A Fault Model with Two Asperities of Different Areas and Strengths

Abstract

A fault with two asperities with different areas and strengths is considered. The fault is treated as a dynamical system with two state variables (the slip deficits of the asperities) and four dynamic modes, for which complete analytical solutions are provided. The seismic events generated by the fault can be discriminated in terms of a variable related with the difference between the slip deficits of the asperities at the beginning of the interseismic interval preceding the event. The effect of the difference between the asperity areas on several features of the model, such as the force rates on the asperities, the slip duration and amplitude, the occurrence of events involving the simultaneous motion of the asperities and the radiation of elastic waves, is discussed. As an application, the $M_w$ 8.0 2007 Pisco, Peru, earthquake is considered: it is modelled as a two-mode event due to the consecutive failure of two asperities, one almost twice as large as the other. The source function and final seismic moment predicted by the model are found to be in good agreement with observations.

Constants in the Solution for Mode 11

1.1 Case $10 \rightarrow 11$

The initial conditions are

$$\begin{aligned} X(0) = {\bar{X}}, \quad Y(0) = {\bar{Y}}, \quad \dot{X} (0)= {\bar{V}}, \quad \dot{Y} (0) = 0, \end{aligned}$$

(A.1)

where ${\bar{X}}$ and ${\bar{Y}}$ are the slip deficits of asperities 1 and 2, respectively, when the asperities start slipping together. Asperity 1, which was already slipping at the onset of mode 11, is associated with a certain rate ${\bar{V}}$, whereas asperity 2 is initially stationary. The slip deficits ${\bar{X}}$ and ${\bar{Y}}$ satisfy the Eq. (24) of Line 2. The constants are

$$\begin{aligned} A= & {} {1 \over \omega _a} \left( {\gamma \over 2} B + {1 \over 1 + \xi } {\bar{V}} \right) ,\end{aligned}$$

(A.2)

$$\begin{aligned} B= & {} {1 \over 1 + \xi } \left[ {\bar{X}} + \xi {\bar{Y}} - \epsilon \left( 1 + \beta \xi \right) \right] , \end{aligned}$$

(A.3)

$$\begin{aligned} C= & {} {1 \over \omega _b} \left( {\gamma \over 2} D - {\xi \over 1 + \xi } {\bar{V}} \right) , \end{aligned}$$

(A.4)

$$\begin{aligned} D= & {} {\xi \over 1 + \xi } \left[ {\bar{X}} - {\bar{Y}} - \epsilon \left( X_P - Y_P \right) \right] . \end{aligned}$$

(A.5)

1.2 Case $01 \rightarrow 11$

The initial conditions are

$$\begin{aligned} X(0) = {\bar{X}}, \quad Y(0) = {\bar{Y}}, \quad \dot{X} (0)= 0, \qquad \dot{Y} (0) = {\bar{V}} , \end{aligned}$$

(A.6)

where ${\bar{X}}$ and ${\bar{Y}}$ are the slip deficits of asperities 1 and 2, respectively, when the asperities start slipping together. Asperity 2, which was already slipping at the onset of mode 11, is associated with a certain rate ${\bar{V}}$, whereas asperity 1 is initially stationary. The slip deficits ${\bar{X}}$ and ${\bar{Y}}$ satisfy the Eq. (23) of Line 1. The constants are

$$\begin{aligned} A= & {} {1 \over \omega _a} \left( {\gamma \over 2} B + {\xi \over 1 + \xi } {\bar{V}} \right) , \end{aligned}$$

(A.7)

$$\begin{aligned} B= & {} {1 \over 1 + \xi } \left[ {\bar{X}} + \xi {\bar{Y}} - \epsilon \left( 1 + \beta \xi \right) \right] , \end{aligned}$$

(A.8)

$$\begin{aligned} C= & {} {1 \over \omega _b} \left( {\gamma \over 2} D - {\xi \over 1 + \xi } {\bar{V}} \right) , \end{aligned}$$

(A.9)

$$\begin{aligned} D= & {} {\xi \over 1 + \xi } \left[ {\bar{X}} - {\bar{Y}} - \epsilon \left( X_P - Y_P \right) \right] . \end{aligned}$$

(A.10)

1.3 Case $00 \rightarrow 11$

The initial conditions are

$$\begin{aligned} X(0) = X_P, \quad Y(0) = Y_P, \quad \dot{X} (0)= 0, \quad \dot{Y} (0) = 0 , \end{aligned}$$

(A.11)

where it was taken into account that P is defined as the state of the fault that satisfies the conditions for the failure of asperities 1 and 2 at the same time. Both asperities are initially stationary and their velocities are null at the beginning of the seismic event. The constants are

$$\begin{aligned} A= & {} {\gamma \over 2 \omega _a} B , \end{aligned}$$

(A.12)

$$\begin{aligned} B= & {} {1 \over 1 + \xi } \left( 1 - \epsilon \right) \left( 1 + \beta \xi \right) , \end{aligned}$$

(A.13)

$$\begin{aligned} C= & {} {\gamma \over 2 \omega _b} D , \end{aligned}$$

(A.14)

$$\begin{aligned} D= & {} {\xi \over 1 + \xi } \left( 1 - \epsilon \right) \left( X_P - Y_P \right) . \end{aligned}$$

(A.15)