Annual Report for 1990 on NASA Grant NAGW-1983 Mapping Regional Freeze/Thaw Patterns with Satellite Microwave Radiometry Submitted by: Anthony W.\England, Principal Investigator Brian Zuerndorfer, Graduate Student John F. Galantowicz, Graduate Student M. Craig Dobson, Co-Investigator Fawwaz T. Ulaby, Co-Investigator The Radiation Laboratory Department of Electrical Engineering and Computer Science The University of Michigan Ann Arbor, Michigan 48109-2122 (313)-763-5534 Period of Performance: January 1, 1990 - December 31, 1990 Reporting date: October 1, 1990

Contents: 1. Summary 3 2. Project Overview 4 2.1 Radiobrightness of freezing soils 4 2.2 Data clustering 5 2.3 Boundary localization and automation 7 2.4 Temporal sequence of Freeze Maps 9 3. Michigan Earth Grid (MEG) 10 4. Modeling Microwave Emission from Moist Soils 10 5. SSM/I Freeze Thaw and Soil Moisture Field Study 10 5.1 Field study measurements and plan 11 5.2 19.35, 37.0 and 85.5 GHz radiometers 11 5.3 Data acquisition and communications 11 6. References 12 7. Figures 13 8. APPENDICES - Copies of papers, a symposium abstract, and a technical report that were produced under the first year of this project Products of this year's effort: England, A.W., Radiobrightness of diurnally heated, freezing soil, IEEE Trans.Geosci. Remote Sensing. 28, pp.464-476, 1990. Zuerndorfer, B., and A.W. England, Radiobrightness decision criteria for freeze/thaw boundaries, submitted to IEEE Trans.Geosci. Remote Sensing, 1990. Zuerndorfer, B., A.W. England, and F.T. Ulaby, An optimized approach to mapping freezing terrain with SMMR data, Proc. of IGARSS'90, College Park, MD, May 21-24, 1990. England, A.W., The radiobrightness measurement of apparent thermal inertia, URSI Com. F Conference on Remote Sensing Signatures, Hyannis, MA, May 15-17, 1990. England, A.W., B. Zuerndorfer, and J.F. Galantowicz, Proposed Michigan Earth Grid (MEG) format for world SSM/I data, Technical Memorandum, June, 1990.

1. Summary Calendar year 1990 was the first year of a 3 year project to develop an operational algorithm for classifying frozen or thawed soils in the northern Great Plains with SSM/I data, and to examine the sensitivity of mesoscale climate models to frozen or thawed soil as a boundary condition. We had shown in an antecedent feasibility project that a combination of the 37 GHz radiobrightness and the 10.7 - 37 GHz spectral gradient of radiobrightness from SMMR would often permit discrimination among frozen and thawed soils. Limitations of the feasibility project were that corroborating ground data were extremely limited, and that the operational flight system is to be the SSM/I with its own spectral channels and overflight times rather than the SMMR. Specific objectives for the current project are: 1. To develop a theoretical model for the radiobrightness of diurnally heated, freezing and thawing soils. 2. To refine the decision criteria for freeze/thaw classification. 3. To use scale-space theory to recover high spatial resolution in the Freeze Map product. 4. To produce a temporal set of Freeze Maps for the northern Great Plains during the fall and winter of 1984. 5. To design and build radiometers for the 19.35, 37.0, and 85.5 GHz SSM/I channels as an SSM/I simulator. 6. To instrument a test site on the northern prairie with the SSM/I simulator and with various temperature and meteorological sensors. We will monitor freezing and thawing of the soil throughout a fall and winter to insure that we understand the process and its signature. 7. To develop an operational algorithm for producing freeze maps of the northern Great Plains from SSM/I data. 8. To investigate the sensitivity of mesoscale climate models to frozen or thawed soil as boundary conditions. During this first year of the current project, we have completed objectives 1 through 4, and have reported our results in several publications. Objective 5 is partially complete: The radiometer design is complete, parts have been ordered, and the remote instrumentation controller for the radiometers and the ground sensors is under development. The system design will permit us to control and record data in our laboratory through a telephone line from a field site in North Dakota. An essential activity, but one that was not part of our original proposal, we have supported the SSM/I user community through the SSM/I Products Working Team (SPWT) by supplying the National Snow and Ice Data Center (NSIDC) with candidate Earth grids for organizing world SSM/I data. We are also in the process of verifying resampling algorithms by converting satellite data to what we call the Michigan Earth Grid - 1 (MEG-1) format, and then resampling to recover, for example, a Mercator projection of a selected region without losing appreciable resolution.

2. Project Overview Using data from the Nimbus-7 Scanning Multichannel Microwave Radiometer (SMMR) for a test area that included North Dakota and parts of the surrounding states and southern Canada, Zuerndorfer et. al. [1] showed that a combination of low 37 GHz radiobrightness, Tb(37), and a low spectral gradient of radiobrightness, aTbh/f, where f is frequency, becomes an effective freeze indicator, or discriminant, for classifying frozen terrain. Frozen surfaces appear cold at 37 GHz, and exhibit a negative spectral gradient that is largely caused by volume scatter darkening at the shorter wavelengths. The spectral gradient is a linear, least-square fit to the 10.7, 18, and 37 GHz SMMR radiobrightnesses. A surface is classified as frozen only if both Tb(37) and Tbh/af are sufficiently low. A freeze map is generated by displaying the FI for each pixel location. A fundamental problem with the FI algorithm was that radiobrightness measurements from different frequency channels with different spatial resolutions were required to estimate a spectral gradient. To make each radiobrightness value refer to a common area on the ground, data from each channel were compensated to a common spatial resolution. Without a priori surface information, the resolution of data at all frequencies were compensated to the (coarse) resolution of the lowest frequency data used in the spectral gradient estimate. Subsequently, all identified freeze/thaw boundaries were localized to the accuracy of the coarse-resolution data. However, by modelling the radiometer beampatterns as Gaussian, we used results from scale-space theory [2] to associate coarse-resolution freeze/thaw boundaries to fine-resolution 37 GHz boundaries (i.e., 37 GHz radiobrightness threshold crossings) [3]. Thus, freeze/thaw boundaries could be estimated at fine-resolution, but the boundary association process required human intervention. A second fundamental problem occurred because diurnal insolation produces temperature gradients within the surface that altered SMMR noon and midnight spectral gradients [4]. In addition, time lags between air and surface temperatures produced offsets in measurements made at noon and midnight. Decision criteria for classifying surfaces, and identifying freeze/thaw boundaries, had to be developed to accommodate these day/night differences. Over this past year, we have developed criteria for classifying frozen and thawed surfaces, and for locating freeze/thaw boundaries, through clustering and a standard unsupervised classification algorithm. The procedure differs from our previous work in that clustering and classification are unsupervised (i.e., autonomous), and the noon and midnight SMMR measurements are clustered separately. We also developed a simple, automated technique for extrapolating freeze/thaw boundary estimates from the coarse spatial resolution of the freeze map to the fine spatial resolution of the 37 GHz channel. The results of this work were presented in [13] and [14], and are summarized below. 2.1 Radiobrightness offreezing soils Freezing and thawing soils exhibit unique radiometric characteristics. To examine these characteristics, diurnal insolation was 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 (the change in internal energy with temperature at constant pressure) 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, particularly clay-rich soils, freeze over a range of temperatures rather than at 0~ 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. The work was reported in reference [4]. 2.2 Data Clustering The decision criteria for detecting freeze/thaw boundaries were based on clustering and unsupervised classification. Unsupervised classification, rather than supervised classification, was used because of the dearth of accurate ground measurements in our test area. All data from SMMR satellite passes that covered more than 67% of the test area were used for clustering, and were incorporated in the scatter diagrams of Figures 1-2. Sixteen noon SMMR passes and thirteen midnight passes met this criterion during our August to December test period. The 18 and 37 GHz averages of vertical and horizontal polarized radiobrightness data were resolution compensated to the (coarse) resolution of the 10.7 GHz channel. Data were re-sampled on a 97.5 km grid (i.e., at the resolution of the 10.7 GHz channel). Scatter diagrams for the noon and midnight data are shown by month in Figures la and lb, respectively. Migrating means clustering determined cluster centroids for the data of Figures la and lb [5,6]. On the basis of data from our preliminary studies, surfaces were classified into three distinct types -- frozen, hot (and dry), and wet (and cool) -- and a fourth type that we call mixed. A frozen surface is characterized by relatively low spectral gradient and a low 37 GHz radiobrightness. Due to the influence of liquid water in the surface, a wet surface is characterized by a high spectral gradient [4] and low 37 GHz radiobrightness [7]. A hot surface has relatively less surface moisture, producing a "dry" surface dielectric constant similar to a frozen surface. Moreover, the relaxation frequency of free-water increases with temperature, further reducing the spectral gradients of hot surfaces. As a result, a hot surface has a relatively low spectral gradient and a high 37 GHz radiobrightness. A mixed surface has a combination of frozen, wet, and hot characteristics. Prior to freezing, a surface region is a combination of wet (and cool) and hot (and dry). As freezing begins, the region includes locally frozen surfaces, and would be classified as mixed. Our freeze/thaw criteria lies within the mixed surface cluster in decision space, and represents maximum Tb(37) and aTb/lf values along the freeze/thaw boundary. That is, any surface point on the freeze/thaw boundary has at least one component, Tb(37) or aTbh/f, equal to that of the freeze/thaw criteria. Equivalently, the FI algorithm [1] requires any surface point classified as frozen to have a 37 GHz radiobrightness and a spectral gradient that are less than those of the freeze/thaw criteria. Cluster centroids determined for the data of Figures la and lb are given in Table 1. Due to SMMR recording problems, limited midnight data were available during December of our test period. Within this limitation, the frozen surface cluster centroid has a lower spectral gradient and 37 GHz radiobrightness at noon than at midnight. Furthermore, because there were few wet surfaces at midnight during the test period, the wet and mixed surface types were inseparable for the midnight data.

Table 1. Cluster centroid in decision space NOON DATA MIDNIGHT DATA Surface Type 37 GHz (K) Gradient (K/GHz) 37 GHz (K) Gradient (K/GHz) Frozen 227 -.43 234 -.35 Hot 277 0.11 258 -.01 Wet 238 0.37 Mixed 250 0.14 243 0.015 Bivariate normal distributions were fit to the cluster data. All data within three standard deviations of a cluster centroid were classified using a Mahanalobis minimum distance classifier (maximum likelihood classification). No preferential weightings of surface types were used. Constant-deviation, single-class ellipses were drawn in decision space for frozen, hot, and wet surfaces (at noon) and for frozen and hot surfaces (at midnight) using the classified data. The freeze/thaw criteria was determined by allowing the deviation of all ellipses to expand equally until all ellipses intersected. The resulting classification ellipses for noon and midnight SMMR data are shown in Figures 2a and 2b, respectively. The corresponding freeze/thaw criteria in decision space are shown in Table 2. Table 2. Freeze/thaw criteria in decision space; a are standard deviations of the data within the ellipses. Refined Deviation c at 37 GHz(K) 37 GHz(K) Gradient(K/GHz) Intersection Noon 252 249 0.0625 3.1 |Midnight 247 244 -0.044 2.55 By viewing the freeze/thaw criteria derived from clustering as initial estimates for determining the freeze/thaw boundary, we refined the boundary criteria by requiring a minimum scatter of Tb(37) along that boundary. This constraint insures that boundaries in Tb(37) images correspond closely to FI boundaries. The process involves adjusting the Tb(37) component of the freeze/thaw criteria, T37, to minimize the sum square error, SSE, in, N SSE= Z [Tbi(37) - T37] i =1 Equivalently, the refined T37 is the average 37 GHz radiobrightness on the boundary. The process is first-order since we do not reiterate SSE minimization with the refined criteria The refined T37 for midnight data from October 24 and for noon data from December 11, are shown in Table 2.

2.3 Boundary Localization and Automation Spectral gradients are regression slopes to SMMR 37 GHz, 18 GHz, and 10.7 GHz radiobrightness measurements. The nominal resolutions of these channels are 30 km, 60 km, and 97.5 kmn, respectively. Without compensating for the resolution differences between the channels, the spectral gradient estimates can be in error. For example, a non-zero gradient estimate can result from radiobrightnesses that are spatially variant but are locally constant over frequency. To avoid such errors, the image data were compensated to one common resolution -- the (coarse) resolution of the lowest frequency channel used in gradient estimation -- prior to clustering. Freeze/thaw boundaries combine 37 GHz threshold crossings and spectral gradient threshold crossings. Corresponding 37 GHz threshold crossings occur in fine-resolution 37 GHz images, but not all 37 GHz threshold crossings represent freeze/thaw boundaries. Some are boundaries between moist and dry terrain. Boundary localization is a three-step process that identifies pixels in fine-resolution, 37 GHz images that correspond to freeze/thaw boundaries at coarse-resolution. Step 1: Uncompensated 10.7 GHz, 18 GHz, and 37 GHz SMMR data are compensated to the resolution of the 10.7 GHz channel. Gaussian spatial filtering is appropriate for resolution compensation if the Fourier transform of the SMMR beampattern is (approximately) Gaussian [8]. Antenna data for the Nimbus-7 SMMR antenna are limited. However, we assume that the Seasat SMMR beampattern [9] approximates the Nimbus-7 SMMR beampattern, and justify Gaussian filtering by showing that the Fourier transform of the Seasat SMMR beampattern is approximately Gaussian (Figure 3). The Gaussian filters used to synthesize compensated data at resolution s2 from uncompensated data at resolution sl are, H(f,S) = e-(2fS)2/2 where the filter width, S, is, 2s - and f is spatial frequency. Values of S for different configurations of resolution compensation are shown in Table 3. Table 3. Filter bandwidths for resolution compensation. Nominal Synthesized Filter Resolution, s Resolution, s2 Bandwidth, S 30 km (Fine) 60 km (Medium) 51.96 km 30 km (Fine) 97.5 km (Coarse) 93.77 km 60 km (Medium) 97.5 km (Coarse) 78.885 km Step 2: Using resolution compensated data, Tb(37) and aTbfaf are calculated for each image pixel at coarse-resolution. Boundaries in coarse-resolution, 37 GHz images are identified

