Final Report February 1, 1990 MESOSCALE MONITORING OF THE SOIL FREEZE/THAW BOUNDARY FROM ORBITAL MICROWAVE RADIOMETRY NASA Grant: NAG5-852 Submitted by: Craig Dobson Principal Investigator Fawwaz T. Ulaby Co-Investigator Brian Zuemdorfer Graduate Student Anthony W. England Co-Investigator Radiation Laboratory Department of Electrical Engineering and Computer Science University of Michigan Ann Arbor, MI 48109

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Page 2 CONTENTS page I Introduction 3 II Summary of the Investigation 3 III Accomplishments and Open Items 9 IV References 30 V Publications (titles follow, copies appear at the back of the report): 2/31 Journal articles England, A.W., 1989, Radiobrightness of diurnally heated, freezing soil, IEEE Geoscience and Remote Sensing, in press. Zuerdorfer, B., A.W. England, M.C. Dobson and F.T. Ulaby, 1989, Mapping freeze/thaw boundaries with SMMR data, J. of Agriculture and Forest Meteorology, in press. Symposia Proceedings and Abstracts England, A.W., 1989, Radiobrightness of periodically heated, two-phase media, Proc. International Geoscience and Remote Sensing Symposium (IGARSS'89), July 10-14, 1989, Vancouver, Canada, p. 163. England, A.W., The radiobrightness measurement of apparent thermal inertia, accepted for URSI com F., May 15-17, 1990, Hyannis, MA. Zuerndorfer, B., A.W. England, and F.T. Ulaby, An optimized approach to mapping freezing terrain with SMMR data, accepted for 1990 International Geoscience and Remote Sensing Symposium (IGARSS'90), May 21-24, 1990, University of Maryland. Zuerdorfer, B., A.W. England, and G.H. Wakefield, 1989, The radiobrightness of freezing terrain, Proc. International Geoscience and Remote Sensing Symposium (IGARSS'89), July 10-14, 1989, Vancouver, Canada, p 2748-2751. Zuerndorfer, B., G.H. Wakefield and A.W. England, Recovery of Fine Resolution Information in Multispectral Processing, December 10, 1989, IEEE International Conference on Acoustics. Speech, and Signal Processing (ICASSP).

Page 3 I Introduction We have recently completed the NASA funded study, "Mesoscale monitoring of the soil freeze/thaw boundary from orbital microwave radiometry." While most of our objectives were met, a few will have to be addressed within the newly begun NASA project, "Mapping regional freeze/thaw patterns with satellite microwave radiometry." The salient scientific products of the initial project are that regional freeze/thaw maps can be extracted from Nimbus 7, Scanning Multichannel Microwave Radiometer (SMMR) data, that diurnal thermal gradients have a small but measurable effect upon the SMMR spectral gradient, and that scale-space filtering can be used to improve the spatial resolution of a freeze/thaw classified image. This Final Report of the initial project contains a summary of the investigation, a discussion of accomplishments and of unresolved issues, and copies of publications and of conference abstracts and proceedings. We very much appreciate the opportunity to have worked upon this exciting, and, we believe, fruitful project. II Summary of the Investigation Soil moisture contributes to the energy exchange between the air and ground through latent heats of fusion and vaporization, and to rainfall runoff through the field moisture deficiency of a drainage basin. Whether or not soil moisture is frozen affects both the rate of energy transfer to the atmosphere, and the infiltration capacity of the soil. The amount and state of soil moisture are regional parameters that one would like to estimate using satellite remote sensing. There is a large body of literature about estimating soil moisture from radiobrightness (e.g. Burke et al., 1979; Wang et al., 1982; Schmugge, 1983; Jackson et al., 1984; Camillo and Schmugge, 1984; and Schmugge et al., 1986). We have developed a technique for mapping the spatial extent of frozen soils from the spectral characteristics of the 10.7-37 GHz radiobrightness. Through computational models for the spectral radiobrightness of diurnally heated, freezing soils, we identified a distinctive radiobrightness signature for frozen soils, and cast that signature as a discriminant for unsupervised classification. Our initial results were reported at the Interdisciplinary Science Land Surface Climatology Program's (ISLSCP) meeting in Las Cruces, NM, November 16-18, 1988 by A. W. England, and will appear as a paper in a special NASA ISLSCP journal publication (Zuemdorfer et al., 1989). Freezing influences the apparent radiobrightness temperature of the ground, Tb, through parameters in the approximation (Ulaby et al., 1981), Tb = e To + (l-e) Tsky, where e and To are the emissivity and surface temperature of the ground, respectively, and Tsky is the effective sky brightness. In this approximation, atmospheric transmissivity is ignored. Frozen ground exhibits signatures of (1) lower thermal temperature, To, (2) higher emissivity, e, and, (3) a decrease in brightness temperature with microwave frequency, f, aTb f <0.0 Signatures (1) and (2) are frequently ambiguous indicators of frozen ground because the variations in radiobrightness that result from freezing are easily confused with the effects of

Page 4 variations in soil moisture. Water molecules in frozen plants and soils are not free to align themselves with microwave electric fields. This constraint upon the rotational freedom of water gives rise to an apparent dryness to microwaves. The consequence is a decrease in the real part of the dielectric constant, and an increase in frozen soil emissivity. For example, the real part of dielectric constants, e', and corresponding emissivities at nadir, e(0), of two, homogeneous, smooth surfaced, 15% moist soils at 10 GHz are (e' from Hoekstra and Delaney, 1974): + 50 C - 50 C Material e(0) lb | C e b _ Tb Goodrich Clay 8.2 0.77 221 4.9 0.86 235 Fairbanks Silt 9.6 0.74 214 4.1 0.89 242 Because of increasing emissivity with freezing, a 100 decrease in the clay and silt soil temperatures, from +50 C to -50 C, would cause an increase in Tb of approximately +14 K and +28 K, respectively. The positive direction of change in Tb with soil freezing will cause confusion in discrimination between moist soils which will appear radiometrically warmer when frozen, and dry soils which undergo little molecular change and will appear radiometrically colder. The shift in emissivity with freezing is most pronounced at the lower microwave frequencies. At 37 GHz, the effect is reduced but not absent. We observe that the 37 GHz radiobrightness correlates relatively well with air temperature (Figure 1). Since soil surface temperature should follow the air temperature, the 37 GHz radiobrightness can be expected to provide a reasonably reliable estimate of soil surface temperature. However, discrimination based only on the 37 GHz radiobrightness would misclassify too often. The third signature of frozen soil occurs because freezing reduces the imaginary part of the dielectric constant, e", proportionally more than it does the real part, e'. The loss tangent, tan ~=e"/E' is a measure of the attenuation per microwave wavelength in emitting media. Reduced loss tangent, or lower attenuation, means that thermally emitted photons originate deeper within emitting media. That is, the effective depth of emission, ze, (l-e'l of the emission originates above ze) becomes a larger fraction of the free-space wavelength, Xo (England, 1974, 1975, 1976, and 1977). For example, Goodrich Clay and Fairbanks Silt exhibit an increase of ze with freezing (dielectric data from Hoekstra and Delaney,1974), + 50C -50 C Material _. tan e | tan 6 ze Goodrich Clay 8.2 3.5 0.43 0.13 Xo 4.9 1.0 0.20 0.36 |o Fairbanks Silt 9.6 5.0 0.52 0.10 =o 4.1 0.02 0.005 15.7 o

Page 5 The effective emission depth of moist soils is typically 10% of the free-space wavelength. Frozen soils have effective emission depths that may be 30% or more of free-space wavelength. The effective emission depth of frozen sandy soils, like the Fairbanks Silt, can be several wavelengths. In the more transparent emitting media, particularly in frozen sandy soil or dry snow, the greater average thermal photon path lengths have two effects: (1) a greater likelihood that thermal gradients affect spectral gradients, and (2) a greater opportunity for volume scattering of photons. (1) Thermally induced spectral gradients occur because longer wavelength photons tend to originate below the optical surface where thermal temperatures may differ by several degrees from surface temperatures. For the lower loss tangents of frozen soil, this difference in average emitting depth is enough to reflect near surface thermal gradients caused by diurnal heating. That is, a positive thermal gradient, aTo/az, will yield a negative spectral gradient, aTb/af. SMMR data are collected at midnight and noon. In the absence of changing weather conditions, midnight thermal gradients will be positive and noon thermal gradients will be negative (Figure 2) so that midnight spectral gradients will be negative, and noon spectral gradients will be positive. An average +0.2 Kelvin/f(GHz) shift in the spectral gradient is observed between midnight and noon for SMMR radiometric brightnesses (Figure 3). While we have developed a computer model of these gradient effects, for the purposes of this report, thermally induced spectral gradients are noise to be filtered out. (2) The second consequence of soil freezing is a greater opportunity for volume scattering - - particularly at shorter microwave wavelengths. This occurs because of the greater average photon path lengths in frozen soil, and because plants and soil appear increasingly heterogeneous at shorter wavelengths. This "law of darkening" means that, for an isothermal, volume scattering halfspace, aTb a <0.0 (England, 1974). Frozen terrain may also be snow covered. Dry snow is exceedingly transparent to microwaves so that snow exhibits significant of darkening (Figure 4, Edgerton et al., 1971). That is, both frozen soil and snow tend to exhibit negative spectral gradients. While neither a low 37 GHz radiobrightness nor a negative spectral gradient is solely adequate as a classifier of frozen soils, particularly at the relatively coarse resolutions of the Nimbus-7 SMMR, a discriminant based upon a combination of these signatures appears to classify correctly most of the time. SMMR radiobrightness data at 6.6 GHz, 10.7 GHz, 18 GHz, and 37 GHz were obtained for August 1, 1984, through December 31, 1984, over an area that included North Dakota, about half of each neighboring state, and part of southern Canada (Figures 7-9). We chose this large, relatively uniform area because of the low spatial resolution of the SMMR instruments -- 150 Km at 6.6 GHz, 100 Km at 10.7 GHz, 60 Km at 18 GHz, and 30 Km at 37 GHz, and because of the importance of soil moisture state to this region's hydrologic processes. The data arrived from the National Space Science Data Center (NSSDC) on 21, high density, SMMR Cell Tapes. Such data are referenced to latitude and longitude in a satellite-centered coordinate system. We produced two types of image products: Single-band, radiobrightness images at the intrinsic resolution of each sensor, and (2) composite, multi-band images at a common resolution based upon local area averaging. Each radiobrightness pixel was referenced to latitude-longitude in a Mercator projection by interpolation and resampling the Cell Tape data. We used a bi-cubic approximation of a sinc function (Moik, 1980) for the interpolation. H and V radiobrightnesses were averaged to produce a single brightness for each pixel for each frequency.

Page 6 In addition to large area images, local area spatial averages of radiobrightness were calculated for each radiobrightness channel at 7 meteorologic sites within our test region —Miles City, MT; Bismark, Fargo, and Williston, ND; and Abileen, Huron, and Rapid City, SD. A local area is defined as a 150x150 Km cell centered on the meteorological site (150 Km is the spatial resolution of the 6.6 GHz channel). Air and ground temperature data for the Fall of 1984 were obtained from NOAA's National Climatic Data Center in Asheville, North Carolina. Air temperature measurements were available for noon and midnight at the meteorologic sites (i.e., simultaneously with the satellite pass), but ground temperature measurements were for 7:00 a.m. and 7:00 p.m. EST, and were not co-located with the meteorologic sites. Ground temperatures are measured at 5 cm depths. Diurnal heating will affect 5 cm temperatures so that there will be differences between those temperatures and the effective soil temperatures at the times of satellite passes. Local area averages at the meteorologic sites were used to define the preliminary boundaries in our Freeze Indicator discriminant. For example, Figure 1 illustrates the correlation between 37 GHz radiobrightness and reported air temperature. The nominal line in these figures is a single best fit linear regression in the least squares sense of all local area averages. Individual linear fits will differ slightly as shown in Figure l(a). We used the nominal line in our discriminant for simplicity, but a more sophisticated discriminant might use the actual least squares fit for the local area and for the time of day. The discriminant boundaries in Figures l(b) and l(c) are merely estimates based upon the nominal regression and a compromise between midnight and noon air temperatures that would imply frozen soil (the lower boundary) and thawed soil (the upper boundary). Remember that diurnal temperature gradients will generally cause midnight, subsurface soil temperatures to be warmer than air temperatures, and noon, sub-surface soil temperatures to be colder. Similarly, local area averages of spectral gradient versus air temperature were the bases for the spectral gradient decision boundaries shown in Figure 3. Note that the midnight freeze boundary in this example is relatively unambiguous, while a more effective noon freeze boundary would be shifted upwards by 0.2 K/GHz. Again, for simplicity in this initial study, we used discriminant boundaries that were time and location independent. Our 2-parameter Freeze Indicator incorporates the single-band, 37 GHz radiobrightness, and a spectral gradient based upon linear regression of 10.7, 18, and 37 GHz radiobrightnesses for each 100x100 Km pixel. Based upon the decision boundaries in Figure l(b) and l(c), the likelihood of frozen ground in a 37 GHz pixel, P37, is estimated as 0 Tb37 > Tbmax P37 =Tax -r Tb37: Tbhmin < Tb37 < Tbmax Tbmax - b min 1: Tb37 < Tbmin where Tb37 is the measured 37 GHz radiobrightness, and the preliminary decision boundaries are Tbmax = 259 K Tbmin = 247 K

Page 7 The likelihood of frozen ground based upon spectral gradient decision boundaries in Figure 3(a) and 3(b) is Psg, and is estimated as 0a Tb_ aTba I ()mTb aTb af af* aTb Tbaf aTb Ng' i l \ <(3 3) \f ax a - Tba I where the preliminary decision boundaries are (f) =0.3 K/GHz max (a ) =-0.3 K/GHz min These boundaries are preliminary in that they were chosen to yield the fewest misclassifications in plots of the type shown in Figure 5(a) and 5(b). More refined discriminants would incorporate area and time specific decision boundaries. This would be relatively straightforward if there were a higher density of weather stations in the test area. As it is, we believe that diurnal temperature modeling well yield effective time dependent boundaries, and, perhaps, requiring sub-region consistency within a classification will yield improved spatially dependent boundaries. The basic sparseness and lack of control of air and ground data should prompt some caution about over-interpreting these results. Our freeze/thaw discriminant, or Freeze Indicator, is the product of p37 and Psg, and is applied at the scale of the 10.7 GHz data. Resolution differences between different frequency channels can produce anomalous composite image results if the data were processed at their original scale. To avoid these problems, the resolution of the data from each channel is compensated to the (coarse) resolution of the lowest frequency channel used in estimating spectral gradients (i.e. 10.7 GHz and 100 Km resolution). Under certain constraints upon the classification process, these images can be referenced to the higher resolution, 37 GHz format for better location of freeze/thaw boundaries (Zuerdorfer, et al., 1989). The effort needed to do this would be justified as a part of an improved classification process. Figures 7 through 12 include images of the Freeze Indicator for various times during the test period. Black in these images indicates a high likelihood of frozen ground.

Page 8 Figure 6(a) and 6(b) show normalized brightness temperatures for midnight and noon, respectively, in the northern Great Plains during the Fall of 1984. Normalized brightnesses are the average regional brightness at each microwave frequency divided by the average regional air temperature. Normalized brightness thus has the dimension of emissivity. Note that there is little systematic ordering among the 10.7, 18, and 37 GHz normalized brightnesses during August through most of November. However, during the latter half of November through December, the normalized brightnesses at midnight are uniformly ordered, 10.7 GHz brightnesses are high, 18 GHz brightnesses are middle, and 37 GHz brightnesses are low. That is, they exhibit negative average spectral gradients. The noon normalized brightnesses for December exhibit a similar trend, but with exceptions. These are, we believe, illustrations of the law of darkening for frozen soils. Soils at midnight in December for the northern Great Plains are very likely to be frozen. Performance of the freeze/thaw discriminant is demonstrated in Figures 7-9 where Freeze Indicator (FI) images are compared with ground and air temperature measurements for midnight on 9/20/84, 10/24/84, and 12/9/84. Midnight FI images are shown as better examples of the potential of a freeze discriminant. Noon FI images are generally less consistent with meteorologic reports because of the contribution of the noontime positive diurnal spectral gradient to the negative frozen ground spectral gradient that we discussed in the last section. Areas not covered by the satellite in a particular pass are shown in white. Tables 1-3 are summaries of the meteorologic reports. On the night of September 20 (Fig. 7), air temperatures throughout the region were near 600 F and had been above freezing for several days. The FI image shows weak, probably false indications of freezing in the prairies of ND, southern Canada, and the rolling glacial terrain east of the Red River Valley in Minnesota. While the dry air of the northern prairies permits nighttime radiation cooling of the ground to temperatures below that of the air, the more likely explanation for the weak freeze indication is short wavelength scattering by the tall prairie grasses in the northern great plains, and by woodland areas in Minnesota. However, there are no strong indications of freezing in the FI image. On the night of October 24 (Fig. 8), air temperatures hovered about freezing throughout the area, but had been below freezing at Williston for several days, and generally above freezing toward the east (see the temperatures for Fargo, Aberdeen, and Huron in Table 2). The FI image shows a strong freeze indication in northwestern ND which is consistent with the temperature patterns. Similarly, the definite thaw indication along the Red River Valley is consistent with the warmer temperatures reported and the generally more moist soil in the Valley. On the night of December 9 (Fig. 9), air temperatures were generally below freezing except at Rapid City, SD, and had been below freezing for several days. There was no more than trace snow on the ground anywhere in the region. The Fl image shows strong freeze indications throughout most of the region with weaker indications near Rapid City, and in the Aberdeen-Fargo sub-region (Aberdeen is not shown on the December 9 map because its temperature report was missing for that date). Again, the Fl image is consistent with the temperature record. 37 GHz radiobrightness and Fl image sequences were produced at midnight and noon for six-day periods in September, October, and December (Figures 10-12). SMMR coverage is based on a 48 hour cycle —midnight (0000 local hours on the date shown), noon (1200 hours on the same date), and then midnight again 36 hours later. However, orbit precession causes gaps in the cycle and variations in the coverage footprint. Within these constraints, our objective was to observe, if possible, weather dynamics reflected in the Fl images.

Page 9 The 37 GHz sequence beginning on September 16 (Figure 10 and Table 1) shows the moist area associated with the Missouri River, Sakakawea and Devils Lakes in ND, and the Missouri River and Lake Oahe in SD. Rain during the night of September 21 appears as a regional darkening of the 37 GHz image for midnight on the 22nd. Note that the rain is not picked up in the FI image. The October sequence (Figure 11) is dominated by a cold front passing through the area from the northwest with rain and snow beginning on October 19. The region is warmer and drier by the 26th. The moisture pattern dominates the 37 GHz image, but only the apparent freeze pattern, which generally lags the cold front, is shown in the FI image. Note that strong freeze indications follow the cold front but weaken in the south with warming on the 26th. The December sequence (Figure 12) is characterized by cold temperatures and snow from December 2 through December 5, followed by daytime warming into the 40s (and even 580 at Rapid City, SD) by the 9th. The FI images reflect this general coldness, but also show daytime thawing toward the end of the period. Accomplishments and Open Items Freeze Indicator images based upon a preliminary, 2-parameter discriminant —37 GHz radiobrightness and 10.7, 18, and 37 GHz spectral gradient —show relatively good correlation with the expected state of moisture in northern Great Plains soils during the Fall of 1984. The discriminant is preliminary in the sense that experimental testing of theoretical models needs to be done to fully understand the spectral radiobrightness signatures of frozen soils. The concept underlying the preliminary discriminant is that frozen soil will exhibit volume scatter darkening at shorter microwave wavelengths much like the effect observed in dry snow. Few other phenomena cause negative microwave spectral gradients. One such phenomenon is diurnal insolation which should cause negative spectral gradients at midnight, but positive spectral gradients at noon. We have modeled diurnal insolation and are in the process of tailoring our discriminant to allow for diurnal gradients. Freeze Indicator images based upon SMMR data effectively map temporal variations in the freeze/thaw pattern for the northern Great Plains at the time scale of days. These patterns are synchronized with weather patterns, but are not identical. We intend to expand our test data set to include several complete seasons. The product would be, in essence, a movie of freeze/thaw patterns as weather fronts sweep through the Great Plains throughout several seasons. The development of these data from SMMR archives should provide one aspect of hydrologic and mesoscale climatic baselines for the region. The one significant objective of our initial project that has not been achieved is the incorporation of the frozen/thawed soil classification map into regional climate climate models. Such classification maps represent new boundary parameters that are not a part of current models. A very high priority of the new investigation is to explore how this freeze/thaw parameter should be incorporated. Ancillary to the scientific achievements, but, nevertheless, extremely important to this investigation, and to continuing freeze/thaw investigations, are the computational and image manipulation tools that have been developed within the Radiation Laboratory partly as a result of this project. The primary computational tool is the Fortran coded model for the spectral radiobrightness of periodically heated, two-phase media (England, 1989). This model guided our

Page 10 examination of SMMR's spectral response of soil to freezing and thawing, and to diurnal insolation. Our image processing tools have evolved from resampling and classification algorithms on a general use, Apollo system, to a dedicated image processing system resident in a Sun/4 Workstation. The Sun/4 system includes two commercial, image processing software packages, and peripherals for reading tapes (and, shortly, CD's), digitizing maps and images, and printing color images to film and to paper.

