Conditional and marginal probabilities in AEM inversions using multivariate Gaussian statistics

Abstract

AEM inversions often involve linearised approximations to the calculation of the Jacobian and Hessian matrices in the forward solution. This results in a second order Taylor series expansion of the estimation of an error surface when calculating the misfit between forward model data and measured data. In the vicinity of a minimum in the error surface, the first-order terms drop out and only second-order terms are present. This guarantees that model parameters resulting from the inversion will be Gaussian distributed with mean model parameter values and model parameter variance terms. The first set is the output of the inversion that is most often used, and we produce conductivity-depth sections from the mean model parameters. However, we are then neglecting the fact that those parameters contain variances which are also of value. In reality, because of the way we have constructed the inversion scheme, the model variance terms contain information about how each model parameter interacts with each other model parameter for a given inversion output result. The collection of variances is most easily assembled in the posterior covariance matrix, where terms on the main diagonal are the autocorrelation values and terms off the diagonal are the cross-correlation or covariance terms. We examine the posterior covariance matrix terms, and exploit the well known characteristics of Gaussian statistics to ask meaningful conditional and marginal probability questions from the inversion data. This is done so that we can ask questions such as: 'Given that the aquifer I am interested in has conductivity ranges between x_low and x_high, where, how deep and how thick is the aquifer?' The result is a probability map that shows the most likely location of the structure of interest given the conditional statements made when posing the question.