where 37 GHz data satisfy the Tb(37) freeze/thaw criteria. Pixels along these 37 GHz image boundaries with aTb/af at or below that of the freeze/thaw criteria are identified as freeze/thaw boundary pixels. Step 3: Fine-resolution, freeze/thaw boundaries are determined by identifying those pixels in fine-resolution, 37 GHz data that satisfy the TB(37) freeze/thaw criteria and correspond to coarse-resolution freeze/thaw pixels of step 2. This process involves tracking boundary locations in 37 GHz images as the amount of resolution compensation is reduced. The resulting boundary locations in the fine-resolution 37 GHz images are best estimates of freeze/thaw boundaries in the sense that they are directly traceable to the coarse-resolution boundaries generated by clustering and maximum likelihood classification. The key to this process is that Gaussian image degradation of step 1 uniquely permits recovery of some fineresolution information, a result derived in scale-space theory [2,10,11]. Unrefined freeze/thaw criteria (Table 2) were applied to SMMR data for midnight October 24 (Figure 4). Refined freeze/thaw criteria were also applied to the October data (Figure 5). The dark pixels in the freeze maps* (Figures 4a and 5a) correspond to surfaces with low FI value -- surfaces which are most likely frozen -- and freeze/thaw boundaries appear as a fuzzy white lines around these frozen regions. The dark pixels in the 37 GHz images (panels b, d, and f in Figures 4 and 5) correspond to surfaces of low 37 GHz radiobrightness. The fuzzy white lines around these dark regions are the boundary pixels that satisfy the Tb(37) freeze/thaw criterion. Some or all of these boundaries correspond with the coarse-resolution freeze/thaw boundaries of the freeze maps. Similarly, the dark pixels in the spectral gradient images (Figures 4c and 5c) correspond to surfaces with low spectral gradient, and the fuzzy white lines are boundary pixels that satisfy the aTb/af freeze/thaw criterion. In all images, regions of no data are shown as white. Comparing the "unrefined" images (Figure 4) with the "refined" images (Figure 5) shows that refined criteria generate coarse-resolution, 37 GHz boundaries that are located more closely to freeze map and spectral gradient boundaries. Moreover, refined fine-resolution, 37 GHz boundaries (Figure 5f) are more consistent with ground data than are unrefined 37 GHz boundaries (Figure 4f). Thus, freeze/thaw boundaries derived from refined criteria should be more accurate that those derived from unrefined criteria. In the refined images of Figure 5, most sections of the coarse-resolution, 37 GHz boundary in the northwest corner of the test area (Figure 5b) correspond with boundaries of the freeze map (Figure 5a). These sections of 37 GHz boundary would be designated as freeze/thaw boundaries. None of the two other boundaries in Figure 5b correspond to any freeze map boundary, and are probably wet/dry boundaries. The freeze/thaw boundary in the coarseresolution, 37 GHz radiobrightness image also corresponds to boundaries in medium-resolution and fine-resolution, 37 GHz images. That is, medium-resolution and fine-resolution freeze/thaw boundaries are the convoluted boundaries in the northwest corner of Figures 5d and 5f, respectively. Some boundaries are formed at fine-resolution that do not correspond to any boundary observed at coarse-resolution. These boundaries appear around dark radiobrightness "islands" in Figure 5f, and cannot be identified on the basis of the available information. Such boundaries are not part of the freeze/thaw boundary estimates. Figure 6 represents automated boundary localization for the October midnight SMMR data. Figure 6a repeats the freeze map of Figure 5a, and Figure 6b shows the associated coarseresolution, 37 GHz radiobrightness image. As before, 37 GHz boundaries are composed of pixels whose 37 GHz radiobrightness equals the Tb(37) component freeze/thaw criteria. However, the 37 GHz boundaries in Figure 6b consist of (fuzzy) white and black sections. Pixels along white boundaries have spectral gradients that are less than or equal to the aTb/'f component of the freeze/thaw criteria. That is, white boundaries are most likely to be freeze/thaw boundaries. Pixels

along black boundaries have larger spectral gradients and are less likely to be freeze/thaw boundaries. In medium-resolution, 37 GHz radiobrightness images (Figure 6c), white boundaries are 37 GHz boundaries that are migrations of white boundaries at coarse-resolution (Figure 6b). Precise distances for boundary migration are calculated from ideal 37 GHz radiobrightness measurements and actual freeze/thaw boundary locations (i.e., radiobrightnesses and boundary locations hypothetically measured at infinitesimal resolution) [12]. While such ideal data is generally unavailable, the midnight and noon SMMR data permits a theoretical migration limit.of (s2-s1)/4 for tracking a boundary from coarser-to-finer resolution images where s and s2 are the resolutions of finer and coarser resolution images, respectively (Table 3). As a result, white boundaries at medium-resolution (Figure 6c) must be within 9.325 km of white boundaries at coarse-resolution (Figures 6b and 6d). Repeating this process, white boundaries at fine-resolution (Figure 6d) must be within 7.5 km of white boundaries at medium-resolution (Figure 6c). Frozen terrain is identified iteratively using fine-resolution data. First, pixels along white, fine-resolution boundaries (Figure 6d) are identified as "frozen" pixels. Second, pixels whose 37 GHz radiobrightnesses are less than or equal to the Tb(37) freeze/thaw criterion, and are contiguous to frozen pixels, are also identified as frozen. Third, the previous step is repeated until no additional pixels are identified as frozen. Fourth, the resulting collections of frozen pixels constitute regions of frozen terrain. Using this procedure, terrain identified as frozen is indicated by whitened regions in the northwest corner of the Figure 6e. Because freeze/thaw boundaries must be closed contours, the final freeze/thaw boundary (i.e., the edge of the identified frozen region) contains boundary sections that did not, previously, show strong freeze/thaw boundary indications. Nonetheless, the final freeze/thaw boundaries of Figure 6e represent the best fineresolution estimates of the actual freeze/thaw boundaries. 2.4 Temporal set of Freeze Maps Automated resolution recovery has been applied to Figures 7a through Figure 7i. Time summaries of the data are given in Table 4. These images show the growth and contraction of ground-freezing from October 24 to November 5, and again from November 27 to December 9. After December 9, the area remains frozen through the end of December. Table 4. Time summary for images of Figure 7a through Figure 7i; measurement interval is the time interval between the current and the previous measurement. Figure Date Time-of-Day Interval (Days) 7a October 24 Midnight - - 7b October 30 Midnight 6 7c November 1 Noon 2.5 7d November 5 Midnight 3.5 7e November 27 Midnight 22 7f November 29 Noon 2.5 7g December 3 Midnight 3.5 7h December 9 Midnight 6 7i December 11 Noon 2.5

1'-O 3. Michigan Earth Grid The Michigan Earth Grid (MEG) is a resampling and storage scheme for DMSP SSM/I satellite data. The project was initiated at the spring meeting of the SSM/I Products Working Team (SPWT) as part of an effort to facilitate the processing, storage and distribution of SSM/I data for users in the geoscience community. The first objective of the MEG project was to design a CDROM storage format-the earth grid-to which SSM/I raw data could be resampled for condensed storage and easy retrieval. This aspect of the project was completed in June and the proposal is summarized in the appendices. SPWT has given tentative approval to the scheme listed in the appendices as MEG 1. The second objective of the MEG project will be the design of an algorithm to map the raw SSM/I data from the original beam centered sample points to points on the earth grid MEG1. The objective of this resampling is to improved upon nearest-neighbor resampling by interpolating in two dimensions to a high sample density before assigning radiometric values to MEG1 grid points. We are currently coding the transformation and will confirm its robustness with sample raw SSM/I data. In addition, since the storage scheme results in a geometry which cannot be directly projected as an image, an algorithm to produce Mercator images from the stored data set is also being developed. This work will be completed by the next SPWT meeting in mid-October. 4. Modeling Microwave Emission from Moist Soils The model of microwave emissions from soil developed by Tony England for freeze-thaw discrimination has been modified in order to examine radiobrightness thermal inertia, or RTI. RTI is in general a measure of the degree to which soil moisture decreases the amplitude of diurnal soil temperature variations. In this study, we have defined the RTI parameter as the twelve hour change in radiobrightness at a particular microwave frequency. The model estimates first the thermal response of the soil to diurnal heating and secondly the soil's microwave emissions. The results of the thermal model are shown in Figures 8 and 9. It is clear that moisture content decreases the range of surface temperatures and temperature gradients. Direct thermal effects and the effect of dielectric constant change with temperature combine to produce distinct diurnal radiobrightness signatures as a function of soil moisture, as shown in Figures 10 and 11. We see that moisture content has a marked effect on the profiles -- especially at 37 and 85.5 GHz. In addition, since the angle of incidence is high (530), the effect of emissivity change in high moisture soils produces an inversion of the diurnal wave at the higher frequencies for horizontally polarized radiation. Figures 12 through 17 show the relationships between soil moisture and the twelve hour radiobrightness difference for all combinations of polarization and frequency. It can be seen that the 2:00 am/pm difference shows maximum sensitivity to soil moisture and that the 85.5 GHz H-polarized signal is the most sensitive overall. A paper to report this work is currently being completed for submission to IEEE Geoscience and Remote Sensing. 5. SSM/I Freeze Thaw and Soil Moisture Field Study A field study slated to begin in the mid to late summer of 1991 will examine the effects of moisture and freezing on microwave emissions from soils. The objective of the study is to verify the theoretical models developed by Tony England and Brian Zuerndorfer as described in previous sections. The following briefly describes the experiment plan and the highlights of current work.

11 5.1 Field Study Measurements and Plan The experiment location is at an agricultural site associated with the University of North Dakota. Since the objectives require a long period of uninterrupted measurement, the entire apparatus will be self-contained except for power and communications connections. In addition, although data acquisition will be automatic, a remote operation capability over telephone lines is being developed so that control can be arrested from the operating computer in North Dakota for trouble shooting and data dumps. As currently envisioned, the types of measurements to be made fall into four categories. Firstly, radiometric measurements will be made with three microwave radiometers at 19.35 GHz, 37.0 GHz, and 85.5 GHz and an infrared radiometer. Secondly, subsoil measurements of temperature will be made at various depths with thermistors. In addition, subsoil measurements of soil moisture will be made in real time by low frequency electromagnetic probes (to be developed), and by conventional soil sample methods for periodic confirmation of probe accuracy. Thirdly, meteorologic measurements such as wind speed and direction, humidity, solar incident radiation, precipitation and snow cover are planned. Lastly, various measurements for internal temperature control are planned in order to maintain precise temperature conditions for the radiometers and the control unit. 5.2 19.35, 37.0 and 85.5 GHz Radiometers The designs for the three microwave radiometers have been completed and the front-end components are currently on order from two vendors, Millitech and Electromagnetic Sciences. Block diagrams of the designs are shown in Figures 18, 19 and 20. The basic Dicke type radiometer was chosen for its simplicity and long term stability. Since we will not be able to calibrate frequently, the temperature of the radiometers must be precisely controlled to maintain gain stability. However, since temperature can be controlled with heaters and proper insulation and there is no constraint on measurement integration time, the overall degree of accuracy and precision of the radiometers should be high. 5.3 Data Acquisition and Communications The remote location of the apparatus requires an elaborate control and communications system for the acquisition of data. We have developed a scheme with the help of a University of Michigan senior, Dan Burkel, who is currently designing software for the controlling computer. The system is outlined in Figure 18. The primary component is a Macintosh II computer which will continuously handle control and data acquisition in the field. Periodically, a Macintosh IIfx located in our offices will arrest control of the field unit from its operating system using remote communications software through high-speed modems. In this mode, an operator will be able to trouble shoot problems, monitor the acquisition of data using real-time displays and acquire data from the field unit's hard disk. The field unit will operate two I/O boards capable of measuring up to 64 voltage inputs and generating TTL and waveform output signals.

1 2 6. References [1] Zuerndorfer, B.W., England, A.W., Dobson, C.M., and Ulaby, F.T., 1990, Mapping freeze/thaw boundaries with SMMR data, J. Agriculture and Forest Meteorology. Vol. 52, pp. 199-225. [2] Witkin, A., 1983, Scale-space filtering, Proc. Int. Joint. Conf. Artif. Intell., Karlsruhe, West Germany, p. 1019-1021. [3] 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. [4] England, A.W., 1990, Radiobrightness of diurnally heated, freezing soil, IEEE Trans. Geosc. and Rem. Sens.. GE-28. No. 4, pp. 464-476. [5] Richards. J.A., 1986, Remote Sensing Digital Image Analysis, Springer-Verlag, Berlin. [6] Clustering and classification was performed on a Sun-4 workstation using EASI software, version 4.1, from PCI, Inc. of Richmond Hill, Ontario (Canada). [7] Hoekstra, P. and Delaney, A., 1974, Dielectric properties of soils at UHF and microwave frequencies, J. Geophys. Res. 79, pp. 1699-1708. [8] Bracewell, R. N., 1986, The Fourier Transform and Its Applications, McGaw-Hill. [9] Njoku, E.G., J.M. Stacey, and F.T. Barath, 1980, The Seasat Scanning Multichannel Microwave Radiometer (SMMR): Instrument description and performance, IEEE Trans. Ocean Engin. OE-5, pp. 100-115. [10] Yuille, A., and T. Poggio, 1986, Scaling theorems for zero crossings, IEEE Trans. Patt. Anal. Mach. Intell.. Vol. PAMI-8, No. 1, p. 15-25. [11] Zuerndorfer, B. and G. H. Wakefield, 1990, Extensions of scale-space filtering to machinesensing systems, IEEE Trans. Patt. Anal. Mach. Intell.. Vol. 12, No.. 9. pp. 868-882. [12] Zuerndorfer, B. and G. H. Wakefield, 1990, Applications of scale-space filtering to signature analysis, IEEE Trans. Acoust. Speech Signal Process., under review. [13] Zuerndorfer, B.W., A.W. England, and F.T Ulaby, 1990, An optimized approach to mapping freezing terrain with SMMR data, 1990 IEEE Int. Geosci. and Remote Sensing Symp., Washington, DC. [14] Zuerndorfer, B.W. and A.W. England, 1990, Radiobrightness decision criteria for freeze/thaw boundaries, IEEE Trans. Geosc. and Rem. Sens., under review.

13 7. Figure Captions Figure 1. Scatter diagram of ITb/af versus Tf(37) throughout North Dakota and the surrounding region. Data were collected from 8/1/84 to 12/31/84. (a) is noon data, and (b) is midnight data. Figure 2. Single class ellipses of aTb/af versus Tb(37) throughout North Dakota and the surrounding region. Data were collected from 8/1/84 to 12/31/84. (a) is noon data, and (b) is midnight data. Figure 3. Radiation pattern of Seasat SMMR antenna versus Gaussian model. Figure 4. A comparison of reported air and soil temperatures with images of North Dakota and the surrounding region. Boundaries were determined using unrefined freeze/thaw criteria. Data were collected at midnight, October 24, 1984. (a) Freeze map at coarse resolution (b) 37 GHz radiobrightness at coarse resolution (c) Spectral gradient at coarse resolution (d) 37 GHz radiobrightness at medium resolution (e) Air and soil temperatures (f) 37 GHz radiobrightness at fine resolution Figure 5. A comparison of reported air and soil temperatures with images of North Dakota and the surrounding region. Boundaries were determined using refined freeze/thaw criteria. Data were collected at midnight, October 24, 1984. (a) Freeze map at coarse resolution (b) 37 GHz radiobrightness at coarse resolution (c) Spectral gradient at coarse resolution (d) 37 GHz radiobrightness at medium resolution (e) Air and soil temperatures (f) 37 GHz radiobrightness at fine resolution Figure 6. Automated images of North Dakota and the surrounding region. Boundaries were determined using refined freeze/thaw criteria. Data were collected at midnight, October 24, 1984. (a) Freeze map at coarse resolution (b) 37 GHz radiobrightness at coarse resolution (c) 37 GHz radiobrightness at medium resolution (d) 37 GHz radiobrightness at fine resolution (e) Classified frozen ground at fine resolution Figure 7. Automated images of classified frozen ground of North Dakota and the surrounding region. Data were collected at irregular intervals from 10/24/84 through 12/11/84. (a) Midnight, 10/24/84 (b) Midnight, 10/30/84 (c) Noon, 11/1/84 (d) Midnight, 11/5/84 (e) Midnight, 11/27/84 (f) Noon, 11/29/84 (g) Midnight, 12/3/84 (h) Midnight, 12/9/84 (i) Midnight, 12/11/84

1 4 Figure 8. July through August diurnal surface temperature variation for 7.5%, 15% and 25% moist soils (compared by weight to dry weight). Based upon a theoretical model of a diurnally heated, homogeneous halfspace having the constitutive properties of a typical sandy loam. Figure 9. July through August diurnal surface temperature gradient variation for 7.5%, 15% and 25% moist soils. Figure 10. July diurnal variation in the 12 hour radiobrightness difference at vertical polarization for 10.7, 19.35, 37.0 and 85.5 GHz radiation and 7.5%, 15% and 25% moist soils. Figure 11. Like Figure 10 except for horizontally polarized radiation. Figure 12. Relationship between the 12 hour change in radiobrightness and soil moisture for noon/midnight, 2:00am/pm, 4:30am/pm and 6:00am/pm observation times and 19.35 GHz, vertically polarized radiation. Figure 13. Like Figure 12 except 19.35 GHz, horizontally polarized radiation. Figure 14. Like Figure 12 except 37.0 GHz, vertically polarized radiation. Figure 15. Like Figure 12 except 37.0 GHz, horizontally polarized radiation. Figure 16. Like Figure 12 except 85.5 GHz, vertically polarized radiation. Figure 17. Like Figure 12 except 85.5 GHz horizontally polarized radiation. Figure 18. Block diagram of the 19.35 GHz radiometer design. Fundamental parameters are listed at the upper left and line power and voltage values are listed next to the line where appropriate. The control unit interface is indicated by the gray box. Figure 19. Like Figure 18 except 37.0 GHz radiometer. Figure 20. Like Figure 18 except 85.5 GHz radiometer.

