0=0 C2 where 0 < s < c. This is identical to p(sll) for a sphere with diameter c. Now, let 0 = 7/2 so the spheroid is viewed along the x-axis. Here, A(O) -* 1 and p(s 1) /2-2s (5.25) 0=7x/2 a2 where 0 < s < a and p(sll) is now identical to that in (5.9). 5.1.2 Square Column The derivation of p(sll) for a rectangular solid is now presented. Mathematically, this shape is convenient to couple with the radiative transfer equations and therefore warrants some consideration. Letting a = b and allowing c = a = 7 -, oo in (2.1) yields a square column. a = b < c gives an oblate square column, a = b > c yields a prolate square column, and a = b = c yields a cube. The radar look direction is assumed to be perpendicular to a face of the column, thereby simplifying the derivation of p(s l1) significantly since

82 C < --- —----— >- -< --- —-----— C a a Case I Case II Figure 5.3: Illustration of the within-crown propagation length for a crown with a rectangular shape factor. Case I occurs for a radar look angle 0 > 0c. Case II occurs for a radar look angle 0 < 0c. The critical angle Oc is determined from tan 0c = a/c. Sm represents the maximum value of s for a given crown. The solids have depth b into the paper. it may now be represented in two dimensions, as shown in Figure 5.3. Two cases must be considered in deriving p(sll). These cases are differentiated through the definition of a critical angle 0c that is related to the crown dimensions by tan Oc -a (5.26) C Case I arises when 0 > Oc, whereas Case II occurs when 0 < 0c The PDF for the entire column may be expressed as an integral over the column volume: p(s 1)= - Pv(s)dV (5.27) where pt,(s) is the PDF describing s within the differential volume dV and VT is the total volume of the column. p(s Il) is partitioned over three regions of interest within

83 which pv(s) varies only with s: p(s l) = p(s)dV + P( Ps)dV2 + p Pv3(s)dV3] = VT [P (S) / dVY + Pv2(s) dV2 + pv3(s) dV3] 1 [p(S)V + p2(s) V + (S)V] VT (5.28) (5.29) (5.30) The volumes V1, V2 shown in Figure 5.4. and V3 and their respective PDFs correspond to the regions z z I dsI ds, x - - - x - - - - Case I Case II Figure 5.4: Geometry used for deriving p(sll) for a square column. For Case I, the maximum value of the within-crown propagation length is Sm= a/ sin 0. Examining the volume V1, The equation of the line of length s is z = - cot Ox + zc (5.31)

84 where z, is the z-intercept and the length of this line is a zc sin 0+ (5.32) Also, ds = dz/ cos 0 where ds = ds1 + ds2. Now, the infinitesimal volume in which s and z are constant is (sb) dl and VI = a2b cot 0. The quantity p(s)ds represents the proportional rate of change of s at a given z. This is simply the ratio of the volume of the strip of thickness dl to the volume V1: 2sb pv (s)ds = 2sbotdl. a2b cot d (5.33) Since dl = dz sin 0 2s sin 0 tan 0 dz pvl(s) = -a2 ds 2s sin 0 tan 0 a2 — cos 0 a2 Through a similar analysis, it can easily be shown that<. ( a ) i < s sh t Through a similar analysis, it can easily be shown that (5.34) (5.35) (5.36) Pv3(s) = pv (S). (5.37) Finally, it is seen straightaway Pv(S) = k(S - Sm) where (5.38) 1 bk(S - Sm) = 0 S = -;otherwise. (5.39) The volumes of interest are: a2b 2 tan 0 (5.40)

85 V2 = Sm(c-a/tanO)bsin0 V3 = V VT= V + V2 + V3= abc. (5.41) (5.42) (5.43) Applying (5.30) and simplifying gives 2 cos 0 sin O 5 + a k (S Sm p s/l)/ > ac c tan 0 o p(s|l) = c —t ^ ' 0; 0 < s < s;otherwise (5.44) where Sm = a/ sin 0. This same technique may be used to derive p(sll) for Case II. Here, Sm = c/ cos 0 and c2b tan 0 ~i = -ta2- (5.45) 2 1/2 = Sm(a-ctan0)bcos0 V3 = V1 VT = V + 1'2 + V3= abc. (5.46) (5.47) (5.48) Therefore pv, (s)ds = 2sb c2b tan 0 (5.49) so Pvl(S) 2sb sin dz c2b tan O ds 2s sin 0 = -; —^ cos O c2 tan 0 = 2 (cos 0)s c). = Pv, (s). (5.50) (5.51) (5.52) (5.53) Furthermore, as in Case I: Pv2 (S) = Sk(S - m ) (5.54)

86 Again, following (5.30) and simplifying: 2 cos sin + (1 ctan ) ( - m) S<Sc l ao P(SIO~~~~~~~1~O p"sli) -; < s < Sm;otherwise (5.55) where m = c/ cos 0. Equations (5.44) and (5.55) represent ramp functions rising from 0 to 2Cos0sin sm as s goes from 0 to Sm. At s = sm, an additional contribution is added to p(sll) to account for the shaded regions in which s = sm, a constant. Figure 5.5 shows p(sll) 1.5 1.0 - C/ 0^ 0.5 0.0 0 ' 0.0 0.5 1.0 1.5 s (meters) Figure 5.5: p(sll) for a square column with a = b three incidence angles. = c = 1. Values are shown for for a square column with a = b = c = 1 at three angles of incidence. For this shape, 0c = 45~ 5.1.3 Cone Setting a = b and a = 0 = 2, in (2.2) yields a right circular cone with apex at z = c, base at z = 0 and basal diameter a, as shown in Figure 5.6. The apex angle

87 Figure 5.6: Geometry of a conical crown with height c, basal diameter a and apex angle c. 3.0 ~ I I l I ' Il l I 0 — =20*,-''' '''*,........ 0=300 2.0 -..0=,,.......... 4 1.0 f.'.~~:':s'2.~ / ^ ^". " -- 0.0 0.0 0.2 0.4 0.6 0.8 1.0 s (meters) incidence angles for a Figure 5.7: p(sll) for several 0.5 and c = 1. right circular cone with a = b =

88 c = 2tan-'(a/2c). Derivation of an analytical expression for p(sll) for this shape is very difficult. Thus, numerical integration is a more straightforward technique for computing the distribution of s. Figure 5.7 was generated with such a technique. In this example, the cone dimensions are a = b = 0.5 and c = 1. The function p(sll) is shown for several incidence angles. 5.1.4 Partial Crown Shapes When examining the specific intensity at an arbitrary depth in the crown layer, it becomes important to consider the within-crown propagation length for partial crown shapes. The PDF p(sll) is now examined as a function of penetration into individual crown shapes. For a square column crown, the partial shape formed by slicing the crown with a plane parallel to the x - y plane is itself a square column. p(sll) for the sliced shape may therefore be computed by applying the appropriate parameters of the sliced crown to (5.44) and (5.55). Other shapes exhibit more complicated behavior with respect to the depth of the slicing plane. Figure 5.8 illustrates the geometry of this problem for spheroidal shapes. The spheroid is sliced at a depth z < 0 where z = 0 represents the spheroid top. Figures 5.9 and 5.10 show p(sll) for a spheroid with a b 2 and c = 10. xz,.......... ~ ~I::~:i~ii~:,i:.~~iiiii5 e~ ~ ~ ~~~~~r::::5::::::::::::: -Xe.: ~ ~ ~ ii~i~~~iiiiiiiii~~~~~ z......~~~~~:~~~~:::f~:I~'t:::: ~~~:I~~....................~,~. ~~.~~..-~~~z............................... Figure 5.8: Geometry of spheroid crowns with propagation depth Izl < c.

89 1.0......,. z z=-2.5 0.8 ---- z =-5.0 0.6 0.4 0.2 0.0 0.0 1.0 2.0 3.0 4.0 5.0 s (meters) Figure 5.9: p(sll) of a spheroid with a = b = 2 and c = 10 for several penetration depths. The incidence angle 0 = 30~. Figure 5.9 shows p(sll) at various depths z for an incidence angle 0 = 30~ and Figure 5.10 shows p(sll) at various incidence angles for z = -c/2 = -5. Note that for z = -c = -10, the resulting p(sll) is identical to that of the entire spheroid.

90 1.0 0.8 0.6 0 - 0 = 20~ -........ 0 = 30~ - = 40~ O = 50~ — ''" ~. --- \ I %Jd I 0.4 0.2 - f, 0.0 L 0.0 1.0 2.0 3.0 4.0 5.0 s (meters) Figure 5.10: p(sll) depth for several incidence angles for an spheroid with propagation z = -c/2. The ellipsoid has a = b = 2 and c = 10.

91 In general, there are two partial crown shapes defined by the intersection of the slicing plane with the crown, one partial shape above the plane and one below. In the case of square columns, both of these partial shapes are themselves square columns. Similarly, in the case of spheroids, both partial shapes are spheroids. For a right circular cone, however, the slicing plane defines a right circular cone in the upper half-space and a frustum of a right circular cone in the lower half-space. p(sll) has already been handled for the former case. For the cone frustum, the geometry of Figure 5.11 may be considered. As before, the cone has a total height c and basal z a Figure 5.11: Geometry for computing p(s l) of a frustum of a right circular cone. diameter a. The height of the frustum is designated by z (shown referenced to the cone base) and depends on the location of the slicing plane. Figure 5.12 shows the results of numerical computation of p(sll) for a cone frustum of several heights at 0 = 30~. This cone has a = b = 0.5 and c = 1. The delta function occurs at values of s = z/ cos(O) which correspond to paths entering the top face of the frustum and exiting its base.

92 3.0.. e II z = 0.75 0.0. * * I. ** I _. \ _ 2.0 '-"a "-: I. " 0.50 1.0 -A, 0.0 0.2 0.4 0.6 0.8 1.0 s (meters) Figure 5.12: p(sl 1) of a cone frustum for several heights at an incidence angle 0 = 30~. The cone has a = b = 0.5 and c = 1. 5.1.5 General Shapes By selecting appropriate values for a,,3 and y and by letting a, b and c vary with quadrant, (2.1) and (2.2) may be used to define a wide variety of crown shapes. As a simple example, if a = 3 = 7 = 2 and a = b = 2 in (2.1) and letting c = 10 for z > 0 and c = 4 for z < 0 the mixed spheroid shown in Figure 5.13 is obtained. The corresponding p(sjl) may be generated numerically and the result is shown in Figure 5.14 for several angles of incidence.

93 Figure 5.13: Geometry of a mixed spheroid. The upper spheroid has a = b = 2 and c = 10. The lower spheroid has a = b = 2 and c = 4. 1.0 0.8 0.6 0.4 0.2 0.0 Lo 0.0 1.0 2.0 3.0 4.0 5.0 s (meters) Figure 5.14: p(sll) of a mixed spheroid shape at several incidence angles.

94 5.2 Within-Crown Propagation Length PDF for Crowns Distributed in Size and Location In order to model p(s) for a wide variety of canopy architectures, the results of the previous section must be generalized to crowns distributed in size and location. Crowns within a given canopy are assumed to have identical shape classes and have specified distributions in center location and crown height. p(sll) will also be considered as a function of penetration depth into the crown layer. The propagation length s through a single tree crown with a specified shape is described in terms of the PDF p(sil;z';c;zi). As illustrated in Figure 5.15, c represents the height of the crown, z, corresponds to the location of the crown center, and z' represents the depth in the crown layer where the value of the propagating intensity is to be estimated. For a collection of crowns distributed in c and zi, the z —.............. Z= Z'... z —d - -i z -d Figure 5.15: Illustration of ellipsoidal crowns in a crown layer of thickness d. the crowns are distributed in both height c and center location zi. z' represents the depth in the layer at which the value of the propagating intensity is to be estimated. PDF of within-crown propagation length for a single crown is p(s l; z') = p(sl1; z'; c; i)p(c)p, (zi)dcdzi (5.56) where pc(c) is the PDF of crown size defined in terms of the random variable c and

95 PZi (zi) is the PDF of crown center location defined by zi. General analytic expressions for p(sll; z') are very tedious to derive for all but the most simple distributions over crown shape, size and location. Thus, it is more convenient to compute the PDF numerically. Some examples of p(s Il; z') are now presented for crown layers consisting of spherical and conical crowns. For crowns having identical center locations zi and a distribution in height described by the PDF pc(c), p(sll; z'; zi) = p(sl; z'; c; zi)pc(c)dc. (5.57) The crown height c and width a may be coupled through the width-to-height ratio k = a/c. For ellipsoidal crowns, k = 1 yields spherical shapes. Figure 5.16 shows p(s) 1.5, -,,. ZI= -8 1.0.- -—.z'=-6 z'= -4 3... --- z'- =-2 0.5 0.0... 0.0 2.0 4.0 6.0 8.0 Within-Crown Propagation Length, s (meters) Figure 5.16: p(s) at various depths z' in the crown layer for a spheric centers located at zi = -4 meters and size uniformly distributed between 4 < c < S with a/c = 1 and 0 = 30~. for an incidence angle 0 = 30~ at various depths z' in the crown layer for spherical crowns centered at zi = -4 meters and with height uniformly distributed between 4m < c < 8m. Note that the maximum propagation length increases directly with

96 z' up to a maximum value of s = 8 at z' = -8. This corresponds to the maximum crown diameter. Figure 5.17 shows p(s) at 0 = 300 for conical crowns centered at zi = -4 with 2.0. [.;",-..~, --— I. —. Z'= -8 1.5 -.......z'= -6 Z' -4 e;~~~~~...... ~z'= -2 1. 10- 0.5 - " 0.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Within-Crown Propagation Length, s (meters) Figure 5.17: p(s) at various depths z' in the crown layer for conical crowns with centers located at z/ = -4 meters and size uniformly distributed between 4 < c < 8 with a/c = 0.5 and 0 = 30~. height uniformly distributed over 4m < c < 8m and with a/c = 0.5. Comparison of Figures 5.16 and 5.17 indicates that the shorter within-crown pathlengths are slightly more important for conical shapes than for spherical shapes. For crowns of identical size with a distribution in center location zi described by the PDF pz,(zi) is given by: p(sl; z'; c) = p(sl; z'; c; z)p, (zi)dzi. (5.58) Figures 5.18 and 5.19 show p(s) at various values of zi for spherical and conical crowns each with constant size c = 6m and center location uniformly distributed with -7m < zi < -3m.

97 1.5 1.0 A, 0.5 - 01F 0.0 L" 0.0 2.0 4.0 6.0 8.0 Within-Crown Propagation Length, s (meters) Figure 5.18: p(s) at various depths z' in the crown layer for a spherical crown with size c = 6 meters, a/c = 1, center location uniformly distributed between -7 < zi < -3 and = 30~. 2.0 1.5 cv 1.0 0.5 0.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Within-Crown Propagation Length, s (meters) Figure 5.19: p(s) at various depths z' in the crown layer for a conical crown with size c = 6 meters, a/c = 0.5, center location uniformly distributed between -7 < zi < -3 and 0= 30~.

98 To consider the effect of crowns distributed in both size and center location, these parameters may be coupled such that small crowns are located low in the crown layer and large crowns are high in the layer. This simulates crown placement in many natural canopies. If zi is distributed between -8 < zi < -3 and c is varied linearly with zi as c = zi + 9 then the smallest crowns have height c = 1 and are located at zi = -8 and the largest crowns have height c = 6 and are located at Zi = -3. Figures 5.20 and 5.21 show p(s) for spherical and conical crowns at various depths z' in the crown layer for such a case at 0 = 30~ with zi uniformly distributed. 1.0 0.8 0.6 0.4 0.2 0.0 L0.0 1.0 2.0 3.0 4.0 5.0 Within-Crown Propagation Length, s (meters) 6.0 Figure 5.20: p(s) at various depths z' in the crown layer for a spherical crown with crown center location uniformly distributed between -8 < zi < -3 and size varying linearly with zi between 1 < c < 6 and a/c = 1 such that the smaller crowns are low in the canopy and the larger crowns are high in the canopy. The incidence angle 0 = 30~. To this point, only uniform distributions in crown size and location have been considered. Another case of interest is crowns with center location lognormally dis

99 1.5 I\~~~~~~ ', ---- z'= -10 1.0 -.... -- z=-8 - - z'= -6 ^ — ^ \ I _'_______ z'' =-3 0.5 -" 0.0 0.0 1.0 2.0 3.0 4.0 Within-Crown Propagation Length, s (meters) Figure 5.21: p(s) at various depths z' in the crown layer for a conical crown with crown center location uniformly distributed between -8 < zi < -3 and size varying linearly with zi between 1 < c < 6 and a/c = 0.5 such that the smaller crowns are low in the canopy and the larger crowns are high in the canopy. The incidence angle 0 = 30~. tributed. A random variable x which is lognormally distributed has PDF px(x) = xv/ — exp { n (5.59) with median= m and mean= mexp(a2/2). It is clear that x is defined over the interval 0 < x < oo. To apply this distribution to a crown layer of finite height, px(x) is offset to -d (the lower boundary of the layer) and normalized so that Pz(Zi)= d )d) (5.60) Poi pd x)dx Figure 5.22 compares the uniform distribution to lognormal distributions with m = 1 and a = 1 and with m = 2.5 and a = 1. In general, the lognormal distribution places more crowns in the lower portion of the crown layer and, when coupled with crown size that varies linearly with center location, accounts for a higher number density of small crowns in the canopy.

100 0.7 0.6 Lognormal Distribution, m=l, o=1 0.5." \- Lognormal Distribution, m=25, o=1 -\ ---- Uniform Distribution 0.4 0.3 0.2 ----------------------- 0.1 0.0 -8. -7. -6. -5. -4. -3. Cown Center Location, zi (meters) Figure 5.22: PDF of crown center height p(zi) for lognormal and uniform distributions. Figures 5.23 and 5.24 show p(s) at 0 = 30~ and z' = -10 meters for spherical and conical crowns with these three distributions in center location. The lognormal distributions yield p(s) that peaks at lower values of s than the uniform distribution.

101 1.0 0.8 0.6 - 0.4 - 0.2 - / 0.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Within-Crown Propagation Length, s (meters) Figure 5.23: p(s) at z' = -10 meters in the crown layer for a spherical crown with center location having three different distributions between -8 < z < -3 and size varying linearly with zi between 1 < c < 6 and a = c such that the smaller crowns are low in the canopy and the larger crowns are high in the canopy. The incidence angle 0 = 30~. 1.5... I I -I - - 1.0 0.5 0.0 L 0.0 1.0 2.0 3.0 4.0 Within-Crown Propagation Length, s (meters) Figure 5.24: p(s) at z' = -10 meters in the crown layer for a conical crown with center location having three different distributions between -8 < zi < -3 and size varying linearly with zi between 1 < c < 6 and a = 0.5c such that the smaller crowns are low in the canopy and the larger crowns are high in the canopy. The incidence angle 0 = 30~.

102 5.3 Calculation of Crown Cross-Sectional Area The expected value of the crown cross sectional area as a function of depth z in a crown layer of thickness d must be considered in order to compute the expected value of the crown layer phase matrix. As illustrated in Figure 5.25, the cross section Ac of an individual crown sliced at depth z depends on the location of the crown 0 - ------------------------------------------------------ Figure 5.25: Cross-sectional area Ac at depth z of a crown centered at zi. center z- along with- the crown shape and size. For a given crown, if zi and crown size are treated as random variables with specified distributions, the expected value of AC is (Ac(z))c= J j Ac(z, zi, t)pz (zi) pt (t) dz dt (5.61) where pz, (zi) and pt (t) are the PDFs describing the distribution of the crown center location z- and crown size parameters t and Ac (z, z-, t) is a function describing crown cross sectional area at depth z for a crown with center at zi and size t. Since, as evident from Equations (4.28), (4.29) and (4.31), the crown layer phase matrix is a linear function of crown cross-sectional area, the behavior of (Ac (z))c is a direct indicator of the behavior of the phase matrix as a function of z.

103 5.3.1 Cross-Sectional Area of Selected Crown Shapes with Specified Size and Location. Given the shape, size and location of a specific crown, the cross-sectional area at a depth z may be determined explicitly by applying the crown shape equations. For a crown with a square column shape of base dimensions a x a and height c with center at zi, the cross sectional area is For a spheroid cr( a2;zi -c < z < Zi + c Ac (zi, a, c)= a2 - 0; otherwise. own with height c, width a, and center at zi: — 7- 1- /2 2 2 na -T1-; z i;-2E_ + i, a,c) =; Zi Z 0;otherwise. (5.62) (5.63) Ac (z, A right circular cone with height c centered at zi, basal diameter a has Ac(z, zi, a,c) 4[1 (; zi - 2 < z zi + 2 0;otherwise. (5.64) Note that the size parameters (a, c) E t. Each of these shapes is non-zero only over the region zc - ~ < z < zt + 2 and may therefore be expressed as an area factor Af times a rectangular pulse function: A (z,, t) = A f(z, zi, t). {u z-(zi )-u [z- (z-+ = A (z, zit) - rect - zi where (a, c) E t, u(z) is the unit step function defined by 1;z> z u(z - Zo) = 0; z<zo (5.65) (5.66) (5.67)

104 and; 1 6< 1/2 rect(z) =;j 1/2 (5.68) 0; otherwise If the crown parameters are known deterministically, (5.66) may be applied straightaway to find Ac. The problem of computing Ac for crowns with statistically specified parameters is examined next. 5.3.2 Cross-Sectional Area for Crowns Distributed in Height If the crown center location has a distribution specified by the probability density function pz, (zi) then (A (z, t)) = Ac (z, zi, t)pz, (zi) dz, (5.69) -= Af (z, zi, t) rect [ ]P, (zi) dzi (5.70) where pzi(zi) is defined over the region z1 < zi < z2 such that PuZ(Zi) =- p.( zi) -rect 2-1+ ) (5.71) Z2 -- Z1 The quantity (Ac (z, t))c may then be written as (A (z, t)) = A1 (z, zi, t) z, (zi) rect - rect[- 2 ] di. (5.72) Jz2 C _ Z2 - Z1 \ Note that the crown centers must be located such that -d + 2 < Z2 < Z1 < -c in 2 - 2 order for all of the the crowns to be completely contained within the layer -d < z < 0. Equation (5.72) is similar to a convolution integral and may be evaluated accordingly. Two cases must be considered. Case I applies when z1 - z2 > c and Case II applies when zl - z2 < C. In each case, the integral must be evaluated over three

105 regions. Equation (5.72) may be expressed as Z. (Z) (Ac (z, t))c = z( Af (z, z, t) pz, (zi) dzi. (5.73) where zu(z) and z((z) represent the upper and lower limits of integration for each of the three regions. These limits are summarized in Table 5.1. The function Table 5.1: Integration limits for computing (Ac (z, t))c. Case I: z1 - z2 > c Range of z Z2 - < Z < Z2 Z2 + ZZ- Z- < + 2n(ZZ) Z + 2 Z + 2 Z1 zl(z) Z2 Z- _ Z- _ Case II: Z1 - Z2 < C Range ofz Z2 - z Z1- Z1 -- < Z < 2+ Z2+2 Z1 +I 2 Zu(Z) Z + Z1- Z1 (Ac (z, t))c may be evaluated for a given crown by applying (5.73) to the specified crown area factor Af and center location PDF Pzi (zi). If the crown center is uniformly distributed in zi then PZ (z.) = rect (-2( )] (5.74) 1- Z2 Z2 - Z1 J and Equation (5.73) may be solved analytically for crowns with areas specified in (5.62) - (5.64). Table 5.2 shows the solution for (Ac (z, t)), for square column, spheroid, and conical crowns as a function of depth z, crown center distribution limits z1 and z2, and crown size (a,c) E t. Figures 5.26 - 5.28 are graphs of (Ac (z, t)), for these crown shapes. Each figure shows cross section as a function of z for a single crown with center uniformly distributed between z2 < Zi < zl and size a = c = 2. It is interesting to note that for each set of crown parameters (A (z, t))cdz = V (5.75) - d where V is the volume of the crown.

106 Table 5.2: Cross sectional area for crowns with centers uniformly distributed between zl and z2. CASE I: z1 - z2 > c Range of z Square Column Spheroid c- < Z < Z2 + c a2(c+2z-2z2) ra2 (c+2z-2z2 2c-z+z2 -2 2 - 2 2(z - 2) 12c.. -(Z- ) Z2 + c < c a2c 7ra2c +2 ^2 Z1-2-z72 6(zl —z2) 2 < Z < + 2c a2(c-2z+2z ) ra2(c+z-z, )(c-2z+2z )2 1- 2 -2 - 2(l -z2) ' 12c2(zl -Z2) Range of z Cone cZ- 2 < Z < c2 + 2 7ra2(c+2z-2z2)( 7c2 -8cz+4z2 +8cz2-8zz2+4z22) 22 - 2 + - 2 ) 2 -Z +2 96c2(Z —Z2) Z2 + c < Z < Z1 2 -ra2c 2 - - 2 12(zl —z2) _ c < z < Z1 + c2 7ra2(c-2z+2zl )3 96c2(z1- z2) CASE II: z1 - z2 < c Range of z Square Column Spheroid c < Z < -Z c a2(c+2z-2z2) 7ra2 (c+2z-22 )2 (c-z+z2 -Z2 2 2 - 2 2(z2i-Z2) 12c2(zl-Z2) -C < Z < Za2 ra2 (3c2 -12z2+12zzl -4zl2+12zz2 4 2-4z 22 ) Z1 — _2, --- __ 12c —a Z2 - c < Z < _ + c a2(c-2z+2zl) 7ra2 c+z-zl )(c-2z+2z )2 2 2 - 2 2(z z -z) 12ca(zi -2) Range of z Cone cZ2 - 2 < Z < Z1 - c ' 7ra2(c+2z-22 )(7c2 -8cz+4z2 +8cz2 -8zz2 +4z22) t2- 2^ S 1- 2 --- —- 96c2(z —Z2) -2 c c ra- < 2+2,r2(3c2-12cz+12z2 +6czl -12ZZ1 +4z1 2+6cz2 -12zz2+4zl Z2+4z22) LZ 2+l 2 - 2$ 1 - - c-2^^222)) — 11-48c2 cL < Z< - c 7ra2(c-2z+2z1 )3 _2 4 _ _2 ____ _1_ + 2 96c2(z-z2)

107 0: an Ic c/3 5.0 4.0 3.0 2.0 1.0 zI= -1, z=-I..... ZI =-1, z2= - I. I - - - - - - 2 - - - - - - z = -1, Z2 = -3 / zI = -1. z2 = -5 /i, I......'- -- ~ ~ // i / ' /. /:. ~ s: o~~~~~~ / s /,' / s /, / I / [ /: ~.. [/n... i... i r r r r r r r r r r r r. r I I 0.0 L._ -6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 Depth in Crown Layer (meters) Figure 5.26: Expected value of crown cross-sectional area at depth z in a crown layer for a single square column crown with center uniformly distributed over 1 < Zi <2 Z2 with a = c = 2. 4.0 I. 0 0 q) c4 c 3 U 3.0 2.0 1.0 0.0 L' -6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 Depth in Crown Layer (meters) Figure 5.27: Expected value of crown cross-sectional area at depth z in a crown layer for a single spheroidal crown with center uniformly distributed over z1 < i < Z2 with a = c= 2.

108 4.0... CI 3.0 z= -1, z2 =-1 <, ------ z -l.z=- - 5 Q U......... z-..- -6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 Depth in Crown Layer (meters) Figure5.28: Expected value of crown cross-sectional area at depth z in a crown layer for a single conical crown with center uniformly distributed over Z1 < Zi < 22 with a = c = 2. 5.3.3 Cross-Sectional Area for Crowns Distributed in Size (A (Z z)) = Ac. (z, z, t) Pt (t) dt (5.76) = J A(z, zi, t) rect [ pt(t) dt (5.77) Expressing the size PDF in terms of the crown height c as = pc(c)- rect [ - -c j (5.79) \ _< i ___ 2 with aC2 - c 2. C2 -- C1

109 where c1 and c2 are the lower and upper limits of crown height, respectively, yields (AC (z, z)) jc A, (z,, t,(c)) p, (c) rect [ rect 2 )] dc. (5.80) JC1 LC C2 - C1 To perform the integration over c, it is convenient to express (Ac (z, c))= c () As (z, zi, t(c)) PC (c) dc (5.81) where the cl(z) and cu(z) represent the lower and upper limits of integration as determined by the product of the pulses in (5.80). Table 5.3 summarizes these limits. The function (AC (z, zi)), may be evaluated by applying (5.81) to specified Table 5.3: Integration limits for computing (Ac (z, i))c. Range zi - < < z < + - + z + - < z < Zi + 2 2- ~- 2 _ -2 - Zu(Z) C2 C2 C2 zl(z) -2(z- zi) C1 2(z- zi) crown area factor Af and size PDF Pc (c). If the crown is uniformly distributed in c such that pc(c) =, (5.82) C2 -C1 then the cross sections may be computed analytically for the crown areas specified in (5.62) - (5.64). Table 5.4 shows the solution for (Ac (z, zi))c for square column, spheroid, and conical crowns as a function of depth in the crown layer z, crown center location zi, and crown height limits c1 and c2. Here, the crown size was assumed to vary with height so that a = c for all values of c. Figures 5.29 - 5.31 are graphs of (Ac (z, Zi)), for these crown shapes.

110 Table 5.4: Cross sectional area for crowns with height uniformly distributed such that c1 < c < C2 and a = c. Range of z Square Column Spheroid -.< < C2- -. (c)3 8(z-z }3 7r(c2+2z-2zi)2(c2-4z+4zi Zi - 2 < - 2 3(c2 -c) 3(c 2-ci 12(C2-c) Z- < Z < Z C +CC cC2 r(cl2+ClC2+c22-12z2+24zzi-12zi2) Z 2 - 2 3 12 Zi +; ~ < < 2+ C23 8(z —zi 3 7r(c2+4z-4zi)(C2-2z+2i)2 Z + 2 2- 1 3(c2-Cl) 3(c2-ci) 12(c2-cl) Range of z Cone Z-C < Z < Zl- 2 7r(c2+2z-2zi)(c22-8c2z+28z2+8c2zi-56zzi+28zi2) 2: 2 - -- 2 48(c2-c ) i- s~ < ~Z1< + 2 117r(cl2+cIc2+C22-6cIz-6c2z+12z2+6c1zi+6c2i -24zz- +12z2) + C < Z < + C2-48 ---j - -- 2 2 48 l7r(c2-2z+2z)3 z + 2-< z <_ l 48(c2-c) 2 - +2 48(c2 -cl 8.0 I-. 0 CD 0 1 -o o u 6.0 F 4.0 P Ir /; \ I r,,. I'%. \ I. I.... I... \ ~; I; \ / ~ \ I; I I;~~~~~~~!:, i.... 2.0 - 0.0 L -5. 0 -4.0 -3.0 -2.0 -1.0 0.0 Depth in Crown Layer (meters) Figure 5.29: Expected value of crown cross-sectional area at depth z in a crown layer for a single square column crown with c uniformly distributed between c1 and c2, a = c and zi = -2.

111 5.0 -~ < 4 --a r: 0 o co u (o 0 U 2 4.0 3.0 2.0 1.0 0.0 L-5.0 -4.0 -3.0 -2.0 -1.0 0.0 Depth in Crown Layer (meters) Figure 5.30: Expected value of crown cross-sectional area at depth z in a crown layer for a single spherical crown with c uniformly distributed between cl and c2, a = c and i- =-2. 5.0.E a) co 0 U Q) 3 U.. 4.0 3.0 2.0 1.0 0.0 L -5.0 -4.0 -3.0 -2.0 -1.0 0.0 Figure 5.31: Expected value of crown cross-sectional area at depth z in a crown layer for a single conical crown with c uniformly distributed between cl and c2, a = c and zi = -2.

112 Each figure shows the cross section as a function ot z tor a single crown with c uniformly distributed between cl < c < c2, crown center at z- = -2 and a = c. It is interesting to note that for each set of crown parameters (A (zz)) dz = V (5.83) J-d where V is the average volume of the crown. 5.3.4 Cross-Sectional Area for Crowns Distributed in Height and Size For crowns with distributions specified in both s and zi (Ac (Z))c = J2 ji Af(z, z, t(c))pi(zi)pc(C) c, 2 -rect[ ziec1 ]ect[C- ( 1 + rect cdc. (5.84) c -- 1 C2 - C1 In this case, analytical solutions are more tedious to derive and numerical techniques become more efficient. Figures 5.32, 5.33 and 5.34 show (A, (z)), for square column, spherical, and conical crowns, respectively. All crowns have been assigned a uniform size distribution with 1 < c < 3 and a = c. The crown centers have been assigned a uniform distribution with z2 < zi < zK. As seen in the cases for which zl = -2, Z2 = -8, the behavior of (A, (z)), for all three shapes becomes similar for crowns whose centers are widely distributed throughout the crown layer.

113 5.0 E 0 <: 0 IU E 4.0 3.0 2.0 1.0 0.0 - -10.0 -8.0 -6.0 -4.0 -2.0 0.0 Depth in Crown Layer (meters) Figure 5.32: Expected value of crown cross-sectional area at depth z in a crown layer for a, single square column crown with center uniformly distributed between z1 and z2 and c uniformly distributed between 1 and 3 meters with a = c. 4.0 E o ct (A C 3.0 2.0 1.0 0.0 '-10.0 -8.0 -6.0 -4 0 -2.0 0.0 Depth in Crown Layer (meters) Figure 5.33: Expected value of crown cross-sectional area at depth z in a crown layer for a single spherical crown with center uniformly distributed between zl and z2 and c uniformly distributed between 1 and 3 meters with a = c.

114 3.0. "s ~ ~ ~~~~~~~~~~~~zi = -5, z2 = -5 --------- =-4, z, =-6 - 1 ------ z =. -.2 =-7......z = -2, z2 = -8 1.0 0 2.. - - - ----- - 0.0 -10.0 -8.0 -6.0 -4.0 -2.0 0.0 Depth in Crown Layer (meters) Figure 5.34: Expected value of crown cross-sectional area at depth z in a crown layer for a single conical crown with center uniformly distributed between zl and z2 and c uniformly distributed between 1 and 3 meters with a = c.

115 In all of these cases, d (AC (z) dz = V (5.85) where V is the average volume of the crown. Applying Equations (4.28), (4.29) and (4.31), the average crown layer phase matrix may be written as (Pc (Os, Os; 0i, oi))1 =d (P( 86; 0, i,; z))cdz (5.86) =- -- ] (A (z))c dz E INTk kZ (OS, s; Oi, i;; sO,,k ) (5.87) k=1 ___= d ^,^(14) (5.88) where Nt is the number density of trees per unit area and 'pl) (0s, 0s; 0i, Xi) is the phase matrix for a unit volume contained in a single crown. For a continuous canopy, V/d = 1/Nt so (Pc (Os, Os; Oi, qi))c = p(l) (0, O; 0i, Xi) (5.89) 5.4 Summary of Crown Shape Statistics This chapter has presented the derivation of shape statistics for specified classes of tree crowns. Given thel distribution for within-crown propagation length p(s1l) for individual crowns from Section 5.1, the techniques discussed in Section 5.2 may be applied to compute p(sll) for distributions of crowns. The PDF p(sjl), together with the cross-sectional area A<(z) discussed in Section 5.3, may be coupled to the radiative transfer solution for an open-crown canopy presented in Chapter IV to solve for backscatter from a. particular canopy. This is done in Chapter VI.

