The scatter of microwave radiation from a natural land surface depends on geometrical properties of the surface (surface roughness) and its electrical properties (dielectric constant) which can be related to moisture content and porosity. Knowledge of these aspects of land surfaces is of interest for many reasons: soil moisture is an important ecological parameter and is also a factor in the energy balance of the interaction between land surfaces and the atmosphere and hence in global change modeling; and surface roughness can be related to a number of geologically interesting parameters such as the age of alluvial fan and lava surfaces. In fact, surface roughness effects are important to the interpretation of most remotely sensed visible and near-infrared (VNIR) and thermal infrared (TIR) images. For these reasons, a significant amount of work has been directed to the problem of determination of surface roughness and dielectric constant from airborne and spaceborne synthetic aperture radar (SAR) data.

Inversion methods have been based on theoretical models such as the small-perturbation model [e.g., van Zyl et al., 1991], and on empirically determined relationships [e.g., Oh et al., 1992; Dubois et al., 1995]. These models prove successful under restricted conditions and often work very well in the geographic and physical circumstances in which they were developed and tested. However, models and approaches of more general applicability have yet to be demonstrated. One difficulty is presented by the high degree of complexity of rough surfaces, which has led to the use of a number of different parameters and combinations of parameters to describe surface roughness. Examples of roughness parameters are rms height (H_r), correlation length (H_c), and slope (M) and offset (C) of surface power spectra. Although all parameters can be shown to be related to the backscatter coefficient in some way [e.g., Ulaby et al., 1978; Mo et al., 1988; Evans et al., 1992], it is not clear under which circumstances a particular parameter becomes important and what represents a complete description of the surface geometric properties of natural surfaces which are important to microwave scattering. An additional problem is the physical difficulty in measurement of surface roughness and the consequent lack of detailed roughness data with which to test inversion models. As a result, most inversions have simply attempted to retrieve the rms height variation and/or dielectric constant [Oh et al., 1992; Dubois et al., 1995].

It is reasonably certain, however, that the mathematical description of complex natural surfaces requires more than one or two parameters [Goff, 1995]. This implies that unique inversion would be possible only from a data set of high dimensionality and introduces the desirability of bringing other types of available data to bear on the determination of surface roughness. Since VNIR reflectances are also dependent on aspects of surface roughness, an approach which utilizes a joint analysis of VNIR data (such as Landsat thematic mapper (TM)) and SAR data should remove some of the ambiguity in inversions. The combination of TM and SAR data has already been shown to aid in interpretation of vegetation and rock signals [Paris and Kwong, 1988; Evans and Smith, 1991]

Death Valley, California, has been a test site for many radar studies that have used the roughness dependence of backscatter as a geological mapping tool [e.g., Schaber et al., 1986; Kierein-Young and Kruse, 1992]. SAR data were used to investigate the distinction of alluvial fan units of differing age [e.g., Evans et al., 1992; Farr, 1992] or to look at the effects of aerodynamic roughness on wind and sand transport [e.g., Greeley et. al., 1991]. Death Valley is also a shuttle imaging radar C (SIR C) supersite and our study forms a part of that program. However, in this study we also use Death Valley as a study area to examine the more general problem of the relation between surface roughness parameters, SAR data, and VNIR data. This is preparation for development of joint SAR/VNIR analysis for surface roughness.

In order to study the relation between SAR and VNIR data and surface roughness parameters, a database has been constructed from extensive field surface roughness measurements on alluvial fans in Death Valley. These field measurements form the basis of a dual analysis in which we first examine the SAR/VNIR roughness signature in a forward modeling sense (from the roughness data) and, second, examine the roughness signature in an inverse sense (from the image data). Using the statistical character of the measured microtopography, we simulate surfaces and model the radar backscatter and VNIR reflectance of the surfaces as the roughness varies. The objective of the analysis is to determine the importance of various roughness parameters in this environment (which roughness parameters determine the signal) as well as to look at the potential uniqueness of inversion methods, and how the addition of VNIR data might contribute toward roughness determination. In our second approach, a method of analysis called foreground/background analysis (FBA) is introduced, that uses adaptive finite impulse response (FIR) filters in an attempt retrieve roughness information from SIR C, airborne SAR (AIRSAR), and TM images. We use FBA as an independent approach to examine the importance of the various surface roughness parameters to SAR/VNIR signature, as well as the uniqueness of the inversion of SAR, and the contribution of VNIR data

The concept behind joint analysis is illustrated schematically in Figure 1 for three types of data: SAR, VNIR, and TIR. For each data type, we have illustrated a set which contains the group of physical parameters that affect its response to a surface. The three sets of physical variables overlap. For example, the observed image signal for both TIR and SAR electromagnetic frequency ranges is related to porosity, and hence it is included in the overlap region between the TIR and SAR sets. Other parameters, such as emissivity (TIR), are not important to the radar signal and hence are not in an overlap region. However, we see that roughness is contained in the overlap of all three of the sets: we obtain a different perspective on surface roughness from each data type, thus giving potential for a more complete description of roughness. Of future interest is the overlap, mentioned above, between TIR data and radar data by virtue of the dielectric constant.

