Resolution of seismic waveform inversion: Bayes versus Occam

In Bayesian inference, probabilistic information about models is posited a priori. This information, which may very well include features in the null space of the forward problem, affects both the computed models and the resulting resolution estimates. In Occam's inversion, on the other hand, the goal is to construct the smoothest model consistent with the data. This is not to say that one believes a priori that models are really smooth, but rather that a more conservative interpretation of the data ought to be made by eliminating features of the model that are not required to fit the data. The length scale associated with the smoothing is an indirect measure of resolution. In some cases the mathematical machinery of Bayesian inference resembles that of Occam's inversion, but the goals and interpretations of the two methods are rather different. To understand better the similarities and differences of these two approaches, we show an application of both methods to the problem of inferring the Earth's subsurface elastic properties from reflection seismic data. On the one hand, by deriving a priori information about the Earth's layering from fine-scale borehole measurements, coupled with information about the noise in the data and the elastic forward modelling operator, we are able to compute the Bayesian a posteriori probability distribution on the space of models. Models pseudo-randomly simulated from this a posteriori probability will exhibit features that are implied by the a priori information as well as the data, even if the former are not well resolved by the data. Then we solve the Occam's inversion problem by determining the maximum model smoothness that allows for the data to be fit, without incorporating a priori information about the models. In this case we estimate the resolution in terms of the degree of model smoothness implied by the data. The main conclusions for the numerical experiments considered in this work are that the subsurface models derived from both techniques are quite similar but error estimates associated with such models are rather different, reflecting the role of the a priori information in the inverse calculation.