(alan@rad.geology.washington.edu

milton@rad.geology.washington.edu

robin@rad.geology.washington.edu

john@rad.geology.washington.edu)

Abstract

Roughness and dielectric constant of the land surface are important parameters for many scientific disciplines. In the geological and geomorphological study of surfaces, surface roughness is related to physical process and to exposure history. The surface roughness of alluvial fans changes as the fans age at a rate that is controlled by climate. Because radar backscatter is sensitive to surface roughness, it is possible to map the relative stage of evolution of fan surfaces using SAR data. The Remote Sensing Laboratory at the University of Washington has sought to address this goal using SIR-C/XSAR data in a project that has three main components: extensive field studies of roughness combined with cosmogenic isotope dating of alluvial fan surfacesl; forward modeling of the effects of the architecture of geologic surfaces on radar backscatter; and development of strategies for inversion from SAR data to surface roughness that are robust and spatially extendible at the regional scale.

Field work has primarily focused on alluvial fans in Death Valley, Owen's Valley, and the Karakox Valley in China. Other field areas have included Queensland Australia and Hawaii. In the field, a close-range stereo photography method was used to characterize roughness, and rock samples were collected for cosmogenic isotope dating (results awaited). Using the field measurements together with forward modeling it was discovered that these geologic surfaces have different backscatter characteristics than the bare agricultural fields analyzed in most inversion studies. The slope and offset of surface power spectrum, together with the dielectric constant, are insufficient to determine the radar backscatter; phase characteristics of the surface power spectrum are also extremely important. Given the number of variables, inversion methods that attempt to obtain the intrinsic surface roughness (roughness at the radar wavelength scale) are highly non-unique even when constrained using field measurements. We have, therefore, examined the range of possible linear solutions for roughness and it is observed that they fall into a small number of domains each with distinct characteristics that are probably associated with physical factors such as the scale of roughness. Solutions determined from the semi-empirical (SEM) and integral equation (IEM) models and our own empirical solutions from the field measurements, fall into a common domain of solutions that are susceptible to contamination by background effects (vegetation, multiple scales of roughness, phase character of the surface) and hence may not be extendible to other geographic locations. A domain of stable solutions that are more extendible does exist; however, this extendibility is achieved at the expense of reduced resolution of roughness levels. In addition it appears that spatial information is likely to be an important component of an inversion proceedure. This year the inversion methods and their extendibility will be tested against cosmogenic dates on alluvial fans in Death Valley, Owen's Valley and China.

(I) Description of goals at project inception.................................... 1

(II) Overall scientific progress and results to date ............................. 1

(III) Additional results anticipated before project termination................. 4

(IV) Summary of activities: 1990 - present...................................... 4

(V) Publications and abstracts.................................................... 6

(VI) References..................................................................... 6

(VII) Figures......................................................................... 7

(VIII) Appendix Ia,b..................................................................... 9

(I) Description of goals at project inception

The objectives of the SIR-C/XSAR study at the University of Washington were to quantify the stage of evolution of geologic surfaces - mostly on alluvial fans - at a regional scale using remote sensing methods. Alluvial fans appear to be created synchronously over large regions, and once a fan depositional episode is concluded, they begin to weather and erode at a rate that, for a given rock type, is largely controlled by climatic factors. The initial effect of weathering from a radar perspective is that these surfaces get smoother with time. Therefore, there is a relationship between the radar backscatter and the age of a given surface that is determined by the rate of weathering and erosion. Figure 1 shows a comparison between field mapped geology (Hunt and Mabey, 1966) and SIR-C image data for part of Death Valley; the older Q2 surfaces are clearly visible in the radar image. Quaternary geologists and soil scientists have long been attempting to use weathering and erosion data to infer details of climatic history. Remotely sensed information on degree of weathering would be of use in correlating regional weathering data. For synchronous alluvial fan surfaces, regional mapping of the relative degree of weathering would also allow extraction of large scale subtle gradients that reflect climatic and paleoclimatic shifts from the clutter of small scale, local idiosyncratic effects.

