G eophy sical Prospecting, 1996' 44, 3 1 3-350 Bayesian inference, Gibbs' sampler and uncertainty estimation in geophysical inversion' Mrinal K. Sen2 and Paul L. Stoffa3 Abstract The posterior probability density function (PPD), o(mldo6,), of earth model rn, where do6" or€ the measured data, describes the solution of a geophysical inverse problem, when a Bayesian inference model is used to describe the problem. In many applications, the PPD is neither analytically tractable nor easily approximated and simple analytic expressionsfor the mean and variance of the PPD are not available. Since the complete description of the PPD is impossible in the highly multi- dimensional model spaceof many geophysical applications, severalmeasuressuch as the highest posterior density regions, marginal PPD and several orders of moments are often used to describe the solutions. Calculation of such quantities requires evaluation of multidimensional integrals. A faster alternative to enumeration and blind Monte-Carlo integration is importance sarnpling which may be useful in several applications. Thus how to draw samples of rn from the PPD becomes an important aspect of geophysical inversion such that importance sampling can be used in the evaluation of these multi-dimensional integrals. Importance sampling can be carried out most efficiently by a Gibbs' sampler (GS). \7e also introduce a method which we called parallel Gibbs' sampler (PGS) based on genetic algorithms (GA) and show numerically that the results from the two samplers are nearly identical. \7e first investigate the performance of enumeration and several sampling based techniques such as a GS, PGS and several multiple maximutn a posteriori (MAP) algorithms for a simple geophysicalproblem of inversion of resistivity sounding data. Several non-linear optimization methods based on simulated annealing (SA), GA and some of their variants can be devised which can be made to reach very closeto the maximum of the PPD. Such MAP estimation algorithms also sample different points in the model space. By repeating these MAP inversions severaltimes, it is possible to sample adequatelythe most significant portion(s) of the PPD and all thesemodels can lPaper presented at the 56th EAEG meeting, June 1994, Vienna, Austria. Received November 1994, revision accepted August 1995. z Institute for Geophysics, The University of Texas at Austin, 8701 N. Mopac Expressway, Austin, TX 78759-8397, USA. ' Institute for Geophysics and Department of Geological Sciences,The University of Texas at Austin, 8701 N. Mopac Expressway, Austin, TX78759-8397, USA. Ct 1996 European Association of Geoscientists & Ensineers ll-)