CHAPTER VI MODELING ANALYSES AND APPLICATIONS This chapter presents some examples in which MIMICS is used to model the scattering characteristics of some vegetation canopies. The first three sections present analyses in which MIMICS I has been applied to model various types of canopies. Section 6.1 presents a modeling analysis in which MIMICS is used to model polarization phase difference for a selection of corn fields in Illinois. In Section 6.2, MIMICS is used to model multi-angle and multi-temporal backscatter from a walnut orchard in Fresno County, California. Section 6.3 presents an analysis of canopy transmissivity and backscatter from various forest stands in the Alaskan boreal forest. In each of these cases, the model is driven with ancillary ground measurements and the output is compared to radar data measured with truck-based or aircraft-mounted systems. Finally, in Section 6.4 MIMICS II is applied to simulate transmissivity and systems. Finally, in Section 6.4 MIMICS II is applied to simulate transmissivity and backscatter for several of these canopies and the results are compared with MIMICS I simulations. 6.1 MIMICS I Corn Canopy Modeling Using SIR-B Data The second Shuttle Imaging Radar (SIR-B) flew aboard the Challenger on mission STS-41G in October of 1984 (Cimino et al., [7]). This instrument provided 116

117 researchers with orbital L-band HH-polarized SAR imagery for various science objectives. One study conducted during the SIR-B flight is described by Dobson and Ulaby, [16] and Ulaby et al., [66]. They used the SIR-B instrument together with aircraft SAR underflights to address such issues as land-cover classification and assessment of near-surface soil moisture content. The test site for this study comprised an area in west-central Illinois of roughly 250 km2 within which approximately 400 agricultural fields were surveyed for ancillary ground measurements of canopy properties. As a first modeling example, MIMICS I is applied to model characteristics of the backscatter from a number of the corn canopies for which ground measurements and SAR data are available. Polarization phase difference distributions were generated from the aircraft SAR data for over 80 different corn fields in the test site (Ulaby et al., [66]). From these distributions, the mean like-polarized phase difference was estimated for each field. For a canopy with small like-to-cross polarized b1ackscatter ratio, the mean like-polarized phase difference relative to VV-polarization, A'HHvv may be approximated by considering the quantities of the modified Mueller matrix defined in terms of the scattering matrix elements (Equation 2.6). If cross-polarized backscatter is small, A>HH-VV - Tan1 [Im(SvShh) (6.1) Rle(Svv hh) [T /,... = Tan1 ['T](43) (6.2) 12 3,3 J where [T](mn) represents the (mn, n.) element of the canopy transformation matrix. From ancillary ground measurements, canopy characteristics representative of an effective average corn canopy were estimated. These characteristics are listed in Table 6.1. The stalks were assigned vertical orientations while the leaves were assigned an orientation distribution uniform in the incremental solid angle dQ = sin OdOdo. Since

118 Table 6.1: Corn canopy parameters for fields observed by the aircraft SAR. Canopy density = 8.77 stalks/m2 Stalk height = 2.5 m Stalk diameter = 2.5 cm Stalk dielectric constant = 6.5 +i 0.5 Soil dielectric constant = 15 +i 2 Soil RMS height = 2 cm Soil correlation length = 26 cm Leaf gravimetric moisture = 0.1 Leaf number density = 200/m3 Leaf diameter = 6.18 cm Leaf thickness = 0.2 mm this experiment occurred late in the growing season, the leaves were senescent and quite dry and therefore had minimal effect on the canopy backscatter. MIMICS I was run at an L-band frequency of 1.2 GHz as a. function of incidence angle. Figure 6.1 compares the measured polarization phase difference to that predicted by MIMICS. The model agrees well with the data, predicting the sharp increase in AO observed for 25~ < 9 < 35~, which is caused by the effect of the Brewster angle on the reflection from the corn stalks. It is important to note that while each circle on this graph represents the observed Ad for a single corn canopy, the model calculation represents the An calculated by MIMICS for the average field conditions represented in Table 6.1, and therefore the results do not incorporate the natural field-to-field variability inherent in the scene. Since a major objective of this study was to assess the ability of orbital SAR for use in soil moisture estimation, the response of the measured SIR-B backscatter to changes in this parameter is now examined (Dobson and Ulaby, [16]). Volumetric soil moisture has been estimated for many of the corn fields in the test site through analysis of soil samples extracted from three locations in each field. Table 6.2 lists the average canopy characteristics that correspond to these fields. Leaf parameters

119 160... |........ O 20 0% O 0 a 1 20. - > 100. -.. 0 - 8 0. 60. S, 40. 1 -0 00 0 0rn~ 0 0 ed 20.O~ 0 0 0 0o Measured From SAR Image 0. -- Calculated(MIMICS) o~ -20. 15. 25. 35. 45. 55. Incidence Angle (degrees) Figure 6.1: Comparison of the polarization phase difference calculated by MIMICS to that extracted from the aircraft SAR imagery. Table 6.2: Corn canopy parameters for fields observed by the SIR-B SAR. Canopy density = 6.58 stalks/m2 Stalk height = 2.8 m Stalk diameter = 2.5 cm Soil RMS height = 2 cm Soil correlation length = 26 cm were the same as those given in Table 6.1. Using the dielectric models described in Appendix A and assigning soil textural components of 10% sand, 30% clay and 60% silt, MIMICS was run as a function of volumetric soil moisture at a frequency of 1.2 GHz for an incidence angle 0 = 30~, corresponding to SIR-B SAR parameters. Figure 6.2 compares MIMICS predictions to data extracted from SIR-B imagery. Three MIMICS simulations are shown in this figure. Each of the first two simulations presents the MIMICS response to changing soil moisture with all other canopy parameters help constant. The uppermost curve

120 5.0 i '. (a) Stalk mg = 0.125 ---—.. (b) Stalk mg = 0.9 0.0- ----- (c) Stalk mg= 1.8 mv - 0.12 0 Measured SIR-B Data. ---m.. —. b~o t o 0_-__ y- -50~t - o - 0 0 0 — -10.0 El b a, -15.0 0.0 0.1 0.2 0.3 0.4 0.5 Volumetric Soil Moisture (mv) Figure6.2: L-band a0 HH-polarized backscatter response to changes in volumetric soil moisture. Data measured by the SIR-B SAR are compared with MIMICS simulations for canopies with (a) dry stalks with gravimetric moisture = 0.125, (b) wet stalks with gravimetric moisture = 0.9, (c) stalks with gravimetric moisture coupled to the soil moisture via mg = 1.8 mv - 0.12. depicts the a~ response for a canopy whose stalks have a gravimetric moisture (mg) of 0.9 (relatively wet) while the lowermost curve depicts a~ for a canopy with stalks of mg = 0.125 (relatively dry). It is seen that MIMICS predicts slightly less sensitivity of aO to changes in soil moisture than was observed by SIR-B. A possible explanation for this phenomenon is that soil moisture may be coupled to other canopy biophysical parameters and, therefore, changes in soil moisture may be coupled to changes in other canopy properties. One such mechanism is illustrated by the third MIMICS simulation in Figure 6.2. While the wet and dry stalk moisture conditions present an upper and lower bound for the SAR data, coupling the stalk moisture (mg) to the soil moisture (mv) via the equation mg = 1.8 mv - 0.12 successfully predicts the

121 appropriate sensitivity of r~ to changes in moisture. Unfortunately, the strategy applied in collecting canopy ground measurements involved using separate teams for measuring soil and vegetation parameters and thus there are very few corn fields for which there are coincident soil and stalk moisture measurements. The insignificant amount of these data make it impossible to verify any discernible relationship between stalk and soil moisture. However, it is instructional to examine the potential for using MIMICS together with SAR measurements to decouple canopy moisture parameters. If one assumes that plant and soil moisture are coupled, the question arises of how to apply measured backscatter parameters together with MIMICS in order to separately estimate the moisture content of the plant and soil. Figure 6.3 shows the co-polarized backscatter response for a corn canopy under wet and dry moisture conditions with other parameters being identical. The polarization responses for these cases are very similar, indicating very little change in the polarization characteristics of the backscatter for the wet and dry conditions. Examination of Figure 6.4 gives a more direct indication of c~ response to changes in stalk and ground moisture content. Here, to assess the sensitivity of canopy backscatter to changes in stalk and soil moisture, MIMICS was run at 0 = 200 varying both moisture parameters. Figure 6.4 (a), (b) and (c) show the response of U7IIT, a VV and ALHH-VV respectively. Roughly speaking, the HH-polarized backscatter exhibits about 4-5 dB of change over the displayed range of stalk moisture and about 10 dB of change over the range of soil moisture. VV-polarized backscatter exhibits about 13 dB of change over the range of stalk moisture while changing by about 10 dB with soil moisture. AO exhibits about 125~ of change with stalk moisture but has virtually no dependence on soil moisture.

122 By combining these three responses, one may illustrate how to predict both moisture contents from a given set of SAR observations. Assuming all other canopy parameters are known, stalk moisture may be estimated from AO. Then, arrHi and aVV may be applied to estimate soil moisture. Given measured values of caHH, aV and AqO for the particular canopy structure, contours that represent families of constant backscatter and phase difference are generated from Figure 6.4. Each contour represents a solution set in stalk and soil moisture for that particular backscatter quantity. For a given canopy state, the intersection of contour families yields the desired moisture values. Figure 6.5 illustrates two examples of this process. Figure 6.5(a) illustrates an example of the intersection of the contours for a canopy with wet conditions and Figure 6.5(b) shows an example for the same canopy with dry conditions. In both cases, AO is constant as a function of soil moisture, and therefore determines the stalk moisture. crH and IyVE then yield the solution for soil moisture. It is important to note that Figure 6.5 represents an ideal modeling case in which aOHH, 7V and AO have been chosen to yield a unique intersection point. In applying this technique to determine canopy moisture status from an actual SAR image, it is highly unlikely that the three families will all intersect at the same point and further analysis involving estimation of canopy parameters would have to be applied for the model to converge on unique estimates of plant and soil moisture.

0 0 z '9. L z 45. " 'Sel&4 - opgle % (ale) Wt cndi ticit A (a) Wet conditions 0.. ti: IO Ori ) C O gle.,,5..itvA gl D) ury conri-lons.. Figure 6.3: Co-polarized L-band response of a corn canopy at 0 = 300 for (a) wet conditions with soil moisture = 0.3 and stalk moisture = 0.6 and (b) dry conditions with soil moisture = 0.08 and stalk moisture = 0.35.

124 ro. *0 -10. -20 S Mosture Stalk Moisture *~~0^:l,~~~~~~0.4 0'4 003 02. 0.6 0.4 ~S 0 0.6 (b) 0 Soloistutre Stalk Moisture (a) co 0 -20.0 --30~ 4)0 (b) 125. o 35. 10. Soil Moi.stue 0.8 Stalk Moisture tigure.4:;orn canopy backscatter response to changes in volumetric soil moisture and gravimetric stalk moisture for an incidence angle 0 = 20~. (a) HH-polarized response, (b) VV-polarized response, (c) polarization phase difference response.

125 0.8 0.7 Io C t~ 0.6 0.5 0.4 0.3 7 0.2 ' 0.0 0.1 0.2 0.3 Soil Moisture (a) Wet conditions 0.4 0.5 0.8 - /3 C/l 0.0 0.1 0.2 0.3 0.4 0.5 Soil Moisture (b) Dry conditions Figure 6.5: Solution set for estimation of canopy moisture conditions (a) wet conditions with soil moisture = 0.3 and stalk moist (b) dry conditions with soil moisture = 0.08 and stalk moisture = 0.35.

126 6.2 Eos Synergism Study MIMICS I is now used to model microwave scatterometer data that were obtained during the August 1987 Eos Simultaneity Experiment (Cimino et al.,[9]; Dobson et al.,[15]; McDonald et al.,[41], [42],[43], [44]). During this experiment, truck-based scatterometers were used to measure radar backscatter from a walnut orchard in Fresno County, California. The modeling of two sets of L- and X-band measurements are discussed. The first set consists of a series of multiangle data for which a set of trees was observed at varying angles of incidence. The second set consists of a series of diurnal measurements in which this same set of trees was observed continuously over several 24 hour periods. With in situ ancillary data describing canopy architecture and moisture conditions used as input, MIMIICS is run at L-band and X-band frequencies of 1.5 GHz and 9.6 GHz. Measured scatterometer data are compared to theoretical data generated by MIMICS. MIMICS is seen to predict the diurnal variations that are observed on 24 hour cycles. -Examinations of backscatter response to changes in canopy dielectric properties are performed to determine the causes of the changes observed in the short term trends and diurnal patterns. This section presents a brief overview of the modeling analysis. A more complete discussion is provided in Appendix G. 6.2.1 Orchard Canopy Characteristics As part of the synergism study, an extensive set of ancillary data was collected in order to characterize the walnut orchard. Data describing canopy architecture (Ustin et al.,[71],[72]), dielectric properties (Dobson,[10]) and canopy water status (Weber and Ustin, [82],[83]) were analyzed to determine canopy density, branch and leaf orientation and size distribution, constituent dielectric properties, and other gross

127 canopy characteristics. To adapt the branch geometry data for input to MIMICS, the orchard is divided into distinct crown and trunk layers with heights of 3.1 m and 1.7 m, respectively. The branches are then divided into the four size classes identified in Table 6.3. Figure 6.6 is a sketch of the geometry of an individual tree, showing the four branch classes and the leaves. The orientation functions are converted into probability distributions for use in MIMICS by dividing each by a normalizing factor given by foi f (0) dO. stem secondary branch 3.1 meters r trunk branches 1.7 meters Figure 6.6: Illustration of a, walnut tree showing the four branch classes and the leaves. Characteristics of the leaves were determined from detailed leaf counts (Ustin et al.,[72]) and are summarized in Table 6.4. The leaves are modeled as thin circular dielectric disks with a specified diameter and thickness. The leaf number density together with the leaf diameter and crown height yield an equivalent canopy leaf area index (LAI) of 3.4.

128 Table 6.3: Canopy Branch Classes. Branch Size Class Constituent Class Trunk Crown Branches Characteristic Branches primary secondary stems Max. Diam. (cm)- 4.0 0.9 0.4 Min. Diam. (cm) 4.0 0.9 0.4 Ave. Diam. (cm) 7.3 1.9 0.6 0.1 Ave. Length (cm) 92.8 35.8 10.9 5.0 Density (#/m3) 0.13 1.25 1.14 250 Orientation f (0) cos6 0 sin4 20 sin 0 sin 0 Table 6.4: Leaf Characteristics. Number density Average diameter Average thickness Leaf area index Orientation Folding angle Radii of curvature 250 leaves per cubic meter 7.47 cm 0.1 mm 3.4 f(0) = sin X = 152~ along midrib pi = 7.7 cm (along midrib) P2 = 10 cm The radii of curvature of the leaves were determined form measurements of the leaf folding angle (Appendix G). The effect of leaf curvature on canopy backscatter was accounted for through the equation (Sarabandi et al. [51]): where ac 1 1 2 ~- (l1(7l)@ (72)2 ar 71 72 F(7) = j exp (iu2) du (6.3) (6.4) is the finite range Fresnel integral, a a /a(6.5) Yi=- 2- 72 = - (6.5) 2n pc V p and ko is the free space wavenumber. This effect was approximated in MIMICS by using flat leaves with effective diameters that depend on the frequency under

129 consideration. Table 6.5 lists the normalized backscatter and corresponding effective diameters for L- and X-bands. At L-band, the effect of leaf curvature is essentially negligible. Table 6.5: Effects of Leaf Curvature at L- and X-Bands. Normalized Backscatter Effective Diameter (ac/la ) (cm) L-Band 0.972 (-0.1 dB) 7.42 X-Band 0.297 (-5.3 dB) 5.52 Flat Leaf 1.000 (0.0 dB) 7.47 A correction factor that accounts for the difference between the actual canopy LAI and the LAI observed with the scatterometer system may be determined by considering the radar measurement volume together with the variation of leaf number density with height (Appendix G). This factor estimates the canopy LAI that is observed by the scatterometer, which is a slowly varying function that has a minimum of 0.35 at 0 = 40~, increasing to a maximum of 0.6 at 51~, and then tails off to 0.55 at 55~. Canopy Dielectric Characteristics Observations of the relative dielectric constant of soil and vegetation were made in situ at 1.2 GHz using a portable dielectric probe. Observations were made of the soil surface and tree trunks. Trunk measurements included both the exterior bark and the interior sapwood. A statistically insignificant amount of dielectric data were recorded for the vegetation in the crown layer. However, the dielectric behavior of these constituents may be inferred from observations of other canopy physiological parameters, and the models applied here to predict the relative dielectric constant do in fact agree with the few recorded observations.

130 The dielectric properties of the tree boles were seen to vary dramatically with time. Figure 6.7 shows a piecewise fit to the measured dielectric constant. This represents the best estimate of the trunk dielectric behavior at L-band. These data 80.0 --- 70.0 - 60.0 o O U 50.0 o f \ o / j\ ---- Re. E,- Piecewise Fit 40.0 / --- —-- Im. Er - Piecewise Fit Q) 300 l \ / \ / o \uRe. eC - Measured Q 30.0 o i ~20. /0 \ f \/ X a | "o Im. E - Measured 10.0. - 0.0 24. 25. 26. 27. Day of August Figure 6.7: Comparison of a periodic piecewise fit to measured L-band trunk dielectric constant data for real and imaginary parts. were recorded during the three day period that coincides with the time during which diurnal scatterometer data were recorded. The numbers on the time axis correspond to midnight on that day of August. Figure 6.8 illustrates the best estimate of the soil dielectric behavior. During these three days, the orchard was irrigated 2.5 hours per day beginning at 6:00 each evening. The irrigation periods are manifest by the jumps in the dielectric constant that begin at 6 p.m. each day. The dielectric continues to increase until the irrigation shuts off. Then,,. decreases as the soil dries. The loss tangent of the soil dielectric was assigned a value of 0.1 at L-band, as was determined from the measured data.

131 40. - 30. --- L-Band —. ---- X-Band CT* 20. 10. 0.00 I 24. T 25. T 26. T 27. (i) (i) (i) Day of August Figure 6.8: Behavior of the soil dielectric constant showing the estimated behavior of the L- and X-band dielectric constant. The symbol (i) indicates the beginning of a 2.5 hour irrigation period.

132 To determine the X-band soil dielectric, the dielectric model presented in Appendix A was inverted using the L-band data, thereby obtaining values for effective soil volumetric moisture. The dielectric model was then applied at 9.6 GHz to estimate the X-band dielectric. Leaf gravimetric moisture content mngi was used to determine the leaf dielectric constant. Analysis of wet and dry leaf weights indicate that the average leaf gravimetric moisture was approximately 0.7. Applying the vegetation dielectric model presented in Appendix A, the relative dielectric constant of leaves were found to be 28.3 + i8.5 and 21.8 + iS.8 at L- and X-bands, respectively. This value was also assigned to the dielectric of the higher order stems. No discernible variation of leaf gravimetric moisture with time was observed. Canopy water status was analyzed to estimate the behavior of the branch dielectric constants. A periodic piecewise fit to the measured leaf water potential was scaled to obtain the branch dielectric behavior shown in Figure 6.9(a). This figure shows the real part of the piecewise fit to the L-band dielectric constant for the three classes of woody vegetation. All measured values of the branch dielectric that were recorded during this time are also shown. The X-band dielectrics were obtained through application of the vegetation dielectric model. This model was numerically inverted at L-band (1.2 GHz) using the dielectric functions shown in Figure 6.9(a), yielding effective values of branch moisture as a function of time. Given the effective moisture, the model was then applied at 9.6 GHz to obtain the real and imaginary parts of the X-band dielectrics. Figure 6.9(b) illustrates the real part of or at X-band.

133 60.0 50.0 40.0 c CZ (L 1: Trunk Layer ------- Primary Branches Secondary Branches o Measured Branches 30.0 20.0 - 10.0 0.0 24. 25. 26. 27. Day of August (a) 60.0 50.0 40.0 St.v) td 30.0 20.0 Trunk Layer ------- Primary Banches ------ Secondary Banches 10.0 - 0.0 2 24 25. 26. 27. Day of August (b) Figure 6.9: Dielectric constants of woody constituents for (a) L-band and (b) X-band.

134 6.2.2 Modeling Analysis As a first step in the modeling analysis, MIMICS was run as a function of radar look angle at L- and X-bands. Table 6.6 lists the canopy dielectric parameters used in this analysis. These values correspond to measurements made at the approximate time that the multi-angle scatterometer data were recorded. Figure 6.10 shows a Table 6.6: Canopy Dielectric Characteristics. Constituent L-Band X-Band Ground Surface 25 + i2.5 20.2 + i7.6 Trunk Branches 45 + i1l.2 35.0 + i14.8 Primary Branches 34 + i8.5 25.9 + ilO.8 Secondary Branches 30 + i7.5 22.7 + i9.4 Leaves and Stems 28.3 + i8.5 21.8 + i8.8 comparison of L- and X-band modeled and measured data over the range 40~ < 0 < 55~ for like- and cross-polarized backscatter. Figure 6.10(a) compares the predicted L-band backscatter with the scatterometer data. This figure demonstrates very good agreement between.MIMICS-generated data and the measured values. The likepolarized backscatter exhibit similar amplitudes with HH being slightly higher than VV in both the measured and modeled data while the cross-polarized backscatter is about 5 dB lower than the like-polarized response. The failure of the model to predict the cross-polarized backscatter at 400 is attributed to the inhomogeneous characteristics of the orchard canopy architecture. Whereas MIMICS I has been derived for a canopy that has a continuous crown layer, it is being used to model backscatter from a canopy with a crown layer that is discontinuous. As incidence angle becomes smaller, a larger proportion of the canopy area observed by the scatterometer consists of smooth, bare soil that is. not covered by the orchard canopy. Since the model predicts backscatter for a canopy

135 -10.0 -15.0 0 -20.0 _-S —A______-, _ __..... -/~ —... —. —... A VV MIMICS ------- HH MIMICS ------ HV MIMICS o VV Measured o HH Measured L-Band A HV Measured -25.0 -30.0 35. 40. 45. 50. 55. 60. Angle of Incidence (degrees) (a) -8.0 -18.0 m C) ts 13 0 pol. A A A A cross Xpol. - X-Band - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -~~~~~~~~~~~ -28.0 -38.0 35.... I.... I........ I.... I i. 40. 45. 50. 55. 60. Angle of Incidence (degrees) (b) Figure 6.10: Comparison of MIMICS results with measured L- and X-band multi-angle data. (a) compares L-band modeled total canopy backscatter to the scatterometer measurements for like- and cross-polarized configurations (HH, VV, HV). (b) compares X-band modeled direct crown backscatter to the scatterometer measurements for these same polarizations.

136 that has a homogeneous crown layer, some error will be introduced in the modeled data. We expect that the model will be more successful in predicting backscatter from this orchard at higher angles of incidence since the scatterometer observes almost no bare soil at these angles. We also expect this effect to be more pronounced for cross-polarized configurations since a smooth soil surface generates very little cross-polarized backscatter compared to that generated by the crown layer. As was found in more detailed analyses (Appendix G), the measured X-band backscatter consists primarily of the direct crown contribution to the total canopy backscatter. Figure 6.10(b) compares the predicted direct crown X-band backscatter with the scatterometer data. Here, MIMICS agrees with the level of the like-polarized backscatter but underestimates the cross-polarized response by as much as 10 dB. The failure of MIMICS to more accurately reproduce the angular dependence of the like-polarized backscatter at 45~ and 50~ may also be attributed to the inhomogeneous nature of the orchard canopy. The effective canopy geometry sampled by the scatterometer measurement volume changes with 0. As radar incidence angle changes, the canopy )a.ckscatter responds to these changes in the sampled canopy volume. The angular dependence of backscatter at X-band has been partially accounted for by applying the LAI correction factor in generating the multi-angle MIMICS data. However, crown layer discontinuities also affect the character of backscatter from the stems and branches. This effect is more prevalent at X-band in part because of the relatively narrow X-band beamwidth and also because the crown layer constituents that contribute most to this effect (leaves and smaller branches) contribute more to X-band scatter than to L-band. As previously noted, the X-band cross-polarized backscatter is significantly underestimated by MIM4 ICS. In general, the effect of higher-order multiple scattering on

137 ra.dar backscatter becomes more important as frequency increases. Ulaby et al. [65] have shown that, at millimeter wave frequencies, a numerical solution to the radiative transfer equations in which higher-order scattering is accounted for may add more than 10 dB to the predicted first-order cross-polarized backscatter while having little effect on the like-polarized backscatter. Since the numerical solution for radiative transfer requires specifying the scattering phase matrix in all incident and scattering directions, determination of the higher-order scattering contribution becomes very computationally intensive. The phase matrix of the walnut orchard crown layer has a very complicated form and determination of the numerical solution is computationally prohibitive. Although an expression for the second-order scattering in the crown layer may be derived (Appendix E), analysis of these higher-order effects is beyond the scope of this study. While Ulaby et al. [65] derived their results at millimeter wave frequencies for which the scattering albedo for vegetation w ^ 0.6-0.9, it is understood that w usually increases with increasing frequency. In light of the study by Ulaby et al.,[65], it is expected that, as frequency increases, higher-order scattering would first be manifest in ternms of its effect on the cross-polarized backscatter. Having established that MIMICS successfully predicts canopy backscatter as a function of angle, the model is now run at a constant incidence angle, 0 = 550, while varying the canopy dielectric parameters so as to simulate the variations seen over the three-day diurnal experiment (Figures 6.8 and 6.9). Figure 6.11 presents the resulting computed backscatter along with the measured values of canopy backscatter for the like- and cross-polarizations. At L-band, MIMICS successfully predicts the appropriate level of the measured data together with the decreasing trend in backscatter observed over the three day period for all three polarization configurations. Furthermore, MIMICS predicts the 1 to 2 dB dip seen in uaVV and aoH in the

138 early afternoon of each day. The X-band MIMICS data presented here represent the direct crown component of total canopy backscatter and have been produced for a canopy with an effective leaf area index of 1.0 in order to account for the variations in effective canopy geometry as a function of incidence angle. An offset of 8 dB has been added to the cross-polarized MIMICS data to approximate the effects of higher-order scattering. Although the measured X-band data exhibit significantly more scatter than does the L-band data, the early afternoon dip in backscatter is present for all three polarizations and is predicted by MIMICS. The variation in the measured data. that is associated with the scatterometer measurement process comes primarily from two sources. The first of these is fading that arises from the coherent nature of the scatterometer. Following the analysis in Ulaby et al. [70], pp.483-486., the uncertainty due to fading is about ~0.2 dB. The other source of variation arises from statistical sampling of the inhomogeneous orchard canopy. This is caused in large part by the partially discontinuous properties of the crown layer. Because of the azimuth scanning technique used to account for the effects of fading, each measured data point represents an average of 30 samples recorded over a single azimuth sweep. The locations sampled within the canopy by each of these 30 samples do not correspond precisely to those observed during other azimuth sweeps. Therefore, some variation will exist simply because the values of a~ do not represent measurements of precisely the same canopy volume. In addition, factors such as wind speed contribute to a time-varying canopy geometry. This effect is readily observed in the measured diurnal data, especially at X-band. Modeling results shown here demonstrate extraordinarily good agreement between measured and predicted backscatter, especially when this measurement variability is taken into

139 *a v %:: 0 24. 25. 26. 27. Day of August (d) -16. - -17. -18. o MIMICS du Meuused Da * eo ~e \e so X-Band e -19. -20.! 24. 25. 26. 27. Day of August Day of August (e) c x To -20. -21. -22. -23. -24. -20.0. - I 24. 25. 26. -J -25. L 27. 24. Day of August (c) 25. 26. Day of August (0 27. Figure 6.11: Comparison of MIMICS results with measured backscatter recorded during the three day diurnal experiment for (a) HH polarized L-band backscatter, (b) VV polarized L-band backscatter, (c) HV polarized L-band backscatter, (d) HH polarized X-band backscatter, (e) VV polarized X-band backscatter and (f) HV polarized X-band backscatter. The X-band HV MIMICS data has been offset 8 dB to account for multiple scatter.

140 account. Figure 6.12 shows the MIMICS-predicted backscatter to changes in canopy biomass for the walnut orchard response for VV and VH polarizations. These data were generated by varying canopy height and generating a~ at each height. The models for canopy biophysical parameters presented in Appendix F were applied to compute the dry canopy biomass. Data are shown for P-band (0.5 GHz), L-band (1.5 GHz), C-band (5 GHz), and X-band (9.6 GHz). For low values of biomass, the backscatter at both like- and cross-polarizations is dominated by the direct-ground component of canopy backscatter whereas at high values the canopy itself dominates a0. Therefore, the a~ value observed at low values of biomass is determined solely from the estimate of direct ground backscatter for both polarizations. In order to achieve a reasonable estimate of both the like- and cross-polarized direct ground backscatter, measured values from Ulaby and Dobson [61] were used to simulate the direct ground backscatter. This approach was necessary because the first-order ground backscatter model implemented in MIMICS does not account for any cross-polarized return. It should also be noted that at high biomass values the X-band cross-polarized response is several dB lower than anticipated because MIMICS does not account for multiple scattering in the crown layer. This analysis demonstrates that the lower radar frequencies (P- and L-bands) are more sensitive to changes in total canopy biomass than are the higher frequencies (C- and X-bands).

141 0. -5. f9 1-% cat 01 s? -10. -15. -20. / -25. L 0.0 2.0 4.0 6.0 Total Canopy Height (meters) 8.0 10.0 I 1.485.188 1.485 0.0 0.0 0.297 0.594 0.891 1 Dry Canopy Biomass (kg/m2) (a) VV-polarized response. -10. -20. 'I 0 -30. -40. 0.0 2.0 4.0 6.0 Total Canopy Height (meters) 8.0 10.0 I I 0.0 0.297 0.594 0.891 1.188 1.485 Dry Canopy Biomass (kg/m2) (b) VH-polarized response. Figure 6.12: Walnut orchard backscatter response to changes in canopy biomass for (a) VV-polarization, (b) VH-polarization. The incidence angle 0 = 30~.

142 6.3 ERS-1 Alaskan Boreal Forest Study In March 1988, a series of airborne SAR data was acquired over the Bonanza Creek Experimental Forest near Fairbanks, Alaska (Way et al., [77], [76]; Dobson et al., [13], [14]). This study was the first in a series of multi-season aircraft experiments flown over selected forest sites for the purpose of understanding the kinds of biophysical properties that may be detected with spaceborne SAR systems such as the C-band SAR to be flown aboard the European Space Agency's Earth Resources Satellite (ERS-1). The purpose of this experiment was to determine if changes in plant fluid status associated with thawing and freezing result in changes in radar backscatter which could be detected by SAR. and to determine if theoretical backscatter models such as MIMICS could predict these changes. This section provides a brief overview of the MIMICS modeling effort that accompanied this study. A more detailed analysis is provided in Appendix H. This analysis focuses on L- C, and X-band data obtained on March 13, March 19 and March 22, 1988. These dates were selected to encompass the range of environmental conditions that occurred over the duration of the experiment. An unseasonably warm period during which thawed conditions prevailed in the forest extended through the evening of March 13. This was followed by more normal subfreezing temperatures for the remainder of the experiment. As liquid water was frozen by the subfreezing temperatures, the dielectric properties of both the vegetation and of the 20-30 cm snow layer that covered the ground were modified, thereby changing the scattering and absorption properties of these constituents.

143 6.3.1 Test Site Description and Canopy Properties Ground Surface Characteristics The ground surface was covered with a snow layer 20-30 cm deep. Below the snow layer, the upper 20 cm of the mineral soil was frozen throughout the entire experiment. The early March thaw caused the snow layer to have a complex wetness structure that varied with stand species (Dobson et al., [13]). Snow wetness varied considerably with spatial location, depth, and time. A Debye-like model presented by Hallikainen et al., [24] was applied to estimate the dielectric properties of the snow. This model, which is reviewed in Appendix A, relates the snow dielectric to snow wetness (volume %), frequency and dry snow density. The modeled values of snow dielectric constant are listed in Table 6.7 at L-, C- and X-bands for frozen and thawed conditions. Table 6.7: Modeled Dielectric Characteristics of Snow for Frozen and Thawed Conditions. Frequency Thawed Conditions Frozen Conditions (Mlarch 13) (March 19-22) L-Band 1.58 + iO.024 1.37 + iO.O C-Band 1.54 + i0.079 1.37 + iO.0 X-Band 1.49 + i0.09 1.37 + iO.0 The dielectric of the frozen mineral soil was measured using portable dielectric probes in a trench cut into the permafrost. The average L-band dielectric constant of the soil was found to be 7.96 + i0.96. Stand Geometry Ground surveys of seven stands were conducted to determine the number of trees per unit area by species and also record their respective diameters at breast height

144 (DBH) (Jaeger, [30]). To estimate above ground biomass for each stand, these data were coupled with allometric equations. The measured DBH, heights, and status of each tree were used to estimate the quantities listed in Table 6.8 on the basis of allometric expressions drawn from the literature for each species (Kirby,[33]; Manning et al.,[40]; Singh,[52]; Yarie and Van Cleve,[84]). Table 6.8: Summary of Stand Biophysical Parameters. Species White Spruce Black Spruce Balsam Poplar Stand Name WS-1 WS-2 WS-5 WS-7 BS-1 BP-2 Density Mean (trunks/hectare) 1248 2073 1484 1123 1975 1615 Standard Deviation 342 576 618 654 1483 407 Basal Area Mean (m2/hectare) 46 41 44 46 12 50 Standard Deviation 16.6 7.0 8.5 12.4 3.3 25.8 Basal Volume Mean (m3/hectare) 442 332 392 442 51 344 Standard Deviation 169 60 100 115 12 190 Dry Biomass - Summer Mean (kg/m2) 21.7 16.7 18.1 21.5 3.7 18.2 Standard Deviation 8.8 3.6 4.8 6.1 0.8 10.9 Dry Biomass - Winter Mean (kg/m2) 21.7 16.7 18.1 21.5 3.7 17.9 Standard Deviation 8.8 3.6 4.8 6.1 0.8 10.7 In addition, trihedral corner reflectors were placed in several stands to estimate canopy transmissivity (Kasischke et al.,[32]). These stands were also characterized with respect to density, height and diameter ( Jaeger,[30]). These stands included a single species of alder and mixed species stands of alder, balsam poplar and white spruce. Table 6.9 summarizes mean DBH, height and basal area for all stands. To characterize the trunk layer geometry in terms of parameters required for MIMICS input, DBH histograms wvere generated from the ancillary ground measurements and coupled with the allometric height equations listed in Table 6.10.

145 Table 6.9: Summary of Mean DBH, Height and Basal Area for All Stands. Stand DBH Height Basal Area Name (cm) (m) (m2/hectare) WS-1 WS-2 WS-5 WS-7 BS-1 BP-2 19.6 14.5 17.9 21.4 8.8 18.0 22.1 20.1 21.3 24.5 7.6 17.6 46 41 44 46 12 50 Stands with trihedral reflectors: Stand Species DBH Height Basal Area Name (cm) (m) (m2/hectare) Alder alder 6.0 6.3 66.5 Balsam Poplar balsam poplar 11.0 12.7 22.9 alder 6.0 6.3 3.1 White Spruce white spruce 7.8 8.6 12.4 balsam poplar 9.4 11.6 10.0 alder 6.1 6.3 5.4 Table 6.10: Equations Defining Height-to-DBH Relationship. Species Equation White Spruce II = -1.7096 + 1.4224(DBH) - 0.016(DBH)2 Black Spruce H = 0.9494 + 0.7657(DBH) Balsam Poplar lI = 1.0526 + 1.143(DBH) - 0.0145(DBH)2 Alder H = 2.871 + 0.5666(DBH) H = height in meters DBHI = diameter in cm measured at breast height

146 Together, these data define the PDF in size required to compute the trunk layer phase matrix for a given stand. All trunks are assumed to have a vertical orientation for purposes of MIMICS simulations. The size and orientations of crown layer constituents have been inferred through a combination of field observations and morphology data from Nelson et al.,[45]. Table 6.11 summarizes the geometry of the crown layer constituents. Each of the - Species White Spruce Black Spruce Balsam Popla Alder Table 6.11: Geometry of Crown Layer Constituents. Constituent Average Average Class Length (cm) Diameter (cm) primary branches 113 2.24 secondary branches 57.16 1.04 needles 1.6 0.1 primary branches 81.3 2.37 secondary branches 51.17 1.06 needles 0.8 0.1 r primary branches 200 1.5 secondary branches 100 0.75 primary branches 200 1.5 secondary branches 100 0.75. - Orientation Function sin4 9 sin9 0 sin 0 sin9(0 - 30~) sin9 0 sin sin9(0 + 60~) sin9(0 + 60~) sin9(0 + 60~) sin9(0 + 60~) I orientation functions is normalized to convert it to a PDF for implementation in MIMICS. Table 6.12 lists the number density of each canopy constituent for each of the seven stands, assuming that each stand may be modeled as a continuous (closed) canopy. The biomass of a single element is computed from the size and dry density parameters of that element as presented in Appendix F.

