4:00 PM - 4:15 PM
[SSS05-08] Rupture propagation modeling based on a stochastic wave equation
★Invited Papers
Keywords:Dynamic rupture simulation, Faulting process, Stochastic differential equation, Mathematical seismologiy
Many seismic inversion results have revealed that on-fault rupture patterns show great variation, implying that modeling them by a deterministic approach with simple boundary conditions is unsuitable. Some kinds of stochastic approaches to represent rupture and slip propagation were suggested: a fractal description[Andrews 1980 BSSA], a spatial random patch model[Ide&Aochi 2005 GJI], a Langevin equation model[Wu&Chen 2018 arXiv], probabilistic cell automaton (PCA)[Ide&Yabe 2019 PAGEOPH], and a boundary integral equation model with random noise[Aso+2019 GRL]. The above models succeeded in representing some nature of earthquake faulting processes informed by inversions; however, how to interpret them in mathematical frameworks that enable us to extract more features thanks to some well-known theories still needs to be investigated. More recently, Hirano[2002 Sci.Rep.] suggested a point source model based on a stochastic differential equation to satisfy many empirical laws of source time functions, including non-negativity, finite duration, ω-2-like spectra, t3-moment growth rate, and the Gutenberg-Richter law. Extending the point source model along a fault surface may deepen our understanding of some nature of rupture propagation in space and time.
To that end, we consider a stochastic wave equation derived from the point source model. The original point source model represented source time functions by convolving two solutions for the Bessel process, a stochastic ordinary differential equation. Instead, we assume that a spatiotemporal distribution of fault slip (D) and slip rate (V:=∂tD) as a random variable and that V is a squared Bessel process with the Laplacian of slip distribution (ΔD) and a non-linear multiplicative noise (√V dW). On a 1-D fault for simplicity, as in a numerical example in Figure 1, this model reproduces 1) macroscopic dynamic rupture propagation speed close to (but less than) the wave velocity, 2) backpropagation of slipping region as reported by inversion analyses[e.g., Hicks+2020 Nat.Geos.], 3) spontaneous termination of the rupture propagation, and 4) k-2-falloff rate of the final slip distribution spectrum, where k is the wavenumber.
We investigate the mathematical aspects and numerical simulation of the model. The equation differs from a PCA model representing slow earthquakes[Ide&Yabe 2019 PAGEOPH]; our model shows rupture propagation rather than diffusion, thanks to the Laplacian term. Given a partial differential equation, our model is also similar to Aso+[2019 GRL], where a space-time Gaussian noise is added to the boundary integral equation. However, in our model, the noise amplitude is proportional to the square root of the slip velocity, which enhances random rupture propagation when the slip velocity accelerates, while the equation of Aso+[2019 GRL] includes stationary noise. We demonstrate how our model behaves, compare the results to empirical laws from inversion, and discuss mathematical backgrounds.
To that end, we consider a stochastic wave equation derived from the point source model. The original point source model represented source time functions by convolving two solutions for the Bessel process, a stochastic ordinary differential equation. Instead, we assume that a spatiotemporal distribution of fault slip (D) and slip rate (V:=∂tD) as a random variable and that V is a squared Bessel process with the Laplacian of slip distribution (ΔD) and a non-linear multiplicative noise (√V dW). On a 1-D fault for simplicity, as in a numerical example in Figure 1, this model reproduces 1) macroscopic dynamic rupture propagation speed close to (but less than) the wave velocity, 2) backpropagation of slipping region as reported by inversion analyses[e.g., Hicks+2020 Nat.Geos.], 3) spontaneous termination of the rupture propagation, and 4) k-2-falloff rate of the final slip distribution spectrum, where k is the wavenumber.
We investigate the mathematical aspects and numerical simulation of the model. The equation differs from a PCA model representing slow earthquakes[Ide&Yabe 2019 PAGEOPH]; our model shows rupture propagation rather than diffusion, thanks to the Laplacian term. Given a partial differential equation, our model is also similar to Aso+[2019 GRL], where a space-time Gaussian noise is added to the boundary integral equation. However, in our model, the noise amplitude is proportional to the square root of the slip velocity, which enhances random rupture propagation when the slip velocity accelerates, while the equation of Aso+[2019 GRL] includes stationary noise. We demonstrate how our model behaves, compare the results to empirical laws from inversion, and discuss mathematical backgrounds.