A Bayesian approach to estimate uncertainty for full-waveform inversion using a priori information from depth migration

Abstract

Full-waveform inversion (FWI) enables us to obtain high-resolution subsurface images; however, estimating model uncertainties associated with this technique is still a challenging problem. We have used a Bayesian inference framework to estimate model uncertainties associated with FWI. The uncertainties were assessed based on an a posteriori covariance operator, evaluated at the maximum a posteriori model. For the prior distribution, we have used a spatially nonstationary covariance operator based on a plane-wave construction with local dips measured from migrated images. Preconditioned frequency-domain FWI was used to estimate the maximum a posteriori model. Efficient manipulation of the posterior covariance was based on a low-rank approximation of the data misfit Hessian preconditioned by the prior covariance operator. The strong decay of the singular values indicated that data were mostly informative about a low-dimensional subspace of model parameters. To reduce computational cost of the randomized singular value decomposition, we have used a Hessian approximation based on point-spread functions. The 2D numerical examples with synthetic data confirmed that the method can effectively estimate uncertainties for FWI. Visual comparisons of random samples drawn from the prior and posterior distributions have allowed us to evaluate model uncertainties. Furthermore, we found out how statistical quantities, such as means and pointwise standard deviation fields, can be efficiently extracted from the prior and posterior distributions. These fields helped us to objectively assess subsurface images provided by FWI.