3.1. Stereo Matching Errors
[29] As noted above, MOC images SP123703 and SP125603 (referred to as MPF 1) were the first pair chosen for mapping because they cover an area near the Mars Pathfinder landing site for which other topographic data sets including “ground truth” were available for comparison. We present a statistical comparison of the slopes in the various data sets rather than a featurebyfeature comparison, both because of the disparate resolutions of the data sources and because this first MOC pair does not include the actual Pathfinder landing point. Figure 2 shows image SP123703, orthorectified, along with color shaded relief representations of both MOLA data and the stereo DEM for the same area. The low relief of this region is readily apparent: from the bottom to the top of Big Crater the total relief is 300 m. The unedited DEM is essentially free from artifacts. The high overall success of the SOCET SET matcher is gratifying, especially given the relatively low contrast of the images, because editing to correct matcher blunders is generally the most timeconsuming and costly step of DEM production.
[30] A portion of the stereomodel to the southwest of and excluding Big Crater was selected for statistical analysis to characterize the flat part of the landing site. Table 2 defines the region(s) used for analysis in each DEM described in this paper, while slopes and other statistics are given in Table 3. The DEM in this area is consistent with our description on the basis of Imager for Mars Pathfinder (IMP) data of the landing site topography as dominated by ridges and troughs with a typical amplitude of a few meters and a wavelength of several tens of meters [Kirk et al., 1999a]. The smallest ridges are not fully resolved in the DEM but a pattern of somewhat larger ridges can be seen. Comparison of the image and DEM tends to support our previous assertion that many of the bright albedo features in the images are associated with local topographic highs. These are probably strips of lightcolored, rockfree sediment similar to those seen (also along ridges) near the lander and interpreted as dunes by Greeley et al. [2000].
Table 2. DEM Subareas AnalyzedSite  Set  Subarea  Used for^{a}  Data Set Geometry^{b}  Start Samp  End Samp  Start Line  End Line 


MPF  1  a  STslopes  Map  101  356  32  665 
MPF  2  a  STslopes  Map  76  203  201  456 
Melas  1  a  STslopes  Map  92  243  16  750 
  b  STslopes  Map  178  340  751  1485 
  c  STslopes  Map  178  243  16  1485 
  d  ST + PC  Ortho  351  900  1501  2300 
  e  PCslopes  Ortho  881  1009  1901  2156 
Melas  2  a  STslopes  Map, rot 7.54°  427  516  2001  3200 
Melas  3  a  STslopes  Map  80  208  155  577 
Eos  1  na  STfit haze  Map  157  352  764  1047 
  nb  STfit haze  Map  71  290  296  404 
  nc  STslopes  Map  185  313  461  1280 
  nd  PCslopes  Ortho  381  1070  1901  2500 
Eos  2  a  STslopes  Map, rot 7.33°  351  440  51  2550 
  b  STfit haze  Map, rot 7.33°  351  440  471  620 
  c  PCslopes  Image^{c}  1  512  1025  1536 
  d  PCslopes  Image^{c}  1  512  2301  2812 
Athabasca  1  a  PCslopes  Image  1  512  12601  13112 
  b  PCslopes  Image  1  512  11571  12082 
  c  PCslopes  Image  25  512  10601  11112 
Athabasca  2  n  STslopes  Map, rot 7.56°  372  427  389  901 
Athabasca  3  a  STslopes  Map, rot 6.59°  524  652  2500  3850 
  b  STfit haze  Map, rot 6.59°  544  633  2261  2510 
  c  PCslopes  Image  1  512  4801  5312 
  d  PCslopes  Image  1  512  3801  4312 
Isidis  1  na  STfit haze  Map  76  285  18  517 
  nb  STslopes  Map  145  285  18  1047 
  nc  PCslopes  Ortho  301  1100  1201  2100 
  sa  STslopes  Map  172  300  65  1264 
  sb  PCslopes  Ortho  401  1200  2151  2850 
Isidis  2  a  STslopes  Map, rot 7.48°  168  274  16  1278 
Isidis  3  a  STfit haze  Image^{d}  1  1024  4392  4992 
  b  PCslopes  Image  1  1024  1801  1350 
Isidis  4  a  STfit haze  Map^{e}     
  b  PCslopes  Image  325  1024  2475  4224 
Elysium  1  a  STslopes  Map, rot 7.53°  172  459  45  1293 
Elysium  2  a  STslopes  Map, rot 6.56°  163  327  21  1370 
  b  STfit haze  Map  226  310  1305  1354 
  c  PCslopes  Image  1  1024  201  1224 
  d  PCslopes  Image  1  1024  2401  3424 
Gusev  1  a  STslopes  Map  92  244  1  650 
  b  STfit haze  Map  153  326  621  400 
  c  STslopes  Map  256  384  1071  2047 
  d  PCslopes  Ortho  236  821  351  1615 
  e  PCslopes  Ortho  641  1240  3551  5150 
