Land and Climate Seminar - Geophysical inversions: data- and physics-based regularized solutions

Speaker: Peiliang Xu

Except for direct measurements, almost all geophysical problems are of inverse ill-posed nature. In this talk, we will first briefly explain some misuse of inverse problem theory in earth sciences. Then, we will briefly discuss three basic approaches to geophysical inverse problems, namely Bayesian methods, frequentists-based regularization and inequality-constrained (or physics-based) solutions. Bearing in mind the fcat that Akaike’s Bayesian information criterion (ABIC) has been widely used in inverse ill-posed problems but little has been done to investigate its statistical aspects, we present an alternative derivation of the marginal distribution of measurements for ABIC under the assumption of normal distributions and show that the principle of ABIC is to statistically estimate the variances of measurements and prior data by maximizing the marginal distribution of measurements. The determination of the regularization parameter with ABIC is essentially equivalent to estimating the relative weighting between measurements and prior data. In the category of data-driven methods, we usually refer them to (biased) regularization from the frequentist’s point of view. Since we often have some physical information on certain geophysical problems, we will show the importance of properly incorporating such information in terms of inequality constraints for geophysical inversions. Geodetic and seismological examples of are used to demonstrate potential problems and advantages of these different types of solution methods.