Multiscale uncertainty quantification for post-stack seismic inversion with wavelet flows

ABSTRACT

Point-estimate statistics for inverse problems, such as the maximum a-posteriori estimator, require the solution of a problem that is typically ill-conditioned for seismic imaging applications, with important implications in terms of computational complexity.

Ill-conditioning is arguably an even bigger challenge for uncertainty quantification, which aims at a more comprehensive characterisation of the posterior distribution. One classical strategy to assuage these issues is the multiscale approach, where the original problem is broken down into a hierarchical sequence of sub-problems with increasing computational complexity.

Each sub-problem describes scale-dependent features of the overall solution for any chosen scale-dependent decomposition. This gives rise to an efficient iterative method that progressively builds a solution from "coarse" to "fine" scales. We propose to leverage recent developments in machine-learning-based variational inference for uncertainty quantification that uses a wavelet-based generative model of the posterior distribution.

The architectural design is based on a normalising flow that generates the scales of a sample sequentially and conditionally based on the coarser scales. As a first application of this framework, we study post-stack seismic inversion here.