147 Table 6.12: Number Density of Canopy Constituents. Stand Name Canopy Density Primary Branches Secondary Branches Needles (trees/m2) (#/m3) (#/m3) (#/m3) Alder 1.36 1.19 9.92 NA BP-2 0.16 0.85 6.69 NA WS-1 0.12 0.44 2.37 12,300 WS-2 0.12 0.48 2.57 13,310 WS-5 0.12 0.50 2.7 14,000 WS-7 0.12 0.48 2.6 13,490 BS-1 0.20 0.25 1.31 18,340 Stand Dielectric Characteristics The dielectric properties of the trees vary as a function of frequency and canopy properties such as constituent dry density and freeze/thaw state. The dielectric properties of the stands were monitored with L- and C-band portable dielectric probes. The dielectrics listed in Table 6.13 were inferred by coupling dielectric measurements to the dielectric models (Dobson et al., [13]). Table 6.13: Relative Dielectric Constant for Tree Constituents. Species Frequency Relative Dielectric (GHz) +5~C -15~C White Spruce 1.25 36.47 + i10.99 5.19 + il.09 5.3 29.01 + i11.97 4.85 + iO.32 9.38 22.78 + i13.20 4.81 + i0.18 Black Spruce 1.25 12.46 + i4.50 3.72 + i0.78 5.3 9.30 + i3.33 3.47 + i0.23 9.38 7.82 + i3.22 3.44 + iO.13 Balsam Poplar and Alder 1.25 30.71 + i9.56 4.95 + il.07 5.3 24.18 + i9.85 4.61 + i0.32 -9.38 19.16 + i10.69 4.57 + i0.17

148 6.3.2 Boreal Forest Transmissivity Analysis Data collected at C- and X-bands on March 22 have been applied to analyze canopy transmissivity (Dobson et al., [13],[14]). To compute the one-way canopy propagation loss, the point target responses of trihedrals that were placed in the forest stands were compared to the response of trihedrals placed in an open area. Stands selected for this analysis included a single-species alder canopy, a balsam poplar stand that contained shorter alder trees, and a white spruce stand that also contained a mixture of balsam poplars and alders. Stand statistics measured in the neighborhood of the targets showed significant local variance in stand geometry, both locally within the neighborhood of individual targets and in comparing different target locations within the same stand. Not only does that within-stand variability affect the estimation of canopy extinction, but also there is an inherent bias toward values of low extinction due to the logistics of placing physically large reflectors in a canopy of large discrete scatterers. The measured extinction values represent realizations over only the few azimuth degrees required to construct the synthetic aperture. It would be best to have a set of infinitesimally small point targets that one could place at a statistically large number of random locations within a given stand. Each set of stand statistics was used as input to MIMICS, applying the dielectric constants for frozen vegetation constituents (Table 6.13) and the trunk height versus DBH equations (Table 6.10). For the mixed-species stands, MIMICS was run separately for each constituent species and the resultant propagation losses were added together to estimate the total net loss. Since only gross estimates of crown biomass were available, and in order to expedite the transmissivity analysis, only extinction through the trunk layer was considered.

149 Figures 6.13 - 6.15 show MIMICS simulations of the maximum and minimum oneway propagation loss for each of the three stands together with the measured values as determined for each trihedral reflector. The maximum and minimum MIMICS simulations correspond to the maximum and minimum biomass conditions for each of the three stands. Figure 6.13 shows these data for the alder stand, Figure 6.14 shows data for the balsam poplar stand and Figure 6.15 presents the white spruce stand simulation. In all cases, the V-polarized extinction is greater than that at H-polarization with the difference being less than 1.5 dB. Figure 6.16 is a plot of the MIMICS-simulated one-way propagation loss versus the measured loss at C-band. Data are shown for all three stands at both polarizations. Each set of stand data were fit with a straight line to help illustrate the combined effects of measurement and model error. Good correlations exist between measured data and model simulations for all three stands, with the correlation coefficient p > 0.75, however MIMICS never predicts 0 dB of loss which may be measured at low values of incidence angle because of placement of the reflectors in canopy gaps. This indicates that more reflectors should be used in this type of study and more careful attention should be paid to random placement of the targets in the canopy. Furthermore, this figure illustrates an underprediction of canopy extinction by MIIICS in the white spruce stand. This illustrates the importance of including the crown layer constituents in canopy transmissivity analyses, especially for foliated species.

150 20.0 15.0 co ta 10.0 c 6 O<~ V Pol Max -- MIMICS -----— H Pol Max -- MIMICS V Pol Min -- MIMICS ------ H Pol Min -- MIMICS o V Pol -- Measured E H Pol -- Measured / I 5.0 - E I 8 I.. R. I 0.0 L 20. 30. 40. 50. 60. 70. Incidence Angle (degrees) (a) C-band. 20.0 cQ '"0 o 03 c~ 6 0 15.0 10.0 5.0 V Pol Max -- MIMICS....... HI- Pol Max -- MIMICS - V Pol Min - MIMICS ------ H Pol Min —MIMICS ' o V Pol — Measured E H Pol — Measured -;- =.......... I........ I.... I.... 0.0 2( D. 30. 40. 50. 60. 70. Incidence Angle (degrees) (b) X-band. Figure 6.13: Transmission loss for one-way propagation through the alder canopy. Measurements are shown for four trihedral targets at (a) C-band and (b) X-band.

151 20.0 o 0 >4 Cl <ru <~ 15.0 10.0 5.0 V Pol Max — MIMICS -- H Pol Max-MIMICS — V Pol Min -- MIMICS ---- H Pol Min — MIMICS V Pol — Measured I HPol —Measured _,,. / 8 r 1'rr f ~ I... I.... I 0.0 1 2( O. 30. 40. 50. 60. 70. Incidence Angle (degrees) (a) C-band. 20.0 15.0 co V Pol Max -- MIMICS ------- H Pol Max — MIMICS -—.- V Pol Min -- MIMICS ----- H Pol Min -- MIMICS o V Pol — Measured 3 HPol — Measured / 10.0 - 5.0 8...,.... I... nn U.U - 20. 30. 40. 50. 60. 70. Incidence Angle (degrees) (b) X-band. Figure 6.14: Transmission loss -for one-way propagation through the mixed balsam poplar/alder canopy. Measurements are shown for seven trihedral targets at (a) C-band and (b) X-band.

152 20.0 '0 c0 o c5 15.0 10.0 5.0 V Pol. Maximum - MIMICS -—. — H Pol. Maximum - MIMICS V Pal. Minimu - MIMICS ------ H Pol. Minimum - MIMICS o V PoL - Measured B HPol. -Measured @ e..'r.,,.,O.,.. 0.0 2( I. 30. 40. 50. 60. 70. Incidence Angle (degrees) (a) C-band. 20.0 0 co (5 15.0 10.0 -— i V Pol. Maximum -- MIMICS -------- H Pol. Maximum — MIMICS ------ V Pol. Minimum - MIMICS ------ H PoL Minimum - MIMICS o VPol. - Measured ' E H Pol. - Measured E., 5.0 'n 20. 20. 30. 40. 50. 60. 70. Incidence Angle (dB) (b) X-band. Figure6.15: Transmission loss for one-way propagation through the mixed white spruce/balsam poplar/alder canopy. Measurements are shown for nine trihedral targets at (a) C-band and (b) X-band.

153 12.0 10.0 HE cX "C, m 4-4 CZ C) V 0-4 8.0 6.0 4.0 2.0 0.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 Measured Data (dB) o Alder Stands, H Pol. o Alder Stands, V Pol. o Balsam Poplar Stands, H Pol. o Balsam Poplar Stands, V Pol. A White Spruce Stands, H Pol. A White Spruce Stands, V Pol. Alder best fit -- p = 0.87 -------- Balsam Poplar best fit -- p = 0.75 - White Spruce best fit -- p = 0.87 Figure 6.16: Comparison of MIMICS simulated and measured transmission loss for oneway propagation through the alder, balsam poplar and white spruce canopies at C-band. The best-fit straight lines are shown for each canopy, together with their respective correlation coefficients p.

154 6.3.3 Boreal Forest Backscatter Analysis Data recorded at L-, C- and X-bands have been applied to analyze backscatter from several single-species stands (Dobson et al., [13],[14]). Of these, only the L-band data has been calibrated to an absolute level. These data were collected on March 13 (frozen canopy conditions) and on March 19 (thawed canopy conditions) (Way et al.,[77]). Since the stands considered in this study were only partially characterized by on-site sampling, information on biomass apportionment and canopy constituent size and density characteristics is only approximate. Errors introduced in the biomass apportionment analysis will have an effect on the backscatter simulated by MIMICS. The 20-30cm thick snow layer also significantly complicated the backscatter analysis. The roughness parameters and other characteristics of the snow-ground interface were not characterized. These parameters could only be estimated by fitting MIMICS to ground backscatter measurements of open areas on sandbars that were outside the tree canopies. Since the roughness of these regions do not correspond to the roughness of a forest floor, and since the goal of this study is to examine model performance without using parameter fitting, the snow substrate was modeled as a half-space of snow. This ignores scattering at the snow-ground interface completely and in some cases reduces the effectiveness of the MIMICS simulations. A simple technique that accounts for the snow-ground interface at L-band was introduced in Section 3.2.2. However, because of lack of adequate characterization of the ground surface, its effectiveness is also somewhat limited. Comparison with Measured Data Table 6.14 lists the MIMICS backscater simulations together with the SAR observations for six stands. Measured data were recorded by the JPL SAR at L-band for

155 both frozen and thawed conditions. This table shows very good agreement for both frozen and thawed canopy states except for VV and VH polarizations for the stands BS-1, BP-2 and Alder under frozen conditions. Figure 6.17 graphically illustrates the Table 6.14: Comparison of MIMICS Estimates to Measured L-band SAR Data (dB). March 13, 1988 March 19, 1988 Thawed Conditions Frozen Conditions Stand Polarization SAR MIMICS SAR MIMICS WS-1 HH -10.0 -9.2 -13.1 -12.2 VV -10.4 -12.1 -14.9 -15.6 VH -15.2 -14.9 -21.0 -22.8 WS-2 HH -8.4 -9.1 -11.4 -12.8 VV -9.9 -12.3 -14.5 -16.4 VH -14.2 -15.0 -20.4 -23.6 WS-5 HH -8.1 -9.1 -11.1 -12.2 VV -9.1 -12.0 -14.8 -15.5 BS-1 HH -12.9 -10.7 -14.9 -16.9 VV -14.4 -15.1 -16.4 -23.2 VH -20.0 -19.5 -23.7 -32.5 BP-2 HH -9.2 -11.7 -12.7 -14.4 VV -10.4 -11.6 -14.8 -22.0 Alder HH -8.7 -9.9 -11.3 -14.6 VV -9.7 -11.4 -14.0 -23.2 effectiveness of MIMICS in predicting the HH- and VV-polarized backscatter. The measured SAR data are plotted against that predicted by MIMICS. Data are shown for all six stands for both frozen and thawed conditions. From here it is seen that MIMICS tends to underpredict backscatter for all stands except for white spruce. The underprediction of VV backscatter for frozen conditions for all stands except white spruce is also evident. The general underestimation of a~ may be attributed to the modeling of the snow surface as an infinite half-space. It is expected that accounting for scatter at the snow-ground interface would increase a~ somewhat and may alleviate this problem. Figures 6.18 and 6.19 present L-band polarization responses for frozen and thawed

156 -5. -10. e A* E 0 * 0 0 0 co v 0 -15. -20. -25. L -25. -20. -15. -10. -5. o~ (dB) MIMICS (a) HH-polarization. -5. -10. F o 4) ~o -v 0 1 -gm S3 G 0 V OA A. * ~ ~ ~ ~ ~ o -15. -20. F -25. L -25. -20. -15. -10. -5. o~ (dB) MIMICS (b) VV-polarization. 0 Whil Spruce -- Thawed * Whie Spruce -- Frozen B Black Spruce - Thawed * Black Spruce - Frozen Balsam Poplar — Thawed A Balsam Poplar -- Frozen v Alder - Thwed v Alder - Frozen Figure 6.17: Comparison of measured canopy backscatter to ter for (a) HH-polarization and (b) VV-polarization.

157 white spruce (WS-5), respectively. Responses are shown for co-polarized and crosspolarized configurations. Figures 6.20 and 6.21 present the measured frozen and thawed L-band responses. Again MIMICS successfully recreates the behavior of the measured data. MIMICS not only correctly reproduces the shapes of each of the surfaces, but MIMICS accounts for the increase in the pedestal observed in going from frozen to thawed states. These figures demonstrate that MIMICS has successfully modeled the backscatter response oT this stand for all of these polarization states. Figure 6.22 shows the linear polarized response of this stand as simulated by MIMICS for frozen and thawed conditions. The character of the responses are very similar for the two environmental states, with the thawed conditions yielding slightly more cross-polarized backscatter. Modeling at C- and X-bands has been complicated by the lack of available calibrated SAR data. Furthermore, data. at these frequencies are available only for frozen canopy conditions. To deal with the uncalibrated data problem, the backscatter values were normalized to that of the white spruce stand WS-1 for each SAR pass. These normalized data are presented in Figure 6.23. For the most part, MIMICS predictions agree with the SAR measurements to within ~1.5 dB. Exceptions to this include some VV-polarization observations of the balsam poplar, black spruce and alder stands. As was the case at L-band, this is probably caused by the method used to model the snow-soil interface.

158 o to N o z 94S itatio A 45 gle v 45. ~5- jiiA'l0. 45. olticity Angle X El4s. iit (a) Co-polarized response. Us-45. _>^ ^^ ^ 5. 0.tion l45. &qgle 45. 0 vt.45. 1ipticiY A0c % (b) Cross-polarized response. Figure 6.18: MIMICS simulated L-band polarization response of frozen white spi WS-5.

159 icei nt' 4 4. 45~~~4s~~~~ ~45. ita0 45 0. ngle ~ s45. ipti ity A (a) Co-polarized response. 0. - (b) Cross-polarized response. Figure6.19: MIMICS simulated L-band polarization response of thawed white spruce stand WS-5.

160.0 to 0 0 N z I Z._ 45. (a) Co-polarized response. 45. (b) Cross-polarized response. Figure 6.20: Measured L-band polarization response of frozen white spruce stand WS-5.

161 01~~.0 -GEM:~.. 0. le 45. jlipticity Ag (a) Co-polarized response. Figure 6.21: Measured L-band polarization response of thawed white spruce stand WS-5. 45. (b) Cross-polarized response. Figure 6.21: Measured L-band polarization response of thawed white spruce stand WS-5.

162 to:2d 90. (a) Frozen Conditions. N._ -a 0 90. (b) Thawed Conditions. Figure 6.22: MIMICS-simulated L-band linear polarization response of thawed white spruce stand WS-5.

163 4. 2. -6. - 0 -8. -6. 4. -2 0. 2. 4. -4. _v -6........, I.,, I., I.,. I... -8. -6.. 4. -2. o. 2. 4. o~/o~ws., (dB) MIMICS (a) C-Band. 2. 0. Ol i:,S -2. -6. -8. I... 1 -8. -6. -4. -2. 0. 2. 4. o~/o~ws (dB) MIMICS (b) X-Band. 0 White Spruce - HH * White Spruce - VV Black Spruce -- HH * Black Spruce -- VV Balsam Poplar - HH A Balsam Poplar - VV v Alder -HH v Alder -VV Figure 6.23: Comparison of measured canopy backscatter to MIMICS simulated backscatter for (a) C-band and (b) X-band. The data have been normalized to the backscatter from white spruce stand WS-1 for each SAR pass.

164 Black Spruce Simulations Having established the ability to model backscatter from these forest stands, MIMICS may now be used to compute backscatter over a wider range of sensor parameters and the resulting simulated backscatter may be examined on a more detailed level. This allows examination of the relative contribution of each of the scattering mechanisms to the net canopy backscatter. Detailed analyses of this type tend to be lengthy and are presented in Appendix H for several of the Alaskan forest stands. However, to gain an understanding of the effect of the snow layer on net backscatter, the approach presented in Section 3.2.2 for modeling the scattering at the snow-soil interface is applied at L-band to the black spruce stand. The black spruce stand (BS-1) is a much more sparsely populated stand than the other species. This stand, in fact, does not represent a closed canopy. However, to simplify this initial analysis, MIMICS I is applied to model a~ for the black spruce stand as if it were indeed a closed canopy. Figures 6.24 and 6.25 compare backscatter from the canopy, with the ground layer modeled as a half-space of snow, with the backscatter for a canopy above a 20 cm thick snow layer over a frozen soil half-space. Figure 6.24 shows this simulation for frozen canopy conditions while Figure 6.25 shows these data for thawed canopy conditions. In both cases, Or~ is higher for the snow-covered soil for all polarizations. The effect is more prevalent for like-polarized backscatter with ao being responding slightly more than Uoh. This demonstrates the effect that the snow layer has on modifying the local angle of incidence at the ground surface.

165 0. -5. -10. m ~0 Pt -15. -20.: —rr-rr --- —r --- r --- — --- --- -- --- --- --- -- --- --- - -25. - -30. -35. 20. 1 30. 40. 50. 60. Incidence Angle (degrees) (a) Snow half-space. 0. -5. -10. t m ts ~o -15. VV Pol......... HH Pol. -VH Pol..... I. I. I... -20. -25. -30. -35. 2C I 30. 40. 50. 60. Incidence Angle (degrees) (b) Snow layer over soil half-space. Figure 6.24: Total canopy backscatter for black spruce stand (BS-1) at L-band under frozen canopy conditions for (a) ground layer consisting of a snow half-space and (b) ground layer consisting of a snow layer on top of a soil half-space.

166 0. -5. ' To 0-1. m ~0 -10. -15. ----------------------------------------------------— ~ --- —---- a~~ -20. __ --- —-— _______= -25. L 20. 30. 40. 50. 60. Incidence Angle (a) Snow half-space. 0. -5. Is ca O -10. -15............... PoL vv Pol. --- HH PoL - VH PoL.... I.... I.... I.... -20. -25. 2( 3. 30. 40. 50. 60. Incidence Angle (b) Snow layer over soil half-space. Figure 6.25: Total canopy backscatter for black spruce stand (BS-1) at L-band under thawed canopy conditions for (a) ground layer consisting of a snow half-space and (b) ground layer consisting of a snow layer on top of a soil half-space.

L67 6.3.4 Boreal Forest Multi-Season Simulation The European Space Agency's Earth Resources Satellite, ERS-1, is scheduled for launch on 16 July 1991. This satellite represents the first in a planned series of remote sensing satellites that will allow monitoring of seasonal phenologic and environmental change of forest ecosystems. ERS-1 includes a C-band, VV-polarized SAR that will operate at an incidence angle of approximately 230. Since the boreal forest study was carried out in support of the ERS-1 mission, it is useful to consider the simulated backscatter response through an annual cycle for sensor parameters corresponding to the ERS-1 platform. Simulated backscatter response to changing environmental conditions has been studied by Way et al.,[80]. Table 6.15 presents a list of environmental and phenologic conditions that exist throughout the course of a typical seasonal cycle. Canopy phenological state is strongly influenced by environmental conditions and may vary with species. The table lists 13 states that may occur through the coarse of a year. The conditions begin in winter with the canopy in a frozen state and progress through a spring thaw, budding, a wet rainy season, a dry season, a flooded period, and autumn freezing. Periods of water stress are manifest by a negative water potential. Applying the dielectric models presented in Appendix A, MIMICS was used to simulate the backscatter from the stand presented in Table 6.15 as a function of phenologic state. The results of this analysis are presented in Figure 6.26. Keeping in mind that for snow-covered ground MIMICS predicted a~ to be slightly lower than SAR observations for both balsam poplar and black spruce, these data perform very much as one might expect. Except for one condition, balsam poplar exhibits the highest backscatter during that part of the season when it is foliated. The exception occurs during the spring thaw, before the leaves come out and when the soil is wet.

168 Table 6.15: Specified Environmental and Phenologic Conditions for MIMICS Simulation. Environment Balsam Poplar Air Soil Snow Leaves Bole Water Condition Temp State State Potential 1 2 3 4 5 6 7 8 9 10 11 12 13 < 0 > 0 > 0 > 0 > 0 >0 > 0 >0 > 0 > 0 >0 < 0 < 0 frozen frozen frozen frozen wet wet wet dry dry dry flooded frozen frozen dry wet none none none none none none none none none none none none none none none none very wet wet wet dry dry wet none none frozen wet wet wet wet wet wet wet dry dry wet wet frozen 0 0 0 0 0 0 0 neg neg neg 0 0 0 White Spruce Black Spruce Needles Bole Water Needles Bole Water Condition Potential Potential 1 frozen frozen 0 frozen frozen 0 2 wet wet 0 wet wet 0 3 wet wet neg wet wet neg 4 dry dry neg dry dry neg 5 wet wet 0 wet wet 0 6 wet wet 0 wet wet 0 7 wet wet 0 wet wet 0 8 wet wet 0 wet wet 0 9 wet wet neg wet wet neg 10 dry dry neg dry dry neg 11 wet wet 0 wet wet 0 12 wet wet 0 wet wet 0 13 frozen frozen 0 frozen frozen 0 Notes: (1) dry soil 10% sand; 30% clay; mvs = 0.1, 5~ (2) wet soil 10% sand; 30% clay; mvs = 0.3, 5~ (3) very wet leaves grav. moisture = 0.8 (4) wet leaves grav. moisture = 0.6 (5) dry leaves grav. moisture = 0.2 C C (6) dry and stressed woody vegetation dielectrics are assumed same as that of frozen woody vegetation (7) ground surface corr. length = 6 cm, rms height = 0.5 cm,

169 -5. -10. 0 ll:: -15. -20. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. Canopy State Figure 6.26: MIMICS simulated canopy backscatter response to environmental state for ERS-1 parameters (C-band, VV-polarization, 8 = 230).

170 During this time, there is little attenuation through the crown layer and the wet soil leads to an accentuated ground-trunk interaction. White spruce and black spruce backscatter respond alike throughout the cycle since both species are coniferous. Black spruce exhibits its highest backscatter when the ground surface is flooded. As with balsam poplar, this arises from an increased ground-trunk interaction. This is not observed in the white spruce or in the poplars because they are more densely populated stands (the poplars are foliated during this time) and C-band extinction through the crown layer is large enough to attenuate this term. 6.4 MIMICS II Simulations of Open-Crown Canopies Now that an understanding of the function and behavior of the closed-crown canopy model has been developed, the simulations may be extended to include the discontinuous crown layer model. In doing this, the response of the models to changes in canopy parameters may be compared and the backscatter response to changes in parameters affecting the discontinuous nature of the canopy may be examined. Only the polarimetric model will be considered in these analyses so that cross-polarized backscatter may be simulated. Three canopies are considered in this analysis. In all three cases, the tree-level parameters have been derived directly from those applied in the Eos synergism study and the ERS-1 Alaskan boreal forest study. The first canopy is a sparsely populated black spruce stand similar to that found in the Alaskan boreal forest. The second canopy is a more densely populated coniferous stand which closely resembles an Alaskan white spruce stand. Finally, a deciduous canopy with parameters derived from those of the the walnut orchard is considered.

171 Since backscatter coming directly from the ground surface is expected to contribute more to the net canopy backscatter in the discontinuous canopy case than in the continuous case, an effort has been made in the MIMICS II simulations to account for a stronger direct ground backscatter mechanism than is usually observed for bare soil. A statistical analysis of backscatter from short vegetation is presented by Ulaby and Dobson [61]. The rough surface parameters applied in the MIMICS II simulations have been inferred from these statistics and are summarized in Table 6.16. The model type, RMS surface roughness, and surface correlation length are Table 6.16: Ground Surface Roughness Parameters for the MIMICS II Simulations. L-Band C-Band X-Band Surface Scattering Model Small Perturbation Physical Optics Geometrical Optics RMS Surface Roughness 1.0 cm 0.5 cm 2.0 cm Correlation Length 5.0 cm 2.0 cm 6.0 cm allowed to vary with frequency so that the like-polarized surface backscatter from a moderately dry soil surface as a function of incidence angle roughly agrees with values measured for short vegetation. 6.4.1 Black Spruce Simulation As mentioned in Section 6.3, the black spruce stand considered in the Alaskan boreal forest study was in fact a sparsely populated open-crown layer canopy. Therefore, this stand (BS-1) represents a reasonable test case for the MIMICS II model. Since the presence of an underlying snow layer complicates model behavior, this analysis will model the surface as a bare soil surface with the roughness parameters listed in Table 6.16. The canopy is modeled as illustrated in Figure 6.27. The crowns are modeled as square columns of identical size with a height of 7.8 meters and side length 1.0

172 Figure 6.27: Canopy geometry used in MIMICS II simulation of black spruce stand BS-1.

173 meters. The crown height was chosen to correspond to the average tree height and is identical to the average trunk height. The tree-level parameters are summarized in Table 6.17. The number densities of the constituents in the crown layer were Table 6.17: Tree level parameters for black spruce stand BS-1. Constituent Mean Mean Number Orientation Length Diameter Density Primary Branches 0.81 m 2.37 cm 1.0 m-3 sin9 (0 - 300) Secondary Branches 0.51 m 1.06 cm 6.55 m-3 sin9 (0) Needles 0.8 cm 0.1 cm 91,700 m-3 sin (0) Trunks 7.8 m 8.9 cm 0.2 m-2 Vertical obtained by increasing those applied in the continuous canopy model in such a way as to keep a constant total number of constituents in the canopy. For a continuous canopy with number density 0.2 trees/m2, the equivalent canopy area per tree is 1/0.2 = 5 m2/tree. If this area is condensed into a square-column crown volume with cross-sectional area 1 m2, the number density of constituents in the crown volume increases by a factor of 5. Thus, the number densities listed in the table represent a factor of 5 increase in those used for modeling the equivalent closed-crown canopy. Constituent dielectric properties are listed in Table 6.18. These parameters are Table 6.18: Black Spruce Canopy Dielectric Characteristics. Constituent L-Band C-Band X-Band Primary branches 14.3 +i 5.1 10.7 +i 4.0 8.9 +i 3.9 Secondary branches 15.7 +i 5.6 11.9 +i 4.5 9.8 +i 4.4 Needles 18.5 +i 6.4 14.1 +i 5.4 11.5 +i 5.5 Trunks 12.5 +i 4.5 9.3 +i 3.3 7.8 +i 3.2 Ground surface 5.6 +i 1.4 6.6 +i 0.9 5.8 +i 1.4 consistent with those found for thawed black spruce. The ground surface dielectric was computed by applying the dielectric models in Appendix A with a volumetric moisture of 0.1.

174 For a single square column crown of height c and side length a, from Equations (5.44) and (5.55): 2cossine + _ a k (S - Sm) ac ctan. m; 0 >c; O < s < Sm;,, = a/ sin O p(511) - 2cos 0sin, + () k (S- Sm) (6.6); OOc; 0 s <c s; Sm m = C/ Cos 0;otherwise. Assuming all of the crowns in the black spruce stand are of identical height and width, it follows that 2sin cos [ -Asm ( a A -Sm acsX [l- e m- (AiSm + 1)] + 1 ctanO eAs 00oo; > 0c, sm a/ sin 0 J p(sl)e'sds= = a/i (6.7) ]0o 2.sif0~os [ [ - eAs (AiSm + 1)] + (1 ct n) e-Asm ac(Ai) a; < Oc, Sm = / Cos 0 for each eigenvalue Ai. Applying this relation to the polarimetric MIMICS II model allows for determination of canopy backscatter. Figure 6.28 shows the simulated L-, C-, and X-band backscatter from the black spruce canopy as a function of radar incidence angle for VV, HH and HV polarizations. Data are shown for the canopy modeled as a discontinuous stand (MIMICS II) and for the canopy modeled as an equivalent continuous canopy, with the crown scattering constituents distributed uniformly throughout the crown layer (MIMICS I). This figure shows that the discontinuous nature of the canopy has a negligible effect on backscatter at all three frequencies, with only a slight effect at C- and X-bands for the cross-polarized return. This phenomenon is also reflected in the one-way crown layer transmissivity, shown in Figure 6.29, where minimal difference is seen between the MIMICS I and MIMICS II models.

175 5.00 -5.00 g3 ~0 O3 MIMICS I - I-band.........MIMICS I - L-band ----- MIMICS I - C-band - - - - MIMICS n - C-band - -. — MIMICS I - X-band ------ MIMICS H - X-band -15.0 -25.0 L20.0 30.0 40.0 50.0 60.0 Incidence Angle (degrees) (a) VV-polarized backscatter. 5.00 ------- ----— B ---B ~* - - ~, AEL X.. - v:la ~O MIMICS I - 1-band......... MIMICS H - L-band ----- MIMICS I - C-band -- -- MIMICS II- C-band —. - -MIMICS I - X-band ------ MIMICS II - X-band -5.00 F -15.0 L20.0 -10.0 30.0 40.0 50.0 60.0 Incidence Angle (degrees) (b) HH-polarized backscatter. -20.0 v m --- —..- '-.. _ ----- *- Q'? MIMICS I - L-band........ MIMICS II - L-band ----- MIMICS I - C-band - - -- -MIMICS II - C-band -- - - MIMICS I - X-band — v —. MIMICS II - X-band -30.0 -40.0 '20.0 30.0 40.0 50.0 60.0 Figure 6.28: Incidence Angle (degrees) (c) HV-polarized backscatter. Comparison of net canopy backscatter from a black spruce canopy modeled with a continuous crown layer (MIMICS I) and a discontinuous crown layer (MIMICS II) at L- C- and X-bands for (a) VV, (b) HH, and (c) HV polarizations.

176 0.000 C-, m 0 1. ca.2 c3 to 0 O ---— ~ --- —----------- -rrr-r ---- ---- -1.00 __2'-QSZB - ur r Ltlp L s s 'b'm s LPI a Zs Z \1 -2.00 1 MIMICS I - L-band.. --- MIMICS I - L-band — o --- MIMICS I - C-band - - o - - MIMICS n - C-band - -- - MIMICS I - X-band ----- MIMICS II - X-band -3.00 F -4.00 ' 20.0 30.0 40.0 50.0 60.0 Incidence Angle (degrees) (a) V-polarized transmissivity. 0.000 ca ECQ >~ 0 c3 cO -1.00 - -2.00 - ~~~~~~~~~~~p ~ ~ ~ ~ ~ ~ ~ ~.~ ~B I. I.S~~~ MIMICS I - L-band --—... - MIMICS II - L-band — o --- MIMICS I - C-band - - - - -MIMICS n - C-band - — A- MIMICS I- X-band ------ MIMICS n - X-band -3.00 F -4.00 20.0 30.0 40.0 50.0 60.0 Incidence Angle (degrees) (b) H-polarized transmissivity. Figure 6.29: Comparison of crown layer transmissivity through a black spruce canopy modeled with a continuous crown layer (MIMICS I) and a discontinuous crown layer (MIMICS II) at L- C- and X-bands for (a) V-polarization and (b) H-polarization.

177 Figures 6.30 and 6.31 illustrate the most dominant contributions to total canopy backscatter for L-band VV and HV polarized backscatter. Simulations are presented for both MIMICS I and MIMICS II. The VV-polarized backscatter is dominated by the ground-trunk interaction mechanism at intermediate incidence angle and by the direct crown backscatter near 0 = 20~ and 0 = 60~. The cross-polarized backscatter is dominated exclusively by direct crown backscatter. Essentially, these figures demonstrate that the crown volumes contain such a sparse distribution of constituents that distributing the constituents in individual crown volumes has an insignificant effect on the net canopy backscatter. To demonstrate this effect further, the number density of scatterers in the crown volumes may be increased while maintaining constant crown size. A density multiplication factor, Md is defined such that the volume density of scatterers in the crowns is multiplied by Md thereby modifying scattering and extinction in the crown volumes. For example, setting Afd = 2 doubles the the phase matrix 'PC while also doubling the extinction matrix fi,. Md = 1 corresponds to the reference canopy. Figures 6.32 and 6.33 illustrate the effect of Md at L-band for incidence angles of 20~ and 600. Figure 6.32 compares the like- and cross-polarized backscatter simulated with the open-crown canopy model with that simulated with the closedcrown canopy model. Little difference is seen between the models for low values of Md whereas as much as 2 dB of difference is observed at higher Md. At both incidence angles, the HH-polarized backscatter modeled with MIMICS II exhibits an enhanced backscatter over that modeled with MIMICS I. However, for VV and HV polarizations, a~ decreases when going from MIMICS I to MIMICS II. This occurs because the HH-polarized backscatter is dominated by the ground-trunk interaction mechanism while direct crown backscatter contributes more significantly to the VV

178 0.000 -4,) O 0 -10.0 -20.0 ):rrr —zr::5....................... " "- -......, \ - i"- J' Total ------- Direct Crown ------ Ground-Trunk -- - Direct Ground - - - - Crown-Ground -30.0 L 20.0 30.0 40.0 50.0 60.0 Incidence Angle (degrees) (a) MIMICS I continuous crown layer. 0.000 O" 1-4 C/, U O" 1-4 '0 0 -10.0 -20.0...........I- - - - - - -- - —..........-..-.., - —..o.,,. ', r~~~~~~~~~~~~~~~~~~~~~~~. Total -------- Direct Crown ------ Ground-Trunk ------ Direct Ground - - - Crown-Ground -30.0 L 20.0 30.0 40.0 50.0 60.0 Incidence Angle (degrees) (b) MIMICS II discontinuous crown layer. Figure 6.30: Comparison of L-band VV-polarized backscatter from a black spruce canopy modeled with (a) a continuous crown layer (MIMICS I) and (b) a discontinuous crown layer (MIMICS II).

179 -5.00 - a O -15.0. * I........ 1.... Total -------- Direct Crown Crown-Ground _______ -_ --- —----------------------—. --- —-_. — - - - - - - - - - - - - - - - - - - - - - - - - - -25.0 -35.0 ' 20.0 30.0 40.0 50.0 60.0 Incidence Angle (degrees) (a) MIMICS I continuous crown layer. -5.00 -4 0 -V~ 0-4 0 o -15.0 F Total -------- Direct Crown. --- — Crown-Ground — ~..... -- - - - - - - - - - - - - -~ — - - - - - - - - - - -25.0 f -35.0 L20.0 30.0 40.0 50.0 60.0 Incidence Angle (degrees) (b) MIMICS II discontinuous crown layer. Figure 6.31: Comparison of L-band HV-polarized backscatter from a black spruce canopy modeled with (a) a continuous crown layer (MIMICS I) and (b) a discontinuous crown layer (MIMICS II).

180 0.000 Cq rn ~ — -10.0 -20.0 >._:.....I. I.. I. i. I... ' "'"'""'" ^ -^-^Et^: ' - - - -.....:1 s ~ ~ ~ ~ ~ ~ 1,,:_ - -' ---' - -- --- 4f.... I.... I I I MIMICS I W pol -o —~....MIMICS II VV pol ------ MIMICS I HHVVpol MIMICS I HH pol - - - - MIMICS I HV pol --—. MIMICS II HV pol — I % 1 I -JU.U 0 0.000 1.00 2.00 3.00 4.00 5.00 Density Multiplication Factor (a) Incidence angle = 20~. 0.000 's 0: m co ~t -10.0, - -... I, I C,"' -.-. Oct..~~-.........;~... // Z...... I....I... I.... MIMICS I W pol ------ MIMICS II VV pol -MIMICS I Il pol -- -- MIMICS II HH pol --- - MIMICS I HV pol ----- MIMICS II HV po -20.0 h -30.0 L O.OC K) 1.00 2.00 3.00 4.00 5.00 Density Multiplication Factor (b) Incidence angle = 60~. Figure 6.32: Comparison of net canopy backscatter from a black spruce canopy modeled with a continuous crown layer (MIMICS I) and a discontinuous crown layer (MIMICS II) at VV, HH and HV polarizations for incidence angles of (a) 20~ and (b) 60~ as a function of the density multiplication factor.

