Efficient stochastic estimation of the model resolution matrix diagonal and generalized cross-validation for large geophysical inverse problems

Abstract

In recent years, larger geophysical data sets and novel model parameterizations have dramatically increased both the data and model space dimensions of many inverse problems. Because of their relatively low computational expense, trade-off curve corner estimation for choosing regularized models and "checkerboard" tests for evaluating model resolution are commonly applied, despite their limitations. We present and demonstrate a low-cost method for accurately estimating the diagonal elements of the model resolution matrix and for implementing generalized cross-validation (GCV) for optimal regularization parameter selection. The ability to estimate the diagonal of the resolution matrix and GCV function thus facilitates the introduction of additional tools for diagonal resolution analysis and regularization evaluation, even for very large inverse problems, with storage and computational costs comparable to those required for obtaining model solutions. We demonstrate the method using a Tikhonov regularized teleseismic body wave velocity inversion example with approximately 260,000 model parameters, where we validate randomly selected Rm diagonal elements against explicitly calculated values and compare GCV-estimated regularized model results to those obtained through traditional methods. Copyright 2011 by the American Geophysical Union.