A Stochastic Dynamics Model for Earthquake Rupture

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姓名

吳宗羲(Tsung-Hsi Wu) 查詢紙本館藏

畢業系所

地球科學學系

論文名稱

(A Stochastic Dynamics Model for Earthquake Rupture)

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摘要(中)

在自然科學領域，隨機模型(stochastic model)被廣泛應用於描述系統的定量與定性關係。由於地震破裂的複雜性與尺度，決定性的力學方法存在著許多限制。有別於決定性的模型，本研究提出一個本質上包含隨機因子的統計力學形式之地震破裂過程之模型，探討地震滑移背後可能的物理機制。
Thingbaijam 及 Mai (2016) 將世界各地的地震破裂模型之樣式公式化，展示地震各地之滑移分布遵守指數截略分布(Truncated Exponential distribution, TEX)，惟迄今仍缺乏物理定律說明此現象。受到該研究的啟發，我們提出一個基於朗之萬方程式(Langevin equations)之隨機過程模型，其所產生之破裂滑移分布在統計性質上與世界各地的地震事件觀測一致，符合Thingbaijam 及 Mai (2016) 研究結果。
本研究主張在一個地震中，穩態破裂過程可由朗之萬方程式描述，並從此觀點重新詮釋Thingbaijam and Mai (2016)的研究結果，如下所列：
TEX擬合參數 u_c 與平均滑移量u_avg相依。
於破裂模型邊緣裁切能增進滑移分布對TEX函數的擬合結果。
地震破裂滑移的TEX分布。

研究結果展示特定朗之萬方程式於有限穩態過程的解析解與地震滑移之指數截略分布(TEX)相同，TEX函數中的擬合參數 u_c 可與朗之萬方程式中的擴散係數與阻尼係數(diffusion and damping coefficient)之比值相關聯。結果暗示著朗之萬方程式可能是主導地震破裂過程的關鍵，而方程式背後存在多種可能的物理意涵，提供我們進一步發掘隨機破裂過程背後物理意義之線索。

摘要(英)

Due to the capability of simplifying chaotic-like dynamics, stochastic modeling approaches have been widely applied in many domains of natural science. In this study, we propose a stochastic dynamics model for earthquake rupture that intrinsically includes fluctuations in the environment as well as uncertainties in the heterogeneity of the faulting plane in the random variable, and reveal that the responsible Langevin equation (LE) may have a fundamental role in interpreting the physics of earthquake rupture process. Specifically, both analytical and numerical solutions of the governing equation meet the empirically observed Truncated Exponential (TEX) feature of rupture slip distribution. We claim that the stationary part of the earthquake rupture process is governed by the Langevin equation, and accordingly explain the phenomena/facts listed below: (1) The scaling relationship between the parameter uc of TEX and the average co-seismic slips uavg. (2) The overall improvement of the goodness-of-fit of TEX resulting from trimming the original rupture model at edges. (3) The commonly observed truncated exponential distribution for earthquake rupture slips.
The result of this study further implies that the fitting parameter uc in TEX is directly related to the ratio of the diffusion and friction coefficient of the Langevin equation, and gives us clues for investigating the physically-based laws underpinning the stochastic rupture process.

關鍵字(中)

★ 地震
★ 隨機
★ 破裂
★ 動力學

關鍵字(英)

★ stochastic
★ earthquake
★ rupture
★ dynamics

論文目次

Table of Contents
中文摘要 i
Abstract ii
List of frequently used symbols vi
Chapter 1 Introduction 1
1-1 Purpose and motivation 1
1-2 Literature review 3
1-2-1 The statistical behavior of earthquake rupture models 3
1-2-2 The study of stochastic process 6
Chapter 2 Stochastic process 8
Frequently used symbols / Brief introduction to terminology in this chapter 8
2-1 Introduction to random process 9
2-1-1 random variables 9
2-1-2 The expectation of random variables 10
2-1-3 Stochastic process 10
2-1-4 Stationary process 11
2-1-5 Ergodicity 11
2-1-6 Wiener Process 12
2-1-7 Markov process 14
2-1-8 Stochastic Differential Equation 14
2-2 Brownian motion 15
2-2-1 The Brownian motion in the Langevin description 15
2-2-2 The general Langevin equation 19
2-2-3 The stationarity of Brownian motion 19
2-3 Equivalence of Langevin equation and Fokker-Planck equation 20
2-3-1 The master equation 21
2-3-2 The Kramers-Moyal expansion of the master equation 22
2-3-3 The 1st and 2nd order of the jump moment 24
2-3-4 The Fokker-Planck equation 31
Chapter 3 The data processing for rupture model 32
3-1 The truncated exponential distribution for earthquake rupture 32
3-2 TEX fitting with rupture models 35
3-2-1 Data selection 35
3-2-2 Method of data fitting 35
Chapter 4 Stochastic Dynamics for Frictions 37
4-1 The governing equations 37
4-1-1 The Langevin equation 37
4-1-2 The Fokker-Planck equation 39
4-2 Solutions of stochastic dynamic frictions 40
4-2-1 The probability distribution 40
4-2-2 Numerical simulation method 42
4-2-3 The time- and ensemble-averaged PDF 46
4-2-4 Numerical simulation constrained by real observations 49
Chapter 5 Discussion and conclusion 53
5-1-1 The data trimming effect in the study of Thingbaijam and Mai 53
5-1-2 Earthquake size, duration, and maximum source slip 54
5-1-3 The meaning of fitting parameter uc in a TEX distribution 55
5-2 Conclusion and future work 57
Reference 58
Appendix 66
Other results of stochastic earthquake rupture 66
Table 1: List of total 177 reference rupture models from SRCMOD database 74

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指導教授

陳建志

審核日期

2018-7-10

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