PAGE 11 Cloud Snow Air Temp Cover Pack 1984(F ~ (x/1Q) Preci. (in.) (in. 1^^ Sft m 12 QQ12 QQ 1224 B m 9/15 Aberdeen 40 63 2 3 0 0 0 0 Bismark 39 64 0 0 0 0 0 0 Fargo 46 63 8 0 0 T 0 Huron 44 64 0 3 0 0 0 0 Miles City 50 65 2 2 0 0 0 0 Rapid City 44 63 4 2 0 0 0 0 Williston 48 60 6 0 0 0 0 9/16 Aberdeen 46 65 1 3 0 0 0 0 Bismark 53 67 0 2 0 0 0 0 Fargo 47 69 0 0 0 0 0 0 Huron 50 64 5 10 0 0 0 0 Miles City 58 74 0 3 - 0 0 Rapid City 52 70 3 6 0 0 0 0 Williston 55 68 0 3 0 0 0 0 9/17 Aberdeen 57 71 2 8 0 0 0 0 Bismark 56 83 0 1 0 0 0 0 Fargo 57 73 0 3 0 0 Huron 58 73 10 3 0 0 0 0 Miles City 61 80 0 0 0 0 Rapid City 64 84 0 0 0 0 0 0 Williston 56 74 0 3 0 0 0 0 9/18 Aberdeen 64 81 0 0 0 0 0 0 Bismark 57 80 0 0 0 0 0 0 Fargo 64 81 00 0 O Huron 62 84 0 0 0 0 0 0 Miles City 60 80 0 0 0 Rapid City 63 90 0 0 0 0 0 0 Williston 67 73 0 0 0 0 0 0 9/19 Aberdeen 59 84 0 1 0 0 0 0 Bismark 55 86 0 0 0 0 0 0 Fargo 65 90 0 0 0 0 0 0 Huron 68 90 0 0 0 0 0 0 Miles City 67 76 0 0 - 0 0 Rapid City 60 90 0 0 0 0 0 0 Williston 57 72 0 0 0 0 0 0 9/20 Aberdeen 62 65 0 9 0 0 0 0 Bismark 58 63 0 9 0 0 0 0 Fargo 54 62 0 6 0 0 0 0 Huron 62 71 0 2 0 0 0 0 Miles City 63 72 7 2 T 0 Rapid City 59 67 2 6 0 0 0 0 Williston 54 51 8 9 0.01 0.15 (R) 0 9/21 Aberdeen 59 82 2 2 0 0 0.1 0 Bismark 58 81 8 7 0 0 0.3 (R) 0 Fargo 59 77 10 9 0 0 0 0 Huron 60 83 0 0 0 0 0 0 Miles City 66 56 10 10 0.18 (R) 0 Rapid City 60 83 2 1 0 0 0.07 (R) 0 Williston 57 54 10 10.02.01 0.3 (R) 0 9/22 Aberdeen 61 65 8 2 T 0 T 0 Bismark 52 60 3 1 0 0 0 0 Fargo 72 66 9 0 T 0 0.02 0 Huron 66 66 10 2.05 0.011(R) 0 Miles City 46 52 4 6 0- 0 Rapid City 45 60 3 4 0 0 0 0 Williston 45 44 10 9 0 0 0 0

PAGE 12 Cloud Snow Air Temp Cover Pack ^Year(~F (x/ 101 Preci. ((in. 1 984 l QQ QQ 1 12 24 am 10/19 Aberdeen 37 44 10 8.01.1 (R) 0 Bismark 36 40 10 10 0 0.04(S) 0 Fargo 40 38 10 10.07 0.2.77 (R) 0 Huron 38 45 10 7 0.1 0.01 0 Miles City 28 39 7 4 - T 0 Rapid City 26 42 2 7 0 T T T Williston 27 34 2 10.1 (S) 0 10/20 Aberdeen 36 44 10 0 0 T 0 Bismark 35 38 9 10 T.01.1 (R) 0 Fargo 37 42 10 10 T.01.12 (R) 0 Huron 35 41 0 10 0 0 T 0 Miles City 31 33 10 10 - -.01 (S) 0 Rapid City 26 43 0 0 0 0 T (S) 0 Williston 33 33 10 10.02 0.08 (S) T 10/21 Aberdeen 37 41 - 10 0 0 T 0 Bismark 36 37 10 10 0 0.03 (S) 0 Fargo 38 39 10 9 0 T.07 (R) 0 Huron 35 44 2 6 0 0 T 0 Miles City 29 32 8 9 - T T Rapid City 33 38 10 10 T 0 T (S) T Williston 32 33 10 9.01 0 T (S) T 10/22 Aberdeen 34 41 7 8 0 0 T T Bismark 33 39 10 8 0 0 0 T Fargo 36 38 1010 T 0 T 0 Huron 31 42 0 3 0 0 0 0 Miles City 25 38 0 7 0 Rapid City 24 43 3 8 0 0 0 0 Williston 27 32 7 10 0 T T (S) 0 10/23 Aberdeen 35 46 10 4 0 0 0 0 Bismark 36 43 10 3 0 0 0 0 Fargo 34 39 10 10 0 0 0 Huron 32 46 3 5 0 0 0 0 Miles City 23 37 1 0.0 0 Rapid City 29 42 0 0 0 0 0 0 Williston 27 40 10 0 0 0 0 0 10/24 Aberdeen 31 54 0 4 0 O O O Bismark 33 48 0 7 0 0 0 0 Fargo 33 49 0 8 0 0 0 0 Huron 36 57 0 1 0 0 0 0 Miles City 35 44 5 10 0 0 RapidCity 36 57 0 3 0 0 0 0 Williston 30 42 5 7 0 0 T 0 10/25 Aberdeen 38 37 10 10 0 0 0 0 Bismark 34 39 10 10 0 0 0 0 Fargo 38 42 10 10 0 0 0 0 Huron 43 40 8 10 0 0 0 0 Miles City 33 44 10 10 T 0 Rapid City 34 54 1 1 0 0 0 0 Williston 30 42 10 10 7 o T O 10/26 Aberdeen 46 56 0 4 0 0 0 0 Bismark 41 57 3 10 0 0 0 0 Fargo 43 53 0 7 0 0 0 0 Huron 46 61 0 0 0 0 0 0 Miles City 44 52 0 10 - T 0 Rapid City 53 65 0 5 0 0 0 Williston 39 50 6 10 0 0 T (R) 0

PAGE 13 Cloud Snow Air Temp Cover Pack C(F) (x/1l o Precio. (in.) 1984 2iM QQ 122 1 12 11 R12e 2 4h h 12/2 Aberdeen 17 10 10 4 T 0 T (S) 2 Bismark 14 19 10 10 T T.01(S) 1 Fargo 8 3 10 10 T T T (S) T Huron 19 14 10 1.01 T.07 (S) 6 Miles City 13 11 10 9.01 (S) 4 Rapid City 19 19 4 4 0 0 T (S) T Williston 11 8 10 8 T 0 T (S) 1 12/3 Aberdeen 4 17 0 4 0 0 T (S) 2 Bismark 15 15 10 7 T T T (S) 1 Fargo 0 10 0 0 0 T 0.1 T Huron 5 14 0 0 0 0 T(S) 6 Miles City -3 -2 10 1 - 0 4 Rapid City 11 27 0 1 0 0 0 T Williston 9 9 10 8 T T T (S) 1 12/4 Aberdeen 13 10 10 0 0 0 0.02 (S) 2 Bismark 8 15 10 1 T 0 T 1 Fargo 9 15 10 0 0 0 T Huron 5 12 6 2 T 0 T (S) 5 Miles City -1 7 0 10 - - 0 4 Rapid City 15 24 2 0 0 0 0 T Williston -5 10 0 10 0 0 0 1 12/5 Aberdeen 14 7 3 5 T 0 T (S) 2 Bismark 19 -1 4 0 0 0 T (S) 1 Fargo 11 -1 10 10 0 0 T (S) T Huron 15 15 4 7 0 T T 4 Miles City 24 13 10 10 - T (S) 3 Rapid City 26 19 8 7 0 T T (S) T Williston 8 -5 3 3 T 0 T (S) 1 12/6 Aberdeen -14 3 0 0 0 0 0 2 Bismark -11 20 0 8 0 0 T 1 Fargo -8 8 0 3 0 0 0 T Huron -6 11 0 4 0 0 4 Miles City 9 23 0 9 - 0 3 Rapid City 6 44 1 1 0 0 0 T Williston -7 24 0 9 0 0 T 1 12/7 Aberdeen 18 42 7 4 0 0 0 2 Bismark 30 43 9 8 0 0 0 1 Fargo 12 43 7 9 0 0 0 T Huron 19 42 3 0 0 0 3 Miles City 35 44 4 0 0 2 Rapid City 43 61 10 0 0 0 0 0 Williston 34 43 3 6 0 0 0 1 12/8 Aberdeen 30 36 2 3 0 0 0 T Bismark 31 42 7 2 0 0 0 T Fargo 30 43 2 5 0 0 0 0 Huron 31 42 0 4 0 0 O 1 Miles City 29 33 3 3 - 0 1 Rapid City 33 51 0 10 0 0 0 0 Williston 31 37 7 8 0 0 0 T 12/9 Aberdeen 46 - 7 0 0 0 T Bismark 31 43 2 3 0 0 0 T Fargo 29 40 0 9 0 0 0 0 Huron 35 39 7 7 0 0 0 T Miles City 26 35 5 4 - 0 1 Rapid City 43 58 6 0 0 0 0 0 Williston 28 37 0 2 0 0 0 0

PAGE 14 290 - --- -- 280 i Nominal 270Actua 7 [XActuail — -- -- ~(a) Midnight. A linear L3,~~a r regression of the Bismark data. 260 -.... is shown ("Actual" curve), ECL! with a regression line for data 250 H- ---- - - - -. collected throughout North 03cni^ t Dakota and the surrounding, o240Q -— r —- - c ur region ("Nominal" curve). m 230,,.z_ 1 __ 2901 280F ~S. L vI~[Nominall: b) Midnight. Shown with C 270- the Bismark data is a linear' n,6G^D, Iregression for data collected I260 L_ _ _ _ ___throughout North Dakota and IC 2~ ~1~'giscrimlinant! the surrounding region O i ("Nominal" curve). Also 250 <o 250' -IOiscriminantl shown are brightness temper~ i o y~ at ~- Iature decision thresholds for a I 240- - - __ frozen or thawed surface. m 2 o _ 2901~: o 9 Nominal 1 280 —0 270 - 270I'(~~~ ~^a^| ~otg~,(c) Noon. Shown with the 260 - -a ________ oBismark data is a linear reE 260.. — -' iscrminan gression for data collected I- -1 CI throughout North Dakota and 250 - 0 the surrounding region I<n~Q, ~ ~i]^~N~iscrlminanttl ("Nominal" curve). Also I-:~ o _ _i3 - shown are brightness temper240 -.=^~~~~~ y/^~~~~~ \'~ ~ature decision thresholds for a a:^ ^._ _ -__ _. I ~jfrozen or thawed surface. 220~ i 240 250 260 270 280 290 300 310 320 245 255 265 275 285 295 305 315 Ar Ter-p (K) Figure 1. 37 GHz SMMR brightness temperature versus measured surface air temperature, Bismark, North Dakota. Data were collected from 8/1/84 to 12/31/84.

PAGE 15 42 - 0 I -2._ -4" -6 0 6 12 18 24 Hour Figure 2. Soil surface thermal gradient. Computed for 10% moist soil at Bismarck, ND, for a standard day in September.

PAGE 16 0. — a —---— " 0.6k~_ 0.4k'- _ -._~' ~ IDiscriminant (12 0.2H — _- 0 — —....~- --- -- -- -- -4 o1, a) i cjp' (a) Midnight 13 _,___ 1 0 —- -- - C~ Q4 - IDiscriminantl -0.4 - - - - - - - mT- n -----—..-; -Q.6~- ~~\t' -TI I i1 240 250 260 270 280 290 300 310 320 245 255 265 275 285 295 305 315 Ar Temp (K) 0.6 - ----------- 0.6C C_] C_ I e i Is scrs mnan t F.-d - - - I i a Dat were colce rm //4t2318.Tefeiqiisuency gradinant ist e -06 \resod f o I~ \~ thresolds9orarozeA or Terhr (K) Figure 3. Frequency gradient versus measured surface air temperature, Bismark, North Dakota. Data were collected from 8/1/84 to 12/31/84. The frequency gradient is the frequency regression slope for simultaneous SMMR brightness temperatures at 37GHz, 18 GHz, and 10.7 GHz. Shown with the Bismark data are frequency gradient decision thresholds for a frozen or thawed surface.

PAGE 17 0 = 450 X = Wavelength H = Horizontal Polarization V = Vertical Polarization 300 300i = 6.0cm. V (Relafve Measurement) 290 280 i = 6.0 cm, H (Relative Measurement) 270 0 X: 260 250 2 240 ",_,3230 __- = 22- cm, V L "- = 2.2cm,H 220 - -3 re0,,81 cm, V 1210 ~-"' 200... i90 180 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 4 MASS PER UNIT AREA (gn cm'2) Figure 4. Brightness temperature versus equivalent dry snow, Crater Lake, Oregon, 22 March 1970 (Edgerton et al., 1971). Note the short wavelength darkening evident for thick snowpacks.

slixld poxnu aJr s x put'p^,qp oe soxoq uodo'uozoig o3 soxoq pnos oip'tpn punaoi uodn paseg 3ooJmJs po~Mq Jo'plXLI'uazog J JoJ sploqsJlp uolsloop uuLIosnlp aJ Mep reMIUSIa ol qju UMOqS'g8/1 /Zl I 8/I/8 moIg P 0llo0 uAO^ ea eC l Moea Uo~N'lmems!^'nueJ odwoi ssomuq. ZHsO hL nI S snsJa^ u)pqre X:ouonba:I S anl S8Z SLZ S9Z SSZ 9Z SZ SZZ 19Z I~ I I i I l 1 9-0^ ______ _____r-__ltell Ipueu'wmjos!1 Z'1C- C) C 0. I0 uooN'(q). - _ _ _ - Io I - >'X- - - 1 L4- - nC _ 2 -k-t L _ ___________.____8*0 (>) da ssa4L4 -8Z SG.Z 99Z SSZ s.z.z 9 06Z 098 OLZ 09Z OZ QOZ OCZ OZ ~I'Q-!u}u leuilosi lqlupTi (C) CX 0 r~ —-~ —- -X- ~ _ _ -lo 4ZO — u- I ~ [ ________________ uoiJ -i~,' 81, 19d 81 39Vd

PAGE 19 0.98:37 GHz 0.96 ~' - l..... 18 GHz v 0.94 - E,...~-. ^'"'.^^'1 -".-,. 10.7 GHz 0.92:. E a) " 0.88 N. I(a) Midnight Iz 0.84 0.82 Aug S ept iOct Nov De Measurement Day 0.98 37 GHz 0.96. 9~ — 18GHz 0.94 - 09 L~ j 10.7 GHz M 0.92. E 0.986 0.88 E 0.862 o (b) Noon 0.84 0.82 Aug I Sept Oct Nov Dec Measurement Day:igure 6. 37 GHz, 18 GHz, and 10.7 GHz SMMR normalized brightness temperatures versus calender day. Measurements were made at irregular intervals from 8/1/84 to 12/31/84. The normalized brightness temperature of a single SMMR frequency channel is the average brightness divided by the average surface air temperature. averages are calculated over North Dakota and the surrounding region.

PAGE 20 Air and Ground Temp Night 9/20/84 ~....~^. ~D ~~ AIR TEMP Thaw O 0 Williston Mixed Freeze i Fargo O Bismarck Miles GND TEMP City ____ E T Thaw E Aberdeen O O Mixed g ~ Freeze B O1 r- 0oHuron Rapid City ~ 0 R. ~!..:::':-.,..:'ii; Figure 7. A comparison of reported air and ground temperatures with the Freeze Indicator for midnight, September 20, 1984.

PAGE 21 Air and Ground Temp Night 10/24/84 ~ E ] AIR TEMP Thaw O 0 Williston Mixed Freeze * Fargo Bismarck Miles City GND TEMP Thaw ] Aberdeen O Mixed' ~ n ~*' mFreeze - OD M 0 Rapid City Huron _ _.'~.,....... Figure 8. A comparison of reported air and ground temperatures with the Freeze Indicator for midnight, October 24, 1984.

PAGE 22 Air and Ground Temp Night 12/9/84 AIR TEMP Thaw 0 Williston Mixed 3 Freeze * Fargo O Bismarck Miles City GNDTEMP ~'....'Thaw aI I U' Mixed E ~' Freeze * Rapid City Huron _= _ Huron Figure 9. A comparison of reported air and ground temperatures with the Freeze Indicator for midnight, December 9, 1984.

PAGE 23 37 GHZ RADIOBRIGHTNESS MIDNIGHT FREEZE INDICATOR ___________________________(9/18/84) /' t lo. 5X55 -' lo. 5 X F Figure 10(a). Midnight 37 GHz and Freeze Indicator image sequences for a 6 day period in September. ~~::~::~~:~~.c~::~:';~:~i:~~(9/18/8:j4):iM.iliii~~~~ii;~j~~~I:S~~~ri:::':::::.::~:~.iliii~~~~~~~~~~i~i~~j~~i~~i~:.......... ~~i:~~~~::~:~:~:~X~:'G:~:i~:~:'i~:i l~~~~~~~~~~~~~~~~................. ~~~~~~:i::::: ~'i~~~~~~~~~~~~~~~~~~~~~~~'~~~~~: ~ ~ ~ ~................ (9/22/84)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Hi.::3~i:~::'::: ~~~~:: -: ~r~5 F:~. ~~:~

PAGE 24 37 GHZ RADIOBRIGHTNESS NOON FREEZE INDICATOR'?~.-.' ~ ~. ~ (9/16/84) A__4755 A__. _F (9/18/84) AU D'4955 AD''iF ( (9/20/84) (9/22/84) A", 5 35 5A. Figare 10(b). Noon 37 GHz and Freeze Indicator image sequences for a 6 day period in September.