Gusev  2  a  STslopes  Map  99  273  20  665 
  b  STslopes  Map  198  347  728  1457 
  c  PCslopes  Ortho  319  915  51  2200 
  d  PCslopes  Ortho  648  1160  2401  4800 
Gusev  3  a  STfit haze  Map  161  327  1201  1450 
  b  STslopes  Map, rot 7.32°  287  414  51  1200 
  c  PCslopes  Ortho^{f}  3725  4236  301  3600 
Gusev  4  a  STslopes  Map, rot 6.53°  209  408  274  1808 
  b  STfit haze  Map  155  314  941  1280 
  c  PCslopes  Image  1  1024  901  2500 
Gusev  5  a  STslopes  Map, rot 7.41°  240  301  901  1220 
Gusev  6  a  STslopes  Map, rot 7.68°  235  437  32  1733 
  b  STfit haze  Map  171  330  901  1220 
  c  PCslopes  Image  1  1024  1701  3500 
Meridiani  2  a  PCslopes  Image  1  512  6001  6513 
  b  PCslopes  Image  1  257  7781  8293 
  c  PCslopes  Image  256  512  4291  4803 
Meridiani  4  a  STslopes  Map, rot 7.35°  150  429  41  1175 
  b  STslopes  Map, rot 7.35°  150  429  451  1175 
  c  STslopes  Map, rot 7.35°  210  429  231  450 
Meridiani  5  a  STslopes  Map, rot 7.38°  200  482  41  1549 
Table 3. Slope StatisticsSite  Set  Subarea  DEM From^{a}  Baseline, m  Stereo Slope Error^{b}^{,}^{c}  RMS Bidir Slope^{b}^{,}^{d}  RMS Adir Slope^{b}^{,}^{e}  99% Adir Slope^{b}^{,}^{e}  Slope Correction to 5 m Base^{f}  99% Adir Slope @ 5 m^{b}  P(Adir ≥15°) @ 5 m  Remarks  Hazard Unit 


MPF  1  a  ST  10  1.50  3.24  4.50  14.57  1.086  15.73  0.012  SW of Big Crater, no jitter  
MPF  2  a  ST  10  1.42  3.26  4.72  17.80  1.098  19.42  0.019  Landing site, filt for jitter  
Melas  1  a  ST  10  2.63  2.71  4.856  14.05  1.000  14.05  0.008  Does not resolve dunes  
  b  ST  10  2.63  1.55  2.66  7.69  1.000  7.69  <0.001  ″  
  c  ST  10  2.63  2.42  4.10  12.41  1.000  12.41  0.004  ″  
  e  PC  3   12.97  15.46  35.83  0.923  33.67  0.289  Dunes resolved!  
Melas  2  a  ST  10  1.81  9.86  12.68  37.16  1.187  41.96  0.233  Layers  
Melas  3  a  ST  10  1.83  11.22  14.08  43.20  1.273  50.09  0.274  ″  
Eos  1  nc  ST  10  1.93  6.24  9.14  30.97  1.092  33.24  0.072   
  nd  PC  3   5.80  7.04  22.30  0.927  21.83  0.029  PC area misses hills  
Eos  2  a  ST  10  1.83  6.03  7.92  23.79  1.189  27.66  0.087   
  c  PC  5.74   8.05  9.52  26.21  1.005  26.31  0.082   
  d  PC  5.74   10.46  13.56  31.71  1.005  31.83  0.239  PC area dominated by hills  
Athabasca  1  a  PC  5.87   1.26  1.72  5.01  1.020  5.11  <0.001  NE of ellipse but similar  
  b  PC  5.87   0.94  1.48  3.76  1.057  3.96  <0.001  ″  
  c  PC  5.87   1.25  1.86  4.84  1.019  4.98  <0.001  ″  
Athabasca  2  n  ST  20  1.88  3.39  4.71  15.30  1.125  17.11  0.019  S of ellipse, higher standing  
Athabasca  3  a  ST  20  1.82  2.48  3.45  10.09  1.409  11.48  0.004   
  c  PC  5.87   3.98  5.33  13.53  1.007  13.62  0.006   
  d  PC  5.87   2.66  3.48  8.44  1.010  8.52  0.001   
Isidis  1  nb  ST  10  2.64  4.65  6.37  24.07  1.202  28.24  0.037  Far outside ellipse  
  nc  PC  3   5.68  7.41  21.28  0.983  20.94  0.027  ″  
  sa  ST  10   4.11  5.78  19.31  1.058  20.34  0.027  ″  
  sb  PC  3   8.43  10.65  28.55  0.987  28.24  0.121  ″  
Isidis  2  a  ST  10  1.86  5.84  8.83  26.18  1.033  26.92  0.088  Dark floored secondary ctrs  Isidis Secondaries 
Isidis  3  b  PC  2.99   2.43  3.13  9.80  0.977  9.58  0.003  Cratered plains  Isidis Crplains 
Isidis  4  b  PC  3.06   3.36  4.52  15.53  0.991  15.41  0.012  Heavily cratered plains  Isidis Heavy Crplains 
Elysium  1  a  ST  10  1.13  3.51  5.06  13.16  1.016  13.37  0.006  Wrinkle ridge  Elysium Ridge 
Elysium  2  a  ST  10  0.68  1.92  2.74  8.20  1.038  8.51  0.001  Plains  
  c  PC  3.29   3.18  4.06  9.72  0.981  9.53  0.003  ″  Elysium Plains 
  d  PC  3.29   3.50  4.41  11.71  0.983  11.52  0.003  ″  
Gusev  1  a  ST  10  1.55  2.79  4.92  15.87  1.076  17.01  0.015  Cratered pl in Thira  
  c  ST  10  1.55  5.62  8.15  23.53  1.066  24.91  0.078  Knobby etched S of Thira  
  d  PC  3   4.19  5.21  14.96  0.982  14.70  0.010  Cratered pl in Thira  
  e  PC  3   9.27  11.51  29.41  0.990  29.16  0.163  Knobby etched pl S of Thira  
Gusev  2  a  ST  10  3.34  8.26  11.22  33.26  1.048  33.12  0.157  Knobby etched pl SW of Thira  
  b  ST  10  3.34  12.55  16.08  40.05  1.049  41.40  0.340  ″  
  c  PC  3   8.93  11.49  28.26  0.989  27.99  0.166  ″  
  d  PC  3   12.05  15.53  36.88  0.985  42.42  0.299  ″  
Gusev  3  b  ST  10  2.63  9.33  12.76  34.55  1.114  37.50  0.233  Smooth pl in ellipse center  
