Minimum support nonlinear parametrization in the solution of a 3D magnetotelluric inverse problem

Abstract

In this paper we describe a new approach to shar p boundary geophy sical inversion. We demon strate that regularized inversion with a minimum support stabilizer can be implemented by using a specially designed nonl inear parametrization of the mode l parameters. Thi s parametrization plays the same role as transformation into the space of the weighted model parameters, introd uced in the origi nal papers on focusing inversion. It allows us to transform the nonq uadrati c minim um support stabilizer into the traditional quadrati c minimum norm stabilizer, which simplifies the solution of the inverse problem. Thi s tran sformation automatically ensures that the solution belon gs to the class of model s with a minimum support. The method is illustrated with synthetic examples of 3D magnetotell uric inversion for an earth conductivity structure. To simplify the calculations, in the initial stage of the iterative inversion we use the quasi-analytic al approximation developed by Zhdanov and Hursan (2000 Inverse Problems 16 1297-322). However, to increase the accura cy of inversion, we apply rigorou s forward modelling based on the integral equa tion method at the final stage of the inversion. To obtain a stable solution of a 3D inverse problem , we use the Tikhonov regularization method with a new nonli near param etri zation. Thi s techn ique leads to the generation of a shar p image of anomalous condu ctiv ity distribution. The inversion is based on the regularized conju gate gradient method .

A Rigorous Three-Dimensional Magnetotelluric Inversion

“…It is also planned to apply our inversion scheme to an experimental data set. However, previous examples from other 3-D MT inversion software developers (see [8,10,14,17]) indicate that successful verification of the inversion technique even on a single practical dataset is a complex task and may take some time.…”

“…This example includes a tilted conductive dyke in a uniform half-space (see [17]). The results presented are encouraging and suggest that the inversion may be successfully applied to solving realistic 3-D inverse problems with real MT data.…”

Applying compactness constraints to differential traveltime tomography

“…The original work of Last and Kubik [1983] suggests that β should have a value close to machine precision (≈ 10 −11 in their case) with λ determined using the discrepancy principle and an assumed, although somewhat arbitrary, noise level. Zhdanov and Tolstaya [2004] advocates the use of a procedure similar to the L-curve technique where β is chosen to be the point of maximum curvature on the trade-off curve relating β to A(m). We have observed that setting β to values near machine precision results in severe instability and that the approach of Zhdanov and Tolstaya [2004] often yields trade-off curves with poorly defined corners.…”

“…Zhdanov and Tolstaya [2004] advocates the use of a procedure similar to the L-curve technique where β is chosen to be the point of maximum curvature on the trade-off curve relating β to A(m). We have observed that setting β to values near machine precision results in severe instability and that the approach of Zhdanov and Tolstaya [2004] often yields trade-off curves with poorly defined corners. We therefore fix β at a reasonable value determined by experience, typically between 10 −4 and 10 −7 .…”

Magnetotellurics in locally-layered random media

“…The resistivity map is then varied iteratively to minimize the objective function, thus giving a trade-off between fitting the data and the assumed nonoscillatory nature of the solution. (Some recent developments in regularization theory have dealt with sharp geoelectric boundaries; see [13], [3], [15].) Paradoxically, it is well known that resistivity does not vary smoothly with depth, but has order one oscillations on very small spatial scales.…”