Inversion of magnetotelluric data in a sparse model domain

The inversion of magnetotelluric data into subsurface electrical conductivity poses an ill-posed problem. Smoothing constraints are widely employed to estimate a regularized solution. Here, we present an alternative inversion scheme that estimates a sparse representation of the model in a wavelet basis. The objective of the inversion is to determine the few non-zero wavelet coefficients which are required to fit the data. This approach falls into the class of sparsity constrained inversion schemes and minimizes the combination of the data misfit in a least-squares a""(2) sense and of a model coefficient norm in an a""(1) sense (a""(2)-a""(1) minimization). The a""(1) coefficient norm renders the solution sparse in a suitable representation such as the multiresolution wavelet basis, but does not impose explicit structural penalties on the model as it is the case for a""(2) regularization. The presented numerical algorithm solves the mixed a""(2)-a""(1) norm minimization problem for the nonlinear magnetotelluric inverse problem. We demonstrate the feasibility of our algorithm on synthetic 2-D MT data as well as on a real data example. We found that sparse models can be estimated by inversion and that the spatial distribution of non-vanishing coefficients indicates regions in the model which are resolved.