This work focuses on the study of two well-known problems in physical geodesy. The first problem concerns the determination of the geoid on a given area on the earth. If the Earth were a homogeneous sphere, the gravity at a point would be entirely determined from its distance to the center of the earth or in terms of its altitude. As the earth is neither spherical nor homogeneous, we must calculate gravity at any point. From a reference ellipsoid, we search to find the correction to a mathematical approximation of the gravitational field in order to obtain a geoid, i.e. A surface on which gravitational potential is constant. The method used is the method of least squares collocation which is the best for solving large generalized least squares problems. In the second problem, We are interested in a geodetic inverse problem that consists in finding a distribution of point masses (characterized by their intensities and positions), such that the potential generated by them best approximates a given potential field. On the whole Earth a potential function is usually expressed in terms of spherical harmonics which are basis functions with global support. The identification of the two potentials is done by solving a least-squares problem. When only a limited area of the Earth is studied, the estimation of the point-mass parameters by means of spherical harmonics is prone to error, since they are no longer orthogonal over a partial domain of the sphere. The point-mass determination problem on a limited region is treated by the construction of a Slepian basis that is orthogonal over the specified limited domain of the sphere. We propose an iterative algorithm for the numerical solution of the local point mass determination problem and give some results on the robustness of this reconstruction process. We also study the stability of this problem against added noise. Some numerical tests are presented and commented.