PAGE 25 37 GHZ RADIOBRIGHTNESS MIDNIGHT FREEZE INDICATOR (1 0/20/84) AN08 1 55 ANO: e -, (1 0/22/84) I,...~^~~~^1 w.......:/..-. ( 7 50/24/84)' —. 1. 1 ( 0/126/1 L -84) AN'.id 755 AN-,,-i Figure ll(a). Midnight 37 GHz and Freeze Indicator image sequences for - a 6 day period in October. ~:.:.::..,............ ~~ AN,875AI,,,,..r F~ue11aM~ng~37Gzan ree:dcao mgesqene o ~:j:ia 6 ypro nOtbr

PAGE 26 37 GHZ RADIOBRIGHTNESS NOON FREEZE INDICATOR "!.?:'... I,.-.______ (10/20/84) /L t:.',JS5 ~AD.)f)-':l- ( 10/24/84) 0/26/84) AL'i8 75 AD(.H.S Figure ll(b) Noon 37 GHz and Freeze Indicator image sequences for a 6 day period in October.

PAGE 27 37 GHZ RADIOBRIGHTNESS MIDNIGHT FREEZE INDICATOR ( 2/3/84) AN 2555- AN1 25FO (12/5/84) ANl 12755 AN275F5 (12/7/84) AN 1 2955. AN'') FG ( 12/9/84" ANI 13155 AN; J' F2 Figure 12(a) Midnight 37GHz and Freeze Indicator image sequences for a 6 day period in December.

'*;Pp 6uLSSLiu q pasneo aJe aePwL'L 8//Z1'UOON aq4; UL sauLL a;LqM pe66ef aqiJ eaqawaa UL pOLad /ep 9 e loj seauanbas abewL 1oPeDLpuI azeaJ pue zHLz UOON (q)ZT ajn6Lj', I,,'.'.1~ SS i Ti O (tb/6/Z I) _ )-i~~~~~~~~~~~r~~..a~ 6.....(17861 1i 1) iS (t7g/~/Z ) OiVDIONI 3Z338AJ NOON SS3NIHl98901OViv ZH9O ~ ~82 39Vld

Page 29 IV References Burke, W.J., T. Schmugge, and J.F. Paris, 1979, Comparison of 2.8- and 21-cm microwave radiometer observations over soils with emission model calculations, JGR 84, p. 287-294. Camillo, P.J., and T.J. Schmugge, 1984, Correlating rainfall with remotely sensed microwave radiation using physically based models, IEEE Trans. on Geosc. and Rem. Sens. GE-22, p. 415-423. Edgerton, A.T., A. Stogryn, and G. Poe, 1971, Microwave Radiometric Investigations of Snowpacks, Final Rept. 1285R-4 of Contract 14-08-001-11828 between Aerojet-General Corp., El Monte, CA, and the U.S. Geological Survey. England, A.W., 1974, The effect upon microwave emissivity of volume scattering in snow, in ice, and in frozen soil, Proc. URSI Spec Mtg on Microwave Scattering and Emission from the Earth, Berne, Switzerland, 23-26 Sept., 1974. England, A.W., 1975, Thermal microwave emission from a scattering layer, JGR 80, p. 44844496. England, A.W., 1976, Relative influence upon microwave emissivity of fine-scale stratigraphy, internal scattering, and dielectric properties, Pageoph 114, p. 287-299. England, A.W., 1977, Microwave brightness spectra of layered media, Geophysics 42, p. 514521. England, A.W., 1989, Radiobrightness of diurnally heated, freezing soil, IEEE Geoscience and Remote Sensing, in press. Hoekstra,- P., and A. Delaney, 1974, Dielectric properties of soils at UHF and microwave frequencies, JGR 79, pp. 1699-1708. Moik, J., 1980, Digital Processing of Remotely Sensed Images, NASA SP-431. Schmugge, T.J., 1983, Remote sensing of soil moisture: Recent advances, IEEE Trans. on Geosc. and Rem. Sens. GE-21, p. 336-344. Schmugge, T.J., 1987, Remote sensing applications in hydrology, Rev. Geophys. 25, p. 148152. Schmugge, T.J., P.E. O'Neill, and J.R. Wang, 1986, Passive microwave soil moisture research, IEEE Trans. on Geosc. and Rem. Sens. GE-24, p. 12-22. Ulaby, F.T., R.K. Moore, and A.K. Fung, 1981, Microwave Remote Sensing, Active and Passive, Addison-Wesley, p. 186-255. Wang, J.R., T.J. Schmugge, W.I. Gould, W.S. Glazar, and J.E. Fuchs, 1982, A multifrequency radiometric measurement of soil moisture content over bare and vegetated fields, Geophys. Res. Let. 9, p. 416-419.

Page 30 Zuerndorfer, B., A.W. England, C. Dobson, and F.T. Ulaby, 1989a, Mapping freeze/thaw boundaries with SMMR data, J. of Agriculture and Forest Meteorology, in press. Zuerndorfer, B., A.W. England, and G.H. Wakefield, 1989b, The radiobrightness of freezing terrain, Proc. of IGARSS'89, Vancouver, B.C., July 10-14, 1989, p. 2748-2751.

Radiobrightness of Diurnally Heated, Freezing Soil A. W. England Radiation Laboratory Department of Electrical Engineering and Computer Science The University of Michigan Ann Arbor, MI 48109-2122 August 15, 1989

Abstract Freezing and thawing soils exhibit unique radiometric characteristics. To examine these characteristics, diurnal insolation is modeled as 1-dimensional heating of a moist soil halfspace during a typical fall at a northern Great Plains site (Bismarck, ND). The 1-dimensional, heat flow equation is non-linear because both the enthalpy and the thermal conductivity of freezing soils are non-linear functions of temperature. The problem is particularly difficult because phase boundaries propagate in time, and because soils that contain clay freeze over a range of temperatures rather than at 00 C —that is, they possess diffuse phase boundaries. A modified Chernous'ko method was used to integrate the heat flow equation to obtain monthly thermal models during a typical September through December period. Diurnal radiobrightness curves at 10.7, 18, and 37 GHz were computed for each month. The 37 GHz radiobrightness best tracks soil surface temperature, the 10.7-37 GHz spectral gradient of thawed soils is strongly positive, the spectral gradient of frozen soils is slightly negative, and the midnight to noon spectral gradient is shifted by approximately +0.1 K/GHz by diurnal changes in the surface temperature and the thermal gradient. These observations support the use of SMMR's 37 GHz radiobrightness and its 10.7-37 GHz spectral gradient as discriminants in a frozen soil classifier for high latitude prairie. Introduction The quantity and state of moisture in soil can be estimated from satellite radiobrightness signatures. There is a large body of literature linking moisture content to radiobrightness [1]-[6]. Whether or not the soil is frozen affects the rate of energy transfer to the atmosphere by limiting evapotranspiration, and affects the rainfall or snowmelt runoff potential by reducing the infiltration capacity of the soil. Zuerndorfer et al. [7] produced freeze/thaw maps of the northern Great Plains

from Nimbus 7 Scanning Multichannel Microwave Radiometer (SMMR) data. This paper is an examination of the theoretical basis for microwave radiometric, frozen soil classification. Figures la and lb illustrate the observational basis for a frozen soil classifier. A 37 GHz radiobrightness below about 247 K indicates frozen soil, and one above about 247 K indicates thawed soil for Bismarck, ND. However, regions near the Missouri River, Sakakawea and Devils Lakes, the Red River Valley, and in the wakes of a regional rainfall often appear anomalously cold at 37 GHz because of exceptional wetness for the northern Great Plains. The confusing factor is that 37 GHz radiobrightness, while least sensitive to moisture of the SMMR frequencies, is darkened by moisture. A second discriminant -- the 10.7-37 GHz spectral gradient -- resolves the ambiguity because frozen soils are observed to have negative spectral gradients, and moist soils are generally observed to have a positive spectral gradient. Figures la and lb also exhibit an upward shift in the spectral gradient between midnight and noon of about 0.1 K/GHz. A valid theoretical model should replicate the behavior of the 37 GHz radiobrightness, the dominant characteristics of the 10.7-18-37 GHz spectral gradient, and the diurnal shifts in the spectral gradient during the September through December months in the northern Great Plains. Model parameters will be chosen for prairie near Bismarck, ND. I-Dimensional Heat Flow The 1-dimensional heat flow equation for temperature T (Kelvin), depth z (cm), and time t (sec) is [8] aE(T) aF(z,t) at 3z (1) where E(T) is enthalpy (cal/cm3) and F(z,t) is heat flux (cal/cm2-K) (T is a function of z and t). Heat flux is linearly related to the temperature gradient, aT/az, by 3

aT F(z,t) = - K(T) a az (2) where K(T) is thermal conductivity (cal/cm-sec-K). That is, aE(T) a K(T)aTl at az\ az With respect to a reference temperature, To, E(T) is generally the linear function E(T) = pp {T-To) (4) where p is density (gm/cm3), and cp is specific heat at constant pressure (cal/gm-K). If K(T) is constant, Equation (3) becomes the parabolic linear differential equation aT a2T t az2 (5) where constant K is thermal diffusivity (cm2/sec) IC= K P P (6) Equation (5) can be solved analytically by either the harmonic or the Laplace methods [8], or by a finite difference numerical method. Watson used the Laplace method [91(10], and Kahle used the finite difference method [11] to develop thermal models for the diurnal insolation of rock and soil. Their models relate day-night differences in thermal infrared temperature to rock and soil type. However, if soils freeze or thaw diurnally, then Equation (3) is not linear and is not amenable to either the Watson or the Kahle methods. Equation (3) looks simpler with the substitution of variables 4

u = K'(r) dt iJ^~~~~~~~~TO~~~~~~~ ~(7) so that u = u(T) and Equation (3) becomes aE(T) a 2u at z2 (8) This I-dimensional, non-linear, heat flow equation is called Stefan's problem [12], and there are three ways to solve it: The finite difference method [13], the moving boundary method [14], and the isotherm propagation or Chernous'ko method [15]. The finite difference method for non-linear problems suffers from uncertain convergence properties. The moving boundary method involves seeking analytical solutions to the linear regions on either side of moving phase boundaries. This method is particularly awkward if the phase change occurs over a depth zone, as in freezing soils, rather than at a plane interface, or if there are multiple freezing isotherms, as in the periodic heating case. The Chernous'ko method has a problem with the periodic heating case. Chernous'ko [15] replaced E(T) with the piecewise constant approximation, H(T), (shown schematically in Figure 2a) H(T) = E(Tn) for Tn T < Tn+, (9) where Tn denotes isotherm n ordered by increasing depth. While these isotherms can be any temperature, integer values of Kelvin are convenient The slope of the linear portions of E(T) below 268 K and above 273 K are the pcp for frozen and moist soils, respectively. The increased slope between 268 and 273 K results from the latent heat of melting of moisture in the soil (linearity of E( in this region is an approximation). In this example, melting occurs over 5 degrees. For sandy soils, the range might be near 1 degree; for some clay soils, it might be more than 10 degrees [16]. 5

In terms of H(T), Equation (9) becomes aH(T) _ 2u at az2 (10) Locally constant H requires that u be a linear function of z between isotherms Tn and Tn+l, i.e., U =n Un Un+lUn (Z - Zn) Zn+1-Zn (11) where Zn is the depth of isotherm Tn. If the value of H(T) at z > Zn is denoted by H+, and H(T) at z < Zn is H-, then Equation (10) can be integrated at constant time along path a-b in Figure 3, fb a JH(T) dz=au (au1 a t azJbt aza t a^~~~~~~~~~~~ ~~~~(12) to yield t[H (b-Zn) + H (Zn-a)] -aZ - ()a (13),t \3z bt \3z/a t (13) From Equation (1 1), because zn is the only function of time on the left of Equation (13), dz,U 1 n+Un IUn-Unr1 dt (H- H+) IZn+1'zn n-Zn-1 (14) Equation (14) is the ordinay, linear differential equation that Chernousko used to propagate each isotherm in time. 6

Equation (14) fails in the periodic heating problem because of an asymmetry in H(T) between heating and cooling. For example, consider the propagation of isotherm zn where Tn- 1 < Tn = Tn+l. In this case, H+> H- and dzn= -1 I (Un-Un-1l d t (H- H+) InZn1 (15) However, if Tn- > Tn = Tn+l, then H+= H- and dzn/dz in Equation (14) is undefined. This asymmetry can be ameliorated by substituting the piecewise constant H(T) shown schematically in Figure 2b, i.e., E(Tn)+E(Tn+l) for Tn< T < Tn+l 2 H(T) E(Tn) for T Tn E(Tn-1)+E(Tn) lE(T4+E(T ) for Tn.1< T < Tn 2 (16) Isotherm propagation with this alternative approximation for E(T) will be referred to as the modified Chernous'ko method. Boundary Conditions Watson [9] and Kahle [11] used boundary conditions for the energy flux, Fnet(z), Fnet(O) = Fsun + Fsky + Fwind - Fground Fnet(o)=- 0 (17) where z = infinity means depths greater than the penetration of the diurnal thermal pulse. The parameters that comprise these boundary conditions for Bismarck, ND, are described in Table 1. FSUn is insolation reduced by cloud cover, atmospheric absorption, albedo, and the cosine of the 7

zenith angle. Fsky is sky brightness plus a small correction for cloud cover. Fwind is a small correction for sensible heat transfer between ground and air used by Kahle [11]. Fground is graybody emission from the soil's surface. Topography and evapotranspiration are ignored. Radiobrightness Consider thermal microwave emission from volume, dV, within a dielectric halfspace (Fig 4). The energy, dQ, arriving at surface element, dA, is dQ = EdV e-2'z/cos O' dA z/cos 8' (18) where 23 is the power loss coefficient, Xo (19) e' is the real part of the dielectric constant, tan 8 is the loss tangent, Xo is free-space wavelength, and E is volume emissive power for dV at thermal temperature, T(z), E =' 23 T(z). (20) With dV = z do dz/cos 0', and with a first order approximation for T(z), the intensity, dI, below the interface in direction 0' is dI- e' 2 (T+(a) z} e-2f zcos'dz/cos e' T (Zo (21) where Tg is soil surface temperature, and I' has been normalized by Planck's coefficient so that its dimension is Kelvin. Integration of Equation (21) yields the upwelling intensity below the interface I'-e' {Tg +cs'() ) 2(3 ao (22) 8

If reflected sky-brightness and atmospheric absorption and emission are ignored, the normalized intensity above the interface (the radiobrightness, Tb) is e(X, 0) r(O')/e', where e(X, 0) is the directional spectral emissivity and the source of the emission polarization. Because of Snell's Law effects, cos 8' below the interface is relatively near to unity for most satellite incidence angles, 8 (typically < 500). With only a slight loss of generality, we shall assume that 0=0 so that polarization can be ignored, and so that brightness temperature can be written Tb= e() {Tg +Ze () 0z o (23) where ze = (23)-1 has the dimension of length. Ze should be called the effective emitting depth. It is equivalent to one optical depth in optics, or to half the skin depth in electromagnetics. The dielectric properties of typical freezing soils at microwave frequencies are given by Hoekstra and Delaney [16]. Figure 5, from their paper, shows the variability in the complex dielectric constant for several moisture percentages in Groodrich clay and in Fairbanks silt. Note that the dielectric properties are essentially constant through freezing for a moisture content of 5%. This insensitivity to temperature for small moisture contents occurs because the water is chemisorbed, or adsorbed, to the clay or sand interfaces within the soil [17] and are not free to rotate with the electromagnetic wave. A reasonable approximation to the complex dielectric properties of moist soil, E*, through freezing is E soil + (m-O.07) p (f Ewater + (1-f) ice} 24) ( 1-rm) (24) where Esoil = dielectric constant of 7% moist soil 2 Kg-K. K.-n2 Ewater =n+ + l+U(jom)1'-a 1+jco2 [18][19] n2 = 1.8 Ks = 295.68 - 1.2283 T + 2.094x10'3 T2 - 1.41x10-6 T3 9

Ke = 4.2 a = 0.012 x1 = 5.62x10-15 e0.188/kT sec 12 = 4.2x10-14 sec T = temperature Kelvin k = Boltzmann's constant = 8.61735x105 eV/K o = angularfrequency, radians/sec Ks-Koo Eice K. + l+j(O' Koo =3.2 Ks = 3.2 + 20715/(T-38) [20] X = 4.76x10-16 eO.577/kT [21]. The Model Appropriate model parameters for prairie near Bismarck are listed in Table 2. Parametric variables in these models are date and moisture content While it might be argued that all parameters should be examined parametrically, the extensive computation required is not warranted by any possibility of inverting radiobrightness data to obtain more than moisture content or state. The initial temperature of the thermal model was estimated from the average, diurnal radiation balance, T2 4 hr (25) where Tog becomes the initial temperature. Tog isotherms were placed at each centimeter of depth, and then propagated subject to the boundary conditions. The propagation time interval was assignable, but typically 6 seconds, and each 24 hour iteration began at midnight. The solution 10

was declared to have converged when the maximum surface temperature difference for each minute between 24 hour iterations was less than 0.001 Kelvins. Convergence required between eight and twelve iterations depending upon the month, the soil moisture content, and the propagation interval. Observations Temperature versus depth profiles at midnight, 6:00 am., noon, and 6:00 p.m. for Bismarck, ND, are shown in Figure 7. The gross features of these profiles are relatively independent of moisture content and month. Among these four profiles, surface temperatures are coldest at 6:00 a.m. (predawn) and hottest at noon as expected, and thermal pulses at depth are most pronounced at 6:00 p.m. Thermal gradients at the surface are always positive at midnight, 6:00 a.m., and 6:00 p.m., and they are always negative at noon. Note that the effect of freezing and thawing during October through December is a general compression of the temperature profile. That is, the apparent thermal inertia would be greater during freezing and thawing. Figure 8 shows diurnal surface temperatures for September through December. The September curves, because temperatures are above freezing, look like the curves for diurnally heated, moist soils [9][10][1 1]. While moisture tends to reduce the day-night temperature difference, the effct is small. The October curves are very different. The daytime peak is lower because of reduced insolation, but nighttime curves are "held up" by the latent heat of fusion of soil moisture. The effects are similarly pronounced in November and December except that daytime peaks appear suppressed. The cause is the same -- daytime radiant heat goes into melting soil ice rather than into raising soil temperature. Again, an observable effect would be a strong increase in the apparent thermal inertia of freezing and thawing soils. Effective emitting depths, Ze, as functions of microwave frequency, moisture content, and time-of-day are shown in Figure 9. The variability with frequency is caused by the Debye 11

relaxation processes in water and ice [16][17]. The September curves show the effects of moisture: The emitting depth of moist soil decreases with frequency, decreases with moisture content, and increases with insolation (maximum soil surface temperature) for frequencies below the primary relaxation frequency of water. The curves for October are dramatically different because ice in soil is effectively transparent to microwaves. Note that, for these model parameters, October near Bismarck is cold enough to completely freeze 10% and 15% moist soils, but not sufficiently cold to completely freeze 20% moist soil. The relatively smooth transitions between frozen and thawed emitting depths result from the assigned 3 degree freezing range of soil in this model. A reduced freezing range would cause a more abrupt transition. The curves for November and December are mutually similar. Their major features are that the widths of the daytime melt period diminish slightly with increased moisture content because more of the daytime insolation must go into melting ice before temperatures rise. The slight increases in emitting depths with water content for the frozen periods occur because ice is more transparent to microwaves than the rock it replaces. The significant features of these curves are that microwave thermal emission originates much deeper in frozen soils than in moist soils, and that, for the SMMR frequencies, frozen soil emitting depths are roughly 1 cm and less. Therefore, frozen soil thermal gradients that are significant over depths of 1 cm should influence the SMMR spectral gradient Radiobrightness curves at the 10.7, 18, and 37 GHz SMMR frequencies, as functions of moisture content and time-of-day, are shown in Figure 10. The September (moist soil) brightness decreases with moisture content, and increases with microwave frequency. The frozen soil brightness curves during October through December are generally high and are relatively independent of microwave frequency. Midnight-noon differences are always positive for moist soils, and generally negative for diurnally thawing soils. While none of these soil models remain completely frozen throughout the day, a choice of parameters that avoids midday thawing would exhibit a positive midnight-noon shift in the spectral gradient. Of the three frequencies, the 37 12

