Impact of DEM uncertainities on direct and indirect topographical effects

Geoid determination using Stokes’ integral requires that all masses above geoid (topography + atmosphere) to be removed. Using 2nd Helmert condensation model, the topographical masses were replaced by a surface layer on the geoid. This replacement caused changes in gravity and equipotential surfaces which are so-called direct topographical effect (DTE) and indirect topographical effect (ITE), respectively. There are two different methods to formulate the Helmert topographical effects: Moritz-Pellinen and Vanicek-Martinec methods. In the Moritz-Pellinen method, the DTE is defined as the gradient difference of topographic potential of topography at the terrain and the potential of the condensation layer at the geoid surface, while in Vanicek-Martinec method both real and condensation potential refer to terrain surface. In study of Jekeli and Serpas (2003) both methods were applied on 1'×1' gravity data and 30''×30'' grids of height to geoid determination of different regions of the USA. The results indicated that the Moritz-Pellinen method is clearly superior to Vanicek-Martinec method. However, in this study, our goal is not to evaluate the effectiveness of these two methods. The subject of this study is the propagation of DEM error in the direct and indirect topographical effects in geoid using the planar approximation of the Moritz-Pellinen method. The integral formulas of standard deviation of the topographical effects were obtained in terms of DEM standard deviation error and error covariance function.    To increase performance, all numerical calculations of all derived integrals were performed by FFT. Numerical investigations of this study are done over the central Alborz mountainous area, as this area is the most rugged terrain in Iran. Two global DEMs, the SRTM and the AW3D30, were freely available with a spatial resolution of one arc second (approximately 30 meters) in the test region. The mean and standard deviation of differences between two DEMs are about 2 and 3 meters, respectively, which produce 0.1 mGal and 1 mm differences in DTE and ITE.Estimation of DTE and ITE error requires the global average error (standard deviation) of the DEM as well as the parameter of correlation length for evaluation of correlated error. Accurate estimation of these parameters needs high-resolution ground control points that were not available in this study. Therefore, using an overall accuracy and a correlation length based on the previous studies, the estimated standard deviation for direct and indirect topographical effects varied from 0 to 0.6 mGal and 0 to 17 mm in the test region, respectively.    The influence of DTE error on the geoid error can be computed by applying the error propagation law on the Stokes’ integral. Our calculations show that the error of SRTM DEM on geoid in central Alborz can exceed from 1 cm, but the values are about 1-2 mm in the flat areas. Therefore, geoid determination with 1 cm accuracy in mountainous areas in Iran requires DEM with better average accuracy than the current available models. Various previous studies indicate that the error of DEMs decreases in mountainous areas.