Green’s Functions for Post-seismic Strain Changes in a Realistic Earth Model and Their Application to the Tohoku-Oki M_w 9.0 Earthquake

Abstract

Based on a spherically symmetric, self-gravitating viscoelastic Earth model, we derive a complete set of Green’s functions for the post-seismic surface strain changes for four independent dislocation sources: strike-slip, dip-slip, and horizontal and vertical tensile point sources. The post-seismic surface strain changes caused by an arbitrary earthquake can be obtained by a combination of the above Green’s functions. The post-seismic surface strain changes in the near field agree well with the results calculated by the method in a half-space Earth model (Wang et al. in Comupt Geosci 32:527–541, 2006), which verifies our Green’s functions. With an increase in the epicentral distance, the effect of the curvature on both the co- and post-seismic strain changes clearly increases, revealing the importance of our spherical theory for far-field calculations. Next, we use our Green’s functions to simulate the post-seismic surface strain changes that were caused by the viscoelastic relaxation of the mantle over the 6-year period after the Tohoku-Oki M_w 9.0 earthquake. Based on continuous Global Positioning System (GPS) observations around Honshu Island of Japan, Northeastern China, South Korea and the Russian Far East, we also deduce the post-seismic strain changes caused by the Tohoku-Oki M_w 9.0 earthquake. Overall, the distributions of the calculated and GPS-derived strain changes agree well each other. Finally, we compare the relative error between the observed and simulated strain changes over the 3.0–4.5-year period after the earthquake in both the near and far field. We find that the relative errors decrease as the epicentral distance increases, which validates our Green’s functions for research in the far field.

Appendices

Appendices

1.1 Appendix 1: Explicit Expressions for the Homogeneous Terms

According to Takeuchi and Saito (1972), one can obtain the numerical solution of Eqs. (6) and (7) when excluding excitation terms.

The explicit expressions for the homogeneous terms are specifically written as

$$\frac{{{\text{d}}\tilde{y}_{1} }}{{{\text{d}}r}} = \frac{1}{\lambda + 2\mu }\left( {\tilde{y}_{2} - \frac{\lambda }{r}\left[ {2\tilde{y}_{1} - n(n + 1)\tilde{y}_{3} } \right]} \right),$$

(53)

$$\begin{aligned} \frac{{{\text{d}}\tilde{y}_{2} }}{{{\text{d}}r}} = & - s^{2} \rho \tilde{y}_{1} + \frac{2}{r}\left( {\lambda \frac{{d\tilde{y}_{1} }}{\text{d}r} - \tilde{y}_{2} } \right) \\ {\kern 1pt} & \quad + \frac{1}{r}\left( {\frac{{2\left( {\lambda - \mu } \right)}}{r} - \rho g} \right)\left[ {2\tilde{y}_{1} - n(n + 1)\tilde{y}_{3} } \right] \\ {\kern 1pt} & \quad + \frac{n(n + 1)}{r}\tilde{y}_{4} - \rho \left( {\tilde{y}_{6} - \frac{n + 1}{r}\tilde{y}_{5} + \frac{2g}{r}\tilde{y}_{1} } \right), \\ \end{aligned}$$

(54)

$$\frac{{{\text{d}}\tilde{y}_{3} }}{{{\text{d}}r}} = \frac{1}{\mu }\tilde{y}_{4} + \frac{1}{r}\left( {\tilde{y}_{3} - \tilde{y}_{1} } \right),$$

(55)

$$\begin{aligned} \frac{{{\text{d}}\tilde{y}_{4} }}{{{\text{d}}r}} = & - s^{2} \rho \tilde{y}_{3} - \frac{\lambda }{r}\frac{{{\text{d}}\tilde{y}_{1} }}{{{\text{d}}r}} - \frac{\lambda + 2\mu }{{r_{2} }}\left[ {2\tilde{y}_{1} - n\left( {n + 1} \right)\tilde{y}_{3} } \right] \\ {\kern 1pt} & \quad + \frac{2\mu }{{r_{2} }}\left( {\tilde{y}_{1} - \tilde{y}_{3} } \right) - \frac{3}{r}\tilde{y}_{4} - \frac{\rho }{r}\left( {\tilde{y}_{5} - g\tilde{y}_{1} } \right), \\ \end{aligned}$$

(56)

$$\frac{{{\text{d}}\tilde{y}_{5} }}{{{\text{d}}r}} = \tilde{y}_{6} + 4\pi G\rho \tilde{y}_{1} - \frac{n + 1}{r}\tilde{y}_{5},$$

(57)

$$\frac{{{\text{d}}\tilde{y}_{6} }}{{{\text{d}}r}} = \frac{n - 1}{r}\left( {\tilde{y}_{6} + 4\pi G\rho \tilde{y}_{1} } \right) + \frac{4\pi G\rho }{r}\left[ {2\tilde{y}_{1} - n\left( {n + 1} \right)\tilde{y}_{3} } \right],$$

(58)

$$\frac{{{\text{d}}\tilde{y}_{1}^{\text{T}} }}{{{\text{d}}r}} = \frac{1}{r}\tilde{y}_{1}^{\text{T}} + \frac{1}{\mu }\tilde{y}_{2}^{\text{T}},$$

(59)

$$\frac{{{\text{d}}\tilde{y}_{2}^{\text{T}} }}{{{\text{d}}r}} = \left[ {\frac{{\left( {n - 1} \right)\left( {n + 2} \right)\mu }}{{r^{2} }} + s^{2} \rho } \right]\tilde{y}_{1}^{\text{T}} - \frac{3}{r}\tilde{y}_{2}^{\text{T}},$$

(60)

where $\lambda (s) = \frac{{\lambda s + {{\mu K} \mathord{\left/ {\vphantom {{\mu K} \eta }} \right. \kern-0pt} \eta }}}{{s + {\mu \mathord{\left/ {\vphantom {\mu \eta }} \right. \kern-0pt} \eta }}},\; \mu (s) = \frac{\mu s}{{s + {\mu \mathord{\left/ {\vphantom {\mu \eta }} \right. \kern-0pt} \eta }}},\; K = \lambda + \frac{2}{3}\mu$, based on the Laplace transform of Maxwell’s constitutive equation (Peltier 1974); and $g = \left| {\varvec{g}(\varvec{r})} \right|$ denotes the magnitude of the gravity (Takeuchi and Saito 1972).

