Given knowledge of geological processes, the age of different surfaces can be constructed from local surface roughness measurements (Farr, 1992; Gillespie et al., 1984). Although SAR measurements have the potential of providing regional roughness estimates, obtaining a consistent, extendible and robust estimate of surface roughness has not been achieved. Our objective is to recover the intrinsic or "textural" roughness related to the scale of the radar wavelength and to quantify the uncertainty associated with this estimate. However, SAR measurements also respond to roughness at scales greater than the radar wavelength, and the sensitivity at all scales varies with imaging / viewing geometry of the SAR instrument. These dependencies must be accounted for if the small-scale textural roughness is to be quantified in SAR images.

Empirical estimates of roughness are typically optimized fits between field roughness measurements and SAR backscatter (Oh et al., 1992). The empirical fits define "good" linear relationships between local measures of roughness and radar backscatter. These empirical quantitative relations, however, are valid only for a small range of specific surfaces and imaging conditions. Extended regionally to entire SAR images, or even to local images acquired at different times, results in inconsistent and inaccurate roughness estimates.

Similarly, theoretical or physically-based approaches such as the integral equation model (IEM) (Shi et al., 1995) and semi-empirical model (SEM) (Dubois et al., 1995) do not account for roughness at scales greater than the measurement wavelength (Smith et al., 1996; Weeks et al., 1996b). Thus, the effect of roughness at these scales is unpredictable in the application of these models. Both empirical and physically-based approaches to parameterize SAR measurements are based solely on what we define as the foreground; i.e, those measurements that are dimensionally and physically associated directly with roughness at the wavelength scale.

We contrast foreground parameters with a background consisting of extrinsic factors also affecting SAR measurements. These extrinsic factors are not explicitly defined or included in techniques estimating roughness from SAR images. Nevertheless, they are significant to determining roughness detectability. For example, estimates of surface roughness are affected by the surface dielectric, spatial phase character of the surface (Weeks et al., 1996b) and local orientation, even though these factors may not be explicitly incorporated in the parameterization of roughness. Previous efforts (Weeks et al., 1996b) indicate that background factors drastically limit the roughness resolution in SAR images.

We have developed a strategy termed foreground / background analysis (FBA) to quantify explicitly the effect of background factors on the interpretation and inversion for a foreground entity such as roughness (Smith et al., 1994, 1996; Weeks et al.,1996b). FBA is particularly suited to testing hypotheses under conditions where no unique invertable solution to the foreground exists. This is the case for Shuttle Imaging Radar (SIR-C) data sets, as the intrinsic dimensionality of these multipolarized SAR measurements is much lower than the number of significant variables.

For geological applications, it is not necessary that estimates of surface roughness be absolute or have an exact field equivalent such as rms height. More important is that the estimate be robust and consistent and that different estimates over space have a precise relative sorting of roughness that can be assigned to specific scales. In most approaches, temporal variation in roughness estimates at a single location may exceed the differences due to variations in imaging geometry or even the differences within a single geologic unit. These effects severely limit interpretation of roughness estimates.

The background has been defined by backscatter variability within a geologic unit calibrated by field measurements, whereas the foreground has been associated with variability among geologic units (Smith et al., 1996; Weeks et al., 1996b). Thus, the foreground and background have been defined at different and somewhat arbitrary scales, although they refer to factors operating at wide overlapping ranges of scales. The choice of scales at which foreground and background were defined has been dictated by local geology.

Our previous efforts have focused on techniques for determining finite impulse response (FIR) filters over the C- and L-band polarizations that minimize propagation of background factors into roughness estimates during FBA (Weeks et al., 1996b). Because of the complex nature of the background, however, even with FBA it is not possible to eliminate background effects from the roughness estimates; i.e., a unique inversion for foreground roughness is not feasible. A necessary tradeoff has been to sacrifice contrast, or resolution in roughness, for minimal propagation of background-induced variability. One of the limitations of FBA as we have applied it to the SIR-C data is that each image requires its own unique FIR filter to provide a consistent set of roughness estimates (i.e., the solutions are not stable temporally). We attribute this to our imperfect knowledge of microwave interactions with natural surfaces. A key component of FBA, which might otherwise appear to be mathematically trivial, is the translation of this conceptual framework into the FIR weights.

