Accuracy Assessment of Digital Elevation Models based on Approximation Theory

Empirical research in DEM accuracy assessment has observed that DEM errors are correlated with terrain morphology, sampling density, and interpolation method. However, theoretical reasons for these correlations have not been accounted for. This paper introduces approximation theory adapted from computational science as a new framework to assess the accuracy of DEMs interpolated from topographic maps. By perceiving DEM generation as a piecewise polynomial simulation of the unknown terrain, the overall accuracy of a DEM is described by the maximum error at any DEM point. Three linear polynomial interpolation methods are examined, namely linear interpolation in 1D, TIN interpolation, and bilinear interpolation in a rectangle. Their propagation error and interpolation error, whose sum is the total error at a DEM point, are derived. Based on the results, the theoretical basis for the correlation between DEM error and terrain morphology and source data density is articulated for the first time. Introduction As a digital representation of a topographic surface, digital elevation models (DEM) are routinely used in various applications such as terrain analysis, hydrological modeling, and energy flux study. Although much research on DEMs has been conducted since the 1950s, there is still a lack of consensus regarding the fundamental question in DEM accuracy assessment: What are the error components of a DEM? How is each component, as well as the overall accuracy, assessed? In the literature, error variance and root mean squared error (RMSE), which are rooted in error propagation theory, have been widely applied (Tempfli, 1980; Li, 1993; Aguilar et al., 2006). However, substantial challenges, both theoretical and practical, present themselves to the applicability of these methods (Wise, 2000; Liu and Hu, 2007). For example, while error propagation theory assumes that DEM errors are random and independent, many empirical studies have observed that DEM errors are actually correlated with terrain morphology and sampling density (Torlegard et al., 1986; Östman, 1987; Fisher, 1991; Hunter and Goodchild, 1995; Kyriakidis et al., 1999; Holmes et al., 2000; Lopez, 2002; Aguilar et al., 2005; Bonin and Rousseaux, 2005; Oksanen and Sarjakoski, 2006). The inability of error propagation theory to account Accuracy Assessment of Digital Elevation Models based on Approximation Theory Peng Hu, Xiaohang Liu, and Hai Hu for the spatial and structural characteristics of DEM errors suggests that an alternative framework for DEM accuracy assessment is necessary. This paper introduces approximation theory adapted from computational science to examine the errors in DEMs interpolated by three linear polynomial interpolation methods, namely linear interpolation in 1D, Triangulated Irregular Network (TIN) interpolation, and bilinear interpolation in a rectangle. These linear interpolation functions can be applied to generate a DEM from the contour lines and spot elevations in a topographic map as well as from lidar point data. In the following sections, we first review the nature and composition of DEM error to lay the foundation of our discussion on approximation theory, then, derive the theoretical formulas for each error component in the three interpolation methods. Based on these formulas, the theoretical reasons underlying the correlation between DEM error and terrain morphology, source data density, and interpolation method are revealed for the first time. DEM Error Components Since an interpolation-generated DEM consists of a set of grid points, an overview of DEM error should first address the point scale. Supposing T is a DEM point whose unknown true elevation is zT. Given an interpolation method, the interpolated elevation using error-free source data is denoted by HT. In reality, HT is rarely the elevation value ZT recorded in a DEM because of the errors in the source data. The relationship between HT and ZT can be written as: ZT HT dT where dT is the impact of the errors in the source data on point T. It can be seen that dT ZT HT. To expedite the discussion on approximation theory, this paper assumes that the gross error and systematic error in the source data have been removed to leave random error only. Under this assumption, dT is equivalent to the amount of random error in the source data propagated to a DEM point through the interpolation function, and will henceforth referred to as the propagation error. The total error of a DEM point T, denoted by ZT, is the difference between the true elevation (zT) and that interpolated by