Knowledge Bank

The topography of the earth plays an important role in solving the geodetic boundary value problem. In this report the effect of topography on gravity gradient data is considered and the effect of topography on the solution of the geodetic boundary value problem by using analytical downward continuation is also investigated. The validity of solving Molodensky's problem by using the analytical downward continuation is inspected. Even though it has been shown that the analytical downward continuation solution is equivalent to Molodensky's solution which is considered theoretically perfect, a very small topographic effect exists. This effect is trivial and can be neglected in the numerical computations. It is also shown that a spherical harmonic expansion cannot exactly represent the disturbing potential outside the Brillouin sphere and nearby the earth at the same time. If the points are near by the earth (between the Brillouin sphere and the earth's surface), there is a topographic effect to the geopotential represented by a spherical harmonic expansion whose coefficients are determined by using the gravity anomalies analytically downward continued onto the ellipsoid. This effect is the same as to the solving of the Molodensky's problem of the analytical downward continuation is also investigated under planar approximation. It is shown that the downward continuation is convergent almost everywhere, except at the infinite point of the circular frequency ω = ∞· This is important for geopotential modeling. Provided the downward continuation is convergent, then the geopotential can be expanded into a spherical harmonic series up to very high degree and order without any theoretical difficulty.