Death Valley was designated a SIR C supersite, one objective being roughness determinations on the vast, mostly vegetation-free alluvial fan surfaces. Over a number of field seasons, surface roughness data have been collected using various methods. Our primary approach has been to devise a technique of close-up stereophotography from which topographic profiles can later be extracted. This technique yields roughness information on spatial scales stretching from millimeters to about a meter. Low-altitude stereophotography from aircraft has enabled us to extend these spatial scales up to tens of meters. The distribution of sites at which measurements have been taken is shown in Figure 2, and sample profiles are illustrated in Figure 3a. These data have been complemented by measurements made using a pin profiler. The sites were selected in order to bracket the range of roughness variation seen on the alluvial fans at the SIR C supersite.

Previous work on natural surfaces has indicated that their one-dimensional (1-D) power spectra often obey a power law [e.g., Sayles and Thomas, 1978; Brown and Scholz, 1985]. Work in and near Death Valley on alluvial fan and lava surfaces has supported this model [Evans et al., 1992; Farr, 1992]. The power spectral representation of surface roughness represents a more complete description than rms height, since it contains the scaling and complete statistical information (if phase is included). The general approach is to fit a power law (F (k)) to the mean power spectral density:

(1)

where k is wavenumber (2p/l) and l is the wavelength. The equation becomes linear in log-log space:

(2)

The two parameters, log(C) and m, are the offset and slope of the power spectra. For natural, random surfaces the slope is usually confined to the range 2 < m < 3. As would be expected, the rms height of the surfaces, being related to the area under the spectrum, is positively correlated with (log(C)).

For Death Valley profiles, we have used a modified (Hanning window) periodogram method of power-spectrum estimation. Other spectral estimators may be more accurate and have less variance when applied to data with a limited number of spatial samples [Austin et al., 1994[. However, this is probably not the case for the Death Valley roughness data, since these spectral estimates are made using about 30,000 points for each surface. For the most part, the calculated spectra seem to fit the power law model over 4 orders of magnitude of spatial scale (Figure 3b). Results for each site are also given in Table 1. Many of the aircraft spectra begin to flatten and deviate from a power law at scales longer than about 5 m, which is about the average wash spacing on these fan surfaces. This flattening reflects the fact that there is little topographic variation on the fans at scales greater than several meters and less than a kilometer.

Examination of the relationship between these field measurements and SAR image data (SIR C data-take 35.1) shows a clear linear relationship between the SAR backscatter coefficient and the surface power spectrum offset. There is almost no relationship with the slope and only a poor relationship with rms height.

The field data from 13 sites of varying roughness in Death Valley are used, in the following sections, as the basis for simulation, modeling and foreground/background inverse analysis of the SAR and VNIR response to Death Valley surfaces. Below we give background details of our methods of simulation, modeling and inversion, and then analyze results from those methods when applied to Death Valley data.

Preliminary simulation of rough surfaces. We have taken as a starting point for the simulation of rough surfaces, the power law nature of their power spectra. Surfaces are simulated using the two-dimensional (2-D) equivalent of the appropriate 1-D power law. The parameters C and M from equation (1) are converted to the 2-D power law using

(3)

(4)

where is the gamma function [Austin et al., 1994].

Random phases are chosen, and the inverse fast Fourier transform (FFT) is used to generate the surface. Example surfaces, generated for slopes and offsets that are within ranges determined for surfaces in Death Valley, are shown in Figure 4. For surfaces with a power law spectra, the rms surface height should vary as the square root of the offset. We have verified that this is the case for these simulated surfaces.

The radar backscatter calculation. Scattering simulations presented here are 1-D (2-D scattering) solutions of Maxwellís equations for electromagnetic wave scattering from a dielectric, rough surface [Li et al., 1994; Tsang et al., 1994]. We will present 2-D (3-D scattering) solutions at a later date, but do not anticipate significant changes in conclusions [Pak et al., 1996]. Unlike the 1-D solution however, the 2-D solution does include cross-polarization terms. Since our approach uses exact solutions of Maxwellís equations, it is not limited, as are most analytic approaches, to a regime of validity such as small rms height compared to the microwave wavelength. The calculation is based on the method of moments [Harrington, 1968]. The scalar integral equation below (here presented for simplicity for a perfectly conducting surface) relates incoming electromagnetic field to the resulting current distribution in the surface

(5)

Where is the surface field, is the surface current, is the surface unit normal, and is a Green's function describing the interaction among elements of the surface. The form of this equation for a dielectric surface is discretized, resulting in the matrix equation

where b is the incoming field. The matrix, Z, of interactions must be inverted to solve for the surface currents x. This is achieved by a computationally efficient matrix solution method called the banded-matrix iterative approach (BMIA) [Li, et. al., 1994]. The current distribution is then used to calculate the field at any location above the surface.

