NUMERICAL STABILITY OF THE LEAST-SQUARES SOLUTION TO THE DISCRETE ALTIMETRY GRAVIMETRY BOUNDARY-VALUE PROBLEM FOR DETERMINATION OF THE GLOBAL GRAVITY MODEL

Spherical harmonic coefficients of the Earth's external gravity held represent important characteristics of the Earth. Satellite tracking data, land-borne gravity observations and altimeter data must be combined to determine these coefficients with a sufficient reliability and resolution. This altimetry-gravimetry problem is governed by the Laplace equation in the space outside the Earth with mixed types of boundary condition. It has been proved that the classical least-squares solution of the continuous altimetry-gravimetry problem is not stable. On the contrary, this paper demonstrates that the least-squares solution of the discrete altimetry-gravimetry problem does not fail and is stable provided that the boundary functionals are discretized in an equal angular grid, the cut-off degree of a global gravity model is not greater than 500, and the covariance matrices of the boundary data are set up in accordance with today's accuracy of geodetic measurements. Moreover, provided that the number of observations is greater than the cut-off degree of the potential series, the approximation error of the least-squares solution does not depend on the number of observations. This paper also treats the numerical stability of the discrete altimetry-gravimetry problem overdetermined by extra gravity data measured by ships over a part of the sea surface. Overlapping boundary data do not have, however, a significant impact on the conditionality of the matrix of normal equations, due to the low accuracy of sea-borne gravity observations. It should be emphasized that the leading motivation of the paper is to show that the classical least-squares solution to the discrete altimetry-gravimetry problem provides us with a reliable tool for constructing global gravity models. The method suggested here avoids converting the altimeter data to gravity disturbances-the preparatory step often made in the construction of a global gravity model-because such a transformation is ill-posed.