The objectives of the project were to use SIR-C/XSAR radar images in conjunction with VNIR and TIR images to study the range of surface modification processes affecting dated chronosequences of alluvial fan surfaces at specific sites in the western Great Basin (see figure 2 for overall site location). The objective was then to extend relationships between weathering state and remote sensing observations to the regional scale. For radar, this amounts to developing the ability to robustly measure relative changes in roughness of geologic surfaces on a regional scale. Extension of spectral mixture analysis (SMA), developed for VNIR data, to SAR data was an important element of the project goals.

(II) Overall scientific progress and results to date

The three major components of the current project at UW are, 1) field work, 2) forward modeling, and 3) inversion for surface roughness. A considerable field effort focused on characterization of the nature, variability and age of alluvial fan surfaces. To determine the age of the fan surfaces, recently developed methods of cosmogenic isotope dating are currently being employed. Forward modeling was aimed at determination of the degree to which accepted roughness parameters control the backscatter for geologic surfaces. It was used to calculate backscatter from the field measured geometry of fan surfaces and compare the results with backscatter from theoretical (random) surfaces used to develop most inversion methods. Finally, inversion methods and SMA-like techniques were examined in an attempt to determine surface roughness from the SAR data. At all stages, considerable developmental work has been required: for example, the field methods for obtaining close-range stereo photographs and extracting roughness information; the sampling strategy most likely to yield success in cosmogenic dating; the ability to model backscatter from real surface profiles and to simulate surfaces with desired characteristics, and finally techniques that explore more extendible methods of inversion for roughness.

Below we cover in detail the three major elements of the work at UW.

1) Field work

Extensive sampling conducted in Death Valley, Owens Valley, and the Karakox Valley, has provided material that is currently being processed for cosmogenic dates. These dates constitute our "ground truth" which will enable relationships between remote sensing imagery and fan surface age to be established. In terms of roughness characterization, our field work has confirmed earlier suggestions that the profiles of geologic surfaces can be described by a power spectrum that conforms to a power law (Figure 3). Using ground based stereo photography for the mm to m scale and low altitude stereo aerial photography for the cm to 100m scale we determined roughness in Death Valley over five orders of magnitude in spatial scale at many sites and for these the power law approximation appears valid. The same measurements were been made at sites in the Owen's Valley and near Kilauea on Hawaii. Conventionally, the surface roughness is specified by the slope and offset of the best fit power law in log-log space. Our studies confirm that radar backscatter is strongly related to the offset of the surface power spectrum. Figure 4 shows typical examples from Death Valley which display strong correlation between C_hh and offset, and much weaker relationships with rms height and spectrum slope. The scatter in all of the relationships, however, suggests that either that the field measurements are imperfect, or that other factors or combinations of factors affect the backscatter.

2) Forward modeling

Forward modeling methods, developed at UW (Pak et al., 1995; Tsang et al., 1994), calculate complete solutions to Maxwell's equations for either 2-D (profiles) or 3-D (surfaces). This modeling was used to calculate backscatter for the field measured 2-D profiles and for theoretical profiles that were simulated assuming power law spectra with the parameters of best fit slope and offset determined from the field measurements. Results (Figure 5) indicate that, in addition to the slope and offset of the surface power spectrum, the non-random phase characteristics of the surface FFT are crucial in determining the radar backscatter. The assumption of a random surface is not valid for geologic surfaces; in the FFT of the surface profile, the role of the phase is to describe the organization of the surface into discrete objects such as pebbles. Our modeling indicates that this phase character controls the gain (slope) of the relationship between roughness parameters (such as offset) and backscatter (Figure 4b). Therefore, entirely different relations between backscatter and roughness are possible for different types of surfaces; agricultural surfaces can not be expected behave like most geologic surfaces. In both cases the surface power spectrum can be adequately fit by a power law, but the slope and offset of that power law do not give enough information to determine the radar backscatter. Variations in phase character of the surface may, therefore, account for much of the scatter in relations between field roughness measurements and radar backscatter. Another factor may be varying dielectric constant and hence penetration, although we do not anticipate significant variations in dielectric on these surfaces.