Zuerndorfer and England 15 SMMR Cluster Data, Noon 1.600 l. -. 1.200 A 0 0.800 0 (9 A A 1' CA 0.4 r 00 0.000 f -0.400....'""O r,~ ~ m September SMMR Cluster Data, Midnight CL -0.800 A October -1.200 _ - 0.600 0 200.0 210.0 220.0 230.0 240.0 250,0 260.0 270.0 280.0 290.0 300.0 37 GHz Brightness (K) Fig. Z a _MMR Cluster Data, Midnight 1.200 ~. 0.800 0 Aus _ 0 As, God Seplerno,, 0.400' "AM'L*. A~ C) w September C._ -1.200 _ o 200.0210.0220.0230. 240.0250.02600 270.0280.0290.0300.0 37 GHz. Brightness (K) Fig. lb

Zuerndorfer and England 16 SMMR Cluster Data, Noon 1.600. 1.200 0.800 C. ~. *o0.800o o. -1.200 -0.600 -1.600;.... l.... I.... I....!.... t.... I.... I.... I.... t.... 200.0 210.0 220.0 230.0 240.0 250.0 260.0 270.0 280.0 290.0 300.0 37 GHz Brightness (K) Fig. 2a SMMR Cluster Data, Midnight 1.600.. 1.200.~. 0.800 N S 0.400 X,, 0.400 0.000.. -1.200. -0.400 - O.'.; C~ -0.800 -1.200 -1.600...'''"......... 200.0 210.0 220.0 230.0 240.0 250.0 260.0 270.0 280.0 290.0 300.0 37 GHz Brightness (K) Fig.:2b

Zuerndorfer and England 17 Fourier Transforms of Beampattern Data Gaussian Model vs. Seasat SMMR 0.9 O Model o.8 [ 0' SMMR "" 0.7 0.6 o'00.5 ~ — " it. ao 0.4 E 0.3 0 0.2 0.1 m m,, 0C ~ — 0 0.2 0.4 0.6 0.8 1 0.1 0.3 0.5 0.7 0.9 Normalized Frequency Fig. 3

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464 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING. VOL. 28. NO. 4. JULY 1990 Radiobrightness of Diurnal eate reezn i ANTHONY W. ENGLAND, SENIOR MEMBER, IEEE Abstract-Freezing and thawing soils exhibit unique radiometric for Bismark, ND. However, regions near the Missouri characteristics. To examine these characteristics, diurnal insolation is Riv modeled as one-dimensional heating of a moist soil half-space during le r, ak ae an evi a t Re ea a t' pical fall at a northern Great Plains site (Bismarck, ND). The onedimensional heat flow equation is nonlinear because both the enthalpy anomalously cold at 37 GHz because of exceptional wet(the change in internal energy with temperature at constant pressure) ness for the northern Great Plains. The confusing factor and the thermal conductivity of freezing soils are nonlinear functions is that 37 GHz radiobrihtness, while least sensitive to of temperature. The problem is particularly difficult because phase moisture of the SMMR frequencies, is. nevertheless, boundaries propagate in time, and because soils. particularly clay-rich d soils, freeze over a range of temperatures rather than at 0~C-that is, da - thes possess diffuse phase boundaries. A modified Chernous'ko method 3 wa,- used to integrate the heat flow equation to obtain monthly thermal diobrightness temperature at frequency f-enerally remodels during a typical September through December period. Diurnal solves the ambiguity between frozen and wet soils beradiobrightness curves at 10.7, 18, and 37 GHz were computed for cause frozen soils have negative spectral gradients.and each month. The 37-GHz radiobrightness best tracks soil surface tem- CO perature: the 10.7-37-GHz spectral gradient of thawed soils is strongly positive; the spectral gradient of frozen soils is slightly negative; an also, that Fig. (a) and (b) exhibit an upward shift in the the midnight-to-noon spectral gradient is shifted by approximately spectral gradient between midnight and noon of about 0. +0.1 K/GHz by diurnal changes in the surface temperature and the K/GHz. A valid theoretical model should replicate the thermal gradient. These observations support the use of the scanning behavior of the 37-GHz radiobrihtness. the dominant multichannel microwave radiometer 37-GHz radiobrightness and its 10.7-37-GHz spectral gradient as discriminants in a frozen soil clas- C sifier for high latitude prairie. and the diurnal shifts in the spectral gradient during the September through December months in the northern Great Plains. I. INTRODUCTION r H UANTITY and physical state (i.e., frozen orI.ONDiESNAH TF W lliquid) of moisture in soil can be estimated from sat-I.ON-MESNA HATFW ellite radiobrightness signatures. A large body of litera- The one-dimensional heat flow equation for temperature links moisture content to radiobrightness [1I]-[6]. tueT()dphz(iantmets)s[8 Whether or not moisture in the soil is frozen affects the rate of energy transfer to the atmosphere by limiting eva- aE( T)/at = -3F(ZI, t) /a-z() potranspiration and affects the rainfall or snowmelt runoff where E( T) is the enthalpy (J m —) and F(-, t) is the potential by reducing the infiltration capacity of the soil. hetfudnsy(W m )(Tiafnconfzadt) Zuerndorfer et al. [7] produced freeze/thaw maps of the heat flux density is lieal relte to th fntemper-atur gra northern Great Plains from Nimbus 7 scanning multichan- dient fT/ux besiyislnayretdtoheem rtuegnel microwave radiometer (SMMR) data. This paper is an dinaTI-zb examination of the theoretical basis for microwave radio- F(z, t) = -K( T) (aT/az) (2) metric, rozen sol classiication.where K( T) is thermal conductivity (W-mn 1-K-'). That For the purposes of this paper, moist soil is defined as is f1rozen when its dielectric properties are essentially those of dry soil. This characteristic of frozen soil is discussed 8E( T) =a a(KT T ~(3 in the section on radiobrightness. Fig. 1(a) and (b) illus- at az ~'az/ trate the observational basis for a frozen soil classifier. A 37-GHz radiobrightness below about 247 K indicates fro- With respect to a reference temperature To, E( T) is zen soil, and one above about 247 K indicates thawed soil generally the linear function E(T) =pc {T - To} (4)

ENGLAND: RADIOBRIGHTNESS OF DIURNALLY HEATED FREEZING SOIL 465 0.3- - -- 02~~~~~ -0.1 - - - - -,,,_ i U. -02 -- - -0: —3-0.4 ~ ~ ~ ~ 230 240 250 260 270 280 290 235 245 255 265 275 285 37 GHz Radlobrghltes (K) (a) 0.4 - - - - - - - - - - -. 0.3~.....-.-...-. - -0 2 —-- -0: -L 2 - _ - 23 240 250 260 270 280 290 235 245 255 265 275 285 37 GNz Rackdofghnee (K) (b) Fig. 1. Frequency gradient versus SMMR 37-GHz brightness temperature. Bismarck, ND. Data were collected from 8/1/84 through 12/31/84. Shown with Bismarck data are clustering decision thresholds for frozen, mixed. or thawed surface. Based upon ground data, solid triangles are frozen, open boxes are thawed, and x's are mixed pixels (personal comniunication from Zuerndorfer. modified from Zuerndorfer et al. (71). (a) Day. (b) Night. where K is thermal diffusivity (in2/S) Equation (3) looks simpler with the substitution of variK = K/pcp. (6) ables Equation (5) can be solved analytically by either the U=~ K(-r) d'r (7) harmonic or the Laplace methods [8], or by a finite dif- T

466 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING. VOL. 28. NO. 4. JULY 1 Chernous'ko method [15]. The finite difference method for nonlinear 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 partic- ularly awkward if the phase change occurs over a finite depth, as in freezing soils, rather than at a plane interface, or if there are multiple freezing isotherms, as in the periodic heating case. The original Chernous'ko method also has difficulty with the periodic heating case. Temp, Chernous'ko [15] replaced E(T) with the piecewise (a) constant approximation H(T), shown schematically in Fig. 2(a), so that H(T) = E(Tn), for Tn < T < T,+, (9) where Tn denotes isotherm n ordered by increasing depth. While these isotherms can be any temperature, integer values of Kelvin were used in the model. The slope of the linear portions of E(T) below 268 K and above 273 K are the pc,, for frozen and moist soils, respectively. The increased slope between 268 and 273 K is a linear approximation to an energy term that includes the latent heat of melting of moisture in the soil. In this example, melt- Temp, K ing occurs over 5 K. For sandy soils, the range might be (b) 1 K; for some clay soils, it might approach 10 K [16]. Fig. 2. Enthalpy versus temperature. E(T) and H(T) are approximaIn terms of H(T), (9) becomes tions to moist soil enthalpy during freezing. (a) HT) represents Chernous ko's, piecewise constant model. Asymmetry of H( T) at each isoHm(T)/lat = a2u/az2. 10) therm produces dissimilar heating and cooling performance. (b) H(T):[ L~~~~~ocl11~~v is modified model. Symmetry in H( T) at each isotherm produces nec-f Locally constant H requires that u be a linear function of essary similarity for heating and cooling. Solid line: E( T). Dotted line z between isotherms T, and T,,. + I i.e.,HT) u = u, + Ufl+1 - Ufl (Z - Zn) for (z, < z < zn,+) n n+ I - ZnTm (11) ca~~~~~~stherins weez is the depth of isotherm Tn. HIf the value of H(T) at z > Zn is denoted by H+, and H(T) at z < Zn, is H-, then (10) can be integrated at I b H+ constant time along path a-b -in Fig. 3, a = -dz(au- (12) a \IJ~~~Z/b.t \LZ a. I Fig. 3. Integration path at constant time. Piecewise constant approximato yield tion to H( T) means that enthalpies, H- and H+, are constant between isotherms..- [ H+(b - Zn) + H-(Zn - a)] = (- a at aZ ~~~ ~~~~b,: az/at Equation (14) fails in the periodic heating problem be(13) cause of an asymmetry in H( T) between heating and cooling. For example, consider the propagation of iso-'From (1 1), bermcaus z is the- only fulnction of time- ont th therm- z, whreT -I<Tn=T +1 P T. Inthis cae H- >r+

ENGLAND: RADIOBRIGHTNESS OF DIURNALLY HEATED. FREEZING SOIL 467 shown schematically in Fig. 2(b), i.e., TABLEI BOUNDARY PARAMETERS E(T,,) + E(T, 4. ) for T,, < T < T,,+I F, solarirradiance = fS( I - A)M(c) cos 2....... rF,k sky irradiance ='Ork: + f2 F,,,nd sensible heat transfer from air to ground H(T) = E(T,,), for T = T, p,,c,,, (w + 2)(Tr- Tgrund) E(T ETne) + E( E~~~' (fo T,, _ - <+E T,.,), solar constant = 1385 W / m2 for T,_i < r < T,, Asabe 2 ~~ ~~~~~~~~A albedo M( 4) approximate atmospheric transmissivity (16) 1.0-0.2 (cos o)-"5 191. [10) 4) zenith angle isotherm propagation with this alternative approximation cos 4 cos X cos 6( -cos (2ar hour/24) + sin X sin 6) if >0. otherwise cos 4) = 0 for E(T) will be referred to as the modified Chernous'ko local latitude method. 6 declination = -23.433~ cos (2w month/ 12) - - ( I - cl ) where cl is average land cover (approximation is III. BOUNDARY CONDITIONS that some is regained through f2) cfndt irradiance from clouds, approximated as half average solar Watson [9] and Kahle [1 1] used boundary conditions irradiance lost in cloud term, f, for the energy flux Fnet() (cl/2)S,(l - A){jM(q) cos dt}/24 a Stefan-Boltzmann constant = 5.6696E-8 W-m-2-K-4 Fnet (0) = Fun + F.%ky + Fwind - Fground Tair average air temperature T.i.r - Tdo cos ( 27r ( hour - 2 )/24) [ 11 ] Fnet ( oo) = 0 (17) Tair monthly average air temperature (e.g., see Fig. 6) where:at infinity means depths greaterthan the penetra- T To - T cos ( 2 - ( month - 01,g ) / 12) e z at infinity means depths greater than the penetra- d, diurnal variation (from meteorological reports) tion of the diurnal thermal pulse. The parameters that Tk, rTar(0.61 + 0.05w~5)0" (Brunt's formula, from Kahle IiI]) comprise these boundary conditions for Bismarck, ND, i water vapor pressure, mmHg p,, air density at surface = 1.25 kg/in3 are described in Table I. Fun is insolation reduced by a' airdeiity at sf dair = E + 3 /kg-K ~ ~~~~~~~~c,, specific heat of dry air = 1.0E + 3 J/kg-K cloud cover, atmospheric absorption, albedo, and the cos- Cd drag coefficient = 0.002 + 0.006(Z/5000). Z is elevation in e of the zenith angle. Fsky is sky brightness plus a small meters [Ii] correctio forclod. Fwn i afr W wind velocity in m/s crction for cloud cover. Fwind is a small correction for e thermal infrared emissivity sensible heat transfer between ground and air used by T, soil surface temp from solution to heat flow equation Kahle [11 ]. Fground is gray-body emission from the soil's surface. pography, evapotranspiration, and sublimation are Topography, evapotranspiration, and sublimation are where e is thermal infrared emissivity and a is the St'efanignored in this model. Of these simplifications, evapo- Boltzmann constant. Initial isotherms at temperature Tog transpiration is potentially most limiting because, under were a nn at In ith interms t depth Tnc some conditions, it can be a large upward term in the en- Prp gato pieitrascudbevre u eetp ergy baac qain oeei sadfiutpoes ically 6 s. That is', every 6 s for 24 h, each of the 50 to model accurately. Until experimental radiometric data isotherms was propagated according to (14). The time inare available for freezing and thawing soils, it is better to terval was chosen to be short enough so that no isotherms note that a potentially important process is being ignored than to model the process poorly.. Some justification exists crossed. If an isotherm propagated to within a preassigned for omitting evapotranspiration. During fall and early distance A (e.g., A = 1 mm), of an identical isotherm winter, lower thermal temperatures and dormant ground (dissimilar isotherms tend to move apart), or to within A vegetation lead to lower evaporation rates. Also, thermal of the surface, that isotherm was dropped, and the reinfrared emissivity e is assumed constant (=0.95). maining isotherms were renumbered. Following isotherm Chanes n mistre ontnt o aler he heral nfrred propagation, the surface thermal gradient was established emnissivity of bare soil. However, the implicit model is of ithrouhal combienation mofsnt (0)lo andth(2) for the depth dormant prairie vegetation over a moist or frozen soil. interva bTween t hemlrdenmot shallowaisothermfand thepsurHow the effective thermal infrared emissivity will vary is fc.Ti hra rdetfxdanwsraetmea not clear. Again, without experimental data, e is best as- ture. If the temperature difference between the new sursumed to be constant. face temperature and the first isotherm was greater than The process of propagating subsurface isotherms is be- bu ul -uo (e.g., > 1 K), then a new isotherm was