We have used this scattering calculation in two modes: in the first mode, Monte Carlo simulations of surface profiles with specified statistics are used. Calculations are repeated until the backscatter converges to a mean backscatter response for a surface of a given statistical type. In the second mode, the scattering calculations are performed on real surface profiles, with the requirement that they be numerous and long enough (with respect to the wavelength) that sufficient backscatter calculations can be performed to ensure convergence for a given surface. Results from the two modes are compared in the results section (section 5) below.

The radiosity calculation for VNIR reflectance. Assuming that we are viewing a scene from vertically overhead, the reflectance of a rough surface deviates from the reflectance of the material which makes up that surface for several reasons: if, as is usually the case, the illumination of a surface by the sun is not perpendicular to the surface, then shadows will be cast by elements of the surface. Shadowing has the effect of making the integrated brightness of a rough surface appear darker than a flat surface of the same material (Figure 5). In addition, varying the slopes of surface elements causes a change in the local angle of incidence of illuminating radiation, and thus there is a variation in energy per unit area according to a cosine law. When integrated, this effect (which we call shading) again darkens a surface (see Figure 5b). An effect which is often ignored is the multiple-scattering interactions that occur between elements of the rough surface. Light may scatter between two or more elements of the surface before it is reflected back to an observing sensor. This "multiple-bounce" effect mitigates the darkening caused by shadowing and shading effects so that the surface, in general, does not appear quite as dark (Figures 5c and 5d). An additional effect arises because the importance of the multiple-interaction effect is dependent on the reflectivity of the surface material. Different frequency ranges will have different degrees of multiple-bounce contributions, and this will distort the spectral response of a surface.

Multiple scattering can be modeled with a radiosity calculation [see Cohen and Wallace, 1993] in which an assumed Lambertian surface is subdivided into geometric elements (S0,..., Sn) with areas (da1,...,dan) and the scattering between elements calculated. The percentage of the light leaving one element (S1) which is intercepted by another element (S2) depends on a geometric factor called the view factor (F_s1s2). This factor is closely related to the solid angle subtended at S1 by S2. For the case when elements are sufficiently distant from one another that their area can be treated as differential, the calculation is simply

(7)

Where q₁ and q₂ are the angles between the normals to the surface elements and the path joining the center of the two elements, and s is the distance between elements. When elements are closer together, however, the view factor calculation involves integration over the areas of both elements

(8)

Using Stoke's theorem, this integral can be solved by conversion to a contour integral. Our view factors have been calculated in this way using code provided by P. Schröder at Princeton University [Schröder and Hanrahan, 1993]. The view factors are then used to form a matrix equation which is solved using Gauss-Seidel iteration (Cohen and Wallace, 1993).

In the simulations reported here, surface digital elevation models (DEMs) are used to divide surfaces into quadrilaterals which form the elements of the radiosity calculation. For computational efficiency, we have chosen an iterative approach which allows us to calculate the upwelling radiation after a specified number of allowed interactions between surface elements. The solution usually converges after three interactions.

The Lambertian assumption is central to the radiosity method. For our present study, measurements on pebbles collected from the field areas in Death Valley show a photometric function which is closely Lambertian. Hence the Lambertian assumption is likely to be accurate for the alluvial fan surfaces as a whole. In addition, we are using the radiosity model to study the impact of aspects of surface geometry on the apparent reflectance of surfaces and not to model observed radiances. Therefore we do not require a high degree of coherence with the Lambertian assumption. The effects of significantly non-Lambertian behavior on the results of the radiosity model are complex and beyond the scope of this article.

Here we describe an adaptive method of inverse analysis in which image data are subjected to finite impulse response (FIR) filters [Smith et. al., 1994]. This represents an extension of spectral mixture analysis that allows any number of spectra to be included in the unmixing process by separating them into two categories of spectra: foreground spectra and background spectra. These spectra are generally selected from the image data: the foreground from regions which express a substantial contribution from a desired physical variable such as roughness, and the background from a region with minimal contribution from that variable (smoother surfaces) and a contribution from undesired variables (such as intermediate slopes or dielectric variation). FBA can be used to maximize the contrast between such a foreground entity (e.g., roughness or moisture) and a given background (e.g. surfaces with various dielectric constants, orientations or vegetation cover). However, inherent in the FBA analytical framework is the concept that the solution for physical parameters such as roughness and moisture is indeterminate, given the SIR C measurement dimensionality. The degree of correlation among SIR C bands is high, and hence the intrinsic dimensionality of SIR C data is small, especially when compared to the number of physical variables. This allows an infinity of solutions to the inversion for a given physical parameter. Our objective for the FBA analysis has therefore become to explore the range of solutions and to select those solutions which minimize uncertainties in estimated parameters (e.g., roughness and moisture) caused by other factors such as instrument calibration, topography, and different types of scattering surfaces (background factors). The validity of these solutions is evaluated using knowledge and experience of the field area (spatial context), comparison to other techniques, and field measurements.

