3d Bayesian Variational Surface Wave Tomography Enables Efficient Uncertainty Quantification in High-Dimensional Problems

Understanding the Earth’s subsurface structure relies heavily on seismic imaging, but obtaining clear and reliable images remains a significant challenge due to inherent data noise and complexities within the planet. Wenda Yang and Xin Zhang, from the State Key Laboratory of Deep Earth Exploration and Imaging at China University of Geosciences, alongside their colleagues, now present a new approach to this problem, employing advanced Bayesian variational techniques to create detailed 3D models of subsurface velocity. This research overcomes limitations of traditional methods by efficiently quantifying uncertainty in the resulting images, a crucial step for accurate interpretation, and delivers more detailed velocity structures than previously possible. By applying these methods to both simulated and real seismic data from Southwest China, the team demonstrates a powerful new tool for investigating the Earth’s hidden depths and promises to advance our understanding of geological processes.

Seismic Waveform Inversion and Tomography Techniques

This research explores techniques used to create images of Earth’s subsurface, focusing on seismic waves and how they travel through different materials. Core methods employed include seismic tomography and full waveform inversion, alongside specialized approaches like double-difference seismic tomography, surface wave tomography, and ambient noise tomography. Researchers also utilize body wave tomography, transdimensional inversion, and rock physics inversion to refine their models. The study incorporates advanced statistical and optimization methods to improve imaging accuracy and reliability. Bayesian inversion, utilizing probability distributions to represent model parameters and their uncertainties, is central to the approach, often employing Markov Chain Monte Carlo for sampling.

Regularization techniques stabilize inversions, while stochastic gradient Langevin dynamics represents a specific variant of Markov Chain Monte Carlo, with boosting and automatic differentiation further enhancing optimization. Variational inference, an alternative to Markov Chain Monte Carlo, plays an increasingly important role. Researchers implemented variational full waveform inversion, boosting variational inference, and physically structured variational inference to improve efficiency and accuracy. These methods, alongside techniques like reweighted variational inversion and Rj-MCMC, represent a powerful toolkit for subsurface imaging.

The research utilizes various data types, including seismic waveforms, surface waves, body waves, and ambient seismic noise. Earthquake and induced seismicity data provide valuable information, while dense seismic arrays and passive seismic data acquisition methods provide necessary data coverage. The study also considers prior knowledge, uncertainty analysis, model parameterization, data assimilation, and the dimensionality of the subsurface model.
D Seismic Tomography via Variational Inference

Scientists developed a new approach to seismic surface wave tomography, extending the technique from two-dimensional to three-dimensional subsurface imaging. This work directly inverts frequency-dependent travel time measurements to construct detailed three-dimensional spatial structures, offering improved accuracy and resolution compared to traditional methods. The study pioneers the application of variational inference, a powerful optimization technique, to address the inherent non-uniqueness and nonlinearity present in tomographic problems. Researchers implemented three distinct variational methods, mean-field automatic differential variational inference, physically structured variational inference, and stochastic Stein variational gradient descent, to solve the inverse problem, rigorously testing them using both synthetic datasets and real-world seismic data collected in Southwest China.

These methods iteratively refine model parameters, minimizing the discrepancy between observed and predicted travel times. Unlike conventional optimization techniques, variational inference provides a means to quantify uncertainty, crucial for robust geological interpretation. While all three variational methods successfully estimate subsurface velocity structures, stochastic Stein variational gradient descent consistently produced more reliable uncertainty estimates than mean-field automatic differential variational inference and physically structured variational inference, due to the Gaussian assumptions inherent in the latter two methods. By directly inverting travel times, the approach incorporates spatial correlations, leading to more accurate results and detailed velocity structures.
D Seismic Imaging via Variational Methods

Scientists have developed a new approach to seismic surface wave tomography, extending the technique from two-dimensional to three-dimensional subsurface imaging. This work addresses the challenge of non-unique solutions often produced by data noise and the inherent nonlinearity of the problem. Researchers directly invert for three-dimensional spatial structures using frequency-dependent travel time measurements, providing more detailed velocity structures than traditional methods. The research focuses on applying three variational methods, mean-field automatic differential variational inference, physically structured variational inference, and stochastic Stein variational gradient descent, to both synthetic and real data collected in Southwest China.

Results demonstrate that all three methods successfully provide accurate velocity estimates of subsurface structures. Stochastic Stein variational gradient descent produced more reasonable uncertainty estimates compared to mean-field automatic differential variational inference and physically structured variational inference, due to the Gaussian assumptions inherent in the latter two methods. Experiments reveal that the variational methods deliver more detailed velocity structures alongside reliable uncertainty estimates when applied to real-world data. This advancement is significant because traditional tomography often struggles to quantify uncertainty, hindering robust decision-making. By treating unknown parameters as random variables, the Bayesian inference framework, implemented through these variational methods, provides a more complete picture of possible solutions and their likelihood.

Variational Inference Improves Subsurface Velocity Imaging

Scientists have developed and applied a suite of variational inference methods to improve three-dimensional surface wave tomography, a technique used to image subsurface velocity structures. Traditional methods often struggle with non-unique solutions and quantifying uncertainty, but this research demonstrates that variational inference offers a powerful alternative. The team successfully applied mean-field automatic differential variational inference, physically structured variational inference, and stochastic Stein variational gradient descent to both synthetic and real data from Southwest China. Results indicate that all three methods accurately estimate subsurface velocities, with stochastic Stein variational gradient descent providing the most reliable uncertainty estimates. The variational methods generated more detailed velocity structures from real-world data than conventional techniques, alongside robust assessments of uncertainty, allowing for more confident interpretation of complex subsurface features. While the study focused on surface wave dispersion data, incorporating other data types, such as body wave travel times and receiver functions, could further refine the results.