468 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING. VOL. 8. NO. 4. JULY 1990 lent temperatures for the previous 24-h iteration. The pro- Rdio cess was repeated until none of the 60 x 24 temperatures Air differed for any 24-h interval by more than 0.001 K. Convergence generally required between 8 and 11 iterations. Suace IV. RADIOBRIGHTNESS Moist Soil dl The radiative transfer equation [22] for the microwave spectral intensity 1A (z..' ) at free-space wavelength X0, at depth z in soil (Fig. 4), and in direction' = cos 0' Fig. 4. Schematic of emission from moist soil. Because of transmissiviy ( p7 positive means upward) is ~~~~of soil, thermal microwave emission originates below its optical surface. ( A' positive means upward) is ~ dl~~iz, 2 2I,( ~ 19) otherm. The Fresnel reflection coefficient for p7 1 (nordz 4rmal incidence() is where the absorption coefficient 20 in terms of the real R + (24) part of the relative permittivity e' and the loss tangent (tan 5) is where complex E* -EC(1 - j tan 6). The emissivity e ( X) $)6~~~~~is I - R whereR is the complex conjugate of R. 2/3 = (27rV tan 6)/X0. (20) The dielectric properties of typical freezing soils at miThevuemsvpwEiteRlgpcrowave volfrequencies are gmissive power E in the Rayleigh-Jeans ap-Hoekstra and Delaney pre oximato eissiepwrEi h alihJasa-[ 161. Fig. 5, from their paper, shows the variability in the proximation moisture percentE = 47rE'2CxT(z) (21in Groodrich clay and in Fairbanks silt. Note that the E = 4-XE'20C,\T~z) (2 dielectric properties are essentially constant through where T(z) is thermal temperature, and CA is a parameter freezing for a moisture content of 5% by weight. This dependent only upon 0-4. If T(z) is replaced by its first- insensitivity to temperature for small moisture contents order approximation, T(z) = Tg + (aT/az)oz, then the ccurs because the water is chemisorbed, or adsorbed, to complete solution to (19) for upwelling radiation, evalu- the clay or sand interfaces within the soi [17 and bound ated at Z, = 0 (just below the interface), is water molecules are not free to rotate with an electromagnetic field. The dielectric properties of moist soil can be lI (0, p)=E'Cx\ Tg + I' (aTy (22) approximated as the sum of the dielectric properties of X 2~~~~~~ & hi soil with bound water, plus the dielectric properties of the 0 ~~~remaining free water. The dielectnic properties of frozen The ntesityjus aboe te inerfce, ormlize by soil are essentially those of soil with only bound water. Cx, is the radiobrightness Tb (measured in Kelvin) and isFothsilin[6,aut7 bywgtofhemsur given by Tb = e ( X, O)1 I(z, ti' )/Ec'Cx where e ( x, 0 ) is is bound water. If we picture free water as occupying soil the directional spectral emissivity and is the source of the ithenseigtie (valtidn ofo 0.7atm<eatraio, whnaresnbeapreomis emission polarization. The apparent brightness T. that thm eihafation ofwte) thenmlealeti prprteasonal appoxi-tsi would be measured by a radiometer located immediately maiontsh ope ilcrcpoete fmitsi above the interface is the sum of Tb and the reflected sky i brightness. From Snell's law, p7' below the interface is =~+ ~ +( )~) (5 relatively near to unity for most satellite incidence angles f l+V ~ae -~e (typically 0 =500 ). With only a slight loss of generality, where we shall assume that A' 1 I50 that polarization can be E* cmlxdeeti osato os ol ignored and brightness temperature can be written sod cmlx ilcrccosato % os ol VW volume fraction of free water or ice, Tb = e( X) T~ +Ze (aTaz)0 (23) Ewaelc complex dielectric constant of water or of ice, respectively,

ENGLAND: RADIOBRIGHTNESS OF DIURNALLY HEATED, FREEZING SOIL 469 2,,,0 TABLE PARAMETERS FOR BISMARCK, ND 100 0.15 8 oI.,""'"'""'' a 0' - X Latitude 47N J<*~~~ 6 0...0.10 oMonths September through December ~~~./..-,.~~~~~~ _Soil moisture 10 15, 20% by weight 04~~ ~ o - _.~~ _~~ 0e-o~.05 * *Cloud cover. cl 20% 2 o0 01.5o~...... Average winds, w /s [S G~~~44 ~ e 0 0 ~~~~o.o'0- Albedo 0.2 0C Lo..~~~~~8~~~-o e 9o- o 0.05 ~ Thermal IR emissivity e0. -20 -10 0 10 20 30 Freeze interval 270-273 K Temperoture, IC M ~Temo~eroture, *~C~~ Dry soil density 1.5E + 3 kg (a) Dry soil specific heat 0.84E + 3 J/k-K Dry soil thermal conductivity.2 W/m-K i2.0, 7% moist soil dielectric constant 3.3 0.I *7% moist soil loss tangent 0.23 o0.0,. Average air temperature. T. 278.3 K Fig. 6 ~ f./~~~~~~~Annual air temperature variation, 16.9 K 8.0o / T K* Temperature phase lag, 0,ag 1.12 months 60 Ad.~ 0.10.. _Diurnal temperature variation, Tt,5 K 4.0 go.-. —— * — */-'0 00Water vapor pressure, t 0.76 mmHg 4.o........ ~150 20 l<' o4...... ~.-. o o,.0.05 models are date and moisture -20 -10 0 10 20 30 argued that all parameters should be examined parametTemglroture, eC Temperature, C rically, the extensive computation required is not war(b) ranted by any possibility of inverting radiobrightness data Fig. 5. Complex dielectric constant at 10 GHz as function of temperature to obtain more th at three water contents. (a) Goodrich clay. (b) Fairbanks silt (from Hoek- an mosure stra and Delaney [16]).. OBSERVATIONS V. OBSER The weight fraction of water m is m = M,./MtOt so that, Temperature versus depth profiles at midnight, 6:00 in termns of m, (25) becomes A.M., noon, and 6:00 P.M. for Bismarck, ND, are shown (m 0.07) ~ ~~~~~~in Fig. 7. The gross features of these profiles are rela= E~3~i+ Psoi fE~ate + (1 f) E~ tively independent of moisture content and month. A~mong (1-m) these four profiles, surface temperatures are coldest at (2)6:00 A.M. (predawn) and hottest at noon, as expected, and 2)thermal pulses at depth are most pronounced at 6:00 P.m. where Thermal gradients at the surface are always positive at + Ks, - K(0, K., - n2midnight, 6:00 A.M., and 6:00 P.m., and they are always E na 2 r = - a +negative at noon. Note that the effect of freezing and water 1 + ( jwr1) + *jW7~2 thawing during October through December is a general ( [ 18 19 ~~~~~compression of the temperature profile. That is, the ap2 = 8,[9)parent thermal inertia would be greater during freezing n= 1. 8, and thawing. Ks= 295.68 - 1.2283T + 2.094 x 103'T2 1.41 Fig. 8 shows diurnal surface temperatures for SeptemX 10-6T', ber through December. The September curves, because Ko= 4.2, temperatures are above freezing, look like the curves for 0= 0.0 12, diurnally heated, moist soils [9]-[1 11. While moisture 7 = 5.62 x 10-'5e30' x~ l0)-2kT 5, tends to reduce the day-night temperature difference, the 7%= 4.2 x i0 - 4 5, effect is small. The October curves are very different. The T temperature (K), daytime peak is lower because of reduced insolation, but k = 1. 3806 x 10-23 j /K, Boltzmann's constant, nighttime curves are "held up"' by the latent heat of fuW angular frequency (rad /s), sion of soil moisture. The effects are similarly pro

~~~~~~~~470 ~IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING. VOL. 28. NO. 4. JULY 1990 300 290 280 * 270 a. E - 260Observations. Tair=278.3+1 6.9'cos(2n(mo-7.1 2)/12) 250,... I.-. 0 2 4 6 8 10 12 Month Fig. 6. Monthly average air temperatures for Bismarck, ND (from National Weather Service data).'"Model" refers to firstorder Fourier component. 102 moist soil 15X Moist Soil 20% Moist Soil 320 -_ _" 300 E 280 0hu 06 hce," V)260 12 hcur - i hCUr 320 ___________________ 300 E 280 05 260 320300 E 280 0 z 260 240 320 300 Q. E 4)

ENGLAND: RADIBRIGHTNESS OF DIURNALLY HEATED. FREEZING SOIL 471 320'- 300 7< E ~-280 - o S moist ~~~~~260 15X molst - 20X moist 240 320 300 E ~-280 0 260 320 Y. 300 E ~-280 z 260 320 X 300 E' 280 Q 260 240 0 6 12 Is 24 Solar Time, hr Fig. 8. Diurnal soil surface temperatures at Bismarck, ND. Curves represent moisture contents by weight. in Fig. 9. The variability with frequency is caused by the for these model parameters, October near Bismarck is cold

472 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING. VOL. 28. NO. 4. JULY 199 10% moist 15X moist 202 moist E 1~~~~~~~~~~~~~~~~~~~~~~~~0.7 3HZ C -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 8 GHz 0.8 -37 GHz C~ 0.4 E 0o.o........_ 1.2 E Uj. 0.8 1'2 E 40 Q0.8 0.4 E w 0.0............ 1.2,, - - - - -- - - - - 0.8 0.4 0.8 0 i2 2 8 d C ~ ~ ~ ~ ~ ~ ~ ~ ~~~SlrTmh 0i.49 fetv mtigdphvru ieo a.Efcieeitigdphi qiaett pia et notc.o o1/ skndphiElcrmgeis uvsrpeetefcieeitn etsfrtremcoaefeunis 0.1,ad3 Gwz Fig.m 9.l Efective eimittingdph vesusgtl wtim hfdy Effctieae qun sfrznol emitting depthiseuvlntopicldthnotcs. ore toug1/2c moisture content because more of the daytime insolation and less. Therefore, frozen soil thermal gradients that are

ENGLAND: RADIOBRIGHTNESS OF DIURNALLY HEATED, FREEZING SOIL 473 l0x moist 15% moist 20% moist 10.7 3HZ 18 GHZ 25 - 37 GHZ.?200 V 180 4) 260 ~)240. 220,J..~200 V 180 0 260 U)240 ~, 220 o200 V 180 0 Z 160- - - - - - - - - - - - ~j 260 In 240 220.0 -2 200 180 Solar Time, hr Fig. 10. Radiobrightness versus time of day. Curves represent brightness for three Moisture contents and at three Microwave frequencies. brightness curves during October through December are That is, the 37-GHZ radiobrightness most closely follows generally high and are relatively independent of micro- thermal temperature. It is this property that justifies its wave frequency. Midinight-noon di~ffere~nces are'ahlways use as one o~f twodsriiatsi nSM Rfoznsi positive for moist soils and geneally negative for diur-classifie (7.... nalythwig ois.Whlenoe f hee oi mdes e- Raiorihtes secra gadens s untinso

~~~~~~~~474 ~IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING. VOL. 28. NO. 4. JULY 1990'.5 1.0 - 10 moist 0.5 O*5^0 ~~~~~ -0. ~15% moist - 20 moist 0.0 September -0.5..... 1.5 N 1:.0 0.5 I — 0.0 October -0.5........... 1.5 N 1.0 0.5 0.0 NoVember -0.5 I- - - - 1.5 1.0 ~0.5 0.0 December -0.5 0 1 18 24 Solar Time, hr Fig. 11. Radiobrightness spectral grdient versus time of day. Curves represent three moisture contents. Gradients are computed as least squares linear regression of radiobrightness at 10.7, 18, and 37 0Hz. chosein as a seco-nd discriminaint in the SUMMR froze#bn soi VIU ISRPACE

ENGLAND: RADIOBRIGHTNESS OF DIURNALLY HEATED. FREEZING SOIL 475 erage +0.2 K/GHz, while the model predicts that thawed 37-GHz spectral gradient is always negative for frozen soil differences should be weakly negative; and the SMMR soils. Therefore, the "and" condition, that the 37-GHz moist soil spectral gradient can be negative on hot (sum- radiobrightness be below some threshold and that the mer) days, while the model predicts positive gradients. spectral gradient be negative, should be an effective clasWhile these discrepancies are ancillary to our objective sifier of frozen soil. of examining the performance of the 37-GHz radiobrightness and the 10.7-18-37-GHz spectral gradient as dis- REFERENCES criminants in a frozen soil classifier, they do suggest that [1] W. J. Burke. T. Schmugge. and J. F. Paris. "Comparison of 2.8- and the model is incomplete. Most importantly, it ignores vol- 21-cm microwave radiometer observations over soils with emission ume scatter darkening by prairie grasses and crop stubble, model calculations." J. Geophys. Res.. vol. 84. pp. 287-294. 1979. and by inhomogeneities within the frozen soil. The scat- 12] J. R. Wang, T. J. Schmugge, W. I. Gould. W. S. Glazar. and J. E. Fuchs, "A multifrequency radiometric measurements of soil moisture tering albedo Co is a measure of the strength of volume content over bare and vegetated fields." Geophys. Res. Let.. vol. 9. scatter darkening. The parameter was developed by Chan- pp. 416-419, 1982. drasekhar [22] to describe darkening in planetary atmo- [3] T. J. Schmugge. "Remote sensing of soil moisture: Recent advances," IEEE Trans. Geosci. Remote Sensing. vol. GE-24. pp. 12spheres, and has been applied by England [23], [24] 22, 1983. among others to describe darkening in frozen soils, ice, [4] P. J. Camillo and T. J. Schmugge. "Correlating rainfall with resnow and dry, planetary regoliths. It has become a pa- motely sensed microwave radiation using physically based models."'snw an drpaeayrglts thsbcm' a IEEE Trans. Geosci. Remote Sensing, vol. GE-22. pp. 415-423. rameter in most theories of wave propagation and scatter- 1984. ing, e.g., Ishimaru [25]. For single scattering, [5] T. J. Schmugge, P. E. O'Neill. and J. R. Wang, "Passive microwave soil moisture research," IEEE Trans. Geosci. Remote Sensing. vol. GE-24. pp. 12-22. 1986. do = Nos/( Nas + 2/5) (29)!6] T. J. Schmugge. "Remote sensing applications in hydrology." Rev. Geophys., vol. 25, pp. 148-152, 1987. where N is the number of scatterers per unit volume, as is [7] B. W. Zuerndorfer, A. W. England, M. C. Dobson, and F. T. Ulaby. "Mapping freeze/thaw boundaries with SMMR data." J. Agriculthe scattering cross section for a single scatterer, and 23 5 tural Forest Meteorol., to be published. is the power loss coefficient defined in (20). For spherical [81 H. S. Carslaw and I. C. Jaeger, Conduction of Heat in Solids, 2nd scatterers whose diameters are small fractions of a wave- ed. New York: Oxford. 1959. [9] K. Watson, "Geologic application of thermal infrared images." Proc. length (Rayleigh scatterers), IEEE, pp. 128-137, Jan. 1975. [10] K. Watson, L. C. Rowan, and T. W. Offield, "Application of thermal ora~ CAx~ ~(30) modeling in the geologic interpretation of IR images." in Remote Sensing, K. Watson and R. Regan, Eds., Geophysics Reprint Series, no. 3. Soc. Exploration Geophysicists, 1983. so that Cwo increases with decreasing wavelength to yield n.3 o.EpoainGohscss 93 so that increases with decreasing wavelength to yield [11] A. B. Kahle, "A simple thermal model of the Earth's surface for a negative spectral gradient of radiobrightness, i.e., a geologic mapping by remote sensing," J. Geophys. Res., vol. 82. "law of darkening." This short wavelength darkening is pp. 1673-1680, 1977. pp. 1673-1680, 1977. [12] G. W. Evans, "A note on the existence of a solution to a problem of the likely cause of the strongly negative spectral gradient Stefan," Quart. Appl. Math., vol. 9, pp. 185-193, 1951. observed in SMMR data for frozen terrain. [131 H. G. Landau, "Heat conduction in a melting solid," Quart. Appl. The thawed soil, midnight-noon spectral gradient dif- Math., vol. 8. pp. 81-94, 1950. ference, and the hot-day spectral gradient may be influ- [14] J. Douglas and T. M. Gallie, Jr., "On the numerical integration of a ~~~~~~~~~~~~~' ~~~~parabolic differential equation subject to a moving boundary condienced by diurnal movement of soil moisture (evapotran- tion," Duke Math. J., vol. 22, pp. 557-571, 1955. spiration?). Such variations in moisture are not included [15] F. L. Chernous'ko, "Solution of non-linear heat conduction problems in either the thermal model nor the radiation model. With- in media with phase changes," Int. Chem. Eng. J., vol. 10, pp. 4248, 1970. out more complete experimental data, any explanation of [16] P. Hoekstra and A. Delaney, "Dielectric properties of soils at UHF these effects would be little more than speculation. and microwave frequencies," J. Geophys. Res., vol. 79, pp. 16991708, 1974. [17] P. Hoekstra and W. T. Doyle, "Dielectric relaxation of surface adVII. CONCLUSION sorbed water," J. Colloid Interface Sci., vol. 36. pp. 513-521, 1971. [18] J. B. Hasted, "Dielectric properties of water and aqueous solutions," in Dielectric and Related Molecular Processes. London: Chem. A modified Chernous'ko method for solving Stefan's Soc., 1972, pp. 121-162. problem yields an acceptably rapid convergence to a so- [19] J. B. Hasted, Aqueous Dielectrics. London: Chapman and Hall, lution for the thermal structure of diurnally insolated, [20] 1973. [20] R. H. Cole and 0. Worz, "Dielectric properties of ice," in Physics moist, and frozen soil. The modification involves an al- of Ice, N. Riehl, B. Bullemer, and H. Englehardt, Ed. New York: ternative piecewise constant enthalpy-temperature ap- Plenum, 1969, pp. 456-554. that has an odd component of temperature in [21] P. R. Camp, W. Kiszenick, and D. Arnold, "Electrical conduction.ro..x.i.mation ice," in Physics of Ice, N. Riehl, B. Bullemer, and H. Englehardt, the vicinity of each isotherm. With this modification, so- Eds. New York: Plenum, 1969, pp. 450-470. lutions exhibit the expected symmetry for heating and [22] S. Chandrasekhar, Radiative Transfer. New York: Dover, 1960, p. 6. cooling. [23] A. W. England. "Thermal microwave emission from a scattering Among the 10.7-, 18-, and 37-GHz SMMR frequen- halfspace," Radio Sci., vol. 9, pp. 447-454, 1974. cies, both the SMMR observations and the models show [24]-, "Thermal microwave emission from a scatterng layer," J. that the 37-GHz radiobrightness best tracks the thermal Geophys. Res., vol. 80, pp. 4484-4496 1975. 1251 A. Ishimam, Wave Propagation and Scattering in Random Media. I. temperature of the soil's surface, and that the 10.7-18- New York: Academic, 1978, p. 11.