181 0.000 m...4 C3 r4 E a 6 -1.00 -2.00 -3.00 -4.00 -5.00 l — 0.000 1.00 2.00 3.00 4.00 5.00 Density Multiplication Factor (a) Incidence angle = 20~. 0.000 -;C.n.E co O -2.00 -4.00 -6.00 -8.00 -10.0 'I 0.000 1.00 2.00 3.00 4.00 Density Multiplication Factor 5.00 (b) Incidence angle = 60~. Figure 6.33: Comparison of one-way transmissivity through a black spruce canopy modeled with a continuous crown layer (MIMICS I) and a discontinuous crown layer (MIMICS II) for V and H polarizations for incidence angles of (a) 20~ and (b) 60~ as a function of the density multiplication factor.

182 and HV-polarized backscatter. As the number density of scatterers in the individual crown volumes increases, the open canopy allows more radiation to penetrate the crown layer than does the closed canopy. This gives rise to a pronounced groundtrunk interaction mechanism and a decrease in the direct crown backscatter. Figure 6.33 illustrates the effect of Md on the canopy transmissivity. Here, the transmissivity of the closed canopy decreases linearly (on the dB scale) with Md while that of the open canopy shows less sensitivity. For low Md, the open and closed canopies have very similar transmissivities, while for high Aid, the open canopy allows more radiation to penetrate than does the closed canopy. This is a direct result of the gaps that are present in the open canopy crown layer. As the crown volumes become more and more opaque, the effect of these gaps becomes more pronounced. For perfectly opaque crowns, the value of crown layer transmissivity becomes P(O), which corresponds to the fraction of incident radiation that intersects no tree crowns. On the other hand, for a perfectly opaque continuous canopy, no radiation penetrates the crown layer. Note that the difference between MIMICS I and MIMICS II diminishes as incidence angle increases. In adapting canopy geometry from the closed-crown to the open-crown case, the importance of the individual crown shapes should be considered. One way which allows this question to be addressed is to examine the effect of varying crown side length, 1. Figure 6.34 shows such an analysis. Here, I is varied from 0.5 m to 2.24 m while holding the total number of crown layer scatterers constant. The maximum value chosen for I yields an individual crown volume in the open canopy that is equivalent to the effective individual crown volume of the closed canopy and is given by Imax = 1//Nt where Nt = 0.2 trees/m2 is the canopy density. As seen in Figure 6.34, a~ of the open canopy at Imax is nearly identical to that of the closed

183 -6.00 -8.00 m O0 O -10.0 -12.0 -14.0 - -16.0 0.500 1.00 1.50 2.00 Crown Side Length (meters) 2.50 Figure 6.34: VV-polarized canopy backscatter as a function of crown side length with total crown layer biomass held constant. Results are for L, C and X bands at an incidence angle of 0 = 20~. Results for the equivalent closed crown canopy (MIMICS I) are also shown.

184 canopy. Making the crown volumes smaller by decreasing 1 while increasing the number density of scatterers in the individual crowns such that the total number of scatterers in the canopy remains constant then indicates the sensitivity of a~ to this parameter. The X-band backscatter exhibits the most sensitivity to 1, with a~ increasing by about 1 dB as I is decreased to 0.5 m. Canopy cover fraction, C, may be defined as the fraction of total canopy area that is seen as covered by vegetation when the canopy is viewed at an incidence angle 0 0. Applying the Poisson distribution, the fraction of ground area not covered by any crown is P(0) = e-N' so that the fraction of covered area is 1 - P(0) or C =1 - e-Nt C (6.8) MIMICS II provides a convenient method for modeling backscatter as a function of cover fraction. Figure 6.35 shows such an analysis at L-band for 0 = 200. The crown volumes are identical to those in the black spruce canopy while canopy density is varied over 0.02 < Nt < 2, giving 0.0198 < C < 0.8647. Backscatter is seen to increase with cover fraction for VV and HV polarizations but decreases slightly for HH polarization. Figure 6.36 shows the components of total VV-polarized backscatter at L-band. This figure shows that for lower values of C the ground-trunk component contributes most to the canopy backscatter whereas at high values the direct crown backscatter comes more into play. The reference canopy has Nt = 0.2 which corresponds to C = 0.18.

185 0.000 o tO -10.0 ' - VV pol. /........ --- — HHpol. /- HHVpol. ~ I I I -20.0 -30.0 0.0 00 0.200 0.400 0.600 0.800 1.00 Canopy Cover Fraction Figure 6.35: L-band backscatter from a black spruce canopy izations as a function of canopy cover fraction. for VV, HH and HV polar -5.00 -10.0 '0 o 0 -15.0 ' I * ' * i ' ' * I ' * 2,,, -", ^ Total ' /,' ""-.. -... Direct Crown,"""""' ---- Ground-Trunk. \ -20.0 -25.0 0.0( 00 0.200 0.400 0.600 0.800 1.00 Canopy Cover Fraction Figure 6.36: L-band VV -polarized backscatter from a black spruce canopy at an incidence angle of 0 = 200 as a function of canopy cover fraction.

186 6.4.2 Coniferous Canopy Simulation Having examined the effects of an open crown layer on backscatter from a sparse stand, the backscatter from a stand of more fully developed coniferous trees is now considered. Total canopy backscatter from the white spruce stand (WS-5) studied in the boreal forest analysis was dominated by the direct crown component. The treelevel parameters of the coniferous stand now simulated by MIMICS II are chosen to be similar to those of stand WS-5 and are summarized in Table 6.19. Table 6.19: Tree level parameters for the coniferous stand. Constituent Mean Mean Number Orientation Length Diameter Density Primary Branches 1.13 m 2.24 cm 3.08 m- 3 sin4 8 Secondary Branches 0.57 m 1.04 cm 16.62 m-3 ~ sin9 0 Needles 1.6 cm 0.1 cm 86,162 m-3 sin Trunks 17.4 m 18.0 cm 0.12 m-2 Vertical For purposes of modeling the crown layer, the crowns are assumed to be conical with identical height h = 10 meters an basal diameter I = 3 meters, yielding a volume of V' = 23.56 m3. Noting that the equivalent volume per tree in the continuous canopy simulation with crown layer thickness d and canopy density Nc was d/NC = 17.4/0.12 = 145 m3, the number density of each crown constituent in Table 6.19 represents an increase of 145/23.56 = 6.15 over the corresponding number density of the continuous canopy. Table 6.20 lists the dielectric parameters for the e coniferous stand at L-, C- and X-bands for both frozen and thawed canopy conditions. These parameters are consistent with those of the white spruce stand. Figure 6.37 shows MIMICS I and MIMICS II simulations of vertically polarized one-way crown layer transmissivity for thawed and frozen canopy conditions at L-,

187 Table 6.20: Coniferous Canopy Dielectric Characteristics. Thawed Conditions Constituent L-Band C-Band X-Band Primary branches 34.78 +i 10.58 27.59 +i 11.34 18.82 +i 12.46 Secondary branches 19.11 +i 6.54 14.57 +i 5.63 11.88 +i 5.78 Needles 22.26 +i 7.40 17.15 +i 6.77 13.84 +i 7.09 Trunks 36.47 +i 10.99 29.01 +i 11.97 22.78 +i 13.2 Ground surface 5.6 +i 1.4 6.6 +i 0.9 5.8 +i 1.4 Frozen Conditions Constituent L-Band C-Band X-Band Primary branches 5.12 +i 1.08 4.78 +i 0.32 4.74 +i 0.18 Secondary branches 4.34 +i 0.97 4.04 +i 0.29 4.00 +i 0.16 Needles 4.53 +i 0.98 4.22 +i 0.30 4.18 +i 0.16 Trunks 5.19 +i 1.09 4.85 +i 0.32 4.81 +i 0.18 Ground surface 7.96 +i 0.96 7.96 +i 0.96 7.96 +i 0.96 C-, and X-bands. The difference between the open-crown and closed-crown cases is more prevalent for thawed conditions. In this case, at 0 = 200 the MIMICS I and MIMICS II transmissivities differ by almost 2 dB at L-band at by about 3 dB at C- and X-band with the open-crown canopy having higher transmissivity than the closed-crown case. These differences decrease as incidence angle increases. For frozen conditions, almost no difference in transmissivity is seen at L-band whereas about 2 dB of difference is observed at 0 = 20~ for the other frequencies. Once again, these differences diminish as 0 increases. Figures 6.38 and 6.39 show simulations of canopy backscatter for thawed and frozen conditions. MIMICS I and MIMICS II simulations are shown for like- and cross-polarized configurations at L-, C- and X-bands. For thawed conditions, the like-polarized backscatter from the open-crown canopy is as much as 6 dB higher than that of the closed-crown canopy with the greatest difference observed at X-band for shallow incidence angles. However, cross-polarized backscatter from the open-crown canopy is less than that from the closed-crown canopy. Backscatter from the frozen

188 canopy exhibits similar behavior but with much less difference between MIMICS I and MIMICS II. Once again, these differences decrease with increasing 0. To gain an understanding of model behavior, the individual contributions to canopy backscatter may be examined. Figures 6.40 and 6.41 show the contributions to total X-band VV-polarized backscatter for thawed and frozen conditions. The open crown layer has been shown to contribute to a higher canopy transmissivity. Figures 6.40 and 6.41 demonstrate that this increase in transmissivity contributes directly to enhanced contributions from the ground-trunk interaction mechanism as well as from the other mechanisms that involve the lower (trunk and ground) layers of the canopy. Furthermore, the open-crown canopy exhibits less direct crown backscatter than does the closed-crown canopy. The effect of applying MIMICS II to model changes in backscatter as a function of canopy parameters that are not directly related to the crown layer constituents is illustrated in Figure 6.42. This figure shows the MIMICS I and MIMICS II like-polarized L-band response to changes in volumetric soil moisture at O = 20~. Figure 6.42(a) shows these simulations for dry trunks with relative dielectric c, = 5.19 + il.09, which is equivalent to that applied for frozen trunks. Figure 6.42(b) shows simulations for wet trunks with Er = 36.4 + i10.99. The remaining canopy dielectrics are assumed to be the same as for a thawed canopy. Soil moisture has been varied from a volumetric fraction of 0.01 which represents a very dry soil surface, to 0.5 which represents a well-saturated surface. For all cases, the increase in crown layer transmissivity has lead directly to an additional 2 dB of increased backscatter sensitivity over this range of soil moisture.

189 0. m [-._ U: E* O Cq* -5. -10. - MIMICS I- L-band -—. ---- MIMICS II - L-band — o — MIMICS I - C-band - - - - - MIMICS II - C-band — A — * MIMICS I- X-band ----- MIMICS II - X-band -15. L 20.0 30.0 40.0 50.0 60.0 Incidence Angle (degrees) (a) Thawed conditions 0. C/..4 E ru 3 O C) -5. _L:..2..'. - - - a,.......!.... I........ MIMICS I - L-band --—... - MIMICS II - L-band — o --- MIMICS I - C-band - - - - MIMICS II - C-band - -A - -MIMICS I - X-band -— v — MIMICS II - X-band 10. 1 S I 20.0 20.0 30.0 40.0 50.0 60.0 Incidence Angle (degrees) (b) Frozen conditions Figure 6.37: One-way V-polarized transmissivity through a white spruce crown layer modeled as continuous (MIMICS I) and discontinuous (MIMICS II) for (a) thawed conditions and (b) frozen conditions.

190 5.00 -5.00 vo MIMICS I - L-band... MIMCS n - LIband ----- MIMICS I - C-band.- * -- MIMICS II - C-band h- - MIMICS I - X-band ------ MIMICS n - X-band -15.0 -25.0 L 20.0 15.0 5.00. 30.0 40.0 50.0 60.0 Incidence Angle (degrees) (a) VV-polarization pq m la I-I *o to MIMICS I - L-band.........MIMICS II- L-band ----- MIMICS I - C-band - -. - -MIMICS I- C-band — A. -. MIMICS I - X-band ------- MIMICS - X-band -5.00 -15.0! 2C -10.0 -20.0 ).0 30.0 40.0 50.0 60.0 Incidence Angle (degrees) (b) HH-polarization ' --- —""""' --- —----— "'",,,,.,,.,,,, — -; I ----,_ — i -BI-. — --& TII 'r; lllh; C L"-",,,,,,.~,__T_-~ -C- — 9 —c MIMICS I - L-band.........MIMICS - L-band — _ — MIMICS I - C-band -- -- MIMICS I - C-band - -A - MIMICS I - X-band ----- MIMICS n - X-band.z % -30.0 -40.0 ' 20.0 30.0 40.0 50.0 60.0 Incidence Angle (degrees) (c) HV-polarization Figure 6.38: Backscatter from a thawed white spruce canopy modeled with a closed crown layer (MIMICS I) and with an open crown layer (MIMICS II) at (a) VV, (b) HH and (c) HV-polarization.

191 -5.00 rr. xL~ i~~~-^ MIMICS I - L-band...... MIMICS U - Lband ----- MIMICS I - C-band - - - MIMICS - C-band —. - MIMICS I - X-bnd ------ MIMICS - X-band -15.0 - 20.0 30.0 40.0 50.0 60.0 Incidence Angle (degrees) (a) VV-polarization m '3 10 MIMICS I - L-band......... MIMICS - L-band —. — MIMICS I - C-band - - -- MIMICS II - C-band - - -~ MIMICS I - X-band ------ MIMICS II - X-band 60.0 Incidence Angle (degrees) (b) HH-polarization -20.0.....,.-.:r::7..., "-W....:i. ~*.-. MIMICS I - L-band........MIMICS II L-band ----- MIMICS I - C-band - - - - MIMICS II- C-band - -- MIMICS I - X-band ------ MIMICS II- X-band m Oa O -30.0 i -40.0 'L 20.0 30.0 40.0 50.0 60.0 Incidence Angle (degrees) (c) HV-polarization Figure 6.39: Backscatter from a frozen white spruce canopy modeled with a closed crown layer (MIMICS I) and with an open crown layer (MIMICS II) at (a) VV, (b) HH and (c) HV-polarization.

192 0.000..... I.... vca -10.0... '~s ' \ " ------ Direct Crown, \..- - Ground-Trunk > -20.0 - > -0 --- —- Direct Ground %*. \ -- - - --- Crown-Ground *'*s-*s.-1.^*v1 \. -30.0.... - 20.0 30.0 40.0 50.0 60.0 Incidence Angle (degrees) (a) Closed crown layer. 0.000 ' Total 3 -10.0 - ' - > -20.0 %% 0U e I ----- DirectGround > 0.0 "-... I - Figure6.40: Contributions to net canopy backscatter at X-band from a thawed white spruce canopy for (a) a closed-crown canopy (MIMICS I) and (b) an opencrown canopy (MIMICS II). -30.0.... ~: I '. I.... 20.0 30.0 40.0 50.0 60.0 Incidence Angle (degrees) (b) Open crown layer. spruce canopy for (a) a closed-crown canopy (MIMICS I) and (b) an opencrown canopy (MIMICS II).

193 0.000 0-4 an sO -10.0 -20.0 % I %L\ /I^ ^ — %: / % *x ''.. t X~~~~~~~~~~~rr Total ---—. — Direct Crown -Ground-Trunk ------ Direct Ground - - —. Crown-Ground -\ - -.. I \ I " I \ \ I! % I I I _ \ - \ \ _\ _I o n nI -JU.U ' 20.0 30.0 40.0 50.0 60.0 Incidence Angle (degrees) (a) Closed crown layer. 0.000 C/ o >, -10.0.-......-... I I,, '.\ N, -\ I I * \I - \ \ / \ _ h i, - N " I I - II \ ~ Total -------- Direct Crown ------ Ground-Trunk -..... Direct Ground - - - -Crown-Ground -20.0 -30.0 - 20.0 30.0 40.0 50.0 60.0 Incidence Angle (degrees) (b) Open crown layer. Figure 6.41: Contributions to net canopy backscatter at X-band from a frozen white spruce canopy for (a) a closed-crown canopy (MIMICS I) and (b) an opencrown canopy (MIMICS II).

194 5.0.... -. -MIMICS W -5.0 X __-' -0- ^ — MIMICS I HH pol A,- -— v --- MIMICSIIvvpoi.V' - -^- -MIMICS II HH pol --- MIMICS II VV pol.8 -------- -10.0 0.0 0.1 0.2 0.3 0.4 0.5 Volumetric Soil Moisture (a) Dry trunks (,o = 5.19 + il.09). 11.0.............. — 0 — MIMICS I VV pol --- --- MIMICS II VV pol 6.0 ----- MIMICS I HHpol -—.- MIMICS II HH pol... 0- 1.0 0 "'*s -<v"' CQ,-'~ ^^ -- — 3 ----4.0 - -- -9.0 0.0 0.1 0.2 0.3 0.4 0.5 Volumetric Soil Moisture (b) Wet trunks (ec = 36.4 + ilO.99). Figure 6.42: Response of L-band like-polarized canopy backscatter to changes in volumetric soil moisture for a closed-crown canopy (MIMICS I) and an open-crown canopy (MIMICS II) for a canopy with (a) dry trunks and (b) wet trunks. Simulations are for an incidence angle of 0 = 20~.

195 6.4.3 Deciduous Canopy Simulation In Section 6.2 and in Appendix G, MIMICS I was applied to model multi-angle and multi-temporal backscatter from a walnut orchard. One conclusion of this study was that the discontinuous nature of the orchard canopy had a significant effect on backscatter at X-band. This hypothesis is now addressed by applying MIMICS II to model a deciduous canopy with tree-level parameters similar to those found in the walnut orchard. The crown volumes are assigned spherical shapes in this analysis. It should be noted that MIMICS II simulates backscatter from natural stands for which the trees are randomly distributed in location and that the orchard canopy represents a well-manicured hedgerow geometry. Therefore, a direct one-to-one comparison of MIMICS II simulations to measurement results is inappropriate. The tree-level parameters of the open-crown canopy are summarized in Table 6.21. In addition, the leaf parameters are identical to those presented in Section 6.2 except for a number density of 308 leaves per cubic meter in each crown volume. This yields an LAI of 3.4 averaged over the canopy. Effects of leaf curvature are ignored in this analysis. Table 6.21: Tree-level parameters for the open-crown deciduous stand. Constituent Mean Mean Number Orientation Length Diameter Density Primary Branches 0.38 m 2.03 cm 1.59 sin4 20 Secondary Branches 0.11 m 0.60 cm 1.39 sin 8 Stems 18 cm 0.1 cm 308 sin 8 Trunks 0.7 m 9.0 cm 308 Vertical To simplify the modeling process, the larger size class of branches that had been placed in the trunk layer in Section 6.2 is now distributed in the crown layer, thereby leaving only the vertical trunks in the trunk layer. This yields an equivalent con

196 tinuous canopy with a trunk layer 0.7 meters tall and a crown layer 4.1 meters tall. For a canopy with density 0.07 trees/m2, the equivalent effective volume per crown in a continuous crown layer is 4.1/0.07 = 58.57 m3 per crown. To model these as spherical crowns, let the spherical diameter d = 4.1 meters. Then the crown volume is =d3= 36.09m3. The number density of crown constituents is then increased by 58.57/36.09 = 1.62 in going from the continuous to the discontinuous crown layer. Table 6.22 lists the L- and X-band relative dielectric constants for the deciduous canopy. These values are consistent with those estimated for the walnut orchard. Table 6.22: Canopy Dielectric Characteristics. Constituent L-Band X-Band Ground Surface 25 + i2.5 20.2 + i7.6 Trunk Branches 45 + ill.2 35.0 + i14.8 Primary Branches 34 + i8.5 25.9 + i10.8 Secondary Branches 30 + i7.5 22.7 + i9.4 Leaves and Stems 28.3 + i8.5 21.8 + i8.8 For spherical crowns of diameter c, j p(sl) e-'sds = (A)ic + 1)] (6.9) for each eigenvalue A-. Applying this relation to the polarimetric open-crown canopy model allows simulation of canopy backscatter. Figure 6.43 shows MIMICS I and MIMICS II simulations of L- and X-band canopy transmissivity through a deciduous canopy as a function of incidence angle. The L-band transmissivity demonstrates less than 0.2 dB of difference between the opencrown and the closed-crown canopies. At X-band, however, one-way transmissivity differs by more than 1 dB between the open- and closed-crown models. Figure 6.44 compares the like-polarized X-band backscatter simulated with MIM

197 ICS I and MIMICS II. At 0 = 20~, there is as much as 2 dB of difference between the two models whereas at 60~ there is essentially no difference. Figures 6.45 and 6.46 show the contributions to X-band canopy backscatter for VV and HH polarizations. In all cases, a significant direct ground contribution is observed. This is an artifact of a combination of the high soil dielectric (cr = 21.8+i8.8 at X-band) together with the enhanced surface roughness used to account for the presence of short vegetation on the ground surface. The open-crown canopy simulations exhibit more direiect ground backscatter than do the closed-crown simulations because of the increased canopy transmissivity. In addition, MIMICS II predicts less direct crown backscatter than MIMICS I. If the underlying soil surface were assumed to be as smooth as that of the walnut orchard, the net canopy backscatter for VV-polarization would in fact be dominated by the direct crown component, and MIMICS II would predict an overall decrease in the net canopy backscatter relative to MIMICS I. Figure 6.47 illustrates the effect of varying the volume of the spherical crown while keeping the total number of scattering constituents in each crown constant. The like-polarized X-band backscatter is shown as a function of crown diameter for incidence angles of 20~ and 60~. These simulations are also compared to the MIMICS I results for the continuous crown canopy. Significant differences between MIMICS I and MIMICS II are seen for the shallower incidence angle especially for small crown volumes whereas essentially no difference exists between MIMICS I and MIMICS II for 0 = 60~ except at crown diameters less than about 2.5 meters.

198 0. m -i;C._ CA3 9 E >t C-~ I 6 <~ -1. -2. -3. -4. — 3 MIMICS I V pol — t3-M — IMICS II Vpol [ — ~4 — MIMICS I H pol _ --- -- MIMICS H pol.!. |.. I I r 20.0 20.0 30.0 40.0 50.0 60.0 Incidence Angle (degrees) (a) L-band 0. m - - Ct.CX 6 -1. -2. -3. -4. -5. 1 20.0 30.0 40.0 50.0 60.0 Incidence Angle (degrees) (b) X-band Figure 6.43: Comparison of MIMICS I and MIMICS II simulations of one-way transmissivity through a deciduous canopy for (a) L-band and (b) X-band.

199 0.000 -5.00 CQ a? -o to -10.0 - -15.0 20.0 30.0 40.0 50.0 60.0 Incidence Angle (degrees) Figure 6.44: Comparison of MIMICS I and MIMICS II simulations of like-polarized Xband backscatter from a deciduous canopy.

200 0.000 -10.0 m t: O oS - --------------------- r-' ~'~-V —^.~~~~~~ 8~~~5 s ''-~': -20.0 Total -—. --- Direct Crown — & --- Ground-Trunk - - - - Direct Ground - -v- - Crown-Ground -30.0 '20.0 30.0 40.0 50.0 60.0 Incidence Angle (degrees) (a) Closed-crown canopy (MIMICS I) 0.000 -10.0 0 -M.0 0> O t~ ~ -----—......\. \, —"'., > _ _ _-H7-_ _* -20.0 Total.o-. --- Direct Crown — o --- Ground-Trunk - - A - -Direct Ground - — W- Crown-Ground -30.0 20.0 30.0 40.0 50.0 60.0 Incidence Angle (degrees) (b) Open-crown canopy (MIMICS II) Figure 6.45: Comparison of contributions to net backscatter for (a) MIMICS I and (b) MIMICS II simulations of VV-polarized X-band backscatter from a deciduous canopy.

201 0.000 -10.0 a- VZ '19,,,,,-r-r_..,__..,______,,;- --- —--— — -----—. —8.-7 rrrrCCCLC ra 0 -o -20.0 Total -— 0 --- Direct Crown — B — Ground-Trunk — & -- Direct Ground - -v- Crown-Ground -30.0 ' 20.0 30.0 40.0 50.0 60.0 Incidence Angle (degrees) (a) Closed-crown canopy (MIMICS I) 0.000 -10.0,""- A _ ----- ^_" ---------- - ------ - - -- _- -- -- --- ---- -- ---- 0 - -,.......-o-........ ~El":''-'"= —''*........... e....... ','"0 --- C0 o I. -20.0 Total -— 0 --- Direct Crown — { --- Ground-Trunk -- -- - Direct Ground - -v — Crown-Ground -30.0 L 20.0 30.0 40.0 50.0 60.0 Incidence Angle (degrees) (b) Open-crown canopy (MIMICS II) Figure 6.46: Comparison of contributions to net backscatter for (a) MIMICS I and (b) MIMICS II simulations of HH-polarized X-band backscatter from a deciduous canopy.

202 0.000 -5.00 0 '.-...,,. --- —---------. --- -. --- --—, -------—, - X-band, VV-pol, 200 -------- X-band, HH-pol, 20~ ---- X-band, VV-pol, 60 ----- X-band, HH-pol, 600 O MIMICS I -10.0 -15.0 2.( 00 3.00 4.00 5.00 Crown Diameter (meters) Figure 6.47: MIMICS II like-polarized X-band backscatter sensitivity to changes in crown diameter. Simulations are shown for incidence angles of 20~ and 60~. The MIMICS I closed-crown canopy simulation is also shown.

203 6.4.4 Summary of MIMICS II Results In this section, MIMICS II has been applied to model three different types of canopy architectures. The first represents a sparsely populated stand of black spruce trees for which the crown layer constituents contributed little to the net canopy backscatter. Accounting for the discontinuities in this canopy through the application of the open-crown canopy model demonstrated that redistributing the crown constituents into individual crown volumes had little effect on Oa unless these volumes were very densely packed with scatterers. The second canopy consisted of a stand of conifers which was much more fully developed. The direct crown component of backscatter from this stand was a significant contributor to the net canopy backscatter. When applying MIMICS II to account for crown layer gaps, an increase was observed in the crown layer transmissivity, and a corresponding increase in the contribution of scattering mechanisms that involve the lower canopy layers (trunk and ground) followed. A decrease in the direct crown backscatter was also observed. The third canopy represented a fully-foliated deciduous stand. Here, MIMICS II predicted a significant difference in the canopy backscatter (compared to MIMICS I) at X-band. In general, the effects of the discontinuous crown layer geometries were found to be most prevalent at high frequencies and at low incidence angles. The canopies under study therefore appear more continuous from a radar perspective at high incidence angles, where a significant number of crown volumes are penetrated by the radar, and at low frequencies, where extinction in the crown volumes is less significant. Implications of accounting for the crown layer gaps were addressed in the analysis of the fully developed coniferous canopy, where an increase was observed in simulated backscatter response to changes in soil moisture. This is a direct result of the increase

204 in crown layer transmissivity which in effect allows the radar to see through to the lower layers of the canopy more easily. As with MIMICS I, MIMICS II models tree canopies as having distinct crown and trunk layers. In this approach, the positioning of the trunks is completely uncorrelated with the placement of the individual crown volumes. While this approximation has a negligible effect on backscatter for many canopy geometries, it may overestimate trunk-ground backscatter for other geometries. Cases in which this may be a problem include coniferous canopies in which the trunks extend a significant distance into the crown volumes. Effects of this limitation may be analyzed by eliminating the trunk layer and distributing the trunks within the crown layer as vertical branches. Thus the canopy is modeled as a single layer of vegetation with one trunk per crown volume. Modeling the canopy in this way allows for the placement of the individual trunks to be directly coupled with the location of the crown volumes.

CHAPTER VII CONCLUSIONS AND RECOMMENDATIONS 7.1 Summary This thesis has presented the development of a first-order radiative transfer model for simulating microwave backscatter from tree canopies (MIMICS). The model is fully polarimetric and accounts for a wide variety of canopy architectures. The fundamental contribution of this work has been the presentation of a. model for tree canopies that have discontinuous crown layer geometries (MIMICS II). The radiative transfer solution derived for the closed-crown canopy geometry (MIMICS I) has been extended to account for the open-crown geometry. Statistics have been developed that describe the scattering and extinction properties of individual crown volumes and these statistics have been introduced into the radiative transfer solution. MIMICS has been very useful in coupling canopy biophysical parameters to measured radar data in a number of modeling studies. In particular, when used to model L- and X-band scatterometer measurements of a walnut orchard, the model not only successfully accounted for the variation in measured backscatter as a function of radar incidence angle, but it also accounted for the variations in a~ observed on 24 hour and longer time scales. Using the model, it has been possible to couple these variations in r~ to variations in soil moisture and canopy water status. When used 205

206 to model backscatter from and transmissivity through Alaskan boreal torest stands, MIMICS successfully accounted for variations occurring as functions of species and canopy freeze/thaw state. In both of these analyses, canopy dielectric properties have been shown to be very important parameters in modeling the canopy scattering and attenuation. Model simulations have been performed in which results computed with the opencrown model were compared to results computed with the closed-crown model for canopies with the same total number of scattering constituents in the crown layer. Results of these analyses have demonstrated that the gaps that occur in the crown layer of a discontinuous canopy may significantly affect the crown layer transmissivity and the canopy backscatter. Generally speaking, for trees whose crowns are not well developed the crown layer has an insignificant contribution to the total canopy backscatter and the difference between the MIMICS I and MIMICS II solutions is small. However, for more well-developed canopies the crown layer constituents may contribute significantly to the net backscatter and therefore the crown layer gaps have an important effect on a~. This effect becomes more pronounced at shallow incidence angles and at higher frequencies. As incidence angle increases, and as frequency decreases the open-crown and closed-crown backscatter solutions become very similar. In spite of the remarkable success obtained in applying MIMICS, there have been some difficulties in simulating some measured data. Most notably, the accuracy of MIMICS is questionable for cross-polarized configurations at X-band. This problem occurred in modeling X-band backscatter from the walnut orchard, and is attributed to the fact that MIMICS, being a first-order model, does not account for multiple scattering contributions.

207 7.2 Recommendations for Future Work Several projects may be considered as natural extensions of this work. MIMICS has been shown to accurately model multi-polarized backscatter at lower frequencies (L-band). However the accuracy of results obtained for cross-polarized backscatter at X-ba.nd have been questionable. For this reason, techniques for obtaining numerically efficient estimates of second- and higher-order scattering should be explored. In addition to studying higher order scattering in the crown layer, a more general second-order model that accounts for multiple scatter between canopy layers should be explored. For example, effects of scattering interactions between the trunk and crown layers may be significant in some coniferous stands in which the trunks extend far into the crowns. In studying sparsely populated canopies and any canopy for which scatter from the underlying ground surface becomes important, the accuracy of the ground surface scattering models comes into question. This is especially true for natural forest canopies in which the underlying surface may have a very complex structure. The characteristics of understory and litter layers should be accounted for. In considering more varied types of forest stand geometries, it may be desirable to account for canopies that consist of more than one tree species. The techniques discussed in Chapter V may be extended to define statistical parameters over a second set of canopy- and tree-level random variables, thereby defining effective phase and extinction matrices for a mixed stand. Applications of this type become important when studying forest succession processes. Beyond these issues of model development, the inversion problem needs to be addressed so that useful information about canopy parameters may be obtained by

208 coupling the model to radar measurements. MIMICS inversion algorithms should attempt to provide estimates of soil moisture, canopy water status and total canopy biomass. This final aspect will provide a significant advance in understanding the role of forest ecosystems in the global carbon cycle.

APPENDICES 209

210 APPENDIX A DIELECTRIC MODELS FOR CANOPY CONSTITUENTS This appendix describes the relationships of the dielectric constants of the various canopy constituents to their respective moisture contents. Section A.1 discusses the variation of vegetation dielectric with gravimetric water content. The same dielectric model is used for all vegetation material, including leaves, trunks and branches. The dielectric behavior is governed by the gravimetric moisture content m9 and the bulk density p, which together define the volumetric moisture content m,. The model used to relate the dielectric coIistait of soil to its volumetric water content is given in Section A.2. This section also provides expressions for the dielectric constant of standing water. In all cases, the dielectric constants are assumed to have the form e = _' - je". Note that in order to apply results derived from these models as inputs to MIMICS, the complex conjugate of e should be used such that the dielectric constants are of the form e = -' + i". A.1 Dielectric Behavior of Vegetation A.1.1 Model in Terms of Volumetric Moisture Ulaby and El-Rayes [63] have shown that the dielectric constant of vegetation material may be modeled by a Debye-Cole dual-dispersion model. This model consists

211 of a free water component that accounts for the volume of the vegetation occupied by water in free form and a bound water component that accounts for the volume of the vegetation occupied by water molecules bound to bulk vegetation molecules. Based on this model, the dielectric constant for vegetation is given by: A+B (4.9+ er _. 22.f74 ) C 2.9+ (A.1) f f(z) (GHz) ++ f(G) fo 0.18 where f(Hz) is frequency in Hz, f(GHz) is frequency in GHz, and co = 4.9 (A.2) e, = 88.045 - 0.4147T + 6.295 x 10-4T2 + 1.075 x 10-5T3 (A.3) fo =(27r)-1 (A.4) (27rT) = 1.1109 x 10-10 - 3.824 x 10-12T +6.938 x 10-4T2 - 5.096 x 10-16T3. (A.5) where T is temperature in ~C and (A.4) gives fo in Hz. Given the gravimetric moisture content mg and the bulk density of the dry vegetation material p, the volumetric water content m, of the vegetation material may be found from m- mOp (A.6) 1 -mg (1 - p) The constants A,B and C are then computed using A = 1.7 + 3.2m + 6.5m2 (A.7) B = m, (0.82m, + 0.166) (A.8) C = 315 m (A.9) 59.5m2 + 1

212 A.1.2 Model for Leaves For leafy vegetation, A, B and C in (A.1) may be computed directly from the moisture gravimetric fraction. The constants become A - 1.7-0.74m +6.16m2 (A.10) B =mg (0.55m, - 0.076) (A.11) 4.64m2 C= 9 (A.12) 7.36m2 + 1 9 The dielectric model given by (A.1) hats been found to give excellent agreement with experimental data over a wide range of moisture conditions and over a frequency range extending from 0.2 to 20 GHz. It is used together with (A.6) through (A.9) to model the dielectric constants of trunks and branches and with (A.10) through (A.12) to model the dielectric constant of leaves. A.2 Dielectric Behavior of the Ground Surface In this section, the dielectric properties for two types of ground surfaces are considered. The first is a soil surface consisting of a mixture of sand, silt and clay. The second is a standing water surface. The dielectric constant for the soil surface is determined using an empirical model whereas a semi-analytic model is used to predict the dielectric of a standing water surface. A.2.1 Soil Hallikainen et al., [25] expressed the dielectric constant of soil consisting of a mixture of sand, silt and clay as Es =I - jeI Cs = El -.,C. (A.13)

213 where the real and imaginary parts each fit a polynomial of the form e = (ao + ao 5 + a2C) + (bo + biS + b2C) mv + (co+ + ciS + cC) m, (A.14) e c' or os. Here, mV is the soil volumetric moisture content while S and C are the sand and clay textural components of the soil in percent by weight. The polynomial coefficients are listed in Table E.1 and the prediction accuracy of the model is given by Hallikainen et al., [25]. Table E.1. Coefficients of Polynomial Expressions Coefficients for Computing c', Frequency (GIz) ao a l a2 bo bl b2 co Cl c2 1.4 2.862 -0.012 0.001 3.803 0.462 -0.341 119.006 -0.500 0.633 4 2.927 -0.012 -0.001 5.505 0.371 0.062 114.826 -0.389 -0.547 6 1.993 0.002 0.015 38.086 -0.176 -0.633 10.720 1.256 1.522 8 1.997 0.002 0.018 25.579 -0.017 -0.412 39.793 0.723 0.941 10 2.502 -0.003 -0.003 10.101 0.221 -0.004 77.482 -0.061 -0.135 12 2.200 -0.001 0.012 26.473 0.013 -0.523 34.333 0.284 1.062 14 2.301 0.001 0.009 17.918 0.084 -0.282 50.149 0.012 0.387 16 2.237 0.002 0.009 15.505 0.076 -0.217 48.260 0.168 0.289 18 1.912 0.007 0.021 29.123 -0.190 -0.545 6.960 0.822 1.195 Coefficients for Computing 'e_ 1.4 0.356 -0.003 -0.008 5.507 0.044 -0.002 17.753 -0.313 0.206 4 0.004 0.001 0.002 0.951 0.005 -0.010 16.759 0.192 0.290 6 -0.123 0.002 0.003 7.502 -0.058 -0.116 2.942 0.452 0.543 8 -0.201 0.003 0.003 11.266 -0.085 -0.155 0.194 0.584 0.581 10 -0.070 0.000 0.001 6.620 0.015 -0.081 21.578 0.293 0.332 12 -0.142 0.001 0.003 11.868 -0.059 -0.225 7.817 0.570 0.801 14 -0.096 0.001 0.002 8.583 -0.005 -0.153 28.707 0.297 0.357 16 -0.027 -0.001 0.003 6.179 0.074 -0.086 34.126 0.143 0.206 18 -0.071 0.000 0.003 6.938 0.029 -0.128 29.945 0.275 0.377 li This model is independent of soil temperature. In general, the dielectric constant of soil changes very little with temperature for soil that is not frozen. For soil temperatures below freezing, however, the temperature dependence becomes more important. Variations in the real and imaginary parts of e, as a function of temper ature and moisture content are shown in Ulaby et al., [70], p. 2099.

