On the solvability of the Stokes pseudo-boundary-value problem for geoid determination

Abstract

A new form of boundary condition of the Stokes problem for geoid determination is derived. It has an unusual form, because it contains the unknown disturbing potential referred to both the Earth's surface and the geoid coupled by the topographical height. This is a consequence of the fact that the boundary condition utilizes the surface gravity data that has not been continued from the Earth's surface to the geoid. To emphasize the `two-boundary' character, this boundary-value problem is called the Stokes pseudo-boundary-value problem. The numerical analysis of this problem has revealed that the solution cannot be guaranteed for all wavelengths. We demonstrate that geoidal wavelengths shorter than some critical finite value must be excluded from the solution in order to ensure its existence and stability. This critical wavelength is, for instance, about 1 arcmin for the highest regions of the Earth's surface.

Furthermore, we discuss various approaches frequently used in geodesy to convert the `two-boundary' condition to a `one-boundary' condition only, relating to the Earth's surface or the geoid. We show that, whereas the solution of the Stokes pseudo-boundary-value problem need not exist for geoidal wavelengths shorter than a critical wavelength of finite length, the solutions of approximately transformed boundary-value problems exist over a larger range of geoidal wavelengths. Hence, such regularizations change the nature of the original problem; namely, they define geoidal heights even for the wavelengths for which the original Stokes pseudo-boundary-value problem need not be solvable.