The application of FBA to image analysis can be made in two ways: in the first method, actual ground data (in this case, roughness data Y_n) are used to optimize the selection of the FIR filter weights by solving the general equation of the FIR filter, using singular value decomposition,

(9)

where Y is the output of the filter (e.g., roughness or moisture estimates), includes the SAR scattering measurements for C and L wavelengths and polarizations, K is a constant which is required because the roughness or moisture is not necessarily zero when the SAR measurements are zero, and e is the resulting error in the estimated parameter (e.g., roughness). The error in solving for w in equation (9), although not shown explicitly, is in units of the driving measure Y_n (e.g., for roughness, this would be units of rms height, slope, offset). By comparing the standard deviation of this error to the range of a given parameter within an image, we can estimate the detectability of the foreground with respect to the background.

In the second method the image is separated into areas of foreground and background. Image-derived spectra, , comprising the foreground and background sets, are used to find the solution vector, w, such that the projection of all background vectors along w is as close as possible to zero, and the projection of all foreground SAR measurement vectors along w is as close as possible to unity. The w vector defines an optimum projection, in the hyperspace defined by the SAR measurement bands, separating the foreground from the background. A set of equations is generated for a collection of background and foreground spectra as follows:

Again, singular value decomposition is used to determine a single w vector and constant K that optimizes both foreground and background equations simultaneously. Solutions can be sought that minimize the effects of surface orientation, intermediate scale topography, or dielectric variation, by choosing spectra from the image that assign such variability to the background.

Clearly, in this form, FBA is a linear analysis. Although nonlinearity can readily be incorporated, we do not feel that this is warranted at the present time, especially since the field roughness data indicate that a linear relationship with SAR backscatter is a good approximation. Our forward modeling supports this conclusion. This issue is also addressed in the discussion section (section 5).

As described in section 4, our preliminary approach to the description of rough surfaces has been to assume that they are random surfaces that can be characterized by a power law power spectrum of a given slope and offset. In modeling, our initial goal has been to examine the sensitivity of VNIR reflectance and radar backscatter to these parameters over the ranges of slope and offset encountered in Death Valley. Part of this goal is to delineate the potential degree of nonuniqueness in inversion attempts that utilize either SAR or VNIR data and hence to assess the degree to which this nonuniqueness might be reduced by using joint VNIR/SAR data sets. Figures 6 and 7 show the results of VNIR and radar calculations illustrating how the scattering varies with slope and offset of the surface power spectra. The radar backscatter is demonstrably more sensitive to offset and slope variation than VNIR reflectance. In fact, the VNIR calculation, while showing a significant variation with offset, shows almost no slope dependency. The dependency of SAR on slope is also considerably weaker than its dependency on offset. Note that the dependency of SAR backscatter on offset is of the opposite sign to that of VNIR reflectance. Isopleths on these surfaces of reflectance or backscatter represent trajectories where no change in reflectance or backscatter is observed with change in slope and offset.

This form of nonuniqueness basically reflects the fact that the dimension of the data is one and that of the parameter space is 2. Given the number of channels present in multifrequency polarimetric SAR, and in most VNIR imagery, the nonuniqueness described above might easily be removed using two or more channels to solve for slope and offset provided the measurements do not vary in the same way. Our simulations for both C band and L band show a differing dependency on offset and slope which would, in principle, allow determination of slope and offset from two bands of SAR data. This assumes that slope and offset are the only parameters causing the backscatter or VNIR reflectance to vary. Unfortunately this is not the case even when considering only surface-roughness effects. Figure 8 shows the backscatter values calculated from three different types of microtopographic profiles. Figures 8a, 8c and 8e illustrate example profiles while Figures 8b, 8d and 8f show the calculated backscatter, with scattering angle, from those profiles. In Figures 8a and 8b results are presented from forty real profiles from a site above Mud Canyon. Next, in Figure 8c & d are results from profiles simulated using the offset and slope of the best fit power law spectrum from the real profiles (Figure 8a) together with random phase. Finally, in Figures 8e and f are results from profiles simulated in the same way as those in Figure 8c but with nonrandom phase information from the real profiles added instead of random phase. In all three experiments therefore the profiles have the same offset, slope, and roughly the same rms height. At the backscatter angle (40°), the scattering coefficient calculated from the simulated profiles (Figure 8d) is greater by a factor of 2 than the coefficient for the real profiles (Figure 8b). Visual inspection of Figures 8a and 8c shows that the simulated profiles do not reproduce the characteristic shapes seen in the real profiles. The simulated profiles have a more distributed roughness that, for the rougher surfaces, is responsible for high slope and a high degree of multiple-scattering interactions, whereas in the real profiles, roughness is more confined to prominent rocks and cobbles. The results shown in Figures 8e and 8f demonstrate that the difference in backscatter is related to the phase character of the surface power spectrum. It appears that surfaces with this phase character allow much more forward scattering than do the random phase surfaces. Describing these rough surfaces clearly requires more than the simple slope and offset of the surface power spectrum or rms height of the surface topography. In fact, we are not sure what the intrinsic dimensionality of the surface really is.

