ON THE SOLVABILITY OF THE STOKES PSEUDO-BOUNDARY-VALUE PROBLEM FOR GEOID DETERMINATION

A new form of boundary condition of the Stokes problem for geoid deter mination is derived. It has an unusual form, because it contains the u nknown disturbing potential referred to both the Earth's surface and t he geoid coupled by the topographical height. This is a consequence of the fact that the boundary condition utilizes the surface gravity dat a that has not been continued from the Earth's surface to the geoid. T o emphasize the 'two-boundary' character, this boundary-value problem is called the Stokes pseudo-boundary-value problem. The numerical anal ysis of this problem has revealed that the solution cannot be guarante ed for all wavelengths. We demonstrate that geoidal wavelengths shorte r than some critical finite value must be excluded from the solution i n order to ensure its existence and stability. This critical wavelengt h is, for instance, about 1 arcmin for the highest regions of the Eart h's surface. Furthermore, we discuss various approaches frequently use d in geodesy to convert the 'two-boundary' condition to a 'one-boundar y' condition only, relating to the Earth's surface or the geoid. We sh ow that, whereas the solution of the Stokes pseudo-boundary-value grav itational problem need not exist for geoidal wavelengths shorter resto red. than a critical wavelength of finite length, the solutions of app roximately transformed boundary-value problems exist over a larger ran ge of geoidal wavelengths. Hence, such regularizations change the natu re of the original problem; namely, they define geoidal heights even f or the wavelengths for which the original Stokes pseudo-boundary-value problem need not be solvable.