3) Unmixing/Inversion

When inverse methods such as the semi-empirical method (SEM - Dubois et al., 1995) and integral equation method (IEM - Shi et al., 1996), developed mostly for bare agricultural fields (see also Oh et al., 1992), are applied to Death Valley geologic surfaces we find that they greatly overestimate the surface roughness (Figure 6). This overestimation may be partly explained, as described above, by the different and non-random phase character of alluvial fan surfaces, but another factor is the multiple scales of roughness present. Washes that descend the alluvial fan surfaces create a second scale of roughness at the m to several m scale in addition to intrinisic roughness at the radar wavelength scale (cm's). The preferential orientation of these washes (downslope) can be seen to cause a change in the backscatter between ascending and descending pass data takes (parallel vs. perpendicular to the wash orientation). Inversion on ascending pass images produces a different estimate of roughness than inversions made on descending pass images (Figure 7).

In natural environments other kinds of complexity enter the picture; for example, the presence of varying mounts of vegetation cover, varying dielectric, and multiple scales of roughness. Given the number of variables, the inversion for roughness is highly non-unique except in special cases. In order to make strides toward the goal of reliable regional mapping of relative roughness on alluvial fan surfaces an inversion method that is extendible must be developed. It must be insensitive to the effects of multiple scales of roughness, varying phase character, and other complicating variables. To this end we have examined the space of possible linear roughness solutions by applying linear FIR filters (see Appendix Ia) to the image data (Figure 8) (note that the relation between field measurements and image backscatter is approximately linear - see Figure 4). Solutions can be evaluated by various criteria such as the separation between surfaces of different roughness (called the foreground - Figure 8a), and the susceptibility to effects of undesired variables (called the background - Figure 8b). In the solution space, empirical roughness solutions and the SEM and IEM solutions, lie in a large domain characterized by high variability. A trough in the solution space delineates a region where solutions are more stable, but offer less resolution of roughness levels (Figure 9). It appears that conventional solutions are very dependent on local variability in background complexity (phase, scales of roughness, vegetation). Indeed there is a striking resemblance between the temporal variance of SAR backscatter, obtained using a sequence of images, and the SEM roughness solution for a given image (Figure 10).

Another method by which more stable solutions may be obtained is through the use of multitemporal image sequences combined with information from image spatial scales. In the case of Death Valley, and alluvial fans in general, the second scale of surface roughness caused by washes poses difficulty. However, its effect on the radar backscatter is anisotropic whereas the effect of intrinsic roughness (at the radar wavelength scale) is isotropic (look direction independent). We have designed filters, acting on spatial scales obtained by wavelet decomposition of the image data, that select for the isotropic component from an image sequence that has varying look directions (see Appendix Ib). When the spatial scale of variability is examined using wavelet transforms of the image data it is found that the temporal (look direction) variance is a maximum at the finer scales and a minimum at the coarsest scales (Figure 11). The filter, therefore, weights spatial scales that are isotropic. This method produces smooth solutions (Figure 12) that have similar appearance to those from the trough of the solution space for single images, but offer apparently greater resolution of roughness levels. In these solutions it is possible to see a downslope gradation (smoothing) in roughness on the fans; this is observed in the field but does not usually stand out in roughness invresions.

It, therefore, appears that spatial information, image sequence variability, and consideration of the sensitivity of solutions to undesired variables, are all factors that can guide us toward robust solutions for relative roughness. These methods will be tested against field data in the coming months.

