Errors of signal processing in digital terrain modelling

Abstract

We present new interpretation of three classes of errors in digital terrain models (DTMs) which can be sources of artefacts in DTM-based studies: (a) errors in interpolation of digital elevation models (DEMs) caused by the Gibbs phenomenon; (b) errors in DTM derivation from DEMs with ‘enhanced’ resolution due to noise increase after DEM differentiation; (c) errors in DTM derivation caused by displacement of a DEM grid. Explanation of artefact roots and ways to avoid them are carried out in the context of the theory of signal processing. The Gibbs phenomenon is a specific behaviour of some functions manifested as over- and undershoots near a jump discontinuity. Any DEM includes jump discontinuities of the elevation, such as escarpments and pronounced errors of DEM generation. There are four main ways to prevent or reduce DEM errors caused by the Gibbs phenomenon: (a) decreasing the jump discontinuity before DEM interpolation; (b) using interpolation functions which do not generate the Gibbs phenomenon; (c) omitting over- and undershoots after DEM interpolation; (d) filtering the Gibbs phenomenon. Derivation of topographic variables from DEMs marked by ‘enhanced’ resolution can lead to artefacts. If a DEM of this kind is interpolated by triangulation-based algorithms, triangular patterns may be revealed on maps of topographic variables. If an ‘enhanced’ resolution of DEM is achieved by the weighted average methods of interpolation, contour ‘traces’ may be seen on maps. This is because partial derivatives used to calculate some topographic variables are very responsive to high-frequency components of a DEM. To prevent these errors one should use a regular DEM with a grid space relating to an average distance between points in an irregular DEM. Displacement of a grid of points, wherein elevation values are interpolated or determined, influences the derivation of topographic variables (e.g. map design of horizontal and vertical curvatures). Some patterns break, merge, and change their width and length. Small dots, lines, and particles of big patterns can appear and disappear on maps. These effects should be taken into account in the application of these maps to DTM-based geological studies.