The choice of the spherical radial basis functions in local gravity field modeling

The choice of the optimal spherical radial basis function (SRBF) in local gravity field modelling from terrestrial gravity data is investigated. Various types of SRBFs are considered: the point-mass kernel, radial multipoles, Poisson wavelets, and the Poisson kernel. The analytical expressions for the Poisson kernel, the point-mass kernel and the radial multipoles are well known, while for the Poisson wavelet new closed analytical expressions are derived for arbitrary orders using recursions. The performance of each SRBF in local gravity field modelling is analyzed using real data. A penalized least-squares technique is applied to estimate the gravity field parameters. As follows from the analysis, almost the same accuracy of gravity field modelling can be achieved for different types of the SRBFs, provided that the depth of the SRBFs is chosen properly. Generalized cross validation is shown to be a suitable technique for the choice of the depth. As a good alternative to generalized cross validation, we propose the minimization of the RMS differences between predicted and observed values at a set of control points. The optimal regularization parameter is determined using variance component estimation techniques. The relation between the depth and the correlation length of the SRBFs is established. It is shown that the optimal depth depends on the type of the SRBF. However, the gravity field solution does not change significantly if the depth is changed by several km. The size of the data area (which is always larger than the target area) depends on the type of the SRBF. The point-mass kernel requires the largest data area.     

Conditionality of inverse solutions to discretised integral equations in geoid modelling from local gravity data

The eigenvalue decomposition technique is used for analysis of conditionality of two alternative solutions for a determination of the geoid from local gravity data. The first solution is based on the standard two-step approach utilising the inverse of the Abel-Poisson integral equation (downward continuation) and consequently the Stokes/Hotine integration (gravity inversion). The second solution is based on a single integral that combines the downward continuation and the gravity inversion in one integral equation. Extreme eigenvalues and corresponding condition numbers of matrix operators are investigated to compare the stability of inverse problems of the above-mentioned computational models. To preserve a dominantly diagonal structure of the matrices for inverse solutions, the horizontal positions of the parameterised solution on the geoid and of data points are identical. The numerical experiments using real data reveal that the direct gravity inversion is numerically more stable than the downward continuation procedure in the two-step approach.