Decoding Earthquakes with AI: A New Approach to Moment Tensor Solutions

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Author: Denis Avetisyan

A novel simulation-based inference method enhances the accuracy and reliability of earthquake source analysis, even with incomplete knowledge of Earth’s internal structure.

The calibration performance of Gaussian likelihood inversions proves robust across a range of Earth structure uncertainties-quantified by fractional perturbations κ-and benefits significantly from balanced experimental networks with full azimuthal coverage, though incorporating shorter periods into the inversion process further enhances accuracy.
The calibration performance of Gaussian likelihood inversions proves robust across a range of Earth structure uncertainties-quantified by fractional perturbations κ-and benefits significantly from balanced experimental networks with full azimuthal coverage, though incorporating shorter periods into the inversion process further enhances accuracy.

This study demonstrates improved moment tensor inversion using Bayesian inference and full-waveform data, addressing limitations of traditional Gaussian likelihood methods under Earth structure uncertainty.

Accurate characterization of earthquake source mechanisms relies on incorporating Earth structure uncertainties, yet traditional Bayesian moment tensor inversions often rely on simplifying assumptions that can introduce bias. This study, ‘Improving moment tensor solutions under Earth structure uncertainty with simulation-based inference’, introduces a robust framework leveraging simulation-based inference (SBI) to model theory errors and improve the reliability of moment tensor solutions. Results demonstrate that SBI significantly reduces bias and better calibrates uncertainty estimates compared to conventional Gaussian likelihood approaches, particularly for shallow events and short-period data. Could this methodology offer a pathway towards more robust earthquake hazard assessment and a deeper understanding of fault rupture processes?

Pinpointing the Earthquake’s Signature: Beyond Location

The precise calculation of an earthquake’s Moment Tensor Solution represents a fundamental step in characterizing seismic events and mitigating their potential hazards. This solution doesn’t simply pinpoint where an earthquake occurred, but also reveals how the Earth deformed during the rupture – detailing the fault’s orientation, the type of movement (compression, tension, or shear), and the magnitude of the energy released. Accurate moment tensor estimations are vital for advancing the understanding of earthquake physics, including the processes driving fault slip and the mechanisms behind complex rupture patterns. Furthermore, these solutions directly feed into seismic hazard assessments, informing ground motion predictions and ultimately contributing to the development of earthquake-resistant infrastructure and effective early warning systems. A robust moment tensor is, therefore, not merely a descriptive tool, but a cornerstone of both scientific inquiry and practical disaster preparedness.

Conventional earthquake source estimation frequently employs simplified Earth models and assumes a Gaussian Likelihood function to determine parameters like fault orientation and magnitude. However, this approach can introduce significant biases, especially when dealing with complex rupture scenarios – events involving heterogeneous fault slip, multiple sub-faults, or intricate velocity structures. The Gaussian Likelihood assumes errors are normally distributed, a condition often not met in real-world seismic data due to noise, incomplete station coverage, and the inherent complexity of wave propagation. Consequently, estimated source parameters may deviate substantially from the true values, impacting our understanding of earthquake physics and potentially leading to inaccurate hazard assessments. These biases are not merely statistical curiosities; they represent systematic errors that can distort interpretations of fault behavior and undermine the reliability of seismic monitoring efforts.

The precision with which earthquake source parameters can be determined is surprisingly sensitive to inaccuracies in models of the Earth’s interior. Recent studies demonstrate that even relatively minor perturbations – as little as a 5% uncertainty – in the established velocity structure of the Earth can introduce substantial errors in calculated seismic moments, fault orientations, and rupture processes. This is because seismic waves travel through the Earth and are affected by variations in material properties; imprecise knowledge of these properties leads to inaccuracies in waveform modeling and, consequently, in the inversion for the earthquake source. The effect isn’t simply additive; these uncertainties can systematically bias the derived parameters, leading to misinterpretations of earthquake physics and potentially flawed hazard assessments. Therefore, continued refinement of Earth models, incorporating data from diverse geophysical sources, is paramount to improving the reliability of earthquake source estimation and a more complete understanding of seismic events.

The Kolmogorov-Smirnov statistic reveals that even small perturbations (<span class="katex-eq" data-katex-display="false">\sim 1\%</span>) to Earth structure significantly degrade the Gaussian likelihood's ability to model observed data variability, as evidenced by discrepancies between expected and empirical <span class="katex-eq" data-katex-display="false">\chi^{2}_{red}</span> distributions.
The Kolmogorov-Smirnov statistic reveals that even small perturbations (\sim 1\%) to Earth structure significantly degrade the Gaussian likelihood’s ability to model observed data variability, as evidenced by discrepancies between expected and empirical \chi^{2}_{red} distributions.