  c  PC  6   3.04  4.07  13.25  1.006  13.32  0.007  Preferred due to jitter in ST  Gusev Crplains 
Gusev  4  a  ST  10  3.09  4.44  6.13  17.94  1.064  19.02  0.028  Mix of etched, cratered plains  
  c  PC  3.13   8.08  10.24  27.96  0.994  27.82  0.047  Etched plains and ejecta  Gusev Etched 
Gusev  5  a  ST  10  1.84  8.26  11.00  31.13  1.200  35.92  0.195  Etched plains and ejecta  
Gusev  6  a  ST  10  3.23  2.26  4.21  9.67  1.035  10.00  0.002  Heavily cratered plains  
  c  PC  3.31   3.03  4.42  13.40  0.990  13.27  0.006  ″  Gusev Heavy Crplains 
Meridiani  2  a  PC  2.9   4.88  9.37  23.05  0.791  18.61  0.037  Albedo variations, not slopes  
  b  PC  2.9   1.25  1.82  4.93  0.946  4.67  <0.001  Bland area, typical  
  c  PC  2.9   2.21  3.38  9.38  0.933  8.76  <0.001  Exposed rougher area  
Meridiani  4  a  ST  10  1.63  1.61  2.25  7.31  1.048  7.65  <0.001  Plains and subdued crater  
  b  ST  10  1.63  1.44  1.92  5.60  1.048  5.87  <0.001  Plains only  Meridiani Plains 
  c  ST  10  1.63  2.52  3.75  10.68  1.061  11.32  0.003  Subdued crater only  Meridiani Crater 
Meridiani  5  a  ST  10  1.13  1.26  2.27  5.18  1.050  5.42  <0.001  Plains and subdued crater  
[31] Figure 3 shows the distribution of bidirectional slopes derived from the plains area of the DEM in Figure 2, for a northsouth baseline of one post (approximately 12 m). The rootmean squared (RMS) slope is 3.24° but, as the histogram shows, the slope distribution has longer tails (i.e., extreme slopes are relatively more common) than for a Gaussian distribution. Slopes on this baseline are in the range ±11.31° for 99% of the test area. For the adirectional slope (maximum slope in any direction, or gradient) over the same baseline, the 99th percentile is 14.57°.
[32] It is also of interest to look at the RMS slopes over a variety of distance scales. Not only do slopes over long baselines and local slopes have different implications for landing safety, rover trafficability, and the geologic processes at work on the surface, but this type of analysis lets us compare the MOC DEM with other topographic data sets for the region. If z(x) is a profile of height (relative to the local average, so that the average of z is 0) as a function of horizontal coordinate, and Δ is a horizontal baseline (“lag”), then the onedimensional autocovariance function ρ(Δ) is given by
where the brackets 〈〉 indicate an ensemble average. The autocovariance is the inverse Fourier transform of the power spectral density, and for an ergodic function, the ensemble average can be approximated by a spatial average over both x and multiple parallel profiles, so that the inverse transform of an appropriate estimate of the power spectrum yields an estimate of the autocovariance [Oppenheim and Schafer, 1975]. The Allan deviation ν, which is defined as the RMS difference in height between two points separated by a distance Δ [Shepard et al., 2001], can be expressed in terms of the autocovariance as
The RMS dimensionless (i.e., rise/run) slope is then s_{RMS} = ν(Δ)/Δ, and the RMS slope in angular units is Θ_{RMS} = tan^{−1}(s_{RMS}). The autocovariance, and hence the RMS slope as a function of baseline, can be obtained efficiently by fast Fourier transform techniques, but care is required because there are multiple approaches available for estimating the power spectral density. The periodogram (squared modulus of the Fourier transform of the profile, padded at the ends with zeros) yields an accurate estimate of the autocovariance when an inverse Fourier transform is applied [Oppenheim and Schafer, 1975] but smoothed estimators of the power spectrum obtained by multiplying the profile by various “window” functions yield distorted versions of the autocovariance and hence inaccurate slope estimates. In our work, we extract profiles from the DEM, subtract the mean elevation of the region (rather than the mean of each profile), pad them with a mirror image of themselves rather than with zeros, and average the squared modulus of the transform of multiple profiles to obtain the power spectral estimate. We have verified by extensive tests that this procedure yields RMS slope estimates in agreement with those obtained by direct calculation in the spatial domain, and it is substantially faster for large data sets. Note that no assumptions other than ergodicity about the topography are required; in particular, the topography need not be fractal for the calculation to be valid.