214 A.2.2 Standing Water The dielectric constant of standing water is, in general, a function of the water salinity S. At frequencies above 5 GHz, however, salinity exercises a negligible influence on c and, therefore, S may be set to zero in the expressions below (Ulaby et al., [70] pp. 2020-2025): sw = Sw-j s (A.15) / EswO -- Cswoo eSw = (SWoo + (2 (A.16) ~~'w = 1 + (2rf7,)sw+ if 27r2f7r, (eswO - s swoo) _i__ sw =21 + (A. 7) SW "1 + (27rfrT)2 2reof where Co is the permittivity of free space, cw,, = 4.9 and f is frequency in Hz. In general, eswO varies with salinity S (parts per thousand) and temperature T (~C) as swo (T, S) = eswo(T, 0) a (T, S) (A.18) where eso (T, 0) = 87.134 - 0.1949T - 0.01276T2 + 2.491 x 10-4T3 (A.19) a (T, S) = 1.0 + 1.613 x 10-5TS- 3.656 x 10-3S +3.210 x 10-5S2 - 4.232 x 10-7S3. (A.20) These expressions are based on data generated for salinities in the range 4 < S < 35. Similarly, the relaxation time Tsr may be expressed as rs (T, S) = rw (T, 0) b (T, S) (A.21) where Tsw(TO) = (2 )(1.1109 x 10-~1- 3.824 x 10-2T \27ir

215 +6.938 x 10-14T2- 5.096 x 10-16T3) b (T, S) = 1.0 + 2.282 x 10-5TS - 7.38 x 10-4S -7.760 x 10-6S2 + 1.105 x 10-sS3. (A.22) (A.23) This expression is based on data for 0 < T < 40"C and 0 < S < 157 for a solution of NaC1. Finally, the ionic conductivity ai is a (T, S) = au (25, S) e (A.24) where ui (25, S) = S[0.18252- 1.4619 x 10-3S +2.093 x 10-5S2 - 1.282 x 10-7S3] > = A[2.033 x 10-2 + 1.266 x 10-4A + 2.464 x 10-6A2 -S(1.849 x 10-5 - 2.551 x 10-7A +2.551 x 108A2)] (A.25) (A.26) with A = 25 - T. These expressions are valid for 0 < S < 40. A.2.3 Snow Layer Hallikainen et al.,[24], have modeled the dielectric constant of a wet snow layer with a Debye-like model. The relative dielectric constant of snow e, = e- je - / is given by E, - A+ rc = 1 + (f/fo )2,, C + (f/fo)mv 1 + (f/fo)2 (A.27) (A.28)

216 where m, is the snow wetness (volume %), f is frequency in GHz, fo is the relaxation frequency of liquid water at 0~C (GHz) and the coefficients A, B, C and x are empirically derived for f < 15 GHz as A = 1.0 + 1.83pd, + 0.02m" 015 B = C =0.073 x = 1.31 where Pds is the dry snow density (g/cm3).

217 APPENDIX B SCATTERING MODELS FOR ROUGH SURFACES This appendix describes three models for backscatter from rough surfaces. A rough surface may be characterized in terms of the surface correlation length, 1, and the standard deviation of surface height, s. The solutions presented here are derived by Ulaby and Elachi [62], Chapter 4, and, although valid only within a limited range of I and s, may still be used quite effectively in many situations. The three models presented are (1) the Kirchhoff model under the stationary phase approximation (also known as the geometrical optics model), (2) the Kirchoff model under the scalar approximation (also known as the physical optics model), and (3) the small perturbation model. Loosely speaking, the geometrical optics model is best suited for very rough surfaces, the physical optics model is suitable for surfaces with intermediate scales of roughness, and the small perturbation model is suitable for surfaces with short correlation lengths. Model forms given here are for surfaces with correlation functions of the form p(g) = exp(-_2/12). The validity conditions for these models are: * Geometrical optics model: kl > 6, > A, ks > V 2.76s 2 cos 0i

218 * Physical optics model: 12 kl > 6, 2 7 > A, m < 0.25 2.76s * Small perturbation model: kl < 0.3, m < 0.3, kl < 3.0 where A is the radar wavelength, k is the wavenumber, Oi is the radar incidence angle and m = v/2s/l is the RMS surface slope. To define the scattering problem geometry, consider a field incident on a rough surface in direction ki and scattered in direction k,. Let v' and h, denote the unit polarization vectors for the vertical and horizontal components of the incident field, and let Vs and hs denote the unit polarization vectors for the scattered field. These unit vectors are given in terms of the inclination and azimuth angles for the incident and scattered fields, (di, Xi) and (Os, s), by the following relations: kf x sinOf cos qf + ysinOf sin Of + z cos Of (B.1) hf = - sinf + cos f (B.2) vf = fx kf = x cos Of cos qf + y cos Of sin f - z sin Of, (B.3) where the subscript f E {i, s} represents either the incident or the scattered wave. For the backscattering case, Os = r- Oi and k, = T + qi. B.1 Geometrical Optics Model For a field with wavenumber k1 incident on a rough surface, the correlation products of scattering matrix elements used to compute the Stokes scattering operator are given by (SpqSn8 = q2 UpqUmn exp -2q j + (B.4) 8~rq n 22q2m Z Z22

219 The polarization dependent quantities are uvv = q — 1 [r-(v(.-ki)(iks) + rh(hs-ki)(h-iks)] (B.) qkiDIv _ qcqz[ Uvh - = z —[rv(vski)(h.-ks) - rh(hs-ki)(v-ks)] (B.6) qzkiD2 Uhv - qkD2 [rv(hs-ki)((h-ks) + rh(vs-ki)(h'-k,)] (B.78) zUhh - k1D2 [r,(h8.k,)(hik,) + rh(r8.ki)(k)] (B.8) q2 2 2 2_ 22 [1 - =i)] (B.9) q q+q+q =2k[-(kki) (B9) q = k 1(sin icos i -sin Ocos Os) (B.10) qy = k1(sin O sin - sin 0s sin q,) (B.11) z = Lki(cos 0 - cosOs) (B.12) D = (k<-.'()2 + (k.-hs)2, (B.13) A is the illuminated area, and rv and rh are the Fresnel reflection coefficients of the surface for v and h polarizations, respectively. B.2 Physical Optics Model In this case, the correlation products of scattering matrix elements are (SpqSmn) = 1612 (IC + I + I). (B.14) The first term, Ic represents coherent scattering from the surface and is present only in the specular direction with respect to the mean surface. The second term, Ii, represents incoherent scattering, and the third term, Is = Isx + Isy+ represents

220 incoherent scattering due to the surface slopes. These terms are given by n00 ( 2n c.2-1 r q212 Ii = 12Aapqa.e-qsp ex (B.15) Is = -rAq l - exp - 4n (B.17) Ic = apqan(2r)2A(q5(q)e-)q2s (B.18) where q,, qy and qz are given by (B.10) - (B.12), qt = q2 + q, A — qx = qy = 0 5(qx) (qy)q- (27qr)7 (B.19) 0 otherwise, Kx = qzs2(bpqan + apqbn) (B.20) y = qzs2(cpqamn apqcmn), (B.21) rvo(cos Oi - cos 0O) cos(q5 - qi ) pq = vv rho(cos Oi cos O - 1) sin(os - qi) pq = vh apq -- (B.22) rVo(1 - cos Ocos 0O) sin($, - 0i) pq = hv rho (cos os 0-,) cos(q, - di) pq = hh bpq = Zpq cos i (B.23) Cpq = Zpqsin i, (B.24) A is the illuminated area, and Zhh = rho[sin 0, - sin Oi cos(q - i)] + rhl(cos Oi - cos Os) cos(qs - i) (B.25) Zh = - sin(5 - qj)[rh0 sini C cos O, + rhl(l - cos 0i cos 0,)] (B.26)

221 Zhv =sin(5, - qi)[rvo sin Oi cos 08 + rvl(l - cos O cos Os)] (B.27) Z, = rvo[sin 0s - sin 0, cos(g, - 0;)] + rvi(cos i - cos OS) cos(Os - Oi). (B.28) The Fresnel coefficients are ro 2 COS Oi + 1 COS t(B.29) rho = (B.29) 7/2 COS Oi - 71i COS Ot r12 sin i + 7r/1 sin Ot rhl - h 2 sin +1 si(B.30) 772 COS Oi - 771 COS Ot lo 1 COS i + 772 COS Ot rTo = -- --- (B.31) 7/1 COS Oi - 772 COS Ot [7r sin Oi - 72 sin Ot - rvo(71 sin Oi + 772 sin 0t)] rvl -- — = --- —------- (B.32) 7/1 COS 0i - 7/2 COS Ot where sin t = - sin Oi (B.33) k2 with k1 being the wavenumber in the medium containing the incident field and k2 being the wavenumber in the rough surface medium. B.3 Small Perturbation Model The unit polarization vectors for the incident, scattered, and transmitted fields are now defined as: kc =- * sin 08 cos gs + y sin 0s sin qs ~ z cos 0s (B.34) h -x sin q, + y cos s, (B.35) v = h, x ki = ~x cos 6 s s 0c sin, - z sin 0, (B.36) ki n = x cosf sin 0 c +y sin f:z cos Of (B.37) lf =-x sinl Of + y cos qf (B.38) Vf hf x kf =:x Cos Of Cos f: y cos Of sin f - sin Of, (B.39)

222 where the subscript f E {i, t} denotes either the incident or the transmitted wave. The + and - superscripts denote upward and downward traveling waves, respectively. In addition, the transverse vector wavenumbers are defined as kl = xkr + y-ky for a general wave and kli = x'kkr, + ykyi for the incident wave. The z components of the vector wavenumbers are klz = k cos s (B.40) klzi = kl cosSi (B.41) k2z = /2-k2 sin2 O (B.42) k2zi = k22-ksin2i. (B.43) The correlation products of the scattering matrix elements for this model are given by (SpqSmn) = A k Cos2 0sfpqfnnW(jk~ - k~-l). (B.44) The spectral density is (sl)2 r12 1 W(jkL - ki) -= 4 exp -4 kl- k-l, (B.45) where Ikl - ki 2 = k2 [sin2 O, + sin2 i - 2 sin 0, sin Oi cos(O -.i)] (B.46) The polarization dependent factors are _ 2krk2klkA(k kI)22- ~~ K2 (B.48) fvh(2k2 t2k2 kzi( k2 - k 2 )(B fv 2- 2klz)(ki ) (B.49) k2z + klz)(k2k2zi - k2 kilzi) hh = 2~k,)( k2kk.,ki(22 - kB- 150 fvh = K1(2 (B.49) f,__ 2kk k2cl -kzi (k2 -kl / l2)r fhv = K1 (B.49) (]k2z + k lz)(k2 2k2zi - k2 kzi) fhh = 2 1 K, (B.50) f-2klz(+ k22- kiz.)

223 with K, 1 kx kxi kyky~ (B.51) kp kpi ky kxi - kx kyi K2 = k- (B.52) kp kpi and k^i = Ik. + ky (B.53) kp= -kI + ky. (B.54)

224 PENDIX C SCATTERING MODELS FOR TRUNKS AND BRANCHES Individual trunks and branches are modeled as homogeneous dielectric cylinders with a specified length lc and diameter d,. Three models are presented for modeling scattering from cylinders. Each model is valid over a specified range of cylinder dimensions. The first model approximates scattering from cylinders whose size is smaller than a wavelength (lc << A) by modeling them as prolate Rayleigh spheroids. This model is appropriate for many size classes of needles and stems. The second model may be applied to long, thin cylinders with length greater than a wavelength but with diameter very small compared to A, i.e. lc >> A and dc << A. This model is appropriate for many types of intermediate size branches. The final model is based on the exact solution for scattering from an infinitely long cylinder and is appropriate for large size classes of branches and trunks. C.1 Scattering Matrix for Prolate Spheroids A solution for scattering from small prolate spheroids with a prescribed orientation has been presented by Tsang et al., [60], pp. 160-162. For cylinders with total length lc and diameter dc, the axis dimensions of the spheroid with equivalent volume

225 are 1 c = 2 ( (C.i) a = (C.2) b = a. (C.3) For spheroids with c > a, define Ac =-1 [2e + ln(1 )] (C4) and Aa = Ab = -A (C.5) abc where e = 1 - -a/c2. For a spheroid oriented with axes specified by xb, Yb and Zb such that its surface is described by 2 2 2 Zb + Yb = (C.6) a2 b2- C2 the scattering matrix elements are S = Q {( Zb) (Zb. Vi) + (Vs ) y V (Vs Zb) (Zb A V) S, Q { (h5-Z)(ZVi)+(h Yb) (y- b Vi)+ (5 Zb) (Zb- - (C.7) 1 + VdAa 1 + VdAb 1 + VdAc I + V~dAa 1 + lvdAb + VdA(C.) Svh =s ({ * hb) ( + b * Y) (hs * yb) (Zb * ) (C.9) h + VdAa I + 1dAb I + VdAc ( (hv Xb) (x. hi) (, b) (b hi) (& Z b) (ib h hi) 1z h Shh = Q 1 + vdA 1 + vdAb 1 + 1 dAc where k2 Q = ~vo^0(c-1), (C.ll) 47r

226 ko is the wave number, vo = 47rabc/3 is the spheroid volume, ~r is the relative dielectric of the spheroid, and abc Vd- (E-1) (C.12) The vectors (vi,,ih) and (vsi, h,) are the directions of the vertical and horizontal polarization vectors of the incident and scattered waves, respectively. C.2 Scattering Matrix for Long Thin Cylinders A solution for scattering from cylinders that are long and thin relative to wavelength has been derived by Sarabandi,[50]. For circular cylinders of cross sectional area A = irdc/4, the elements of th polarizability tensor 7 are PXX = 2A'- 1 (C.13) Pyy = P~ (C.14) PzZ = A(E,- 1) (C.15) For -a non-magnetic cylinder with the direction of the incident electric field specified by a = adx + dyy + azi, the far field scattered amplitude is 2k A sin U <S(a) = _ko [k5 x k, x (IP a)] U (C.16) o)- 47r U where U = 0.5kolc(kl Z- ki z), (C.17) the vectors ki and ks are the directions of propagation of the incident and scattered fields and ko is the wavenumber. For vertical and horizontal polarization vectors (vi, h') incident on the cylinder and vectors (v', h') scattered from the cylinder, where

227 the prime indicates that the vectors are in the local coordinate system of the cylinder, the scattering matrix elements are Svv = S(V). V (C.18) ShV = (0) h (C.19) Svh = S(h').*; (C.20) Shh = S(hi) hl. (C.21) C.3 Scattering Matrix for Large Cylinders For an finite-length dielectric cylinder oriented vertically, (Ruck et al.,[49]): - iEoo _0 ( 1) n ine' Eoo (_l)n CTEein91 S' (1~, qY) = Q (17b;, ~ [ -iZs-)-1)~ }C.E2),; - n= —oo On 1) Cne 2^n=-oo _n E where O' is the azimuth scattering angle in the plane perpendicular to the cylinder axis, hi is the angle formed between this plane and the unit vector along the direction of propagation of the incident field and s), is the angle formed between this plane and the unit vector along the direction of propagation of the scattered field. The summation coefficients are the same as those for an infinitely long homogeneous cylinder with relative dielectric constant er and diameter d: VPn - q2J(xo)H,(1x)(xo)J2(x1)( cTEM _ Mn n __ n-qn~nxzO) Hnl) Jn x, ) (C.23) PnNn- [qnHn()(xo)Jn(xl)]2 MnNn — qN, - O(20 nl) (Xo)Jn2(xl) Cn E- --- -2qJ(r)H,7'(xoJ(- (C.24) PnNn- [qnH1)(Xo)Jn(xl)]2 Cn 2= - - S (C.25) -X-0 PnNn- [qnHnl)(xo)Jn(l)] where kod cos 4i xo = (C.26)

228 xl= k -d sin(C.27) " sin ( 1 1 )i n2 'E - sin2 i Cos2 i (C.2 Vn = SlJn(xo)Jn(xi) - SoJn(xo)Jn() (C.29) en = rlHl1)(xo)Jn(xl) -soHl)(xo)Jn(xl) (C.30) n, = slH(1)(xo)Jn(xl) - soHn')('xo)Jn(xl) (C.31) Mn = rlJn(xo)JZ/(xl)- soJ(xo)Jn(xl) (C.32) and SO =, r (C.33) cos Oi 'r _ sin. Here, Jn( ) and Jn( ) represent the Bessel functions of the first kind of order n and their derivatives and H(1)( ) and H'()( ) represent the Hankel functions of the first kind of order n and their derivatives. The correction factor Q (4ii, Os) transforms the infinite cylinder solution to the finite cylinder case. In so doing, it is assumed that the length of the cylinder is such that,l >> A or that the cylinder dimensions are such that the relations 0.5 < kod 10 and,l >> d hold (Ruck et al.,[49]). In this case, Qi,, cos \ {sin [ko (sin Oi + sin Os) (C34) Q (Oz,,Os)=- is r cos Oi [ko (sin i + sin,s) ] f The scattering matrix of an arbitrarily oriented cylinder may be expressed as S = 7R S' T (C.35) where 'R= - (C.36) (h. v.) (h. h)

229 and 9T = (i I.)(v. (C.37) (hi?) (hi h?) The unit vectors in (C.36) and (C.37) represent the directions of the polarization vectors for the scattered and incident fields, respectively. The vectors (v,, h,) are directed along the scattered vertical and horizontal polarization directions in the reference coordinate system of the radar. The vectors (v, hc) are directed along the scattered vertical and horizontal polarization directions of the coordinate system local to the cylinder. The vectors (vi, hi) and (vc, h) represent a similar set of directions for the incident field. Given the previously stated constraints on the cylinder dimensions, the only region of error for this model is at angles of incidence at or near end-on (i c^ 2 ).

230 APPENDIX D SCATTERING MODELS FOR LEAVES Individual leaves are modeled as homogeneous dielectric disks with specified thickness T and diameter d. Two models are presented for modeling scatter from such disks each of which is valid over a specified range of disk dimensions. The first model approximates scattering from a disk whose diameter is small compared to a wavelength (d << A) by modeling it as an oblate Rayleigh spheroid. The second model is a physical optics approximation of scattering matrix elements. Generally speaking, the Rayleigh model is appropriate for low frequencies while the physical optics model is appropriate for high frequencies. D.1 Scattering Matrix for Oblate Spheroids A solution for scattering from small oblate spheroids with a prescribed orientation has been presented by Tsang et al., [60], pp. 160-162. For disks with thickness r and diameter d, the axis dimensions of the spheroid with equivalent volume are T 3 3 c = 2 ) (D.1) a = () (D.2) b = a. (D.3)

231 For spheroids with c < a, define [ x/Va2 - c2 /a2 v — 2] Ac (a2 - c2)3/2 an (D.4) and Aa Ab — AC. (D.5) abc For a spheroid oriented with axes specified by Xb, Yb and ib such that its surface is described by a2 + b2 + (D.6) a2 b+ c1 the scattering matrix elements are 5 { ( *b) (xb Vi) (0s b) (Yb * Vi) (Vs ) ( Vi)} (D.7) svv + Q A + 1 A + l A (D.7) 1 + VdAa 1 + VdAb 1 + UdAc Svh = Q{(V,-b)(xb-/) +(08'Yb) (iab')+ (O'~b) 'hi)} (D.9) Sh,,h (i, * ( b) (xb it) (, yb) (b i hi) (t * i) (bi ) 1 Shh = j 1V4dAa + l+vAb + 1+vA J (DO) Q= So (Er-l), (D.11) 47r co is the wave number, vo = 4rabc/3 is the spheroid volume, 6. is the relative dielectric of the spheroid, and vd = -1). (D.12) The vectors (vi, /i) and (05, h8) are the directions of the vertical and horizontal polarization vectors of the incident and scattered waves, respectively.

232 D.2 Physical Optics Model Here, the disk is modeled as an a x b square resistive plate with area equivalent to that of the disk. For a disk of thickness r and diameter d, the dimensions of the square plate are a = 2 (D.13) b = a. (D.14) A physical optics solution for scattering from an arbitrarily oriented plate has been obtained by Sarabandi, [50]. The resistivity of the plate is izo R kor(- (D.15) where ko and Zo are the propagation constant and intrinsic impedance, respectively, of free space. Let the plate be oriented such that the spherical coordinate angles (Oj, Oj) specify the direction of the unit vector normal to the surface of the plate. Furthermore, let the directions of propagation of the incident and scattered fields be specified by the spherical coordinate angles (Oi, Xi) and (0s, ^S), respectively. Then, define the reflection coefficients 2R rH(1i) = (1 + -sec 1)-1 (D.16) 2R rE(1) = (1 + cos l )- (D.17) zo where cos <1 = -[sin Oj sin Oi cos(qj - qi) + cos Oi cos Oj > 0. (D.18) The scattering matrix elements are -iab sin U sin V Svv = A U V P2{cos cos P[(sin O sin 0j + cos 0 cos0j cos(,i - bj)) A U V

233 *(sin 0, sin Oj + cos 0, cos 0 cos(g,$ - qj)) + cos Oi sin(i- - qj) cos 0, sin(qS - /j)] (rH - FE) +[cos(Oi - qj)(sin 0, sin 0j + cos 0, cos 0j cos(q, - Oj)) + cos 0j sin(45 - 0j) CosS 0 sin(q, - jj)] * (rH - cos2 /3 cos2 rE)} (D.19) -iab sin U sin V Svh = U V P{cos cos [- cos0j sin(qi - 0j) *(sin 0, sin 0j + cos 0, cos 0j cos(0q - j)) +cos(qi - $)cos 0sin(q$ - q5)] (rH - FE) +[- cos Oi sin(i- - j )(sin 0, sin 0j + cos 0s cos 0j cos(bs - 0j)) +(sin Oi sin 0j + cos Oi cos 0j cos(Oi - j)) *cos 0 sin(o-j - )] (rH - cos2 /cos2 oE)} (D.20) -iab sin U sin V S Shy =c A U V P cos cos[(sin 0 sin j + c os O cos 0j cos(i - j)) * cos j sin(j - q-,) + cos Oi sin(qi - $j-) cos(O, - ~j)](rF - FE) +[cos(i - (j) cos 0j sin(j- - q,) + cos 0j sin(Oi - 0j) cos(O, - j)] (rH - cos2 p Cos2 qrE)} (D.21) -iab sin U sin V Shh P= { P{cos cos [- cos qj sin(q0 - j) cos j sin(qj - -,) A U V + cos(qi - j) cos(q$ -j) )] (Hc - rE) +[- cos 6O sin(q- - 0j) cos 0j sin(qj - -5s) +(sin Oi sin 0j + cos O6 cos Oj cos(/i - j5)) cos(,s - j)] *(rH - cos2 /, cos2 OrE)} (D.22) where P = (1 - cos2 2 cos2 ))-1/2 (D.23)

234 and U = k(sin - sin ) (D.24) 2 kob V = (sin P cos - sin ' cos q'). (D.25) 2 The angles /3,,fl' and q' are defined through sin q = sin Oi sin(i - j) (D.26) cos 0 = [1- sin2 0i sin2(q! - oj)]1/2 (D.27) sin = q[cos Oi sin Oj - sini cos j cos(i - 0)] (D.28) cos / = -q[co cos cos + sin Q{ sin j cos(i - qj)] (D.29) sin ~' = sin 0, sin(O., - qj) (D.30) cos =' (1 - sin2 0, sin2(q, - (j))1/2 (D.31) and cos 0, sin Oj - cos Oj sin 0, cos(O - (D.32) 1 - sin2 0 sin2 (5 - j) where q = [1 - sin2 0i sin2(oj - i)]-1/2. (D.33)

235 APPENDIX E SECOND ORDER SCATTERING IN THE CROWN LAYER This appendix presents the derivation of the second-order radiative transfer solution for specific intensity in the crown layer of a tree canopy. The coupled radiative transfer equations in this layer are I+ (,;, z) = e-+ (z+d)/lI+ (. i _ -d) +/e-+ (z-z')/If,+ (, z') dz' (E.1) I- (-,l, q, z) = ecTz/'II I(-i_, 01, ) + eIc-(z-z')/I'F- (_(-1,, z') dz' (E.2) where rc+ and ic; are the crown layer extinction matrices for positive and negative propagating intensities, I+ and I, respectively, and F+ and F7 are the crown layer source functions. Applying the canopy boundary conditions yields the solutions for the specific intensities: I+(1, ~, z) = e-~ (z+d)/e-+Ht/I'7L (y) e-t H'/te-ICd/tI Io (-Io, 0o)6 (y - /,o)~ ( - 0o) + e-(z+d)/ e-KHt/' ( ) H e-tt -H'/ /* e";c-(-d-z')/L- ( ') dz'~ J d

236 + e-c (z+d)/e-K+H'/PZ (P). e~;7 ^(-(a+H,)-z')/,t-(-p,, z') dz' J-(d+Ht) + ec(z+d)/l11 e *(d+z')/IF+(i., 4, Z') dz' - ( d+ t ) /_;e-~ \(-z')/It +(p, z') dz' ~-d (E.3) IC-(-, u,q$Z)= eKCZ/"Io ( —o, o0) 6 (, - 1o) 5 (4 -_ o) + j~ eK'c-(z-z')hl-;(-,, z') dz' Z (E.4) The solutions for the first-order positive- and negative-going specific intensities at a depth z in the crown layer are I(,l+(,, Z) = icd (-1, 0, z) -L [j e- (z ')/ PC(, Kq;-1o, ko)ea;z'/4~ dz'] Io (-po, 4o) pCL/A~~~~~ d ( -J-dy~ J(E.5) (E.5) eK' Z/l" (P - -o) - (O - 0o) + [ eK (Z-z')/"Pc(-P,; -ilo, qo)eK;c -z'/ dz'] Io (-ito, 0o) (E.6) where Io (-slo, <o) is the intensity incident on the layer in the direction (-11o, 0o) with po = cos 0o and P, is the crown layer phase matrix. The first-order crown layer source functions are Fc(1, ( ~,z) = H [j7l AJ 'Pc(, 1 ', )I(i +(I ', [ Z) dQ' (+-.|(,zz) - - - p110, 1 ',+ q')I, (1', z) dQ'] +f27r T1 )- z),;-'-~p^,^ /a~ (-,^,z)df'] cF~ c(-,~,,z1) = - I0' Pc(-P, ~; p1i q')Il+(', '), Z )dQ + j|r f 'PC(-P, );-Pi ', q-', z) dQ'. (E.7) (E.8)

237 Substituting the first-order intensities into (E.7) and (E.8) and rearranging yields F^ ( z) = { I jorj lpC(H;. 2 ) i* [ e-c +(z-Z')/A' c(p', 4; -o0, qo)eK'z'/o0 dz' dQ' d + -Pc(,;; —/o, o)eKCrz/o 1 2rO 1, +-/ - ~-'P(~; -,V) [ e K~e( -z')/'(-_T C '; _-yo, o)e Cz'/1/ dz'] dQ'} ~Io (-/o, 0o) (E.9) r(1)- 1 cl1 C1z (.) (-t,, ~, z) VJ ( ~' e-(,,)-z ')/c('. '; -jUo o)eK 7'/' dz' dQ'.-d +- c(-#, 4; -to, Oo)eK~- z/.o it -;'rC( —,,,/; ~-fl ) - { o JJ o pe '(z ji j '4 ) + [ ~e I(z- ' )/(- ', ';-;o, -Oe o)e'ZtC c /z' dz' d' Io (-o, 0o). (E.10) Substituting these first-order source functions into (E.3) and neglecting interactions with the ground gives the second order solution for positive-going intensity in the crown layer: -d JO O *,dc (it,;-, z) )eKc 0 dz" d(E. dz' d ) \ +1 - e-C (Z-Z )/,7(; - po. O)eKc zl'~ dz' q- -d

238 +i - e-I e (Z-Z')/ [ l i -P(; -,u, q ) /. J-d [JO JO F \Je' ( -go, O)e/4d/ dQ' dz' Io (-Po, qo) (E.12) The positive-going intensity emerging from the crown is then: Id(e ) c= Idc( [,4O) (E.13) = { i/|i f e1 clL, L ) * (|e-; < e ( Z'-~"1' p (/;-UO, X)eK z/"~dz") dQ'] dz' }J-d + - eI'Z'//PTc(,I, q; -Po, Co)ec zl'/"o dz' /~ J-a 1 0 l 1 + ec J- (,; J ') 0K1 f, [ 2ir [ 1 1/:: + [!o2-Po' 10 +) e '; / P (-/;-P'o)e' z / dz") d' d)z' ~Io (-oo, 4o) (E.14) For backscatter, take sL = Po and s> = no + 7r. Making these substitutions and rearranging the equations gives the second-order solution for backscattered intensity: is(2)(fto O+ ) = {_- o eK +z./o(~Pc(o, o + 7r;;-/o, o)eK;'z/.o dz' ro d + -e /e- C r 1 Pc(Po, ~o + 7r; /, ~') [10 -d 0O ~ tode ( K )/lP(,;- o)e Z/d) dQ' dz' 1 e-' Z+ [fo-1'focL' j O; -go) 1o)elzz II1 " d d1 dz J-d 01 ~ er I 1 _ + — i / o [o c(,o, - + 7o;+ -, ') * (, e 'C(..".. )/,(-,; -Po, o)eKc Z"/ dz") dQ] dz' ~Io (-Po, 0o) (E.15) This solution represents the sum of three scattering mechanisms. The first term corresponds to the first-order solution for direct crown backscatter. The remaining

239 two terms correspond to second order scattering mechanisms. Each of these terms 1 2 3 0 -d_ -d - z --- " I2(,)Crown Layer Figure E.1: Terms contributing to direct crown backscatter in the second-order solution. is illustrated in Figure E.1. The mechanisms involved in each process are: * Term 1: This is the first-order term that represents an intensity incident on the layer at an angle 00, propagating to a depth z' and scattered directly back to the radar. * Term 2: This is a second-order term representing an intensity that is incident on the layer at an angle 00, propagates to a depth z", is then scattered in an upward-going direction defined by (,', q'), propagates to a depth z', and is then scattered back toward the radar. * Term 3: This is a second-order term similar to term 2. It corresponds to an intensity that is incident on the layer at an angle 00, propagates to a depth z", is

240 then scattered in a downward-going direction defined by (-u'/, 4), propagates to a depth z', and is then scattered back toward the radar. It is important to note that for a given radar incidence angle, computation of term 1 requires knowledge of the crown layer phase matrix only for a single scattering configuration. Similarly, knowledge of the extinction matrix is required only for intensity propagating in the (-,uo, 4o) and (Io, qo + ir) directions. However, computation of terms 2 and 3 requires knowledge of the phase and extinction matrices for scatter and propagation in all directions. For complicated phase and extinction matrices, this greatly increases the time required to compute the second order solution.

241 APPENDIX F CONNECTING MODELS FOR CANOPY BIOPHYSICAL PARAMETERS This appendix discusses some connecting models that relate fundamental canopy parameters to the parameters that drive MIMICS. Section F.1 defines the volume fractions of leaves, branches and trunks in a canopy, section F.2 defines the leaf area index of a canopy and section F.3 defines the specific and total water densities and biomasses of the canopy constituents. All three sections relate their defined parameters to MIMICS inputs. F.1 Elemental Volume Fractions The volume fraction of particles with an average volume V, (cubic meters) distributed with a density Np (particles per cubic meter) is vp = NpV. (F.1) Therefore, the volume fraction of leaves within the crown layer is given by v= Nr abfd (a, b) da db (F.2) where NI is the number density of leaves within the crown volume, r is the leaf thickness, a and b are the dimensions of the equivalent flat plates that model the

242 leaves and fd (a, b) is the probability distribution function (PDF) of leaf sizes. Here, the leaf thickness is assumed to be constant for all of the leaves. Similarly, the volume fraction of branches within the crown layer is given by vb= 4 Jl lcdcfc (l, d) dcl ddc (F.3) where Nb is the number density of branches within the volume, lc is the branch length, dc is the branch diameter and fc (lc, d) is the PDF of branch sizes. Finally, the volume fraction of boles within the trunk layer is -irNt 2 fc Vt= 7 JJ HdfH, (Ht, d)dHt ddt (F.4) the average height of the trunk layer, H[ and d1 ar the bole height and diameter, respectively, and fc(Ht, dt) is the PDF that describes the distribution of bole size. Here, a landscape patch is defined as a section of land that consists of some identifiable stand that is homogeneous with respect to the spatial distribution of the canopy elements. F.2 Leaf Area Index The leaf area index (LAI) is defined by Ulaby et al., [70], p. 1563 as the total single-side surface area of all the leaves contained in the canopy over a unit area of ground. This quantity is given by LAI = d-v (F.5) where vj is the leaf volume fraction, d is the thickness of the crown layer and r is the leaf thickness.

243 F.3 Biomass Parameters F.3.1 General Definitions The gravimetric moisture content of a specific plant part is defined as mg= =- M (F.6) MM + Md where M, is the mass of the water in the plant part and Md is its dry mass. The dry mass for a particle with volume Vp(cm3) is Md = p4Vp (grams) (F.7) where pp is the dry density of the particle in grams per cubic centimeter. Knowing Md and ma, the mass of water in the particle is given by AL = -aMd (grams) (F.8) 1- mg and the specific water density within such a particle is MW Ds, = V (F.9) mg -PPg ( ) (F.10) 1 -m 9cm Knowing the density of water in a typical particle, it is of interest to find the total area density of water in the landscape patch that contains the distribution of such particles. For particles distributed with a volume fraction VP, the total area density of water within the landscape patch is Dt = 1000 D vphp (F.11) = 1000 m ppvphp ( ) (F.12) where hp is the vertical extent (in meters) of the particle distribution.