GHz radiobrightness is least affected by moisture content or by freezing and thawing. That is, the 37 GHz radiobrightness most closely follows thermal temperature. It is this property that justifies its use as one of two discriminants in a SMMR, frozen soil classifier [7]. Radiobrightness spectral gradients, as functions of moisture content and time-of-day, are shown in Figure 11. The moist soil model gradients are always strongly positive, and the frozen soil model gradients are weakly negative. This unambiguous correlation is the reason that spectral gradient was chosen as a second discriminant in the SMMR, frozen soil classifier. SMMR observations are not universally unambiguous -- summertime, hot day SMMR data often exhibit negative gradients. However, summertime exceptions are unlikely to confuse a frozen soil classification. The midnight and noon surface thermal gradients in Figure 7 are typically +1.5 and -3.5 degree/cm, respectively. This -5 degree/cm midnight-noon shift in the thermal gradient contributes about +0.1 K/GHz to the model's +0.5 to +1.4 K/GHz shift in the spectral gradient for the frozen soils shown in Figure 11. The preponderance of the shift is caused by the basic moist soil/frozen soil differences in spectral gradient. Note that the midnight-noon shift for moist soils (September) is negative. Discrepancies Model results are highly consistent with the SMMR observations reported by Zuerndorfer et al. [7]. However, there are three discrepancies: The SMMR frozen soil spectral gradient tends to be around -0.3 K/GHz, while the model predicts something like -0.1 K/GHz; the SMMR midnight-noon differences in spectral gradient for thawed soils average +0.2 K/GHz, while the model predicts that thawed soil differences should be weakly negative; and the SMMR moist soil spectral gradient can be negative on hot, summertime days, while the model predicts positive gradients. 13

While these discrepancies are ancillary to our objective of examining the performance of the 37 GHz radiobrightness and the 10.7-18-37 GHz spectral gradient as discriminants in a frozen soil classifier, they do suggest that the model is incomplete in that it ignores volume scatter darkening by prairie grasses and crop stubble, and by inhomogeneities within the frozen soil. The scattering albedo, coo, is a measure of the strength of volume scatter darkening. The parameter was used by Chandrasekhar [22] to describe darkening in planetary atmospheres, and applied by England [23][24] to describe darkening in frozen soils, ice, snow, and dry, planetary regoliths. It has become a parameter in most theories of wave propagation and scattering, e.g., Ishimaru [25]. For single scattering [25], o Na = a Nau + 2J3 (26)where N is the number of scatterers per unit volume, a is the scattering cross section for a single scatterer, and 2p is the power loss coefficient defined in Equation (19). For spherical scatterers whose diameters are small fractions of a wavelength (Raleigh scatterers), a a 4 (28) so that oo increases with decreasing wavelength to yield a negative spectral gradient of radiobrightness -- a "law of darkening". This short wavelength darkening is the likely cause of the strongly negative spectral gradient observed in SMMR data for frozen terrain. The hot-day, negative spectral gradient may occur when heat drives most water from the canopy and from the upper layers of soil. Dry soils have an increased effective emitting depth so that volume scatter darkening may again cause a negative spectral gradient. The discrepancy between observed and modeled, midnight-noon shifts in moist soil, spectral gradients may be caused by moisture (dew) on the plant canopy at midnight. However, without experimental studies, such explanations are only speculation. 14

Conclusions A modified Chernousco method for solving Stefan's problem yields an acceptably rapid convergence to a solution for the thermal structure of diurnally insolated, moist and frozen soil. The modification involves an alternative, piecewise constant enthalpy-temperature approximation. With that modification, solutions exhibit the expected symmetry for heating and cooling. Among the 10.7, 18, and 37 GHz SMMR frequencies, both the SMMR observations and the model show that the 37 GHz radiobrightness best tracks the thermal temperature of the soil's surface, and show that the 10.7-18-37 GHz spectral gradient is always negative for frozen soils. Therefore, the "and" condition, that the 37 GHz radiobrightness be below some threshold and that - the spectral gradient be negative, should be an effective classifier of frozen soil. Acknowledgement This study was supported by NASA Interdisciplinary Research Program Grant NAG5852. 15

References [1] Burke, W.J., T. Schmugge, and J.F. Paris, "Comparison of 2.8- and 21-cm microwave radiometer observations over soils with emission model calculations," J. Geohvs. Res.. 84, pp.287-294, 1979. [2] Wang, J.R., T.J. Schmugge, W.I. Gould, W.S. Glazar, and J.E. Fuchs, "A multifrequency radiometric measurement of soil moisture content over bare and vegetated fields," Geophys. Res. Let.. pp. 416-419, 1982. [3] Schmugge, T.J., "Remote sensing of soil moisture: Recent advances," IEEE Trans. Geosci. Rem. Sensing. GE-24, pp.12-22, 1983. [4] Camillo, P.J., and T.J. Schmugge, "Correlating rainfall with remotely sensed microwave radiation using physically based models," IEEE Trans. on Geosci. and Rem. Sens.. GE22, pp. 415-423, 1984. [5] Schmugge, T.J., P.E. O'Neill, and J.R. Wang, "Passive microwave soil moisture research," IEEE Trans. on Geosci. and Rem. Sens.. GE-24 pp. 12-22, 1986. [6] Schmugge, T.J., "Remote sensing applications in hydrology, Rev. Geohvs.. 25, pp. 148-152, 1987. [7] Zuerndorfer, B.W., A.W. England, M.C. Dobson, and F.T. Ulaby, "Mapping freeze/thaw boundaries with SMMR data," in press, J. Agricultural and Forest Meteorology, 1989. [8] Carslaw, S., and JC. Jaeger, Conduction of Heat in Solids, 2nd Ed., Oxford, 1959. [9] Watson, K., "Geologic application of thermal infrared images," Proc. of IEEE, pp. 128137, Jan., 1975. [10] Watson, K., L.C. Rowan, and T.W. Offield, "Application of thermal modeling in the geologic interpretation of IR images," Rem Sensing, K. Watson and R. Regan ed., Geophysics Reprint Series, no. 3, Society of Exploration Geophysicists, 1983. [11] Kahle, A.B., "A simple thermal model of the Earth's surface for geologic mapping by remote sensing," J. Geophys. Res.. 82, pp. 1673-1680, 1977. 16

[12] Evans, G.W., "A note on the existence of a solution to a problem of Stefan," Ouarterly of Anplied Mathematics., pp. 185-193, 1951. [13] Landau, H.G., "Heat conduction in a melting solid," Quarterly of Applied Mathematics. 8, pp. 81-94, 1950. [14] Douglas, J., and T.M. Gallie, Jr., "On the numerical integration of a parabolic differential equation subject to a moving boundary condition," Duke Mathematical J.. 22, pp.557-571, 1955. [15] Chernous'ko, F.L., "Solution of non-linear heat conduction problems in media with phase changes," International Chemical Engineering 10, pp.42-48, 1970. [16] Hoekstra, P., and A. Delaney, "Dielectric properties of soils at UHF and microwave frequencies," J. Geophvs. Res.. 79, pp. 1699-1708, 1974. [17] Hoekstra, P., and W.T. Doyle, "Dielectric relaxation of surface adsorbed water," L Colloid and Interface Sci. 36, pp. 513-521, 1971. [18 Hasted, J.B., "Dielectric properties of water and aqueous solutions," Dielectric and Related Molecular Processes. The Chemical Society, London, pp. 121-162, 1972. [19] Hasted, J.B., Aqueous dielectrics, Chapman and Hall, London, 302p., 1973. [20] Cole, R.H., and 0. Worz, "Dielectric properties of ice," Physics of Ice N. Riehl, B. Bullemer, and H. Englehardt ed., Plenum, NY, pp. 456-554, 1969. [21] Camp, P.R:, W. Kiszenick, and D. Arnold, "Electrical conduction in ice," Physics of Ice, N. Riehl, B. Bullemer, and H. Englehardt ed., Plenum, NY, pp. 450-470, 1969. [22] Chandrasekhar, S., Radiative Transfer, Dover, NY, p. 6, 1960. [23] England, A.W.,'Thermal microwave emission from a scattering halfspace," Radio Science 9, pp. 447-454, 1974. [24] England, A.W.,'Thermal microwave emission from a scattering layer," J. Geophvs. Res.. &Q, pp. 44844496, 1975. [25] Ishimaru, A., Wave Propagation and Scatterin in Random Media. I, Academic, NY, p. 11, 1978. 17

iTable 1. Boundary Parameters Fsun Solar irradiance = fl So (1-A) M(O) cos ) Fsky Sky irradiance = Tskv4 + f2 Fwind Sensible heat transfer from air to ground Pa ca Cd (W + 2) (Tair - Tground) Fground eaT So Solar Constant = 0.03313 cal/sec A A- - edo M() Approximate atmospheric transmissivity 1.0 - 0.2 (cos ))-0.5 [9][10] ~^~~ ~ Zenith angle cos ) cos X cos 6 (-cos(2ic hour/24)+sin X sin 6) if > 0, otherwise cos q = 0, local latitude 5 declination = - 23.4330 cos(2R month/12) fl I (l-cl) where cl is average cloud cover. Approximation is that some is regained through f2. f2 Irradiance from clouds, approximated as half the average solar irradiance lost in the cloud term, fl. (cl/2) So (1-A) (J M(O) cos 0 dt)/24 d~~~~a Stefan-Boltzmann constant = 1.3533x10'Tair Average air temperature Toair - Tdel cos(2x(hour-2)/24) [1 1] Toair Monthly average air temperature (e.g., see Fig. 6) To — TI cos(2x(month-8lag)/12) Tdel -Diurnal variation ( f meteorological reports) Tsky Tair(O.61 +0.05 w.5)025 (Brunt's formula, from Kahle [11]) w.. Water vap pressue, mmHg Pa Air density at surface = 1.25x10- gm/cm Ca ca _______Specific heat of dry ar =.24 cal/gm-K Gd..........Drag coeffiient =0.02K.006(..Z500),Z is elevation in meters [11] JW -_d vel-oci&ty_ in laiim/sec- - - ---- e T Lermalnrat en ssivlt Tg Soil surface temp fm solution to eat flow equation 18

Table 2. Parameters for Bismarck, ND I Latitude _____ 470 N Months_ September through December (i.e., 9- 12) Soil moisture 10%, 15%, 20% by weight Cloud cover, cl 20% Average winds, w 5 m/sec Albedo 0.2 Thermal IR 0.95 emissivity, e Freeze interval 270-273 K Dry soil density 1.5gm/cm Drsoil secific heat 0.2 ca/gm Dry soil thermal )0.C 5 ca cm-sec-deg conductivity 7% moist soil "3. dielectric constant 7% moist soil 0.23 loss tangent Average air 278.3-K(Fig.6) temperature, To Annual air temp 16.9 K deg variation, T1 Temp phase lag, Olag.2 mon Diurnal temp 5 K deg variation, Tdel Water vapor 0.76 mmHg pressure, w 19

Captions: Fig. 1. Frequency gradient versus SMMR 37 GHz brightness temperature, Bismarck, North Dakota. Data were collected from 8/1/84 through 12/31/84. Shown with the Bismarck data are clustering decision thresholds for a frozen, mixed, or thawed surface. Based upon ground truth, the solid boxes are frozen, open boxes are thawed, and x's are mixed pixels (personal communication from Zuerdorfer, modified from Zuemdorfer et al. [7]). Fig. 2. Enthalpy versus Temperature. E(T) and H(T) are approximations to moist soil enthalpy during freezing. H(T) in 2a represents Chernous'ko's, piecewise constant model. The asymmetry of H(T) at each isotherm produces dissimilar heating and cooling performance. H(T) in 2b is the modified model. The symmetry in H(T) at each isotherm produces the necessary similarity for heating and cooling. Fig. 3. Integration path at constant time. The piecewise constant approximation to H(T) means that the enthalpies, H- and H+, are constant between isotherms. Fig 4. Schematic of emission from a moist soil. Because of the transmissivity of soil, thermal microwave emission originates below its optical surface. If ) is the beam angle out of the page, q =' = 0. do) is the solid angle of the beam, dco = dO d). Fig. 5. The complex dielectric constant at 10 GHz as a function of temperature at three water contents for (a) Goodrich clay and (b) Fairbanks silt (from Hoekstra and Delaney [16]). Fig. 6. Monthly average air temperatures for Bismarck, ND (from National Weather Service data). "Model" refers to the first order Fourier component. Fig. 7. Temperature versus depth profiles for Bismarck, ND. Curves represent midnight, 6:00 a.m., noon, and 6:00 p.m. for three moisture contents. Fig. 8. Diurnal soil surface temperatures at Bismarck, ND. Curves represent moisture contents by weight. Fig. 9. Effective emitting depth versus time of day. Effective emitting depth is equivalent to optical depth in optics, or to 1/2 the skin depth in electromagnetics. Curves represent effective emitting depths for 3 microwave frequencies -- 10.7, 18, and 37 GHz. Fig. 10. Radiobrightness versus time of day. Curves represent the brightness for 3 moisture contents and at 3 microwave frequencies. Fig. 11. Radiobrightness spectral gradient versus time of day. Curves represent 3 moisture contents. The gradients are computed as a least squares regression of the radiobrightness at 10.7, 18, and 37 GHz. 20

Decision Space la Bismarck (Day) 0.4, 0.3 ~. X ~ 3 _ Thawed 03 33 C | I l il X ^ ~02 ~~ _ J ^__ I: ^ ~02 <~ e e Mood~ ~ —Mixed 0.1 A Frozen.o- 0 ~1 -0.1 ~... Lt. 0.2 -0.3.... -0.4 230 240 250 260 270 280 290 235 245 255 265 275 285 37 GHz Radobrightneu (K) ib Bismarck (Night) 0.4 -. -.. -. -....... Thawed X.^ 0.2 ~ ~ ~[J.....~~ IMixed 0.1 - - ~:zz:~ zzzz3 Frozen LL. I040 240 250 260 270 280 290 235 245 255 285 275 285

Figure 2a C X Xsi ( Temp, K Figure 2b...... ~ E(T) -m"',~J -----— MI H(T)..''' I.....,''' 260 265 270 275 280 Temp, K 22 22

Figure 3. Time ~> b H+' Figure 4 Radiom ter Surface / dA 1 Moist Soil dV4 3' 23

Figure 5a 12.0,, 100 *0.15 8.0- / K' <' 6.0 - ~ 4.0 **?* -_-~ ~~0~~ *-* 0.05 - G~~0~~ 0 oOQi5 1 0 ~ P K -0 00-O- o 0 oo -— o 0.105 0 2.0 I 10.0 4._0 _ 0 o -00 0 0 0 0.05 220 -10 0 10 2030 Temperature, ~C Figure 5b 10.0'^^ ~~~~/8.0 8/~0.0 -M 6.0 - *.10 ^__ ~~^~ 0 30 20 -10 0 10 20 Temperaturel oC

Figure 6 300 290 $ 280 e 270 1 E 1^ ^ - -- Model i- 260 Observations Tair=278.3+16.9*cos(2;(mo-7.1 2)/12) 250 -.,., ~,.,. 0 2 4 6 8 10 12 Month 25

10% Moist Soll 15% Moist Soil 20% Moist Soil 320 300 240 300 E 280 0 260 240 J20 300 E 280 I260 2.40 320 300 E 280' 260 240.. 3 0 C 20 3C 40' 3 C 2C 3C 5C3 10 2C 30 C 5C Depth, cm Fig. 7 26

320 Sept 300 280 10o moist 260 15 moist 20% moist 240 320 Oct 300 280 260 240 - 320 Nov 300 280 260 240.................... 320 Dec 300 280 260 240 0 - - - - -. I.24.,..Flig. 8 Solar Time, hr 27

Dec EmiLLangDepLh, cm Nov EmiLtini Dep cm OctmiLtnligDehth cm SeptEmniL^ing DepIth3cm 9 o - ^ ~ c. I I *3 * -~~ ~ I f 0 I- 0 3 0 C,) U,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~c u~~~~~~~~~~~~.~~~~ ~. -"^ N~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~N r~~~~~~~~~~~~~~~~~~~ II! J___________IJ_________9 1 _________I______-_ C) C) *-JI J 1 I III II I I r c~~~~~~~~~~~~1 1 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ aD~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Dec RadlobrlghLneM.c K Nov Radiobrightness, K Oct Radiobrightness. K Sept Radiobrightness, K - 0 0 0)- N N oh CD C) N) b 0 0% ) O0 ( ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~00 -'. - i. ~~... A A. 0~~ -13 0~~~~~~~~~~~~~~~~~~~~~~~ q~~~~~~~~~~~~~~~~~~~~~~~~~~,1.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~, o~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~o J~~~~~~~~~ U) -40 N)._LJ_ II / l rN ~'~ ^~^ ~ ~'~~~""" ~ 0 N) ii~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~U w C)j N N N

1.5 N 1.0 Ile'.lox____ ^~^ " _ v - 1 0% r10% moist 0.5 0 ~- ~ 15% moist _Q I ~~~ —-~..20% moist 1 0.0 September 1.5 N 1.0 0.0 Ho.o October -0.5 1......................... 1.5 10.5 I J o.o November -0.5 -- -.............~ 1.5 N 1.0 05 0.0 Io.o December 0.5o 12 1 24 Fig. 1 1 Solar Time, hr 30

MAPPING FREEZE/THAW BOUNDARIES WITH SMMR DATA B. W. Zuemdorfer, A. W. England, M. C. Dobson, and F. T. Ulaby Radiation Laboratory Department of Electrical Engineering and Computer Science The University of Michigan Ann Arbor, MI 48109-2122

ABSTRACT Nimbus 7 SMMR data are used to map daily freeze/thaw patterns in the upper Midwest for the Fall of 1984. The combination of a low 37 GHz radiobrightness and a negative 10.7, 18, and aTb 37 GHz spectral gradient, —, appears to be an effective discriminant for classifying soil as frozen or thawed. The 37 GHz emissivity is less sensitive to soil moisture than are the lower frequency emissivities so that the 37 GHz radiobrightness appears to track soil surface temperature relatively well. The negative gradient for frozen ground is a consequence of volume scatter darkening at shorter microwave wavelengths. This shorter wavelength darkening is not seen in thawed moist soils. INTRODUCTION Soil moisture contributes to the energy exchange between the air and ground through latent heats of fusion and vaporization. Whether as boundary conditions for mesoscale climate modelling, or as inputs to an agricultural productivity model, the amount and state of soil moisture are regional parameters that one would like to estimate through satellite remote sensing. There is a large body of literature that addresses the estimation of soil moisture from remotely sensed radiobrightness (e.g. Burke et al., 1979; Wang et al., 1982; Blanchard and Chang, 1983; Schmugge, 1983; Jackson et al., 1984; Camillo and Schmugge, 1984; Schmugge et al., 1986; and Grody, 1988). We present evidence that moisture state can also be inferred from radiobrightness. Freezing influences the measured radiobrightness temperature of the ground, Tb, through parameters in the approximation (Ulaby et al., 1981), Tb = e To + (l-e) Tsky, where e and To are the emissivity and surface temperature of the ground, respectively, and Tsky is the effective sky brightness. In this approximation, atmospheric transmissivity is ignored. Frozen ground exhibits signatures of (1) lower thermal temperatures, To, (2) higher emissivity, e, and, as we will demonstrate, (3) a decrease in brightness temperatures with microwave frequency, aTb -<0. af Signatures (1) and (2) are frequently ambiguous indicators of frozen ground because the changes in radiobrightness that result from freezing may be either positive or negative, depending upon the soil moisture content. Water molecules in frozen plants and soils are not free to align themselves with microwave electric fields. This constraint upon the rotational freedom of water gives rise to an apparent dryness to microwaves. The consequence is a decrease in the real part of the dielectric constant, e', and an increase in frozen soil emissivity. For example, the real part of dielectric constants, E', and corresponding emissivities at nadir, e(0), of two, homogeneous, smooth surfaced, 15% moist soils at 10 GHz are (e' from Hoekstra and Delaney, 1974):