[33] Figure 4 is a plot of RMS slope calculated by fast Fourier transform techniques for six independent DEMs covering different parts of the Mars Pathfinder landing ellipse at a variety of scales. Each of the curves shows a characteristic “dogleg” shape, with a steep section with RMS slope proportional to Δ^{−1} for large Δ and a shallower loglog slope at small Δ (the highest resolution data set does not show an extended steep section but the beginning of a rolloff is nonetheless visible). Several factors contribute to the rolloff, the most important being edge effects that make the autocovariance and hence slope estimates increasingly inaccurate at large baselines. The scale at which edge effects become important depends somewhat on the data, but simulations indicate it is generally 10–20% of the horizontal size of the data set. Detrending of the data has a similar impact on the slope curve and also tends to become important at 10–20% of the data set length [Shepard et al., 2001]. The simplified approach to control used for the Viking stereo and photoclinometry data [HowingtonKraus et al., 1995; Tanaka, 1997], in which the whole DEM is tied to a single average elevation and regional slopes are not represented, is equivalent to detrending of the data and is probably responsible for the rollover at ∼10% of the data set length (∼1 km). The rollover in the Viking photoclinometry data at ∼4 km, however, represents a real change in the slope properties of the terrain, as the MOLA data show a corresponding breakpoint and this scale is only ∼2% of the MOLA data set width. As noted above, an early version of our MOC stereo data [Kirk et al., 2001a] was controlled in a way that effectively detrended it, but the version shown here was tied to MOLA at multiple points with differing elevations and hence has an accurate regional slope.
[34] The similarity of the shallow portions of the slope curves at comparable baselines is striking, especially given that no two of these data sets cover precisely the same area, and supports the validity of our results. The MOLA data set, extracted from the 64 pixel/degree EGDR gridded product [Smith et al., 2001] covers a 2° × 2° region approximately centered on the landing site. The photoclinometry data are taken from Viking image 004A72, which contains only smooth plains similar to those near the lander and south of Big Crater. The Viking stereo data cover the central, planar portion of the 100 × 200km Pathfinder landing ellipse, excluding nearby rugged craters and streamlined islands [HowingtonKraus et al., 1995; Tanaka, 1997] and agree well with the other results; stereo data from the full ellipse including more rugged features near the ends such as craters and streamlined islands yielded higher overall slopes as we previously reported [Kirk et al., 2001a, 2002b] but a similar trend with baseline. The curve for IMP data is derived from a DEM extending 10 m on each side of the lander [Kirk et al., 1999a] that was resampled from very irregularly spaced data. Slopes calculated directly from the unresampled data points by A. Haldemann (personal communication, 2003) agree well with those from the DEM, though they show more structure. Slope estimates over larger baselines can be computed from coarser IMP DEMs extending farther from the lander, but the RMS slope is systematically underestimated in these data sets, because much of the distant landing site was hidden behind ridges and the DEMs therefore contain (unrealistically) smooth patches interpolated from actual data for the visible areas. We have therefore not included these estimates in Figure 4, but their lower RMS slopes are consistent with the value of 4.7° at 1m baselines quoted by Kirk et al. [1999a].
[35] There is no fundamental reason for the shape of the slope distribution to be independent of baseline, but it is likely that this assumption is at least approximately correct over modest baseline variations, and we have found in practice that slope distributions for many areas of Mars evaluated at various baselines—including all DEMs described in this paper—are always longtailed. (Slope distributions from stereo typically have slightly longer tails than those from photoclinometry, perhaps reflecting blunders in the stereomatching process.) Because the distribution shape is unlikely to change greatly, the curves in Figure 4 can also be used to scale estimates of percentile slopes measured at one baseline to a slightly different baseline. On this basis the 99th percentile adirectional slope for the MOC stereo DEM can be extrapolated to a baseline of 5 m, giving a value of 15.73°. The corresponding value for a second pair covering the landing point itself, discussed below, is 19.42°. Five meters is approximately the lengthscale at which the MER (and Pathfinder) airbag system “feels” the topography on which it lands, and consequently one simplified criterion for a safe landing site is a small (e.g., ≤1%) probability of encountering a slope in excess of 15° over this baseline.