In this paper we use a short time series of images of an area in Death Valley to test the significance of background factors, dependent on spatial scales, to optimize roughness estimates. The images are decomposed into dyadic scales (i.e.., spatial scales increasing by powers of two) using a wavelet transform. We assume that backscatter is dominantly affected by roughness and that dielectric effects are minimal. For the purpose of this analysis, we assume the roughness to be proportional to the backscatter. We further assume surface roughness on the ground to be constant over the short time series and to be isotropic at the wavelength scale. Evidence is provided for background effects when the data-set backscatter is inconsistent (i.e., the backscatter measurements for a given wavelength and polarization do not agree in the data set). Our goal in FBA is to determine a single FIR filter that when applied to SAR images decomposed into the dyadic scales, estimates roughness and minimizes background effects.

We apply FBA to a short time series of four SIR-C images. The images are decomposed using a wavelet transform into nine different scales. To optimize estimates of roughness relative to both foreground and background factors, we split the standard FIR equation into two equations that are solved simultaneously using singular value decomposition. The roughness estimates and associated uncertainties are evaluated with respect to the image context and field measurements.

The short time series of the four co-registered SIR-C images ordered by acquisition times are: DT19.0, DT35.1, DT120.3, DT8.3. The first two images were acquired in ascending orbits while the last two images were acquired in descending orbits. This data set was used to quantify the impact of extrinsic background factors on roughness estimates of the alluvial surfaces emanating from the Kit Fox Hills and the Panamint Mountains in the lower right section of Figure 1. The look angle range was from 36.1^o to 46.8^o. DT8.3 was acquired on the second SIR-C mission in October 1994 while the remaining three images were acquired in April 1994. The value of the backscatter coefficient, s_o, ranges from 0.001 for the smooth playa sand surfaces on the valley floor to 0.06 for Grotto Canyon. This corresponds to a roughness range of 0.5 to 2.0cm rms height (Weeks et al., 1996a).

To determine the significance of background factors at different spatial scales, a 512² subset of each SAR image was decomposed into nine separate dyadic scales using a six-element Symmlet wavelet (Benedetto and Frazier, 1994). The wavelet forward and inverse transform software WAVELAB from Stanford University was used to perform the image decomposition (anonymous ftp to wavelab@ playfair.stanford.edu).

The following equations define the objectives for deriving a single FIR filter for the multi-temporal data set:

Background:
+ C (1)

Foreground:
+ C (2)

Because the wavelet basis is orthogonal, the sum of the nine images decomposed into each of the dyadic scales yields the original SAR image. We assume roughness is only textural when the range of differences in s_o in a multitemporal pixel is within s_o = +0.003. Otherwise, the w_s vector would be all one's, yielding the sum of the nine different scales, and thus yielding the original SAR image. Background factors are indicated by the temporal differences in backscatter of the four images (Equation 1). As our study was limited to the alluvial fans and valley floor, the area from the Panamint Mountains was omitted from our analysis.

Field observations show a decrease in surface roughness from the rougher upper ends of the alluvial fans to the smoother distal ends, not apparent in the SEM estimate illustrated in Figure 1a. Also significant in Figure 1a is roughness variation laterally across the fans. In the SEM approach, roughness at scales larger than the wavelength and surface roughness phase differences are not assumed to affect backscatter. Roughness estimated from optimized regressions of s_o to field measurements is temporally and spatially variable. In a general sense, the roughness recovered using the IEM and SEM methods or by empircal regressions, are similar even though the models are distinctly different.

Figure 1

a) Death Valley roughness estimates by the SEM model applied to a subset of DT35.1.G marks Grotto Canyon Fan and K the Kit Fox Hills. Black = rough; White = smooth.

b) Standard deviation of Lhh from four SIR-C images. The image has been inverted so that a high standard deviation is dark.

The significance of spatial scale to surface roughness is illustrated in Figure 2a, which shows that temporal correlation between images falls off steeply with spatial scale. In contrast, the coherence between wavelengths or polarizations at any single time is high (>0.80) until the spatial scale becomes finer than 100 - 200m scale (scale=7). The temporal variability of the coherence at each scale is most pronounced for scales ranging from 400 - 800m (e.g., scale = 5) as shown in Figure 2b. The temporal change in coherence at spatial scales much larger than the radar wavelength is indicative of background influences which are dependent on the conditions of imaging and surface roughness scales much larger than the radar wavelength. Given the high temporal coherence and lower coherence variability illustrated by Figure 2, we find the best roughness estimates to be associated with the coarsest scales. In contrast, background influences dominate the finer scales, and intermediate scales have mixed contributions of foreground and background. The near-zero coherence at the finest scale in Figure 2a can in part be due to slight misregistration of the image data set, but is most likely due to speckle noise.

