Identifying Central Tendencies in Ensemble Solutions to Geophysical Inverse Problems

Model space exploration methods, including Markov Chain Monte Carlo and other sampling schemes, have been applied across diverse geophysical inverse problems, including electrical resistivity sounding, seismic tomography, gravity inversions, and mantle rheology. These methods yield an ensemble of solutions, representing samples from the posterior on the model given our prior knowledge and the data used in the inversion. In the presence of significant non-linearity and uncertainty, ensemble solutions can sample multi-modal probability distributions, and can capture tradeoffs between model parameters. In traditional, optimization-based inversions, practitioners can communicate a single preferred solution such as a 2D image of subsurface structure or a 1D profile of a parameter such as resistivity or viscosity. How to best communicate properties of the ensemble solutions is less straightforward, particularly for non-linear inverse problems. Here, we use 1D electrical resistivity sounding as a model problem to analyze methods for extracting meaningful representative solutions from ensembles generated by transdimensional, hierarchical Bayesian inversions. As the level of data noise increases, the model space average or median is itself a poor solution in the sense that it produces much higher misfit than the solutions within the ensemble. This problem is especially pronounced for transdimensional inversions, in which the number of free parameters is not set a priori and the ensemble contains solutions with varying levels of model complexity. We assess the effectiveness of applying clustering algorithms to ensembles, which generate more accurate representative solutions when the data have large uncertainties.