A parallel effort at UW has examined the inversion of VNIR data for surface roughness in Death Valley. Using a forward radiosity model together with measured spectra from alluvial fan surfaces, an inverse FIR filter was trained. Results show that the filter is insensitive to albedo variation and that field roughness measurements are retrieved with surprising success.

(III) Additional results anticipated before project termination

One of the key elements in the program at UW relates to the connection between age of alluvial fan surfaces and radar backscatter. Although it has been possible to do this for Owen's Valley using dating of interbedded lavas, for most fans the determination of age rests on the complex process of cosmogenic isotope dating. Large numbers of samples from alluvial fans in Death Valley, Owens Valley and China are presently undergoing the dating procedure and results are awaited within the next few months.

Inversion/unmixing procedures can then be trained and tested against these results. Of special interest is the extendibility of inversions; we intend to examine extendibility firstly within Death Valley where most of our development has focused on the Stovepipe Wells area. Dates will also become available for fans further to the south such as Trail Canyon fan. The process will then expand to include Owen's Valley and the Karakox Valley in China. Although it might be anticipated that different processes, or process rates, should give rise to different age roughness relationships, the relative weathering state should be accurately reproduced, after which other information can be used to calibrate the method.

We intend to develop our inversion methods to include spatial scale information as well as temporal variability. We have already used the wavelet decomposition of images to help isolate the isotropic portion of the radar backscatter. The combination of spatial and temporal data is one way to address the inherent ambiguity inversions from in an image data set of low intrinsic dimensionality.

In parallel with the effort on inversion, we are extending the forward modeling to look at the decomposition by scattering mechanisms. The current calculation is converted into an iterative model in which the contribution from single scattering, double bounce, and multiple scattering can be examined for a variety of simulated and real surfaces. We will then look at the validity of current scattering mechanism decompositions.

(IV) Summary of Activities 1990 to present

A. Field:

The UW group conducted five field seasons in Death Valley and Owens Valley, one in Hawaii, two in China (Karakox Valley), and one in Australia. This work was dedicated to characterization of roughness, age and soil development on geologic surfaces (mostly alluvial fans and lavas). A field method of making close-up stereo photographs of the ground surface to produce high resolution surface profiles was used for roughness characterization.

Death Valley and Owen's Valley

(1) Combined close-range stereo photography, close-range multispectral VNIR images, and field spectrometer data at 13 sites in Death Valley. Measurements designed to characterize roughness and to enable development of joint SAR/VNIR analysis of roughness.

(2) 50 low altitude (1:1000 scale) stereo pair aerial photographs obtained over alluvial fans in the Stovepipe Wells area of Death Valley.

(3) Roughness estimates obtained at two spatial scales from the field stereo photography: the sub-centimeter to meter scale, and the 10's cm to 10's meters scale. Available roughness spectra now span four orders of magnitude in length scale at ten sites in Death Valley

(4) Organized field workshop in Death Valley, California, attended by members of various science teams.

(5) Collected samples for cosmogenic dating from numerous alluvial fan surfaces of various ages in Death Valley and Owen's Valley. In conjunction with LLNL cosmogenic dating of the ages of fan surfaces is under way; results already available for Owen's Valley.

(6) Requested and obtained day/night flights of the C130 aircraft to collect TIMS, NS-001 data and color IR photography over Death Valley and Owens Valley. Comparison of thermal and SAR data intended.

Karakox Valley, China

(1) Participated in field work and sample collection in China with SIR-C PI Tom Farr for characterization of fan surface in the Karakox Valley, Western China.

(2) Collected samples from alluvial fan surfaces of various ages In conjunction with LLNL cosmogenic dating of the ages of fan surfaces is under way.

Hawaii

(1) Close-up stereo photography for determination of roughness parameters at 18 sites on the island of Hawaii in the vicinity of Kilauea.

Trafalgar Hills, Australia

(1) Soil moisture study - measurements made at 10 sites near the Trafalgar Hills in Queensland.

(2) Collation of low altitude helicopter gamma ray emission survey (for soil moisture) over the area.

