Filters for geodesy data based on linear and nonlinear diffusion

The article deals with filtering of data on closed surfaces by using the linear and nonlinear diffusion equations. The linear diffusion filtering is given by the Laplace-Beltrami operator representing linear diffusion along the surface. For the nonlinear diffusion filtering, we introduce nonlinear diffusion equations with diffusion coefficient depending on surface gradient and/or surface Laplacian of solution. This allows adaptive filtering respecting edges and local extrema in the data. For numerical discretization we develop a surface finite-volume method to approximate the partial differential equations on surfaces like sphere, ellipsoid or the Earth surface. The surfaces are approximated by a polyhedral mesh created by planar triangles representing subdivision of an initial icosahedron or octahedron grids. Numerical experiments illustrate behaviour of the linear and nonlinear diffusion filters on testing data and on real measurements, namely the GOCE satellite observations and the satellite-only mean dynamic topography. They show advantages of the nonlinear filters which, on the contrary to the linear one, preserve important structures in processed geodesy data.