A general model for modifying Stokes’ formula and its least-squares solution

Abstract

Today the combination of Stokes’ formula and an Earth gravity model (EGM) for geoid determination is a standard procedure. However, the method of modifying Stokes’ formula varies from author to author, and numerous methods of modification exist. Most methods modify Stokes’ kernel, but the most widely applied method, the remove compute restore technique, removes the EGM from the gravity anomaly to attain a residual gravity anomaly under Stokes’ integral, and at least one known method modifies both Stokes’ kernel and the gravity anomaly. A general model for modifying Stokes’ formula is presented; it includes most of the well-known techniques of modification as special cases. By assuming that the error spectra of the gravity anomalies and the EGM are known, the optimum model of modification is derived based on the least-squares principle. This solution minimizes the expected mean square error (MSE) of all possible solutions of the general geoid model. A practical formula for estimating the MSE is also presented. The power of the optimum method is demonstrated in two special cases.