Article,
Efficient estimation of nonlinear posterior model covariances using maximally sparse cubature rules
Geophysics, 77 (5): ID1-ID8 (2012)
DOI: 10.1190/geo2011-0400.1
Abstract
We estimated posterior model covariances by randomly perturbing data and start model vectors with samples drawn from prior probability distributions, solving the corresponding deterministic inversion for each sample, and integrating the resulting model vectors over all realizations. The computational burden of this method is the number of samples required for the integration because each sample involves a solution to a nonlinear inverse problem. For certain classes of smooth prior probability functions, we present a new approach to the model covariance estimation problem using grids based on maximally sparse cubature integration rules. The N+1 (degree-two) and 2N (degree-three) cubature rules exist to exactly integrate symmetric Gaussian distributions in N stochastic dimensions. These represent the two sparsest integration rules possible and thus provide the most efficient estimation method for model covariances using integration. In contrast to Monte-Carlo (MC) integration, which requires hundreds of random samples to estimate covariances and has ill-defined convergence, sparse-cubature (quadrature) rules have determined numbers of samples and definite convergence. The trade-off is that they also have limited degrees of accuracy. Using a simple 1D electromagnetic example with 67 stochastic variables, we demonstrated that this sparse integration is three to five times more efficient than MC methods in terms of numbers of required samples.