[36] The loglog slope of the curves in Figure 4 can be interpreted in terms of fractal geometry [Turcotte, 1997]: if s_{RMS}(Δ) ∝ Δ^{H−1}, where 0 ≤ H ≤ 1 is the Hurst exponent, the fractal dimension of the surface is D = 3 − H. The MOLA, Viking stereo and photoclinometry and MOC stereo data sets show a similar dimension D ∼ 2.3, whereas for the IMP data, D ∼ 2.4. The transition occurs at a baseline near 10 m, which is roughly the wavelength of the system of ridges in the landing site. The change in Hurst exponents may thus reflect the structural features (fluvial or eolian ridges vs. rocks) that dominate relief at different scales, but it should be borne in mind that the slope estimates are also affected by the noise properties of the data and method used to produce the DEM. In any case, we find it interesting that a straightline extrapolation of the Viking curves from baselines ≥80 m, or even of the MOLA curve from 1–2 km baselines, yields slopes at centimeter scales that are within a factor of two of those measured in situ. Such nearconstancy of the Hurst exponent over many orders of magnitude is definitely the exception rather than the rule, but this behavior at the Pathfinder site is clearly established by multiple data sets. Other DEMs described in this paper exhibit a variety of characteristics in their slopebaseline relations, such as single or multiple straight sections and even curved (nonfractal) sections. The values of the Hurst exponent, and the ranges of scales over which they apply, may provide clues to the geologic processes that shaped the terrains and are certainly superior to measures of the roughness at a single scale for characterizing the surface [Shepard et al., 2001]. Unfortunately, a discussion of such characteristics of the candidate MER landing sites is beyond the scope of this paper. We limit our discussion of the MER mainly to slope estimates at the single baseline of 5 m and use the slopebaseline curve mainly to correct our results to this scale, not because we believe that this lengthscale fully characterizes the surface, but because of the MER safety criterion for slopes at 5 m baseline. Readers interested in roughness at other scales can obtain slopebaseline tables and slope probability distributions for all units (as well as the DEMs from which they were generated) at http://webgis.wr.usgs.gov/mer/moc_na_topography.htm.
[37] Figure 4 also contains data from a second MOC stereopair, M1102414/E0402227 (MPF 2). As seen in Figure 5, this pair extends northward from Big Crater and hence includes the actual Pathfinder landing point. It also overlaps the previously discussed pair MPF 1, allowing all four images to be controlled in a single adjustment calculation so that the DEMs register precisely. Slope statistics for the plains around the landing point, derived from this model, are consistent with both the earlier MOC model to the south and with the Pathfinder IMP data, though extreme slopes appear slightly more abundant. A direct comparison between the two MOC DEMs in their area of overlap provides an indication of the errors in the stereomatching process, though this comparison is complicated somewhat by the systematic errors in the second DEM revealed in Figure 5. In addition to the “washboard” pattern of ridges and troughs crossing the model, caused by spacecraft jitter, the DEM is systematically elevated along the edges relative to its midline. A similar pattern is seen in many MOC stereo DEMs, including those produced by other institutions [Ivanov and Lorre, 2002; Ivanov, 2003; Caplinger, 2003], with the edges sometimes elevated, sometimes depressed relative to the center. These artifacts result from uncorrected optical distortion in the MOCNA camera. The amplitude of the topographic error depends on the distortion itself (which was found to be positive or pincushion, with a maximum amplitude of ∼1% at the edges of the field of view), the stereo convergence angle, and the miss distance between the centerlines of the two images. Thus MPF 1 (Figure 2) was not visibly affected by this problem because the image tracks were fortuitously close together. The SOCET SET scanner sensor model is capable of correcting for known optical distortion, but, unfortunately, the DEMs presented here were all generated before the phenomenon was understood and they could not be regenerated and reedited in the time available. To minimize the effect on slope estimates, stereo DEMs were generally rotated to align the ground track with the line axis and highpass filtered with a boxcar spanning the full alongtrack extent of the model, in order to remove the acrosstrack curvature before calculating slopes.
[38] The RMS deviation between the two MOC DEMs of the Pathfinder site, after filtering to suppress both jitter and distortionrelated artifacts, is 4.8 m. This height discrepancy can be used to estimate the stereomatching error, because it should equal the rootsummedsquare of the expected vertical precisions (EP) of the two pairs. The EP of a single stereopair is estimated as
where IFOV is the instantaneous field of view of the image pixels in meters on the ground, the parallax/height ratio is calculated from the threedimensional intersection geometry but reduces to tan(e) for an image with emission angle e paired with a nadir image, and Δp is the RMS stereomatching error in pixel units. By a simple statistical argument, if two images of differing IFOV are paired, the RMS of their IFOV values is used in the above equation. Applying this formulation to the two MOC pairs of the Pathfinder site, we estimate that Δp = 0.30 pixel. This value is slightly higher than the 0.2 pixel often used as a rule of thumb [Cook et al., 1996] and confirmed with our matching software for several other planetary image data sets [Kirk et al., 1999a; HowingtonKraus et al., 2002b]. Other comparisons (below) give smaller values of Δp, so the matching error in this case may be increased by the blandness of the site and by the especially low contrast of the images in the first pair.