Embracing Uncertainty: A Principled Approach to Inference

Bayesian inference is utilized within full-waveform moment tensor inversions to formally address the impact of Earth structure uncertainty. Traditional methods often assume a fixed, known Earth model; however, this approach introduces error due to real-world velocity variations. Bayesian inference treats the Earth model itself as a probability distribution, allowing for the simultaneous estimation of both the moment tensor and the parameters defining the Earth structure. This is achieved through the application of Bayes’ theorem, which combines a prior probability distribution representing existing knowledge of Earth structure with a likelihood function quantifying the fit of the waveform data given a particular Earth model and moment tensor. The resulting posterior probability distribution then provides a more complete characterization of solution uncertainty, accounting for both data misfit and model parameters.

Traditional full-waveform moment tensor inversions often rely on a Gaussian likelihood function, which assumes the Earth model parameters are fixed but unknown. This assumption is limiting as it does not account for inherent uncertainties and variability in Earth structure. Bayesian inference, conversely, explicitly models the Earth model as a random variable, characterized by a prior probability distribution reflecting existing geological knowledge. This probabilistic treatment allows for the quantification of uncertainty in the Earth model parameters themselves, and propagates that uncertainty through the inversion process. By treating the Earth model as a probability distribution rather than a single fixed value, the Bayesian approach provides a more complete and realistic representation of the solution space and associated uncertainties.

Traditional moment tensor inversions often assume a fixed, known Earth structure, leading to inaccuracies when this assumption is violated. Bayesian inference directly addresses this limitation by treating Earth structure parameters as random variables with associated probability distributions. This allows the inversion process to integrate over a range of plausible Earth models, weighted by their probability, rather than relying on a single, potentially inaccurate model. Consequently, the resulting moment tensor solution reflects the uncertainty in the Earth structure, providing more realistic estimates of source parameters, including magnitude, depth, and fault mechanism, and quantifying the associated uncertainties through the posterior probability distribution of the moment tensor.

Calibration analysis of three inference methods, performed over 400 inversions, reveals that score and ML-based compression SBI methods produce more faithful and unbiased posterior contours than the Gaussian likelihood approach, with uncertainties estimated via bootstrapping.
Calibration analysis of three inference methods, performed over 400 inversions, reveals that score and ML-based compression SBI methods produce more faithful and unbiased posterior contours than the Gaussian likelihood approach, with uncertainties estimated via bootstrapping.

Modeling Complexity: Simulation-Based Inference in Action

Simulation-Based Inference (SBI) is employed to model the interrelationship between Earth structure parameters, observed seismic data, and the resulting Moment Tensor Solution, which describes the magnitude and orientation of a seismic event. This approach circumvents the need for pre-defined parametric relationships, instead learning the complex mapping directly from simulation outputs. SBI treats the forward problem – simulating seismic data given Earth structure – as a probabilistic model, allowing for quantification of uncertainty in both the Earth structure and the estimated Moment Tensor Solution. By framing the inference problem within a probabilistic framework, SBI enables a comprehensive assessment of model fit and the exploration of multiple plausible Earth structures consistent with the observed data.

Traditional methods for determining Earth structure and the associated Moment Tensor Solution often rely on simplifying assumptions to render the inverse problem tractable. These assumptions, while computationally necessary, introduce potential biases and limit the accuracy of the resulting models. Simulation-Based Inference (SBI) circumvents this limitation by directly learning the complex relationship between Earth structure, seismic data, and the Moment Tensor Solution through numerical simulations. By foregoing the need for pre-defined parametric models or linearized approximations, SBI allows for a more comprehensive and potentially more accurate representation of the underlying physical processes, enabling the incorporation of complex geological features and data characteristics without requiring a priori constraints.

Full-Waveform Data, while containing significant information for characterizing Earth structure, presents computational challenges due to its high dimensionality. To address this, techniques such as Optimal Score Compression and Deep Learning Compression are employed to reduce the data’s dimensionality while retaining key information relevant to parameter estimation. The Deep Learning Compression model utilized in this process required 12 hours for training, after which it efficiently facilitates tractable computations for subsequent analyses. These compression methods allow for more efficient simulation-based inference by reducing the computational burden associated with processing large waveform datasets.

This work introduces two score-based inversion (SBI) frameworks: one utilizing optimal score compression by projecting station residuals onto error-weighted sensitivity kernels, and another employing a deep convolutional neural network and axial transformer to learn a non-linear compression from full waveforms, both leveraging normalizing flows to model the posterior distribution over source parameters.
This work introduces two score-based inversion (SBI) frameworks: one utilizing optimal score compression by projecting station residuals onto error-weighted sensitivity kernels, and another employing a deep convolutional neural network and axial transformer to learn a non-linear compression from full waveforms, both leveraging normalizing flows to model the posterior distribution over source parameters.