+ 50 C - 50 Material a, Tlb i aL Tb Goodrich Clay 8.2 0.77 221 4.9 0.86 235 Fairbanks Silt 9.6 0.74 214 4.1 0.89 242 Because of increasing emissivity with frequency, a 100 decrease in the clay and silt soil temperatures, from +50 C to -50 C, would cause an increase in Tb of approximately +14 K and +28 K, respectively. The positive direction of change in Tb with soil freezing will cause confusion in discrimination between moist soils which will appear radiometrically warmer when frozen, and dry soils which undergo little molecular change and will appear radiometrically colder. The shift in emissivity with freezing is most pronounced at the lower microwave frequencies. At 37 GHz, the effect is reduced but not absent. We observe that the 37 GHz radiobrightness correlates relatively well with air temperature (Figure 1). Since soil surface temperature should follow the air temperature, the 37 GHz radiobrightness can be expected to provide a reasonably reliable estimate of soil surface temperature. However, discrimination based only on the 37 GHz radiobrightness would misclassify too often. Our data suggest a third signature of frozen soil. Freezing reduces the imaginary part of the dielectric constant, e", proportionally more than it does the real part, e'. The loss tangent, tan6=e"/e', is a measure of the attenuation per microwave wavelength. Reduced loss tangent, or lower attenuation, means that thermally emitted photons originate deeper within emitting media. That is, the effective depth of emission, ze, (l-e-1 of the emission originates above ze) becomes a larger fraction of the free-space wavelength, Xo (England, 1974, 1975, 1976, and 1977). For example, Goodrich Clay and Fairbanks Silt exhibit an increase of ze with freezing (dielectric data from Hoekstra and Delaney,1974), + 50C - 50 C Material ~e' e tan5 z Goodrich Clay 8.2 3.5 0.43 0.13,o 4.9 1.0 0.20 0.36 3 FairbanksSilt 9.6 5.0 0.52 0.10 o 4.1 0.02 0.005 15.7 Xo The effective emission depth of moist soils is typically 10% of the free-space wavelength. Frozen soils have effective emission depths that may be 30% or more of free-space wavelength. The effective emission depth of frozen sandy soils, like the Fairbanks Silt, can be several wavelengths. In the more transparent emitting media, particularly in frozen sandy soil or dry snow, the greater average thermal photon path lengths have two effects: a greater likelihood that thermal gradients affect spectral gradients, and a greater opportunity for volume scattering of photons. Thermally induced spectral gradients occur because longer wavelength photons tend to originate below the optical surface where thermal temperatures may differ by several degrees from surface temperatures. For the lower loss tangents of frozen soil, this difference in average emitting depth is enough to reflect near surface thermal gradients caused by diurnal heating. That is, a 2

positive thermal gradient, 3To/az, where z is depth in the soil, will yield a negative spectral gradient, aTb/af, where f is microwave frequency. SMMR data are collected at midnight and noon. In the absence of changing weather conditions, midnight thermal gradients will be positive and noon thermal gradients will be negative (Figure 2) so that midnight spectral gradients will be negative, and noon spectral gradients will be positive. An average +0.2 Kelvin/f(GHz) shift in the spectral gradient is observed between midnight and noon for SMMR radiometric brightnesses (Figure 3). We are developing a computer model of these gradient effects. For now, thermally induced spectral gradients are noise to be filtered out. The second consequence of soil freezing is a greater opportunity for volume scattering -- particularly at shorter microwave wavelengths. This occurs because of the greater average photon path lengths in frozen soil, and because plants and soil appear increasingly heterogeneous at shorter wavelengths. This "law of darkening" means that, for an isothermal, volume scattering halfspace, dTb <0 af (England, 1974). Frozen terrain may also be snow covered. Dry snow is exceedingly transparent to microwaves so that snow exhibits significant of darkening (Figure 4, Edgerton et al., 1971). That is, both frozen soil and snow tend to exhibit negative spectral gradients. While neither a low 37 GHz radiobrightness nor a negative spectral gradient is solely adequate as a classifier of frozen soils, particularly at the relatively coarse resolutions of the Nimbus-7 SMMR, a discriminant based upon a combination of these signatures offers considerable promise. Our objective under NASA Interdisciplinary Research Program Grant NAG5-852 has been to determine whether such a discriminant is feasible. RADIOBRIGHTNESS AND GROUND TEMPERATURES, AND THE CLASSIFICATION DISCRIMINANT Nimbus 7 SMMR (Scanning Multichannel Microwave Radiometer) radiobrightness data at 6.6 GHz, 10.7 GHz, 18 GHz, and 37 GHz were obtained for August 1, 1984, through December 31, 1984, over an area that included North Dakota, about half of each neighboring state, and part of southern Canada (Figures 7-9). We chose this large, relatively uniform area because of the low spatial resolution of the SMMR instruments -- 150 Km at 6.6 GHz, 100 Km at 10.7 GHz, 60 Km at 18 GHz, and 30 Km at 37 GHz, and because of the importance of soil moisture state to this region's hydrologic processes. The data arrived from the National Space Science Data Center (NSSDC) on 21, high density, SMMR Cell Tapes. Such data are referenced to latitude and longitude in a satellite-centered coordinate system. We produced two types of image products: Single-band, radiobrightness images at the intrinsic resolution of each sensor, and (2) composite, multi-band images at a common resolution based upon local area averaging. Each radiobrightness pixel was referenced to latitude-longitude in a Mercator projection by interpolation and resampling the Cell Tape data. We used a bi-cubic approximation of a sine function (Moik, 1980) for the interpolation. H and V radiobrightnesses were averaged to produce a single brightness for each pixel for each frequency. In addition to large area images, local area spatial averages of radiobrightness were calculated for each radiobrightness channel at 7 meteorologic sites within our test region —Miles 3

City, MT; Bismark, Fargo, and Williston, ND; and Abileen, Huron, and Rapid City, SD. A local area is defined as a 150 Km cell centered on the meteorological site (150 Km is the spatial resolution of the 6.6 GHz channel). Air and ground temperature data for the Fall of 1984 were obtained from NOAA's National Climatic Data Center in Asheville, North Carolina. Air temperature measurements were available for noon and midnight at the meteorologic sites (i.e., simultaneously with the satellite pass), but ground temperature measurements were for 7:00 a.m. and 7:00 p.m. EST, and were not co-located with the meteorologic sites. Ground temperatures are measured at 5 cm depths. Diurnal heating will weakly affect 5 cm temperatures so that there will be some differences for the times of the satellite pass. Local area averages at the meteorologic sites were used to define the preliminary boundaries in our Freeze Indicator discriminant. For example, Figure 1 illustrated the correlation between 37 GHz radiobrightness and reported air temperature. The nominal line in these figures is a single best fit linear regression in the least squares sense of all local area averages. Individual linear fits will differ slightly as shown in Figure l(a). We used the nominal line in our discriminant for simplicity, but a more sophisticated discriminant might use the actual least squares fit for the local area and for the time of day. The discriminant boundaries in Figures l(b) and l(c) are merely estimates based upon the nominal regression and a compromise between midnight and noon air temperatures that would imply frozen soil (the lower boundary) and thawed soil (the upper boundary). Remember that diurnal temperature gradients will generally cause midnight, subsurface soil temperatures to be warmer than air temperatures, and noon, sub-surface soil temperatures to be colder. Similarly, local area averages of spectral gradient versus air temperature were the bases for the spectral gradient decision boundaries shown in Figure 3. Note that the midnight freeze boundary in this example is relatively unambiguous, while a more effective noon freeze boundary would be shifted upwards by 0.2 K/GHz. Again, for simplicity in this preliminary study, we used discriminant boundaries that were time and location independent. Our 2-parameter Freeze Indicator incorporates the single-band, 37 GHz radiobrightness, and a spectral gradient based upon linear regression of 10.7, 18, and 37 GHz radiobrightnesses for each pixel. Based upon the decision boundaries in Figure l(b) and l(c), the likelihood of frozen ground in a 37 GHz pixel, p37, is estimated as 0 Tb(37) > Tbma Tbmax - Tb(37) P37 < BaxTmin Tbmn < Tb(37) < Tbmax = Tbmax - Tbmin 1 Tb(37) < Tbmin where Tb (37) is the measured 37 GHz radiobrightness, and the preliminary decision boundaries are Tbmax = 259 K Tbmin = 247 K The likelihood of frozen ground based upon spectral gradient decision boundaries in Figure 3(a) and 3(b) is Psg, and is estimated as 4

o0 > max Tbh aTb Ps =max af aT aTb yTb psg < __) (^b.( min f max max min af af where tpem rdmin where the preliminary decision boundaries are ) =0.3 K/GHz max (^ = -0.3 K/GHz min These boundaries are preliminary in that they were chosen to yield the fewest misclassifications in plots of the type shown in Figure 5(a) and 5(b). More refined discriminants would incorporate area and time specific decision boundaries. This would be relatively straightforward if there were a higher density of weather stations in the test area. As it is, we believe that diurnal temperature modeling well yield effective time dependent boundaries, and, perhaps, requiring sub-region consistency within a classification will yield improved spatially dependent boundaries. These refinements are to be part of our continuing project. However, it is the basic sparseness and lack of control of air and ground data that should prompt some caution about over-interpreting our results. Our freeze/thaw discriminant, or Freeze Indicator, is the product of p37 and Psg' and is applied at the scale of the 10.7 GHz data. Resolution differences between different frequency channels can produce anomalous composite image results if the data were processed directly at their original scale. To avoid these problems, the resolution of the data from each channel is compensated to the (coarse) resolution of the lowest frequency channel used in estimating spectral gradients (i.e. 10.7 GHz and 100 Km resolution). Under certain constraints upon the classification process, these images can be referenced to the higher resolution, 37 GHz format for better location of freeze/thaw boundaries (Zuerndorfer, et al., 1989). The effort needed to do this would be justified as a part of an improved classification process. Figures 7 through 12 include images of the Freeze Indicator for various times during the test period. Black in these images indicates a high likelihood of frozen ground. 5

OBSERVATIONS Figure 6(a) and 6(b) show normalized brightness temperatures for midnight and noon, respectively, in the northern Great Plains during the Fall of 1984. Normalized brightnesses are the average regional brightness at each microwave frequency divided by the average regional air temperature. Normalized brightness thus has the dimension of emissivity. Note that there is little systematic ordering among the 10.7, 18, and 37 GHz normalized brightnesses during August through most of November. However, during the latter half of November through December, the normalized brightnesses at midnight are uniformly ordered, 10.7 GHz brightnesses are high, 18 GHz brightnesses are middle, and 37 GHz brightnesses are low. That is, they exhibit negative average spectral gradients. The noon normalized brightnesses for December exhibit a similar trend, but with exceptions. These are, we believe, illustrations of the law of darkening for frozen soils. Soils at midnight in December for the northern Great Plains are very likely to be frozen. Performance of the freeze/thaw discriminant is demonstrated in Figures 7-9 where Freeze Indicator (FI) images are compared with ground and air temperature measurements for midnight on 9/20/84, 10/24/84, and 12/9/84. Midnight Fl images are shown as better examples of the potential of a freeze discriminant. Noon FI images are generally less consistent with meteorologic reports because of the contribution of the noontime positive diurnal spectral gradient to the negative frozen ground spectral gradient that we discussed in the last section. Areas not covered by the satellite in a particular pass are shown in white. Tables 1-3 are summaries of the meteorologic reports. On the night of September 20 (Fig. 7), air temperatures throughout the region were near 60~ F and had been above freezing for several days. The FI image shows weak, probably false indications of freezing-in the prairies of ND, southern Canada, and the rolling glacial terrain east of the Red River Valley in Minnesota. While the dry air of the northern prairies permits nighttime radiation cooling of the ground to temperatures below that of the air, the more likely explanation for the weak freeze indication is short wavelength scattering by the tall prairie grasses in the northern great plains, and by woodland areas in Minnesota. However, there are no strong indications of freezing in the FI image. On the night of October 24 (Fig. 8), air temperatures hovered about freezing throughout the area, but had been below freezing at Williston for several days, and generally above freezing toward the east (see the temperatures for Fargo, Aberdeen, and Huron in Table 2). The Fl image shows a strong freeze indication in northwestern ND which is consistent with the temperature patterns. Similarly, the definite thaw indication along the Red River Valley is consistent with the warmer temperatures reported and the generally more moist soil in the Valley. On the night of December 9 (Fig. 9), air temperatures were generally below freezing except at Rapid City, SD, and had been below freezing for several days. There was no more than trace snow on the ground anywhere in the region. The FI image shows strong freeze indications throughout most of the region with weaker indications near Rapid City, and in the Aberdeen-Fargo sub-region (Aberdeen is not shown on the December 9 map because its temperature report was missing for that date).- Again, the FI image is consistent with the temperature record. 37 GHz radiobrightness and FI image sequences were produced at midnight and noon for six-day periods in September, October, and December (Figures 10-12). SMMR coverage is based on a 48 hour cycle —midnight (0000 local hours on the date shown), noon (1200 hours on the same date), and then midnight again 36 hours later. However, orbit precession causes gaps in the cycle and variations in the coverage footprint. Within these constraints, our objective was to observe, if possible, weather dynamics reflected in the FI images. 6

The 37 GHz sequence beginning on September 16 (Figure 10 and Table 1) shows the moist area associated with the Missouri River, Sakakawea and Devils Lakes in ND, and the Missouri River and Lake Oahe in SD. Rain during the night of September 21 appears as a regional darkening of the 37 GHz image for midnight on the 22nd. Note that the rain is not picked up in the Fl image. The October sequence (Figure 11) is dominated by a cold front passing through the area from the northwest with rain and snow beginning on October 19. The region is warmer and drier by the 26th. The moisture pattern dominates the 37 GHz image, but only the apparent freeze pattern, which generally lags the cold front, is shown in the FI image. Note that strong freeze indications follow the cold front but weaken in the south with warming on the 26th. The December sequence (Figure 12) is characterized by cold temperatures and snow from December 2 through December 5, followed by daytime warming into the 40's (and even 580 at Rapid City, SD) by the 9th. The FI images reflect this general coldness, but also show daytime thawing toward the end of the period. CONCLUSIONS Freeze Indicator images based upon a preliminary, 2-parameter discriminant —37 GHz radiobrightness and 10.7, 18, and 37 GHz spectral gradient —show relatively good correlation with the expected state of moisture in northern Great Plains soils during the Fall of 1984. The discriminant is preliminary in the sense that both theoretical and experimental work needs to be done to fully exploit the diurnal radiobrightness signatures of frozen soils. The concept underlying the preliminary discriminant is that frozen soil will exhibit volume scatter darkening at shorter microwave wavelengths much like the effect observed in dry snow. Few other phenomena cause negative microwave spectral gradients. However, one such phenomenon is diurnal insolation which should cause negative spectral gradients at midnight, but positive spectral gradients at noon. We are in the process of tayloring our discriminant to allow for these diurnal gradients. Freeze Indicator images based upon SMMR data effectively map temporal variations in the freeze/thaw pattern for the northern Great Plains at the time scale of days. These patterns are synchronized with weather patterns, but are not identical. We intend to expand our test data set to include several complete seasons. The product would be, in essence, a movie of freeze/thaw patterns as weather fronts sweep through the Great Plains throughout several seasons. The development of these data from SMMR archives should provide one aspect of a meso-scale climatic baseline for the region. 7

REFERENCES Blanchard, B.J., and A.T.C. Chang, 1983, Estimation of soil moisture from Seasat SAR data, Water Res. Bull. 19, p. 803-810. Burke, W.J., T. Schmugge, and J.F. Paris, 1979, Comparison of 2.8- and 21-cm microwave radiometer observations over soils with emission model calculations, JGR 84, p. 287-294. Camillo, P.J., and T.J. Schmugge, 1984, Correlating rainfall with remotely sensed microwave radiation using physically based models, IEEE Trans. on Geosc. and Rem. Sens. GE-22, p. 415-423. Edgerton, A.T., A. Stogryn, and G. Poe, 1971, Microwave Radiometric Investigations of Snowpacks, Final Rept. 1285R-4 of Contract 14-08-001-11828 between Aerojet-General Corp., El Monte, CA, and the U.S. Geological Survey. England, A.W., 1974, The effect upon microwave emissivity of volume scattering in snow, in ice, and in frozen soil, Proc. URSI Spec Mtg on Microwave Scattering and Emission from the Earth, Berne, Switzerland, 23-26 Sept., 1974. England, A.W., 1975, Thermal microwave emission from a scattering layer, JGR 80, p. 44844496. England, A.W., 1976, Relative influence upon microwave emissivity of fine-scale stratigraphy, internal scattering, and dielectric properties, Pageoph 114, p. 287-299. England, A.W., 1977, Microwave brightness spectra of layered media, Geophvsics 42, p. 514521. Grody, N.C., 1988, Surface identification using satellite microwave radiometers, IEEE Transactions on Geoscience and Remote Sensing, V. 26, p. 850-859. Hoekstra, P., and A. Delaney, 1974, Dielectric properties of soils at UHF and microwave frequencies, JGR 79, pp. 1699-1708. Moik, J., 1980, Digital Processing of Remotely Sensed Images, NASA SP-431. Schmugge, T.J., 1983, Remote sensing of soil moisture: Recent advances, IEEE Trans. on Geosc. and Rem. Sens. GE-21, p. 336-344. Schmugge, T.J., 1987, Remote sensing applications in hydrology, Rev. Geophys. 25, p. 148152. Schmugge, T.J., P.E. O'Neill, and J.R. Wang, 1986, Passive microwave soil moisture research, IEEE Trans. on Geosc. and Rem. Sens. GE-24, p. 12-22. Ulaby, F.T., R.K. Moore, and A.K. Fung, 1981, Microwave Remote Sensing. Active and Passive, Addison-Wesley, p. 186-255. Wang, J.R., T.J. Schmugge, W.I. Gould, W.S. Glazar, and J.E. Fuchs, 1982, A multifrequency radiometric measurement of soil moisture content over bare and vegetated fields, Geophvs. Res. Let. 9, p. 416-419. 8

Watson, K., L.C. Rowan, and T.W. Offield, 1983, Application of thermal modeling in the geologic interpretation of IR images, Remote Sensing, SEG Reprint Series, no. 3, p. 345-369. Zuerndorfer, B., A.W. England, and G.H. Wakefield, 1989, The radiobrightness of freezing terrain, Proc. of IGARSS'89, Vancouver, B.C., July 10-14, in press. 9

