Abstract
As the KTH method for geoid determination by combining Stokes integration of gravity data in a spherical cap around the computation point and a series of spherical harmonics suffers from a bias due to truncation of the data sets, this method is based on minimizing the global mean square error (MSE) of the estimator. However, if the harmonic series is increased to a sufficiently high degree, the truncation error can be considered as negligible, and the optimization based on the local variance of the geoid estimator makes fair sense. Such unbiased types of estimators, derived in this article, have the advantage to the MSE solutions not to rely on the imperfectly known gravity signal degree variances, but only the local error covariance matrices of the observables come to play. Obviously, the geoid solution defined by the local least variance is generally superior to the solution based on the global MSE. It is also shown, at least theoretically, that the unbiased geoid solutions based on the KTH method and remove–compute–restore technique with modification of Stokes formula are the same.
Highlights
Today ultra-high Earth Gravitational Models (EGMs) allow detailed geoid determination all over the Earth
Due to truncations of the EGM series and the area of Stokes integration, such solutions have an inherent bias, and, in case of the least-squares modification of Stokes’ formula (LSMSF) technique, the optimum solution is provided for the minimum of the expected global mean square error (MSE)
While the LSMSF method implies that the EGM and gravity anomaly data are combined by spectral
Summary
Today ultra-high Earth Gravitational Models (EGMs) allow detailed geoid determination all over the Earth. The higher-order harmonics of the EGMs are typically much less accurate than the low-to-medium wavelengths, which calls for improving the geoid estimator by using additional terrestrial gravity data in a combined local/regional solution This is the case for some versions of the remove–compute–restore (RCR) method (e.g., Forsberg 1993; Sansò and Sideris 2013) and the least-squares modification of Stokes’ formula (LSMSF; Sjöberg 1980, 1984a, b, 1991, 2003, 2005a). We will consider the LSMSF method in the case that Stokes’ integral covers a sufficiently large region, such that the remote zone effect becomes negligible, in which case the solution can be regarded as unbiased (provided that all data are unbiased) from a statistical point of view This implies that each solution (not necessarily the least-squares combination) will be (at least practically) unbiased, so that the MSE solution can be replaced by a local minimum variance solution.
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