Truncation errors in the vertical extension of gravity anomalies by poisson’s integral theorem

Abstract

When the values of gravity anomalies are given at the geoid, Ag can be calculated at altitude by application of Poisson’s integral theorem. The process requires integration of Δg multiplied by the Poisson kernel function over the entire globe. It is common practice to add to the kernel function terms that will ensure removal of any zeroth and first order components of Δg that may be present. The effects of trancating the integration at the boundary of a spherical cap of earth central half angle ψ_o have been analyzed using an adaptation of Molodenskii’s procedure. The extension process without removal terms retains the correct effects of inaccuracies in the constant term of the gravity reference model used in the definition of Δg. Furthermore, the effects of ignoring remote zones or unmapped areas in the integration process are very much smaller for the extension without removal terms than for the commonly used formula with removal terms. For these reasons the Poisson vertical extension process without removal terms is to be preferred over the extension with the zeroth order term removal. Truncation of this process at the point recommended for the Stokes integration, namely, the first zero crossing of the Stokes kernel function, leaves negligible truncation errors.