Our VNIR radiosity calculations illustrated in Figure 6 seem to be less affected by the observed organizational differences between simulated and real surfaces. In comparison with the one real surface DEM available to us (Figure 5), the simulated surfaces are about 10% darker. One similarity occurs in the degree of multiple interactions occurring in both types of surface: about 13% of the real surface reflectance is caused by multiple interactions, whereas more than 20% of the simulated surface reflectance is due to multiple interactions (for a surface with a reflectivity of 0.35). Hence the differences between the simulated and real surfaces results in less multiple scattering (compared to the simulated surfaces) in both the VNIR and radar wavelengths.

Especially for radar scattering, the two-parameter power law description of Death Valley surfaces is inadequate to explain the observed signal. Examination of the general appearance of the simulated surfaces suggests that this type of description may be somewhat more successful at describing soil surfaces, but even in this case there appears to be a significant deficiency (K. Pak et al., manuscript in preparation, 1996). For more complex surfaces, the information contained in the phase of the Fourier transform is certainly important. The use of slope and offset (or fractal dimension) does not take this information into account. It is the phase characteristics that control the organizational or shape characteristics of a surface. This is illustrated in Figure 9, where phase is taken from the FFT of the real profile shown (Figure 9a) and combined with the amplitude spectrum of another real profile (Figure 9b), chosen at random, to produce the simulated profile shown in Figure 9c. The same basic shapes as seen in Figure 9a are seen in profile 9c, and it is obvious that the phase has exerted a dominating influence in controlling the appearance of the profiles. These shape characteristics, and hence the phase of the FFT, are apparently very capable of influencing the radar backscatter and must affect our ability to invert for roughness from SAR and VNIR data. Only over surfaces where the phase character is constant can we expect to accurately invert for offset and slope, or rms height.

Unfortunately, the phase aspect of the surface power spectra proves very difficult to characterize. The variation in phase with frequency is nonrandom and exhibits a degree of correlation that is highly variable from one profile to the next.

We have examined FIR-based inversions of SAR and TM data for roughness. In addition to the use of field data, we have compared the FBA results with results obtained using the semi-empirical roughness (SER) model of Dubois et al. [1995]. The SER model is primarily empirical and was developed using extensive roughness and moisture data from agricultural fields. The FBA solutions are initially optimized using the surface roughness measurements as input. We first compare TM-derived roughness estimates with those derived from SAR data, and examine the degree of success in recovery of the field-measured roughness. We then look at solutions obtained from SAR data for various combinations of the available bands in SIR C and AIRSAR.

For TM data, the basic physical idea is that the shade component of the image is related to the roughness. In Figure 10, we compare the TM shade fraction derived from spectral mixture analysis [e.g., Adams et al., 1989] to the roughness estimate from the SER inversion of SIR C data. There are some gross similarities between the two images: for instance, the alluvial fans appear to be rougher than the valley floor and playa. However, the TM image includes the effects of albedo variation, which can be mistaken for shade variation, and the SER solution evidently contains effects of intermediate-scale slope variation (finer than 7.5 minute DTM resolution) and vegetation abundance variation. For example, in the TM image, the alluvial fan labeled A in Figure 10b, which appears to be significantly smoother than surrounding fans is, in fact, a fan of different composition. Spectral differences have leaked through into the shade image. The detailed differences between the two images are significant. One of the objectives of FBA, when applied to the TM image, is to attempt to diminish the undesired albedo effect to obtain a more accurate rendition of roughness variation. Likewise, the objective in SAR analysis is to find a solution less sensitive to the effects to vegetation, dielectric, and intermediate-slope variation.

