Laplace’s Equation in Cartesian Coordinates and Satellite Altimetry

Here we are interested in anomalies due to local structure. Consider a patch on the Earth having a width and length less than about 1000 km or 1/40 of the circumference of the Earth. Within that patch we are interested in features as small as perhaps 1-km wavelength. Using a spherical harmonic representation would require 40,000 squared coefficients! To avoid this enormous computation and still achieve accurate results, we will treat the Earth as being locally flat. Here is a remove/restore approach that has worked well in our analysis of gravity and topography: