Summary
Basically two different evaluation methods are available to compute geoid heights from residual gravity anomalies in the inner zone: numerical integration and least squares collocation.
If collocation is not applied to a global gravity data set, as is usually the case in practice, its result will not be equal to the numerical integration result. However, the cross covariance function between geoid heights and gravity anomalies can be adapted such that the geoid contribution is computed only from a small gravity area up to a certain distanceψ o from the computation point. Using this modification, identical results are obtained as from numerical integration.
Applying this modification makes the results less dependent on the covariance function used. The difference between numerical integration and collocation is mainly caused by the implicitly extrapolated residual gravity anomaly values, outside the original data area. This extrapolated signal depends very much on the covariance function used, while the interpolated values within the original data area depend much less on it.
As a sort of by-product, this modified collocation formula also leads to a new combination technique of numerical integration and collocation, in which the optimizing practical properties of both methods are fully exploited.
Numerical examples are added as illustration.
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de Min, E. A comparison of Stokes' numerical integration and collocation, and a new combination technique. Bulletin Géodésique 69, 223–232 (1995). https://doi.org/10.1007/BF00806734
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DOI: https://doi.org/10.1007/BF00806734