In Figures 10c and 10d we show results obtained using the first of the FBA modes in which the field roughness measurements are used to control and optimize the FIR solution. Solutions obtained using SIR C data (all bands) and the TM data are compared. Considering the fact that these solutions were obtained from such vastly different types of data, the most striking thing about them is their similarity. They are quite comparable in their overall ordering of surfaces with respect to roughness and, in fact, retrieve the field roughness measurements with about equal precision (see Table 2). Indeed, the error is somewhat less than that obtained with the SER model. Regressions of the measured offset against the SAR and VNIR estimated offsets are displayed in Figure 11. It appears that the TM estimates are more accurate for the rougher surfaces and show more error at the smoother offsets. This result is predicted by the forward modeling results shown in Figure 6, where the VNIR reflectance becomes more sensitive to offset change at the lower offset (higher roughness) values. The SAR inversions show about equal accuracy throughout the roughness offset range.

In detail, there are relative differences between the SAR and TM derived images. Marked roughness variation, seen in the SER solution along the fans emanating from the Kit Fox Hills, is not seen to the same extent in the TM solution. In this respect, the SIR C FIR solution (Figure 10c) appears to fall between these solutions. In all solutions there are still changes in roughness due to either vegetation, intermediate-scale topography, or illumination geometry.

As a cautionary note, we would like to point out that the ability of VNIR data to retrieve the surface roughness measurements (and hence produce similar roughness images to those obtained from SAR data) does not constitute proof that roughness information is being isolated. This is because there may in some instances be a relationship between surface composition (hence spectral character) and surface roughness so that it is not possible to be sure that the roughness solution is not simply a reflection of subtle compositional information masquerading as roughness variation.

We have examined the retrieval of measured surface roughness parameters for a variety of combinations of bands in SIR C and AIRSAR data. The errors in retrieval are given in Table 2. In all cases, rms height is estimated with larger errors than offset or slope. Estimates for rms height and slope based only on C band (C_hh and C_vv) have less error than those based only on L band (L_hh and L_vv). When all the FIR solutions are examined, it is found that the filters are markedly different for each of the solutions. Many different FIR filters lead to similar degrees of fit to the field data, and this illustrates that the field data do not uniquely constrain solutions. This is because the solutions, which are only constrained at the locations of the field sites, are extremely sensitive to the background. Hence, the field measurement driven solutions are not particularly robust. In particular, best detections require a separate FIR for the AIRSAR and for the SIR C data. This is probably due to slight differences in look angle, resolution, and azimuth between SIR C and AIRSAR data. We see below that, by taking the second FBA approach, and by using all wavelengths and polarizations, it is possible to derive a single, FIR which can be applied to either image to obtain levels of roughness detection near to those obtained from just the L_hhand L_vv or C_hh and C_vv channels (solution h in Table 2).

Using the second mode of FBA analysis in which the image data are used to optimize the solution, the simplest initial approach is to designate the roughest surface (Grotto Canyon) to the foreground and the smoothest surface (the valley floor/playa) to the background. In addition, we can require that this solution return smooth roughness variation for these surfaces by choosing many foreground and many background vectors from each surface so that the observed backscatter variability (mostly speckle) within each surface is represented. This method yields a single FIR filter that can be applied to both the AIRSAR and SIR C image to obtain relative estimates of the roughness. This solution is much more robust than the ones driven by the field measurements: small changes to w have little effect on the outcome. Figures 12a and 12b show these results. Note that the light-dark anomalies present along the Kit Fox Hills in Figure 10a are not present in these images. In addition, the images of Figure 12a and 12b express less local roughness variation. In terms of the retrieval of roughness measurements (Table 2), this solution is marginally less successful than the previous ones. In addition, the resolution of more subtle roughness variation appears to have been sacrificed for this stable solution, and the images seem to display only the first-order roughness variations.

In an attempt to obtain increased roughness resolution, a third FIR solution to roughness was obtained by optimizing the solution with three image-derived roughness types. We have placed the Grotto Canyon fan in the foreground, Kit Fox fans intermediate between the foreground and background, and the playa data in the background. Results from TM and SIR C data are shown in Figures 12c and 12d. The rms errors in retrieving the roughness measurements for all solutions are given in Table 2. Despite the fact that the filters are all very different, all models retrieve the surface roughness measurements with about the same precision. The final model, shown in Figure 12d, is marginally the most successful, and this model, in our estimation, shows the most accurate spatial variation of roughness, while being least sensitive to other variables.

