Low Cost Stochastic Estimation of Optimal Regularization Parameter and Model Resolution Matrix Diagonal in Large Geophysical Inverse Problems

The expanding availability of high quality geophysical data, such as those from EarthScope, are increasing the size of many linear inverse problems, both in the number of data equations and model parameters. Simultaneously, advances in system parameterization and regularization, such as adaptive grids, spherical wavelets, and finite frequency seismic rays, are fundamentally changing the resolution of linear inversions. These advances highlight the need for an unbiased computationally cheap resolution estimator. Recent developments in the stochastic estimation of the diagonal of large unavailable matrices now make such resolution estimation possible for large regularized linear geophysical inverse problems. We present a stochastic algorithm to estimate the generalized cross validation (GCV) optimal regularization weighting parameter, and the diagonal of the resolution matrix for regularized linear least squares problems. The method is applied to a linear seismic velocity inversion of over 260,000 model parameters and nearly 20,000 equations. We find that regularization parameter selection through GCV is strongly dependent on accurate estimates of data noise. Resolution estimates are validated against explicitly calculated resolution kernels for randomly selected model parameters. The ability to estimate the GCV function and the diagonal of the resolution matrix are two important inversion tools that are now available for large linear inverse problems.