On the numerical solution of parameters of the least squares modification of Stokes’ formula

Abstract

In regional gravimetric geoid determination, it has become customary to utilize the modified Stokes formula, which combines local terrestrial data with a global geopotential model (GGM). A modification method, proposed by L.E. Sjoberg in 1984 (with later developments), allows least squares minimization of the influence of any error source in geoid modelling. In this approach, depending on the local gravity data quality, the chosen radius of integration, and the characteristics of the used GGM, the modification parameters sn vary. New satellite gravity missions are expected to improve significantly the accuracy of geopotential models. Of particular interest of this study is to evaluate the impact of future (i.e. post-GOCE) geopotential coefficients. A set of least squares modification parameters is determined from the system of linear equations, aiming at minimizing the global mean square error of geoid estimator. Some difficulties may be encountered when practically computing the modification parameters. In particular, for certain parameters the design matrix suffers from numerical ill-conditioning. Importantly, Tikhonov regularization is satisfactory in providing a solution for the modification parameters. Numerical results are presented to illustrate the applicability of the obtained parameters in geoid modelling by comparing with GPS-levelling data.