476 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING. VOL. 28. N 4. JULY 99 Anthony W. England (M'87-SM'89) received NASA Scientist-Astronaut in 1967 served as Mission Scientist for Apo the B.S. degree in earth sciences, the M.S. degree's 13 and 16 and flew as a Mission Specialist on Space Shuttle Cha in geology and geophysics in 1965. and the Ph.D. lenger's Spacelab 2 Mission in 1985-a solar astronomy and plasma phyI degree in geophysics in 1970 from the Massachu- ics mission. He has been Program Scientist for NASA's Space Station. an setts Institute of Technology. Cambridge. an Adjunct Professor at Rice University. He is currently Professor of Elec He has undertaken heat flow measurements trical Engineering and Computer Science at the University of Michigan throughout the Southwest: participated in and led Ann Arbor. where he teaches and conducts research in Earth and planetar geomagnetic and gravimetric studies in Montana microwave remote sensing. He was an Associated Editor for the Jourim and Antarctica. respectively: participated in radar of Geophysical Research. He served on the Earth Science Panel of th / ~~~~~sounding studies of temperature glaciers in Wash- NRC's Space Science Board and on several federal committees concerne, ington and Alaska. and of polar glaciers in Ant- with Antarctic policy, nuclear waste containment, and federal science ani arctica, and has investigated theoretically and experimentally the micro- technology. wave radiometric signatures of snow, ice. and frozen ground and of Dr. England is a member of the American Geophysical Union. and th, planetary surfaces and atmospheres. He was Deputy Chief of the Office of Administrative Committee of IEEE's Geoscience and Remote Sensing So Geochemistry and Geophysics of the U.S. Geological Survey. He was a ciety.

5AiR141j 4 BEES Fr 4d L8 ~ RADIOBRIGHTNESS DECISION CRITERIA FOR FREEZE/THAW BOUNDARIES B. Zuerndorfer and A. W. England Radiation Laboratory Deparntment of Electrical Engineering and Computer Science The University of Michigan Ann Arbor, MI 48109 September 14, 1990 Abstract -~A freeze indicator, based on a low 37 GHz radiobrightness and a low 10.7, 18, and 37 GHz radiobrightness spectral gradient, has been used to classify frozen surfaces in the northern Great Plains. By modeling the radiometer beamnpatterns as Gaussian, freeze/thaw boundaries can be located at the (fine) resolution of the 37 GHz channel. The promne of the freeze indicator, and subsequent boundary location' estimate, depends on the accuracy of the boundary decision criteria. We show that decision criteria based upon clustering and unsupervised classification yield good pefrace. We also prsent a simple algorithm for registering coarse-resolution freeze indicator boundaries to equivalent boundaries in fne-resolution 370GHz, radiobrightness images.

1 Zuerodorfer and England INTRODUCTION Soil moisture contributes to the energy exchange between the air and the ground through latent heats of fusion and vaporization. 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 [e.g., 1-7]. In addition, moisture state can also be inferred from radiobrightness. Using data from the Nimbus-7 Scanning Multichannel Microwave Radiometer (SMMR) for a test area that included North Dakota and parts of the surrounding states and southern Can, Zuerndorfer et. al. [9] showed that a combination of low 37 GHz radiobrightness, TB(37), and a low spectral gradient of radiobrightness, aT(f)/af where f is frequency, becomes an effective freeze indicator, or discriminant, for classifying frozen terrain. Frozen surfaces appear cold at 37 GHz, and exhibit a negative spectral gradient that we shall argue is largely caused by volume scatter darkening at the shorter wavelengths. This two parameter freeze indicator (El) has been applied to SMMR data from the greater North Dakota test area. The spectral gradient is a linear, least-square fit to the 10.7, 18, and 37 GHz radiobrightnesses. A surface is classified as frozen only if both TB (37) and aT,(f)/af are sufficiently low. A freeze map is generated by displaying the FI for e ach pixel location. A fundamental problem with the El algorithm is that radiobrightness measurements from different frequency channels with different spatial resolutions are required to estimate a spectral gradient. To make each radiobrightness value refer to a common area on the grund, data from each channel are compensated to a cotmmon spatial resolution so that spata averaging is identical at all frequencies. Without a priori surface information, the resolution of data at all frequencies are compensated to the (comre) resolution of the lowest fr-equency data used'in the spectral gradient estimate, and fine-resolution iformation is lost. Subsequently,,all identifiedfreeze/thaw boundaries are localized to the accuracy of the comre-resolution data. However, by modellig the radiometer

2 Zuerndorfer and England 2 beampatterns as Gaussian, results from scale-space theory [24] can be used to register coarse-resolution freeze/thaw boundaries to fine-resolution 37 GHz boundaries (i.e., to 37 GHz radiobrightness threshold crossings) [101. A second fundamental problem occurs because diurnal insolation produces temperature gradients within the surface that alter SMR noon and midnight spectral gradients [ 11 ]. In addition, time lags between air and surface temperatures, or poorly compensated heating/cooling of the SMMR instrument by sunlight and Earth shadow [12], may also produce offsets in measurements made at noon and midnight. Decision criteria for classifying surfaces, and identifying freeze/thaw boundaries, must accommodate these day/night differences. In this paper, we first examine the characteristics of frozen and thawed surfaces in decision space. Frozen and thawed surfaces are classified, and freeze/thaw boundaries located, through clustering and a standard unsupervised classification algorithm The procedure differs from our previous work in that clustering and classification are unsupervised, and the noon and midnight SMMR measurements are clustered separately. Second, we discuss the appropriateness of Gaussian beamnshape modelling, and we demonstrate a simple, automated technique for extrapolating freeze/thaw boundary estimates from the coarse spatial resolution of the freeze map to the fine spatial resolution of the 37 GHz channel. 2 BACKGROUND 2.1 Freeze Indicator Freezing influences the radiobrightness temperature of the surface, TB, through parameters in the approxiaton [11], T,9(f) = e (f)TO +AT, (f)1 where,

3 Zucrndorfer and England3 ATB,(f)-z' ((az o t e (J) is the emissivity at microwave frequencyf, To is the surface temperature, (aT/az)o is the surface thermal gradient, and z,(f) is the optical thickness of the surface at frequency f. When compared to thawed surfaces, frozen surfaces exhibit signatures of { 1 ) lower thermal temperatures, To, (2) higheremissivity, ef), (3) larger optical thickness, z,(f), and (4) decreasing radiobnghtness, TB(f). Signatures (1) and (2) are well understood, but are generally ambiguous indicators of frozen surfaces. Ambiguities arise because changes in radiobrightness that result from freezing the surface may be either positive or negative, depending upon the surface moisture content. For example, a very dry soil emissivity of 0.9 will yield a 90 decrease in radiobrightness for a decrease in soil temperatures from +50 C to 5 C. Because the soil is dry, there is relatively little change in soil emissivity with fezing. In moist soils, freezing causes an increase in soil emissivity because 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 causes a decrease in the real part of the dielectric constant., e'(f). A typical soil emissivity would increase from 0.8 to 0.9 with freezing, so that a decrease in soil temperatures f-rom +50 C to -50 C would produce a 190 increase in TB(f). Because T,(f) can either increase or decrease with fr-eezing, misclassification will results if TB(f) at a single firequency were solely used to discrimination between fr-ozen soils and soils that are warmer or drier. These variations in emissivity with fr-eezing are most pronounced at lower microwave frequencies. Signature (3) arises because freezing reduces the ianary part of the dielectric constant, "fproortioaly more than it does the real part, e(f). Reduced e~f') means reduced absorption, so that temlyemitted photons originate deeper within emittig media. Thus, the effective depth of emission,1 or optical thickness z,(f), becomes a larger fr-action of the free-space wavelength, ~

Zuerndorfer and England 4 [13-16]. The effective emission depth of moist soils is typically 10% of the fre-sace wavelength. Frozen soils have effective emission depths that may be 30% or more of free-space wavelengths. As a result, the subsurface temperature gradient of frozen soil contributes more to radiobrightness than does an equivalent gradient in thawed soil. The contribution can be several degrees at the lower microwave frequencies. Table 1 shows the contribution of thermal gradient on obrightness, ATB(f), and on spectral gradient, A(aT,/af), as a function of soil moisture content and time of day (noon la midnight). The data of Table 1 are derived from a model of typical soil near Bismark, North Dakota for September 22 [11]. The model used an incidence angle of 53.1~ and radiometer frequencies of 10.7 GHz, 19.35 GHz, and 37.0 GHz. ATB(f) is calculated at these frequencies, and A(Tf) is the least-square regression slope to the three ATB(f) values. In the model, the soil contains 7% bound water, so that a soil moisture content of 10% has a mixing ratio of 0.03 free water. We see that thermal gradients exert the strongest influence on emissions from frozen, or dry, soil. The thermal gradient produced noon-to-midnight shift in spectral gradient, A(aTB/af)mt is 0.12 K/GHz for frozen soil and, as will be shown, is consistent with SMMR observations.

5 Zuerndorfer and England5 Table 1. Contributions of thermal gradient to radiobrightness and spectral gradients for noon and midnight data Radiobrightnesses, AT, are in K and spectral gradients, A(aTg/af), are in K/GHz; A(aTBaf)M is A(T/af) at noon minus A(aTB/af) at midnight. A,(y), NOON AT(/'), MIDNIGHT moisture 10.7 19.35 37.0 10.7 19.35 37.0 content, % GHz GHz GHz 4r) GHz 0Hz 0Hz 4r) 7(Frozen) -3.5 -.8 -1.0 0.09 1.3 0.7 0.4 -0.028 0.12 10 -2.3 -1.0 -0.5 0.061 0.6 0.4 0.3 -0.013 0.07 15 -1.4 -0.6 -0.3 0.039 0.3 0.2 0.1 -0.007 0.05 20 -1.0 -0.4 -0.2 0.029 0.2 0.1 0.1 -0.005 0.03 25 -0.8 -0.3 -0.1 0.023 02 o.1 0.1 -0.003 0.03 As a consequence of signatures (1), (2), and (3), the 37 GHz SMMR radiobrightness is more strongly corelated with air temperature than are the 10.7 GHz and 18 GHz SMMR radiobrightnesses. That is, the 37 GHz radiobrightness should serve effectively as one discriminant among fr-ozen and thawed soils. If soil is modeled as a diurnally heated, homogeneous halfspace, the spectral gradients of radio'brightness are always positive for thawed soils, and are slightly negative (typically -0. 1 K/GHz for frozen soils) [ 11 ]. However, observed spectral gradients in frozen soils -- signature (4) -- may be more negative than -1.0 K/GHz. The likely cause of such strongly negative gradients is an increased darkening at shorter wavelengths caused by volume scattering within the frozen soil. In the more transparent emitting media, such as frozen soil or dry snow, the grater average thermal photon path length provides a greater opportunity for volume scattering of photons. Scattering is more severe at shorter wavelengths because soils and plants appear increasingly heterogeneous at the scales of these wavelengths [14,17]. Thus, a negative spectral gradient should correlate with frozen soil, and the radiobrightness spectral gradient should serve as a second discrminant among frozen and thawed soils.

Zuerndorfer and England6 2.2 SMMR Data The data in Figures 1-3 are derived from SMMR radiobrightness measurements made from August 1984 to December 1984 over seven meteorological sites. These sites -- Miles City, MT; Bismarck, Fargo, and Williston, ND; and Aberdeen, Huron, and Rapid City, SD -- are within a test area that includes North Dakota, about half of each neighboring state, and part of Canada. The radiobrighmess data are averages of horizontally and vertically polarized SMMR radiobrightness measurements. All data have been compensated to the (coarse) resolution of the 10.7 GHz channel, and are spatially interpolated to the latitude and longitude of each meteorological site using a bi-cubic approximation to a sinc function [34]. Scatter diagrams of TB(37) as a function of meteorological air temperature (Figures la and Ib) show a nominal tracking of air temperature by TB(37). However, there is an approximate 40 decrease between the noon and the midnight regression lines caused by air temperatures that lag surface temperatures. A simple regression model for T. (37) would be, TB (37) = e (37) [TMjR + Tms(t)J (2) where, e (37) S 37 GHz Emissivity T~J *Air Temperature (K) Tms*) Temperature Bias (K) t.Time, so that, 4 e(37)

Zuernidorfer and England 7 s of Z)T are shown in ~~Figures 37Gz Equivalent scatter diagrams of T~f as a function of air temperature are shown in Figs 2a and 2b for noon and midnight SMMR data,, respectively. The values of DT8/Zf are the slopes of linear-least-square regressions, as functions of frequency, to SMMR 10.7, 18, and 37 GHz radiobrightnesses at each meteorological site. The data of Figures 2a and 2b show the predicted decrease in spectral gradient with decreasing air temperature (i.e., as surfaces freeze). There are also the anticipated 0.1 K/GHz increase in noon gradients relative to midnight gradients caused by diurnMal heating and cooling. In addition, a negative tendency in aTB/af is observed at noon for high air temperatures. This may be caused by volume scattering by dried surface vegetation. Heat and plant senescence in late-summer decrease the moisture in the vegetation canopy. This dry vegetation will act as a scattering layer -- particularly at higher microwave frequencies. These data (Figures 1 and 2) yield scatter diagrams of aT~laf as a function of TB(37) (Figures 3a and 3b). In Figures 3a and 3b, data labelled as "frozen" have air temperatures less than 270 K, and "thawed" data have air temperatures greater than 274 K. Data labelled as "mixed" have air temperatures between 270 K and 274 K, inclusive. While air temperature is an imperfect indicator of frozen terrain, we see that low TB(37) and oTIaf correspond to frozen surfaces. 2.3 Freezing Terrain and Snow Pack Snow packs produce emission darkening and negative spectral gradients [17], and trace amounts of snow were often observed whenever fr-eezing occurred. It might be argued that our observed negative spectral gradients are caused purely by the snow pack. Figures 4a and 4b show aTBcf)Iaf as a function of TB(37) and snow cover for those noon and midnight SMMR data that correspond to air temprtrs less than 271 K. The figures show the expected trend towards decreasing oTB(f)/af and decrasi'ng TB(37) with increasing snow depth. However, there is such variability with snow depth that other phenomena must occur.

