Resolution and uncertainty in 1D CSEM inversion: A Bayesian approach and open-source implementation

Resolution and uncertainty in controlled-source electromagnetic (CSEM) inversion is most naturally approached using a Bayesian framework. Resolution can be inferred by hierarchical models with free parameters for effective correlation lengths ("Bayesian smoothing"), or model-choice frameworks applied to variable resolution spatial models (Bayesian splitting/merging). Typical 1D CSEM data can be modeled with quite parsimonious models, typically O(10) parameters per common midpoint. Efficient optimizations for the CSEM problem must address the challenges of poor scaling, strong nonlinearity, multimodality and the necessity of bound constraints. The posterior parameter uncertainties are frequently controlled by the nonlinearity, and linearized approaches to uncertainty usually are very poor. In Markov Chain Monte Carlo (MCMC) approaches, the nonlinearity and poor scaling make good mixing hard to achieve. A novel, approximate frequentist method we call the Bayesianized parametric bootstrap (sometimes called randomized maximum likelihood) is much more efficient than MCMC in this problem, considerably better than linearized analysis but tends to modestly overstate uncertainties. The software that implements these ideas for the 1D CSEM problem is made available under an open-source license agreement.