a) Dyadic Scale	b) Dyadic Scale

To provide a scale-dependent roughness estimate we determined an FIR filter for a specific frequency or polarization by fitting Equations 1 and 2 to the multitemporal image data set decomposed into dyadic scales. Because the coherence for the finer spatial scales was near zero, we eliminated these scales 8 and 9 (i.e, w_s=0.0). Approximately 25% of the pixels in the 512² image used Equation 2 and the remainder used Equation 1. No significant improvement in reducing temporal backscatter variability was obtained by adding other channels of image data. These filters applied over different polarizations at a given frequency provided statistically similar relative roughness estimates. The main differences in roughness were noted between frequencies (e.g., C and L bands).

a)	b)

a)	b)

The resulting FIR roughness images for C and L bands (Figure 3) themselves are smooth because of the uncertainties introduced at finer spatial scales. These uncertainties act to reduce the w_s in FBA. If the assumption that backscatter is dominated by the surface roughness is valid, then the inference that backscatter is equivalent to roughness is legitimate. Figure 3b reduces the temporal variability seen in Figure 1b by a factor of four for the alluvial fans and valley floor. Both images in Figure 3 exhibit roughness gradients on the alluvial fans below Kit Fox Hills which are validated by visual field observations. These gradients are more apparent in L-band than C-band. Roughness gradients are not as pronounced at Grotto Canyon.

Figure 4 depicts the uncertainty remaining after application of the single FIR filter to each of the images. Figure 4b is the same as Figure 1b except that now the standard deviation is computed after application of the FIR filter. The spatial patterns in Figure 4 correspond to topographical features, indicating the FIR filter does not remove all effects at larger scales, and that greater uncertainties still persist in areas with large-scale variability such as the Kit Fox Hills. Not all areas of high uncertainty are associated with rough surfaces; for example, the black area in the middle of the playa sand dunes on the valley floor in Figure 4a and b. It is likely that this feature is associated with a change in surface moisture (e.g., a change in dielectric).

The temporal uncertainties of Figure 4 for the fan surfaces increase in proportion to roughness. These uncertainties are converted to rms height roughness using the field data of Weeks et. al. (1996a) to define a simple calibration. For L-band, the uncertainty ranges from +0.1 to +0.2 for smoothest and roughest fan surfaces, respectively. For C-band, the uncertainty for the roughest surface is nearly double that of L-band (e.g., +0.3).

To isolate the SAR signal due to roughness at the wavelength scale, we have defined the foreground as part of the backscatter measurement that is invariant and independent of acquisition geometry. Conversely, the background is that part of the SAR signal that exhibits changes and that varies with image acquisition geometry. This distinction is reasonable because textural roughness does not change over short time periods. However, this will not be perfect, because there are some differences in s_o that are due to differences in look angles (36 - 46^o) and azimuth (e.g., ascending and descending orbits). The largest of these differences is caused by variations in the look azimuth.

We interpret the results to imply that the spatial scale of roughness has a significant effect on the backscatter coefficient. The role of scale has not been considered in the development of the SEM or IEM models or in the empirical regressions between backscatter and field roughness measurements. In this study we find a significant tradeoff between roughness resolution and spatial scale. The present time series study indicates the potential of ten or more significant roughness levels with a single FIR filter operating on spatial scales compared to the three or four roughness levels obtained from FIR filters applied only in the polarization / frequency domain (Smith et al., 1996; Weeks et al., 1996b).

In this paper the validity of the SAR-estimated surface roughness is assessed for all pixels by the change of s_o in any single channel. Changes in the surface dielectric and spatial phase character, if present, may propagate through the FIR filter causing unknown effects in roughness estimates. The roughness uncertainty images (Figure 4) will not detect the presence of these factors. However, dielectric SEM images do show subtle changes on the fan surfaces and larger changes near water-saturated salt beds on the valley floor (Dubois et al., 1995).

A major unresolved issue is the determination of the objective functions used to construct FIR filters to estimate roughness. Future efforts to develop a set of objective functions that include effects of both dielectric and spatial scale may allow significant increases in detection of surface roughness over FIR filters optimized independently for the background factors. The overall complexity of the background, however, limits the utility of a single FIR filter. A hierarchical application of filters tuned to work with specific ranges of look angles may be required to remove specific background influences to achieve greater extendibility of this technique.

A single FIR filter applied over the dyadic spatial scales of four SAR images increased the detectability of roughness by roughly a factor of four over estimates obtained from a single image. The reduction in uncertainty achieved with the FIR filter is evidence of anisotropic scattering from scales much larger than the radar wavelength. The reduced temporal variation achieved from a single FIR filter is indicative of the common background factors arising from conditions dependent on the imaging conditions.

This work was supported by NASA/JPL contract No. 958450. We thank W.M. Keck Foundation for computer equipment used to analyze the data.

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