TABLE 1 Cloud Snow Air Temp Cover Pack ^YS L~F) l. ~101 Precio. in) (in.a.1984 QQ. 2 Q 12 Q 12 2 Bamz 24Ah 9/15 Aberdeen 40 63 2 3 0 0 0 0 Bismark 39 64 0 0 0 0 0 0 Fargo 46 63 8 0 0 0 T 0 Huron 44 64 0 3 0 0 0 0 Miles City 50 65 2 2 0 0 0 0 Rapid City 44 63 4 2 0 0 0 0 Williston 48 60 6 0 0 0 0 0 9/16 Aberdeen 46 65 1 3 0 0 0 0 Bismark 53 67 0 2 0 0 0 0 Fargo 47 69 0 0 0 0 0 0 Huron 50 64 5 10 0 0 0 0 Miles City 58 74 0 3 0 0 Rapid City 52 70 3 6 0 0 0 0 Williston 55 68 0 3 0 0 0 0 9/17 Aberdeen 57 71 2 8 0 0 0 0 Bismark 56 83 0 1 0 0 0 0 Fargo 57 73 0 3 0 0 0 Huron 58 73 10 3 0 0 0 0 Miles City 61 80 0 0 0 0 Rapid City 64 84 0 0 0 0 0 0 Williston 56 74 0 3 0 0 0 0 9/18 Aberdeen 64 81 0 0 0 0 0 0 Bismark 57 80 0 0 0 0 0 0 Fargo 64 81 00 0 0 0 0 Huron 62 84 0 0 0 0 0 0 Miles City 60 80 0 0 - - 0 Rapid City 63 90 0 0 0 0 0 0 Williston 67 73 0 0 0 0 0 0 9/19 Aberdeen 59 84 0 1 0 0 0 0 Bismark 55 86 0 0 0 0 0 0 Fargo 65 90 0 0 0 0 0 0 Huron 68 90 0 0 0 0 0 0 Miles City 67 76 0 0 0 0 Rapid City 60 90 0 0 0 0 Williston 57 72 0 0 0 0 0 0 9/20 Aberdeen 62 65 0 9 0 0 0 0 Bismark 58 63 0 9 0 0 0 0 Fargo 54 62 0 6 0 0 0 0 Huron 62 71 0 2 0 0 0 0 Miles City 63 72 7 2 T 0 Rapid City 59 67 2 6 0 0 0 0 Willston 54 51 8 9 0.01 0.15 (R) 0 9/21 Aberdeen 59 82 2 2 0 0 0.1 0 Bismark 58 81 8 7 0 0 0.3 (R) 0 Fargo 59 77 10 9 0 0 0 0 Huron 60 83 0 0 0 0 0 O Miles City 66 56 10 10 0.18 (R) 0 Rapid City 60 83 2 1 0 0 0.07 (R) 0 Williston 57 54 10 10.02.01 0.3 (R) 0 9/22 Aberdeen 61 65 8 2 T 0 T 0 Bismark 52 60 3 1 0 0 0 0 Fargo 72 66 9 0 T 0 0.02 0 Huron 66 66 10 2.05 0.011(R) 0 Miles City 46 52 4 6 - - 0 0 Rapid City 45 60 3 4 0 0 0 0 Williston 45 44 10 9 0 0 0 0

TABLE 2 Cloud Snow Air Temp Cover ^Year (OF-) (x/1o Q Preci, in in. 1984 iQ 12 12 1 i 10/19 Aberdeen 37 44 10 8.01 0 1 (R) 0 Bismark 36 40 10 10 0 0.04(S) 0 Fargo 40 38 10 10.07 0.2.77(R) 0 Huron 38 45 10 7 0.1 0.01 0 Miles City 28 39 7 4 T 0 Rapid City 26 42 2 7 0 T T Williston 27 34 2 10 - -.1 (S) 0 10/20 Aberdeen 36 44 - 0 0 0 T 0 Bismark 35 38 9 10 T.01.1 (R) 0 Fargo 37 42 10 10 T.01.12(R) 0 Huron 35 41 0 10 0 0 T 0 Miles City 31 33 10 10 -.01 (S) 0 Rapid City 26 43 0 0 0 0 T (S) 0 Williston 33 33 10 10.02 0.08 (S) T 10/21 Aberdeen 37 41 - 10 0 0 T 0 Bismark 36 37 10 100 0.03 (S) 0 Fargo 38 39 10 9 0 T.07 (R) 0 Huron 35 44 2 6 0 0 T 0 Miles City 29 32 8 9 T T Rapid City 33 38 10 10 T 0 T (S) T Williston 32 33 10 9.01 0 T (S) T 10/22 Aberdeen 34 41 7 8 0 0 T T Bismark 33 39 10 8 0 0 0 T Fargo 36 38 10 10 T 0 T 0 Huron 31 42 0 3 0 0 0 0 Miles City 25 38 0 7 0- 0 Rapid City 24 43 3 8 0 0 0 0 Williston 27 32 7 10 0 T T (S) 0 10/23 Aberdeen 35 46 10 4 0 0 0 0 Bismark 36 43 10 3 0 0 0 0 Fargo 34 39 10 10 0 0 0 0 Huron 32 46 3 5 0 0 0 Miles City 23 37 1 0 - 0 Rapid City 29 42 0 0 0 0 0 0 Williston 27 40 10 0 0 0 0 0 10/24 Aberdeen 31 54 0 4 0 0 0 0 Bismark 33 48 0 7 0 0 0 0 Fargo 33 49 0 8 0 0 0 Huron 36 57 0 1 0 0 O 0 Miles City 35 44 5 10 - - 0 Rapid City 36 57 0 3 0 0 0 0 Williston 30 42 5 7 0 T 0 10/25 Aberdeen 38 37 10 10 0 0 00 Bismark 34 39 10 10 0 0 0 0 Fargo 38 42 10 10 0 0 0 Huron 43 40 8 10 0 0 Miles City 33 44 10 10 - T 0 Rapid City 34 54 1 1 0 0 0 0 Williston 30 42 10 10 T 0 T 0 10/26 Aberdeen 46 56 0 4 0 0 0 0 Bismark 41 57 3 10 0 0 0 0 Fargo 43 53 0 7 0 0 0 0 Huron 46 61 0 0 0 0 0 0 Miles City 44 52 0 10 - - T 0 Rapid City 53 65 0 5 0 0 0 0 Williston 39 50 6 10 0 0 T (R) 0

TABLE 3 Cloud Snow Air Temp Cover Pack Ye ~ O Pr(~1 (x/10 eia. (inin.) 1984 Sfts oQ 12 00 12 12 24h Rem 24h 12/2 Aberdeen 17 10 10 4 T 0 T (S) 2 Bismark 14' 19 10 10 T T.01(S) 1 Fargo 8 3 10 10 T T T (S) T Huron 19 14 10 1.01 T.07 (S) 6 Miles City 13 11 10 9 -.01 (S) 4 Rapid City 19 19 4 4 0 0 T (S) T Williston 11 8 10 8 T 0 T (S) 1 12/3 Aberdeen 4 17 0 4 0 0 T (S) 2 Bismark 15 15 10 7 T T T (S) 1 Fargo 0 10 0 0 0 T 0.1 T Huron 5 14 0 0 0 0 T(S) 6 Miles City -3 -2 10 1 - 0 4 Rapid City 11 27 0 1 0 0 0 T Williston 9 9 10 8 T T T (S) 1 12/4 Aberdeen 13 10 10 0 0 0 0.02 (S) 2 Bismark 8 15 10 1 T 0 T 1 Fargo 9 15 10 0 0 0 0 T Huron 5 12 6 2 T 0 T (S) 5 Miles City -1 7 0 10 - 0 4 Rapid City 15 24 2 0 0 0 0 T Williston -5 10 0 10 0 0 0 1 12/5 Aberdeen 14 7 3 5 T 0 T (S) 2 Bismark 19 -1 4 0 0 0 T (S) 1 Fargo 11 -1 10 10 0 T (S) T Huron 15 15 4 7 0 T T 4 Miles City 24 13 10 10 - T (S) 3 Rapid City 26 19 8 7 0 T T (S) T Williston 8 -5 3 3 T 0 T (S) 1 12/6 Aberdeen -14 3 0 0 0 0 0 2 Bismark -11 20 0 8 0 0 T 1 Fargo -8 8 0 3 0 0 0 T Huron -6 11 0 4 0 0 0 4 Miles City 9 23 0 9 - 3 Rapid City 6 44 1 1 0 0 T Williston -7 24 0 9 0 0 T 1 12/7 Aberdeen 18 42 7 4 0 0 0 2 Bismark 30 43 9 8 0 0 0 1 Fargo 12 43 7 9 0 0 0 T Huron 19 42 3 0 0 0 0 3 Miles City 35 44 4 0 0 2 Rapid City 43 61 10 0 0 0 0 0 Williston 34 43 3 6 0 0 0 1 12/8 Aberdeen 30 36 2 3 0 0 0 T Bismark 31 42 7 2 0 0 0 T Fargo 30 43 2 5 0 0 0 0 Huron 31 42 0 4 0 0 0 1 Miles City 29 33 3 3 0 1 Rapid City 33 51 0 10 0 0 0 Williston 31 37 7 8 0 0 0 T 12/9 Aberdeen - 46 - 7 0 0 0 T Bismark 31 43 2 3 0 0 0 T Fargo 29 40 0 9 0 0 00 Huron 35 39 7 7 0 0 0 T Miles City 26 35 5 4 - - 1 Rapid City 43 58 6 0 0 0 O 0 Williston 28 37 0 2 0 0 0 0

280 [Nominall 270 Actual j 2 ^ Actuall ~(a) Midnight. A linear 3 y^^ ~' regression of the Bismark data 260 _ _ _ _ is shown ("Actual" curve), wE a rgsi l with a regression line for data 1 250- - - - - - --- — _ _ collected throughout North,'Q, Dakota and the surrounding ) 240 - - --- -_ —-- _ _ region ("Nominal" curve). CO 230 _ _ ~ _ _ _ _ _ - ~~~~_ -_ ~ 290! 280 —-- ~ L v[Nominal] b) Midnight. Shown with 270 ------------ the Bismark data is a linear j!,b i,^0 i regression for data collected 26^ 9 -- ------— ~ - - -iscrimi -nant' throughout North Dakota and. jscri~nant the surrounding region E I ("Nominal" curve). Also a, 250- -~- -_ _ ji lfiscrimnantl shown are brightness tempera I ^G ^Xature decision thresholds for a 1 240 - -'~ -..- - _ frozen or thawed surface..1 1 __i 290 | Q, INominall 280..... 270 _ _ - 2 70 ~, ~(c) Noon. Shown with the ~ ____o! ____ Bismark data is a linear re" 260 - __J lscrminantl gression for data collected 0 c rO _ throughout North Dakota and, 250 - --- scriminant the surrounding region ~~u~ ~ "^~\ ml \^ ^^\ ("Nominal" curve). Also 240 - - -- -I shown are brightness temper~=, y aature decision thresholds for a azc t _ _____.__. — - __~~~ I frozen or thawed surface. 2201 240 250 260 270 280 290 300 310 320 245 255 265 275 285 295 305 315 Ar TeOp (K) Figure 1. 37 GHz SMMR brightness temperature versus measured surface air temperature, Bismark, North Dakota. Data were collected from 8/1/84 to 12/31/84.

l - sun's doclnation _j so:D i uToj: a. C 6 12 8 LOCAL SOLAR TIME (HRS) Figure 2. Diumal surface temperature variation for different seasons at 30~ N (from Watson, et al., 1983). Subsurface temperatures will exhibit a reduced amplitude and a phase lag with respect to the surface temperature. That is, midnight thermal gradients will be negative, and noon thermal gradients will be positive.

0.8 r- 0.61 0.84 —... Y' cm.I U' IDiscriminant - 02a — 4 —--- --------- --- -__ 240 250 260 270 280 290 300 310 320 245 255 265 275 285 295 305 315 Ar Terr (K) 0.2 -- - __ 0.4k0. 8 \. iD scriminantl 0 2 —- -—.- - C i-0.21 ~~ ~ ~~(b) Noon -a<2~ - - c- - -.... — -o.6t 1 i "1-' i l" i' I I t I I I I 240 250 260 270 280 290 300 310 320 245 255 265 275 285 295 305 315 Ari Tep (K) Figure 3. Frequency gradient versus measured surface air temperature, Bismark, North Dakota. thresholds for a frozen or thawed surfac. a., j Iiscriminanti O — —.. -- --- - 4ZL- - - -m -04Data were collected from 8/1/84 to 12731/4. The frequency gradient is the frequency and 10.7 GHz. Shown with the Bismark data are frequency gradient decision

e = 45~ x = Wavelength H = Horizontal Polarization V = Vertical Polarization 300 i6.0 cm, V(Relative Measurement) 290 2.X = 6.0 cm, H (Relative Measurement) 270 a: 260,, 250 240.. I —1'^ =^ 2.2 cm,V 0 230 __ "'" ~... X. = 22cm,H - 220~ l o'ft, C s~~X 2t~~~~t____ _os 0.8 1 cm, V r 2-0 2 6 -.-=- -- -4 80 2 4 6 8 MASS PER UNIT AREA (gm cm2) Figure 4. Brightness mperahot wavelength darkening evident fo 1970 (Edgerton e al., 197sopacs). Note e s snowpacks.

slpaxd paxiuu awT s pue'pa, eqlp ae saxoq uado'uazog ae saxoq pllos oap'iln puno.g uodn pasrg'asJJys pO^ 10p o'poxTu'uOzo.J B JOJ sploqsanJl uo0lso3p guuaisnpl ae Mp 3l mst aI qlm UAoq' S -'18/! 1 UA m8/1/8 wao P103llo 0;as m 0ea lo' Cea LoN'lmeus Sa' illmaduai sssotlU uq ZHD Li AIS nsaA luatpe 2,^ uanbad S Jns!l 0 8Z SZ 599 Z S 5>Z SZ sZ! 06Z OZ OLZ 09% O0- Or (XZ OZ~ i i! I I I I _ I i.i I 9'o"U u r IWU...... I. | =~~~~~- - --:o CC ) ___ —----— 0 II n a lueuludlJ:s!i - ax —.'L_~N )(x -D — ) X CX 190 i i I, Im 1 1, 1 1912~~~~ —l l l ^*______~^lUBUIJOS!Ql_ r' - --.... _ x _ _._........,.,..,-.,,,.... x _ 190

0.98 37 GHz 0.96 A! | ---- 18 GHz E 0.84 - N 0.82' 1 11 11 1 1 1 ^ ~U X l l' Aug Sept Oct \ Nov Dec7 Measurement Day 0.98 317 GHz ~ 0.96 G v ~ —- 18GHz ca 0.928 0.84 _ E 0.9 a) 0.88 N 0.86 E O 0.84 Aug Sept Oct Nov Dec Measurement Day Figure 6. 37 GHz, 18 GHz, and 10.7 GHz SMMR normalized brightness temperatures versus calender day. Measurements were made at irregular intervals from 8/1/84 to 12/31/84. The normalized brightness temperature of a single SMMR frequency channel is the average brightness divided by the average surface air temperature. averages are calculated over North Dakota and the surrounding region. averages are calculated over North Dakota and the surrounding region.

Air and Ground Temp Nignt 9/20/84 ~ ~~ "~ OAIR TEMP Thaw O O Williston O Mixed Freeze ~ Fargo 0 Bismarck Miles GND TEMP City -0 Thaw E Aberdeen 0 0 Mixed 3 O -' Freeze * 0"1 ~ r "iO 0 Huro Rapid City ~ Huron _..: midnight...September20..98...................'".~'"~~~~~~~~~~~~~~~~~' X,.,.,.I::::....:....'....:~-.:...;.:~~~~~~~..'w S,::....:............::.....-.....:::.."...:...........:..'..:....:;..................:::::-.::. ~...........-::.:.. -.......:'?.........-..............::.i...::.........:.:..:...:.'........y~~ ~"..........-......................i...:..::;..... _ _ X........................................... -.~:'ri: iiiii~iii'ii. rril':. i::]:::::l~:i..........: r..........: -..: Fiure 7. A comparison of reported air and ground temperatures with the Freeze Indicator....'or... midnight, September 20, 1984.

Air and Ground Temp Nignt 10/24/84 ~~O \~ |AIR TEMP Thaw O 0 Williston lMixed Freeze * Fargo M Bismarck Miles City GND TEMP _B3 Thaw E Aberdeen O H! Mixed 5 ~* i Freeze * 0 El o O Rapid City Huron,~,, N D....,...... Figure 8. A comparison of reported air and ground temperatures with the Freeze Indicator for ~~~~~~~~~~~~~~~~midnight., Octobr 24,..984....... midnight, October 24, 1984.

Air and Ground Temp Nignt 12/9/,84 ~E3} g- AIR TEMP Thaw O * Williston Mixed 0 Freeze ~ Fargo M Bismarck Miles City GND TEMP l ]|Thaw a * I. Mixed g ~ [* l lFreeze Rapid City Huron Fiure 9. A comparison of reported air and ground temperatures with the Freeze Indicator or midnight, December 9, 1984.

5:::~.~~~~~~~~~~~~~~~~~~~~~;';:~:~:~:~:;;i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~,CX.:~:~:~:~:~:~: ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~''. —::::;;,<:,,.;,:,,!'''I I ~~~~~~~~~~~~"(9/'1 6/84)"::":::':: i?":I AN047S5 - ~~~~~~~~~~~~~~~AN047FG - ~~~~~(9/1 8/84)-...........i~' ~-: ~: ~ ~.....::i::': ~ ~ ~ ~ ~ ~ ~ ~~~~~ ~........ (9/20184)+~~~~(92084 AN047 S5 AN047 FG AN053FG AN051S5 (9/22/84)''"' i:!:!'i.;'i~~~~~~~~~~~~~i:;:;:::::: ~ ~ ~ ~ ~ ~ ~~::::a::::::::i::g:::::cI:!:i::::::::;~.~~~~~~~~~~~~~~~~~~~~~~":!!i~ii:: ";:I: I' ii:!!'ii':ii ":;:i:":'~,iii:.~.-..~....: ~...~.t...~~sz:i:::2:::: ~:..~.~.~.~.~.........; ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~ ~~.;::::::.:...,,:i::~:I:~~~~:~x;.:., ~::i~~~~~~~~~~:i~~~~~~f:~~~~ ~ ~ ~...;:.!.!.'!!:. ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~!..'~. -.:::~~. "~'.''~. -- -::~:.......~:~: 5~~~~~~....:;::5i::iiiii::iiiii:;~.i~':-~:::::::~., i~.!'"'''..~:.!'.~i~i~:~ AN047S5 ~~~~~~~~~~~~~~~~~AN053FG sure O~a)Midnight 37 OHz and Freeze Indicator image sequences for a 6dapeidn September.

",1...0 ~,. (9/1 6/84) AD047S5 AD047FG (9/18/84) AD049S5., AD049FG (9/20/84) ^-..^ —~-'~~~~~~~~~~~~......................................'...... ADO54FG (9122184) AD053S5 A 5 Figure 10(b). Noon 37 GHz and Freeze Indicator image sequences for a 6 day period in September.

10/20ro/84) 84)l:. o:.~r b y 1 1 ~ 1 i ]a~; ~~~~~~~"~~~~~~~~~~~~z I I I ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~iiii.? AN081 S5 AN081 FG (10/22/84) ~;~~~~s~.... ~~i~:::::::,,. ~. 10)24/84) ANC87FG AN087S5 Figure I11(a). Midnight 37 GHz and Freeze Indicator image sequences for a 6 day period In October. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-'.. ~~~~~~~~~~~~~~~~~~~~~~i~~~~~~~~~si:~~~~~~~~~~~~~~~~~~~~~s;:~;~~~~~~~~~~~~~~~~~~~~~~;: ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i.:...:::. ~. hii~iii:,i~iii~'m ANC87FG AN087S5 Figure l l~~~~~~(a0228). iih 7GzadFez niao mg eune o a eid~ October.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

AD081 S5 AD081FG.''<................ AD083S5 AD083FG (10/24/84).... B id10/26/84) I I;i..:'..:i1 11 11........ Px:::::~':. I ADC37s5 Aoc8FGX Figure I Ib). Noon 37 GHz and Frceze Indicator image sequences for a 6 day penod in

(12/3/84) AN 125S5 AN125FG (12/5 /84) 1 2 27S5 AN127FG (12/7/84) AN129S5 AN12FG (12/9/84) ss Are 31 FG Figure 12(a). Midnight 37 GHz and Freeze Indicator image sequences for a 6 day period in December.

J/ UMI HAUbHlIUMlNtbb NOON hHtEZE INDICATOR..(1,2/3/84 _ (12/5/84) DD127S5 AD127FG (12/7/84) ED ^~~~~0~~~1 293S~'5 AD1 29FG kD D 29S5 (12/9/84) AD-3-FG A0131 S5 Figurel2(b). Noon 37 GHz and Freeze Indicator image sequences for a 6 day period in December. The jagged white lines in the Noon, 12/5/84,image are caused by missing data.

