The accuracy of grid digital elevation models linearly constructed from scattered sample data

In this paper, a theoretical-empirical model is developed for modelling the accuracy of a grid digital elevation model (DEM) linearly constructed from scattered sample data. The theoretical component integrates sample data accuracy in the model by means of the error-propagation theory. The empirical component seeks to model what is known as information loss, i.e. the sum of the error due purely to sampling the continuous terrain surface with a finite grid interval and the interpolation error. For this purpose, randomly spaced data points, supposed to be free of error, were converted into regularly gridded data points using triangulation with linear interpolation. Original sample data were collected with a 2x2m sampling interval from eight different morphologies, from flat terrain to highly rugged terrain, applying digital photogrammetric methods to large-scale aerial stereo imagery (1:5000). The DEM root mean square error was calculated by the true validation method over several sets of check points, obtaining the different sampling densities tested in this work. Several empirical models are calibrated and validated with the experimental data set by modelling the DEM accuracy by combining two variables such as sampling density and a descriptive attribute of terrain morphology. These empirical models presented a morphology based on the product of two potential functions, one related to the terrain roughness and another related to the sampling density. The terrain descriptors tested were average terrain slope, standard deviation of terrain slope, standard deviation of unitary vectors perpendicular to the topographic surface (SDUV), standard deviation of the difference in height between adjacent samples in the grid DEM (SDHD), and roughness estimation by first-, second-, or third-degree surface fitting error. The values obtained for those terrain descriptors were reasonably independent from the number and spatial distribution of the sample data. The models based on descriptors SDHD, SDUV, and standard deviation of slope provided a good fitting to the data observed (R-2> 0.94) in the calibration phase, model SDHD being the one that yielded the best results in validation. Therefore, it would be possible to establish a priori the optimum grid size required to generate or store a DEM of a particular accuracy, with the saving in computing time and file size that this would mean for the digital flow of the mapping information in GIS.