Computation of spherical harmonic coefficients and their error estimates using least-squares collocation

 Equations expressing the covariances between spherical harmonic coefficients and linear functionals applied on the anomalous gravity potential, T, are derived. The functionals are the evaluation functionals, and those associated with first- and second-order derivatives of T. These equations form the basis for the prediction of spherical harmonic coefficients using least-squares collocation (LSC). The equations were implemented in the GRAVSOFT program GEOCOL. Initially, tests using EGM96 were performed using global and regional sets of geoid heights, gravity anomalies and second-order vertical gravity gradients at ground level and at altitude. The global tests confirm that coefficients may be estimated consistently using LSC while the error estimates are much too large for the lower-order coefficients. The validity of an error estimate calculated using LSC with an isotropic covariance function is based on a hypothesis that the coefficients of a specific degree all belong to the same normal distribution. However, the coefficients of lower degree do not fulfil this, and this seems to be the reason for the too-pessimistic error estimates. In order to test this the coefficients of EGM96 were perturbed, so that the pertubations for a specific degree all belonged to a normal distribution with the variance equal to the mean error variance of the coefficients. The pertubations were used to generate residual geoid heights, gravity anomalies and second-order vertical gravity gradients. These data were then used to calculate estimates of the perturbed coefficients as well as error estimates of the quantities, which now have a very good agreement with the errors computed from the simulated observed minus calculated coefficients. Tests with regionally distributed data showed that long-wavelength information is lost, but also that it seems to be recovered for specific coefficients depending on where the data are located. Received: 3 February 2000 / Accepted: 23 October 2000     

Computation of spherical harmonic coefficients from gravity gradiometry data to be acquired by the GOCE satellite: regularization issues

The issue of optimal regularization is investigated in the context of the processing of satellite gravity gradiometry (SGG) data that will be acquired by the GOCE (Gravity Field and Steady-State Ocean Circulation Explorer) satellite. These data are considered as the input for determination of the Earths gravity field in the form of a series of spherical harmonics. Exploitation of a recently developed fast processing algorithm allowed a very realistic setup of the numerical experiments to be specified, in particular: a non-repeat orbit; 1-s sampling rate; half-year duration of data series; and maximum degree and order set to 300. The first goal of the study is to compare different regularization techniques (regularization matrices). The conclusion is that the first-order Tikhonov regularization matrix (the elements are practically proportional to the degree squared) and the Kaula regularization matrix (the elements are proportional to the fourth power of the degree) are somewhat superior to other regularization techniques. The second goal is to assess the generalized cross-validation method for the selection of the regularization parameter. The inference is that the regularization parameter found this way is very reasonable. The time expenditure required by the generalized cross-validation method remains modest even when a half-year set of SGG data is considered. The numerical study also allows conclusions to be drawn regarding the quality of the Earths gravity field model that can be obtained from the GOCE SGG data. In particular, it is shown that the cumulative geoid height error between degrees 31 and 200 will not exceed 1 cm. AcknowledgmentsThe authors thank Dr. E. Schrama for valuable discussions and for computing the orbit used to generate the long data set. They are also grateful to Prof. Tscherning and two anonymous reviewers for numerous valuable remarks and suggestions. The orbit to generate the short data set was kindly provided by J. van den IJssel. Computing resources were provided by Stichting Nationale Computerfaciliteiten (NCF), grant SG-027.     

Two different methods of geoidal height determinations using a spherical harmonic representation of the geopotential, topographic corrections and the height anomaly–geoidal height difference

 It is suggested that a spherical harmonic representation of the geoidal heights using global Earth gravity models (EGM) might be accurate enough for many applications, although we know that some short-wavelength signals are missing in a potential coefficient model. A `direct' method of geoidal height determination from a global Earth gravity model coefficient alone and an `indirect' approach of geoidal height determination through height anomaly computed from a global gravity model are investigated. In both methods, suitable correction terms are applied. The results of computations in two test areas show that the direct and indirect approaches of geoid height determination yield good agreement with the classical gravimetric geoidal heights which are determined from Stokes' formula. Surprisingly, the results of the indirect method of geoidal height determination yield better agreement with the global positioning system (GPS)-levelling derived geoid heights, which are used to demonstrate such improvements, than the results of gravimetric geoid heights at to the same GPS stations. It has been demonstrated that the application of correction terms in both methods improves the agreement of geoidal heights at GPS-levelling stations. It is also found that the correction terms in the direct method of geoidal height determination are mostly similar to the correction terms used for the indirect determination of geoidal heights from height anomalies. Received: 26 July 2001 / Accepted: 21 February 2002