(None) = RL-2001 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 0~ C —that is, they possess diffuse phase boundaries. A modified Chemrnous'ko method was used to integrate the heat flow equation to obtain monthly thermal models during a typical September through December period. Diumrnal 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 estimatd 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. 1-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) a t oaz (1) where E(T) is enthalpy (cal/cm3) and F(z,t) is heat flux (cacm2-K) (T is a function of z and t). Heat flux is linearly related to the temperature gradient, DT/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) - aK(T)aT) at az az (3) With respect to a reference temperature, To, E(T) is generally the linear function E(T) = p p (T-T (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 at (5) where constant K is thermal diffusivity (cm2/sec) =_ K pCP (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 [9][10], and Kahle used the finite difference method [1 1] 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'(X) de To (7) so that u = u(T) and Equation (3) becomes aE(T) = a2u az2 (8) This 1-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() with the piecewise constant approximation, H(T), (shown schematically in Figure 2a) H(T) = E(Tn) for Tn < T < Tn+1 where Tn denotes isothem n by increasing dem ordered by ireasig 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(T) 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) a2u at az2 (10) Locally constant H requires that u be a linear function of z between isotherms Tn and Tn+l, i.e., U Un +U n+-Un (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, fib a H(T)dz (au - (au at a,t aa,t /a (12) to yield H(b-n)+ H'(z-a)] a()b -I ) t az,t az,t (13) From Equation (11), because zn is the only function of time on the left of Equation (13), dZ, 1 Unn+1-Un) |Un-Un-1 | dt (H- H+) 1\4Zn+l Zn Zn -1 (14) Equation (14) is the ordinary, 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 (Un-Un-1 dt (H — H+) Zn-Zn-1l (15) However, if Tn-l > 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(T)+E(T ) for Tn< T < Tn+1 2 H(T)= E(Tn) for T - Tn E(Tn)E(Tn) for Tn-I< T < T 2 (16) Isotherm propagation with this alternative approximation for E(T) will be referred to as the modified Chemous'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-25z/cos O' dA z/cos O' (18) where 2p is the power loss coefficient, 2P = 2xEktani ko (19) e' is the real part of the dielectric constant, tan 8 is the loss tangent, Do is free-space wavelength, and E is volume emissive power for dV at thermal temperature, T(z), E = e' 23 T(z). (20) With dV = z dco dz/cos ', and with a first order approximation for T(z), the intensity, dI', below the interface in direction 0' is dl e' 2 T+( aT z} e-2p zcos O'dz/cos 0' Tg o (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' '{Tg + ao (2) 2P (22) 8

If reflected sky-brightness and atmospheric absorption and emission are ignored, the normalized intensity above the interface (the radiobrightness, Tb) is e(^, 0) I'(0')/E', where e(X, 0) is the directional spectral emissivity and the source of the emission polarization. Because of Snell's Law effects, cos 0' below the interface is relatively near to unity for most satellite incidence angles, 0 (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(X) {Tg + ze () } Oz o (23) where ze = (213)-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 - + m-O.07) = soil + ) 'f w ater + (1-f) ice (24) (l-m) (24) where ~soil ' dielectric constant of 7% moist soil 2 Ks-K. K.-n2 water - n + + l+(jcotl)1-a 1 +ja2 [18][19] n2 = 1.8 Ks = 295.68 - 1.2283 T + 2.094x10-3 T2 - 1.41x10-6 T3 9

Koo = 4.2 a = 0.012 x1 = 5.62x10-15 e0.188/kT sec t2 = 4.2x10-14 sec T = temperature Kelvin k = Boltzmann's constant = 8.61735x10-5 eV/K o = angular frequency, radians/sec Ks-K. eice Klo + jw +J(O Koo =3.2 Ks = 3.2 + 20715/(T-38) [20] T = 4.76x10-16 e0.577/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, ea To -1 (Fsun+Fsky) dt 2T4 24 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 effect 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 1 1

