Improving moment tensor solutions under Earth structure uncertainty with simulation-based inference

Abstract

Bayesian inference represents a principled way to incorporate Earth structure uncertainty in full-waveform moment tensor inversions, but traditional approaches generally require significant approximations that risk biasing the resulting solutions. We introduce a robust method for handling theory errors using simulation-based inference (SBI), a machine learning approach that empirically models their impact on the observations. This framework retains the rigour of Bayesian inference while avoiding restrictive assumptions about the functional form of the uncertainties. We begin by demonstrating that the common Gaussian parametrisation of theory errors breaks down under minor ($1-3 \%$) 1-D Earth model uncertainty. To address this issue, we develop two formalisms for utilising SBI to improve the quality of the moment tensor solutions: one using physics-based insights into the theory errors, and another utilising an end-to-end deep learning algorithm. We then compare the results of moment tensor inversion with the standard Gaussian approach and SBI, and demonstrate that Gaussian assumptions induce bias and significantly under-report moment tensor uncertainties. We also show that these effects are particularly problematic when inverting short period data and for shallow, isotropic events. On the other hand, SBI produces more reliable, better calibrated posteriors of the earthquake source mechanism. Finally, we successfully apply our methodology to two well studied moderate magnitude earthquakes: one from the 1997 Long Valley Caldera volcanic earthquake sequence, and the 2020 Zagreb earthquake.

Figures (12)

• Figure 1: Panel (a) shows the effect of $\kappa=3\%$ fractional perturbations to the layer depths, $V_p$ and $V_s$ of the Southern California 1-D velocity model dreger1990broadband. Panel (b) shows the resulting uncertainty in the observations for a single station component, as well as the variance in the seismograms (i.e. the diagonal of $\mathbf{C}_t$ in (c)). Panel (c) shows the three independent contributions to the Gaussian covariance $\mathbf{C}$. $\mathbf{C}_t(\mathbf{m})$ is computed for an event and receiver configuration similar to the LV2 earthquake studied in this work (see Sections \ref{['sec:synthetic_inversions']} & \ref{['sec:LV2']}). Note the differences in the colourbar scales, where the theory errors dominate for this moderate magnitude event.
• Figure 2: Illustration of the two SBI frameworks introduced in this work. Panel (a) shows optimal score compression, which projects the residuals $\mathbf{D} - \boldsymbol{\mu}_*$ for stations ST01, ST02, ST03, etc. onto error weighted sensitivity kernels $\mathbf{G}$ to produce compressed observations $\mathbf{t}$. Panel (b) instead uses a deep learning algorithm to learn a flexible, non-linear compression operation. A shared CNN processes full-waveforms into dense features with T time-steps and C feature channels. These station-level features are passed to an N-block axial transformer to aggregate information across the station array before producing a final compressed representation. Panel (c) visualises a normalizing flow architecture, used by both compression frameworks to perform empirical density modelling. The normalizing flow is trained to model the posterior over source parameters $q_\phi(\mathbf{m} \mid \mathbf{t})$ by transforming samples from a simple latent distribution $p_0(z)$ using learnable transformations $f^i_\phi$.
• Figure 3: Earthquake location (gold star), focal mechanism (red beachball), and station configuration (brown triangles) for the two events studied in this manuscript. Panel a) shows the LV2 volcanic event in California originally studied in dreger2000dilational, with focal mechanism solution from phạm2021toward. We also use this source-receiver configuration for the synthetic experiments in the main text. Panel b) shows the 2020 tectonic event near Zagreb, Croatia with focal mechanism solution from hu2025bayesian.
• Figure 4: A comparison between the analytically expected $\chi^2_\mathrm{red}$ statistic distribution against their empirical distributions under varying levels of Earth structure uncertainty ($\kappa \in [0.1,1,3,5]$). The degree of agreement between the analytic and empirical distributions probes how well the Gaussian likelihood models the observed variability in the data observations $\mathbf{D}$. We quantify this with the KS statistic in \ref{['eq:ks_stat']}. We find that even under very minor Earth structure perturbations ($\sim1\%$) a Gaussian likelihood becomes a very poor approximation.
• Figure 5: Examples of the per-trace $\chi^2_\mathrm{red}$ statistic under Earth structure perturbations for $\kappa =3\%$. Higher values of $\chi^2_\mathrm{red}$ are denoted by a brighter-copper colour, with significant departure from those expected under a Gaussian likelihood. Even under modest perturbations in the Earth structure, there are significant non-Gaussian effects in the observations.