Least-squares modification of extended Stokes’ formula and its second-order radial derivative for validation of satellite gravity gradiometry data

Abstract

To cite this version:Mehdi Eshagh. Least-squares modification of extended Stokes' formula and its second-order radial derivative for validation of satellite gravity gradiometry data. Journal of Geodynamics, Elsevier, 2010, 49 (2) This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Abstract The gravity anomalies at sea level can be used to validate the satellite gravity gradiometry data. Validation of such a data is important prior to downward continuation because of amplification of the data errors through this process. In this paper the second-order radial derivative of the extended Stokes' formula is employed and the emphasis is on least-squares modification of this formula to generate the second-order radial gradient at satellite level. Two methods in this respect are proposed: a) modifying the second-order radial derivative of extended Stokes' formula directly, b) modifying extended Stokes' formula prior to taking the second-order radial derivative. Numerical studies show that the former method works well but the latter is very sensitive to the proper choice of the cap size of integration and degree of modification.

Deterministically-Modified Integral Estimators of Gravitational Tensor

“…Sprlak and Novak (2014a) used the deflections of vertical for generating the gravity gradients at satellite level, also Sprlak and Novak (2014b) found integral relations between the GOCE and GRACE types of data which can be used for validation purpose of their data. The stochastic modification was used by Eshagh (2010a) to modify the second-order radial derivative of the extended Stokes formula. He proposed two methods of a) modification prior to derivative and b) derivative prior to modification.…”

“…The truncation coefficients of the estimators are simply generated by their spectral forms. Eshagh (2010a) showed that because of the inclusion of the parameters 0 n  , 1 n  and 2 n  the series will be convergent due to the elevation of the satellite, but the series should be used to high degrees. He showed that a maximum degree of 5000 should be enough for this purpose.…”

“…As can be seen, Figure 1 shows this issue for the modified kernels, by comparing them to the original ones, especially for the modified kernels of the VH and HH estimators. This can somehow show that the modification processes have been successful.The system of equations from which the modification parameters are obtained is ill-conditioned for the VH and HH estimators (ESHAGH, 2010a). This is due to inclusion of i k u in the mathematical expressions of the elements of the coefficient matrix of the systems.…”

Inversion of satellite gradiometry data using statistically modified integral formulas for local gravity field recovery

“…It is simple because its mathematical formulation is easier than the other gradients, and it is important as it has the most power with respect to the others. Let the following estimator for this gradient at satellite level (Sjö berg, 2003;Eshagh, 2009aEshagh, , 2010a:…”

From Tensor to Vector of Gravitation

“…Visser (2009) tried to estimate the biases and scale factors of the accelerometers used in the GOCE spacecraft by using its precise orbit. The stochastic modification was used by Eshagh (2010a) to modify the second-order radial derivative of the extended Stokes formula. Eshagh (2011a) also modified the second-order radial derivative of the Abel-Poisson formula in a least-squares sense to generate the second-order radial gradient at satellite level using an EGM and geoid model.…”