The compatibility conditions in altimetry-gravimetry boundary value problems

The role of compatibility conditions for altimetry-gravimetry boundary value problems (AGBVPs) is quite complicated by way of: guaranteeing the existence of the solution; smoothing the data (boundary conditions) at the boundary; and providing a higher level of regularization and smoothness to the solution. Pseudo-differential operators (PDOs) can be applied not only for the reformulation of AGBVPs in a simple way, but also for imposing compatibility conditions at the coastline. It is shown here that spherical PDOs can be combined with the theory of spherical harmonics and spherical wavelets to obtain a numerical solution for AGBVPs with compatibility conditions along the coastline. The role of compatibility conditions, is summarized and emphasized and compatibility conditions are given in an explicit form based on the combined representation of functionals of the disturbing potential in terms of spherical harmonics, spherical PDOs and wavelets. Numerical aspects of the application of the compatibility conditions to a case-study area in Canada are discussed.