BAYESIAN NEURAL NETWORK PARAMETERS PROVIDE INSIGHTS INTO THE EARTHQUAKE RUPTURE PHYSICS

Abstract

I present a simple but informative approach to gain insight into the Bayesian neural network (BNN) trained parameters. I used 2000 dynamic rupture simulations to train a BNN model to predict if an earthquake can break through a simple 2D fault. In each simulation, fault geometry, stress conditions, and friction parameters vary. The trained BNN parameters show that the network learns the physics of earthquake rupture. Neurons with high positive weights contribute to the earthquake rupture and vice versa. The results show that the stress condition of the fault plays a critical role in determining its strength. The stress is also the top source of uncertainty, followed by the dynamic friction coefficient. When stress and friction drop of a fault have higher value and are combined with higher weighted neurons, the prediction score increases, thus fault likely to be ruptured. Fault's width and height have the least amount of uncertainty, which may not be correct in a real scenario. The study shows that the potentiality of BNN that provides data patterns about rupture physics to make an additional information source for scientists studying the earthquake rupture.

1. INTRODUCTION

Because of the limited observational data and computational cost, geoscientists often rely on simple low-resolution simulations to study physical systems such as dynamic earthquake rupture, long-term tectonic process, etc. Such simplified models are indeed a powerful tool beside the observational data but sometimes cannot capture the proper physics of the system. As a result, it becomes difficult to accurately identify and understand the underlying causes. Machine learning (ML) approaches have been successfully used to solve many such geophysical problems with limited data and require computational overhead. For example, Ahamed & Daub (2019) used neural network and random forest algorithms to predict if an earthquake can break through a fault with geometric heterogeneity. The authors used 1600 simulated rupture data points to train the models. They identified several patterns responsible for earthquake rupture. Machine learning approaches are also used in seismic event detection (Rouet-Leduc et al., 2017) , earthquake detection (Perol et al., 2018) , identifying faults from unprocessed raw seismic data (Last et al., 2016) and to predict broadband earthquake ground motions from 3D physics-based numerical simulations (Paolucci et al., 2018) . All the examples show the potential application ML to solve many unsolved geophysical problems. However, the machine learning model's performance usually depends on the quality and quantity of data. Bad quality or insufficient data increases the uncertainty of the predictions (Hoeting et al., 1999; Blei et al., 2017; Gal et al., 2017) . Therefore, estimating the source of uncertainty is vital to understanding the physics of earthquake rupture and seismic risk. On top of that black-box nature of the ML algorithms inhibits mapping the input features with model output prediction. As a result, it becomes challenging for scientists to make actionable decisions. To overcome insufficient earthquake rupture data, I used the Bayesian neural network algorithm to develop a model reusing the simulations data of Ahamed & Daub (2019) . I present an exciting approach to learning the patterns of earthquake ruptures from the trained model parameters. Unlike regular neural networks, BNN works better with a small amount of data and provides prediction uncertainty. The approach gives more information on rupture physics than the traditional geophys-ical methods. I also describe the workflow of (1) developing a BNN and (2) estimating prediction uncertainty. Figure 1a shows the zoomed view of the original domain for better visualization of the fault barrier.

Nucleation

Rupture is nucleated 10 km to the left of the barrier and propagates towards the barrier. In each simulation, eight parameters were varied: x and y components of normal stress (sxx and syy), shear stress (sxy), dynamic friction coefficient, friction drop (µ sµ d ), critical slip distance (d c ), and width and height of the fault. The fault starts to break when the shear stress (τ ) on the fault exceeds the peak strength τ s = µ s σ n , where µ s and σ n are the static friction coefficient and normal stress, respectively. Over a critical slip distance d c , the friction coefficient reduces linearly to constant dynamic friction µ d . 1600 simulation data points were used to train, and 400 were used to test the model performance. The training dataset has an imbalance class proportion of rupture arrest (65%) and rupture propagation (35%). To avoid a bias toward rupture arrest, I upsampled the rupture propagation examples. Before training, all the data were normalized by subtracting the mean and dividing by the standard deviation.

3. BAYESIAN NEURAL NETWORK

In a traditional neural network, weights are assigned as a single value or point estimate. In a BNN, weights are considered as a probability distribution. These probability distributions are used to estimate the uncertainty in weights and predictions. Figure 2 shows a schematic diagram of a BNN where weights are normally distributed. The posterior network parameters are calculated using the following equation: P (W |X) = P (X|W )P (W ) P (X) (1) Where X is the data, P (X|W ) is the likelihood of observing X, given weights (W ). P (W ) is the prior belief of the weights, and the denominator P (X) is the probability of data which is also known as evidence. The equation requires integrating over all possible values of the weights as:



Figure1: (a) A zoomed view of the two-dimensional fault geometry. The domain is 32 km long along the strike of the fault and 24 kilometers wide across the fault. The rupture starts to nucleate 10 km to the left of the barrier and propagates from the hypocenter towards the barrier, (b) Linear slip-weakening friction law for an earthquake fault. The fault begins to slip when the shear stress reaches or exceeds the peak strength of τ s . τ s decreases linearly with slip to a constant dynamic friction τ d over critical slip distance (d c ). The shear strength is linearly proportional to the normal stress σ n , and the friction coefficient varies with slip between µ s and µ d . I used the simulated earthquake rupture dataset created byAhamed & Daub (2019). The simulations are a two-dimensional rupture, illustrated in figure. 1. The domain is 32 km long and 24 km wide. Figure1ashows the zoomed view of the original domain for better visualization of the fault barrier. Rupture is nucleated 10 km to the left of the barrier and propagates towards the barrier. In each simulation, eight parameters were varied: x and y components of normal stress (sxx and syy), shear stress (sxy), dynamic friction coefficient, friction drop (µ sµ d ), critical slip distance (d c ), and width and height of the fault. The fault starts to break when the shear stress (τ ) on the fault exceeds the peak strength τ s = µ s σ n , where µ s and σ n are the static friction coefficient and normal stress, respectively. Over a critical slip distance d c , the friction coefficient reduces linearly to constant dynamic friction µ d .

