2:45 PM - 3:00 PM
[SCG55-10] Bayesian first-arrival traveltime tomography using physics-informed neural networks and particle-based variational inference
Keywords:Seismic tomography, Physics-informed neural network, Particle-based variational inference
First-arrival traveltime tomography (FAT) using refraction seismic data is important for understanding seismic wave velocity structure at depth. Quantifying uncertainty of seismic velocity estimates in tomography is essential to ensure the estimation reliability. For this purpose, Bayesian estimation has been introduced to seismic tomography to estimate the posterior probability distribution function (P-PDF) of velocity structure based on errors in travel time data and prior information (e.g., Ryberg et al. 2018). These previous studies are based on a mesh discretization of the analysis domain, calculating the travel time using a numerical method, and parametrizing the velocity for Bayesian estimation. Since this estimation problem is nonlinear, sampling methods such as Markov chain Monte Carlo (MCMC) are used.
Physics-informed neural networks (PINN) (Raissi et al. 2019), which solves partial differential equations and inverse problems with neural networks (NNs) constrained from the equations, has attracted much attention. It has been also applied to seismic tomography (e.g., Waheed et al. 2021). This is a mesh-free framework leveraging continuous functions represented by NNs. In previous Bayesian tomographic methods, to reduce the number of parameters in the P-PDF sampling, adaptive discretization using reversible-jump (rj) MCMC is often introduced, where the number of parameters is also a subject of Bayesian estimation, leading to highly grid-dependent individual samples. As a result, only statistics such as the mean and variance of the resulting velocity samples are discussed. By contrast, samples consisting of physically-consistent velocity models based on smooth functions represented by NNs may allow geophysical discussion for individual samples. Considering this advantage, this study develops a Bayesian estimation framework for PINN-based seismic tomography.
In PINN-based seismic tomography, two NNs are used: one predicting seismic velocities from coordinates and the other predicting travel time from the source and receiver coordinates. The weight parameters of the two NNs are optimized so that the predicted travel time is close to the observed one, while the seismic velocity and travel time satisfy the Eikonal equation at random evaluation points. In Bayesian seismic tomography, the goal is to estimate the posterior predictive distribution of seismic wave velocities predicted by the NNs. Such a approach is classified as Bayesian neural network (BNN), in which the weight parameters that compose NNs are Bayesianly estimated. We introduce a Bayesian estimation method called particle-based variational inference (ParVI, e.g., Liu et al. 2016), known for its high parallelism and good approximation efficiency. In ParVI, a functional expressing the difference between the PDF approximated by many particles and the true one is minimized by updating each particle sequentially. However, when the network structure is complex, multimodality of the P-PDF of the weight parameters becomes stronger, and the BNN based on ordinary ParVI underestimates the uncertainty. Therefore, we formulate ParVI not in the space of weights, but in the space of continuous functions predicted by the NN (Wang et al. 2019), i.e., seismic wave velocities. This approach leverages the assumption that the shape of the PDF in the function space is simpler.
We conducted numerical experiments on the observation arrangement and velocity structure simulating refraction FAT (Agata et al., arXiv, 2023). The results of estimating the P-PDF were found to be reasonable for the observation arrangement. This result suggests that the proposed method is applicable to real seismic data. In addition, each particle (sample) of the velocity distribution that constitutes the approximate P-PDF had a physically natural smooth distribution represented by NN. This is in contrast to those based on adaptive discretization by rjMCMC, which have discontinuous velocity distributions.
Physics-informed neural networks (PINN) (Raissi et al. 2019), which solves partial differential equations and inverse problems with neural networks (NNs) constrained from the equations, has attracted much attention. It has been also applied to seismic tomography (e.g., Waheed et al. 2021). This is a mesh-free framework leveraging continuous functions represented by NNs. In previous Bayesian tomographic methods, to reduce the number of parameters in the P-PDF sampling, adaptive discretization using reversible-jump (rj) MCMC is often introduced, where the number of parameters is also a subject of Bayesian estimation, leading to highly grid-dependent individual samples. As a result, only statistics such as the mean and variance of the resulting velocity samples are discussed. By contrast, samples consisting of physically-consistent velocity models based on smooth functions represented by NNs may allow geophysical discussion for individual samples. Considering this advantage, this study develops a Bayesian estimation framework for PINN-based seismic tomography.
In PINN-based seismic tomography, two NNs are used: one predicting seismic velocities from coordinates and the other predicting travel time from the source and receiver coordinates. The weight parameters of the two NNs are optimized so that the predicted travel time is close to the observed one, while the seismic velocity and travel time satisfy the Eikonal equation at random evaluation points. In Bayesian seismic tomography, the goal is to estimate the posterior predictive distribution of seismic wave velocities predicted by the NNs. Such a approach is classified as Bayesian neural network (BNN), in which the weight parameters that compose NNs are Bayesianly estimated. We introduce a Bayesian estimation method called particle-based variational inference (ParVI, e.g., Liu et al. 2016), known for its high parallelism and good approximation efficiency. In ParVI, a functional expressing the difference between the PDF approximated by many particles and the true one is minimized by updating each particle sequentially. However, when the network structure is complex, multimodality of the P-PDF of the weight parameters becomes stronger, and the BNN based on ordinary ParVI underestimates the uncertainty. Therefore, we formulate ParVI not in the space of weights, but in the space of continuous functions predicted by the NN (Wang et al. 2019), i.e., seismic wave velocities. This approach leverages the assumption that the shape of the PDF in the function space is simpler.
We conducted numerical experiments on the observation arrangement and velocity structure simulating refraction FAT (Agata et al., arXiv, 2023). The results of estimating the P-PDF were found to be reasonable for the observation arrangement. This result suggests that the proposed method is applicable to real seismic data. In addition, each particle (sample) of the velocity distribution that constitutes the approximate P-PDF had a physically natural smooth distribution represented by NN. This is in contrast to those based on adaptive discretization by rjMCMC, which have discontinuous velocity distributions.