1.2 Appendix 2: The Solution of the Differential Equation

According to Tanaka et al. (2006, 2007), the expressions for $F_{v}^{i}$ (i = 1,…,4), $F_{u}^{i}$ (i = 1,…,4) and $F_{t}^{i}$ (i = 1, 2) can be expressed as follows:

$$\begin{aligned} F_{u}^{1} \left( {t;n} \right) = - & \frac{1}{2\pi i}\frac{G}{{g_{0} a}}\oint {\left[ {\frac{{3\lambda (r_{s} ,s) + 2\mu (r_{s} ,s)}}{{\lambda (r_{s} ,s) + 2\mu (r_{s} ,s)}}\frac{1}{{r_{s} }}X^{{\text{Press}}} \left( {r_{s} ,s;n} \right)} \right.} \\ \quad & \left. { + \frac{{\lambda (r_{s} ,s)}}{{\lambda (r_{s} ,s) + 2\mu (r_{s} ,s)}}x_{2}^{{\text{Press}}} \left( {r_{s} ,s;n} \right)} \right]\frac{{{\text{e}}^{st} }}{s}{\text{d}}s, \\ \end{aligned}$$

(61)

$$F_{u}^{2} \left( {t;n} \right) = - \frac{1}{2\pi i}\frac{G}{{g_{0} a}}\oint {x_{2}^{{\text{Press}}} \left( {r_{s} ,s;n} \right)} \frac{{{\text{e}}^{st} }}{s}{\text{d}}s,$$

(62)

$$F_{u}^{3} \left( {t;n} \right) = - \frac{1}{2\pi i}\frac{G}{{g_{0} a}}\oint {x_{3}^{{\text{Press}}} \left( {r_{s} ,s;n} \right)} \frac{{{\text{e}}^{st} }}{s}{\text{d}}s,$$

(63)

$$F_{u}^{4} \left( {t;n} \right) = - \frac{1}{2\pi i}\frac{G}{{g_{0} a}}\oint {x_{4}^{{\text{Press}}} \left( {r_{s} ,s;n} \right)} \frac{{{\text{e}}^{st} }}{s}{\text{d}}s,$$

(64)

$$\begin{aligned} F_{v}^{1} \left( {t;n} \right) &= - \frac{1}{2\pi i}\frac{G}{{g_{0} a}}\oint {\left[ {\frac{{3\lambda (r_{s} ,s) + 2\mu (r_{s} ,s)}}{{\lambda (r_{s} ,s) + 2\mu (r_{s} ,s)}}\frac{1}{{r_{s} }}X^{{\text{Shear}}} \left( {r_{s} ,s;n} \right)} \right.} \\ & \quad \left. { +\, \frac{{\lambda (r_{s} ,s)}}{{\lambda (r_{s} ,s) + 2\mu (r_{s} ,s)}}x_{2}^{{\text{Shear}}} \left( {r_{s} ,s;n} \right)} \right]\frac{{{\text{e}}^{st} }}{s}{\text{d}}s, \\ \end{aligned}$$

(65)

$$F_{v}^{2} \left( {t;n} \right) = - \frac{1}{2\pi i}\frac{G}{{g_{0} a}}\oint {x_{2}^{{\text{Shear}}} \left( {r_{s} ,s;n} \right)} \frac{{{\text{e}}^{st} }}{s}{\text{d}}s,$$

(66)

$$F_{v}^{3} \left( {t;n} \right) = - \frac{1}{2\pi i}\frac{G}{{g_{0} a}}\oint {x_{3}^{{\text{Shear}}} \left( {r_{s} ,s;n} \right)} \frac{{{\text{e}}^{st} }}{s}{\text{d}}s,$$

(67)

$$F_{v}^{4} \left( {t;n} \right) = - \frac{1}{2\pi i}\frac{G}{{g_{0} a}}\oint {x_{4}^{{\text{Shear}}} \left( {r_{s} ,s;n} \right)} \frac{{{\text{e}}^{st} }}{s}{\text{d}}s,$$

(68)

$$F_{t}^{1} \left( {t;n} \right) = \frac{1}{2\pi i}\frac{G}{{g_{0} a}}\oint {x_{1}^{\text{T}} \left( {r_{s} ,s;n} \right)} \frac{{{\text{e}}^{st} }}{s}{\text{d}}s,$$

(69)

$$F_{t}^{2} \left( {t;n} \right) = - \frac{1}{2\pi i}\frac{G}{{g_{0} a}}\oint {x_{2}^{\text{T}} \left( {r_{s} ,s;n} \right)} \frac{{{\text{e}}^{st} }}{s}{\text{d}}s.$$

(70)

In the above equations, the relationship between X, x₁ and x₃ can be found in Okubo (1993) as

$$X^{{\text{Press}}} = 2x_{1}^{{\text{Press}}} - n(n + 1)x_{3}^{{\text{Press}}},$$

(71)

$$X^{{\text{Shear}}} = 2x_{1}^{{\text{Shear}}} - n(n + 1)x_{3}^{{\text{Shear}}}.$$

(72)