عنوان مقاله [English]
Geoid determination using Stokes’s integral requires that all masses above geoid (topography + atmosphere) be removed. Using 2nd Helmert condensation model, the topographical masses replaced by a surface layer on the geoid. This replacement caused change in gravity and equipotential surfaces which 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 formulation, both real and condensation potential refers to terrain surface. Jekeli and Serpas (2003) applied both methods on 1^'×1' gravity data and 〖30〗^''×〖30〗^'' grids of height to geoid determination of different regions of the USA. They indicated that the Moritz-Pellinen method is clearly superior to Vanicek 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 planar approximation of the Moritz-Pellinen method. The integral formulas of standard deviation of the topographic 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 DEM are about 2 and 3 meters respectively, which produce 0.1mGal and 1 mm differences in DTE and PITE.
Estimation of DTE and ITE error requires the global average error (standard deviation) of the DEM as well as the parameter of correlation length to evaluation of correlated error. Accurate estimation of these parameters need high-resolution ground control points that were not available in this study. Therefore, the overall accuracy σ=6.5 m was adopted from (Kiamehr and Sjoberg, 2005) and the correlation length was also considered based on the study (McCobby et al., 2017). Based on these values, the estimated standard deviation for direct and indirect topographical effects varies 0-0.6 mGal and 0-17 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 in respect than current available models particularly the various previous studies were indicated that the error of DEMs decreases in mountainous area.