An accurate non-iterative algorithm for computing the length of the position vector to a subsatellite point

Abstract

The radial distance (length of a position vector) from the geocenter to the geoid as defined by the spherical harmonic potential coefficients is needed e.g. in the process of adjusting satellite altimeter data. The geocentric latitude and longitude associated with this distance are assumed known—in this case derived from satellite altimetry. Typically, the radial distance can be computed to a desired accuracy in an iterative process. Even if a crude initial value is adopted, a sub-meter accuracy is achieved on the second iteration, while the third iteration yields a sub-millimeter accuracy. If the best possible initial value is taken, such as the radial distance to the mean earth ellipsoid, the iterative process may be accelerated by one iteration. But even then two iterations will be needed in most cases. However, an algorithm has been designed that yields excellent results, characterized by a sub-centimeter accuracy, already from the first iteration. It results in important computer savings, considering that in real data reductions of satellite altimetry, the radial distance needs to be computed at thousands of locations.