# On the Solution of the Inverse Stokes Problem Including Ellipsoidal Effects

• Bernhard Heck
Conference paper
Part of the International Association of Geodesy Symposia book series (IAG SYMPOSIA, volume 122)

## Abstract

The inverse Stokes problem deals with the determination of gravity anomalies Δg from given disturbing potential T of geoidal height data, which are based on the processing of satellite altimetry data. By introducing a couple of simplifications —the so-called spherical approximation and the constant radius approximation—a very simple solution is achieved, which can be represented by a spherical integral formula (inverse Stokes integral) in the space domain. This type of approximation has often been applied in the past for the calculation of gravity anomalies from altimetric sea surface heights, and is implicity contained also in Least Sqares Collocation and FFT procedures. A more accurate solution of the inverse Stokes problem is achieved by retaining the first order ”ellipsoidal effects” in the boundary condition relating Δg and T, neglecting terms of order f 2 (f= flattening of the reference ellipsoid approximating the earth). These ellipsoidal effects are due to
• the non-radial direction of the derivative appearing in the boundary condition,

• terms in the reference potential depending on J 2 (zonal harmonic of degree 2) and ω (angular velocity of the earth’s rotation), and

• deviations of the true boundary surface from a sphere.

As a consequence of these ellipsoidal terms the solution of the inverse Stokes problem can no longer be represented by a simple spherial integral formula. Starting from an approximation of the boundary condition of order 0(f) the solution of the inverse Stokes problem is provided in the frequency domain, using spherical harmonics as base functions. Special attention is given to the null space of the boundary operator as well as to small eigenvalues related to the harmonics ot first degree. The results are transformed from the frequency domain into the space domain, resulting in two alternatives for the procedure of evaluation. In the first alternative the altimetry-derived disturbing potential (or geoidal height) data are inserted into the inverse Stokes formula, and a correction term is applied to the resulting gravity anomalies. In contrast, in the second alternative a correction term is added to the disturbing potential data, and this reduced data is inserted in the inverse Stokes formula. A numerical evaluation based on the EGM96 geopotential model results in estimates of the order of magnitude of the correction terms, which is ±3 • 10-6 ms-2 and ±2m, respectively; these numbers mirror the approximations existent in the ”spherical” inverse Stokes problem. The correction terms show a low-frequency behaviour, dominated by spectral terms of degree ≤ 20.

## Keywords

Correction Term Gravity Anomaly Space Domain Geoidal Height Reference Ellipsoid
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