3D Bayesian Variational Surface Wave Tomography and Application to the Southwest China

Seismic surface wave tomography uses surface wave information to obtain velocity structures in the subsurface. Due to data noise and nonlinearity of the problem, surface wave tomography often has non-unique solutions. It is therefore required to quantify uncertainty of the results in order to better interpret the resulting images. Bayesian inference is the most widely-used method for this purpose. However, the commonly-used Monte Carlo methods require huge computational cost and remains intractable in high-dimensional problems. Variational inference uses optimization to solve Bayesian inverse problems, and therefore can be more efficient in the case of large datasets and high-dimensional parameter spaces. Variational inference has been applied to 2-D surface wave tomographic problems. In this study, we extend the method to 3-D surface wave tomography by directly inverting for 3-D spatial structures from frequency-dependent travel time measurements. Specifically, we apply three variational methods, mean-field automatic differential variational inference (mean-field ADVI), physically structured variational inference (PSVI) and stochastic Stein varational gradient descent (sSVGD) to surface wave tomographic problems using both synthetic data and real data in the Southwest China. The results show that all methods can provide accurate velocity estimates, while sSVGD produces more reasonable uncertainty estimates than mean-field ADVI and PSVI because of Gaussian assumption used in the later methods. In the real data case, the variational methods provide more detailed velocity structures than those obtained using traditional methods, along with reliable uncertainty estimates. We therefore conclude that variational surface wave tomograph can be applied fruitfully to many realistic problems.

💡 Research Summary

Surface‑wave tomography aims to infer subsurface shear‑wave velocity structures from measured travel‑time data, but the inverse problem is highly nonlinear and contaminated by noise, leading to non‑unique solutions. Bayesian inference provides a principled framework for quantifying the uncertainty of the recovered models, yet conventional Markov‑chain Monte‑Carlo (MCMC) sampling becomes computationally prohibitive when the parameter space expands to three dimensions with thousands of voxels. This study addresses the scalability issue by employing variational inference (VI), which approximates the posterior distribution with a tractable family of distributions and turns Bayesian inference into an optimization problem. Three VI algorithms are investigated: mean‑field automatic differential variational inference (mean‑field ADVI), physically structured variational inference (PSVI), and stochastic Stein variational gradient descent (sSVGD).

Mean‑field ADVI assumes each model parameter follows an independent Gaussian distribution, dramatically reducing the number of variational parameters and enabling fast convergence even for large 3‑D grids. However, the independence assumption neglects correlations among neighboring voxels, often resulting in overly narrow uncertainty estimates. PSVI incorporates geological priors—such as smoothness, layer continuity, and known structural constraints—directly into the variational family, thereby producing a more realistic Gaussian approximation that respects physical expectations. Nevertheless, PSVI still relies on a Gaussian form and may struggle to capture multimodal or highly skewed posteriors. sSVGD, by contrast, is a particle‑based method that evolves a set of samples (particles) under a Stein‑gradient flow. This approach does not impose a specific parametric shape on the posterior, allowing it to represent non‑Gaussian, multimodal, and asymmetric distributions while automatically accounting for inter‑parameter dependencies through particle interactions.

The authors first validate the three methods using synthetic data. A synthetic 3‑D velocity model is generated, and frequency‑dependent travel‑time data are simulated with added Gaussian noise. All three VI techniques recover the true velocity field with comparable root‑mean‑square errors and structural similarity indices, demonstrating that the variational approximations are sufficiently accurate for the forward problem. When assessing uncertainty, sSVGD yields the broadest credible intervals that align with the known noise level and data sparsity, whereas ADVI and PSVI produce narrower intervals due to their Gaussian assumptions, potentially under‑representing true uncertainty.

The methodology is then applied to real data from Southwest China, a tectonically complex region characterized by the interaction of the Indian and Eurasian plates, numerous fault systems, and heterogeneous crustal structures. Travel‑time measurements in the 0.5–2.0 Hz band are collected from a dense network of broadband stations. The inversion domain is discretized into a 3‑D grid with 50 km spacing in all three dimensions, resulting in several thousand unknown velocities. All three variational schemes generate velocity models that display finer lateral and vertical resolution than those obtained with conventional normalized least‑squares tomography. Notably, high‑velocity anomalies associated with the rigid Indian plate and low‑velocity zones linked to the overlying sedimentary basin are more sharply delineated. The accompanying uncertainty maps reveal that sSVGD assigns larger posterior variance in regions where data coverage is sparse or where the signal‑to‑noise ratio is low, providing a realistic assessment of model reliability. ADVI and PSVI, while computationally cheaper, tend to underestimate variance in these critical zones.

In summary, this work demonstrates that variational Bayesian surface‑wave tomography can efficiently handle high‑dimensional 3‑D inverse problems while delivering credible uncertainty quantification. Mean‑field ADVI and PSVI are attractive for rapid, first‑order assessments, but sSVGD offers superior uncertainty characterization by relaxing the Gaussian restriction. The authors suggest future extensions such as integrating body‑wave data, employing hierarchical priors, or developing real‑time monitoring pipelines that exploit the computational speed of variational methods. The results underscore the potential of variational inference to become a standard tool for modern seismic imaging and hazard assessment.