11:45 AM - 12:00 PM
[STT43-05] Acquisition of a Stochastic Differential Equation Representation for Slow Earthquakes via Deep Learning to Deepen Phenomenological Understanding
Keywords:Slow earthquakes, Signature, Stochastic differential equations, Deep learning, Tremors
Slow earthquakes are a sliding phenomenon that occur at speeds much slower than those of typical earthquakes, and they have been suggested to be associated with plate-boundary earthquakes. Consequently, they have been studied from various perspectives. Owing to their diffusive nature, slow earthquakes are thought to be mathematically modeled by stochastic differential equations (SDEs) driven by Brownian motion. In this study, we focus on the low-frequency tremors—primarily within the 2–8 Hz band—that are observable by seismometers to develop a mathematical model.
A pioneering model for low-frequency tremors is the Brownian Slow Earthquakes (BSE) model proposed by Ide [1]. The BSE model simplifies the slip area between plates that causes low-frequency tremors as a single circular seismic source, whose radius varies randomly according to an SDE, thereby generating seismic waves. This model not only reproduces the spectrum of low-frequency tremors but also, by estimating its parameters based on observational data from multiple sites, contributes significantly to our understanding of slow earthquakes by elucidating their regional characteristics [2].
On the other hand, the BSE model is described by a very simple SDE—specifically, an Ornstein-Uhlenbeck (OU) process with two parameters. While it can approximate the central features of low-frequency tremors, it does not account for influences such as fluid effects, which are considered important in the context of slow earthquakes. Furthermore, although the BSE model is designed for a single observation point, it is known—and actively studied—that the envelopes of slow earthquake waveforms exhibit correlations across multiple observation points, rendering models that incorporate multiple sites highly meaningful.
Therefore, to flexibly extend the BSE model, we propose an approach that obtains an SDE model in a data-driven manner from observed low-frequency tremor waveforms. Specifically, we model the drift term (which represents the average behavior) and the diffusion term (which represents the magnitude of fluctuations) of the SDE using neural networks, and we employ deep learning based on real data to capture their dynamics flexibly. We then discuss the physical properties of the resulting model and examine in detail the differences between models obtained at different locations. Moreover, we extend the one-dimensional SDE model—corresponding to a single observation point—to a multi-dimensional SDE model that accounts for clusters of waveforms from several nearby observation points, and we discuss the correlations between the drift and diffusion terms across different locations. Additionally, our study adopts an SDE learning algorithm based on the Signature method, which was proposed by Issa et al. (2024) [3] and excels in feature extraction from time-series data. By independently modifying this algorithm to ensure that it maintains physical consistency during training, we have successfully developed a method that captures the detailed characteristics of slow earthquakes and is readily applicable to multi-dimensional time-series data.
[1]Ide, S. (2008), A Brownian walk model for slow earthquakes, Geophys. Res. Lett., 35, L17301, doi:10.1029/2008GL034821.
[2] Ide, S., & Maury, J. (2018). Seismic moment, seismic energy, and source duration of slow earthquakes: Application of Brownian slow earthquake model to three major subduction zones. Geophysical Research Letters, 45(7), 3059-3067.
[3] Issa, Z., Horvath, B., Lemercier, M., & Salvi, C. (2024). Non-adversarial training of Neural SDEs with signature kernel scores. Advances in Neural Information Processing Systems, 36.
A pioneering model for low-frequency tremors is the Brownian Slow Earthquakes (BSE) model proposed by Ide [1]. The BSE model simplifies the slip area between plates that causes low-frequency tremors as a single circular seismic source, whose radius varies randomly according to an SDE, thereby generating seismic waves. This model not only reproduces the spectrum of low-frequency tremors but also, by estimating its parameters based on observational data from multiple sites, contributes significantly to our understanding of slow earthquakes by elucidating their regional characteristics [2].
On the other hand, the BSE model is described by a very simple SDE—specifically, an Ornstein-Uhlenbeck (OU) process with two parameters. While it can approximate the central features of low-frequency tremors, it does not account for influences such as fluid effects, which are considered important in the context of slow earthquakes. Furthermore, although the BSE model is designed for a single observation point, it is known—and actively studied—that the envelopes of slow earthquake waveforms exhibit correlations across multiple observation points, rendering models that incorporate multiple sites highly meaningful.
Therefore, to flexibly extend the BSE model, we propose an approach that obtains an SDE model in a data-driven manner from observed low-frequency tremor waveforms. Specifically, we model the drift term (which represents the average behavior) and the diffusion term (which represents the magnitude of fluctuations) of the SDE using neural networks, and we employ deep learning based on real data to capture their dynamics flexibly. We then discuss the physical properties of the resulting model and examine in detail the differences between models obtained at different locations. Moreover, we extend the one-dimensional SDE model—corresponding to a single observation point—to a multi-dimensional SDE model that accounts for clusters of waveforms from several nearby observation points, and we discuss the correlations between the drift and diffusion terms across different locations. Additionally, our study adopts an SDE learning algorithm based on the Signature method, which was proposed by Issa et al. (2024) [3] and excels in feature extraction from time-series data. By independently modifying this algorithm to ensure that it maintains physical consistency during training, we have successfully developed a method that captures the detailed characteristics of slow earthquakes and is readily applicable to multi-dimensional time-series data.
[1]Ide, S. (2008), A Brownian walk model for slow earthquakes, Geophys. Res. Lett., 35, L17301, doi:10.1029/2008GL034821.
[2] Ide, S., & Maury, J. (2018). Seismic moment, seismic energy, and source duration of slow earthquakes: Application of Brownian slow earthquake model to three major subduction zones. Geophysical Research Letters, 45(7), 3059-3067.
[3] Issa, Z., Horvath, B., Lemercier, M., & Salvi, C. (2024). Non-adversarial training of Neural SDEs with signature kernel scores. Advances in Neural Information Processing Systems, 36.