RADIOBRIGHTNESS OF PERIODICALLY HEATED, TWO-PHASE MEDIA A. W. England Radiation Laboratory Department of Electrical Engineering and Computer Science The University of Michigan Ann Arbor, Ml 48109 (313)-763-5534 December 19, 1988 Abstract Soils that contain liguids or gases that freeze during diurnal insolation will appear radiometrically distinctive. That is, they will appear to have anomalously high thermal inertias caused by the latent heats of fusion or sublimation. The effect should be observable in the diurnal variation of the radiobrightness of freezing moist soils, or in the radiobrightness of Mars soils that are saturated with CO2 ice. The 1-dimensional, heat flow equation for moist soils, E(T) a K(T) aT ot o z az is non-linear because both the enthalpy, E(T), and the thermal conductivity, K(T), are non-linear functions of T at freeze/thaw phase boundaries. Furthermore, diurnal insolation may cause phase boundaries at more than one depth, z. The problem is particularly difficult because these phase boundaries propagate and, occasionally, cancel themselves, and because soils that contain clay freeze over a range of temperatures rather than at 00 C —that is, they possess diffuse phase boundaries.

2 The problem of periodic heating of two-phase media has come to be known as Stefan's problem. There are several numerical techniques for its solution. The Chernous'ko method was most readily modified for diffuse phase boundaries, and was developed as a modeling tool for examining the radiobrightness of diurnally heated soils. These models exhibit diurnal radiobrightness spectral gradients similar to those computed from the 10.7, 18, and 37 GHz radiobrightness temperatures from the Scanning Multichannel Microwave Radiometer (SMMR) on Nimbus-7 (reported in a separate abstract, England, et al), and may explain the anomalously flat radiobrightness spectrum of Mars.

The Radiobrightness Measurement of Apparent Thermal Inertia A. W. England Radiation Laboratory Department of Electrical Engineering and Computer Science The University to Michigan Ann Arbor, MI 48109 (313)-936-1340 Abstract Thermal inertia is a measure of a material's resistance to change in temperature. If a material's thermal conductivity, K, and its volumetric heat capacity, pc, are independent of temperature, then its thermal inertia, P, is (Kpc)1/2. Thermal inertia, as a geologic mapping tool, has been estimated from the differences in pre-dawn and afternoon thermal infrared (TIR) brightnesses. Materials that have high thermal inertias exhibit low day-night, TIR brightness differences. Discrimination among geologic materials based upon thermal inertia has been demonstrated both by aircraft experiments, and by the Heat Capacity Mapping Mission (HCMM) satellite experiment. In principle, thermal inertia can as easily be estimated from day-night differences in microwave radiobrightness temperatures. The advent of the Special Sensor Microwave/Imager (SSM/I) class of satellite instruments, with their 6:00 am and 6:00 pm daily coverage over most of the Earth, invites thermal inertia mapping. However, the spatial scales of appropriate microwave targets are very different from the spatial scales of HCMM targets. HCMM had a spatial resolution of 600 m, and SSM/I resolutions vary from 15 km at 85.5 GHz to 69 km at 19.35 GHz. For the coarse resolutions of the SSM/I instrument, appropriate thermal inertia targets would have to be regionally extensive such as soil moisture or snow wetness. To test thermal inertia's sensitivity to soil moisture, thermal models of diurnally heated, prairie soils containing 10%, 15% and 20% by weight moisture were developed for a typical September through December period near Bismarck, North Dakota. Thermal profiles from these models were incorporated in radiobrightness models for each of the SSM/I frequencies to produce expected diurnal radiobrightnesses and spectral gradients for the study period. This combination of models has also been used to explain the day-night shift in the spectral gradient of radiobrightness that is observed in Scanning Multichannel Microwave Radiometer (SMMR) data, and to guide development of an algorithm for classifying frozen soils. Prior to the onset of night-time freezing, the day-night radiobrightness differences show only weak sensitivity to moisture content. However, once diurnal freezing and thawing begins, the latent heat of fusion of moisture in soil greatly enhances the soil's apparent thermal inertia, and this enhancement is strongly dependent upon the quantity of available moisture. This enhanced sensitivity of apparent thermal inertia to moisture content is most evident in the October and November profiles. The full effects of freezing and thawing are not realized during September because soils are only partially frozen during a clear, September night, nor are they realized in December because soil surfaces are only partially thawed during a typical December day. Furthermore, the models assume a snow-free surface and, thus, are less appropriate for late fall in North Dakota.

AN OPTIMIZED APPROACH TO MAPPING FREEZING TERRAIN WITH SMMR DATA B. Zuerndorfer, A. W. England, F. T. Ulaby Radiation Laboratory Department of Electrical Engineering and Computer Science The University of Michigan Ann Arbor, MI 48109 (313)-763-5534 November 17, 1989 Abstract Soil moisture contributes to the energy exchange between the air and the ground through latent heats of fusion and vaporization. Consequently, the processes of thawing frozen ground or of evaporating soil moisture cause soil thermal inertias to appear anomalously high. There is a large body of literature about deriving soil moisture from radiobrightness. Moisture state can also be inferred from radiobrightness [1]. Frozen soil classification is based upon a combination of 37 GHz radiobrightness and spectral gradient, aTb(Cf)Af, where Tb(f) is the radiobrightness at frequency f. Frozen soils appear cold at 37 GHz, and exhibit a negative spectral gradient that is largely caused by volume scatter darkening at the shorter wavelengths. This two parameter "freeze indicator" has been applied to data from the Scanning Multichannel Microwave Radiometer (SMMR) on Nimbus-7. For these data, the spectral gradient is a linear, least-square fit to the 10.7, 18, and 37 GHz radiobrightnesses. Conceptually, a surface is classified as frozen only if both the 37 GHz radiobrightness and the spectral gradient are sufficiently low. A freeze map is generated by displaying the freeze indicator for each pixel location. However, data processing is complicated by the very different spatial resolutions of the different SMMR frequency channels. Resolution compensation (equalization) must be performed prior to classification so that spatial averaging is similar at all frequencies. Assuming that no a priori surface information is available, common practice for resolution compensation is to degrade the high frequency (fine resolution) data to the resolution of the low frequency (coarse resolution) data. As a result, fine resolution information is lost.

2 Under certain constraints, fine resolution information can be recovered in the location estimate of the freeze/thaw boundaries [2]. If the constraints are met, the coarse resolution freeze/thaw boundaries can be registered to fine resolution, 37 GHz boundaries (i.e., to 37 GHz radiobrightness threshold crossings). These 37 GHz boundaries become better estimates of freeze/thaw boundary locations than those at coarse resolution. In this paper, we show that boundary registration can be optimized through clustering. Specifically, 37 GHz radiobrightnesses and spectral gradients from SMMR measurements are grouped into frozen and thawed clusters for an area that includes North Dakota, about half of each neighboring state, and part of Canada for the Fall of 1984. From the intersection of the clusters, fine resolution 37 GHz radiobrightness boundaries are defined that register with the freeze map boundaries. In addition, fine resolution, 37 GHz radiobrightness boundary widths (regions of boundary uncertainty) are also defined. [1] Zuerndorfer, B. W., England, A. W., Dobson, C. M., and Ulaby, F. T. (1989). "Mapping freeze/thaw boundaries with SMMR data," J. Agriculture and Forest Meteorology, under review. [2] Zuerndorfer, B. W., England, A. W., and Wakefield, G. H. (1989). "The radiobrightness of freezing terrain," 1989 IEEE Int. Geosci. and Remote Sensing Symp., Vancouver, Canada.

THlE RADIOBRIGHITNESS OF FREEZIN( TERRAIN B. Zuerndorfer, A. W. England, and G. H. Wakefield Radiation Laboratory Department of Electrical Engineering and Computer Science The University of Michigan Ann Arbor, MI 48109 ABSTRACT observed at fine resolution. In the sections that follow, we review The combination of a low 37 GHz radiobrightness and a the Freeze Indicator work of Zuerndorfer et. al. [1989], and discuss negative 10.7, 18, and 37 GHz spectral gradient appears to be an the use of Gaussian filtering for resolution compensation and its apeffective discriminant for classifying frozen ground. The spatial plication to freeze/thaw boundary mappings. resolution of lower frequency, satellite borne, microwave radiome- FREEE I CA ters is typically relatively coarse, e.g., 100 Km for the 10.7 GHz REEZE INDICATOR channel of SMMR. With certain restrictions, scale-space theory can Freezing influences the measured radiobrightness temperabe used to map freeze/thaw boundaries that are identified at the ture, Tb, f the ground, as observed by a satellite microwave rapoorest resolution of a multiband image to images having the highest te, b, th rou d as observed by a satellite microwave raresolution. That is, the process permits classification at lower res- diometer, through parameters in the approximation [Ulaby et. al., olution, but boundary location at higher resolution. 19811, Tb eTo+(1 - e) Tsky, INTRODUCTION where e and To are the emissivity and surface temperature of the Soil moisture contributes to the energy exchange between the ground, respectively, and Tsky is the effective sky brightness. Atair and the ground through latent heats of fusion and vaporization. miospheric transmissivity is ignored in this approximation. Frozen The processes of thawing frozen ground or of evaporating soil ground exhibits signatures of (1) lower thermal temperatures, To, moisture cause soil thermal inertias to appear anomalously high. (2) higher emissivity, e, and (3) a decrease in brightness temperaMicroscale and mesoscale climate models benefit from estimates of tures with microwave frequency, the amourT and state of soil moisture as part of their boundary conditions. Tb F < 0. There is a large body of literature about deriving soil moisture from radiobrightness [e.g. Burke et. al., 1979; Wang et. al., Signatures (1) and (2) are well understood, but are generally 1982; Blanchard and Chang, 1983; Schmugge, 1983; Jackson et. ambiguous indicators of frozen ground. Ambiguities arise because al., 1984; Camillo and Schmugge, 1984; Schmugge ct. al., 1986; changes in radiobrightness that result from freezing the ground may and Grody, 1988]. In addition, there is strong evidence that mois- be either positive or negative, depending upon the soil moisture ture state can also be inferred from radiobrightness. Using data from content. A typical dry soil emissivity of 0.8 will yield a 80 radiolhe Nimbus-7 Scanning Multichannel Microwave Radiometer the Nimbus-7 Scanning Multichannel Microwave Radiometer brightness decrease for a 100 decrease in thermal temperature (SMMR) for a test area that included North Dakota and parts of the (enough to completely freeze the ground), with relatively little surrounding states and southern Canada, Zuerndorfer et. al. [19891 change in soil emissivity. However, in moist soils, freezing causes showed that a combination of low 37 GHz radiobrightness and a an increase in soil emissivity, and a subsequent increase in radionegative spectral gradient of radiobrightness offers an encouraging brightness. Freeze Indicator, or discriminant, for classifying frozen terrain. Water molecules in frozen plants and soils are not free to A fundamental problem with the Freeze Indicator algorithm align themselves with microwave electric fields. This constraint developed by Zuerndorfer et. al. [1989] is that radiobrightness mea- upon the rotational freedom of water gives rise to an apparent drysurements from different frequency channels having different spatial ness of frozen plants and soils. The consequence is a decrease in the resolutions are required to estimate a spectral gradient. To make each radiobrightness value refer to a common area on the ground, data real part of the dielectric constant, e', and an increase in soil emisfrom each channel is compensated to a common, coarse resolution sivity. For example, the real part of dielectric constants, e', and cor(i.e., the resolution of the lowest frequency SMMR channel that responding emissivities at nadir, e(0), of two, homogeneous, contributes to the gradient estimate). As a result, identified sooth surfaced, 15 moist soils at 10 Hz are (' from Hoekstr freeze/thaw boundaries are localized only to the accuracy of the res- n a 1 olution of the lowest frequency channel that was used. and Delaney [1974): In this paper, we present a technique for using the fine res- + 50 C - 50 C olution in the high frequency channel to improve the localization of Material c' c(0) TI e' e(0) Tb freeze/thaw boundaries. By using Gaussian convolution, or Gaussian filtering, to perform resolution compensation, it becomes pos- Goiricth Clay 8.2 0.77 221 4.9 0.86 235 sible to register boundaries observed at coarse resolution to those Fairbtanks Sill 9.6 0.74 214 4.1 0.89 242

Because of increasing emissivity with freezing, a 100 de- SMMR. However, a 2-parameter, freeze signature comprised of the crease in the clay or silt soil temperatures, from +50 C to -50 C, 37 GHz radiobrightness and the 10.7, 18, and 37 GHz spectral grawould cause an increase in Tb of approximately +14 K or +28 K, dient, offers a promising initial discriminant for the classification of respectively. The positive direction of change in Tb with soil freez- frozen soil. The preliminary decision boundaries that the soil is ing will cause confusion in discrimination between soils that are frozen are: frozen and cold, and soils that are warm and dry. (1) Tb(37 GHz) < 247 K, The shift in emissivity with freezing is most pronounced at - 0.3K the lower microwave frequencies. At 37 GHz, the effect is reduced, (2) 3-frequency, spectral gradients < f(GHz) although not absent. Because the 37 GHz radiobrightness is less dependent upon soil moisture, it does exhibit a stronger correlation OUNARY A AT with air temperature than do the lower frequency radiobrightnesses. U DAY L ALIA This higher correlation also occurs because air and surface tempera- ife 1989], tures tend to agree, while air and subsurface temperatures (i.e., n the Freeze Indicator of Zuerndorfer et. al. [1989, spectral those temperatures that influence longer wavelength radiobrightness) gradients were estimated using linear regressions of SMMR 37 often differ. That is, the 37 GHz radiobrightness offers a more reli- GHz, s1 Glz, and 10.7 GHz radiobrightness measurements. The.able estimate of sub-zero soil surface temperatures than do lower nominal resolutions of these channels are 30 Km, 60 Km, and 100 frequency radiobrightnesses. However, discrimination based only Km., respectively. Without compensating for the resolution differon 37 GHz radiobrightness would misclassify too often. ences between the channels, the spectral gradient estimates can be in error. For example, a non-zero gradient estimate can result from Zuerndorfer et. al. [1989] suggest a third sinature of frozen surfaces that have radiobrightnesses that are spatially variant but are uedoer et. al. suggest a third signature constant with frequency. To avoid anomalous gradient estimates, the soil. Freezing reduces the imaginary part of the dielectric constant, image data were compensated to a common resolution -- that is, the e", proportionally more than it does the real part, e'. The loss tan- (coarse) resolution of the lowest frequency channel used in gradient gent, tan 6 = c/c', is a measure of the attenuation per microwave estimation. (Such resolution compensated data is available with the gent, tan 5 = e"/e', is a measure of the attenuation per microwave; ^^ ^ g ^^ ^ ^ ^ ^ ^ p^^ ^ ^ wavelength. Reduced loss tangent, or lower attenuation, means that SMMR data [NASA, 1978], and was used in the Freeze Indicator thavelength. Remed p losstangents o or lower wattenuaetion medansat results of Zuerndorfer et. al. [1989]). However, at this poorer spathermally emitted photons originate deeper within emitting media. tial resolution, the Freeze Indicator does a relatively poor job of loThat is, the effective depth of emission, ze, (l-e- of the emission cating freeze/thaw boundaries. That is, the fine resolution informaoriginates above ze) becomes a larger fraction of the free-space tion of the 37 GHz channel is lost. wavelength, Xo [England, 1974, 1975, 1976, and 1977], For ex-betteruse fineresolutioninformat ofthe37 ample, Goodrich Clay and Fairbanks Silt exhibit the following in- To better use the fine resolution information of the 37 GHz ample, Goodrich Clay and Fairbanks Silt exhibit the following in- channel, we note that freeze/thaw boundaries exist in images genercrease of ze with freezing (data from Hoekstra and Delaney [1974]): ated from only the higher resolution, 37 GHz data. However, as previously discussed, these boundaries are ambiguous because of classification errors between frozen and dry soils. We would like to + 50 C 50C identify those boundaries in 37 GHz images that correspond to the.tcn.-ian si Z L i~ X boundaries between frozen and thawed surfaces that were classified ____ _____, _____ ____ ___ _____ aLt coarse resolutions. Goodrich Clay 8.2 3.5 0.43 0.13. 1 4.9 1.0 0.20 0.36. 0 Our boundary identification and localization process for Fairbanks Silt 9.6 5.0 0.52 0. 10 41 0.02 0.005 15.7 X SMMR data requires three steps. First, the uncompensated 10.7 0 o, 18 GHz, and 37 GHz SMMR data is read and compensated to....... ~t~~~~~~~~ ~~~* the resolution of the 10.7 GHz channel. Due to the inverse relation between spatial resolution and spatial filter bandwidth [Bracewell, The effective emission depth of moist soils is typically 10% 19861, resolution compensation can be achieved by spatial filtering. of th e free-space wavelength. Frozen soils have effective emission We use Gaussian filtering for resolution compensation of the 18 depthe free-space waffectivelengths. The GHz and 37 GHz channels (i.e., 37 GHz and 18 GHz data are depths that may be 30% or more of free-space wvets synthesized at the resolution of the 10.7 GHz channel by Gaussian effective emission depth of frozen sandy soils, like the Fairbanks synthesized at the resolution of the 10.7 GHz channel by Gaussian Silt, can be several wavelengths. In the more transparent emitting filtering). The Gaussian filter bandwidths used in resolution commedia, such as frozen soil or dry snow, the greater average thermal pensation are calculated from the uncompensated resolutions of each photon path l ength has the effect of providing a greater opportunity channel. Second, resolution compensated data are used in the Freeze photon path length has the effect of provng a greater opportunity ndicator algorithm to classify frozen soil surfaces, and to identify for volume scattering of photons. freeze/thaw boundaries in images generated from compensated Volume scattering occurs because soils and plants appear in- (coarse resolution) 37 GHz data. In our preliminary work, creasingly heterogeneous at the scales of shorter microwave wave- freeze/thaw boundaries for the 37 GHz parameter in the Freeze lengths. These heterogeneities scatter thermally emitted photons be- Indicator are located where the 37 GHz brightness crosses a threshfore they escape through the soil surface, and the scattering is in old of 247 K, corresponding to a nominal Tb for a -50 C surface in creasingly severe at shorter wavelengths. This "law of darkening" the test area (i.e., a frozen surface). Third, freeze/thaw boundaries means that for an isothermal volume scattering halfspace, identified in compensated (coarse resolution) 37 GHz images are registered to boundaries observed in uncompensated (fine resolu~3Tb_ -0~ t~i~~tion) 37 GHz images. This is done by tracking boundary locations -.- < 0 in 37 GHz images as the amount of resolution compensation is reduced. The resulting boundary locations that are registered in the uncompensated 37 GHz images are used as estimated locations of (England, 1974]. Frozen terrain may also be snow covered. Be- freeze/thaw boundaries.on the surface. cause snow is exceedingly transparent and relatively heterogeneous to microwaves, snow exhibits significant of darkening [Edgerton et. In the boundary identification process, resolution al., 1971]. That is, both frozen soil and snow can cause negative compensation is achieved by Gaussian filtering. Only Gaussian filspectral gradients. tering can guarantee that all boundaries observed in the (coarse resolution) compensated 37 GHz images can be registered to boundNeither a low 37 GHz radiobrightness nor a negative spec- aries observed in the (fine resolution) uncompensated 37 GHz imtral gradient is solely adequate as a discriminant for classifying ages. This result follows from scale-space filtering theory in comnfrozen soils at the relatively coarse resolutions of the Niimbus-' puter vision (Witkin, 1983; Yuille and Poggio, 1986].