B. Image data

(1) Obtained all SIR-C scenes taken over the Death Valley supersite, scenes also obtained for Owens Valley, the Amazon Basin, Hawaii, and Mt. Rainier. AIRSAR images from Death Valley, Owens Valley, Australia, and Mt. Rainier also obtained.

(2) Landsat TM images obtained for Death Valley, Owens Valley, Trafalgar Hills (Australia), Hawaii, and China (Karakox Valley)

(3) Requested and obtained day/night flights of the C130 aircraft to collect TIMS, NS-001 data and color IR photography over Death Valley and Owens Valley.

(4) Obtained gamma ray emission survey, in image form, taken over the Trafalgar Hills in Australia. The gamma ray emission should be related to soil moisture.

C. Modeling and inversion studies

(1) Developed and installed software designed to simulate natural rough surfaces and to calculate the radar backscatter from them. The method uses complete solutions of Maxwell's equations and can be used to compute predicted backscatter from measured surface profiles. This software complements our currently available radiosity models which calculate VNIR reflectance for the same rough surfaces, including the effects of multiple scattering interactions.

(2) Exchanged roughness/dielectric models, results, and information with Pascale Dubois at JPL, and J.C. Shi at UC Santa Barbara.

(3) Examined the applicability of existing roughness inversion algorithms (developed primarily for soil surfaces) to geologic surfaces in Death Valley. Discovered problems in extendibility of these algorithms

(4) Looked at alternative solution methods and developed a method for analysis of the space of possible solutions in order to place existing algorithms in a larger context. We determined that more extendible solutions will trade roughness resolution for extendibility.

(5) Examined use of temporal sequences of images with different imaging parameters to produce robust roughness solutions.

(6) Continued to study the potential use of VNIR data in inversion for surface roughness.

(V) Publications and Abstracts (see references section)

(1) Attendance and presentations made at all SIR-C/XSAR team meetings.

(2) Presentation at Spring AGU in Baltimore.

(3) Article published in SIR-C special issue of JGR planets (1996).

(4) Article published in SIR-C/XSAR special issue of RSE (1997).

(5) Related articles submitted to IJRS and published in Microwave Opt. Technol. Lett.

(6) Invited talks at the IEEE conference in Toulouse, France on "Retrieval of bio- and geophysical parameters from SAR data for land applications".

(7) Invited talk presented at the AIRSAR workshop at the University of New South Wales, Australia (see appendix II) in November 1995.

(8) Talk given at Airborne RS conference in San Francisco 1996.

(9) Article published in conference proceedings of Airborne RS conference 1996

(10) Two manuscripts are in preparation.

(VI) References

Dubois, P.C., van Zyl, J.J., and T. Engman (1995), Measuring soil moisture with imaging radars, IEEE Trans. Geosci. Remote Sensing, 33(4), 915-926.

Hunt, C. B. and D.R. Mabey (1966), Stratigraphy and structure, Death Valley, California, U.S. Geol. Survey Prof. Paper, 494-A, 162pp.

Li W., Weeks R., and A.R. Gillespie (1996), Multiple scattering in the remote sensing of natural surfaces, Inter'l. Jour. Rem. Sens., submitted.

Pak, K., Tsang.L., Chan, C.H., and J.T. Johnson (1995), Backscattering enhancement of vector electromagnetic waves from two-dimensional random rough surfaces based on Monte Carlo simulations, J. Opt. Soc. Am. A, 12(11), 2491-2499.

Oh, Y, K. Sarabandi, and F.T. Ulaby, An empirical model and an inversion technique for radar scattering from bare soil surfaces, IEEE Trans. Geosci. Remote Sensing, 30, no.2, 370-381, 1992.

Shi, J.C., Wang, J., Hsu, A., O'Neill, P., and E.T. Engman, Estimation of bare soil moisture and surface roughness parameters using L-band SAR image data, IEEE Trans. Geosci. Remote Sensing, IGARSS'95 special issue, in press.

