Full-waveform earthquake source inversion using simulation-based inference

TL;DR

The paper tackles the problem of uncertain, non-Gaussian seismic noise in full-waveform inversion for earthquake source parameters. It adopts simulation-based inference (SBI) with neural density estimators to learn an empirical noise model and directly model the posterior $p(\boldsymbol{m}\mid \boldsymbol{D})$, using optimal score compression to make high-dimensional data tractable. Across synthetic tests and two real Atlantic earthquakes, SBI yields well-calibrated posteriors that better reflect parameter uncertainties than standard Gaussian-likelihood inversions, while reducing the number of forward-model evaluations by roughly an order of magnitude. The approach is shown to be directly applicable to real-source inversions with complex priors, offering more reliable uncertainty quantification and paving the way for broader SBI deployment in seismology.

Abstract

This paper presents a novel framework for full-waveform seismic source inversion using simulation-based inference (SBI). Traditional probabilistic approaches often rely on simplifying assumptions about data errors, which we show can lead to inaccurate uncertainty quantification. SBI addresses this limitation by building an empirical probabilistic model of the data errors using machine learning models, known as neural density estimators, which can then be integrated into the Bayesian inference framework. We apply the SBI framework to point-source moment tensor inversions as well as joint moment tensor and time-location inversions. We construct a range of synthetic examples to explore the quality of the SBI solutions, as well as to compare the SBI results with standard Gaussian likelihood-based Bayesian inversions. We then demonstrate that under real seismic noise, common Gaussian likelihood assumptions for treating full-waveform data yield overconfident posterior distributions that underestimate the moment tensor component uncertainties by up to a factor of 3. We contrast this with SBI, which produces well-calibrated posteriors that generally agree with the true seismic source parameters, and offers an order-of-magnitude reduction in the number of simulations required to perform inference compared to standard Monte Carlo techniques. Finally, we apply our methodology to a pair of moderate magnitude earthquakes in the North Atlantic. We utilise seismic waveforms recorded by the recent UPFLOW ocean bottom seismometer array as well as by regional land stations in the Azores, comparing full moment tensor and source-time location posteriors between SBI and a Gaussian likelihood approach. We find that our adaptation of SBI can be directly applied to real earthquake sources to efficiently produce high quality posterior distributions that significantly improve upon Gaussian likelihood approaches.

Figures (11)

• Figure 1: Upper panel: realistic observations $(\mathbf{m}_i, \mathbf{D}_i)$ (red crosses) define a joint distribution (contoured background). Lower panel: the posterior distribution can be modelled empirically from this dataset, allowing for direct evaluation $\hat{p}(\mathbf{m} \mid \mathbf{D}_{1,2})$ for a given observation.
• Figure 2: Schematics of the Masked Autoencoder for Distribution Estimation (MADE) and Masked Autoregressive Flow (MAF). MAFs are built by chaining together MADEs, each representing a transformation $T_j$, and permuting the order of conditioning to ensure suitably flexible density estimation. Each MADE $\varphi$ models a chain of conditional Gaussian distributions, parametrised by $\{\mu_k, \sigma_k\}$ where $k$ indexes the element of $\mathbf{x}$, to produce estimates of the density. Training can be thought of as learning to transform observations of the posterior $p(\mathbf{m} \mid \mathbf{D})$ to a simple base distribution $p(\mathbf{u}_0)$. Density estimation (left-to-right) repeatedly applies the change of variables formula in Eq. \ref{['eq:change_of_variables']} by computing $\text{det}J_T(\mathbf{u}_j)$ for each intermediate transformed variable $\mathbf{u}_j$. Sampling (right-to-left) successively applies transformation $T_j$ to samples from the base distribution $\mathbf{u}_0 \sim p(\mathbf{u}_0)$. See the text in Section \ref{['sec:ndes']} for more details.
• Figure 3: Diagram of the SBI workflow for earthquake source inversion. Top: seismic sources are sampled from a prior $p(\mathbf{m})$, simulated, and combined with real noise data. This emulates sampling from a realistic likelihood, $\mathbf{D}_i \sim p(\mathbf{D} \mid \mathbf{m} )$. The resulting observations $\mathbf{D}_i$ are compressed with compression method $\phi$ to create a dataset of summary statistics $\mathbf{t}_i$ and model parameters $\mathbf{m}_i$. A NDE $\psi$ is then trained on this data to model the associated probability density function. Once trained, inference (bottom) can be performed without access to the forward model by compressing a new observation $\mathbf{D}_\text{obs}$ and passing the compressed observation $\mathbf{t}_\text{obs}$ directly through the NDE. See the text in Section \ref{['sec:data_compression']} for more details.
• Figure 4: Main panel: UPFLOW ocean bottom seismometer array located in the Azores-Madeira-Canaries region in the mid-Atlantic. This consists of 50 ocean-bottom seismometers that were deployed in July-August 2021, denoted by the labelled markers. 49 OBSs were recovered in August-September 2022, with UP14 being lost. Of these, a subset of 43 stations could be used after data quality checks tsekhmistrenko2024performance. GCMT focal mechanism solutions are plotted as beachballs for two regional moderate magnitude earthquakes that we study in this work. Lower left: A zoom-in on the Azores islands, with a number of labelled permanent land stations operated by Instituto Português do Mar e da Atmosfera (IPMA) that are used in this study.
• Figure 5: Solutions for the artificial strike-slip event with known Gaussian noise considered in this study. Panel a) shows the analytic posterior distribution (purple), MCMC posterior samples produced using a correctly specified Gaussian likelihood (red), and the SBI approach using a MAF trained on $4,000$ simulations (blue). Inner and outer contour rings correspond to $[68,95]\%$ credibility contours. The dashed lines represent the true solution. The SBI maximum a posteriori solutions and 1D uncertainty estimates are given at the top of each column. Moment tensor component units are given in the bottom left. Panel b) shows the simplified station configuration used for the artificial examples, as well as the location and focal mechanism of the artificial event. Panel c) compares the beachball solutions of the SBI sample ensemble against the analytic posterior, showing very little physical difference between the results.