Beyond Simplification: Characterizing the Nuances of Faulting

The standard depiction of earthquakes relies on the ‘double-couple’ mechanism, envisioning fault rupture as a simple shear along a plane. However, many earthquakes exhibit complexities beyond this model, and this methodology provides a robust means to characterize the ‘non-double-couple component’ of the seismic moment tensor. This component arises from forces beyond pure shear – such as tensile stresses, bending moments, or volumetric changes – and its accurate assessment reveals previously hidden details of the faulting process. By quantifying these non-double-couple contributions, researchers gain insight into the diverse physical mechanisms driving earthquakes, potentially linked to fluid interactions, heterogeneous stress fields, or the geometry of fault junctions, thereby refining our understanding of earthquake rupture physics and improving seismic hazard assessments.

A robust moment tensor solution isn’t solely about finding a best-fit answer; it critically depends on accurately quantifying the uncertainty surrounding that solution. Recent advancements in statistical methodology, specifically through the implementation of Sequential Bayesian Inference (SBI), demonstrate a significant improvement in posterior calibration. This means the estimated uncertainties in the moment tensor solution more faithfully reflect the true range of plausible values. Compared to traditional methods relying on Gaussian likelihoods, SBI achieves substantially better statistical coverage – a key metric for evaluating the reliability of uncertainty estimates. This enhanced calibration is crucial for applications like seismic hazard assessment and earthquake source analysis, as it provides a more trustworthy basis for interpreting earthquake characteristics and predicting future seismic activity. Essentially, SBI offers not just a solution, but a quantified level of confidence in that solution, moving beyond simply identifying the most probable earthquake mechanism.

Characterizing shallow, isotropic seismic events – those lacking a clear directional source – presents a significant challenge in understanding near-source rupture physics, but recent advances are yielding improved insights. This work demonstrates a method, SBI, capable of analyzing these complex events with high fidelity, leading to more accurate moment tensor solutions even when faced with substantial model uncertainty – up to a 5% perturbation. By reducing bias in these solutions, researchers gain a clearer picture of the forces at play during earthquakes, refining models of fault behavior and ultimately contributing to more robust seismic hazard assessments. This enhanced capability is crucial for understanding rupture initiation and propagation in the immediate vicinity of the earthquake source, filling a critical gap in current seismological understanding.

Inversions of shallow isotropic sources using Gaussian likelihood (red), score-compression SBI (blue), and ML-based compression SBI (purple) successfully recover the true source mechanism (indicated by the orange diamond) within <span class="katex-eq" data-katex-display="false">\pm[1,2]\sigma</span> contours, as shown by the lune plots and accompanying fractional isotropic, CLVD, and double-couple decomposition.
Inversions of shallow isotropic sources using Gaussian likelihood (red), score-compression SBI (blue), and ML-based compression SBI (purple) successfully recover the true source mechanism (indicated by the orange diamond) within \pm[1,2]\sigma contours, as shown by the lune plots and accompanying fractional isotropic, CLVD, and double-couple decomposition.

The pursuit of accurate moment tensor solutions, as detailed in this work, demands acknowledging inherent uncertainties within Earth’s structure. Traditional methods often struggle with this complexity, relying on simplifying assumptions. This research demonstrates a shift towards embracing that complexity through simulation-based inference. It echoes a sentiment expressed by Ernest Rutherford: “If you can’t explain it simply, you don’t understand it well enough.” The study’s success lies not in ignoring structural uncertainty, but in systematically addressing it, ultimately yielding more reliable solutions. The core concept of Bayesian inference, coupled with SBI, provides a pathway to clarity amidst complexity.

Where Do We Go From Here?

The exercise, as presented, reveals a familiar truth: the more precisely one attempts to model a complex system, the more acutely one feels the absence of perfect knowledge. This work rightly prioritizes acknowledging Earth structure uncertainty-a step beyond merely estimating parameters, and towards embracing the inherent limitations of the estimation itself. They called it ‘simulation-based inference,’ but it often feels more like simulation-based honesty. The improvement over Gaussian likelihoods is not merely numerical; it’s a shift in perspective.

Yet, the problem isn’t solved, merely reframed. The computational expense of truly exhaustive simulation remains a considerable hurdle. One suspects future efforts will focus less on perfecting the forward problem-the wave propagation itself is, after all, just physics-and more on intelligent sampling strategies. A truly elegant solution will likely involve distilling the essential information from vast simulations, identifying the features that most strongly constrain the moment tensor, and discarding the rest. Simplicity, after all, is not a compromise; it’s the goal.

Ultimately, this work points toward a broader trend. The field is moving away from seeking ‘the’ solution, and towards quantifying the range of plausible solutions. It’s a subtle but vital distinction. The earthquake doesn’t care about one’s best estimate; it cares about the possibilities. And a clear-eyed assessment of those possibilities-however unsettling-is the most valuable information one can obtain.

Original article: https://arxiv.org/pdf/2603.18925.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2026-03-22 12:45