Bayesian Earthquake Location with a Neural Travel-Time Surrogate: Fast, Robust, and Fully Probabilistic Inference in 3-D Media

TL;DR

This work presents a Bayesian earthquake location framework that replaces costly 3-D travel-time calculations with a physics-informed neural surrogate and performs fully probabilistic inference via MH-within-Gibbs sampling. By adopting a heavy-tailed Student-$t$ likelihood with latent weights, the method robustly handles outliers and velocity-model errors while yielding complete posterior distributions for hypocenter coordinates and origin time. Application to the Luding aftershock sequence demonstrates comparable location accuracy to established methods such as NonLinLoc, with substantially reduced computation time and explicit spatial uncertainty maps; incorporating the Student-$t$ misfit further improves robustness under extreme-noise conditions. The approach provides a scalable, real-time, uncertainty-aware tool for seismic monitoring and opens avenues for joint velocity–hypocenter inversion and integration with automated phase-picking pipelines.

Abstract

We present a Bayesian earthquake location framework that couples a Deep Learning Surrogate with Gibbs sampling to enable uncertainty-aware hypocenter estimation. The surrogate model is trained to reproduce the three-dimensional first-arrival travel-time field by enforcing the Eikonal equation, thereby removing the need for computationally intensive ray tracing. Within a fully probabilistic formulation, Gibbs sampling is used to explore the posterior distribution of source parameters, yielding comprehensive uncertainty quantification. Application to the 2021 Luding aftershock sequence shows that the proposed approach attains location accuracy comparable to that of NonLinLoc while reducing computational cost by more than an order of magnitude. In addition, it produces detailed posterior probability maps that explicitly characterize spatial uncertainty. This integration of physics-informed learning and Bayesian inference provides a scalable, physically consistent, and computationally efficient solution for real-time earthquake location in complex velocity structures.

Figures (8)

• Figure 1: Processing workflow of the earthquake location framework. A neural-network-based travel-time surrogate replaces traditional travel-time computation and is trained and evaluated using the CSNCD dataset.
• Figure 2: P- and S-wave (PS) travel-time accuracy evaluation. (a) Mean residuals and (b) standard deviations of the predicted PS travel times for 32-bit (solid lines) and bfloat16 half-precision computations (dashed lines).
• Figure 3: Spatial distribution of earthquake hypocenters in the Luding region obtained from four different catalogs. (a) The original CENC catalog, which contains routinely reported hypocentral solutions. (b) The Loc3D relocation results of Liu et al., representing a conventional deterministic relocation approach. (c) Bayesian relocation results obtained using our method with a Gaussian error model, yielding a more coherent seismicity structure. (d) Bayesian relocation results obtained using our method with an additional Student-$t$ data misfit, further improving robustness against outliers and producing a tighter clustering of events. All subplots use hypocentral depth as the color attribute (0--30 km) and share a common colorbar for direct comparison across methods.
• Figure 4: Distributions of the horizontal (left column) and depth (right column) 5--95% uncertainty-interval widths for different location algorithms. Panels (a) and (b) show results for our baseline Bayesian location method; (c) and (d) show results for our method with a Student--t prior on the data misfit; and (e) and (f) show results for the NLLoc algorithm. For each panel, blue bars denote all located events within the study time window (5--15 September 2022), and orange bars denote the subset of events that can be matched to the reference catalog in space and origin time. Histograms are computed using equal-width bins; the rightmost bin (labeled "$>50$") contains all events with 5--95% interval widths larger than 50 km.
• Figure 5: Posterior sampling results obtained from four independent MCMC runs initialized at distinct starting locations. Panels (a)--(d) show the horizontal (top row) and vertical (bottom row) projections of the Markov chains. Light gray curves represent burn-in trajectories, and colored points denote retained posterior samples. The blue cross marks the posterior mean of each run, while the red triangle indicates the respective initial state. Despite the large differences in initialization, all chains converge toward the same high-probability region, demonstrating the robustness of the Bayesian location inference and the stability of the sampling procedure.