Non-smooth gravity problem with total variation penalization functional

Summary This work deals with reconstruction of non-smooth solutions of the inverse gravimetric problem. This inverse problem is very ill-posed, its solution is non-unique and unstable. The stable inversion method requires regularization. Regularization methods commonly used in geophysics reconstruct smooth solutions even though geological structures often have sharp contrasts (discontinuities) in properties. This is the result of using a quadratic penalization term as a stabilizing functional. We introduce the total variation of the reconstructed model as a stabilizing functional that does not penalize sharp features of the solution. This approach permits reconstruction of (non-smooth) density functions that represent blocky geological structures. An adaptive gradient scheme is shown to be effective in solving the regularized inverse problem. Numerically simulated examples consisting of models with several homogeneous blocks illustrate the behaviour of the method.