Bayesian seismic inversion by residual flow

ABSTRACT Sampling from the posterior distribution of a Bayesian problem can pose a computational challenge. Normalizing flows offer a framework for learning complex distributions, making them a candidate for approximating the posterior distributions in Bayesian problems. Differences in flow architecture offer distinct attributes. Most flows are engineered with a structured Jacobian matrix for efficient probability calculation. However, this can limit the transformation’s flexibility and affect the flow’s inductive bias, potentially altering learned patterns. We use residual flow with an unconstrained transformation form in an unsupervised framework for sampling from the posterior of a nonlinear Bayesian amplitude-variation-with-offset inverse problem. The aim is to invert elastic properties (P-wave velocity, S-wave velocity, and density) using seismic data and sparsely available well logs. We investigate the method’s applicability using synthetic and field cases. Furthermore, we compare residual flow features with two other prevalent flows, specifically, real-valued nonvolume preserving (real-NVP) and masked autoregressive flow (MAF), as well as an adaptive version of Hamiltonian Monte Carlo. Results show residual flow reasonably approximates the posterior of the inverse problem and generates acceptable samples. We assess the flows’ ability to approximate a target distribution by restricting the number of trainable layers in two tests. Residual flow provides superior results in both cases. Real-NVP fails to generate sufficiently accurate samples in the architecture with minimal trainable layers. However, adding more layers enhanced the metrics and accelerated the convergence rate. MAF results fall between the other flows but struggle with balancing variance and accuracy in the density log. MAF has a threefold speed advantage over residual flow. Real-NVP can converge faster than residual flow when it has sufficient capacity to model the target distribution.