Smith, M.O., Roberts, D.A., Hill, J., Mehl, W., Hosgood, B., Verdebout, J., Schmuck, G., Koechler, C., and J.B. Adams (1994), A new approach to determining spectral abundances of mixtures in multispectral images, IEEE Trans. Geosci. Remote Sensing, Proc. IGARSS'94, JPL, Pasadena, CA.

Smith M.O., Weeks R., and A.R. Gillespie (1996), Using background factors to optimize roughness from multipolarized SAR images, Presented 2nd Int'l. Airborne Rem. Sens. Conf. Exhib., San Francisco 24-27 June.

Tsang L., Chan, C.H., and K. Pak (1994), Backscattering enhancement of a two-dimensional random rough surface (three-dimensional scattering) based on Monte Carlo simulations, J. Opt. Soc. Am. A, 11, 711-715.

Tsang, L., Pak, K., Weeks, R.J., Shi, J-C., and H. Rott, Electromagnetic wave scattering from real-life rough-surface profiles and profiles based on an averaged spectrum, Microwave Opt. Technol. Lett., 12(5), 258-262.

van Zyl, J.J., C.F. Burnette, and T.G. Farr, Inference of surface power spectra from inversion of multifrequency polarimetric radar data, Geophys. Res. Lett., 18, no.9, 1787-1790, 1991.

Weeks, R.J., Smith, M.O., Pak, K., Wen-Hao Li, Gillespie, A. R., and B. Gustafson (1996), Surface roughness, radar backscatter and VNIR reflectance in Death Valley, California, J. Geophys. Res., 101(E10), 23077-23090.

Weeks, R.J., Smith, M.O., Pak, K., and A.R. Gillespie (1997), Inversions of SIR-C and AIRSAR Data for the Roughness of Geological Surfaces, Remote Sens. Environ., in press.

(VII) Figures

(1) Comparison of (a) field mapped geology (Hunt and Mabey, 1966) with (b) SIR-C image data for Death Valley. The older Q2 surfaces shown in (a) are clearly visible in the SIR-C image, but younger Q3 and Q4 surfaces are harder to distinguish from one another in (b).

(2) Location of the Great Basin.

(3) (a) Location of field sites within the Stovepipe Wells area of Death Valley. (b) Examples of mean power spectra for some of the sites shown in (a), obtained from surface profiles extracted from ground based stereo photography and low altitude aerial stereo photography.

(4) Relationship of ground roughness measurements to SIR-C image backscatter (DT35.1). Backscatter for C_hh is compared to (a) power spectrum offset, (b) rms height, and (c) power spectrum slope.

(5) Results of forward modeling using complete solutions of Maxwell's equations (Tsang et al., 1994) for surface profiles. (a) Scattering coefficient with scattering angle for three different types of surface profile (length approximately 1m): the solid line shows averaged results from 40 real profiles at a site above the Kit Fox Hills in Death Valley; the dotted line shows results from profiles simulated using the best-fit power-law to the mean spectrum of the above real profiles together with random phase; and the dashed line shows results for profiles also simulated using the best-fit power law but combined with the real phases taken from spectra of the 40 real profiles. The arrow indicates the monostatic backscatter value (-40°). (b) The calculated backscatter coefficient against rms height for each of the sites in Death Valley together with linear best fits: solid symbols and solid line show the results for profiles simulated as in (a) above (dotted line); open symbols and dashed line show results for the real profiles (20 per site).

(6) Results for DT35.01 of three different inversions for surface rms height and a comparison with surface data: (a) a linear empirical model (the FIR model - optimized using the field rms height data) (S = location of Stovepipe Wells); (b) the SEM solution (Dubois et al., 1995); (c) the IEM solution (Shi et al., 1996). (d) Comparison of FBA results with field rms data. (e) Comparison of SEM results with field rms data. (f) Comparison of IEM results with field rms data. The scales on the y axes are different.

