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

**Zdenek Martinec** ** & **
**Ctirad Matyska**
### Summary

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.

J. Geod., **71** (1997), 103-112.