244 The specific biomass of a particle is defined in terms of mass per unit volume of the specific plant part. The total biomass is defined in terms of the total mass per unit area over the appropriate landscape patch. Both biomasses may be defined in terms of wet and dry plant material. The specific dry biomass of a particle is equivalent to its dry density. The specific wet biomass of the particle is Bsp = Pp + Dp (F.13) = [-2m] (-sg. (F.14) The total dry biomass of a class of particles is B = 1000 ppvh (m2 (F.15) and the total wet biomass of a class of particles is Btp = B + D (F.16) = 1000 ppvphp ( - ) (F.17) = 1000 vp Bphp (m.2 (F.18) F.3.2 Constituent Biomasses and Water Contents Table F.1 lists the specific water contents and the specific biomasses for the three canopy constituents. Each of these parameters is given in units of grams per cubic centimeter and represents the density of water or material within the plant part itself. The variable p represents the dry density of the vegetation material and the variable m9 represents the gravimetric moisture of the constituent part as defined by (F.6). Table F.2 lists the total water density and total biomasses of the indicated canopy constituents. These parameters are defined in units of kilograms per square meter

245 and account for the constituent mass per unit area over the appropriate landscape patch. The variable he represents the crown the thickness of the crown layer and d represents the thickness of the trunk layer. The total biomass of leaves is also known as the total foliar biomass and defines the mass of leaves or needles per unit volume of canopy crown. For deciduous trees in temperate climates, this quantity is closely related to the net primary production. Table F.1. Specific water content and biomasses of the individual canopy constituents. Constituent Specific Water Specific Dry Specific Wet W ___________Content ( jBiomass () [ Biomass (jfi) s Leaves D,= -mg Bs = = -Bs p 4 1-m gPl VI dW 1 -1 Branches D~ mgb B -p BS Trunks D - mgt Pt Bst Pt -m Trukwts Dj -mt Pt = Pt Bt = a Table F.2. Total water content and biomasses of the canopy constituents. Constituent Total Water Total Dry Total Wet l _. | __Density (X-) Biomass (X-) Biomass (X) Leaves Dt = 100ODlvi, h Bt = 1000pivlh, Bt = 1000Btlvih, Branches Dtb = lOOODWbvbhc Bb = l00pbvbh Btb = 1000Bbvbh Trunks Dtt 1= OOODvtd Bt 1000ptvtd Btt = 1OOBvttd Total Woody Material Dt = D wt B+ D Bt + B t B + B^ tb + Bw Bd w - Bwb + Bwt Total Crown Material D =- Dt b + Dt = Bt B + Bt Bt = B t + Bt Total Canopy Material Dt + Dtb + Dt Bt1 + Btb + B l B + Bt d wb + BwtT U

246 APPENDIX G MODELING MULTIANGLE AND MULTITEMPORAL BACKS CATTER FROM A WALNUT ORCHARD - THE EOS SYNERGISM STUDY In this appendix, MIMICS I is used to model microwave scatterometer data that were obtained during the August 1987 Eos Simultaneity Experiment (Cimino et al.,[9]; Dobson et al.,[15]; McDonald et al.,[41], [42],[43], [44]). During this experiment, truck-based scatterometers were used to measure radar backscatter from a walnut orchard in Fresno County, California. Two sets of L- and X-band measurments are modeled. The first set consists of a seies of multiangle data for which a set of trees was observed at varying angles of incidence. The second set consists of a series of diurnal measurements in which this same set of trees was observed continuously over several 24 hour periods. With in situ ancillary data describing canopy architecture and moisture conditions used as input, MIMICS is run at L- and X-band frequencies of 1.5 GHz and 9.6 GHz. Measured scatterometer data are compared to theoretical data generated by MIMICS. MIMICS is seen to predict the diurnal variations that are observed on 24 hour cycles. Examinations of backscatter response to changes in canopy dielectric properties are performed to determine the causes of the changes observed in the short

247 term trends and diurnal patterns. G.1 Study Objectives and Test Site Description One of the objectives of the Eos Synergism Study is to characterize short-term variations that occur in vegetation canopies as a result of processes that change on rapid temporal scales (Cimino et al.,[9]; Way et al.,[81]). To this end, a concern of this analysis is to model the diurnal variations observed in the microwave backscatter from a tree canopy. Successfully coupling the measured canopy backscatter to modeled data will significantly aid in understanding the diurnal and short-term changes in canopy properties, thereby allowing one to infer physiological proce'sses occurring in the plant. This in turn will influence the ability to monitor changes in vegetation that occur on both seasonal and year-to-year time scales. A field experiment was performed at the Kearney Agricultural Center in Fresno County, California during the summer of 1987 as part of the synergism study. The test site consisted of a stand of over 200 six-year-old black walnut trees. The evapotranspiration of these trees has been monitored for several years, thus providing a somewhat controlled environment in which the synergism study could be performed. The orchard consisted of several individual water-stressed and unstressed tree plots. This analysis focuses on one individual plot of 48 unstressed trees. The average spacing between orchard rows was 6.7 meters and the average spacing between trees of a given row was 3.3 meters. These trees were irrigated in the evening with the amount of water equivalent to 100% of the water that evaporated from the canopy during the day as determined by meteorological measurements made at a nearby weather station. This experiment occurred during the trees' second year of controlled water treatment.

248 A hedgerow pruning technique has been used on these trees for several years. Consequently, the orchard does not represent a canopy with a true continuous crown layer. However, the tree crowns represent a significant enough fraction of the crown layer to allow for the canopy to be modeled as having a continuous crown layer. This is accomplished in the modeling analysis by re-distributing the crown layer constituents uniformly throughout the crown layer as if this layer were indeed continuous, thereby applying effective constituent volume densities in computing the phase and extinction matrices. G.2 Scattero temer Measurement Procedure Several sets of microwave scatterometer data were recorded over the course of the synergism experiment (Cimino et al.,[9]; Dobson et al.,[15]). The scatterometer experiments were designed to investigate the possibility of diurnal variations in radar backscatter from tree canopies. This analysis focuses on two of these data sets. The first is a set of multi-angle data in which the same set of trees were observed at incidence angles ranging from 40~ to 55~. This measurement set was recorded over a time span of about two hours during the mid-afternoon. The second data set consists of a three-day measurement series during which this same set of trees was observed continuously over a three day period at a 550 incidence angle. G.2.1 System Description and Operation L- and X-band data were recorded using the University of Michigan scatterometer system, POLARSCAT (Tassoudji et al.,[58]), which was mounted on a boom platform for observation of the orchard. POLARSCAT is a calibrated polarimetric radar system capable of measuring the amplitude and phase of the signal backscat

249 tered from a scene illuminated by its antenna for each of the four linear polarization configurations (HH, VV, HV, VH). The L- and X-band channels operated at 1.5 and 9.6 GHz, respectively. An HP 8753 Vector Network Analyzer functions as the transmit source and the primary signal processor. Automatic control of the equipment is accomplished with an HP 9000 computer. The computer is interfaced with the network analyzer and peripheral equipment including a disc drive and printer to provide for real-time data reduction, hard-copy output, and data storage. Other functions of the system are controlled by a network of DC control lines which originate from a manually operated control panel. This network controls the scatterometer RF sections, the antenna positioner, the polarization switching, and boom movement. The antennas were mounted on a rotatable positioner at the top of a rotatable boom that allowed pointing along any direction in azimuth and elevation. A video camera was mounted next to the antennas to allow the operator to view antenna pointing direction with a video monitor. The HP 9000 computer, HP 8753 Vector Network Analyzer, peripheral equipment, and other controlling instrumentation were located inside a motor home to provide for cooling and a clean working environment. The boom was elevated to a height of 12.2 meters and the system was parked next to the orchard. Each canopy measurement was recorded for a fixed incidence angle and polarization configuration by rotating the antennas in azimuth and recording 30 independent samples over the azimuth extent of the tharget. Te 30 samples were then averaged together to yield an average radar backscatter over the azimuth sweep. Thus, each backscatter data point corresponds to an average of thirty spatially independent backscatter measurements. This type of processing allows for the effects for signal

250 fading and target inhomogeneities to be accounted for. Figure G.1 illustrates the scatterometer measurement geometry. The region probed by the scatterometer contained three rows of trees. The shaded areas representing the tree crowns correspond to estimated volumes that contain foliage. The sensed volume V shown in Figure G.1(a) may be determined at a slant range R given the radar beamwidth /3 and look angle 0. The volume V has a length that depends on the range resolution of the scatterometer system. Since the vegetation canopy has a deterministic row structure, the fraction of V that contains vegetation may be readily computed through geometrical analysis. Considering Figure G.l(b), as the scatterometer scans through an azimuth angle a, V traces out an arc centered at the slant range R. The foliage fraction may be defined as the fraction of this scanning arc volume at the slant range R that intersects the foliage volume. Figure G.2 compares the uncalibrated backscattered power as a function of slant range for L- and X-bands to the foliage fraction computed using an X-band antenna beamwidth/3 = 2.8~. Through a straightforward geometrical analysis (Paris,[46]), it is possible to determine the sources of backscatter that contribute to the total backscattered power at a given slant range. Useful information exists at ranges where the power is above the noise floor. From this standpoint, useful data exist for 11 < R < 35 at L-band and for 13 < R < 18 at X-band. Comparing these traces to the foliage fraction indicates that significant sources of L-band backscatter exist at ranges for which the foliage fraction does not exist, thereby indicating the presence of significant scattering interactions between the canopy and ground surface. On the other hand, the X-band backscattered power contains essentially no information outside the foliage fraction region, indicating a lack of such scattering interactions. The reason for the apparent lack of backscatter interaction mechanisms

< * CD n C CD CD ^)?- CDD P 0 - 3 v ^ 1 o3 (D- cn "< - - > * D C CD 1 o r S - ( PcJ C, 1> co OT P9 O F S. O O 0. > i 0:> 00' o o0 C: o / '.':': tc en i —J 3o to.~..~..~..~..~..~..~. 0

252 -5.00 -15.0 I I L Band ------ XBand ------ Foliage Fraction ca OS T3 To -25.0 -35.0 -45.0 - -55.0 5.00 I 15.0 25.0 35.0 45.0 Slant Range (m) Figure G.2: Uncalibrated L- and X-band canopy backscatter versus slant range at 0 = 55~. The foliage fraction was computed for the estimated X-band sensing volume and scaled to fit on the dB axis.

253 at X-band is twofold. First, the occurrence of such interactions at ranges outside the immediate crown volume is masked by the system noise. That is, the signal level of these mechanisms is below -40 dB. Second, the relatively narrow X-band antenna beamwidth (/ = 2.80) limits the size of the sensing volume V to such an extent that much of these interaction mechanisms are lost simply because measurement geometry prevents them from being observed. If the scatterometer were positioned on a much higher boom, the measurement footprint within the canopy would be large enough to observe these interactions. X-band modeling in this analysis will focus on the direct crown component of canopy backscatter. It is interesting to note that the foliage fraction predicts the slight dip in backscattered power that occurs at R - 16 m as a result of the hedgerow canopy structure and that this drop is more significant at X-band than at L-band. G.2.2 Calibration The scatterometers were calibrated to an absolute level using a set of wire meshes and a Luneberg lens. In order to calibrate all four polarization configurations, a polarimetric calibration target was developed. The target consisted of an array of parallel wires oriented 45~ with respect to the antenna polarization vectors. This configuration generates like- and cross-polarized returns for each of the two transmit polarizations. One calibration sequence was performed using the wire targets before the multiangle data set was recorded. Two calibrations were performed against the polarimetric targets during the diurnal data series between August 24 - 26. One calibration was performed immediately before the beginning of the diurnal series and the other was performed immediately after the diurnal series was completed. Between these

254 calibrations, system stability was monitored by measuring the backscatter from the Luneberg lens several times a day. G.3 Orchard Canopy Characteristics As part of the synergism study, an extensive set of ancillary data was collected in order to characterize the walnut orchard. Data describing canopy architecture (Ustin et al.,[71],[72]), dielectric properties (Dobson,[10]) and canopy water status (Weber and Ustin, [82],[83]) were analyzed to determine canopy density, branch and leaf orientation and size distribution, constituent dielectric properties, and other gross canopy characteristics. This section describes the canopy architecture and other properties that have been adapted for input to MIMICS. G.3.1 Canopy Architecture Because of the row structure of the orchard and the hedge-row pruning practices, statistical sampling of the tree crown geometry was not possible. Instead, the length, diameter at mid-length and zenith and azimuth orientation angles for all branches with diameters greater than 2 cm were measured for eight trees. The number and size classes of all lateral branches were also recorded. All branch segments were numbered so that the tree skeletons could be reconstructed from these observations. In addition, the smaller branches with diameters less than 4 cm were statistically sub-sampled by class size. Four sample classes were considered (0-1, 1-2, 2-3, 3-4 cm diameters). To adapt the branch geometry data for input to MIMICS, the orchard is divided into distinct crown and trunk layers with heights of 3.1 m and 1.7 m, respectively. These heights correspond to the observed canopy architecture. The branches are

255 then divided into the four size classes identified in Table G.1. Figure G.3 is a sketch of the geometry of an individual tree, showing the four branch classes and the leaves. Figure G.3: Illustration of a walnut tree showing the four branch classes and the leaves. The larger branches tend to be located in the lower portion of the canopy and are therefore considered to be part of the trunk layer. This layer consists of the trunkbranch size class and includes the tree trunks and all branches with diameters greater than 4.0 cm. The remaining three branch size classes are distributed throughout the crown layer. As shown in Table G.1, the primary branch class consists of all branches with diameters less than 4.0 cm and greater than 0.9 cm. The secondary branch class includes those branches whose diameter is less than 0.9 cm and greater than 0.4 cm. The high order stems constitute the smallest size class and have diameters less than 0.4 cm. Most of these stems represent the green petioles that are attached directly to the leaves. These size parameters, are summarized in the table along with their

256 average diameter and length, volume density (branches per cubic meter), and the functional form of branch orientation for each class. In general, the larger branches tend to have mostly vertical orientations whereas the smaller branches exhibit no preferred orientation. Plots of the branch orientation PDFs are shown in Figure G.4. The branches in the trunk layer were assigned a cos6 0 2.0 1.5 L. 1.0 0.5 0.0 0 7x/2 3ir/4 0 (radians) Figure G.4: Branch orientation probability distribution functions (PDFs). distribution so that the mean value of 0 is 0. The primary branches were assigned a distribution of sin4 20 so that 0 has a mean value of 45~. The secondary branches and higher order stems were assigned spherical distribution functions (sin O)so that they show no preferred pointing direction in the elemental solid angle dQ = sin Od0dq. This implies that these branches are oriented such that their axis directions are uniformly distributed on a spherical surface. Trigonometric functions were chosen to describe these distributions since they provide a reasonable description of canopy

257 architecture and their forms allow for convenient numerical integration. These functions are converted into probability distributions for use in MIMICS by dividing each by a normalizing factor given by fo f (0) dO. Note that the PDFs describing primary and secondary branch orientations are completely described over the domain 0< 0 < Table G.1: Canopy Branch Classes. Branch Size Class Constituent Class Trunk Crown Branches Characteristic Branches primary secondary stems Max. Diam. (cm) - 4.0 0.9 0.4 Min. Diam. (cm) 4.0 0.9 0.4 Ave. Diam. (cm) 7.3 1.9 0.6 0.1 Ave. Length (cm) 92.8 35.8 10.9 5.0 Density (#/m3) 0.13 1.25 1.14 250 Orientation f (0) cos6 0 sin4 20 sin 0 sin 0 Characteristics of the leaves were determined from detailed leaf counts (Ustin et al.,[72]) and are summarized in Table G.2. Leaves were assigned a sin 0 orientation such that the direction in which the normal to the leaf surface is oriented is uniformly distributed over a spherical surface. The leaves are modeled as thin circular dielectric disks with a specified diameter and thickness. The leaf number density together with the leaf diameter and crown height yield an equivalent canopy leaf area index (LAI) of 3.4. For purposes of defining the folding angle X, the leaf is modeled as a square plate of area a x a equivalent to the area of the leaf modeled as a disk. The angle X is measured along the leaf midrib as illustrated in Figure G.5(a). Knowing X and a determines s, the distance between the opposite edges of the leaf. The leaf radius of curvature p is then defined as shown in Figure G.5(b). If the square plate conforms to

258 Table G.2: Leaf Characteristics. Number density 250 leaves per cubic meter Average diameter 7.47 cm Average thickness 0.1 mm Leaf area index 3.4 Orientation f (0) = sin 0 Folding angle X = 1520 along midrib Radii of curvature pi = 7.7 cm (along midrib) P2 = 10 cm a a/2 a/2 5e a S8 I, I I I I I X (a) (b) Figure G.5: Geometry used to model an a x a leaf folded along its midrib. The folding angle X shown in (a) defines the distance s between the opposite edges of the leaf. Chord length s and arc length a define the sector of a circle with radius p shown in (b).

259 a cylindrical surface such that a now defines a circular sector with chord length s, then p is the radius of the cylinder. pi corresponds to the radius of curvature determined from the leaf folding angle along the midrib. p2, the radius of curvature along the other leaf axis, has not been derived from measured data but rather represents an estimated value. The effect of leaf curvature on canopy backscatter may be accounted for through an analysis based on the model for a curved leaf introduced by Sarabandi et al. [51]. The backscattering cross section of a curved leaf (a,) normalized to that of a flat leaf (ouf) of equivalent area may be approximated by ac' 71 7 where ~(7)- = fjexp (iu2) du (G.2) is the finite range Fresnel integral, a a o (G.3) 7=- ~-, Y2= 2\- (G.3) 2 Vpi 2 Vp2 and ko is the free space wavenumber. A plot of (G.1) as a function of frequency is shown in Figure G.6. This plot illustrates the effect leaf curvature has on backscatter from a single leaf. Normalized backscatter of a leaf conformed to a cylindrical surface with pi = 7.7 cm and of a leaf conformed to an ellipsoidal surface with pi = 7.7 cm and p2 = 10 cm are shown. For a given curvature geometry and radar frequency, an effective leaf diameter de may be defined that corresponds to the diameter of a flat disk-shaped leaf having backscatter identical to the curved leaf. Since ac A2 where Ae is the effective leaf area, the effective diameter is de =(-) ~df (G.4) Orf

260 1.0.iI '' I 0.8 0.6 'U 0.4 z t -- Cylindrical Leaf, pl = 7.7 cm 0.2 Ellipsoidal Leaf, = 7.7 cm, 2 = 10cm ------—. Ellipsoidal Leaf, p. = 7.7 cm, p2 10 cm 0.0 I... I 0. 2. 4. 6. 8. 10. Frequency (GHz) Figure G.6: Backscatter from a curved leaf normalized to a flat leaf of equal area. Backscatter is shown for a leaf curved to fit a cylindrical surface with radius of curvature pi = 7.7 cm and for a leaf curved to fit an ellipsoidal surface with pi = 7.7 cm and p2 = 10 cm.

261 where df is the diameter of the flat leaf. The effect of leaf curvature on canopy backscatter may be approximated in MIMICS by using flat leaves with effective diameters that depend on the frequency under consideration. Table G.3 lists the normalized backscatter and corresponding effective diameters for L- and X-bands. At L-band, the effect of leaf curvature is essentially negligible. Table G.3: Effects of Leaf Curvature at L- and X-Bands. Normalized Backscatter Effective Diameter (ac/of) (cm) L-Band 0.972 (-0.1 dB) 7.42 X-Band 0.297 (-5.3 dB) 5.52 Flat Leaf 1.000 (0.0 dB) 7.47 A correction factor that accounts for the difference between the actual canopy LAI and the LAI observed with the scatterometer system may be determined by considering the foliage fraction together with the variation of leaf number density with height. The effective leaf area index may be computed by integrating the product of the average area of a single leaf Al and the leaf number density per unit volume NI over the slant range extent of the crown layer d/ cos 0: LAI(O) = ' Ni(s, 0)AI cos Ods (G.5) The value of LAIe represents an estimate of the leaf area actually observed by the scatterometer. The number density N1 for a given slant range s and incidence angle 0 is given by fN(s, 0) = f(s, O)Nlc(s) where f(s, 0) is the foliage fraction and Nl(s) is the number density of leaves at s. The correction factor Q(O) by which the actual canopy LAI is modified is given by the ratio of the effective LAI to the actual canopy LAI, Q(0) = LAIe(O)/LAI. By making use of canopy ground measurements and the scatterometer measurement geometry illustrated in Figure G.1, Q(O) may be

262 computed over the range 40~ < 0 < 55~. The factor Q(0) represents a slowly varying function that has a minimum of 0.35 at 0 = 40~, increases to a maximum of 0.6 at 51~, and then tails off to 0.55 at 55~. G.3.2 Canopy Dielectric Characteristics Observations of the relative dielectric constant of soil and vegetation were made in situ at 1.2 GHz using an Applied Microwave PDP1.2 field portable dielectric probe. Observations were made of the soil surface and tree trunks. Trunk measurements included both the exterior bark and the interior sapwood. The soil measurements were made using vendor-supplied attachments while the vegetation measurements were made using coaxial probe tips designed specifically for insertion into the tree boles. A statistically insignificant amount of dielectric data were recorded for the vegetation in the crown layer. However, the dielectric behavior of these constituents may be inferred from observations of other canopy physiological parameters, and the models applied here to predict the relative dielectric constant do in fact agree with the few recorded observations. Dielectric Properties of Tree Trunks To measure the relative dielectric constant of tree trunks, several probe tips were inserted at various depths into one of the tree boles in the 100% treatment plot. The probe could then be attached to any of these probe tips for observing the dielectric constant. The sensing probes were 0.141" in diameter, yielding a sensing volume for a dry medium that extends a maximum of 0.18 cm from the tip. The dielectric properties of the tree boles were seen to vary dramatically with time and exhibit a diurnal pattern which depended upon the insertion depth of the probe

263 into the tree trunk. Figure G.7(a) shows a piecewise fit to the measured values of the real part of the dielectric constant. Data are shown for two insertion depths during a three day period that coincides with the time during which diurnal scatterometer data were recorded. The numbers on the time axis correspond to midnight on that day of August. The dielectric constant is seen to reach a peak near daybreak at about 6:00 am. Shortly thereafter, the values decrease rapidly until a minimum is reached at about 12:00 noon. Beginning at about 7:00 pm, the dielectric constant increases until the maximum is again reached. These trends are consistent with data observed throughout the entire coarse of the experiment. The dielectric constant attained its highest values at about 2 cm depth inside the bole. Figure G.7(a) shows values recorded for this depth along with values observed at a 4-cm depth. In general, although the dielectric constant exhibits similar periodicity for all insertion depths, the values observed at other depths do not attain the extremely high maximum values observed at 2 cm nor do they they reach the same minima. Although the maximum values obtained at the 2 cm depth approach that of water or sap, no bleeding of sap occurred around the probe tip. The piecewise fit was chosen to represent an estimate of average dielectric for the outer bole layers since these layers generally exhibit high loss and therefore limit propagation of the incident field into the inner trunk layers. Figure G.7(b) compares this piecewise fit to the real and imaginary parts of bole dielectric constant for a 2-cm probe depth. To be consistent with measured data, a loss tangent of 0.25 was used to compute the imaginary part of the piecewise fit.

264 80.0 70.0 60.0 50.0 40.0 30.0 20.0 10.0 0.0 24. Re. er - Piecewise Fit 0 Re. Er - 2 cm depth B Re. ~r- 4 cm depth 25. 26. 27. Day of August (a) 80.0 0 t) -4 70.0 60.0 50.0 40.0 30.0 20.0 10.0 Re. er - Piecewise Fit --—... - Im. er - Piecewise Fit o Re. E - Measured l Im. Er - Measured 0.0 - 24. 25. 26. 27. Day of August (b) Figure G.7: Comparison of a periodic piecewise fit to measured L-band trunk dielectric constant data for (a) two insertion depths and (b) real and imaginary parts.

265 Dielectric Properties of Soil The soil was monitored on an approximate hourly basis. Each observation consisted of a minimum of 15 independent, randomly selected samples used to calculate the mean r of the surface layer. Because of spatial variations associated with the locations of the sprinkler heads used to irrigate the soil, the measured dielectric data exhibited a high degree of scatter that made it difficult to estimate an effective soil dielectric. Therefore, the soil surface area that received irrigation and remained mostly wet was analyzed separately from the area that received no irrigation and remained mostly dry. The overall effective dielectric behavior was then estimated by combining these results. Figure G.8 illustrates the process used to estimate effective soil dielectric. Figure G.8(a) shows the real part of the measured dielectric data for the irrigated and non-irrigated areas. The fit to the measurements recorded for the irrigated areas was derived by considering the measured dielectric values together with the orchard irrigation schedule. During these three days, the orchard was irrigated 2.5 hours per day beginning at 6:00 each evening. The irrigation periods are manifest by the jumps in the dielectric constant that begin at 6 p.m. each day. The dielectric continues to increase until the irrigation shuts off. Then, Er decreases as the soil dries. The measurments of,r of the non-irrigated area were essentially constant with a real part of 3.2. The loss tangent of the soil dielectric was assigned a value of 0.1 at L-band, as was determined from the measured data. Figure G.8(b) shows the-real part of the L- and X-band effective soil dielectric used to model the combination of irrigated and non-irrigated areas. The X-band soil dielectric was estimated by applying the model developed by Hallikainen et al. [25]. This model expresses the real and imaginary parts c' and E" of the dielectric constant I ~~~~~~~ ~~of h ielecrccn

266 40.0 1 1 1 r 1 1I I I I 0 0 Fit to irrigated area ------ Fit to non-irrigated area o Measured irrigated area X Measured non-irrigated area i Cr o c) Vo 5T 30.0 20.0 10.0 F BS El El0........... ----—. —..-} --- —--- - - - - o -e - - - D ---- ---------- n ( 24. 24. T 25. (i) T 26. (i) Day of August 1 27. (i) (a) 40. 30. - C4 20. 10. - 0.00 24. T 25. T 26. (i) (i) Day of August (b) T 27. (i) Figure G.8: Behavior of the soil dielectric constant showing (a) the fits to the measured L-band dielectric constants of the irrigated and non-irrigated areas and (b) the estimated behavior of the L- and X-band dielectric constant for the combination of irrigated and non-irrigated areas. (i) indicates the beginning of a 2.5 hour irrigation period.

267 of soil consisting of a mixture of sand, silt and clay in terms of the polynomial e = (ao + ajS + a2C) + (bo + bIS + b2C) nz + (co + clS + c2C) m (G.6) where e = ec or c", my is the soil volumetric moisture content and S and C are the sand and clay textural components of the soil in percent by weight. To determine the X-band soil dielectric, (G.6) was solved for mv using the L-band dielectrics to find an effective value for mv and then applied at 9.6 GHz to estimate the X-band dielectric. The general decreasing trend in ~r occurred because the canopy had been irrigated very heavily during the previous week. Whereas the canopy was irrigated 2.5 hours each day during the diurnal experiment, it had been irrigated 5 hours per day during the previous week. As the irrigated soil dried out form this saturated state, the total surface area of dry soil increased, thus leading to a general decrease in the effective soil dielectric over the three-day period. By applying the wet soil dielectrics model, a correspondence may be drawn between the change in soil dielectric and the change in soil volumetric moisture. According to this dielectric model, the change in soil dielectric from a maximum of Er - 22 + i2.2 to a minimum of or ^ 5 + iO0.5 approximately corresponds to a decrease in soil volumetric moisture from 0.32 to 0.1, or 68%. Dielectric Properties of the Crown Constituents Very few dielectric measurements were obtained for tree crown constituents. However, the dielectrics of the leaves and branches may be inferred from other ancillary data. Leaf gravimetric moisture content mgi may be used to determine the leaf dielectric constant. This quantity is defined in terms of the fresh and dry weights of the leaves, W/ and Wd, as mg = wI. Analysis of wet and dry leaf weights

268 indicate that the average leaf gravimetric moisture was approximately 0.7. Applying the vegetation dielectric model proposed by El-Rayes, [18], and Ulaby and El-Rayes [63] at L- and X-bands, the relative dielectric constant of leaves were found to be 28.3 + i8.5 and 21.8 + i8.8, respectively. This value was also assigned to the dielectric of the higher order stems. No discernible variation of leaf gravimetric moisture with time was observed. In order to assess the plant water status, a substantial amount of leaf water potential data was recorded (Weber and Ustin [82], [83]). The dielectric constants of the tree branches have been inferred by examining these leaf water potential measurements. Water potential is a complex characteristic of all plant tissue that defines the thermodynamic state of water in the plant (Bradford and Hsiao,[3]; Kramer,[34]; Passioura,[47]). Two components of water potential are of concern in this study. These are turgor pressure and osmotic potential. Turgor pressure is the pressure exerted by the cell contents on the cell walls. Osmotic potential is a measure of the ability of a solution to draw water from pure water through a semi-permeable membrane, and can be measured as the pressure that must be exerted on the solution such that no net flow of water occurs from pure water. Water potential is the sum of osmotic potential and turgor pressure. Measurement of water potential provides a sensitive means of assessing plant water status. The movement of water through a plant is along a water potential gradient from the soil, through the roots and stems, to the leaves, and, finally, to the air. Evaporation of water from the leaf through the process of transpiration increases the concentration of solutes thereby decreasing the water potential. This leads to the movement of water from the stem to the leaf and so on down to the roots. In a steady-state situation, if the plant had adequate water, the system dynamics would

269 lead to a constant water potential at any point in the system. Water potential reaches a minimum when the plant reaches the period of highest water demand or when it is water stressed. Figure G.9 shows the measured values of leaf water potential for the three day experiment duration. A periodic piecewise fit to these data is also shown. Each 0.0 rA 43 cr..4 o.,.. -5.0 -10.0 -15.0 -20.0 -25.0 ' 24. 25. 26. Day of August 27. Figure G.9: Comparison of measured leaf water potential to piecewise fit. measured data point represents the water potential measured for one individual leaf. More negative values of water potential indicate a stronger draw of water by the leaf from the plant. That is, the pressure with which the leaves are drawing water from the plant increases as the water potential becomes more negative. This phenomenon should have some effect on branch dielectric constant. Specifically, the branch di

270 electric should decrease as leaf water potential becomes more negative. Similarly, as water potential becomes less negative, the leaves draw less water from the plant, and branch dielectric should increase. Therefore, the behavior of branch dielectric should be similar to that of the piecewise fit shown in Figure G.9. Figure G.10 shows the real part of the piecewise fit to the dielectric constant for the three classes of woody vegetation. All measured values of the branch dielectric that were recorded during this time are also shown. These measured data were found by measuring the dielectric constant of a secondary branch at the point where it branched into higher order stems. The L-band piecewise fits shown for the branches were obtained by scaling the leaf water potential fit to match the measured data. Figure G.9 was first scaled to match the measured values of secondary branch dielectric. The primary branches were then assigned a dielectric function that had slightly more dynamic range than that used with the secondary branches. The maximum values of the secondary branch dielectrics were chosen to be close to the leaf dielectric. The loss tangent of all woody constituents, e" /e, is assigned a value of 0.25, as was measured for the bole and secondary branches. Since all dielectric measurements of this orchard were performed at L-band, Xband dielectrics were obtained through application of the model developed by ElRayes,[18] and Ulaby and El-Rayes [63] for estimation of vegetation dielectric constant. The vegetation dielectric is modeled with a Debye-Cole dual-dispersion model consisting of a free water component that accounts for the volume of the vegetation occupied by water in free form and a bound water component that accounts for the volume of the vegetation occupied by water molecules bound to bulk sucrose molecules. Assuming the bulk density of the dry vegetation material p = 0.4 and a constant

271 60.0 50.0 40.0 cw P0 C4~ s^, Trunk Layer -----— Primary Branches -Secondary Branches 0 Measured Branches 30.0 20.0 10.0 0.0 24. 25. 26. 27. Day of August (a) 60.0 50.0 40.0 -4 c3 <S 30.0 20.0 Trunk Layer -P --- — Primary Banches ------ Secondary Banches 10.0 - 0.0 24. 25. 26. 27. Day of August (b) Figure G.10: Dielectric constants of woody constituents for (a) L-band and (b) X-band.

272 temperature T = 13.6~ C, the dielectrics model was numerically inverted at L-band (1.2 GHz) using the dielectric functions shown in Figure G.1O(a), yielding effective values of mg as a function of time. Given the effective mg, the model was then applied at 9.6 GHz to obtain the real and imaginary parts of the X-band dielectrics. Figure G.1O(b) illustrates the real part of ~r at X-band. In general, the X-band dielectrics tend to be less than the L-band values and have less dynamic range. This frequency behavior has, in fact, been observed for other tree species (Dobson et al.,[11]). The diurnal variations shown in Figure G.10 are identical for all three days. That is, these functions were chosen to represent an average dielectric response of the vegetation over the duration of the diurnal experiment. G.4 Modeling Analysis As a first step in the modeling analysis, MIMICS was run as a function of radar look angle at L- and X-bands. Table G.4 lists the canopy dielectric parameters used in this analysis. These values correspond to measurements made at the approximate time that the multi-angle scatterometer data were recorded. Figure G.11 Table G.4: Canopy Dielectric Characteristics. Constituent L-Band X-Band Ground Surface 25 + i2.5 20.2 + i7.6 Trunk Branches 45 + ill1.2 35.0 + i14.8 Primary Branches 34 + i8.5 25.9 + i10.8 Secondary Branches 30 + i7.5 22.7 + i9.4 Leaves and Stems 28.3 + i8.5 21.8 + i8.8 shows a comparison of L- and X-band modeled and measured data over the range 40~ < 0 < 55~ for like- and cross-polarized backscatter. Figure G.ll(a) compares the predicted total canopy L-band backscatter with the scatterometer data. This

273 figure demonstrates very good agreement between MIMICS-generated data and the measured values. The like-polarized backscatter exhibit similar amplitudes with HH being slightly higher than VV in both the measured and modeled data while the cross-polarized backscatter is about 5 dB lower than the like-polarized response. Results similar to these have been obtained for like-polarized configurations by Chauhan and Lang [6] by using a distorted Born approximation to model the like-polarized backscatter from this data set. The failure of the model to predict the cross-polarized backscatter at 40~ is attributed to the inhomogeneous characteristics of the orchard canopy architecture. Whereas MIMICS I has been derived for a canopy that has a continuous crown layer, it is being used to model backscatter from a canopy with a crown layer that is discontinuous. As incidence angle becomes smaller, a larger proportion of the canopy area observed by the scatterometer consists of smooth, bare soil that is not covered by the orchard canopy. Since the model predicts backscatter for a canopy that has a homogeneous crown layer, some error will be introduced in the modeled data. We expect that the model will be more successful in predicting backscatter from this orchard at higher angles of incidence since the scatterometer observes almost no bare soil at these angles. We also expect this effect to be more pronounced for cross-polarized configurations since a smooth soil surface generates very little cross-polarized backscatter compared to that generated by the crown layer. As has been shown in Figure G.2, measured X-band backscatter consists primarily of the direct crown contribution to the total canopy backscatter. Figure G.ll(b) compares the predicted direct crown X-band backscatter with the scatterometer data. Here, MIMICS agrees with the level of the like-polarized backscatter but underestimates the cross-polarized response by as much as 10 dB. The failure of MIMICS

274 -10.0 -15.0 F a' 0 -20.0 F E E........ ---- ---- ---- VV MIMICS -------- HH MIMICS ---- HV MIMICS o VV Measured o HH Measured L-Band A HV Measured -25.0 - -30.0 L 35. 40. 45. 50. 55. 60. Angle of Incidence (degrees) (a) -8.0 -18.0 F /at ' I I.. I. I.... 0 0 ED like -- —.-...-.. - po-.. A A A A cross pol. - X-Band - - - - - - - - - - - - - - - - - - - - - - - - - - - - — ~~~~~~~~~~~~ -28.0 F -38.0 L 35. 40. 45. 50. 55. 60. Angle of Incidence (degrees) (b) Figure G.11: Comparison of MIMICS results with measured L- and X-band multi-angle data. (a) compares L-band modeled total canopy backscatter to the scatterometer measurements for like- and cross-polarized configurations (HH, VV, HV). (b) compares X-band modeled direct crown backscatter to the scatterometer measurements for these same polarizations.

