Regularization of linear and non-linear geophysical ill-posed problems with joint sparsity constraints

In this paper, we deal with the solution of linear and non-linear geophysical ill-posed problems by requiring the solution to have sparse representations in two appropriate transformation domains, simultaneously. Geological structures are often smooth in properties away from sharp discontinuities (i.e. jumps in 1-D and edges in 2-D). Thus, an appropriate 'regularizer' function should be constructed so that recovers the smooth parts as well as the sharp discontinuities. Sparsity inversion techniques which require the solution to have a sparse representation with respect to a pre-selected basis or frames (e.g. wavelets), can recover the regions of smooth behaviour in model parameters well, but the solution suffers from the pseudo-Gibbs phenomenon, and is smoothed around discontinuities. On the other hand, requiring sparsity in Haar or finite-difference (FD) domain can lead to a solution without generating smoothed edges and the pseudo-Gibbs phenomenon. Here, we set up a regularizer function which can be benefited from the advantages of both wavelets and Haar/FD operators in representation of the solution. The idea allows capturing local structures with different smoothness in the model parameters and recovering smooth/constant pieces of the solution together with discontinuities. We also set up an information function without requiring the true model for selecting optimum wavelet and parameter beta which controls the weight of the two sparsifying operators in the inverse algorithm. For both linear and non-linear geophysical inverse problems, the performance of the method is illustrated with 1-D and 2-D synthetic examples and a field example from seismic traveltime tomography. In all of the examples tested, the proposed algorithm successfully estimated more credible and high-resolution models of the subsurface compared to those of the smooth and traditional sparse reconstructions.