Computation of three stochastic modifications of Stokes's formula for regional geoid determination

In the regional geoid studies, the modified Stokes formula is often used nowadays. This method combines local terrestrial data with an appropriate global geopotential model in a truncated form of Stokes's integral. This paper is devoted to three stochastic least-squares (LS) modifications, which were originally proposed by Sjöberg (Least squares modification of Stokes and Vening-Meinesz formulas by accounting for errors of truncation, potential coefficients and gravity data. Report No. 27, Department of Geodesy, University of Uppsala, 16pp.) in 1984 (with later developments). The main principles of the LS modifications and some spectral models of the gravity field characteristics are reviewed. Certain difficulties may be encountered when computing the modification parameters from a system of linear equations. In particular, the design matrices of the unbiased and optimum LS modifications suffer from numerical ill-conditioning. Two mathematical regularization strategies are selected in order to find a practical solution for the sought modification parameters. Typical numerical outcome of the regularization and the applicability of the obtained LS parameters are discussed. The present contribution tackles the LS modification-related problems in the context of a specially designed Matlab software package. The core quantities of the stochastic LS modifications can be computed by this software, which is made available on the Computers & Geosciences server.