275 to more accurately reproduce the angular dependence of the like-polarized backscatter at 45~ and 500 may be attributed to the inhomogeneous nature of the orchard canopy. Whereas MIMICS has been derived for canopies with continuous crown layer geometries, it is applied here to a canopy with a discontinuous crown layer. Furthermore, as can be seen from Figure G.1, the effective canopy geometry sampled by the scatterometer measurement volume changes with 0. This fact is further illustrated by the dip in backscattered power and in the foliage fraction at R _ 16 m (Figure G.2). As radar incidence angle changes, the canopy backscatter responds to these changes in the sampled canopy volume. The angular dependence of backscatter at Xband has been partially accounted for by applying the LAI correction factor Q(0) in generating the multi-angle MIMICS data. However, crown layer discontinuities also affect the character of backscatter from the stems and branches. This effect is more prevalent at X-band in part because of the relatively narrow X-band beamwidth and also because the crown layer constituents that contribute most to this effect (leaves and smaller branches) contribute more to X-band scatter than to L-band. As previously noted, the X-band cross-polarized backscatter is significantly underestimated by MIMICS. In general, the effect of higher-order multiple scattering on radar backscatter becomes more important as frequency increases. Ulaby et al. [65] have shown that, at millimeter wave frequencies, a numerical solution to the radiative transfer equations in which higher-order scattering is accounted for may add more than 10 dB to the predicted first-order cross-polarized backscatter while having little effect on the like-polarized backscatter. Since the numerical solution for radiative transfer requires specifying the scattering phase matrix in all incident and scattering directions, determination of the higher-order scattering contribution becomes very computationally intensive. The phase matrix of the walnut orchard

276 crown layer has a very complicated form and determination of the numerical solution is computationally prohibitive. Although an expression for the second-order scattering in the crown layer may be derived, analysis of these higher-order effects is beyond the scope of this study. While Ulaby et al. [65] derived their results at millimeter wave frequencies for which the scattering albedo for vegetation w c^ 0.6- 0.9, it is understood that w usually increases with increasing frequency. In light of the study by Ulaby et al.,[65], it is expected that, as frequency increases, higher-order scattering would first be manifest in terms of its effect on the cross-polarized backscatter. Figure G.12 shows the relative contributions of the three most important contributors to the total backscatter for the like- and cross-polarization configurations at L-band. Whereas in both like-polarized cases the same three mechanisms dominate the total backscatter, MIMICS predicts that the crown-ground and trunk-ground interaction terms are the more dominant mechanisms for the HH backscatter while the direct crown backscatter is the more dominant for VV backscatter. The major contributors to the cross-polarized backscatter are those mechanisms generated by the crown layer. This is expected since the crown layer branches are oriented such that they depolarize more than the trunk layer constituents. Having established that MIMICS successfully predicts canopy backscatter as a function of angle, the model is now run at a constant incidence angle, 0 = 55~, while varying the canopy dielectric parameters so as to simulate the variations seen over the three-day diurnal experiment (Figures G.8 and G.10). Figure G.13 presents the resulting computed backscatter along with the measured values of canopy backscatter for the like- and cross-polarizations. At L-band, MIMICS successfully predicts the appropriate level of the measured data together with the decreasing trend in backscatter observed over the three day period for all three polarization configura

277 -10.0 -15.0 —, 0 -20.0 -— " cL r,.rr' ~L C C C C C C -~ -10.0 -15.0 0 -20.0 -20.0,,_,,,___.. __....,....!.... I.... I.,.. I -25 n -25 n I 35. 40. 45. 50. 55. Incidence Angle (Degrees) 60. 35. 40. 45. 50. 55. 60. Incidence Angle (Degrees) -15.0 -20.0 -o -25.0 -25.0................... Total........Crown-Ground - Direct Crown ------ Trunk-Ground -30.0 I 35. 40. 45. 50. 55. 60. Incidence Angle (Degrees) Figure G.12: Components of canopy backscatter for HH, VV and HV polarizations.

278 tions. Furthermore, MIMICS predicts the 1 to 2 dB dip seen in aVV and o4HV in the early afternoon of each day. The X-band MIMICS data presented here represent the direct crown component of total canopy backscatter and have been produced for a canopy with an effective leaf area index of 1.0 in order to account for the variations in effective canopy geometry as a function of incidence angle. An offset of 8 dB has been added to the cross-polarized MIMICS data to approximate the effects of higher-order scattering. Although the measured X-band data exhibit significantly more scatter than does the L-band data, the early afternoon dip in backscatter is present for all three polarizations and is predicted by MIMICS. The variation in the measured data that is associated with the scatterometer measurement process comes primarily from two sources. The first of these is fading that arises from the coherent nature of the scatterometer. Following the analysis in Ulaby et al. [70], pp.483-486., the uncertainty due to fading is about ~0.2 dB. The other source of variation arises from statistical sampling of the inhomogeneous orchard canopy. This is caused in large part by the partially discontinuous properties of the crown layer. Because of the azimuth scanning technique used to account for the effects of fading, each measured data point represents an average of 30 samples recorded over a single azimuth sweep. The locations sampled within the canopy by each of these 30 samples do not correspond precisely to those observed during other azimuth sweeps. Therefore, some variation will exist simply because the values of a~ do not represent measurements of precisely the same canopy volume. In addition, factors such as wind speed contribute to a time-varying canopy geometry. This effect is readily observed in the measured diurnal data, especially at X-band. Since X-band backscatter is a great deal more sensitive to changes in the geometry of the leaves

279 -1120 - e e V e ~ ~ -16. L-Band X-Band -15.0 t19.- M M. CS. -16.0 -20. 24. 25. 26. 27. 24. 25. 26. 27. Day of August Day of August -11.0 - -1(. ---- MIMICS daM - MIMICS Dai -l -16. _ e Mewund Data -12.0 0 e Measured Da e.-13.0 -. 0.. t. - | -. -0. 24. 25. 26. 27. 24. 25. 26. 27. -16.0 00.. # * > -14.0 - -19. - 9 * ~ ~ L-Band gX-Band -15.0.-_ _ _ _ _ _ _ _ _ _ _ _ _ _ _._._ -20...__ _ _._ I.I 24. 25. 26. 27. 24. 25. 26. 27. Day of August Day of August (b) (e) -16.0 ae - 20. C ---- MIMICS dDat ~a Me eeofs Md oatt eDo -18.0 i. -2. -- -- — *- -*-o-*-'- -- - - -23. -- -- - * — -*- - *-*0- -— 2 L-Band *e * Ge -19.0 a MIMICS4Du-2. 020 0 Msured Da X-Band -20.0 -25.1 24. 25. 26. 27. 24. 25. 26. 27. Day of August Day of August (c) (0 Figure G.13: Comparison of MIMICS results with measured backscatter recorded during the three day diurnal experiment for (a) HH polarized L-band backscatter, (b) VV polarized L-band backscatter, (c) HV polarized L-band backscatter, (d) HH polarized X-band backscatter, (e) VV polarized X-band backscatter and (f) HV polarized X-band backscatter. The X-band HV MIMICS data has been offset 8 dB to account for multiple scatter. Day of August Day of Augu; (C)~ ~ ~~~o ~ 0 ~~ Fiue:.3,~omaio MMC eut i maue akcte crd uin th he a ducleprmetfr()H plrzdL:ndbcsatr (b Vplaie -an akcttr c H oa'edLbn acsatr (d)~~~~~~~ ~Hplrzd)-adbcicte,()V oaie -adb~kcte '\ ~:H oaizdXbadbcscte.Tt.~an VMMISdt o?~ ~~a benofe Bt con o: utpesatr

280 and stems than is L-band, and since these constituents exhibit more time-varying behavior than do the larger branches, a higher degree of scatter is observed in the measured X-band data. Modeling results shown here demonstrate extraordinarily good agreement between measured and predicted backscatter, especially when this measurement variability is taken into account. Comparison of the modeled and measured canopy backscatter allows some insight into the sources of backscatter variation that are caused by changes in the physiological state of the vegetation and soil. The trends observed in L-band backscatter over this three day period may be explained in part by Figure G.12. At 0 = 55~, the HH polarized backscatter is dominated by the trunk-ground backscatter component. This term responds directly to changes in both trunk and soil dielectric. The VV polarized backscatter, however, is dominated by the direct crown component which should depend strongly on changes to the dielectrics of the crown layer constituents. The components which interact with the ground do have a measurable effect on aov, exhibit some response to changes in soil and trunk dielectric, but not as much as caHH. The cross-polarized backscatter, however, is dominated by both the crownground and direct crown components. The trunk-ground component is more than 10 dB below these other two and should have almost no effect at 0 = 55~. a7V should therefore respond mostly to changes in both the crown and ground dielectric properties. The X-band response consists only of the direct crown component of canopy backscatter. The diurnal variations observed here are therefore attributed to changes in crown layer dielectrics. The MIMICS simulations shown in Figures G.14 through G.15 explain how Lband backscatter responds to changes in soil and vegetation dielectric constants.

281. — m c la I., I 0 10 0. (a) HH-polarized response. - To 0. (b) VV-polarized response. - m Ox - 60. 4020. Tl20.) ye ~(Primary Branchs) Re k) 0 Re -r (primary BralcheS) 0. (c) HV-polarized response. FigureG.14: L-band MIMICS response to changes in dielectric constant. trunk and primary branch

282 cs go m ID I 0. (a) HH-polarized response. ~0 -1 0. Re (Soil) (eTrunk) (b) VV-polarized response. ~T CQ -18. -20. 20 is. ~~~~~20. 0. 1 0.. 40. Re r (Soil) Re (Trurk) (c) HV-polarized response. Figure G.15: L-band MIMICS response to changes in soil and trunk dielectric constant.

283 Each figure shows c~ as a function of the real part of the dielectric constant for the like- and cross-polarization configurations. Figure G.14 shows backscatter as a function of trunk and primary branch dielectrics while Figure G.15 shows net canopy backscatter as a function of soil and trunk dielectrics. The imaginary parts are varied as well, keeping a constant loss tangent for each constituent (0.25 for vegetation and 0.1 for soil). These figures give a direct indication of the sensitivity of backscatter to changes in the canopy parameters. In each case, the dielectric parameters were varied over a range appropriate to the measured dielectric data. The HH-polarization response indicates that varying the soil dielectric constant from 20 + i2 to 2 + i0.2 corresponds to a maximum change of approximately 6 dB in HH*. Comparing this to the changes observed in aHH vs. trunk and branch dielectrics confirms that this quantity is most sensitive to changes in soil dielectric. The responses of caov and ajV seen in Figure G.14 (b) and (c) indicate almost no sensitivity to changes in trunk dielectric. However, these quantities are very sensitive to changes in primary branch dielectric over the range from cr = 20 + i5 to 4 + il. This sensitivity gives rise to the dips in backscatter seen in the early afternoons in Figure G.13 (b) and (c). The responses of auV and cHV to soil dielectric seen in Figure G.15 (b) and (c) show about a 2-3 dB change in a~ over this range of soil dielectric values, indicating some sensitivity to soil moisture. The decreasing trend in measured c~ may be attributed to the decreasing soil dielectric constant. All three polarizations respond to this change but with varying degrees of sensitivity. As shown in Figure G.13, the absolute level of CHH decreases by about 2 dB over the three day period while aVV and uaV each change by about 1 dB. Since a 68% decrease in effective soil volumetric moisture (mvy) has been observed, which includes the effect of irrigated as well as non-irrigated soil surfaces,

284 cHH exhibits a sensitivity to changes in soil moisture of about 2 dB/0.68 = 2.94 dB while ayV and cV have sensitivities of about 1 dB/0.68 =1.47 dB. Since aHH is twice as sensitive to changes in mvy, HH would be the the most effective polarization for determining changes in soil moisture. The VV and HV polarizations are more sensitive to changes in the dielectric properties of the crown constituents. This is substantiated by the dips observed in acV and cav during the early afternoons. The VV and HV backscatter dip from between 0.5 dB to 1.5 dB each afternoon. However, aHH which is much less sensitive to branch dielectric (Figure G.14(a)) exhibits no such dips. It is probable that the branch dielectrics are related to the water potential via the plant water status. Cov and crHV are therefore seen to yield some indication of water status, thereby containing some information about the physiological state of the plant. In this analysis it was found that accurate estimates of leaf parameters are important in modeling the X-band backscatter. Figure G.16(a) shows the X-band direct crown MIMICS response to changes in LAI and leaf gravimetric moisture for HH polarization. This gives a direct indication of how direct crown backscatter varies with leaf parameters. About 6 dB of sensitivity is observed over the indicated range of parameters. This underscores the importance of accurately estimating leaf dielectric, curvature and LAI in analyzing X-band radar data. Figure G.16(b) shows HH polarized X-band direct crown backscatter response to changes in primary and secondary branch gravimetric moisture. The range of moisture values shown here corresponds to the range of effective values applied in estimating the X-band dielectric constants. crHH shows only about 1.2 dB of sensitivity over changes in moisture values for the primary branches. However, significantly less sensitivity is observed to changes in secondary branch moisture. This indicates

285 -14.3 -19.0 -~20.0 -01100.7 0,8 0 (a) -17.0 -0o -19. SC..ndary l.aa., o.6 0.7 0.snchhg dpiarysranchM8 (c) -17.0 --18.0 --19. 0.8. 0.5 0.6 SeCOndary ]Bra 04 Bh Mg primary Branch Ug (C) Figure G.16: X-band HH-polarized direct crown backscatter response to changes in (a) leaf gravimetric moisture and leaf area index, (b) primary and secondary branch gravimetric moisture and (c) primary and secondary branch gravimetric moisture with primary branches assigned a sin 0 orientation function.

286 straightaway that the dips observed in the X-band diurnal backscatter are caused by variations in the dielectrics of the primary branches. Figure G.16(c) illustrates the effect of modifying the branch orientation on Xband HH polarized direct crown backscatter. The primary branches here have been assigned an orientation PDF of the form sin 0 which is identical to that assigned to the secondary branches and stems. Although the sensitivity to branch moisture here is similar to that shown in Figure G.16(b) where these branches had an orientation PDF of the form sin4 20, there is about a 0.5 dB decrease in backscatter over the entire range of primary and secondary branch moisture. This indicates a direct response of HH-polarized backscatter to changes in branch orientation. Figure G.17 shows the MIMICS-predicted backscatter to chnges in canopy biomass for the walnut orchard response for VV and VH polarizations. These data were generated by varying canopy height and generting a~ at each height. The models for canopy biophysical parameters presented in Appendix F were applied to compute the dry canopy biomass. The actual height of the canopy was 4.8 meters. Data are shown for P-band (0.5 GHz), L-band (1.5 GHz) C-band (5 GHz) and X-band (9.6 GHz). For low values of biomass, the backscatter at both like- and cross-polarizations is dominated by the direct-ground component of canopy backscatter whereas at high values the canopy itself dominates a~. Therefore, the a~ value observed at low values of biomass is determined solely from the estimate of direct ground backscatter for both polarizations. In order to achieve a reasonable estimate of both the like- and cross-polarized direct ground backscatter, measured values from Ulaby and Dobson [61] were used to simulate the direct ground backscatter. This approach was necessary because the first-order ground backscatter model implemented in MIMICS does not account for any cross-polarized return. It should also be noted that at

287 0. -5. -10. -15. 9s 0 10 -20. / -25.0.0 0.0 0.0 2.0 4.0 6.0 Total Canopy Height (meters) I 8.0 10.0.188 1.485 [.188 1.485 0.297 0.594 0.891 1 Dry Canopy Biomass (kg/m2) (a) VV-polarized response. -10. -20. S~<f1 -30. -40. 0.0 2.0 4.0 6.0 Total Canopy Height (meters) 8.0 10.0 I I I I I 0.0 0.297 0.594 0.891 1.188 1.485 Dry Canopy Biomass (kg/m2) (b) VH-polarized response. Figure G.17: Walnut orchard backscatter response to changes in canopy biomass for (a) VV-polarization, (b) VH-polarization. The incidence angle 6 = 30~.

288 high biomass values the X-band cross-polarized responce is several dB lower than anticipated because MIMICS does not account for multiple scattering in the crown layer. This analysis demonstrates that the lower radar frequencies (P- and L-bands) are more sensitive to changes in canopy biomass than are the higher frequencies (Cand X-bands).

289 APPENDIX H MODELING MICROWAVE ATTENUATION AND BACKSCATTER FROM ALASKAN BOREAL FOREST CANOPIES In March 1988, a series of airborne SAR data was acquired over the Bonanza Creek Experimental Forest near Fairbanks, Alaska (Way et al., [77], [76]; Dobson et al., [13], [14]). This study was the first in a series of multi-season aircraft experiments flown over selected forest sites for the purpose of understanding the kinds of biophysical properties that may be detected with spaceborne SAR systems such as the C-band SAR to be flown aboard the European Space Agency's Earth Resources Satellite (ERS-1). The purpose of this experiment was to determine if changes in plant fluid status associated with thawing and freezing result in changes in radar backscatter which could be detected by SAR and to determine if theoretical backscatter models such as MIMICS could predict these changes. Two aircraft-mounted SAR systems were deployed during this study. The Jet Propulsion Laboratory's (JPL) P-, L- and C-band quad-polarized SAR, mounted aboard a NASA/Ames Research Center DC-8, operated at center frequencies of 450 MHz, 1.26 GHz and 5.3 GHz, respectively. The Naval Air Center/Environmental Research Institute of Michigan's (ERIM) L-, C-, and X-band quad-polarized SAR, mounted on a Navy P-3, operated at center frequencies of 1.25 GHz, 5.26 GHz and

290 9.38 GHz, respectively. An array of passive reflectors and active radar calibrators (ARCs) was deployed in the vicinity of the Fairbanks International Airport and imaged during the overflights to provide for external calibration of these systems. This analysis focuses on the data obtained by the JPL SAR on March 13 at 15:03 and March 19 at 23:17 and by the ERIM/NADC SAR on March 22. All of these passes have a common look direction. These dates were selected to encompass the range of environmental conditions that occurred over the duration of the experiment. An unseasonably warm period during which thawed conditions prevailed in the forest extended through the evening of March 13. This was followed by more normal subfreezing temperatures for the remainder of the experiment. As liquid water was frozen by the subfreezing temperatures, the dielectric properties of both the vegetation and of the 20-30 cm snow layer that covered the ground were modified, thereby changing the scattering and absorption properties of these constituents. H.1 Test Site Description and Canopy Properties The Bonanza Creek Experimental Forest is 30,000 hectares in size and is contained within the Tanana Valley State Forest west of Fairbanks, Alaska, within a zone of discontinuous permafrost along the Tanana River. The wide diversity of forest successional stages that exist in the forest include mono- and mixed-species stands of aspen, birch, white spruce, balsam poplar, black spruce, willows and alders. To minimize the effects of surface slope on the radar backscatter, this study focuses on stands that are on the relatively flat islands along the Tanana River.

291 H.1.1 Stand Selection A series of forest stands were selected on the basis of forest cover conditions and accessibility for use in canopy extinction studies using point targets (Kasischke et al., [32]). For measurement of canopy transmissivity at C- and X-band, arrays of passive reflectors were deployed both in forested stands and on a nearby unforested sandbar. These stands consisted of a mono-species alder stand, along with mixed-species white spruce and balsam poplar stands. In addition to the stand selected for the canopy extinction studies, several other stands were selected for backscatter analyses. Each of these stands is of uniform age with a single-species composition and encompasses at least 4 hectares in area. Of these stands, 19 were white spruce, 12 were balsam poplar and 11 were black spruce. Ancillary data has been obtained and summarized for seven of these stands (Jaeger, [30]). The measured and derived stand characteristics provide the basis for the MIMICS simulations of canopy backscatter and transmissivity. H.1.2 Temperature Conditions An unseasonably warm period prevailed in early March 1988. During this time, thawed conditions prevailed in the vegetation and a melt zone formed in the surface layer of the snow. These conditions persisted through the March 13 JPL SAR overflight. Air temperatures during this flight ranged between 2.0~C and 9.50C. Subfreezing temperatures returned on March 14 for the remainder of the experiments. The air temperature on March 19 ranged between -14.0~C to -14.50C and was less than -20.0~C during the flight of the ERIM/NADC SAR on March 22.

292 H.1.3 Ground Surface Characteristics The ground surface was covered with a snow layer 20-30 cm deep. Below the snow layer, the upper 20 cm of the mineral soil was frozen throughout the entire experiment. Detailed measurements were made for one white spruce and one black spruce stand at the times of the overflights. These measurements included air and snow temperature profiles, snow depth, snow density profiles and snow wetness. Wetness measurements included a hydrochloric acid detection technique (Davis et al., [8]) and in situ dielectric measurements at L- and C-bands. The early March thaw caused the snow layer to have a complex wetness structure that varied with stand species (Dobson et al., [13]). Although the average dry density of the snow pack was found to be 0.2 g/cm3 throughout the experiment, snow wetness varied considerably with spatial location, depth, and time. The average volumetric moisture of the snow pack was found to be 2% during the SAR overflight on March 13. A Debye-like model presented by Hallikainen et al., [24] was applied to estimate the dielectric properties of the snow. This model relates the snow dielectric to snow wetness (volume %), frequency and dry snow density. The modeled values of snow dielectric constant are listed in Table H.1 at L-, C- and X-bands for frozen and thawed conditions. Table H.1: Modeled Dielectric Characteristics of Snow for Frozen and Thawed Conditions. Frequency Thawed Conditions Frozen Conditions (March 13) (March 19-22) L-Band 1.58 + iO.024 1.37 + iO.0 C-Band 1.54-+ iO.079 1.37 + iO.O X-Band 1.49 + i0.09 1.37 + iO.O The dielectric of the frozen mineral soil was measured using portable dielectric

293 probes in a trench cut into the permafrost. The average L-band dielectric constant of the soil was found to be 7.96 + iO.96. H.1.4 Stand Geometry Ground surveys of seven stands were conducted to determine the number of trees per unit area by species and also record their respective diameters at breast height (DBH) (Jaeger, [30]). Within each stand, a line transect was drawn along the longest stand axis. Ten sample plots were selected at random distances along the transect. Within each plot, the DBH and species were recorded for all trunks with diameter greater than 1 cm and taller than 30 cm. The status of each tree (alive/dead, broken/unbroken) was noted along with the height of at least four live trees per plot. To estimate above ground biomass for each stand, the enumerated stand data were coupled with allometric equations. The measured DBH, heights, and status of each tree were used to estimate quantities including basal area, biomass volume, and biomass on the basis of allometric expressions drawn from the literature for each species (Kirby,[33]; Manning et al.,[40]; Singh,[52]; Yarie and Van Cleve,[84]). These expressions have been derived for stands in Alaska and in the Canadian Northwest Territories and Yukon. The specific equations applied here are listed by Jaeger [30]. These estimates were summed over all trees in a stand to yield the estimates shown in Table H.2 as averaged over the net areas of all sample plots in each stand. The standard deviations listed in the table are based on the plot-to-plot variance within each stand. In addition to the six stands listed in Table H.2, the stands containing the trihedral corner reflectors were also characterized with respect to density, height and diameter (I(asischke et al.,[32]; Jaeger,[30]). These stands included a single species

294 Table H.2: Summary of Stand Biophysical Parameters. Species White Spruce Black Spruce Balsam Poplar Stand Name WS-1 WS-2 WS-5 WS-7 BS-1 BP-2 Density Mean (trunks/hectare) 1248 2073 1484 1123 1975 1615 Standard Deviation 342 576 618 654 1483 407 Basal Area Mean (m2/hectare) 46 41 44 46 12 50 Standard Deviation 16.6 7.0 8.5 12.4 3.3 25.8 Basal Volume Mean (m3/hectare) 442 332 392 442 51 344 Standard Deviation 169 60 100 115 12 190 Dry Biomass - Summer Mean (kg/m2) 21.7 16.7 18.1 21.5 3.7 18.2 Standard Deviation 8.8 3.6 4.8 6.1 0.8 10.9 Dry Biomass - Winter Mean (kg/m2) 21.7 16.7 18.1 21.5 3.7 17.9 Standard Deviation 8.8 3.6 4.8 6.1 0.8 10.7 of alder and mixed species stands of alder, balsam poplar and white spruce. Table H.3 summarizes mean DBH, height and basal area for all stands. To characterize the trunk layer geometry in terms of parameters required for MIMICS input, DBH histograms were generated from the ancillary ground measurements and coupled with the allometric height equations listed in Table H.4. Together, these data define the PDF in size required to compute the trunk layer phase matrix for a given stand. Measurements of orientation angles were performed to characterize the PDF for characterizing trunk orientation. However, the number of non-vertical trunks in each stand was very small and the lean angles of these trunks was also small. Hence, all trunks are assumed to have a vertical orientation for purposes of MIMICS simulations. The thickness of the crown layer for each stand was defined based on field observations. For both white spruce and black spruce, the crown layer thickness is assumed to be equal to trunk height. For alders and balsam poplars, the crown layer thick

295 Table H.3: Summary of Mean DBH, Height and Basal Area for All Stands. Stand DBH Height Basal Area Name (cm) (m) (m2/hectare) WS-1 WS-2 WS-5 WS-7 BS-1 BP-2 19.6 14.5 17.9 21.4 8.8 18.0 22.1 20.1 21.3 24.5 7.6 17.6 46 41 44 46 12 50 Stands with trihedral reflectors: Stand Species DBH Height Basal Area Name (cm) (m) (m2/hectare) Alder alder 6.0 6.3 66.5 Balsam Poplar balsam poplar 11.0 12.7 22.9 alder 6.0 6.3 3.1 White Spruce white spruce 7.8 8.6 12.4 balsam poplar 9.4 11.6 10.0 alder 6.1 6.3 5.4 Table H.4: Equations Defining Height-to DBH Relationship. Species Equation White Spruce H = -1.7096 + 1.4224(DBH) - 0.016(DBH)2 Black Spruce H = 0.9494 + 0.7657(DBH) Balsam Poplar H = 1.0526 + 1.143(DBH) - 0.0145(DBH)2 Alder H = 2.871 + 0.5666(DBH) H = height in meters DBH = diameter in cm measured at breast height

296 ness was assumed to be 25% of the average trunk height. To estimate the number density and sizes of crown layer constituents, the total biomass of each stand listed in Table H.2 was apportioned among the various constituent classes using allometric equations reported in the literature (Manning et al.,[40]; Singh,[52]; Van Cleve and Viereck,[73]; Yarie and Van Cleve,[84]). Typically, these equations provide a statistical breakdown of biomass apportionment for dry biomass of the trunk bark, trunk wood, the branches and the twigs and foliage. Since most of these equations were empirically derived for stands outside of the Bonanza Creek Experimental Forest, perhaps with differing local site conditions, they may produce errors in characterizing the biomass apportionment of the Bonanza Creek stands. However, they are the best available sources of information. Additional apportionment error arises in partitioning the branches and foliage into component parts required by MIMICS (i.e. various branch size classes and foliage). In this case, apportionment of biomass was accomplished on the basis of destructive sampling undertaken at the timet of the overfights. Results of the biomass apportionment are summarized for each species in Table H.5. Table H.5: Dry Biomass Fractions of Canopy Components as Percent of Total. White Black Balsam Alder Spruce Spruce Poplar Trunk 85.79 86.45 90.0 90.0 Primary Branches 6.13 4.94 4.9 4.9 Secondary Branches 5.48 5.06 5.1 5.1 Foliage 2.60 3.55 NA NA The size and orientations of crown layer constituents have been inferred through a combination of field observations and morphology data from Nelson et al.,[45]. Table H.6 summarizes the geometry of the crown layer constituents. The orientation functions are specified in terms of the inclination angle 0 where 0 = 0 corresponds

297 Species White Spruce Table H.6: Geometry of Crown Layer Constituents. Constituent Average Average Class Length (cm) Diameter (cm) primary branches 113 2.24 secondary branches 57.16 1.04 needles 1.6 0.1 primary branches 81.3 2.37 secondary branches 51.17 1.06 needles 0.8 0.1 r primary branches 200 1.5 secondary branches 100 0.75 primary branches 200 1.5 secondary branches 100 0.75 Black Spruce Orientation Function sin4 0 sin9 0 sin 0 sin9(0 - 30~) sin9 0 sin 0 sin9(0 + 60~) sin9(0 + 60~) sin9 ( + 600) sin9(0 + 60~) Balsam Poplai Alder I I to a vertical cylinder. Each of these functions is normalized to convert it to a PDF for implementation in MIMICS. Table H.7 lists the number density of each canopy constituent for each of the seven stands, assuning that each stand may be modeled as a continuous (closed) canopy. Under this assumption, the number density Nk of a given constituent may be computed from net stand biomass x biomass apportionment fraction biomass of a single element x crown layer thickness x stand area The biomass of a single element is computed from the size and dry density parameters of that element. Table H.7: Number Density of Canopy Constituents. Stand Name Canopy Density Primary Branches Secondary Branches Needles (trees/m2) (#/m3) (#/m3) (#/m3) Alder 1.36 1.19 9.92 NA BP-2 0.16 0.85 6.69 NA WS-1 0.12 0.44 2.37 12,300 WS-2 0.12 0.48 2.57 13,310 WS-5 0.12 0.50 2.7 14,000 WS-7 0.12 0.48 2.6 13,490 BS-1 0.20 0.25 1.31 18,340

298 H.1.5 Stand Dielectric Characteristics The dielectric properties of the trees vary as a function of frequency and canopy properties such as constituent dry density and chemistry and amount of liquids in the constituent. As environmental temperature changed from warm to sub-freezing, the chemistry and volume fraction of liquids in the canopy constituents changes dramatically between thawed and frozen states. These changes were reflected in the dielectric properties of the canopy elements. The dielectric properties of the stands were monitored with L- and C-band portable dielectric probes. The dielectric behavior of the stands was modeled by applying these in situ data together with dielectric models. The dielectrics listed in Table H.8 were inferred by coupling trunk dielectric profiles to dielectric models (Dobson et al., [13]). Table H.8: Relative Dielectric Constant for Tree Constituents. Species Frequency Relative Dielectric (GHz) +5~C -15~C White Spruce 1.25 36.47 + i10.99 5.19 + il.09 5.3 29.01 + ill.97 4.85 + i0.32 9.38 22.78 + i13.20 4.81 + i0.18 Black Spruce 1.25 12.46 + i4.50 3.72 + i0.78 5.3 9.30 + i3.33 3.47 + iO.23 9.38 7.82 + i3.22 3.44 + iO.13 Balsam Poplar and Alder 1.25 30.71 + i9.56 4.95 + il.07 5.3 24.18 + i9.85 4.61 + i0.32 9.38 19.16 + i10.69 4.57 + iO.17 H.2 Boreal Forest Transmissivity Analysis Data collected by the ERIM/NADC SAR have been applied to analyze canopy transmissivity (Dobson et al., [13],[14]). These data were recorded at C- and X-bands during a total of six passes over the test site on March 22. To compute the one-way canopy propagation loss, the point target responses of the trihedrals that were placed

299 in the forest stands was compared to the response of those trihedrals placed in the open on the sandbar. The background clutter was removed for each target response and the image intensity was normalized to correct for known effects such as range fall-off and antenna variations. The confidence interval associated with this process and with uncertainties in trihedral alignment is estimated to be ~1 dB (Dobson et al.,[13]). Three stands were selected for the deployment of the reflector arrays. Four targets were placed in a single species alder stand, seven targets were placed in a balsam poplar stand that contained shorter alder trees, and nine reflectors were placed in a white spruce stand that contained a mixture of alders and balsam poplars. At several reflector site, ancillary data were recorded that included tree species, location relative to the trihedral, diameter and height. Table H.9 summarizes the stand statistics in the neighborhood of each target for the region of the stand toward which the reflector was bore-sited. Statistics were recorded for only six of white spruce stands. These data, together with more extensive tabular summaries (Kasischke et al.,[32]), show significant local variance in stand geometry, both locally within the neighborhood of individual targets and in comparing different target locations within the same stand. Not only does the within-stand variability affect the estimation of canopy extinction, but it should also be noted that in examining the measured transmissivity data there is an inherent bias toward values of low extinction due to the logistics of placing physically large reflectors in a canopy of large discrete scatterers. The measured extinction values represent realizations over only the few azimuth degrees required to construct the synthetic aperture. It would be best to have a set of infinitesimally small point targets that one could place at a statistically large number of random locations within a given stand.

300 Table H.9: Sta.nd Characteristics in the Neighborhood of the Trihedral Reflectors. Target Species Trees/hect. T06017A T06020A T06018A T06076A T06026P Alder Alder Alder Alder Alder Balsam Poplar T06008P Alder Balsam Poplar T06006P Alder Balsam Poplar T06021P Alder Balsam Poplar T12201P Alder Balsam Poplar T12202P Alder Balsam Poplar T12203P Alder Balsam Poplar T12201S Alder Balsam Poplar White Spruce T12202S Alder Balsam Poplar White Spruce T06005S Alder Balsam Poplar White Spruce T06011S Alder Balsam Poplar White Spruce T12203S Alder Balsam Poplar White Spruce T09101S Alder Balsam Poplar White Spruce 18333 17778 30000 40000 1375 1250 1125 3125 476 2540 1665 1905 606 1313 2063 4286 273 1091 0 600 2200 938 1250 469 2333 778 1556 947 1900 1700 1375 750 1500 1647 2235 4235 are Average DBH (cm) 6.09 (1.46) 7.00 (1.89) 5.78 (1.49) 5.49 (1.72) 8.13 (1.44) 8.49 (1.60) 5.38 (1.46) 10.64 (3.33) 5.43 (0.29) 8.86 (4.02) 6.59 (2.50) 11.50 (5.22) 5.63 (2.22) 11.85 (4.17) 4.28 (0.95) 11.55 (4.34) 7.50 (0.73) 14.66 (4.91) 0.00 (0.0) 11.40 (3.89) 10.45 (4.72) 6.55 (2.10) 11.54 (3.32) 15.60 (2.86) 5.66 (3.37) 9.29 (3.74) 8.32 (3.99) 6.73 (4.77) 8.03 (4.04) 7.28 (3.80) 5.77 (1.96) 10.85 (2.82) 7.36 (4.49) 6.41 (2.24) 8.69 (3.52) 5.62 (2.38) Average Height (m) 6.32 (0.83) 6.84 (1.07) 6.15 (0.84) 5.98 (0.98) 7.47 (0.81) 11.84 (1.80) 5.92 (0.83) 12.62 (2.16) 5.95 (0.16) 11.00 (2.95) 6.60 (1.41) 12.34 (3.60) 6.06 (1.26) 13.71 (2.18) 5.29 (0.54) 12.95 (2.73) 7.12 (0.42) 14.45 (2.71) 0.00 (0.0) 12.80 (2.26) 10.86 (3.86) 6.58 (1.19) 13.35 (2.22) 15.10 (1.82) 6.08 (1.91) 11.63 (2.81) 9.17 (3-61) 6.69 (2.70) 10.30 (3.94) 8.23 (2.72) 6.14 (1.11) 12.75 (1.48) 8.25 (3.96) 6.50 (1.27) 11.26 (2.73) 6.72 (2.15) Numbers in parentheses are standard deviations.

301 To simulate one-way propagation loss for these three stands, each set of stand statistics recorded in the neighborhood of the targets was used as input to MIMICS, applying the dielectric constants for frozen vegetation constituents (Table H.8) and the trunk height versus DBH equations (Table H.4). For the mixed-species stands, MIMICS was run separately for each constituent species and the resultant propagation losses were added together to estimate the total net loss. When modeling each of these stands, only extinction through the trunk layer was considered. This approach was taken because (1) the trunk layer statistics were well characterized by the ancillary ground measurements, (2) the apportionment of biomass in the crown of each species was only approximate and (3) the computation time required to model each species in the neighborhood of each reflector was prohibitively long. Furthermore, since over 85% of the dry biomass for each of these species is in the trunk layer (Table H.5), the canopy extinctions should be dominated by the trunk layer. Given the computation time constraint and in light of the limitations of the biomass apportionment for the crowns, MIMICS was run for the three stands modeled only as trunk layers. Figures H.1 - H.3 show MIMICS simulations of the maximum and minimum oneway propagation loss for each of the three stands together with the measured values as determined for each trihedral reflector. In all cases, trunk layer extinction increases smoothly with incidence angle and the X-band extinction is generally higher than that at C-band. The maximum and minimum MIMICS simulations correspond to the maximum and minimum biomass conditions for each of the three stands. Figure H.1 shows these data for the alder stand, Figure H.2 shows data for the balsam poplar stand and Figure H.3 presents the white spruce stand simulation. In all cases, the Vpolarized extinction is greater than that at H-polarization with the difference being

302 less than 1.5 dB. If all extinction were indeed caused by the trunk layer, then the MIMICS simulations for the maximum and minimum biomass conditions would be expected to bound the measured propagation loss values. For the alder stand, MIMICS overestimates extinction by as much as 3 dB at each frequency. However, MIMICS does predict the general polarization and incidence angle behavior. The low values in the measured data may be related to the high variance in the tree density in the neighborhood of the target boresite direction and the natural tendency to place the reflectors in local clearings within the dense alder canopy. The measured values observed for the balsam poplar stand are well bounded by the MIMICS simulations for both frequencies. However, the balsam poplar stand generally contained larger and fewer trees than did the alder canopy, thereby yielding individual measurements that were more dependent on the locations of fewer individual trees in front of the reflectors. This effect increased the spread in the measured data. This same effect is observed in the white spruce canopy. In this case, MIMICS underestimated a number of observed values. This effect may be attributed to the lack of a crown layer in the modeling analysis. This is expected to have more of an effect in the white spruce canopy since these trees have a high number density of needles within the crown whereas the deciduous species were not foliated. Figure H.4 is a plot of the MIMICS-simulated one-way propagation loss versus the measured loss at C-band. Data are shown for all three stands at both polarizations. Each set of stand data were fit with a straight line to help illustrate the combined effects of measurement and model error. Good correlations exist between measured data and model simulations for all three stands, with the correlation coeffi

303 20.0 15.0 r cn v I 10.0 F V Pol Max -- MIMICS.- H Pol Max — MIMICS V Pol Min — MIMICS / H Pol Min - MIMICS /, o VPol — Measured / ' ' ' H Pol —Measured /....,.. -.............. 5.0 0.0 L 20 1. 30. 40. 50. 60. 70. Incidence Angle (degrees) (a) C-band. 20.0 I.. c/ cu a 0 — Cfl 15.0 10.0 5.0 V Pol Max -- MIMICS —.. --- H Pol Max -- MIMICS - V Pol Min -- MIMICS ------ H Pol Min -- MIMICS o V Pol -- Measured H Pol -- Measured! 0 El l 0.0 L 20. I I 1 30. 40. 50. 60. 70. Incidence Angle (degrees) (b) X-band. Figure H.1: Transmission loss for one-way propagation through the alder canopy. Measurements are shown for four trihedral targets at (a) C-band and (b) X-band.

