THE ABEL-POISSON KERNEL AND THE ABEL-POISSON INTEGRAL IN A MOVING TANGENT-SPACE

The upward-downward continuation of a harmonic function like the gravi tational potential is conventionally based on the direct-inverse Abel- Poisson integral with respect to a sphere of reference. Here we aim at an error estimation of the ''planar approximation'' of the Abel-Poiss on kernel, which is often used due to its convolution form. Such a con volution form is a prerequisite to applying fast Fourier transformatio n techniques. By means of an oblique azimuthal map projection/projecti on onto the local tangent plane at an evaluation point of the referenc e sphere of type ''equiareal'' we arrive at a rigorous transformation of the Abel-Poisson kernel/Abel-Poisson integral in a convolution form . As soon as we expand the ''equiareal'' Abel-Poisson kernel/Abel-Pois son integral we gain the ''planar approximation''. The differences bet ween the exact Abel-Poisson kernel of type ''equiareal'' and the ''pla nar approximation'' are plotted and tabulated. Six configurations are studied in detail in order to document the error budget, which varies from 0.1% for points at a spherical height H = 10 km above the terrest rial reference sphere up to 98% for points at a spherical height H = 6 .3 x 10(6) km.