[39] A similar analysis relating DEM differences to EP and thence to Δp may also be performed with independently produced DEMs from the same stereopair. We compared our DEM of pair E0200270/E0501626 (Melas 1) in Melas Chasma with corresponding models produced by Ivanov and Lorre [2002] and by the Harris Corporation for Malin Space Science Systems [Caplinger, 2003]. SOCET SET was used to resample the latter two DEMs via a sevenparameter transformation (translation, rotation, and isotropic scaling) to bring them into best registration with the USGS DEM. Elevation differences after this registration step were dominated by an alongtrack “arch” with an amplitude of a few tens of meters. This arch, as well as a slight sidetoside tilt in the difference DEMs, clearly results from the USGS model being controlled to MOLA with cameraangle adjustments allowed to vary quadratically with time as described above, the other models being uncontrolled. Difference statistics were collected after highpass filtering with a boxcar the full width of the DEM and 50 lines high to remove the overall arch, and removing a handful of “blunders” (individual DEM posts with wildly discrepant heights) from the Ivanov and Lorre [2002] (JPL) model. The RMS difference between the USGS and Harris/MSSS models was 4.2 m, corresponding to Δp = 0.22 pixel provided the errors in the two matching algorithms are equal in magnitude and uncorrelated with one another. The RMS difference between the USGS and JPL models was only 1.8 m, strongly suggesting that the SOCET SET matcher used by USGS and the JPL matching algorithm are similar enough that they are subject to partially correlated errors. An alternative but much less likely explanation would require very small matching errors in the JPL data set, intermediate errors of ∼0.13 pixel in the USGS model, and large ∼0.28 pixel errors for the Harris/MSSS model, all statistically independent from one another.
[40] The USGSJPL comparison raises some doubt about the assumption of independent errors in the USGS and Harris/MSSS DEMs. If these data sets also contain partially correlated errors, Δp might be greater than 0.22 pixel. We therefore carried out a final assessment of stereo matching errors specific to SOCET SET by using simulated data. The nadir image E0200665 in Gusev crater was resampled by an affine transformation (translation, rotation, and anisotropic scaling) to register as closely as possible with its stereo partner E0201453. The transformed nadir image was then input to SOCET SET as if it were actually the oblique image. In this way, we were able to simulate a stereopair with typical MOC imaging geometry and a range of surface textures typical for the Gusev site and the candidate MER sites as a whole, but with no local topographic parallax. Any deviations of the DEM recovered from this pseudostereo pair from a smooth surface could thus be attributed to matching error. The result obtained indicates Δp = 0.22 pixel, supporting the assumption of equal and independent errors in the USGS and Harris/MSSS models and agreeing closely with the longstanding 0.2pixel rule of thumb. Table 1 gives EP estimates calculated for each MOC pair by using Δp = 0.22 pixel. Values range from 0.8–5.8 m, with almost half between 1 and 2 m.
[41] The spatial pattern of matcher errors revealed by this test is also of considerable interest, because it influences the error in slopes calculated from stereo data. As seen in Figure 6, the matching errors take the form of thin strips extending across the DEM, so that errors between points some distance apart are partially correlated. A similar error pattern can be seen superimposed on the real topography in smoother areas of many of our DEMs, so we believe this behavior is universal. Fourier analysis of the “slopes” that would be inferred from this erroneous “topography” indicates that the elevation errors become uncorrelated at acrosstrack distances exceeding 81 DEM posts, and their correlation at smaller distances is approximately described by a Hurst exponent H = 0.5. This simple functional form allows us to predict the slope error σΘ as a function of baseline Δ, ground sample distance (GSD) between DEM posts, and EP:
Note that if elevation errors were uncorrelated between DEM posts, the slope error would be a factor of 9 greater at the limit of resolution (Δ = GSD), and would be proportional to Δ^{−1}. Stereo slope errors at the resolution limit, calculated by the above formula, are given for each model in Table 3. Values range from 0.7°–3.3° with the majority between 1°–2°. Slope errors at the limit of resolution are substantially smaller than the recovered RMS slope except in the Meridiani Planum “Hematite” site, where the estimated slopes are comparable to or less than σ_{Θ} by this formula, and the true stereo errors may be even greater, as discussed below. Because the Δ^{−0.5} dependence of the errors is steeper than the typical variation of the estimated slopes with baseline, the errors become increasingly unimportant at longer baselines.
