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Geoelectric modeling with Kernel methods

Electrical resistivity variations measured along vertical boreholes are usually very rough, with amplitudes of the shortest wavelengths being about equal to those of the longest ones. On the other hand, surface resistivity measurements are usually interpreted in terms of models that contain only long wavelengths. The usual approach consists of applying Tikhonov's regularization, incorporating model roughness penalizers for regularization. The roughness measures ordinarily considered are the first and second order derivatives of the model. Short wavelengths are avoided because they can not be recovered uniquely from the data. However, they do in fact influence the overall scale of the resulting model, so they must be taken into account somehow. In this paper we present an attempt to deal with this problem by way of constructing resistivity models that are at the same time smooth and rough. Our work is based on the kernel method which we have adapted for nonlinear inversion. We keep the original connection between regularization operators and support vector kernels, so the algorithm still possesses the regularization properties of kernel methods. By incorporating the kernel method as penalizer we are able to generate a variety of resistivity variations and include some desired properties for the resulting models.