This final section will highlight the most significant portions of the work that has been accomplished, identify directions in which this research could be pushed, and place this work into some sort of context. The first topic is the characterization of the terrain surface using digital data. This work has suggested that the approach to surface characterization is important, and that a point solution is preferable. The second revolves around the question of determining the "best" comparison. The best fit between the two DEM sources was characterized by constant shifts in the easting. Finally, the relationship between DEMs and the real terrain they attempt to portray is recapitulated. Systematic trends in the distribution of 3 arc second DEMs were identified, along with particular instances of wide discrepancies between the DEMs. How can one relate such differences to accuracy?

Any investigation using terrain data requires a decision concerning the model for surface representation. Terrain may be digitally modeled in a number of plausible ways, and this topic has been a subject of research since the early days of digital elevation data. Peucker (1978) developed two levels for surface data structures; explaining these will be helpful for the coming discussion. The top level distinguishes between point structures, line structures, and patch structures. The term "patch structures" indicates that mathematical functions determine the surface for particular areas of the model. As a second level, these categories may be further divided according to whether the elements are arrayed in regular or irregular distributions. It is important to note that topological relations between data elements in point structures may be implicit, explicit, or algorithmic. In this research, I treated elevation data as collections of regularly distributed point elevations, based on USGS specifications for DEM data. However, the USGS does not make any reference to the treatment of topological relations; elevations not at sampled points are not defined by the specifications.

In the absence of any widely held geomorphological theory, I opted for a surface model that would be both conceptually and computationally simple and would make use of available data in a straightforward manner. The terrain model surface was defined to be bilinear. This bilinear surface was specified by the grid of sampled points from the DEM. As a result of this definition, every location on the surface possessed an implicit elevation that could be calculated rapidly. This quality was used to develop a more sensitive comparison technique than most examples from the research literature. In this work, DEM data were kept separate from the surface modeling question until the stage of comparison itself. That is, processing occurred only upon the data locations as the data were projected and subjected to a datum shift. Elevation values for the locations were not affected by any processing; they remain identical to the elevations in the original DEM file.

In contrast to the approach adopted here, research into the comparison of alternative DEMs generally does not dwell on the surface representation issue. Isaacson & Ripple (1990), for example, treat 3 arc second and 30m data as pixel values and performed all processing in a raster environment. Bolstad & Stowe (1994) likewise employ raster structures uncritically. They use field surveyed points to assess the accuracy of the DEM and a SPOT-derived elevation model; apparently these point values were compared with the raster cell elevation into which they fell. In contrast to these approaches, a body of work has been developed on the ability of various data structures to model terrain surface (Kumler, 1992; Weibel & Heller, 1991; McCullagh, 1988). A standard raster approach is to assign the entire area of each cell to one elevation value. The resulting surface consists of level squares or rectangles discontinuous at cell boundaries. When raster elevation data are transformed to another projection, elevation data must be resampled. If the 3 arc second - 30 meter DEM comparison were attempted employing a standard raster model, the raster elevation data would be altered from the original gridded DEM files.

Another potential structure is the triangulated irregular network (TIN). Fundamentally the TIN is a point structure. Therefore, the coordinates may be reprojected and transformed without affecting the elevation values. The TIN model may specify that the triangular facets be defined as planar; in this case, the elevation surface is continuous but the first derivative surface is characterized by breaks along facet edges. Elevations for areas within the facets of a TIN may also be characterized by some polynomial function, however, which generates continuous slope surfaces.

Regular and irregular point models may use a variety of spatial interpolation approaches to estimate elevation at specific locations, but in all events some assumption about the character of the underlying surface must be adopted (Lam, 1983). A potentially fruitful research direction is the comparison of alternative topographical characterization methods, that is, alternative assumptions about the surface between the data points. For example, how much would the comparison statistics have changed if a bicubic method were used instead of bilinear interpolation to estimate the 30m elevation at the 3 arc second points? The answer might depend upon the data set being tested and the nature of the terrain itself.

The second question involves making the comparison itself. For this work I had to characterize the sameness or difference between two terrain surfaces. The technique developed here used RMSE to provide a single value as an estimate of difference. However, differences at all points in a particular comparison file were examined to provide a more complete view of the comparison. This was done for all DEMs in this study. The question becomes more complicated when the ability to transform one of the data sets is included. The 3 arc second points could be shifted using an affine transformation prior to locating the bounding 30m points and calculating the interpolated 30m estimate. This transformation process is enacted by the program operator in an interactive fashion. Coefficients are changed, the data is processed, and an RMSE for the comparison is displayed. The operator can then change the coefficients and rerun the processing. I performed this hundreds of times in attempting to minimize the RMSE for each of the study DEMs. My assumption was that the lower the RMSE, the better the fit. The best fit, presumably, meant that any systematic positional (or vertical; the code also permits changes to the elevation) discrepancy had been eliminated. A preferable way to accomplish this might be to develop an automated optimization routine that would determine the best values for the 7 coefficients in the affine transformation.