In our forward modeling, we have not considered the complications introduced by variables other than the roughness parameters (offset, slope, phase, and rms height) defined at or near the scale of the radar wavelength . These complications might include, for example, albedo variation that mimics the VNIR radiance signal one might expect from roughness variations. In SAR data, variations in dielectric constant, intermediate-scale topography (e.g., stream channels), and the amount of vegetation cover can masquerade as variations in roughness. However, we anticipate that differing dependencies of the various bands on these several parameters, and in particular, the differing sensitivity of SAR and VNIR, can be used to tailor inversion methods to the character of surfaces in a given environment. This is particularly true for the phase character of surfaces which controls the organization of surface elements. VNIR reflectance is much less sensitive to this aspect of surface roughness than the radar backscatter. Therefore the joint inversion of VNIR and SAR data holds much promise in separating the various complexities of surface roughness.

In FBA analysis the advantage in using the image data to select w, using field knowledge of spatial roughness variation, is that, given intelligent choices of foreground and background, the solution will to some degree account for the complexity . The FIR filters will exploit the number of available degrees of freedom to best account for the chosen variability. This approach also introduces the possibility of the using the spatial scale of variance as additional information. Solutions which are driven using the field data, while perhaps more objective, will have a tendency to suffer from the effects of masking. Though constrained to fit at the locations of the field sites, the solution is free to deviate at all other locations because of background factors not included in the field measurements.

Some of the complexity in the FIR solutions may have arisen from the use of a linear analysis. However, given the complexity of the physical processes, the degree of true non-linearity is not clear. Our simulations do show a slightly nonlinear dependency over these roughness ranges (as does the SER model), but our forward modeling based on real surface data suggests that a linear relationship is adequate. It is possible, however, to incorporate nonlinearity into the FIR approach, and this will be the subject of further analysis. In particular, we hypothesize that nonlinearity may affect the stability of solutions.

The inversions for the roughness parameters show, in general, only a fairly coarse resolution of surface roughness variation. This results from the inherent indeterminacy of this inversion problem. In this respect, we found a strong trade-off between resolution and stability. A single inverse model, which would be applicable in all circumstances, could only have extremely coarse resolution because there are so many physical variables for which we must account. In this context, the multifrequency, polarimetric nature of SIR C is obviously important, but the more intrinsic dimensionality of the data (which is less than the number of measurement bands) is not clear. Our results suggest that a hierarchical approach to joint SAR/VNIR analysis might exploit additional degrees of freedom in the combined data sets to reduce ambiguity and achieve a greater accuracy and resolution of roughness variation.

Here, there is an additional possibility that TIR data (in particular, the thermal inertia) can be used to aid in retrieval of the dielectric constant. Other avenues for improvement may be offered by the use of images taken at different illumination angles: our modeling shows that illumination angle dependence especially for VNIR, shows much more sensitivity to roughness than does the absolute magnitude of reflectance. The use of multiple images from a particular area gives an additional opportunity to remove albedo and dielectric variations which might masquerade as roughness variation. These topics will be the subject of future articles.

1. Surface roughness measurements in Death Valley indicate that power spectra of the surface microtopography can be described using a power law. In some cases this is true over 4 orders of magnitude of spatial scale. However, in other cases there is a deviation from the power law at scales greater than several meters.

2. The power law description of these surfaces is not sufficient to enable complete characterization of the effects of surface scattering on incident radar/VNIR radiation. The power spectrum offset appears to be especially important, but forward modeling suggests that at least as significant as offset are phase aspects that control the organization and characteristic shapes of the surface. This is especially true for radar scattering, for which this characteristic exerts a strong influence on the amount of multiple scattering occurring at the surface.

3. The application of FBA, which provides a useful way of analyzing SAR and VNIR data and of incorporating field data and knowledge into inversions for roughness, demonstrates that roughness can be retrieved from VNIR data in Death Valley. Using field data to optimize the FIR solution, we obtain strikingly similar results from SAR and TM data, but we still cannot be sure whether subtle albedo variations masquerade as roughness. The field roughness measurements can be recovered with almost equal precision by inversion from VNIR (TM) data, SIR C data, and AIRSAR data.

4. The susceptibility of solutions to background factors, and the coarseness of their roughness resolution, reflect the indeterminate nature of the inverse problem. Robust solutions, which might have general applicability, display only the first-order roughness variation. This suggests that inversions should be tailored to the background complexity in any given scene. There is evidence to suggest that some background factors have different characteristic spatial scales of variation than surface roughness and that this might be exploited to enhance resolution in roughness determination.

5. VNIR data exhibit a different sensitivity than SAR data to roughness parameters. In particular, they do not appear to be sensitive to the slope of the surface power spectra, and are not as sensitive to the phase character of the power spectra. It is therefore possible to reduce the indeterminacy of the inversion for roughness using joint data sets. We suggest that a hierarchical, joint analysis of SAR and VNIR data will improve accuracy and resolution in inversion for surface roughness.

This work was supported by NASA/JPL contract 958450. We would also like to thank Steve Couthern for his help with computations, P. Schröder for supplying the view factor code, and Anthony England and an anonymous reviewer for thoughtful and constructive criticism.