Zuerndorfer and England 8 The results of McFarland et. al. [26] indicate that snow depths of 2 cm or less show only slight darkening, particularly over dry soil. Some of our data show rather strong darkening at snow depths of under 2.5 cm. Burke et. al. [27] used emissive darkening at 37 GHz to locate snow boundaries on bare soil, but could not determine snow boundaries over regions of frozen ground. These observations suggest that emission darkening of frozen terrain is occurring. While the snow studies of [26] and [27] were not explicitly concerned with frozen terrain, Wegmflller measured soil temperatures and the radiobrightness of frozen terrain (at 3.1 GHz, 4.5 GHz, 7.2 GHz, and 10.6 GHz) [33]. During his study period, the ground was snow-free and the soil froze to a depth of roughly 5 cm during early-morning hours. The 7.2 and 10.2 GHz penetration depths in frozen soil were calculated as 2.0 cm and 1.2 cm, respectively, so that the early-morning 7.2 GHz and 10.6 GHz radiobrightnesses were produced by frozen terrain. A plot of vertically polarized radiobrightness versus time-of-day [33, Figure 1] shows the 10.6 GHz radiobrightness falling below the 7.2 GHz, radiobrightness during the early-morning. Moreover, the temperature differential between the 10.6 and 7.2 GHz radiobrightnesses'increased from midnight to dawn (as the ground froze). Clearly, freezing terrain is a source of emission darkening. However, snow packs, frozen soil, and dry or frozen vegetation are all potential sources of volume scattering. We hope to differentiate amoung these through on-going research. 3 PROCESSING Clustering deemns decision criteria for coarse-resolution classification, and the location of freeze/thaw boundaries. The location of these freeze/thaw boundaries can be refined by imposing spatial constraints.

9 Zuerndorfer and England 9 3.1 Data Clustering The decision criteria for T1(37) and aT/Baf were based on clustering and unsupervised classification. Unsupervised classification, rather than supervised classification, is used because of the dearth of accurate ground measurements in our test area. Seven air temperature and eleven soil temperature recording sites provided the ground data for the entire test area. Soil temperatures ~~w~~~~~~~s daa were measured at 5 cm depth and were made at dawn and dusk, whereas SMMR overflights were at noon and midnight. To increase the number of data available for clustering beyond that used for Figures 1-3, all data from SMMR satellite passes that covered more than 67% of the test area were incorporated in the scatter diagrams of Figures 5-6. Sixteen noon SMMR passes and thirteen midnight passes met this criterion during our August to December test period. As before, the 18 and 37 GHz averages of vertical and horizontal polarized radiobrightness data were resolution compensated to the (coarse) resolution of the 10.7 GHz channel, but unlike the data of Figures 1-3, these compensated data were re-sampled on a 97.5 k~m grid (i.e., at the resolution of the 10.7 GHz channel). Spectral gradients were computed as the slopes of least-square linear regressions to the 10.7, 18, and 37 GHz radiobrightnesses. Scatter diagrams for the noon and midnight data are shown by month in Figures 5a and 5b, respectively. Migrating means clustering determined cluster centroids, for the data of Figures 5a and Sb [18,19]. On the basis of ground measurements and the data of Figures 1-3, surfaces were classified into three distinct types — frozen, hot (and dry), and wet (and cool) -- and a fourth type that we call mixed. A frozen surface is characterize by relatively low spectral gradient and a low 37 GHz. radiobrightness. Due to the influence of liquid water in the surface, a wet surface is characterized by a high spectral gradient [1 1] and low 37 GHz radiobrightness [20]. A hot surface has relatively less surface moisture, producing a "dry" surface dielectric constant simiar to a frozen surface. Moreover,_ the relaxat - Ion -— __ — _ frequncy o freewate inrese ittmertue furhe reucn ta

Zuerndorfer and England 10 spectral gradients of hot surfaces. As a result, a hot surface has a relatively low fpectral gradient and a high 37 GHz radiobrightness. A mixed surface has a combination of frozen, wet, and hot characteristics. Prior to freezing, a surface region is a combination of wet (and cool) and hot (and dry). As freezing begins, the region includes locally frozen surfaces, and would be classified as mixed. A freeze/thaw criteria lies within the mixed surface cluster in decision space, and represents maximum TB(37) and aTT/1)f values along the freeze/thaw boundary. That is, any surface point on the freeze/thaw boundary has at least one component. TB(37) or DfT8/af, equal to that of the freeze/thaw criteria. Equivalently, the FI algorithm [9] requires any surface point classified as frozen to have spectral gradient and 37 GHz radiobrightness less than those of the freeze/thaw criteria. Cluster centroids determined for the data of Figures 5a and 5b are given in Table 2. Due to SMMR recording problems, limited midnight data were available during December of our test period. Within this limitation, the frozen surface cluster centroid has a lower spectral gradient and 37 GHz radiobrightness at noon than at midnight. Furthermore, because there were few wet surfaces at midnight during the test period, the wet and mixed surface types were inseparable for the midnight data. Table 2. Cluster centroid in decision space _____ l NOON DATA MIDNIGHT DATA 37 GHz Gradient 37 GHz Gradient Surface Type (K) (K/GHz) (K) (K/GHz) Frozen 227 -.43 234 -.35 Hot 277 0.11 258 -.01 Wet 238 0.37 Mixed 250 0.14 243 0.015 Bivariate normal distributions were fit to the cluster data All data within three standard deviations of a cluster centroid were classified using a Mahanalobis, minimftum, distance classifier

Zuerndorfer and England 1 Constant-deviation, single-class ellipses were drawn in decision space for frozen, hot, and wet surfaces (at noon) and for frozen and hot surfaces (at midnight) using the classified data. The freeze/thaw criteria was determined by allowing the deviation of all ellipses to expand equally until all ellipses intersected. The resulting classification ellipses for noon and midnight SMMR data are shown in Figures 6a and 6b, respectively. The corresponding fteeze/thaw criteria in decision space are shown in Table 3. Table 3. Freeze/thaw criteria in decision space; a are standard deviations of data within the ellipses. -. =, I I I l I,, Refined Deviation at 37 GHz(K) 37 GHz(K) Gradient(K/GHz) Intersection ~~~~~~~~~~~~~~~~i. i. i 111 Noon 252 249 0.0625 3.1 a Midnight | 247 244 -0.044 2.55 a 3.2 Refining the Decision Criteria The TB(37) boundary pixels have 37 GHz radiobrightnesses that equal the TB(37) component of the freeze/thaw criteria. If we view the freeze/thaw criteria derived from clustering as initial estimates for determining the freeze/thaw boundary, we suggest the boundary can be refined by equiring a minimum scatter of TB(37) along that boundary. This constraint ensures boundaries in TB(37) images correspond closely to Fl boundaries. The TB(37) component of the freeze/thaw criteria, T37, is adjusted to miiiethe sum square error, SSE, in, N SSE = ~ [T(37)-T J,(4) where,

~~~~Zuerndorfer and England ~12 N number of TB(37) boundary pixels T(37) 37 GHz radiobrightness at the ia boundary pixel. Equivalently, the refined T37 isthe average 37 GHz radiobrightness on the boundary. The process is first-order since we do not reiterate SSE minimization with the refined criteria. The refined T37 for midnight data from October24 (Figures 8 and 10) and for noon data from December 11 (Figures 9 and 1 1), are shown in Table 3. 3.3 Boundary Localization Spectral gradients are regression slopes to SMMR 37 GHz, 18 GHz, and 10.7 GHz radiobrightness measurements. The nominal resolutions of these channels are 30 km, 60 kmn, and 97.5 kin, respectively. Without compensating for the resolution differences between the channels, the spectral gradient estimates can be in error. For example, a non-zero gradient estimate can result from radiobrightnesses that are spatially variant but are locally constant over frequency. To avoid such errors, the image data were compensated to one common resolution -- the (coarse) resolution of the lowest frequency channel used in gradient estimation -- prior to clustering. Freeze/thaw boundaries combine 37 GHz threshold crossings and spectral gradient threshold crossings. Corresponding 37 GHz threshold crossings occur in fine-resolution 37 GHz images, but not all 37 GHz threshold crossings represent fireeze/thaw boundaries. Some, as previously noted, are boundaries between moist and dry terrain. Boundary localization is a three-step process that identifies pixels in fine-resolution, 37 GHz images that correspond to freeze/thaw boundaries at coarse-resolution. ~teIJ: Uncompensated 10.7 GHz, 18 GHz, and 37 GHz SMMR data are compensated to

~~~Zuer~ndo~rfer and England ~13 [22]. Antenna data for the Nimbus-7 SMMR antenna are limited,) and some investigators have assumed an antenna pattern based upon a uniformly illuminated SMMR aperture [23]. Such an illumination would produce beamwidths that are much narrower than those specified for the SMMR [21]. Alternatively, we assume that the Seasat SMMR beamnpattern [30] approximates the Nimbus-7 SMMR beampattern, and justify Gaussian filteringby showing that the Fourier transform of the Seasat SMMR beampattern is approximately Gaussian (Figure 7). The Gaussian filters used to synthesize compensated data at resolution s2 from uncompensated data at resolution s, are, f,95= e 2 (5) where the filter width, S, is, = 2_ 21/2 (S(2-s1) 9,2 I andf is spatial frequency. Values of Sfordifferent configurations of resolution compensation are shown in Table 4. Table 4. Filter bandwidths for resolution compensation Nominal Synthesized Filter Resolution, s, Resolution, s2 Bandwidth,, $ 30 km(Fine) 60 km(Medium) 51.96 k 30 km(Fine) 97.5 km(Comre) 93.77 k 60 km(Medium) 97.5 km(Comre)- 78.885 k

Zuerndorfer and England 14 SI 2: Using resolution compensated data, TB(37) and aTB/af are calculated for each image pixel at coarse-resolution. Boundaries in coarse-resolution, 37 GHz images are identified where 37 GHz data satisfy the TB(37) freeze/thaw criteria. Pixels along these 37 GHz image boundaries with DT,/~f at or below that of the freeze/thaw criteria are identified as freeze/thaw boundary pixels. ~tep 1~: Fine-resolution, freeze/thaw boundaries are determined by identifying those pixels in fine-resolution, 37 GHz data that satisfy the TB(37) freeze/thaw criteria and correspond to coarse-resolution freeze/thaw pixels of step 2. This process involves tracking boundary locations in 37 GHz images as the amount of resolution compensation is reduced. The resulting boundary locations in the fine-resolution 37 GHz images are best estimates of freeze/thaw boundaries in the sense that they are directly traceable to the coarse-resolution boundaries generated by clustering and maximum likelihood classification. The key to this process is that Gaussian image degradation of step 1 uniquely permits recovery of some fine-resolution information, a result derived in scale-space theory [24,25,32]. 4 RESULTS 4.1 SMMR Images Unrefined fireeze/thaw criteria (Table 3) were applied to SMMR data for midnight October 24 (Figure 8) and for noon December 1 1 (Figure 9). Refined freeze/thaw criteria were also applied to the October and Deccmber data (Figures 10 and 1 1, respectively). The dark pixels in the freeze ma3(Figures 8a, 9a, 10a, and 1 la) correspond to surfaces with low Fl value -- surfaces which are most likely frozen -- and fr-eeze/thaw boundaries appear as a fuzzy white lines around these frozen

~~Zuernm~dorfer and England ~15 regions. The dark pixels in the the 37 GHz images (panels b, d, and f in'Figures 8, 9, 10, and 11) correspond to surfaces of low 37 GHz radiobrightness. The fuzzy white lines around these dark regions are the boundary pixels that satify the TB(37) freeze/thaw criterion. Some or all of these boundaries correspond with the coarse-resolution freeze/thaw boundaries of the freeze maps. Similarly, the dark pixels in the spectral gradient images (Figures 8c, 9c, 0lc, and 1 lc) correspond to surfaces with low spectral gradient, and the fuzzy white lines are boundary pixels that satify the TB(f)/af freeze/thaw criterion. In. all images, regions of no data are shown as white. Comparing the "unrefined" images of (Figures 8 and 9) with the "refined" images (Figure 10 and 11) shows that refined criteria generate coarse-resolution, 37 GHz boundaries that are located more closely to freeze map and spectral gradient boundaries. Moreover, refined fine-resolution, 37 GHz boundaries (Figures l0f and 1 If) are more consistent with ground data than are unrefined 37 GHz boundaries (Figures 8f and 9f). Thus, freeze/thaw boundaries derived from refined criteria should be more accurate that those derived from unrefined criteria. In the refined images of Figure 10, most sections of the coarse-resolution, 37 GHz boundary in the northwest corner of the test area (Figure l0b) correspond with boundaries of the freeze map (Figure 10a). These sections of 37 GHz boundary would be designated as freeze/thaw boundaries. None of the two other boundaries in Figure l0b correspond to any freeze map boundary, and are probably wet/dry boundaries. The freeze/thaw boundary in the coarse-resolution, 37 GHz radiobrightness image also corresponds to boundaries in medium-resolution and fine-resolution', 37 GHz images. That is, medium-resolution and fine-resolution freeze/thaw boundaries are the convoluted boundaries in the northwest corner of Figures l0d and 10f, respectively. Some boundaries are formed at fine-resolution that do not correspond to any boundary observed at coarse-resolution. These boundaries appear around dark radiobrightness "islands" in Figure l0f, and cannot be identified on the basis of the available information. Such boundaries are not part of the fiteeze/thaw boundary estimnates.

~~~Zuerndorfer and England ~16 In Figure 11, all coarse-resolution7 GHz radiobrightness boundaries (Figure lib) correspond the freeze map boundaries (Figure 1 la), and all fine-resolution boundaries (Figure 1 if) become freeze/thaw boundary estimates. dnight and noon fine-resolution freeze/thaw boundaries (Figures Of and 1 f) are consistent with midnight and noon ground data (Figures 1 Oe and 1e). 4.2 Automation Figures 12 and 13 represent automated boundary localization for the October midnight and December noon SMMR data, respectively. Figures 12a and 13a repeat the freeze maps of Figures i0a and 1 ia, and Figure i2b and i3b show associated coarse-resolution, 37 GHz radiobrightness images. As before, 37 0Hz boundaries are composed of pixels whose 37 GHz radiobrightness equals the TB(37) component freeze/thaw criteria. However, the 37 GHz boundaries in Figures 12b and i3b consist of (fuzzy) white and black sections. Pixels along white boundaries have spectral gradients that are less than or equal to the aT,()/af component of the freeze/thaw criteria. That is, white boundaries are most likely to be freeze/thaw boundaries. Pixels along black boundaries have larger spectral gradients and are less likely to be fireeze/thaw boundaries. In medium-resolution, 37 0Hz radiobrightness imgs (Figures i2c and i3c), white boundaries are 37 0Hz boundaries that are migrations of white boundaries at coarse-resolution (Figures i2b and i3b). White boundaries at medium-resolution are (spatially) near white boundaries at coarse-resolution. Precise distances for boundary migration are calculated from ideal 370GHz radiobrightness measurenents and actual freeze/thaw boundary locations (i.e., radiobrightnesses and boundary locations hypothetically measured infinitesima resolution) [31]. Such ideal data is generally unavailable. However, using the midnight and noon SMMR data, a migration limit of (s2 - sl1)4 was detemndfor trackig a boundaries from coarser-to-finer resolution images; s, and sare the resolutions of the finer and coarser resolution images, repectively (Table 4). As a result, white boundaries at medium-resolution (Figures 12c and 13c) must be within 9.325 kmof white

~~~~Zuer~dorfer and England ~17 process, white boundaries at fine-resolution (Figures 12d and 13d) are 37 GHz boundaries that are within 7.5 km of white boundaries at medium-resolution (Figures 12c and 13c); s1 = 30 km and s2 =W6 km. Frozen terrain is identified iteratively using fine-resoluton, 37 GHz data. First, pixels along white, fine-resolution boundaries (Figures 12 and 13d) are identified as "frozen" pixels. Second, pixels whose 37 GHz radiobrightnesses are less than or equal to the TB(37) freeze/thaw criterion, and are contiguous to frozen pixels, are also identified as frozen. Third, the previous step is repeated until no additional pixels are identified as frozen. Fourth, the resulting collections of frozen pixels constitute regions of frozen terrain. Using this procedure, terrain identified as frozen are indicated by whitened regions in the northwest coer of the Figure 12e and the all but the southeast corner if Figure 13e. Because freeze/thaw boundaries must be closed contours, the final freeze/thaw boundary (i.e., the edge of the identified frozen region) contains boundary sections that did not', previously, show strong freeze/thaw boundary indications. Nonetheless, the final freeze/thaw boundaries of Figures 12e and B3e are the best fine-resolution estimates of the actual fr-eeze/thaw boundaries using available data. The images of Figure 14a through Figure 14i are the results of automation applied to data obtained at irregular intervals from October 24, 1984 through December 1 1, 1984. Time summaries of the data are given in Table 5. These images show the growth and contraction of ground-freeze from October 24 to November 5, and again fr-om November 27 to December 9. After December 9, the area remains frozen through the end of December.

able 5. Time summary for images of Figure 14a through Figure 14i; measurement interval is the time interval between present and previous measurements. Measurement Measurement Measurement Figure Date Time-of-Day Interval (Days) 14a October 24 Midnight - 14b October 30 Midnight 6 14c November 1 Noon 2.5 14d November 5 Midnight 3.5 14e November 27 Midnight 22 14f November 29 Noon 2.5 14g December 3 Midnight 3.5 14h December 9 Midnight 6 14i December 11 Noon 2.5 5 CONCLUSIONS In Zuerndorfer et. al. [9], a freeze indicator algorithm was developed that used a low 37 GHz radiobrightness and low spectral gradient to classify frozen soil. In this paper, results of the freeze indicator were reviewed using SMMR radiobrightness data collected over the northern Great Plains from August to December, 1984. Effects of snow pack on radiobrightness are similar to that of frozen ground. However, our observations could not be explained by the presence of snow pack alone, and frozen ground was indicated. location estimtes of freeze/thaw boundaries.

