Nonlinear inversion: Regularization as a priori information

Abstract:

Regularization is an approach to ill-posed or ill-conditioned inverse problems, and using it can be viewed as adding prior information about the parameters of interest. There are many approaches to choosing a regularization parameter, but it is still an open question as to how to weight the additional information. The discrepancy principle considers the residual norm to determine the regularization parameter or weight while a similar approach, the chi-squared method, uses the regularized residual in a statistical test to find the weight. Using the regularized residual has the benefit of giving a clear chi-squared test with a fixed noise level. Recently, this approach has been developed for nonlinear problems. We will give the appropriate chi-squared tests in the Gauss-Newton and Levenburg Marquart algorithms, and these tests are used to find a regularization parameter or weight on initial parameter estimate errors. This algorithm is applied to two benchmark problems in nonlinear geophysics: two-dimensional cross-well tomography, and one dimensional subsurface conductivity estimation.