3.2. Photoclinometry Simulations
[42] The errors involved in the process of calibrating images for photoclinometry against a priori topographic data were discussed above. In order to assess some of the errors intrinsic to the clinometric method itself, we created a series of simulated images from known DEMs, then attempted to recover these DEMs by photoclinometry. Selfaffine fractal topography [Turcotte, 1997] is convenient for such tests because it is easy to produce, contains roughness at a range of lengthscales, and its roughness can be controlled to crudely approximate the properties of various natural terrains [Shepard et al., 2001; Shepard and Campbell, 1998]. We generated a series of random fractal DEMs with 1025 × 1025 posts, which we used to simulate 1024 × 1024 pixel images. The algorithm used to produce these models differs from the Fourierfiltering approach described by Turcotte [1997]. A series of DEMs of size 3 × 3, 5 × 5, 9 × 9, and so on, are filled with statistically independent “white” noise, interpolated to the full size of the desired DEM, scaled to give the desired power law relation between relief and horizontal scale, and added together. Each DEM was also resampled to a 1024 × 1024 array of elevations at pixel centers for comparison with clinometric results in the same form. (As shown by Beyer et al. [2003], slopes measured between adjacent pixel centers will be less than slopes across individual pixels in the same model; our twodimensional photoclinometry method returns the former whereas their point or “zerodimensional” photoclinometry estimates the latter. Fortunately, the difference is much less significant for natural surfaces than for fractal models, which have significant pixelscale roughness.) Models were generated with Hurst exponents H = 0.2, 0.5, and 0.8. Hurst exponents for our Martian DEMs fall near the upper end of this range. The models were scaled to have a post spacing of 3 m and RMS slope of 1° between adjacent pixels. Versions of the H = 0.8 highpass and lowpass filtered at 16pixel scale were also produced with the same scaling and hence somewhat smaller slopes at given baseline. An H = 0.8 model with 10° RMS slopes was also generated.
[43] Images were then simulated from these DEMs, with an assumed emission angle of 0°, incidence angle of 45°, and Minnaert photometric model with k = 0.72, appropriate for Mars at this phase angle [Kirk et al., 2000b]. For each model, images were produced both with illumination along the sample axis and at 22.5° (roughly typical of MOC images used in this study) to this axis. Photoclinometry was then performed on the synthetic images in a manner identical to that used for the MOC images. Figure 7 shows the image, the exact DEM, and the error in the clinometric DEM for a subset of cases. To characterize the original and photoclinometrically recovered DEMs, we calculated the RMS elevation, and RMS bidirectional slope in the sample direction, as well as the RMS error in photoclinometric topography and RMS clinometric slope, as summarized in Table 4. The raw RMS elevation errors are a substantial fraction of the relief, but it is evident from Figure 7 that these errors, which essentially result from a lack of boundary conditions on the photoclinometric reconstruction, are far from random. They almost entirely concentrated between, as opposed to along, profiles aligned with the sun, so that their impact on pixeltopixel slopes in the downsun direction or at small angles to this direction is minimized. The relative errors both for elevations along profiles and for slopes are typically <1% for the models with 1° nominal slope and <10% for models with 10° slope. Thus, if the surface and atmospheric photometric behavior are known perfectly (i.e., if the calibration process were completely accurate), twodimensional photoclinometry can be expected to give quite accurate slope estimates.
Table 4. Photoclinometric Analysis of Simulated Images With Uniform Albedo^{a}Nominal RMS Slope^{b}^{,}^{c}  Hurst Exponent H  Filter^{d}  Sun Azimuth^{e}  Exact RMS Height  RMS Height Error^{f}  RMS Height Error (Destriped)^{g}  Exact RMS Slope^{h}  PC RMS Slope  % Error in PC Slope 


1  0.2   0  0.2518  0.6399  0.0182  0.9999  0.9971  −0.28 
1  0.2   22.5  0.2518  0.6078  0.0469  0.9999  1.0132  1.33 
1  0.5   0  0.8772  0.7763  0.0174  1.0000  0.9976  −0.24 
1  0.5   22.5  0.8772  0.7411  0.0411  1.0000  1.0100  1.00 
1  0.8   0  3.0545  2.2273  0.0214  1.0000  0.9976  −0.24 
1  0.8   22.5  3.0545  2.1370  0.0415  1.0000  1.0031  0.31 

1  0.8  Lowpass  0  3.0510  2.1977  0.0160  0.9006  0.8986  −0.22 
1  0.8  Lowpass  22.5  3.0510  2.1350  0.0228  0.9006  0.9064  0.64 
1  0.8   0  3.0545  2.2273  0.0214  1.0000  0.9976  −0.24 
1  0.8   22.5  3.0545  2.1370  0.0415  1.0000  1.0031  0.31 
1  0.8  Highpass  0  0.2717  0.3448  0.0141  0.5203  0.5196  −0.13 
1  0.8  Highpass  22.5  0.2717  0.3320  0.0301  0.5203  0.5143  −1.15 

1  0.8   0  3.0545  2.2273  0.0214  1.0000  0.9976  −0.24 
1  0.8   22.5  3.0545  2.1370  0.0415  1.0000  1.0031  0.31 
10  0.8   0  30.5451  40.9136  1.5424  9.9917  9.4691  −5.23 
10  0.8   22.5  30.5451  37.1268  2.2806  9.9917  9.2364  −7.56 
