Uncertainty quantification for regularized inversion of electromagnetic geophysical data-Part I: motivation and theory

We present a method for computing a meaningful uncertainty quantification (UQ) for regularized inversion of electromagnetic (EM) geophysical data that combines the machineries of regularized inversion and Bayesian sampling with a 'randomize-then-optimize' (RTO) approach. The RTO procedure is to perturb the canonical objective function in such a way that the minimizers of the perturbations closely follow a Bayesian posterior distribution. In practice, this means that we can compute UQ for a regularized inversion by running standard inversion/optimization algorithms in a parallel for-loop with only minor modification of existing codes. Our work is split into two parts. In Part I, we review RTO and extend the methodology to estimate the regularization penalty weight on the fly, not unlike in the Occam inversion. We call the resulting algorithm the RTO-TKO and explain that it samples from a biased distribution which we numerically demonstrate to be nearby the Bayesian posterior distribution. In return for accepting this small bias, the advantage of RTO-TKO over asymptotically unbiased samplers is that it significantly accelerates convergence and leverages computational parallelism, which makes it highly scalable to 2-D and 3-D EM problems. In Part II, we showcase the versatility and computational efficiency of RTO-TKO and apply it to a variety of EM inversions in 1-D and 2-D, carefully comparing the RTO-TKO results to established UQ estimates using other methods. We further investigate scalability to 3-D, and discuss the influence of prior assumptions and model parametrizations on the UQ.