Geodetic Boundary-Value Problems and the Height Datum Problem

Abstract

The analysis of boundary-value problems has always been used in geodesy as a frame to understand the nature of the problem of determining the gravity disturbing potential, with respect to well-posedness (existence, uniqueness, continuous dependence of the solution on boundary data). The first historical problem of this kind is Stokes’s, followed by Molodensky’s and other more and more sophisticated versions of boundary-value problems of mixed type, in an attempt of approaching more realistic (Closer to measurements) conditions. Formulations for realistic problems have invariably displayed “nice” mathematical properties. As shown in this paper, also a realistic boundary-value problem for gravity and geopotential numbers as boundary data on continental areas and marine geoid, as derived from altimetry and stationary sea surface heights, on the oceans, can be demonstrated to have such properties for solutions lying in suitable function spaces. In order to establish realistic boundary conditions for this problem it is necessary to take into account that 1) for different continental areas that are not connected by levelling lines one must assume different reference geopotential values, and even on very large individual continents, because of error propagation in levelling, it is not possible to impose a unique reference value with the required accuracy; 2) ocean circulation models enable us to compute the sea surface topography but for an additive constant; furthermore, such models cannot be applied close to coasts, so the marine geoid cannot be used to connect different continental geoids; 3) moreover, there are oceanic regions (for example, equatorial areas) where stationary SST models do not hold, so that the marine geoid too has to be computed separately for a number of different patches that cannot be directly connected. Consequently, a number of different unknown constants must be introduced in the boundary conditions, and correspondingly suitable constraints have to be imposed. The type of unknowns to account for different height datums and the correspondent conditions to be imposed to the solutions are thoroughly discussed.