Fast probabilistic inversion of AEM data using approximate rejection sampling

Abstract:

Probabilistic inversion of airborne EM (AEM) data I appealing as 1) it allows quantifying the full posterior variability/uncertainty, and 2) allow incorporating complex (1D/2D/3D) prior information. In principle a sample of the posterior distribution can be generated using Monte Carlo sampling techniques, from which the posterior distribution can be analysed. In practice however, the use of Monte Carlo sampling methods is related to high computational demands. Inversion of a single 1D AEM sounding can be done with a 1D prior using Monte Carlo sampling methods in minutes. However, when 2D prior constraints are considered the computational demands increase dramatically.

As an alternative to these computational heavy Monte Carlo sampling algorithms, we propose a method based on approximate rejection sampling. It is based on first generating a large set of Np corresponding models (from the prior) and data (computed using a forward model), i.e. the reference list. This set will be used as a proxy for the full forward, such that for a new model, the data will be chosen as the data corresponding to the most similar model in the reference list. This is naturally an approximation, and therefore the corresponding modelling error is estimated using a probabilistic model. Then, to sample the posterior for a specific data set, the likelihood (accounting for the modelling error) for each of the Np sets of model and data in the reference list is computed from which any posterior statistics such as the mean, mode, quantile can be estimated.

The main attraction of the method is that it is computationally efficient, and that the full space of uncertainty is explored. The prior and the forward model need only be evaluated Np times, and can be arbitrarily complex. The actual computation of the local properties of the posterior distribution, can be done fast and in parallel, only evaluating the likelihood!

The drawback is that full realizations of a full 3D model, cannot be achieved in this manner. Instead, it can be applied to localized problems, and used to obtain local statistics such as the mean mode and quantiles.

1D and 2D examples will be shown and compared to full probabilistic inversion.