Hybrid Error Estimation Methods and Dimension Reduction for Large-scale Bayesian Inverse Problems

Abstract
Many inverse problems in geophysics are solved within the Bayesian framework, in which a prior probability density function of a quantity
of interest  is optimally updated using newly available observations. For large-scale problems, the maximum likelihood of the posterior
distribution can be estimated using techniques such as least-squares minimization (e.g., 4D-Var), or ensemble approaches (e.g., EnKF). Both
methods rely, either explicitly (ensemble) or implicitly (Hessian estimate in minimization subspace) on low-rank approximation of the
second-order moment statistics. In this presentation,  deterministic and stochastic approaches to estimate the posterior error covariance
matrix for large-scale inverse problems will be presented and discussed. First, in the context of 4D-Var, the standard Monte-Carlo
method will be compared with Hessian-based estimates from a randomization approach and from the deterministic BFGS algorithm. New
preconditioning techniques for the BFGS method that significantly improve the posterior error estimates will be presented and
illustrated with numerical examples. Secondly, posterior error approximations will be discussed in the broader context of dimension
reduction for large-scale inverse problems. The formalism of dimension reduction for Bayesian problems will be introduced and different
optimality criteria presented. An optimal method that maximizes the degree of freedom for signal of the inversion will be proposed. Its
numerical efficiency will be demonstrated with an Observation Simulation System Experiment consisting of a methane source inversion
over North America using space-based observations from GOSAT and the GEOS-Chem chemistry-transport model. To conclude, interesting
theoretical  links between this optimal posterior covariance low-rank approximation and existing variational and ensemble data assimilation
system will be discussed.