(7) Results of the SEM inversion for rms height for two different SIR-C images. (a) DT35.01 is an ascending-pass image (essentially perpendicular to the mean wash orientation). (b) DT120.30 is a descending pass image (parallel to the mean wash orientation). (c) and (d) show a comparison of the SEM results with the field rms height data for DT35.01 and DT120.30, respectively.

(8) Surfaces generated from linear FIR filters designed to estimate rms height. Each 36 x 36 surface is formed by generating FIR filter weights (e.g., w_n vector) that represent all possible linear combinations of C and L band co-polarized channels. The x-axis corresponds to C-band and the y-axis to L-band. The z-axis is a measure of the performance of the linear filter in the following ways: a) foreground contrast - the absolute value of the mean difference in solution value between image data points from fan surfaces of contrasting roughness; b) background - the weighted average of the standard deviations of pixels from surfaces of constant roughness; (c) the foreground divided by the background.

(9) Example of the type of solution that occurs within the trough of the foreground/background surfaces (Figures 8). Such a solution appears to minimize the local variance. (a) Application to AIRSAR data using L-band co-polarized channels. (b) Application to SIR-C data take 35.01 (same FIR filter as (a)) using L-band co-polarized channels.

(10) (a) The SEM solution for rms height using DT35.1. (b) The variance from a sequence of 4 coregistered SIR-C images. The image has been inverted so that a high standard deviation is dark.

(11) The mean (a) and standard deviation (b) of the coherence in backscatter amplitude in a sequence of four SIR-C images at different spatial scales for Chh, Lvv and Lhh. The dyadic scales are: 1= 6.4 - 12.8km, 2= 3.2 - 6.4km, 3= 1.6 - 3.2km, 4= 0.8 - 1.6km, 5= 400 - 800m, 6= 200 - 400m, 7= 100 - 200m, 8= 50 - 100m, 9= 25 - 50m.

(12) Relative roughness estimated from an FIR filter applied to a sequence of images decomposed into dyadic scales for a) Chh and b) Lhh. The bright surfaces are smooth; dark surfaces are rough. In order of increasing scale number, w_s for examples a and b respectively are: (0.81, 0.58, 0.57, 0.42, 0.29, 0.17, 0.06) and (0.82, 0.64, 0.59, 0.42, 0.30, 0.23, 0.09).

(VIII) Appendix Ia: - FIR filters

FIR filters are linear filters of the form

(1)

where Y_n is the output of the filter, s is the input signal which consists of nb channels or bands, w is the filter, and K a constant. In our case, the input signal is the SAR image data (s_q - consisting of as many as six bands of data per pixel) to which the filter weights (w_q) are to be applied in order to produce Y_n, the estimate of the desired quantity (intrinsic roughness). The filter, operating in the spectral domain (wavelength and polarization), is simply a linear transformation of the image bands. Solution surfaces can be constructed by cycling through possible linear transformations.

Appendix Ib: - analysis of image sequences

The following equations define the objectives for deriving a single FIR filter for the multi-temporal (image sequence) data set which has been decomposed by spatial scale s using the wavelet transform. The filter is applied across spatial scales in order to minimize variance within the image sequence:

(2)

(3)

where w_s is the FIR filter vector we desire to find; s^o_s,t1 and s^o_s,t2 are the calibrated radar backscatter at a given frequency/polarization acquired at two different times; _s is the mean calibrated backscatter for the four images for each scale s; the mean calibrated SAR backscatter measurement from the original four images; s the scale; and N the number of scales. For the 512² images, N=9. For each pixel in the image the choice to use Equation 1 or 2 depends on the similarity in backscatter measurements among the four data sets; if the range of s_o in a pixel for all four images was within + 0.003 then Equation 2 was applied. If any image pair exceeded this range for a given pixel, Equation 1 was applied.

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. The influence of background factors is 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.