These investigations found that substantial improvements in RMSE for each 7.5’ quadrangle could be generated by adjusting the transformation parameters. In all but two cases, an absolute shift to the east of 50-130 meters improved the fit. This improvement ranged from three to five percent in flat areas to ten to 35 percent in high relief areas. Relative shifts in either easting or northing did not improve the fit, and, aside from one quadrangle, neither did absolute shifts in the northing. It is unclear whether these easting shifts are due to a registration or systematic production problem, or whether they relate to an incorrect specification of the datum in the metadata.

A related issue for improving the fit is to attempt to relate other topological variables such as slope to the quality of fit. Empirical work dating back to the early part of the century has related elevation accuracy to slope (Shearer, 1990; Guth, 1992; Bolstad & Stowe, 1994). I calculated a version of slope using just the 4 bounding 30m points but did not identify a strong relationship between DEM difference and slope. However, due to the number of points used in the elevation comparison, the common slope algorithms could not be implemented (Srinivasan & Engel, 1991, present a survey of 4 slope calculation methods). A relatively straightforward extension to the code would allow one to generate slope values in a number of ways. It is possible that a differently calculated slope would show a relationship with elevation difference. Another possibility is that DEM difference is not directly related to DEM accuracy; in this case, any particular calculated value for slope might not correlate with elevation difference.

For the last theme of this conclusion, it is worthwhile to revisit some of the thoughts from section VI, since they relate to the place of this research within the broader area of DEM accuracy. In earlier paragraphs, I treated the subject of the surface representation and focused particularly on the nature of the surface. Assuming that the bilinear surface representation for the 30m data is adequate, the relationship between corresponding spot elevations in each DEM and the actual elevation will now be discussed. This relationship is not directly decipherable in the absence of ground truth information. One approach is to assume that the different elevation datasets are independent, equally likely samples of the actual surface. In this case, average differences and variance of differences provide some idea about the distribution from which the samples were drawn. Statistics about the spatial structure of the differences could be combined with knowledge about the distribution to model the actual elevation surface.

A second approach is to assume that the 30m data is more likely to be more accurate, and in any event is adequate enough for a given application. Systematic trends in the distribution of the 3 arc second data were identified and investigated in this research. A regression model of the difference revealed trends between the primary contour intervals of the source 3 arc second data. Differences between the two DEMs could be used to generate models to improve 3 arc second data over areas for which 30m data is not available (Ehlschlaeger & Shortridge, 1996). Alternatively, systematic patterns of difference, for example, the rising trend in the inter-spike range, could be modeled and removed. The result of this would be a 3 arc second data set that is more similar to 30 meter data. Finally, specific instances of large patches of sizable differences between the DEMs can indicate problems with particular 3 arc second datasets. This research identified a particular portion of the Camarillo quadrangle with large systematic error. This error was related to alterations to the source contour lines for the 3 arc second DEM.

One research direction is to characterize differences between the two DEMs in terms of their respective accuracy specifications. Such an effort seems difficult in light of the nature of those specifications. One might attempt to model the accuracy specifications for 30m data in isolation and 3 arc second data in isolation following the approaches of Fisher (1991) and Hunter & Goodchild (1995). The resulting uncertainty distributions characterize the spread of increasingly implausible elevations for each location in both datasets. The amount of overlap of the two distributions would then indicate the degree of correspondence, accounting for uncertainty in each DEM estimate. A deeper problem with the accuracy specifications for these DEMs is the lack of any information about the spatial structure of the error. RMSE in isolation cannot provide an idea about the nature of error in an elevation model, and how that error might affect applications using the DEM.

This work has developed a set of techniques and a methodology for the comparison of different but collocated elevation datasets. In the case studies, qualities of the relationship between 3 arc second and 30 meter DEMs were identified and discussed, systematic errors were modeled, and regions with large discrepancies isolated. Prospects for modeling differences and improving existing data using these techniques seem bright. As more accurate and more highly resolved terrain data become available, perhaps from satellite or global positioning systems, these methods can be used to come closer to identifying the relationship between ground truth and readily available digital elevation models.