Earthquakes

Abstract

Describing earthquake fault motion is indispensable to understanding the mechanism of tsunami generation. Moreover, seismic waves, excited by the fault motion, are analyzed in order to estimate the magnitude and location of earthquakes. The information is used to perform rapid tsunami calculations and predictions. At the same time, we should note that seismic waves sometimes function as noise among tsunami signals. This chapter introduces earthquake seismology, which is closely related to tsunami phenomena, and illustrates a practical method of seismic wave simulation. Section 4.1 explains a mathematical representation of an earthquake fault as a point source in order to quantitatively describe the relation between the fault motion and seismic waves. Section 4.2 explains an empirical scaling law representing the fault size from small to large earthquakes. We also introduce the idea of earthquake stress change (stress drop) as a mechanism behind the scaling law. Section 4.3 illustrates the finite difference method as a practical method of seismic wave simulation. By using this numerical method, we investigate seismic waves, ocean acoustic waves, and the permanent displacement caused by an earthquake. The simulation results can be used in the simulation of tsunami propagation.

Appendix A: Seismic Wave Propagation in 2-D Space: P-SV Problem

Seismic wave propagation in 2-D space is unrealistic. We should not directly compare the seismograms simulated in 2-D space with observed seismograms because the observed seismograms are in 3-D space. Nevertheless, the wave propagation in 2-D space is very useful for studies of the physical mechanism of seismic wave propagation. Since the 2-D and 3-D propagations have different dimensions, we need to carefully define each parameter and its units. This appendix illustrates a 2-D seismic wave propagation problem by deriving the propagation equations and source parameters from those of the 3-D simulation.

At the beginning, the 2-D space (x, z) is defined based on the 3-D space (x, y, z) as shown in Fig. 4.24. An earthquake fault is supposed in 3-D space with the strike and rake defined as ϕ_s = π/2 and λ = π/2, respectively (see also Fig. 4.14). Also, we assume that the fault length L is much longer than the fault width W. Then, the fault is located in the 3-D space (x, y, z) as shown in Fig. 4.24. In the (x, z) space, the fault is represented as a line segment characterized by the length (width) W and the dip δ. In the (x, y) space, the fault ranges from y = − L/2 to y = L/2 where L ≫ W. The moment time function is given by

$$ M(t)=\mu LWD(t) $$

(A.1)

where D(t) is the slip on the fault. Defining the 2-D space (x, z) at the plane y = 0 in the 3-D space (x, y, z), we consider the wave propagation in the 2-D space (x, z).

Fig. 4.24

Defining the 2-D (x, z) space from the 3-D space (x, y, z). We set the strike and rake to be ϕ_s = π/2 and λ = π/2, respectively, in Fig. 4.14. Also, the fault length L is much longer than the fault width W. We define the 2-D space (x, z) as the plane y = 0 in the 3-D space (x, y, z)

Due to the symmetry of the problem (now we suppose λ = π/2), the wavefield in the 2-D space (x, z) is represented by (v_x, v_z) and v_y = 0, and the spatial derivative with respect to the y-axis is zero ∂/∂y = 0. Then, the equation of motion (Eq. 4.51) is reduced to

$$ {\displaystyle \begin{array}{l}\rho \frac{\partial {v}_x}{\partial t}=\frac{\partial {\tau}_{xx}}{\partial x}+\frac{\partial {\tau}_{xz}}{\partial z}+{f}_{x,}\\ {}\rho \frac{\partial {v}_z}{\partial t}=\frac{\partial {\tau}_{zx}}{\partial x}+\frac{\partial {\tau}_{zz}}{\partial z}+{f}_z.\end{array}} $$

(A.2)

The stress components τ_xx, τ_xz = τ_zx, and τ_zz are given by the constitutive law Eq. (4.52) as

$$ {\displaystyle \begin{array}{c}\frac{\partial {\tau}_{xx}}{\partial t}=\left(\lambda +2\mu \right)\frac{\partial {v}_x}{\partial x}+\lambda \frac{\partial {v}_z}{\partial z},\\ {}\frac{\partial {\tau}_{zz}}{\partial t}=\lambda \frac{\partial {v}_x}{\partial x}+\left(\lambda +2\mu \right)\frac{\partial {v}_z}{\partial z},\\ {}\frac{\partial {\tau}_{zx}}{\partial t}=\mu \left(\frac{\partial {v}_z}{\partial x}+\frac{\partial {v}_x}{\partial z}\right).\end{array}} $$

(A.3)

Note that in our definition of the 2-D space, the dimensions of all the parameters in (4A.2) and (4A.3) are common with those in the 3-D space. For example, the unit of density ρ is [kg/m³], and that of the body force (f_x and f_z) is [N/m³] also in the 2-D problem.