19 6 ACKNOWLEDGEMENT appreciate his friendly and stimulating correspondences.

~~Zuerndorfer and England ~20 REFERENCES Burke, W.J. T. Schmugge, and. Paris, 1979, Comparison of 2.8- and 21-cm microwave radiometer observations over soils with emission model calculations, JGR 84, pp. 287-294. [2] Wang, J.R., T.J. Schmugge, W.I. Gould, W.S. Glazar, and J.E. Fuchs, 1982, A multi-frequency radiometric measurement of soil moisture content over bare and vegetated fields, Geophys. Res. Let. 9, p. 416-419. [3] Blanchard, B.J., and A.T.C. Chang, 1983, Estimation of soil moisture from Seasat SAR data, Water Res. Bull. 19, pp. 803- 8 1 0. [4] Schmugge, T.J., 1983, Remote sensing of soil moisture: Recent advances, IEEE Trans. on Geosc. and Rem. Sens. GE-21, pp. 336-344. ~~~~~~~~oel[5] 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. 415-423. [6] 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, pp. 12-22. [7] Grody, N.C., 1988, Surface identification using satellite -microwave radiometers, IEEE Trans. Geosc. and Rem. Sens., V. 26, pp. 850-859. [8] Schmugge, T.J., 1987, Remote sensing applications in hydrology,Rev. Geophys. 2S, pp. 148-152. [9] Zuerndorfer, B.W., England, A.W., Dobson, C.M., and Ulaby, F.T., 1989, Mapping freeze/thaw boundaries with SMMAR data J. Agriculture and Forest Meteorology, in press. [10O] 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, Cana. [ 11] England, A.W., 1990, Radiobrightness, of diurnally heated, fr-eezing soil, IEEE Trans. Geosc. and Rem. Sens., GE-28, No. 4, pp. 464-476. [ 12] Millman, A. 1988, Personnnal co~mmuicaton, Juneimi 27, 1098.9

~~~~Zuemdorfer and England ~21 13] England, A.W., 1974, The effect upon microwave emissivity of volume scattering in snow, in ice, and in fzen soil, Proc. URSI Spec Mtg on Microwave Scattering and Emission from the Earth, Bee, Switzerland, 23-26 Sept., 1974. [14] England, A.W., 1975, Thermal microwave emission from a scattering layer, JGR 80, pp. 4484-4496. [15] England, A.W., 1976, Relative influence upon microwave emissivity of fine-scale stratigraphy, internal scattering, and dielectric es, Pageoph 114, pp. 287-299. [16] England, A.W., 1977, Microwave brightness spectra of layered media, Geophysics 42, pp. 5 14-521. [17] 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. [18] Richards. J.A., 1986, Remote Sensing Digital Image Analysis, Springer-Verlag, Berlin. [19] Clustering and classification was perfrmed on a Sun-4 workstation using EASI software, version 4. 1, from PCI, Inc. of Richmond Hill, Ontario (Canad). [20] Hoehstra, P. and Delaney, A., 1974, Dielectric properties of soils at UHF and microwave fre~quencies, J. Geophys. Res. 79, pp. 1699-1708. [21] NASA., 1978, The Scanning Multichannel Microwave Radiometer (SMMR) experiment, The Nimbus-7 Users Guide., The Landsat/Nimbus Project, Goddard Space Flight Center, NASA, p. 213-245. [22] Bracewell, R. N.,, 1986, The Fourier Transform and Its Applications, McGaw-Hill. [23] Chin, R.T., Yeh, C., and Olson, W.S., 1985,, Restoration of multichannel microwave radiometric images, IEEE Trans. Patt. Anal. Mach. Intell., PAMI-7, pp. 475-484. [24] Witkin, A., 1983, Scale-space filtering, Proc. Int. Joint. Con!. Aruif. Intell., Karlsruhe, West Germany, p. 1019-1021.

Zuerndorfer and England 22 [25] Yuille, A., and T. Poggio, 1986, Scaling theorems for zero crossings, IEEE Trans. Pan. Anal. Mach. Intell., Vol. PAMI-8, No. 1, P. 15-25. [26] McFarland, M.J., G.D. Wilke, and P.H. Harder, 1987, Nimbus 7 Investigation of snowpack properties in the northern great plains for the winter of 1978-1979, IEEE Trans. on Geosc. and Rem. Sens. GE-2S, pp. 35-46. [27] Burke, H.-H.K., C.J. Bowley, and J.C. Barnes, 1984, Determination of snowpack properties from satellite passive microwave measurements, Remote Sensing Environ., Vol. 15, pp. 1-20. [28] Kunzi, K.F., P. Subash, and H. Rott, 1982, Snow-cover parameters from Nimbus 7 Scanning Multichannel Microwave Radiometer (SMMR) data, IEEE Trans. on Geosc. and Rem. Sens. GE-20, pp. 452-467. [29] Barton, D.K. and H.R. Ward, 1969, Handbook of Radr Measunnents, Prentice-Hall, Inc., Englewood Cliffs, NJ. [301 Njoku, E.G., J.M. Stacey, and F.T. Barath, 1980, The Seasat Scanning Multichannel Microwave Radiometer (SMMR): Instrument desription and prormance, IEEE Trans. Ocean Engin. OE-S, pp. 100- 115. [31] Zuerndoffer, B. and G. H. Wakefield, 1990, Applications of scale-space filtering to signature analysis, IEEE Trans. Acoust. Speech Signal Process., under review. [32] Zuerndorfer, B. and G. H. Wakefield, 1990, Extensions of scale-space filtering to machine-sensing systems, IEEE Trans. Pant. Anal. Mach. Intell., in press. [33] Wegmd~ller, U., 1990, The effect of a frozen soil layer on the microwave signatures of bare soil, Remote Sensing Environ., in press. [34] Moik, J., 1980, Digital Processing of Remotely Sensed Images, NASA, NASA SP-43 1.

~Zuerndorfer and Engi~and ~23 FIGURE CAPTIONS Figure 1. TB(37) versus measured air temperature at meteorological sites in North Dakota and the surrounding region. Data were collected from 8/1/84 to 12/31/84. (a) Noon data (b) Midnight data Figure 2. aT,/af versus measured air temperature at meteorological sites in North Dakota and the surrounding region. Data were collected from 8/1/84 to 12/31/84. (a) Noon data (b) Midnight data Figure 3. aT5/af versus TB(37) at meteorological sites in North Dakota and the surrounding region. Data were collected from 8/1/84 to 12/31/84. (a) Noon data (b) Midnight data Figure 4. aTBIa~f versus TJ7(37) at meteorological sites in North Dakota and the surrounding region, for measured air temperature less than 271 K. Data were collected from 8/1/84 to 12/31/84 and sorted by snow depth. (a) Noon data

Zuerndorfer and England 24 Figure 5. Scatter diagram of cT/laf versus TB(37) throughout North Dakota and the surrounding region. Data were collected from 8/1/84 to 12/31/84. (a) Noon data (b) Midnight data Figure 6. Single class ellipses of aTB/af versus T3(37) throughout North Dakota and the surrounding region. Data were collected from 8/1/84 to 12/31/84. (a) Noon data (b) Midnight data Figure 7. Spatial frequency response of Seasat SMMR beampattern versus Gaussian model. Figure 8. A comparison of reported air and soil temperatures with images of North Dakota and the surrounding region. Boundaries were determined using unrefined freeze/thaw criteria. Data were collected at midnight, October 24, 1984. (a) Freeze map at coarse-resolution (b) 37 GHz rdobrightness at coarse-resolution (c) Spectra gradient at coorse-resolution (d) 37 GHz radiobrightness at meimrsolution (e) Air and soil temperatures (t) 37 GHz radiobrightness at fine-resolution

~~~Zuerndorfer and England ~25 Figure 9. A comparison of reported air and soil temperatures with images of North Dakota and the surrounding region. Boundaries were determined using unrefined freeze/thaw criteria. Data were collected at noon, December 11, 1984. In (a) Freeze map at coarse-resolution (b) 37 GHz radiobrightness at coarse-resolution (c) Spectral gradient at coarse-resolution (d) 37 GHz radiobrightness at medium-resolution ~~~-(e) Air and soil temperaturesution (f) 37 GHz radiobrightness at fine-resolution Figure 10. A comparison of reported air and soil temperatures with images of North Dakota and the surrounding region. Boundaries were determined using refined fireeze/thaw criteria. Data were collected at midnight,, October 24, 1984. (a) Freeze map at coarse-resolution (b) 37 GHz radiobrightness at coarse-resolution (c) Spectral gradient at coorse-resolution (d) 37 GHz radiobrightness at medium-resolution (e) Air and soil temperatures (f) 37 GHz radiobrightness at fine-resolution

~Zuern~dorfer and England ~26 Figure 11. A comparison of reported air and soil temperatures with images of North Dakota and the surrounding region. Boundaries were determined using refined freeze/thaw criteria. Data were collected at noon, December 11, 1984. (a) Freeze map at coarse-resolution (b) 37 lHz radiobrightness at coarse-resolution (c) Spectral gradient at coarse-resolution (d) 37 GHz radiobrightness at medium-resolution (e) Air and soil temperatures (f 370GHz radiobrightness at fine-resolution Figure 12. Automated images of North Dakota and the surrounding region. Boundaries were determined using refined freeze/thaw criteria. Da ta were collected at midnight, October 24, 1984. (a) Freeze map at coarse-resolution (b) 37 0Hz radiobrightness at coarse-resolution (c) 370GHz radiobrightne'ss at medium-resolution (d) 37 0Hz radiobrightness-at fine-resolution (e) Classified fr-ozen ground at fine-resolution

~Zuernm~dorfer and England ~27 Figure 13. Automated images of North Dakota and the surrounding region. Boundaries were determined using refined freeze/thaw criteria. Data were collected at noon, December 11, 1984. (a) Freeze map at coarse-resolution (b) 370GHz radiobrightness at coarse-resolution (c) 37LHz radiobrightness at medium-resolution (d) 37 Hz radiobrightness at fine-resolution (e) Classified frozen ground at fine-resolution Figure 14. Automated images of classified frozen ground of North Dakota and the surrounding region. Data were collected at irregular intervals from 10/24/84 to 12/11/84. (a) Midnight, 10/24/84 (b) Midnight, 10/30/84 (c) Noon., 11/1/84 (d) Midnight, 11/5/84 (e) Midnight,, 11/27/84 (1 Noon, 11/29/84 (g) Midnight, 12/3/84 (h) Mfidnight, 12/9/84 (i) Mfidnight, 12/11/84

~Zuern~dorfer and England ~28 FIGURES 37 GHz Radiobrightness Noon Data 300 290-4 ___. —.1....-a = ~~~~~280 290 300 30 32 ~260 260 C~~~~~~~C 240 1 230 220___ 210 240 5 260.. 270 280 290 300 310 320 Air T rpefture (K) ~ Data -Regression Fig. I a 3 7 GHz Radiobrightness Midnight Data 300 290 ~280'270 260 250 240 Cs__ 230 220 __ ___ _ 210 240 250 20 270i 280 290 3600 3101 320 Air Temprwature (K) 0 Data -Regresin Fig. lb

~~Zuer~dorfer and England ~29 ISpectral Gradient Noon Data ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~1.4 _____:~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~. a~~~~~~~~~~~~,.,n 1.2 - __ N 06_____ ___ I I 00.4 0.2 *. 0 35 -0.4 _I _ -0.8- - - - -1~~ ~ ~ ~~~~~~ - - -!!~ -1.2 __-1.4- -................. ~~~~280 290 30260 270 2 290 0 320 Air Tperature (K) Fig. 2a Spectral Gradient Midnight Data 1.6 - - 1.4: 1.2 - - -0.4 -0.6 -0.8 0-1 -1.2-1. 240 250 260 270 280 290 300 310 320 Air Temperature (K) Fig. 2b