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 Zuemrndorfer 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], co-= No N a + 2[5 (26) Na+2P3 (26)where N is the number of scatterers per unit volume, car is the scattering cross section for a single scatterer, and 253 is the power loss coefficient defined in Equation (19). For spherical scatterers whose diameters are small fractions of a wavelength (Raleigh scatterers), -4 a a ~4 (28) so that co 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 Chemrnousrko 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 NAG5 -852. 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,". Gohys 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.. 9, pp. 416-419, 1982. [3] Schmugge, T.J., "Remote sensing of soil moisture: Recent advances," IEE Tran 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. Geohys.. 25, pp. 148-152, 1987. [7] Zuerdorfer, 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, H.S., and J.C. Jaeger, Conduction of Heat in Solids, 2nd Ed., Oxford, 1959. [9] Watson, K., "Geologic application of thermal infrared images," Proc. of IEEE, pp. 128 -137, Jan., 1975. [10] Watson, K., L.C. Rowan, and T.W. Offield, "Application of thermal modeling in the geologic interpretation of IR images," Remote Sensine, 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," Quarterly of Applied Mathematics. 9, pp. 185-193, 1951. [13] Landau, H.G., "Heat conduction in a melting solid," Ouarterly 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 Enineering J.. 10, pp.42-48, 1970. [16] Hoekstra, P., and A. Delaney, "Dielectric properties of soils at UHF and microwave frequencies," J. Geophys. 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 ce 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.. 80, pp. 4484-4496, 1975. [25] Ishimaru, A., Wave Proagation and Scatterin in Random Media. I, Academic, NY, p. 11, 1978. 17

Table 1. Boundary Parameters Fsun Solar irradiance = fi So (1-A) M(O) cos | Fsky Sky irradiance =o Tsky4 + f2 Fwind Sensible heat transfer from air to ground..Pa ca Cd (W +2) (Tair Tground) Fground eoT4 So Solar Constant = 0.03313 calsec A Al o M(O Approximate atmosphec transmissivity..___.__... 1.0 - 0.2 (cos 4)-0.5 [9][10] -: -:-...... --- —---—...... - Zenith angle cos Co cos os 8 (-cos(27C hour/24)+sin X sin 8) _________ if > 0, otherwise cos ) = 0 local latitude 8 declination = - 23.4330 cos(27 month/12) fl =(1-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) (I M()) cos ) dt)/24; Stefan-Boltzmann constant = 1.3533x101 Tair Average air temperature Toair - Tdel cos(2i(hour-2)/24) [1 1] Toair Monthly average air temperature (e.g., see Fig. 6).__________ To - T1 cos(22(month-Ola )/12) Tdel Diurnal variation ( meteoroogical reports) Tsky Tair(0.61 +0.05 w 5). (Brunt's formula, from Kahle [11]) w atr ssur Pa Air density at surface = 1.25xl 13gmcm3 ca c eatofdryair=.24 gmCd_ Drag g-coe cnt =.Zis elevation in meters [11 W Nmsd veloatm m/secn T g- - i -- ae t emp s ution to et ow equationITg........Soil surface temp from soludion- to- heat flo-w equaton 18

