Exploring and Summarizing Ensemble Solutions to Geophysical Inverse Problems

Posterior sampling algorithms have been applied to many geophysical inverse problems, including electrical resistivity sounding, seismic tomography, and gravity and geodetic inversions. Unlike optimization‐based approaches that yield a single best‐fit solution, these algorithms produce an ensemble of solutions that sample the posterior probability distributions of the model parameters given the data and prior knowledge. These distributions can be multi‐modal in the presence of significant non‐linearity in the forward model and uncertainty in observations. Communicating the richness of information contained in the ensemble solutions is not straightforward, so practitioners often present a single preferred solution such as a 2D image of subsurface structure or a 1D profile of a parameter. Ensemble solutions generated by transdimensional, hierarchical, Bayesian methods, in which the ensemble samples solutions with different levels of complexity, present particular challenges. As a model problem, we use 1D electrical resistivity sounding to assess methods for extracting representative solutions from ensembles. Due to the problem's non‐linearity and non‐uniqueness, the model‐space average is not a good predictor of the data and thus not representative of the solutions within the ensemble. We apply clustering and sorting algorithms to ensemble solutions in order to extract information about central tendencies. We illustrate how manifold learning and dimensionality reduction techniques can provide insight into ensemble solutions even in the presence of nonlinearity and uncertainty. Finally, we show that the direct application of K‐medoids clustering in model space can select solutions that are representative of the ensemble in both model space and data space.