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(Received October 18, 1995; revised April 17, 1996;
accepted April 18, 1996)

Copyright 1996 by the American Geophysical Union.

Paper number 96JE01247.

0148-0227/96JE-01247$9.00

Figure 1. Schematic representation of the physical parameter space for three types of remotely sensed data. Each data type is shown as a set that includes all physical variables that affect its response to a surface. Variables that affect more than one data type are displayed in the overlap region between sets. M, moisture; P, porosity; A, albedo; E, emissivity; and R, roughness. M and P are related to the dielectric constant.

Figure 2. Site locations of roughness measurements in Death Valley, California. At each site, the microtopographic profiles have been measured using a close-up photogrammetric technique. Low altitude photogrammetry at these sites has extended the scale of topographic measurements to tens of meters.

Figure 3. Microtopographic data from sites in Death Valley: (a) example profiles and, b) example mean power spectra. In both Figures 3a and 3b, results from different sites have been offset from one another. Spectra labeled a1, b1, c1, and d1 were obtained from aerial photogrammetry. Spectra labeled a2, b2, c2, and d2 were obtained from close-up ground-based photogrammetry at the same sites.

Figure 4. Examples of surfaces simulated from random phase power spectra with varying offset and slope.

Figure 5. The results of radiosity calculations for a 1x1 m DEM (resolution 1 cm) of the Grotto Canyon wash surface. (a) Grayscale image of the DEM, lighter colors being higher. (b) Radiosity results for single scattering draped over the DEM. (c) Radiosity results for multiple scattering draped over the DEM. (d) Difference between Figure 5c and 5b. The grayscale is in units of reflectance normalized to the reflectance of a flat surface. The multiple interactions illuminate the shadowed regions of Figure 5b so that they appear lighter.

Figure 6. (a) Results of 3-D radiosity calculations for relative reflectance (compared to a flat surface) for simulated rough surfaces of various spectral slope and offset.

Figure 7. Results of radar backscatter (s_hh) calculations on simulated surface profiles of varying spectral slope and offset.

Figure 8. Microtopographic data from above Mud Canyon, Death Valley, and backscatter values calculated from them. (a) Sample of a real microtopographic profile. (b) Calculated radar backscatter with look angle for this microtopography (40 real profiles used), using a fixed incidence angle of 40°. (c) Sample of a simulated profile having the same spectral slope and offset and rms height as the real profile example shown in Figure 8a together with random phase. (d) Results of radar backscatter calculations for these profiles (40 profiles used to form the average) for an incidence angle of 40°. (e) Sample of a profile simulated as in Figure 8c, using the phase information from the real profiles in Figure 8a instead of random phase. Scattering is mostly into the forward direction for cases in Figures 8a and 8e whereas the case in Figure 8c shows much more backscatter. At the backscatter angle of -40° (monostatic case), relevant to SAR data, the values of so are similar in Figures 8b and 8f and much greater in Figure 8d.

Figure 9. (a and b) Examples of real topographic profiles from the Kit Fox Hill fans. (c) A profile generated from an inverse FFT, using the phase of the FFT of the profile shown in Figure 9a together with the amplitude of the profile shown in Figure 9b (chosen at random). The form of the profile in Figure 9a is almost completely preserved in Figure 9c, demonstrating the controlling influence of phase on the nature of the profiles.

Figure 10. Images depicting roughness variation in Death Valley as determined by various methods: (a) SER model, (b) the shade component derived from TM data, (c) FBA analysis of SIR C data using the field roughness data to optimize the solution and, (d) FBA analysis of TM data using the field roughness data to optimize the solution.

Figure 11. Plots of FBA estimates of roughness (power spectrum offset) against the field roughness data. (a) Regression for the SIR C based FBA estimate. (b) Regression for the TM based FBA estimate. Axes for both Figures 11a and 11b have been multiplied by -1.

Figure 12. Results of the application of FBA in which the image data are used to optimize the solution. (a and b) The foreground is designated by the roughest areas of the image, and the background by the smoothest areas in the image. (a) application to AIRSAR data. (b) Application to SIR C data (same FIR filter as in Figure 12a). (c and d) The Kit Fox Hills fans are designated as a third element of intermediate roughness lying between the foreground (Grotto Canyon) and the background (playa). (c) Application to TM data. (d) Application to SIR C data.

The offset and slope of the surface power spectrum are given as well as the rms height. Sites are labeled in Figure 2.

The detectabilities given are normalized to the range of the roughness measure so that errors in different roughness parameters are comparable. The inverse of these error estimates is a measure of the resolution of roughness (how many scales of roughness can be resolved).

* These solutions are displayed in Figures 10 and 12

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