Truncation Error of Poisson’s Integral in Downward Continuation of Residual Gravity Anomalies

The global gravity models (GGM) are combined with the surface gravity data to geoid determination in remove-restore scheme. In the remove step, the residual gravity anomalies are computed by subtracting the long wavelength signal of gravity anomalies, computed from GGM, as well as the gravitational effect of topographic masses. In next step, the residual anomalies are downward continued (DWC) into the geoid/ellipsoid surface for solving the Stokesian boundary value problem. In restore step, the long wavelength of geoid and indirect effect of topography are restored. The main goal of the present paper is to study the truncation error of spheroidal Poisson’s integral. The comparison  of truncation coefficient of full and spheroidal kernel shows that the truncation error of spheroidal kernel is at least 500 times of full kernel. As a result, modification of the kernel using spheroidal Poisson kernel is vital for DWC. Since the Poisson kernel depends on height, the modification must be computed for individual observation height. The computation of modification coefficients for all observations needs long computational time. To overcome this problem, they can be interpolated using suitable pre-computed coefficients of few reference altitudes. To escape from time consuming modification process, we proposed a fast and accurate method based on the full kernel. This method uses the orthogonal property of Legendre polynomial. For numerical test, the proposed method was applied in Iran within latitude band of and longitude band of . To test the effect of the truncation error on DWC accuracy, Helmert gravity anomalies corresponding to spherical degree 281-2160 were synthesized using EGM2008 and spherical harmonics of the topography on both Earth’s surface and geoid. The truncation error of full, spheroidal and modified spheroidal (using Molokensij method) were evaluated for integration radius = 0.5 and 1 arc-deg. Our results show that for both radii, truncation error of full and modified kernel is about hundreds Gals, whereas these values can reach to several mGals for spheroidal kernel. Numerical results show that large truncation error yields the wrong results of DWC with spheroidal Poisson kernel. Also, the results show the good performance of proposed method in comparison with Molodenskij modified kernel.