For the 3-D space (x, y, z), the body force f_p(x, t) corresponding to the moment tensor M_pq(t) is given by (4.20):

$$ {f}_p\left(\mathbf{x},t\right)=-\frac{\partial }{\partial {x}_q}\left[{M}_{pq}(t)\delta \left(\mathbf{x}-\boldsymbol{\upxi} \right)\right]. $$

(4.20)

As shown in Fig. 4.25, the fault is divided into N subfaults along the fault length (along the y-axis). For the jth subfault, the centroid is located at (ξ_x, ξ_yj, ξ_z). The y coordinates ξ_yj change according to the index j. Each subfault is ΔL = L/N in length and W in width. The equivalent body force for the jth subfault f_{p, j}(x, t) is given by

$$ {f}_{p,j}\left(\mathbf{x},t\right)=\frac{1}{N}{f}_p\left(\mathbf{x},t;{\xi}_{yj}\right), $$

(A.4)

where

$$ {\displaystyle \begin{array}{ll}{f}_p\left(\mathbf{x},t;{\xi}_{yj}\right)=& -\frac{\partial }{\partial x}\left[{M}_{px}(t)\delta \left(x-{\xi}_x\right)\delta \left(y-{\xi}_{yj}\right)\delta \left(z-{\xi}_z\right)\right]\\ {}& -\frac{\partial }{\partial y}\left[{M}_{py}(t)\delta \left(x-{\xi}_x\right)\delta \left(y-{\xi}_{yj}\right)\delta \left(z-{\xi}_z\right)\right]\\ {}& -\frac{\partial }{\partial z}\left[{M}_{pz}(t)\delta \left(x-{\xi}_x\right)\delta \left(y-{\xi}_{yj}\right)\delta \left(z-{\xi}_z\right)\right].\end{array}} $$

(A.5)

Fig. 4.25

Dividing a fault into N small subfaults along the y-axis. The centroid of the jth subfault is located at (ξ_x, ξ_yj, ξ_z)

Summing up the contributions from each subfault, we obtain

$$ {\displaystyle \begin{array}{c}{f}_p\left(\mathbf{x},t\right)=\sum \limits_{j=1}^N{f}_{p,j}\left(\mathbf{x},t\right)=\sum \limits_{j=1}^N\frac{1}{N}{f}_p\left(\mathbf{x},t;{\xi}_{yj}\right)\\ {}=\sum \limits_{j=1}^N\frac{M(t)}{N}\frac{f_p\left(\mathbf{x},t;{\xi}_{yj}\right)}{M(t)}\\ {}=\mu DW\sum \limits_{j=1}^N\frac{L}{N}\frac{f_p\left(\mathbf{x},t;{\xi}_{yj}\right)}{M(t)}\\ {}=\mu DW\sum \limits_{j=1}^N\varDelta L\frac{f_p\left(\mathbf{x},t;{\xi}_{yj}\right)}{M(t)}.\end{array}} $$

(A.6)

Limiting N → ∞ gives

$$ {\displaystyle \begin{array}{l}{f}_p\left(\mathbf{x},t\right)=\mu DW{\int}_{-L/2}^{L/2}d{\xi}_y\frac{f_p\left(\mathbf{x},t;{\xi}_y\right)}{M(t)}\\ {}=\mu DW{\int}_{-L/2}^{L/2}d{\xi}_y\left\{-\frac{\partial }{\partial x}\left[\frac{M_{px}(t)}{M(t)}\delta \left(x-{\xi}_x\right)\delta \left(y-{\xi}_y\right)\delta \left(z-{\xi}_z\right)\right]\ \right.\\ {}-\frac{\partial }{\partial y}\left[\frac{M_{py}(t)}{M(t)}\delta \left(x-{\xi}_x\right)\delta \left(y-{\xi}_y\right)\delta \left(z-{\xi}_z\right)\right]\\ {}\left.-\frac{\partial }{\partial z}\left[\frac{M_{pz}(t)}{M(t)}\delta \left(x-{\xi}_x\right)\delta \left(y-{\xi}_y\right)\delta \left(z-{\xi}_z\right)\right]\right\}.\end{array}} $$

