Accelerated Bayesian Full Waveform Inversion with Multifidelity Surrogate Modeling

Once trained, surrogate models can emulate costly physics-based forward solvers at a negligeable computational cost, making them attractive tools to accelerate computationally expensive Markov chain Monte Carlo (MCMC) inversions. In the context of waveform modeling, it is nevertheless challenging to derive accurate surrogate models over the support of a realistically chosen prior probability density function (pdf). To circumvent this issue, one solution is to identify a region of high posterior densities with a somewhat inaccurate surrogate solver and then retrain a new surrogate model using samples drawn from this approximate and inflated posterior pdf. The resulting surrogate model has less coverage, but also higher accuracy in the posterior region of interest. Based on these ideas, we introduce a multifidelity MCMC formulation in the context of crosshole ground-penetrating radar (GPR) full-waveform inversion. To reduce the dimensions of the input and output domains as needed for efficient surrogate modeling, we rely on parameterization in the form of principal components inferred from training data, while surrogate modeling is performed with polynomial chaos expansions. To initialize the algorithm, a surrogate model is first learned between larger-scale features of the input domain and lower-frequency characteristics of the output domain using samples from the prior pdf. The associated modeling error is quantified and parameterized by a covariance matrix that is included in a model-error-aware likelihood function. An MCMC inversion is then performed using this first low-fidelity surrogate model to obtain a first approximation of an inflated posterior pdf. As this tempered posterior pdf has less support than the prior pdf, samples from it can be used to train a higher-fidelity surrogate model with larger input (finer-scale features) and output (higher frequencies) dimensions. This new surrogate model is then used in a second MCMC inversion with an updated likelihood function, and so on. In a test example with four fidelity levels, in which we move from initially 15 input principal components to 100, the posterior pdf is estimated at two-orders-of-magnitude lower computational cost than if using the full-physics forward solver only. The mean of the estimated posterior pdf is unbiased, which is not the case for an algorithm in which the surrogate model is learned using samples from the prior pdf only. The methodology could be adapted to other applications beyond crosshole GPR, such as seismic or tracer test data inversions.