f Sharp boundaries in multi-layer models via Minimum Gradient Support regularization

We present a focusing regularization technique utilizing a multi-layer model with fixed vertical discretization, while preserving the capability to reproduce sharp vertical transitions. The method relies on minimizing the number of layers of non-negligible resistivity gradient, instead of minimizing the norm of the model variation itself. AEM methods are capable of producing extremely large datasets that are conveniently inverted for smoothly varying 1D models of fixed vertical discretization. The vertical smoothness of the obtained models stems from the application of Occam type regularization constraints, meant for addressing the ill-posedness of the problem. An important side-effect of such regularization, however, is that sharp vertical layer boundaries can no longer be accurately reproduced as the model is required to be smoothly varying. This issue can be overcome by inverting for fewer model layers using variable layer thicknesses, but having to decide on a particular and constant number of layers for inversion of a large survey can be equally problematic. Here, we present a focusing regularization technique for getting the best of both methodologies. It allows for accurate reconstruction of resistivity distributions using a fixed vertical discretization, while preserving the capability to reproduce sharp vertical transitions. The formulation is flexible and can be coupled with traditional lateral/spatial smoothness constraints, in order to resolve interfaces in stratified soils with no additional hypothesis about the number of layers. This approach ensures model results that are consistent with the measured data while favouring, at the same time, sharp vertical transitions. The formulation is general and can also be applied in a horizontal direction, in order to promote sharp lateral transitions such as faults. We present the theoretical framework of our regularization methodology and illustrate its capabilities by means of both field and synthetic datasets. We further demonstrate how the concept has been integrated in our existing Spatially Constrained Inversion (SCI) formalism and show its application to large scale inversions of airborne time-domain electromagnetic data.