304 20.0 15.0 _s m o;Y cX 3 O rr 6 s? V Pol Max — MIMICS -.H Pol Max — MMICS / V Pol Min -- MIMICS H PolMin — MIMICS V Pol — Measured HPol — Measured ' " _ _ _ _ _ __ --- —---...'" ' Eo 10.0 I 5.0 } a 0 ~ ~ ~~~~0:, 0 err rrrrrrr -.. I.. I. I 0.0 L 2C I. 30. 40. 50. 60. 70. Incidence Angle (degrees) (a) C-band. 20.0 I I 15.0 F ca c5 10.0 5.0 V Pol Max -- MIMICS....... H Pol Max — MIMICS/ -. --- V Pol Min — MIMICS...... HPol Min — MIMICS o V Pol — Measured H Pol -- Measured / 8 8 F '."... I'.. I.... I.. 0.0 L 2C. 30. 40. 50. 60. 70. Incidence Angle (degrees) (b) X-band. Figure H.2: Transmission loss for one-way propagation through the mixed balsam poplaralder canopy. Measurements are shown for seven trihedral targets at (a) C-band and (b) X-band.

305 20.0 15.0 o 10.0 CA 0 10.0 6.0 5.0 0.0 2( V Pol. Maximum — MIMICS -—. --- H PoL Maximum -- MIMICS --- V Pol. Minimum - MIMICS..-. — H Pol. Minimum - MIMICS V Pol. - Measured E H Pol. - Measured,. S/"'t"",,, D. 30. 40. 50. 60. 70. Incidence Angle (degrees) (a) C-band. 20.0 v CA -. C) CE 6 15.0 10.0 5.0 *....,....,....,..... I... i.. i V Pol. Maximum -- MIMICS —.. — H Pol. Maximum -- MIMICS -V Pol. Minimum -- MIMICS -.. — H Pol. Minimum - MIMICS o V Pol. - Measured /* ] o H Pol. - Measured o,' o.. 1-A ---, ---".1~~~~~~~~ 0.0 2!0. 30. 40. 50. 60. 70. Incidence Angle (dB) (b) X-band. Figure H.3: Transmission loss for one-way propagation through the mixed white sprucebalsam poplar-alder canopy. Measurements are shown for nine trihedral targets at (a) C-band and (b) X-band.

306 cient p > 0.75, however MIMICS never predicts 0 dB of loss which may be measured at low values of incidence angle because of placement of the reflectors in canopy gaps. This indicates that more reflectors should be used in this type of study and more careful attention should be paid to random placement of the targets in the canopy. Furthermore, this figure illustrates the underestimation of canopy extinction by MIMICS in the white spruce stand. This illustrates the importance of including the crown layer constituents in canopy transmissivity analyses, especially for foliated species. Figures H.5 and H.6 show MIMICS simulations of one-way canopy transmissivity at C- and X-bands for a mature white spruce stand (WS-2). This simulation is compared to the mean value observed by the trihedral reflector measurements for the young mixed-species stand. The total canopy transmissivity is shown along with the individual contributions from the crown and trunk layers, for H- and V-polarizations. The total transmissivity is shown to be dominated by the transmissivity through the crown layer, which is comprised of needles and branches. This is caused in a large part by the high number density of needles in the crown layer (13,310/m2). At the higher incidence angle (0 = 56~), The MIMICS simulations agree very well with the average measured transmissivity, predicting values to within one standard deviation of the mean. However, at the steeper incidence angle (0 = 36~), MIMICS underestimated the transmissivity (overestimates extinction) by at least 1 dB. In fact, the mature canopy exhibits more extinction than the mixed-species stand in nearly all cases. This is because the younger stand has a less developed crown layer than the mature stand. Having established confidence in the ability of MIMICS to simulate extinction for these canopies, a variety of similar architectures and differing environmental condi

307 12.0 10.0 3s C) C0.. 8.0 6.0 4.0 2.0 0.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 Measured Data (dB) o Alder Stands, H Pol. o Alder Stands, V Pol. E Balsam Poplar Stands, H Pol. o Balsam Poplar Stands, V Pol. A White Spruce Stands, H Pol. A White Spruce Stands, V Pol. Alder best fit -- p = 0.87 -------- Balsam Poplar best fit -- p = 0.75 -- - White Spruce best fit -- p = 0.87 Figure H.4: Comparison of MIMICS simulated and measured transmission loss for oneway propagation through the alder, balsam poplar and white spruce canopies at C-band. The best-fit straight lines are shown for each canopy, together with their respective correlation coefficients p.

308 0. 1/ E 8 -2. -4. -6. -8. -10. L_ 20. 30. 40. 50. 60. Incidince Angle (degrees) (a) H-polarization. 0.._, E 6 8 -2. -4. -6. -8. -10. L20. 30. 40. 50. 60. Incidince Angle (degrees) (b) V-polarization. Figure H.5: Comparison of MIMICS simulated one-way canopy transmissivity for a mature white spruce stand (WS-2) at C-band to the average measured transmissivity of the mixed-species white spruce stands. Error bars are based on the mean value ~ one standard deviation.

309 0. "I-, *s P9 Crt 4-a CA ~rA E s I -2. -4. -6. -8. -10. L 20. 30. 40. 50. 60. Incidince Angle (degrees) (a) H-polarization. 0. v 'I CA C.E IY -2. -4. -6. -8. -10. L 20. 30. 40. 50. 60. Incidince Angle (degrees) (b) V-polarization. Figure H.6: Comparison of MIMICS simulated one-way canopy transmissivity for a mature white spruce stand (WS-2) at X-band to the average measured transmissivity of the mixed-species white spruce stands. Error bars are based on the mean value ~ one standard deviation.

310 tions are now considered. Figures H.7 and H.8 present simulations of total one-way canopy transmissivity for mature stands of white spruce and black spruce, respectively. V-polarized data are shown for L-, C- and X-bands for frozen and thawed canopy conditions. In all cases, transmissivity decreases as the canopy changes from the frozen to the thawed states. Furthermore, transmissivity is inversely related to canopy biomass, i.e. transmissivity decreases with increasing canopy biomass. Behavior as a function of frequency demonstrates that transmissivity also decreases as frequency increases. H.3 Boreal Forest Backscatter Analysis Data recorded by both the JPL SAR and the ERIM/NADC SAR have been applied to analyze canopy backscatter (Dobson et al., [13],[14]). Data was extracted from the SAR imagery for large single-species stands and spatially averaged to obtain the mean backscatter from a single stand. C- and X-band data were recorded with the ERIM/NADC SAR on March 22 (frozen canopy conditions) (Kasischke, et al.,[32]). Since the antenna gain pattern of this system contains significant ripple that is not fully characterized, the data are not fully calibrated to an absolute level. \Valid data comparison is therefore limited to data recorded for stands at a common range within a given pass. Comparison of data recorded on different passes and at different incidence angles are only approximate since the passes may have different biases. Similarly, comparison of data between frequencies can only be performed on a relative scale. Therefore, MIMICS simulations of data recorded by this system are restricted to comparing those data recorded at a common range location on a given SAR pass.

311 0. *-b * CA To -2. -4. -6. -8. -10. ' 20. 30. 40. 50. 60. Incidence Angle (degrees) (a) Frozen conditions. 0..v H 3 -2. -4. -6. -8. -10. '20. 30. 40. 50. 60. Incidence Angle (degrees) (b) Thawed conditions. Figure H.7: MIMICS simulated one-way canopy transmissivity for a mature white spruce stand (WS-2) at L-, C- and X-bands for (a) frozen canopy conditions and (b) thawed canopy conditions.

312 0. gs,I r I. CT.41 >lb - C13:2 8 -2: -4. -6. -8. -10. - 20. 30. 40. 50. 60. Incidence Angle (degrees) (a) Frozen conditions. 0. m3 s-._.>_ E Ci. s cu -2. -4. -6. -8. -10. L 20. 30. 40. 50. 60. Incidence Angle (degrees) (b) Thawed conditions. Figure H.8: MIMICS simulated one-way canopy transmissivity for a black spruce stand (BS-1) at L-, C- and X-bands for (a) frozen canopy conditions and (b) thawed canopy conditions.

313 L-band data were collected with the JPL SAR on March 13 (frozen canopy conditions) and on March 19 (thawed canopy conditions) (Way et al.,[77]). The antenna gain pattern of this system is relatively smooth and the data is easier to correct for gain variations. The two passes of L-band data were calibrated against the SAR response to several 182 cm trihedral corner reflectors located at the Fairbanks International Airport. MIMICS simulations of canopy backscatter is limited to those stands that were characterized by on-site sampling. Since the stands considered in this study were only partially characterized by on-site sampling, information on biomass apportionment and canopy constituent size and density characteristics is only approximate. Errors introduced in the biomass apportionment analysis will have an effect on the backscatter simulated by MIMICS. The 20-30cm thick snow layer also significantly complicated the backscatter analysis. The roughness parameters and other characteristics of the snow-ground interface were not characterized. These parameters could only be estimated by fitting MIMICS to ground backscatter measurements of open areas on sandbars that were outside the tree canopies. Since the roughness of these regions do not correspond to the roughness of a forest floor, and since the goal of this study is to examine model performance without using parameter fitting, the snow substrate was modeled as a half-space of snow with estimated RMS roughness of 1.2 cm and 24 cm correlation length. This ignores scattering at the snow-ground interface completely and in some cases reduces the effectiveness of the MIMICS simulations. A simple technique that accounts for the snow-ground interface at L-band was introduced in Section 3.2.2, however, because of lack of adequate characterization of the ground surface, its effectiveness is also somewhat limited.

314 H.3.1 Comparison with Measured Data Table H.10 lists the MIMICS backscatter simulations together with the SAR backscatter observations for six stands. Measured data were recorded by the JPL SAR at L-band for both frozen and thawed conditions. A one-to-one comparison of the measured and simulated values shows very good agreement for both frozen and thawed canopy states. MIMICS, however, significantly underestimates VV-polarized backscatter from stands BS-1, BP-2 and Alder under frozen conditions. The crosspolarized backscatter from frozen BS-1 is also significantly underestimated. Figure Table H.10: Comparison of MIMICS Estimates to Measured L-band SAR Data (dB). March 13, 1988 March 19, 1988 Thawed Conditions Frozen Conditions Stand Polarization SAR MIMICS SAR MIMICS WS-1 HH -10.0 -9.2 -13.1 -12.2 VV -10.4 -12.1 -14.9 -15.6 VH -15.2 -14.9 -21.0 -22.8 WS-2 HH -8.4 -9.1 -11.4 -12.8 VV -9.9 -12.3 -14.5 -16.4 VH -14.2 -15.0 -20.4 -23.6 WS-5 HH -8.1 -9.1 -11.1 -12.2 VV -9.1 -12.0 -14.8 -15.5 BS-1 HH -12.9 -10.7 -14.9 -16.9 VV -14.4 -15.1 -16.4 -23.2 VH -20.0 -19.5 -23.7 -32.5 BP-2 HH -9.2 -11.7 -12.7 -14.4 VV -10.4 -11.6 -14.8 -22.0 Alder HH -8.7 -9.9 -11.3 -14.6 VV -9.7 -11.4 -14.0 -23.2 H.9 graphically illustrates the effectiveness of MIMICS in predicting the HH- and VVpolarized backscatter. The measured SAR data are plotted against that predicted by MIMICS. Data are shown for all six stands for both frozen and thawed conditions. From here it is seen that MIMICS tends to underestimate backscatter for all stands except for white spruce. An underestimation of VV backscatter for frozen conditions

315 -5. -10. 0 cD O0 -15. v. 0.. I.... I. * *~~~~~~c -20. - -25. -25. -20. -15. -10. -5. o~ (dB) MIMICS (a) HH-polarization. -5. -10. F I * I * *. 0 A 0 a 0 ~0 0 ~o -15. - -20. -25. L -25. -20. -15. -10. -5. o~ (dB) MIMICS (b) VV-polarization. o White Spruce - Thawed * White Spruce - Frozen El Black Spruce -- Thawed * Black Spruce -- Frozen A Balsam Poplar -- Thawed A Balsam Poplar - Frozen v Alder - Thawed v Alder - Frozen Figure H.9: Comparison of measured canopy backscatter to MIMICS simulated backscatter for (a) HH-polarization and (b) VV-polarization.

316 for all stands except white spruce is also clearly evident. Since the alder and balsam poplar stands have no foliage in their crown layers, and since the black spruce stand has a fairly sparse crown layer as compared to the white spruce canopies, scattering mechanisms involving interaction with the ground surface contribute more readily to the net canopy backscatter than they do for the white spruce stands, since they are fully foliated. The general underestimation of o~ may therefore be attributed to the modeling of the snow surface as an infinite half-space. It is expected that accounting for scatter at the snow-ground interface would increase acr somewhat and may alleviate this problem. Table H.11 compares the measured and predicted L-band polarization ratios OHH/av and aCHH/VH for the six stands under frozen and thawed conditions. Here Table H.11: Comparison of Measured and Modeled Polarization Ratios at L-band. | HaH/'VV (dB) March 13, 1988 March 19, 1988 Stand Thawed Conditions Frozen Conditions SAR MIMICS SAR MIMICS WS-1 0.4 2.9 1.8 3.4 WS-2 1.5 3.1 3.1 3.6 WS-5 1.0 2.9 3.7 3.3 BS-1 1.5 4.4 1.5 6.3 BP-2 1.2 -0.1 2.1 7.6 Alder 1.0 1.5 2.7 8.6 aHH/O \H (dB) WS-1 5.2 5.7 7.9 10.5 WS-2 5.8 5.9 9.0 10.8 BS-1 7.1 8.8 8.8 15.6 MIMICS performs nicely with the exception of frozen conditions where underestimation of VV- and VH-polarized backscatter on account of the snow/soil model becomes apparent. MIMICS successfully predicts the the polarization ratios are smaller for the thawed canopy state for all of the stands.

317 Table H.12 illustrates the effect of changing freeze/thaw state on the canopies. Measured and simulated L-band values of the ratio hawed ozen are presented for thawed /Cfrozen are presented for each stand. MIMICS successfully predicts the observed increase in a~ in going from Table H.12: Comparison of the Effects of Freeze/Thaw State on L-band Backscatter. a- hwed /a-froen (dB) HH VV VH Stand SAR MIMICS SAR MIMICS SAR MIMICS WS-1 3.1 3.1 4.5 3.6 5.8 7.9 WS-2 3.0 3.7 4.6 4.2 6.2 8.5 WS-5 3.0 3.1 5.7 3.6 - 7.9 BS-1 2.0 6.2 2.0 8.1 3.7 13.0 BP-2 3.5 2.7 4.4 10.4 - 15.1 Alder 2.6 4.7 4.3 11.8 - 16.2 a frozen to a thawed state. For the white spruce stands, the MIMICS estimates are in very good agreement with the measurements. Notable discrepancies exist again at VV and VH polarizations for the more sparse stands. Figures H.10 and H.11 present L-band polarization responses for frozen and thawed white spruce (WS-5), respectively, as simulated by MIMICS. Responses are shown for co-polarized and cross-polarized configurations. Figures H.12 and H.13 present the measured frozen and thawed L-band responses. Again MIMICS successfully recreates the behavior of the measured data. MIMICS not only correctly reproduces the shapes of each of the surfaces, but it accounts for the increase in the pedestal observed in going from frozen to thawed states. These figures demonstrate that MIMICS has successfully modeled the backscatter response of this stand for all of these polarization states. Figure H.14 shows the linear polarized response of this stand as simulated by MIMICS for frozen and thawed conditions. The character of the responses are very similar for the two environmental states, with the thawed conditions yielding slightly more cross-polarized backscatter.

318 to N z 0 45. (a) Co-polarized response. 45. (b) Cross-polarized response. Figure H.10: MIMICS simulated L-band polarization response of frozen white spruce stand WS-5.

'*-SAA puls aunlds az!Mm paveip jo asuodsal uo!wz!Iqulod pu-q-,r pallnu!s SIIINI T:11 H91nq! asuodsai pazm!elod-ssoIa (q); ls! d.; alB ^ir — 0'0 a'lO '0dti0 cl osuodsa- paz!ijlod-oD (v) RIN;0 'SP~~~~~~~~~~~~~ 61~

320 0. -'90 45 -(a) Co-polarized response. 0 Z,5. 4b,, '., gi^^ol~ll0. (b) Cross-polarized response. Figure H.12: Measured L-band polarization response of frozen white spruce stand WS-5.

321 (a) Co-polarized response. 0 0. '44. (b) Cross-polarized response. Figure H.13: Measured L-band polarization response of thawed white spruce stand WS-5.

322 (a) Frozen Conditions. (b) Thawed Conditions. -90. Figure H.14: MIMICS-simulated L-band linear polarization response of thawed white spruce stand WS-5. Aqje~0.glee~ polarizatio n (b) Thawed Conditions. spruce stand WS-5.

323 s -o ~0 4. 2. 0. -2. -4. -6. A v A eG * @0 0e *Z 0 o _ A V I...,V.. I.. I. I..... I.. -8. 4. 2. I -8. -6. -4. -2. 0. 2. 4. O~/~WSl (dB) MIMICS (a) C-Band. 1. I -....... I... I... 1 -o 4Q -o 0 O ra 0. -2. -4. -6. 0 A a t v * El 00 V * v a -8. L -8. -6. -4. -2. 0. 2. 4. o~/owsa (dB) MIMICS (b) X-Band. o White Spruce - HH * White Spruce - VV Black Spruce -- HH * Black Spruce -- VV Balsam Poplar - HH A Balsam Poplar - VV v Alder - HH v Alder- VV Figure H. 15: Comparison of measured canopy backscatter to MIMICS simulated backscatter for (a) C-band and (b) X-band. The data have been normalized to the backscatter from white spruce stand WS-1 for each SAR pass.

324 Modeling at C- and X-bands has been complicated by the lack of available calibrated SAR data from the ERIM/NADC. SAR. Furthermore, since this SAR flew only during times for which the canopies were frozen, only this environmental state is considered. To deal with the uncalibarted data problem, the backscatter values were normalized to that of the white spruce stand WS-1 for each SAR pass. These normalized data are presented in Figure H.15. For the most part, MIMICS predictions agree with the SAR measurements to within ~1.5 dB. Exceptions to this include some observations of the balsam poplar, black spruce and alder stands at VV-polarization. As was the case at L-band, this is probably caused by the method used to model the snow-soil interface. H.3.2 White Spruce Simulations Having established the ability to model backscatter from these forest stands, MIMICS is now used to simulate backscatter over a wider range of sensor parameters and the backscatter is examined on a more detailed level. White spruce and black spruce stands are considered in these simulations. Since the best estimates of backscatter were obtained for white spruce canopies, these stands are considered first Figure H.16 shows the simulated backscatter from stand WS-5 at L-, C- and Xbands for frozen conditions. Figure H.17 shows backscatter for thawed conditions. In general, MIMICS predicts that a~ increases at L-band as the canopy moves from a frozen to a thawed state. However, a decrease is observed in c~ at C- and X-bands for these conditions. Furthermore, for frozen conditions, ca increases with frequency whereas the reverse is true for thawed conditions. The scattering contributions to the net canopy backscatter are depicted in Fig

325 -7. -9. = — - * —.. — C _;7~:~7........ -.... band ________...., -- -- x.b ~id S -11. I -13. - -15. L 20. 30. 40. 50. 60. Incidence Angle (degrees) (a) HH-polarization. -o 00 SO Incidence Angle (degrees) (b) VV-polarization. 60. Incidence Angle (degrees) (c) VH-polarization. Figure H.16: MIMICS simulated canopy backscatter for a white spruce stand (WS-5) at L-, C- and X-bands under frozen canopy conditions for (a) HH-polarization, (b) VV-polarization and (c) VH-polarization.

326 -5. -7. v 00 -9. -11..... C-.b d __...... X-!nd -13. -i. 20. -8. -10. -12. _ -14. -16. 30. 40. 50. 60. Incidence Angle (degrees) (a) HH-polarization. -18. '..... I.... 20. 30. 40. 50. 60. Incidence Angle (degrees) (b) VV-polarization. -10. m C 3, -30. -... I....- I. 20. 30. 40. 50. 60. Incidence Angle (degrees) (c) VH-polarization. Figure H.17: MIMICS simulated canopy backscatter for a white spruce stand (WS-5) at L-, C- and X-bands under thawed canopy conditions for (a) HH-polarization, (b) VV-polarization and (c) VH-polarization.

327 ures H.18 through H.21 for L- and C-bands for frozen and thawed canopy states. In all cases, canopy backscatter is dominated by the direct crown contributions to r~, thereby negating effects that the snow-covered soil may have on the net backscatter. Treatment of the ground surface becomes important only at L-band, for HHpolarization, where the ground-trunk interaction is significant at high incidence angles. Figures H.22 and H.23 present net L- and C-band backscatter for all white spruce stands for frozen and thawed states. As expected, a~ is seen to respond similarly for all three canopies. At L-band, for all polarizations ao increases as the canopy thaws while at C-band, a~ decreases. The C-band cross-polarized backscatter, however, remains relatively constant.

328 -5.0 -10.0 -15.0 \ -— 4 —. Dian on ----- Tnmk-Gmomd... I....I.... I.... -20.0 -25.0' 20 D. 30. 40 50. 60. Incidence Angle (degrees) (a) HH-polarization. -15.0 m o>., -20.0 Taul ----- DircCrow ---- Tn mk-Groud / -25.0 -30.0 ' -' - - 20. 30. 40. 50. 6C Incidence Angle (degrees) (b) VV-polarization..20.0....... O. a 0o 0 0v r) -30.0 I I - - - I -, - Toul -—.o ---. Dirt own _ __ —,___ CrownGrond -—.- -- --— ~ - ---- ~ -40.0 30. 40. 50. 60. Incidence Angle (degrees) (c) VH-polarization. Figure H.18: Canopy backscatter components for white spruce stand (WS-5) at L-band under frozen canopy conditions for (a) HH-polarization, (b) VV-polarization and (c) VH-polarization.

329 -5.0 -s.o -10.0 -15.0 S^.~~~~~~~ —.1 — Dint Cown — 0 --- Tnmnk-Groad -20.0 -25.0' - - 20. 30. 40. 50. 60. Incidence Angle (degrees) (a) HH-polarization. -5.0.. -10.0.-15.0 0 > ---- Total -20.0 -.... Di Crow — I — Tik-Gnmd -25.0. 20. 30. 40. 50. 60. Incidence Angle (degrees) (b) VV-polarization. -10.0........... -20.0 m, -30.0 "~o "~'~~~. —4.~-.-. oDimt Crown —. — Crown Ground -40.0... 20. 30. 40. 50. 60. Incidence Angle (degrees) (c) VH-polarization. Figure H.19: Canopy backscatter components for white spruce stand (WS-5) at Lband under thawed canopy conditions for (a) HH-polarization, (b) VV polarization and (c) VH-polarization.

330 -5.0 -10.0 l: 3 0 -15.0 - Toi ---- Tnmk-Gmmun,_,,.....t --- —-U....... —.... _ -20.0 )0. -25.0 L 20 ). 30. 40. 50. Incidence Angle (degrees) 6 (a) HH-polarization. -5.0 -10.0 c m 0> 0 Total 4-. — Direa Crown —. — Tnk.-Gmd -15.0 -20.0 -25.0 L20. 30. 40. 50. Incidence Angle (degrees) 60. (b) VV-polarization. -15.0 i T3 m 0 0 -25.0 - -35.0 1 - Tota -— 4 --- Direct Cion ---- Crown Grond -__ — __ '"'""""ffi...*.* ",..,g,. -45.0 20. 30. 40. 50. 60. Incidence Angle (degrees) (c) VH-polarization. Figure H.20: Canopy backscatter components for white spruce stand (WS-5) at C-band under frozen canopy conditions for (a) HH-polarization, (b) VV-polarization and (c) VH-polarization.

331 -10.0 C m qi -15.0 _ Toua.-..- Direct Crown ---- Tnmk-Grrmd I i'- -- -. — -_____ -20.0 _ -25.0............... 20. 30. 40. 50. 60. Incidence Angle (degrees) (a) HH-polarization. _C.n... -10.0 -C c t) w 0 Eo -15.0 F -- ToWl -—.. —. Direa Crow —. — Tink-Grmund.... i... i. i.. I.. -20.0 - 20. 30. 40. 50. 60 Incidence Angle (degrees) (b) VV-polarization. _inn. I -20.0 ar tv o > -30.0 0 - Total — 4 ---. Direct Crown — r — Cwn Gound t 30. 40. 50. 60. Incidence Angle (degrees) (c) VH-polarization. Figure H.21: Canopy backscatter components for white spruce stand (WS-5) at Cband under thawed canopy conditions for (a) HH-polarization, (b) VVpolarization and (c) VH-polarization.

332 -5. -10. *g,0 21 (5 -15. -- -- T — d --- —-2 - — 4 — Tbawed WS-I —. bawedWS-2 — M~* wFnmoWS-5 ---- Fnm WS-2 ----- Fmu WS-5 -25. L 30. 35. 40. 45. 50. 55. Incidence Angle (degrees) (a) HH-polarization. -10. -15. g1 S m tO - Thawed WS-I -— 4 — Thawed WS2 ----- bawedWS-5 — o — FlaM WS-1 - - - -. FZnC L3-1-2 -— F ---- Fmmn WS-2 —.Frnwc WS-5 -25. 30. 35. 40. 45. 50. 55. Incidence Angle (degrees) (b) VV-polarization. -10. -15. C m -1 0 o 0 7- - - -f - - - ----—,.43 - -20. -25. -30.1... 30. 35. 40. 45. 50. 55. Incidence Angle (degrees) (c) VH-polarization. Figure H.22: Total canopy backscatter for frozen and thawed white spruce stands at Lband for (a) HH-polarization, (b) VV-polarization and (c) VH-polarization.

333 - m m o.......... —.. -15. T he WS-I -—. ---- Thswo wS-2 — 4 --- ThawedWS20 --- PmzmWS-5 -20. Fmzm WS-2 -- PRozmWS-2.-2.i o 30. 30. 35. 40. 45. 50. 5: Incidence Angle (degrees) (a) HH-polarization. -5. -10., " r --- a _ m c % -15. --- Thwed WS----- Thawed WS-2 — _ — F'mznadWS-S -.- F- -rozen WzS— B —. FrofmWS-2 — a — Fmzmn WS-S. i.... I I - I - - -,sc LE -23. ' 30. 35. 40. 45. 50. Incidence Angle (degrees) (b) VV-polarization. S -15. 1 -m U ~o 00 I- 0 'Iawd WS-1 -25. 1 -30. - —.... Thawed WS-2 ---- Tbawed WS-5 --. — Frotm WS-1 —. -. Fnrom WS-2 -— _-. FP WS-5 i irzw -35. L 30. 35. 40. 45. 50. 55. Incidence Angle (degrees) (c) VH-polarization. Figure H.23: Total canopy backscatter for frozen and thawed white spruce stands at Cband for (a) HH-polarization, (b) VV-polarization and (c) VH-polarization.

334 H.3.3 Black Spruce Simulations The black spruce stand considered here is much more sparsely populated stand than the white spruce. This stand, in fact, does not represent a closed canopy. However, to simplify this initial analysis, MIMICS I is applied to model a~ for the black spruce stand as if it were indeed a closed canopy. Figures H.24 and H.25 show a~ at L-, C- and X-bands for frozen and thawed conditions. Backscatter from this stand exhibits more complex behavior than does that from white spruce. For frozen conditions, a~ shows a general increase with frequency. However, for thawed conditions, backscatter decreases with frequency for VV and VH polarizations whereas the HH-polarized backscatter increases with frequency. As the canopy moves from a frozen to a thawed state, L-band backscatter increases for all three polarizations. Examination of the individual contributions to the net canopy backscatter lends some insight into the behavior of a~. Figures H.26 and H.27 show the major contributors to L-band backscatter for frozen and thawed states while Figures H.28 and H.29 show those for C-band backscatter. At L-band, the trunk-ground interaction mechanism is a significant contributor to a~ at HH-polarization for both frozen and thawed states. However, the direct crown mechanism is the dominant term for VV and VH polarizations. Similar trends are observed at C-band. The decrease in scattering contributions that involve the ground surface is probably responsible for MIMICS underestimating the black spruce backscatter. To gain an understanding of the effect of the snow layer on net backscatter, the approach presented in Section 3.2.2 may be applied at L-band to model the scattering at the snow-soil interface. Figures H.30 and H.31 compare backscatter from the canopy with the ground layer modeled as a half-space of snow to this layer

335 -9.,.. -11.,,. -~.-.bmd -13. - -. -15. -17. -19.I _ _ -: -....... -. - -- _= 20. 30. 40. 50. 60. Incidence Angle (degrees) (a) HH-polarization. -16. - -1S...........,..... -18. -. -- X-.b-d -22. C -24. I.. 20. 30. 40. 50. 60. Incidence Angle (degrees) (b) VV-polarization. -15. i -20. o ~ I -2-5. -30. X-bbdd -35.... 20. 30. 40. 50. 60. Incidence Angle (degrees) (c) VH-polarization. Figure H.24: MIMICS simulated canopy backscatter for a white spruce stand (WS-5) at L-, C- and X-bands under frozen canopy conditions for (a) HH-polarization, (b) VV-polarization and (c) VH-polarization.

336 -5. -7. -9. -11. -13. -15. 20 -12. -14. F - L-bd....... b d X- Dd ). 30. 40. 50. 60. Incidence Angle (degrees) (a) HH-polarization. -16.........%...................................... ".... L-bod. ~.. C-band -18. -20. -22. 2( X-baod O. 30. 40. 50. 60. Incidence Angle (degrees) (b) VV-polarization. -15. F ----.... ------ ---- X —an -30. 20. 30. 40. 50. 60. Incidence Angle (degrees) (c) VH-polarization. Figure H.25: MIMICS simulated canopy backscatter for a black spruce stand (BS-1) at L-, C- and X-bands under thawed canopy conditions for (a) HH-polarization, (b) VV-polarization and (c) VH-polarization.

337 30. 40. 5 Incidence Angle (degrees) (a) HH-polarization. -20.0 -25.0 C. m m Opi -30.0 -35.0 \, _ %\ T-oul / \ -—.. --- Dirct Cown s\ —. — Tnk-Gromd - -- Cowo-Gnoud / /.... I../ %s 30. 40. 50. 60. Incidence Angle (degrees) (b) VV-polarization. c m -30.0 0 0 -40.0 '.... 20. 30. 40. 50. Incidence Angle (degrees) 60. (c) VH-polarization. Figure H.26: Canopy backscatter components for black spruce stand (BS-1) at L-band under frozen canopy conditions for (a) HH-polarization, (b) VV-polarization and (c) VH-polarization.

338 m-0.0.... T.....[nM.d..-....... x -10.0 -15............................. ---- 23 -20.0 -125.0 m —. —_ —C-.........A --- — -30.0 * 20. 30. 40. 50. 60. Incidence Angle (degrees) (a) HH-polarization. -1._.................. Toua -20.0 t -25.0 -30.0 -- I — K- -- - I -!20. 30. 40. 50. 60. Incidence Angle (degrees) (b) VV-polarization. -15.0 r- --- —... |..... I-.... C. ----, - -20.0 -25.0 >..-.. Dirmct Crown -30.0 - -o --- Tnk-GnMd - a - - Crown-Goxid -35.0. '.... '. *... 20. 30. 40. 50. 60. Incidence Angle (degrees) (c) VH-polarization. Figure H.27: Canopy backscatter components for black spruce stand (BS-1) at L-band under thawed canopy conditions for (a) HH-polarization, (b) VV-polarization and (c) VH-polarization.

339 C 0 0 ~! 60. Incidence Angle (degrees) (a) HH-polarization. -o m C) -e 0D 20. 30. 40. 50. 60. Incidence Angle (degrees) (b) VV-polarization. -20.0 -25.0 0 ). -30.0 -35.0 - Toul -— 4 --- Direct Cwn — TF — Tmn-Groumd ~- - - Crown-Ground I A -40.0 L 20. 30. 40. 50. 60. Incidence Angle (degrees) (c) VH-polarization. Figure H.28: Canopy backscatter components for black spruce stand (WS-5) at C-band under frozen canopy conditions for (a) HH-polarization, (b) VV-polarization and (c) VH-polarization.

340 0 Ci 0 Incidence Angle (degrees) (a) HH-polarization. c 0 m 0> 60. Incidence Angle (degrees) (b) VV-polarization. -20.0 -e co C O 0 CJ Toual -—...- Direa Crown — F — Tnim-Gmand -- - - Crowa-Grotnd -25.0 -30.0 -35.0 L 20. 30. 40. 50. 60. Incidence Angle (degrees) (c) VH-polarization. Figure H.29: Canopy backscatter components for white spruce stand (WS-5) at Cband under thawed canopy conditions for (a) HH-polarization, (b) VVpolarization and (c) VH-polarization.

341 modeled as a 20 cm thick snow layer over a frozen soil half-space. Figure H.30 shows this simulation for frozen canopy conditions while Figure H.31 shows these data for thawed canopy conditions. In both cases, a~ is higher for the snow-covered soil for all polarizations. The effect is more prevalent for like-polarized backscatter with aV. being responding slightly more than rhh%.

342 0. -5. -10. -l ~G -15. VV Pol. HH Pol. -. — VH Pol. ~.. ----------- - - - - -- - - - - --. --- —--—. - -.~ ~, I........ I..., -20. -25. -30. -35. 20. 30. 40. 50. 60. Incidence Angle (degrees) (a) Snow half-space. 0. -5. 0 m-1 at -10. -15. -20. VV Pol. -----—. HHPol.. --- — VH Pol. --------— ~ ------ ---- --- ---- --- —.,.. _ -25. -30. -35. 20. 30. 40. 50. 60. Incidence Angle (degrees) (b) Snow layer over soil half-space. Figure H.30: Total canopy backscatter for black spruce stand (BS-1) at L-band under frozen canopy conditions for (a) ground layer consisting of a snow half-space and (b) ground layer consisting of a snow layer on top of a soil half-space.

343 0. -5. ~o ot: -10. -15. -20. -25. - 20. 30. 40. 50. 60. Incidence Angle (a) Snow half-space. 0. -5. 0 0/ Ot -10. -15. -20. -250.30. 20. 30. 40. 50. 60. Incidence Angle (b) Snow layer over soil half-space. Figure H.31: Total canopy backscatter for black spruce stand (BS-1) at L-band under thawed canopy conditions for (a) ground layer consisting of a snow half-space and (b) ground layer consisting of a snow layer on top of a soil half-space.

BIBLIOGRAPHY 344

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