Uncertainty quantification in time-lapse full-waveform inversion of vertical seismic profiling data: contrasting the Hamiltonian Monte Carlo and Stein Variational Gradient Descent methods

Abstract Time-lapse full-waveform inversion (TL-FWI) is increasingly important in energy-transition applications, particularly for monitoring CO2 geological storage. Because TL-FWI is highly nonlinear and baseline and monitor surveys commonly differ in acquisition, illumination, and noise conditions, uncertainty quantification is essential for reliable interpretation. Two promising Bayesian approaches are Hamiltonian Monte Carlo (HMC) and Stein variational gradient descent (SVGD), which represent the posterior in fundamentally different ways. HMC constructs the posterior through stochastic sampling with gradient-based proposals, whereas SVGD approximates it using a deterministically evolving particle ensemble. Here, we investigate the feasibility of 2D VSP-based TL-FWI using these two approaches under a matched computational budget. Both methods recover the main injection-related velocity anomaly, but they express posterior uncertainty differently. Under this resource-limited comparison, HMC yields uncertainty fields that remain comparatively diffuse and become increasingly difficult to interpret in deeper and off-well regions where illumination is weak. SVGD, in contrast, produces uncertainty patterns that are more strongly organized by acquisition geometry, placing larger uncertainty more explicitly in poorly constrained regions, although this also causes the posterior means and profiles to inherit stronger illumination-shaped artifacts. These results suggest that the practical distinction between HMC and SVGD is not simply one of better or worse uncertainty quantification, but rather how limited computational effort is converted into posterior information. The preferred method is therefore budget- and objective-dependent. Specifically, HMC remains attractive when more exhaustive posterior exploration is computationally feasible, whereas SVGD provides a more scalable and practically interpretable route for rapid uncertainty assessment in TLFWI monitoring applications.