~Zuern~dorfer and England ~30 Decision Space Noon Data, 6' 4 250 27 1.4 -IA- - 0.2 - 0.2 -- o-0.6 - -1 - -1.4~ -r....... 200 210 220 230 240 250 260 270 280 290 300 37 GHz Rodiobrightness (K) i Thawed X xed A Frozen Fig. 3a Decision Space Midnight Data -14 -1.2 -1.4 -1.6 200 210 220 2310 240 250 260 270 2830 2'90' 300 37 GHz Rodlobrightness (K) 0 Thawed X Wixed A Frozen Fig. 3b

~~~Zuefrni~dorfer and England 31 Gradient vs. 37 GHz Temp. (NOON) Sorted by Snow Depth ~~~0.6 -,___ _ 0.5 -____________ 0.4 -_ _ __ _ _ 0.3 - _ _ _ _ _ _ _ _ _ _ _ _ ~,0.2 I ~0.1 __ _x__. 0_ Y -0.1 Vi -0. 4 ___ -0.5 ______ -0.6______ _ -0.7______ _ ~~22~240 25 240 260 37 GHz Rodiobrightness (K) No Snow ~<2.5 cm Snow * 2.5 cm Snow ~x 5 ~-10 cm Snow 7.5-0cmSw >10 cm Snow Fig. 4a Gradient vs. 37 GHz Temp. (MIDNIGHT) Sorted by Snow Depth 0.6 _ _ _ _ _ _ 0.5 ______ 0.4__ _ _ _ _ __ _ _ _ _ 0.3 _ _ _ _ _ _ 0.2 _ _ _ _ _ _ _ _ _ _ _ 0.1 _ _ _ _ *-0.2 jj-0.43_ __ _ -0.5 ______ _ -0.7-N —-______ 220. 37 GHz Rodlobrightress (K) 03 No Snow 0 <2.5 cm Snow + 2.5 cm Snow x S cm Snow A 7.5 -lO0Snow * >lI crn Snow Fig. 4b

Zuemdofer and England 32 SMMR Cluster Data, Noon 1.600................... A 1.200 A -~. 0.800 - 0 X N A 0 &i A C),`2 0.400 0. *0.400, -0.800 - % -1.200 St -1.600....,..........,..,...|... 200.0 210.0 220.0 230.0 240.0 250.0 260.0 270.0 280.0 290.0 300.0 37 GHz Brightness (K) Fig. 5a SMMR Cluster Data, Midnight 1.600 1.200 0.400 A A o ~~~~A * 0.000 -0.400 --- -1.200 vim___ __ _ 200.0 210.0 220.0 230.0 240.0 250.20.0 270.0802.000 37 GHz Brightness K

Zuemdorfer and England 33 SMMR Cluster Data, Noon 1.600... 1.200 -~. 0.800 -~ 10.400 -.a. 0.0 -Oo4oo,CL 0) -0.800 0.. -1.2600 200.0 210.0 220.0 230.0 240.0 250.0 260.0 270.0280.0290.0300.0 37 GHz Brightness (K) Fig. 6a SMMR Cluster Data, Mi 1.600. 1.200 0.800 0 040 *0.600.1200fi

~~Zuer~dorfer and England ~34 Fourier Transforms of Beampattern Data Gaussian Model vs. Seasat SMMR Model o 0.6 —-~ —- U _ SMMR 0.7- - 0~~~~~. 0~~~~~. U.~~~~. 0.4 —-- 0~~~~ 0 0.2 0.4 0.6 0.8I 0.1 0.3 0.5 0.7 0.9 Normnidized Frequency Fig. 7

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AN OPTIMIZED APPROACH T6 MAPPINGJFREEZINGTERRAIN WITH SMMR DATA B. Zuemdorfer, A. W. England, F. T. Ulaby Radiation Laboratory Depmnent of Electrical Engineering and Computer Science The University of Michigan Ann Arbor, MI 48109 Abstract — A freeze indicator, based on a 37 GHz boundaries then provide improved location estimates of radiobrightness and a radiobrightness spectral gradient, has been freeze/thaw boundaries. shown to be an effective discrimninant for classifying frozen surfaces [1] and in locating freeze/thaw boundaries [2]. The Diurnal variations cause SMMR measurements of a single performance of this discriminant depends upon the accuracy surface to be different at noon and midnight. For example, with which decision boundaries can be established in decision diurnal heating and cooling of the soil produce offsets in the space. In this paper, we show that decision boundaries that are measured spectral gradients between noon and midnight [3]. In based upon clustering and unsupervised classification yield addition, poorly compensated heating and cooling of SMMR by good performance. sunlight and Earth shadow may produce offsets in measurements made at noon and midnight [4]. Decision criteria INTRODUCTION for classifying soil, and identifying freeze/thaw boundaries, must accommodate these day/night differences. Soil moisture contributes to the energy exchange between the air and the ground through latent heats of fusion and In this paper, we show that frozen and thawed surfaces can vaporization. Consequently, the processes of thawing frozen be classified, and freeze/thaw boundaries located, through ground or of vapoing soil oistur caus soil ths i s clustering and unsupervised classification. The procedure is to appear anomalously high. There is a large body of literature "opumized" in the sese that noon and midnight SMMR about deriving soil moisture from radiobrighmtess [e.g., 8,9]. masueents are clustered separaely. Specifically, th 37 GHz Moisture state can also be inferred from radiobrightness r rand spectral gradient from SMMR Frozen soil classification is based upon a combination of 37 measurements are grouped into frozen and thawed clusters for GI-z radiobrighmness and spectral gaint, a)Tb(.f)/f, where GHz radiobrightness and spectral gradient, aTb(J)/af, where midnight and noon. The intersection of cluster regions for each Tb(f) is the radiobrightness at frequency f. Frozen soils appear time-of-day provides the necessary freeze/thaw boundary. _...... infformadon to perform resolution extrapolation. cold at 37 GHz, and exhibit a negative spectral gradient that is largely caused by volume scatter darkening at the shorter CLUSTER DATA wavelengths. The data used for clustering was collected from August 1984 This tWO parametegfreez inicator ha been applied to data to December 1984 over a test area that includes North Dakota, from the Scanning Multichanntel Microwav Radiometir about half of each neighboring state, and part of Canada. (SMMR) on Nimbus-7. For these data, the spectral gradient is a Unuevsdcaifatorhrtan upvsd linear, least-square fit to the 10.7, 8, and 37 GHz Unassification, is used clbecause of the dearth of accurate ground radiobrightnesses Conceptually, a surface is classified as froze mwcn i sed asfcatrequires Its in our test area. Su prie classification requires only if both the 37 GHz radiobrightness and the spect gral dient tang g d da a refece ] e success of such mdnin with groud data as a r eeec 5].'Me success of such ame sutfficiently low. A freeze map is g rateot by displaying the a bydsplangth training depends, in pa, onthe ad eq uac of the ground data. In freeze indicator for each pixel location. However, data our study, seven air tempeture and el een soil temperature processing is complicated by the very different spatial recoring sprvided tgd fo tem tur recodingsits prvidd th grnroddat forthe entire test area. resolutions of the different SMMR frequncy channels, and by Soil temperat wee measured at 2 inches, or cm, depth SoUl temperatu wres weasre d at 2 inches, or 5 cm, depth differences between night and day SMMR radiobrightness and were made at dawn and dusk, whereas SMMR overflights measeenta were at noon and midnight The sparseness of ground sites and the differences between the times of soil temperature The different spatial resolutions of the SMMR channels meairn s and satellite overpassesprecludes the ground data require resolution c ai (equalization) tobe perfomid from being an accraate rWfence for the state of surface moisture prior t c eficao, so t al averaging is similar at all m supervised classification n frequencies. Assuming that no a prio surface information is available, cannon practice for resolution compensation is to Data were selected from those SMMR satellite passes that degrade the high frequency (fine-resolution) data to the covered more than 67% of the test area Sixteen noon SMMR resolution of the low frequency (coarse-resolution) data As a passes and ddirteen midnight passes met this criterion during our result, fine-resolution information is lost. However, by using August to December test period. The 18 and 37 GHz gaussian filtering in resolution compensation, the coarse- radiobrighness data were resolution compensated to the (coars:e resolution freeze/thaw boundaries can be registered to fme- resolution of the 10.7 GHz channel. The compensated surface resolution 37 GHz boundaries (i.e., to 37 GHz radiobrightness data were re-sampled on a 97.5 Km grid (ie., the resolution,t, threshold crossings) [2]. The fine-resolution, 37 GHz the 10.7 GHz channel). Spectral gradients were computed.'

CONCLUSIONS REFERENCES We have shown that the separate clustering of SMMR noon [ 1 ] Zuemrdorfer, B.W., England, A.W., Dobson,.M., and and midnight data yields an optimized estimate of the location of Ulaby, F.T. (1989); "Mapping freeze/thaw boundaries with freeze/thaw boundaries. While limited further improvements SMMR data," J. Agriculture and Forest Meteorology-, in press. might be attained with more data in the scatter diagrams, significant improvements will depend upon better controlled air [2] Zuemdorfer, B.W., England, A.W., and Wakefield, G.H. and soil temperanture measurements. (1989); "The radiobrightness of freezing terrain, 1989 IEEE Int. Geosci. and Remote Sensing Symp.; Vancouver, Canada. SMMR Cluster Data, Noon SMMR Cluster Data, Midnight 1.600~~~~.....,....,,.'....'.......- 1..600,......'....'.. 1.8..00 1.600 W, 1.200 1.200 1.00 —,w3 0.6.00 1.200 1.200..1.600 _ _Au" _0.400_ __ o.20- -1.200 -3.ooo -1.0 200.0210.0220.0230.0240.0250.0260.0270.0260.0290.0300.0 200.0210.0220.0230.0240.0250.0 260.0 270.0260.0 290.0 300.0 371Hz Bihtne (K) 370GHz Brightne (K) Figure 1. Scattrdarmo specta grdin x37 GHz Figure 2. Scatte diagram of spcrlgrdetx 37 GHiz raio"Wibrightes for noon SMMRdata 8/1/84- radiobiightnss for — dnight SMMR data, 8/1/8412/31/84. 12/31/84. SMMR Cluster Data, Noon SMMR Cluster Data, Midnight 1.200 1.200 0.600.-.00 0.000 ~~~~~~~~~~~~~~~~~0.6000 0.0 210.0 220.0 230.0 240.0 250.0 2e0.0 270.0 280.0 290.0 300.0 200 37jO.000 ightnns (K) 37 G~z Br~o20.000 370(Hz Brightness (K) 30zBihns K F rdibiguh.S cm elpses for noon SMM R data, 8/1/84- Fg rea.dingecaselpe or mdnihtneMsdt,8//4:2/31/8 4. ~ ~ ~ ~~~~~~~~12/31/84.

Michigan Earth Grid version 1 (MEG 1) UNIFORM SPACING FORMAT 6/27/90 Characteristics: 1 MEG 1 grid point spacing is uniform along rows of constant latitude and equals the spacing between rows, but individual points within each row and corresponding points in adjacent rows will not fall on a common meridian. That is, a simple rectilinear display of the data will not yield a Mercator-like image. Every useful display of the data will require a resampling program to map MEG 1 into the desired projection. 2 s. MEG acoordinates are designed to be nearly 25 km apart at the equator. 1/297 flattening of the oblate ellipsoidal Earth will render the grid points at the poles somewhat closer together. 3. MIEGi is based upon a polar, right-hand coordinate system with 0 = 0 the north pole, and 0 the 1800 meridian. The 1800 meridian was chosen because most people can visualize its location, and because this seam in our grid does not cross a major land mass. The first point in each row will be on the 180~ meridian, and successive points will progress toward the east (right-hand system). The spacing between the last point in a row and the first will vary between 0 and 25 kmn among rows. This will result in a ragged seam along the 1800 meridian. 4. Assuming an equatorial radius of 6,378.388 km, the Earth equatorial circumference is 40,076.594 km. A 25 km spacing yields 1604 grid points (including the zero point). We suggest that this be rounded to 1600 equally spaced points which yields a 25.048 km spacing along the equator. If M400 is number of points on the equatorial row (designated row n = 400), and mn is the individual point designation, then 0 <= mn < M400 = 1600. For other than the equatorial row, Mn = M400 - INT[M400(1-sin On)] = - LNT[ - M400 sin On]t where On = ic (n/800) radians (effectively the complement of latitude), and function INT means the largest integer that is less than or equal to the argument. Note that under this scheme, the pole rows are null sets. 6. We choseC an ideantic-al spcingrf betweensan rwsxy so that, if N is them inumbehr of 1rowsY N = 801

2 7. The location of each MEG1 point, Pnm, is uniquely mapped from nm to lat-lon by: Lat(Pnm) = 90 ( 1 - n/400), degrees N if > 0, degrees S if < 0. Lon(Pnm)= 180 (1 - 2nm/Mn), degrees W if > 0, degrees E if <0. These equations can be inverted to provide a unique mapping from lat-lon to nm. 8 Assuming an 8 byte word per grid point for each satellite pass (7 bytes for the brightness channels and 1 byte for a time tag), and that we record at least 2 passes per day, then there are 16 bytes associated with each grid point per day. Total bytes per day becomes: 16 x Ntot where Ntot is the total number of grid points in MEG 1. For an Earth's radius at the equator of r = 6,378.388 km, Ntot can be approximated: N approx. area of Earth's surface area associated with each grid point 25.048 2 4 IL 6378.3882 627.402 =814,865 grid points That is, the total daily byte requirement becomes:- 16 x 814,865 =13.038 Mbytes/day. If a CD holds 650 Mbytes, then 1 CD covers about 50 days. If northern and southern hemisphere data are produced on separate disks, and with some data compression so that null data (gaps of no coverage or 1 pass coverage) do not take up the full 16 byte allotment for each grid point, then it seems reasonable that 4 months of 1 satellite, two pass cove rage could be recorded on 1 CD using the MEG I format.

MEG1 6'N.......ze~~~~~~.. SO - p ~ ~ O0 4 ~~~~~~~ 180W 00~~~~~80' 0a 1 0 Example grid points. ~~~~~0 O 0 _.... ~~~~~~~~~~~~ 0 OO-...

Michigan Earth Grid version 4 (MEG4) MERCATOR-LIKE FORMAT 6/28/90 Characteristics: MEG4 coordinates are along constant latitude and longitude. 2. MEG4 coordinates are designed to be nearly 25 km apart at the equator. 1/297 flattening of the oblate ellipsoidal Earth will render the meridional grid points at the poles somewhat closer together. The density of grid points along a constant latitude decreases by 112 above 600 N and S (cos 60 = 0.5) and again by 1/2 above 750 N and S (cos 75 = 0.26) to limit the oversampling. These shift locations are convenient because Alaska falls almost entirely within the 60~-75~ range, and 60~ N is a major division among the Canadian Provinces. There are no grid points above 87.60 degrees latitude because the SSM/I does not cover the last 2.40. 3. MEG4 is based upon a polar, right-hand coordinate system with 0 =0 the north pole, and 4) = 0 the 1800 meridian. The 1800 meridian was chosen because because most people can visualize its location, and because this seam in our grid does not cross a major land mass. 4. Point spacing is based upon the equatorial radius of 6,378.388 km and an equatorial circumference of 40,076.594 km. A 25 km spacing yields 1603.06 grid points. To preserve a simple relationship with the degree (assuming we don't want to use grads), we can resample either at 1/4.degree (15 minute) intervals which yields 1440 points at 27.83 1 km spacing at the equator, or at 1/5 degree (12 minute) intervals which yields 1800 points at 22.265 km spacing at the equator. We chose a hybrid of 15 minute sampling along constant latitude and 12 minute sampling along the meridians. This means that the aspect ratios of the latitude and longitude spacing between sample points are equal at 360, 660, and 780 N and S latitude which permits easy, natural looking projections at these latitudes from MEG4 without resampling.

2 1440 points per row. Between +/-600 and +/-750 latitude, there is 30 minute-of-longitude sampling so that M60J75 = 720 points per row. Above +/-750 latitude, there is 1 degree-oflongitude sampling so that M>75 = 360 points per row. In each row, the point designation, m, spans the range 0 <= m < Mx: x = <60, 60-75, >75. 6. We chose 12 minute-of-latitude spacing between rows so that, if N is the number of rows, N = 901, and, if n is the row number, 0 <= n<= 900, n = 0 is the north pole, n = 450 is the equator, and n = 900 is the south pole. Because the Defense Meteorological Satellites don't actually fly over the poles, the actual range of n would be 12 <= n <= 888, which corresponds to the highest latitude row being 2.4~ from the poles. 7. Therefore, the location of each MEG point, Pnm, is uniquely mapped from n to lat-lon by: Lat(Pnm) = 90 (1 - n/450), degrees N if > 0, degrees S if <0. Lon(Pnm) = 180 (1 - mn/720), degrees W if > 0, degrees E if <0, for 150 <= n <= 750. Lon(Pnm) = 180 (1 - m/360), degrees W if > 0, degrees E if <0, for 75 <= n < 150 & 750 < n <= 825. Lon(Pnm) = 180 (1 - m/180), degrees W if > 0, degrees E if < 0, for 12 <= n < 75 825 <n <= 888. Each of these equations can be inverted to provide a unique mapping from lat-lon to nm. 8. Assuming an 8 byte word per grid point for each satellite pass (7 bytes for brightness channels and 1 byte for a time tag), and that we record at least 2 passes per day, then there are 16 bytes associated with each grid point per day. Total bytes per day becomes: 150 <= n <= 750: 16 x 1440 x 601 = 13.847 Mbyte/day 75<=n<150&750<n<=825: 2x16x720x75 = 1.728 Mbyte/day 12 <= n<75 &825 <n <= 888: 2x16x360x63 = 0.726 Mbyte/day TIOTAL =16.301 Mbyte/day If a CD holds 650 Mbytes, then 1 CD covers about 40 days. If northern and southern hemisphere data are placed upon separate disks, and if some efficiency is realized by, e.g., compressing null data (gaps of no coverage or 1 pass coverage where all 16 bytes are not

MEG4 90N 75N 60N 30N 00 30S 60S 75S 180W 00 10 Example grid points lie at line intersections.

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