Downward continuation of gravity field

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Gravity is observed on the earth's surface but for the geoid computation it is needed at the sea level (on the geoid): it has to be "continued downward" onto the sea level through topographical masses. In our approach to geoid determination, the best approximation of topographical masses is subtracted from actual topography and we consider the result (Earth without topography) to represent the situation in the "NT space". In fact, we replace topography by a condensation layer on the geoid and call this new space the "Helmert space". Helmert's space is thus the NT space augmented by the condensation layer. In both the NT and Helmert's space, the downward continu-ation is carried out through a part of space that is as void of mass as we can possibly make it. The downward continuation process is the inverse of upward continuation. While upward continuation is a smoothing operation, downward continuation is an operation that boosts up high frequencies. This spells trouble from the numerical point of view: the process of downward continuation is bound to be unstable. From the mathematical point of view, the upward continuation in a space void of mass is described by the classical Poisson integral. The inverse problem is described by Fredholm's integral equation of first kind, the solution of which is known to be unstable. Yet, the integral equation represents a well-posed problem according to the accepted definition that has a unique, finite but unstable solution. In our geoid estimation we needed the downward continuation evaluated in Helmert's space and we use the Jacobi iterative inversion of the system of Fredholm's linear equations (ref. 463). Even though this approach gives reasonable results for the geoid computation, it is devoid of physical meaning. We wish to carry on with this research, adding some physically meaningful constraints (requirement of harmonicity, minimization of 2D Laplacean on the geoid, etc.) to add some physical sense to the solution.