Efficient physics-informed Bayesian inversion of VSP and hydrofrac data

Abstract

Bayesian inversion provides the same information as regularized inversion of seismic data, except it also supplies a probability estimate of the solution throughout model space. The cost, however, is that Bayesian inversion is orders-of-magnitude more expensive than regularized inversion by a gradient optimization method. To mitigate this cost, we present an efficient physics-informed Bayesian inversion method that combines regularized inversion to get both the optimal solution and the posterior probability functions in model space. A gradient-optimization method is used to efficiently estimate the maximum a posterior (MAP) solution, and so function evaluations are only needed around the MAP point in model space. This efficiently provides the posterior probability in that neighborhood, and therefore avoids the tremendous expense of sampling points throughout the high-dimensional model space. We present two applications of this physics-informed Bayesian inversion: VSP traveltime inversion and migration of passive seismic data. For 4D monitoring of hydrofrac operations, reuse of the previously computed traveltimes for probability estimates is orders-of-magnitude less expensive for computing migration and posterior images.