(A.7)

We calculate

$$ {\displaystyle \begin{array}{l}{f}_p\left(\mathbf{x},t\right)\\ {}=\mu DW\left\{-\frac{\partial }{\partial x}\left[\frac{M_{px}(t)}{M(t)}\delta \left(x-{\xi}_x\right)\delta \left(z-{\xi}_z\right)\right]\right\}{\int}_{-L/2}^{L/2}d{\xi}_y\delta \left(y-{\xi}_y\right)\\ {}+\mu DW\left\{-\left[\frac{M_{py}(t)}{M(t)}\delta \left(x-{\xi}_x\right)\delta \left(z-{\xi}_z\right)\right]\right\}{\int}_{-L/2}^{L/2}d{\xi}_y\frac{\partial }{\partial y}\delta \left(y-{\xi}_y\right)\\ {}+\mu DW\left\{-\frac{\partial }{\partial z}\left[\frac{M_{pz}(t)}{M(t)}\delta \left(x-{\xi}_x\right)\delta \left(z-{\xi}_z\right)\right]\right\}{\int}_{-L/2}^{L/2}d{\xi}_y\delta \left(y-{\xi}_y\right)\end{array}} $$

Finally, setting L → ∞, we obtain

$$ {f}_p\left(\mathbf{x},t\right)=-\frac{\partial }{\partial x}\left[\mu DW\frac{M_{px}(t)}{M(t)}\delta \left(x-{\xi}_x\right)\delta \left(z-{\xi}_z\right)\right]-\frac{\partial }{\partial z}\left[\mu DW\frac{M_{pz}(t)}{M(t)}\delta \left(x-{\xi}_x\right)\delta \left(z-{\xi}_z\right)\right]. $$

(A.8)

Since, in our problem, the subscript p takes the value p = x and z, substituting λ = π/2 and ϕ_s = π/2 into Eq. (4.62), we obtain

$$ {\displaystyle \begin{array}{l}\frac{M_{xx}}{M(t)}=-\sin 2\delta, \\ {}\frac{M_{xz}}{M(t)}=-\cos 2\delta, \\ {}\frac{M_{zz}}{M(t)}=\sin 2\delta .\end{array}} $$

(A.9)

Substituting Eqs. (A.9) into (4A.8) and comparing the resultant equation with Eq. (4.20), the moment tensor in the 2-D space (x, z) should be defined as

$$ {\displaystyle \begin{array}{l}{M}_{xx}^{2D}=-\upmu DW\sin 2\delta, \\ {}{M}_{xz}^{2D}={M}_{zx}^{2D}=-\upmu DW\cos 2\delta, \\ {}{M}_{zz}^{2D}=\upmu DW\sin 2\delta, \end{array}} $$

(A.10)

when the fault width is W and the dip is δ (as shown in Fig. 4.24). We should note that the dimension of the 2-D moment tensor defined in Eq. (A.10) is different from that in 3-D space. In the 2-D space, the moment is defined as M^2D = μDW. The units of M^2D are given by [N], whereas the moment in 3-D space is [N m].

In summary, for the 2-D space (x, z), the earthquake fault is described by the fault width W and the dip δ. The moment tensor [N] is given by Eq. (A.10). The equivalent body force [Nm⁻³] in the 2-D space is given by

$$ {f}_p\left(\mathbf{x},t\right)=-\frac{\partial }{\partial {x}_q}\left[{M}_{pq}^{2D}\left(\boldsymbol{\upxi}, t\right)\delta \left(\mathbf{x}-\boldsymbol{\upxi} \right)\right], $$

(A.11)

which is in the same form as Eq. (4.20) in 3-D space, but the delta function is defined in 2-D space, δ(x − ξ) = δ(x − ξ_x)δ(z − ξ_z). The equation of motion and the constitutive law are given by Eqs. (A.2) and (A.3), respectively.