A Hierarchical Bayesian Framework for Seismic Velocity Inference in Heterogeneous Media

Thesis Abstract

Seismic velocity models in heterogeneous subsurface media are critical for accurate imaging and hazard assessment, yet traditional inversion approaches often yield non-unique solutions and biased estimates when confronted with complex lithologies, anisotropy, and sparse data. This study addresses the fundamental problem of robust velocity inference in heterogeneous media by developing a hierarchical Bayesian framework that integrates multi-physics observations, prior geological knowledge, and spatially varying parameter fields to quantify uncertainty and improve interpretability. The aim is to formulate a probabilistic model that simultaneously estimates posterior distributions of velocity fields, texture indicators, and hyperparameters governing spatial correlation, while accommodating measurement noise, model error, and undersampling. Specific objectives are (i) to construct a hierarchical Bayesian model that links seismic travel-time and ambient noise-derived velocity estimates to a latent velocity field with spatially varying priors informed by geological facies; (ii) to incorporate a probabilistic forward model that accounts for ray-theory travel-time, finite-difference-based wave propagation approximations, and anisotropic effects through a tensor-valued velocity parameterization; (iii) to develop scalable inference strategies using integrated nested Laplace approximation (INLA) and Hamiltonian Monte Carlo (HMC) to obtain posterior distributions over velocity, uncertainty metrics, and hyperparameters; (iv) to validate the framework on synthetic benchmarks with controlled heterogeneity and on field datasets from complex sedimentary and metamorphic terrains; and (v) to compare hierarchical Bayesian results with conventional deterministic inversions in terms of accuracy, uncertainty quantification, and sensitivity to data sparsity. The methodology adopts a sequential mixed-methods design grounded in Bayesian inference. The population comprises subsurface models with heterogeneous lithologies and anisotropic properties. A two-stage sampling strategy is employed synthetic models to stress-test the framework, followed by a real-world dataset consisting of 1) 1200 travel-time picks from controlled-source seismic surveys and 2) continuous ambient seismic noise correlations yielding 60 reference velocity profiles. Data collection instruments include high-precision geophone arrays, shallow-offset and deep-offset seismic sources, and borehole-derived shear-velocity logs for ground-truth calibration. The hierarchical model comprises a latent velocity field V(x) defined over a 3D grid, a facies indicator field F(x), and hyperparameters governing spatial correlation (e.g., Matérn covariance) and measurement error. The forward model integrates ray-tracing travel times, finite-difference time-domain (FDTD) simulations for representative scenarios, and an anisotropy tensor ? to capture velocity as a function of direction. Priors for V(x) are spatially smoothed Gaussian processes conditioned on F(x), with non-stationary hyperparameters to reflect facies transitions. Inference combines INLA for approximate marginal posteriors in the latent Gaussian structure and HMC for sampling non-linear components, ensuring computational tractability for high-resolution grids. Validity and reliability are addressed via cross-validation on held-out field segments and sensitivity analyses to prior choice and grid resolution. Model specification tests include posterior predictive checks and coverage probability assessments of velocity estimates. The analysis plan emphasizes regression-style diagnostics to examine the relationships between inferred velocities, facies indicators, and known geology, and employs Bayesian model comparison metrics such as Bayes factors and WAIC to adjudicate competing anisotropy representations and covariance structures. Ethical considerations encompass transparent reporting of uncertainty, nondisclosure of proprietary data sources, and reproducibility through open-code repositories and data-sharing agreements where permissible. Expected findings include (i) spatially coherent velocity fields with quantified uncertainties that reflect lithological variability and anisotropy; (ii) improved velocity estimates in data-sparse regions owing to information transfer from neighboring facies and prior geological knowledge; (iii) clear probabilistic delineation of regions where traditional inversions are prone to misinference due to non-uniqueness or noise, with explicit posterior uncertainty bounds. The study contributes to knowledge by advancing a unified probabilistic framework for seismic velocity inference that explicitly models heterogeneity, anisotropy, and data limitations, bridging geophysical theory with practical inversion workflows. It offers methodological innovations in hierarchical modeling, covariance structure learning, and scalable Bayesian inference for geophysical imaging. The main conclusion is that a hierarchical Bayesian approach provides superior uncertainty-aware velocity estimation in heterogeneous media compared with conventional methods, particularly under data sparsity and complex lithologies, and the recommendations include adopting hybrid inference pipelines in standard seismic workflows and extending the framework to joint inversion with elastic parameters and attenuation for enhanced subsurface characterization.