Table 2. Parameters for Bismarck, ND Latitude 470 N Months September through December (i.e., 9- 12) Soil moisture 15% -- 2 e t Cloud cover, cl 20% Average winds, w 5 m/sec Albedo. Thermal IR 0.95 emissivity, e Freeze interv al 270273 K Dry soil density 1.5 gmcm3 Dry soil speci heat 2 cal/g Dry soil thermal. 5 cacm-sec-deg conductivity 7% moist soil 3.3 dielectric constant 7%moistsoil 0.23 loss tangent 2.3 K..6 Average air78.3 K (F temperature, To Annual air temp 16Kdeg variation, T1 Temp phase lag, lag 1.1 monh s Diurnal temp 5 Kdeg variation, Tdel Water vapor -.7 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 Zuerndorfer, modified from Zuerndorfer 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 Chemous'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 4 is the beam angle out of the page, 4 = )' = 0. dwo is the solid angle of the beam, do = dO dO. 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 Bismarck (Day) la I L9 VA^ -- -- -- -- -- -- -- --- -- --- -._ 0.3 -. 02 E 0.1 X >k: ES 0 -0.1 -02 -0.3 --0.4. 230 240 250 260 270 280 290 235 245 255 265 275 285 37 GHz Radlobrightness (K) lb Bismarck (Night) 0.4 - | -- - - 0.3 -- - - 0.2 ----- — i LI00 0.1 - - 10.3 -0.1 - I -- o 1 -Thawed X Mixed Frozen Frozen " {(5 Thawed X Mixed A Frozen '.: A\ 240 235 245 250 60u z7u 255 265 37 GHz Radiobightness (K) Otu z9V 275 285 21

Figure 2a 0.t wL - E(T) *..uusomme H(T) 280 260 265 270 275 Temp, K Figure 2b CL ow 260 - E(T) ~mminmmm H(T) 265 270 275 280 Temp, K 2 2

Figure 3 Time - - CL 0. I Figure 4 23

Figure Sa 30 Temperoturel, oC Figure Sb 12.0 10.0 8.0 6.0 4.0.. - -. 0,11 49 0 0 - 2.01 O-M0 0 — 01 1 ~ o00 0O.O 5 1O 20 0 -20 -10 0 Teml peroturel, *C 2 4

Figure 6 300 290 D 280 ~ 270 a. E - - Model I- 260 * Observations Tair-278.3+1 6.9*cos(2i(mo-7.1 2)/12) 250-. 4...1 0 2 4 6 8 10 12 Month 25

10o Moist Soil 320 i 300 I I i 280 Cam W 260 240 320 C 300 E 280 260 240 320 ----- o 00 E 0 280 260 240 320 300 E 0 280 Lo 0 5 260 240 3 40 5C FIg. 7 15 Mlolst Soil 20X Most Sol...-~ _ - 00 hour 06 hour ~ 12 hour 1 8 hour 3 Depth, cm 26

320 Sept 300 280 - 10% 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, 2. Fig. 8 Solar6 12 Tim 1 24 hr. Solar Time, hr 27

I ox moist 15X moist 2OX moist 1.2 U Ch 0.8 1~ MMA) I i E a gm Idw I".0 1.2 0.8 0.4 0.0 1.2 I F, ....,.....,..... I -.... I I 00011/I...... - - - - - - - - - I E u.e ~04 2 S E U CL 0.8 S t 0.4 - hI. I U * I I I I I I i - - - - - - - - - - I i - - - - 11, 1-4 11 L V.V......... 6' 6 12 24 Solar Time, hr Fig. 9 28

10 moist 15% moist 201 moist. i 260 In In w 240 '. 220 La o 200 o 180 m 160 y 260 -c 240 C e 220 L,~ 200 I 180 -e-a U 0 160 L 260 l 240 C220, 200 = 180 o Z 160 I 260 e 240 220 L ~ 200 180 160. ~........ 10.7 GHz 18 GHz 37 GHz - I II r.. < h\ I.. - - - - - - - - - - - - - - - - - - - - I I - ^- I V 6 12 18 2 - - - - - - - - - c 5 12 18 -- - - 2 ^ 6 12 1 Solar Time, hr -w --- 2 ' Fig. 10 29

1.5 N 1.0 I0O moist 5- 15X moist _- 20% moist I- 0.0 September -0.5 1.5 N 1.0 October -0 5............,a -0.5 1.5 ~ I N 1.0 0o.o November -0.5...................... -0.5 o.............. 1.5 -------- 2 Fig. I I December -0.5 6 12 18 24 F 1 1 Solar Time, hr 30