Figures 1-4 demonstrate preliminary results of our boundary Ed AT., A. Stogryn,nd G. Poe, 1971, identification process. Figure 1 shows the Freeze Indicator map of iogeric ton, A.T, A. Stogryn, and G. Poe, 1971 Microwave Rof the test area of Zuerndorfer et. al. [1989] for SMMR data of mid- iomerc Investig^on of bew Aks e Final 1285R-4 of night September 20, 1984. A geographical map of the test area -- Contract 14-08-001-11828 between Aerojet-General Corp., El North Dakota and the surrounding regions -- is included. The Freeze Monte, CA, and the U.S. Geological Survey. Indicator shows dark pixels for surfaces that are more likely frozen. England, AW., 1974, The effect upon microwave emissivity of Figure 2 shows the 37 GHz radiobrightness map of the same re- Englande s g, The effect upon microwave emissivity of gion, where the resolution has been compensated to that of the 10.7 o e scatteing in snow, i ce and in fron soil, Proc. URS GHz channel. Regions of darker pixels represent surface regions of Spec Mtr on Microw a nd Emission from th arth, lower radiobrightness. Shown as white pixels are the estimated freeze/thaw boundaries. Figure 3 shows a 37 GHz radiobrightness England, A.W., 1975, Thermal microwave emission from a scatmap of the region at a resolution corresponding to the 18 GHz er ayer, IGR pp. 4484-4496. channel. Figure 4 shows an uncompensated 37 GHz radiobrightness map. Figures 2-4 all show the freeze/thaw boundary. Englad, A.W., 1976, Relative influence upon microwave emissivity of fine-scale stratigraphy, internal scattering, and dielectric Based upon the Freeze Indicator map of Figure 1, the dark properties, Paeoh 4, pp. 287-299. regions in the 37 GHz radiobrightness map of Figure 2 would be classified as frozen. The localization improvement afforded by the England, AW., 1977, Microwave brightness spectra of layered higher resolution data is apparent in tracking the freeze/thaw bound- media, Geohvsis 42, pp. 514-521 aries from Figures 2-4 (i.e., as resolution in improved). In addition, new regions and boundaries appear as resolution improves, as in the Grody, N.C., 1988, Surface identification using satellite microwave dark islands in the northeast coer of Figure 4. These new regions radiometers IEEE Transactions on Geoscience and Remote are distinct from those classified at coarse resolution, and cannot be V. 26, pp. 850-859. ascertained from the Freeze Indicator. Thus, classification of the new regions must be determined by other means. Hoekstra, P., and A. Delaney, 1974, Dielectric properties of soils at UHIF and microwave frequencies, JGR 79, pp. 1699-1708. CONCLUSIONS Moik, J., 1980, Digital Processing of Remotely Sensed Images, NASA, NASA SP-431. In Zuerndorfer et. al. [1989], a multiple frequency, Freeze Indicator algorithm was developed which shows promise as a NASA, 1978, The Scanning Multichannel Microwave Radiometer classifier of frozen soil. However, the common (coarse) resolution (SMMR) experiment, The Nimbus-7 Users Guide, The Landneeded in the classification process results in a Freeze Indicator that sat/Nimbus Project, Goddard Space Flight Center, NASA, p. has coarse resolution and, subsequently, poor localization of 213-245. freeze/thaw boundaries. Use of scale-space theory permits the mapping of freeze/thaw boundaries from the coarse resolution of the Schmugge, T.J., 1983, Remote sensing of soil moisture: Recent compensated images to the finer resolution of the 37 GHz image. advances, IEEE Trans. on Geosc. and Rem. Sens. GE-21, pp. 336-344. A significant limitation to the improvements available in freeze/thaw boundary localization is the performance of the Freeze Schmugge, T.J., 1987, Remote sensing applications in hydrology, Indicator in correctly classifying surfaces. Work in determining Rev. Geophys. 25, pp. 148-152. frozen soil using radiometer data is in its preliminary stages. For the results presented in this paper, simple threshold crossings were used Schmugge, T.J., P.E. O'Neil, and J.R. Wang, 1986, Passive mito determine freeze/thaw boundaries in the high frequency radio- crowave soil moisture research, IEEE Trans. on Geosc. and Rem. brightness maps. Such criteria may not be optimal. As work on the Sens. GE-24, pp. 12-22. Freeze Indicator develops, a better understanding of freeze/thaw boundaries will ensue and, subsequently, produce better criteria for Ulaby, F.T., R.K. Moore, and A.K. Fung, 1981, Microwave Redetermining freeze/thaw boundaries in the higher frequency, finer mote Sensing. Active and Passive, Addison-Wesley. resolution maps. Wang, J.R., T.J. Schmugge, W.I. Gould, W.S. Glazar, and J.E. Fuchs, 1982, A multi-frequency radiometric measurement of soil REFERENCES moisture content over bare and vegetated fields, Geophys. Res. Let. 9, p. 416-419. Blanchard, B.J., and A.T.C. Chang, 1983, Estimation of soil moisture from Seasat SAR data, Water Res. Bull. 19, pp. 803- Witkin, A., 1983, Scale-space filtering, Proc. Int. Joint. Conf. Ar810. tif. Intell., Karlsruhe, West Germany, p. 1019-1021. Bracewell, R. N., 1986, The Fourier Transform and Its ApDlica- Yuille, A., and T. Poggio, 1986, Scaling theorems for zero crosstions, McGraw-Hill. ings, IEEE Trans. Patt. Anal. Mach. Intell., Vol. PAMI-8, No. 1, p. 15-25. Burke, W.J., T. Schmugge, and J.F. Paris, 1979, Comparison of 2.8- and 21-cm microwave radiometer observations over soils Zuerndorfer, B.W., A.E. England, M.C. Dobson, and F.T. Ulaby, with emission modelcalculations, JGR 84. pp. 287-294. 1989, Mapping freeze/thaw boundaries with SMMR data, submitted to J. Agriculture and Forest Meteorology. Camillo, P.J., and T.J. Schmugge, 1984, Correlating rainfall with remotely sensed microwave radiation using physically based models, IEEE Trans. on Geosc. and Rem. Sens. GE-22, pp. 415423.

igure. Freeze Indicator map of the test site, includingIFigure 2. GHz rdiobghtness map o the -est site at he graphical map. gre. Hz radionghtness map of the-esi se at the (coarse) resolution of the SMMR 10.7 GHz channel. Figure 3. 37 GHz radiobrightness map of the test site at the Fire 4. 37 GHz radiobnghtness map of the test site at the (medium) resolution of the SMMR 18 GHz channel. (fine) resolution of the SMMR 37 GHz channel. I I I ~~~~~I

Recovery of Fine Resolution Information in Multispectral Processing' Brian Zuerndorfer, Gregory H. Wakefield, and Anthony W. England Department of Electrical Engineering and Computer Science University of Michigan, Ann Arbor, MI 48109-2122 Email address: bwz @ caen.engin.umich.edu Scale-space filtering represents the signal i(x) by the twoABSTRACT dimensional function r(x,s) and draws inferences about variations of i(x) from the threshold crossing contours of e(x,s). The In this paper, we consider multiple-sensor processing and location of these thresholdcrossings, x(s), is dependent on scale develop a unified method forrepresenting multiple-sensor data. and calculated from When resolutionvaries between sensors, such amultiple-sensor system can be viewed as samples of a scale-space signal e(x(s),s)=x(s)), (1) representation. We show that if the spatial transfer function of the sensors are Gaussian, then scale-space filtering can be used for some specified threshold function ^.). The threshold to recover small scale (fine resolution) information through crossings form contours in the x-s plane. In the nomenclature extrapolation in scale. As an example of multiple-sensor pro- of scale-space theory, the x-s plane is the scale-space, the cessing, weconsidermultispectralprocessingofremote sensing function e(.,.) is the scale-space image, and the threshold in which images of surface scenes are simultaneously generated crossing contours, x(s),are thefingerpints. at different (center) frequencies. Although Zuerndorfer and Wakefield [7] have shown that INTRODUCTION the requirments can be relaxed while still preserving major points of the theory, the strongest theorems of scale-space signal Scale-space filtering was introduced in the early 1980's as representation [2,8] assume Gaussian kernels of the form a technique for signal analysis over multiple scales [1,2]. The origins of scale-space filtering liein the edge detection concerns i lx2 of computer vision but have since been applied to other prob- h(x,s) = exp - (2) lems in computer vision [3,4] as well as a model for multire- 2 solution systems [5]. In this latter context, scale-space analysis provides a mathematical framework for data integration in and a Laplacian operatorfor 0{.}. In this case, the fingerprints multiresolution systems that are characterized by classes of present a continuous track of the inflecion points of the signal sensors varying along a dimension called scale. For example, i(.) as it is filtered over scale. The inflection points can be used multispectral analysis in remote sensing requires integration of for locating "edges" in the signal [9]. data from constant-Q bandlimited sensors that vary in center frequency [6]. A simple approximation of such a muliresolu- Given the use of Gaussian kernels, Gaussian filtering can ion system is that of a multiscale system in which a single be used to degrade the resolution (broaden the PSF) of fineparamter, e.g., bandwidth, chracterizes the primary resolution sensors in multiresolution systems to match that of differences among each of the bandlimited sensors. Under this a coarse-resolution sensor. Alternatively, given the features of approximation, the data from each sensor represents samples a signal measured by a coarse-resolution sensor, it is useful to of the scale-space representation of the imaged object register their locatios with signal featues measured by fineresolution sensors. Modelled as a multiscale system, these two Formally, scale-space filter theory describes the effects of problems represent interpolaion and extrapolation, filter scale on functions e(xs) of the form [1,2,7], respectively, of the sampled fingeprints In the following, we present a formal development of extrapolation in scale-space;cxs)~O {rxJs)}, ^and then apply extrapolation to multispectral processing of remote sensing [6]. where O(. } is a linear opeatorand r(x,s) is a filter output signal given by, EXTRAPOLATION IN SCALE-SPACE r(x, ) a h(x,s)*i x). Extapolaion concens determining the threshold crossing contours, x(s), given by, The function i(x) is the input signal and h(x,) is a family of filters which is parameteized by a continuous variable. Tis h(x(s),)*i,(x(s))r(x(s),s) = a(x(s)), (3) variable s is inversely popo l to filter bandwidth and denote the scale of the w e.) is a thshol funcon. The functon i) is an indicator composed of a linear combination of single sensor l Woik supported by NSF MIP-8657884 and the Shell Oil dlma Compay Foundation

a. aiO(.), x (s)= (9a) J1 r.(x0,so) where i(.) is the ideal output signal from the ja sensor, and the and (8b) can be tten as and (8b) can be written as, a,'s are coefficients. The function i.) is the image output from thej sensor for a device havinng tesimal spatial resolution. X( (9b Without loss of generality, the sensor data are ordered by) = increasing scale of the sensor. Comparing (3) with (1), the, operator O(. in (3) is the identity. r=(o,so) ('(so)) + soso'(so) + srxO) A - r,(xo, so) The fingerprints x(s) in (3) are estimates of boundary locations in i(.). True boundaries occur between surface regions Repeating the steps above for u(s) yields, having different i.) values, so that boundaries occur at x values where i(.) crosses the threshold function, ( )2 )u(s)=u(s=)+(s -s)u'(so)+ 2 u"(s)+..., (10) i,^) = orx). (4) where, The threshold function a(.) is a linear function of x. ( so=,(u0 so) u'($o) —' - (1la) pT(Uo, So) In general, i(.) cannot be processed directly due to the finite p( _ (1 b) PSF of the sensor. However, in the absence of noise, the u (s ) = threshold crossings of r(.,s) approach those of i,(.) as s -00, since the kernel h(.,s) approximaes a delta function as s - 0. P((O, SO) (u'(s0)) + 2spX(uo, sOu'(so) + s(2)P(Uo, So) To better approximate boundary locations, we seek x(s) for as B -p,(uo,so) small a scale as possible. If sj is the finest scale at which data from the j* channel is available, and s <... < s, then x(s) can and uo a U(So). The threshold function 0(.) is linear, so that only be determined fors > N. However si<sn, and the boundary,(.) =B for some constant B. Thus, for u(s) to approximate estimate is improved if there exists a threshold function 0(.) x(s), it is necessary that, such that, x'(O = u'() and x"(so) = u"() h(u(s),s)*i,(u(s)) p(u(s),s) = (u(s)), (5) or, h(x,s)* r i(Cx) where u(s) are the fingerprints derived from (5), and h(,)* x (2) geIPI (12) u(s) -x(s) for SN s St. (6) A h(x,s)* i(x) Note that u(s) for a particular threshold function [(.) need only h(x,s)*^il(x) approximate x(s) over part ofa single contour, and thatdiffet for n=1,2,3. threshold functions are used to approximae x(s) for diff B h(,s) *i(X) contowL EXAMPLES To demosrate the signal and threshold requimeat o achieve (6), consider the Taylor expansion of x(s) about s,.. ahe (6), con r t' T r We apply the results of the previous section to a multi(s ^~~- ~spectral system that integrates data fm N sensors, where each x(^S=-x(s+^(s -S*x(s+( 2x(S)+. ( sensoroperatsa tdiffrnt(center)frequency. Inthis system, x()- x(so)+(s -OSx'(S)+ + -2 x"(s+.... (7) each sensorreceives energyusing an apere,e.g., devices such as lenses in optical applicatio, antennas in microwave By the implicit function theorem [10], application, and rrays in soar pplications that collect radiated energy that is emitted or lected from a subject of r. ~0 so) interest. In this system as the sensoris scanned over a subject, x'(sft) s= * (8i) the output signal is the convolution of the spatially varying ax)-r,(xso so) radiation intensity of the subject with the radiation pattern X"(s) = -8b) (antenna pattern) of the aperture The radiation intensity of the subject is equivalent to the scale-ce input for the jP sensor, (.). The radiation pattrn of the apture is equivalent to a (r^Xoso) - ca(X)^),(rX(s))2 + 2r=(rxo s (z) + r4,(0so). scale-space fierimpul seehe j sensor h(.,s), where OCxo - r,( X )'the scale of the impulse respo, is the width (beamwidth) of the radiation pattern; is proportional to the wavelength of the sensor. In this system, ru regional boundaries occur at where xomx(SQ); subscripted variables'mdin cat partial diflevel crossingsofi(.), and are pproimaed by the fingerprints entiation. In (3) the threshold function a(.) is a liner, so that x(s). o(.) = A for some consta ant A, and a(.) =. Bysolution to heat equation, the use of a Gaussian kernel yield A class of functions for which (12) holds is that where r,(rx~s) = Sar(x,.S. As a result, (8a) can be writen ~,4 different sensor input, i.), are scaled versions of each other. In this case, thi intior s given by,

a, (x) -(a,) (13a and threshold function, and the corresponding fingerprint is -400 ~(13a) j shown in Figure 4. Comparing the fingerprints of Figures 2 and 4 shows that the fingerprint of il(.) (Figure 4) is a reasonable and the finest scale (highest frequency) input signal is given by, match to the fingerprint of i(.) (Figure 2), particularly at smaller scales. Since the fingerprints of ii(.) cannot be calculated at il(x) = aK(x), (13b) small scales, the fingerprints of il(.) can be used to approximate the fingerprints of i,(.) at small scales. foran arbitrary function K(.). The significance of such functions CONCLUSION can be seen in multiplicative models, For the cases of Gaussian filtering and linear threshold ij(x) = I(x)R(x), crossings, we've demonstrated that extrapolation of scale can be performed in multispectral processing for signals that satisfy as are often used in image processing and remote sensing. In (12). The fingerprints of extrapolated signals approximate the active remote sensing systems, I(.) and R(.) are surface illu- actual multispectral fingerprints at small scales, and can be used mination and reflection functions, respectively. In passive when the multispectral fingerprints are not available. systems, I(.) and Rl(.) are surface temperature and surface emissivity, respectively. In both systems, RJ(.) is a function of In showing the approximation of extrapolation fingerprints the surface type and is dependent on frequency. As a result, the to multispectal fingerprints, only three terms of the Taylor indicator is given by, expansion were exploited (i.e., (12) was satisfied for 3 terms). It can be shown that N>3 terms of a Taylor expansion can be i4(x) = I(x)R,(x). (14) used if (12) is satisfied for N terms. As a result, in the absence ji of noise, extrapolation fingerprints that match the actual multispectral fingerprints at N>3 terms of a Taylor expansion will An example of a surface satisfying (13a) and (13b) is one provide a better approximation at small scales. consisting of a single surface type, and a spatially varying illumination intensity or surface temperature. REFERENCES Another class of functions that satisfies (12) is quadratic [1] A Witkin; "Scale-space filtering," Proc. Int. Joint Conf functions. In remote sensing applications, such signals occur in Artif. Inte.; Karsruhe, West Germany; 1983; PP. 1019-1021. passive systems where surface- temperature and surface emissivity change linearly in the vicinity of a boundary [11]. Inthis 2] A. Yuille and T. Poggio; "Scaling theorems for zero case, lJ(.) and Rt(.) are linear, crossings," IEEE Trans. Part. Anal. Mach. Intell., Vol. PAMI-8, No. 1; Jan. 1986; PP. 15-25. I(x) = ax + P, [3] S. Barnard; "Stochastic stereo matching over scale," Proc. R,(x) = acx + DARPA Image Understanding Workshop; 1988. so that, so, (ra2:+[ b )z (+(c) [4] A. Witkin, D. Terzopoulis, and M. Kass; "Signal matching ()= (a+(t +( ) t(16)hrough scale space," nt. J. Computer Vision, Vol. 1.; 1988; and PP. 134144. ia(x) =ax2+b~x+ c, (16b) [5] B. Zuemdorfer, A. England, and G. Wakefield; "The where, radiobrightness of freezing terrain," Proc. Int. Geosci. Remote a, = a0t Sensing Symp., Vancouver, B.C.; 1989; PP. 2748-2751. b =at p, +6,^ (V[6] J. Richards; Remote Sensing Digital Image Analysis; (X1i'~~~ ^~ HSpringer-Verlag, Berlin; 1986. cj = AIN 1e. c } * = [7] B. Zuerndorfe and G. Wakefield; "Extensions of scalespace filtering to machine sensing," submitted to IEEE Trans. A simplified example of a one-dimensional surfaces model Pant. Anal. Mach. Intell.. that satisfies (16a) and (16b) is shown in Figures 1-4 (this model is derived from [ 11]). In the figures, the indicator is composed [8] J. Babaud, A. Witkin, M. Baudin, R. Duda; "Uniqueness of of data from two sensors, the Gausian kernel for scale-space filtering", IEEE Trans. Patt. Anal. Mach. Intell., Vol. PAMI-8, No. 1; Jan 1986; PP. itlx) = il(x) - i(x), (17) 26-33. where the scale of sensor 1 is less than the scale of sensor 2. In [9] D. Mar and E. Hildreth;'Theory of edge detection. Proc. Figures 1-4, the functions il(.) and i2(.) are quadratic in the R. Soc. London B, Vol. 207; 198, PP. 187-217. vicinity of regional boundarie The surface type changes at [ 10] C. Ooffa Calcu of Several Variables, Harper & x=125, so that i1(.) and i2(.) exhibit different behavior in the0] C Go Ccu of S Vribl Harper Row, New York; 196I vicinity of boundaries for x<125 and x>125 (Figure 1). A R New Y 196 threshold level is selected to locate a boundary around the surface region with a low i () value. The crrponding fin- [11] B. Zerndorfer, A. Englan, F. Ulaby, and C. Dobon; surface region with shw low in 1(. ) value2. Th e 3 spthein' n- -Mapping freeze/thaw boundaris with SMMR daa'. ugerprint is shown in Figure 2. Figure 3 shows the i(.) afuncto M. Ariuturnd Forest Meteorology. ~ ~~mined m J. Agrttundmr and Forest Mettorolo~.

3.0,. I l 2.5 - - _ 2.0 1.5 0.5 o0.0 0.5..............-.. S' -..... -i.... S'' 2, 0. 25. 50. 75. 100. 125. 150. 175. 200. 225. 250. X Position X Position Figue 1. Sensor input signals, indictor, and threshold. Figure 2. Indicator fingprints. 3.0 I..... fill 2.5 2.0 1.5 -9, 1.0......... 7so 0.5 0.0 -0.5.0. 25. 50. 75. 100. 125. 150. 175. 200. 225. 250. X Position X Position Figure 3. High frequency (fine scale) sensor input signal and Figure 4. High frequency sensor fingerprints. threshold.

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