Determining the optimal focusing parameter in sparse promoting inversions of EMI surveys

If the magnetic field caused by a magnetic dipole is measured, the electrical conductivity of the subsurface can be determined by solving the inverse problem. For this problem a form of regularisation is required as the forward model is badly conditioned. Commonly, Tikhonov regularisation is used which adds the $\ell_2$-norm of the model parameters to the objective function. As a result, a smooth conductivity profile is preferred and these types of inversions are very stable. However, it can cause problems when the true profile has discontinuities causing oscillations in the obtained model parameters. To circumvent this problem, $\ell_0$-approximating norms can be used to allow discontinuous model parameters. Two of these norms are considered in this paper, the Minimum Gradient Support and the Cauchy norm. However, both norms contain a parameter which transforms the function from the $\ell_2$- to the $\ell_0$-norm. To find the optimal value of this parameter, a new method is suggested. It is based on the $L$-curve method and finds a good balance between a continuous and discontinuous profile. The method is tested on synthetic data and is able to produce a conductivity profile similar to the true profile. Furthermore, the strategy is applied to newly acquired real-life measurements and the obtained profiles are in agreement with the results of other surveys at the same location. Finally, despite the fact that the Cauchy norm is only occasionally used to the best of our knowledge, we find that it performs at least as good as the Minimum Gradient Support norm.