The Topographic Potential with Respect to the Reference Ellipsoid

Abstract

The masses between the topographic surface and a reference surface which approximates the mean sea surface of the earth, i.e. the geoid, are called topographic masses and the potential they create in a certain point under consideration is called topographic potential. Also it is well known that an ellipsoid of revolution is a very good analytic approximation of the geoid and a slightly better one than a sphere. Therefore such an ellipsoid is chosen here as a reference surface.

The idea to derive the analytical formulas for the topographic potential in an exact form by means of a rigorous integral representation was already applied by (Grafarend and Engels, 1993) to a sphere as a reference surface. The technique they used slices the topographic masses into infintesimal spherical shells with no global, but with interrupted support. Here we present the same idea applied to an ellipsoidal reference surface.

Though the reference surface is an ellipsoid of revolution it is convenient to apply spherical coordinates. Their big advantage is the separability of the threedimensional Laplacean in contrast to geodetic coordinates (Grafarend, 1988). That makes it possible to expand the kernel of the Newton-Integral, which is the formal solution of the topographic potential, into a series of solid spherical harmonics. Then the integration with respect to the radial component can be done very easily with the single drawback that both the upper and the lower limit of integration are functions of latitude and longitude. But that is only a minor problem since the integrals with respect to the surface coordinates must be discretized for numerical evaluation anyway.

By applying spherical coordinates and having an ellipsoidal reference surface one has to distinguish between four different cases depending on the position of the point under consideration and the location of the topographic masses with respect to the sphere of convergency containing this point. After comparison of the four cases one sees that only three different types of integrals occur in all solutions. With the introduction of Heaviside-functions (Walter, 1974) we can summarize the four cases and end up with a sum of six integrals, two of each type. This result is very suitable for numerical evaluation because it keeps the series truncations to a minimum.

In order to compare this solution with already existing solutions an additional series expansion of the radial component must be applied. The reason for this is the fact that all six integrals have only interrupted support and not a global one as the existing solutions have. By introducing this series expansion we get a sum of seven integrals with one integral that has global support and corresponds to the classical solutions.

The other six integrals still depend on Heaviside-functions. Hence they have only interrupted support and can be regarded as correctional terms to the classical solution.

Both solutions are rigorous integral representations of the topographic potential in an exact form as long as numerical analysis is not taken into account. The main advantage of the solution with the global integral is that it is directly comparable to classical solutions whereas the advantage of the first solution is definitely the numerical evaluation because of the minimum of series truncations.

The interested reader in the Ph.D thesis of the author which will be published in the near future.