[44] In addition to imperfectly corrected atmospheric haze, real images are subject to spatial variations of surface photometric properties (“albedo”) that are not included in the clinometric modeling process at all. To get some idea of the impact of such albedo variations, we created an albedo map for our fractal terrains that mimicked some of the properties of smallscale albedo variations on the Martian surface. The albedo distribution was obtained by stretching and clipping the highpassfiltered DEM, yielding a pattern in which local topographic minima are slightly darker than the rest of the surface. The pattern was then smoothed with a 2pixel Gaussian filter, smeared slightly with the Adobe Photoshop “wind” filter, and scaled to have a RMS variation of 0.63% of the mean. This amplitude of albedo variation was chosen because it gives the same contrast variation that would be obtained from a (uniform albedo) surface with 0.5° RMS slope under the assumed illumination. Synthetic images of the unfiltered H = 0.8 surface with illumination azimuths of 0°, 22.5°, and 45° were multiplied by the albedo map and analyzed by photoclinometry (Figure 8). The results are summarized in Table 5 and Figure 9. In the case with illumination along the sample axis, the recovered RMS slope along this axis is increased from 1° to 1.12°, which is precisely what is expected if the real slopes and albedorelated “slopes” are independent (the albedo distribution is locally correlated with elevation, but not with slope) so that the corresponding variances add linearly: . (Beyer et al. [2003] state the identical conclusion in terms of the slightly different slope values measured within rather than between pixels.) The albedorelated slope error increases dramatically with the angle between the illumination direction and slope baseline, however. This behavior results from the particular form taken by albedorelated artifacts. The photoclinometry algorithm interprets each lowalbedo patch on the surface as a region tilted away from the sun. To preserve continuity of the DEM, each such patch is joined by a ridge extending indefinitely on the upsun side and a trough on the downsun side. Similarly, bright patches would be tilted toward the sun and flanked by a trough and a ridge. These ridges and troughs have no effect on slopes in the illumination direction, but a strong effect on slopes at right angles to this direction. Fortunately, the pattern of stripes (ridges and troughs) can be strongly suppressed or even removed by appropriate spatial filtering of the DEM. The model must first be rotated so the stripes are aligned with the sample axis. Then an estimate of the stripes is generated by lowpass filtering the DEM along the sample direction and highpass filtering at right angles to isolate features with a particular minimum length and maximum width. This stripe estimate is then subtracted from the DEM, and the process is generally repeated to remove stripes of various sizes before rerotating the model to its original orientation. Figure 8 shows the effects of such spatial filtering on the DEM recovered from one of our variablealbedo simulated images. As seen in Figure 9, the filtering process does not entirely eliminate the slope error, but reduces the errors at all angles so that they are comparable to the error in slopes measured downsun.
Table 5. Photoclinometric Analysis of Simulated Images With Variable Albedo^{a}Sun Azimuth  RMS Slope Constant Albedo  RMS Slope Variable Albedo  RMS Slope Variable Albedo (Destriped) 


0  0.9976  1.1212  1.1043 
22.5  1.0031  1.3585  0.9753 
45  0.9959  2.0827  0.9520 
67.5  0.9850  2.0799  0.9900 
90  0.9471  2.5250  1.1200 
[45] The spatial filtering to remove stripes also affects some spatialfrequency components of the real topograph. Relief (ridges or troughs) that is highly elongated and aligned with the sun direction will be strongly suppressed, but the effect on slopes is minimal if the topography is close to isotropic. The filtering removes only components with high spatial frequency in the crosssun direction and low spatial frequency downsun. As a result, slopes in the downsun direction are unaffected. Crosssun slopes are reduced, but such slopes are poorly estimated by photoclinometry anyway, because the image contains little information about them. These assertions were verified by tests in which fractal topography with H = 0.8, and an estimate of that topography recovered by photoclinometry, were subjected to the destriping process. Downsun slopes were recovered almost perfectly by photoclinometry and were unaltered by filtering. Crosssun slopes were underestimated by about 5% by photoclinometry and reduced an additional 10% by filtering. The apparent Hurst exponent for baselines from the minimum to 10% of the data set width varied by less than 0.006 between the differently processed data sets.
[46] From the foregoing simulations we can draw two useful conclusions about the errors that surface albedo variations may introduce into our slope estimates for candidate landing sites. First, because all DEMs were filtered to suppress albedorelated stripes before slope estimates were calculated, they do not contain the amplified slope errors seen in Figure 8 for baselines crossing the illumination direction. Second, the magnitude of albedo variations and hence of slope artifacts will vary from region to region, but where the albedo variations contribute a small fraction of the total contrast in the image, they will contribute only minor errors to the slopes. Inspection of the MOC images indicates that albedo variations are small in relation to topographic contrast in all sites described here, with the exceptions of Mars Pathfinder and parts of Melas Chasma (where photoclinometry was not attempted) and Meridiani Planum. Photoclinometric slope results for Meridiani are likely to be mildly to severely corrupted by albedo variations and are thus upper limits on the actual slopes, but, despite this, the Meridiani site is clearly the smoothest studied.