Gravitational spectra from direct measurements

A simple rapid method is described for determining the spectrum of a surface field from harmonic analysis of direct measurements along great circle arcs. The method is shown to give excellent overall trends to very high degree from even a few short arcs of satellite data. Three examples are taken with perfect measurements of satellite tracking over a planet made up of hundreds of point-masses using (1) altimetric heights from a low orbiting spacecraft, (2) velocity residuals between a low and a high satellite in circular orbits, and (3) range-rate data between a station at infinity and a satellite in highly eccentric orbit. In particular, the smoothed spectrum of the Earth's gravitational field is determined to about degree 400(50 km half wavelength) from 1 D x 1 D gravimetry and the equivalent of 11 revolutions of Geos 3 and Skylab altimetry. This measurement shows there is about 46 cm of geoid height remaining in the field beyond degree 180.