Bayesian full waveform inversion using deep learning surrogate models

Summary Full waveform inversion (FWI) is among the most accurate seismic imaging techniques, yet its Bayesian formulation remains computationally prohibitive. The gold-standard method, Markov chain Monte Carlo (McMC), requires hundreds of thousands of expensive forward simulations. Variational inference (VI) approaches offer improved efficiency but require gradient computations and may converge to incorrect posteriors. We introduce a probabilistic surrogate-assisted framework that replaces most expensive forward evaluations during sampling with predictions from a misfit-guided surrogate based on Deep Learning (DL) models, instantiated here using a convolutional neural network (CNN). The framework supports three regimes: (1) an offline regime in which the surrogate remains fixed after training, (2) an adaptive regime in which the surrogate is refined during sampling using a self-supervised active learning mechanism, and (3) a two-stage regime in which the surrogate screens proposals prior to exact evaluation. The two-stage regime preserves the exact posterior by construction, while the adaptive regime is asymptotically exact in the infinite-sample limit, provided that the surrogate approximation error vanishes. Because the target subsurface features in our study are spatially localised, we adopt a rock-physics-guided, shape-based parameterisation that represents each target by a small number of control nodes. This results in a low-dimensional inverse problem with a dimensionality of order 10 in the examples considered here, at which a misfit-guided DL surrogate becomes tractable. We demonstrate our framework on synthetic models of increasing dimensionality, including a CO2 plume monitoring scenario. The adaptive regime achieves a 7.1 × wall-clock speedup and reduces GPU-hour cost by approximately 84% relative to McMC with the exact forward solver, while recovering posteriors in close agreement with the reference. Comparison with Stein variational gradient descent (SVGD) shows the surrogate-assisted McMC sampling is approximately an order of magnitude more efficient in the examples considered here. This framework applies to any surrogate paired with any forward solver and objective function producing a scalar misfit.