Slip-Weakening Models of the 2011 Tohoku-Oki Earthquake and Constraints on Stress Drop and Fracture Energy

Abstract

We present 2D dynamic rupture models of the 2011 Tohoku-Oki earthquake based on linear slip-weakening friction. We use different types of available observations to constrain our model parameters. The distribution of stress drop is determined by the final slip distribution from slip inversions. As three groups of along-dip slip distribution are suggested by different slip inversions, we present three slip-weakening models. In each model, we assume uniform critical slip distance eastward from the hypocenter, but several asperities with smaller critical slip distance westward from the hypocenter. The values of critical slip distance are constrained by the ratio of deep to shallow high-frequency slip-rate power inferred from back projection source imaging. Our slip-weakening models are consistent with the final slip, slip rate, rupture velocity and high-frequency power ratio inferred for this earthquake. The average static stress drop calculated from the models is in the range of 4.5–7 MPa, though large spatial variations of static stress drop exist. To prevent high-frequency radiation in the region eastward from the hypocenter, the fracture energy needed there is in the order of 10 MJ/m², and the average up-dip rupture speed cannot exceed 2 km/s. The radiation efficiency calculated from our models is higher than that inferred from seismic data, suggesting the role of additional dissipation processes. We find that the structure of the subduction wedge contributes significantly to the up-dip rupture propagation and the resulting large slip at shallow depth.

Appendix

The amplitude spectrum of slip rate in a rupture model with process zone (Fig. 4) can be expressed as a function of final slip $D$, the first corner frequency $f_{\text{ris}} = 1/t_{\text{ris}}$ and the second corner frequency $f_{\text{pz}} = 1/t_{\text{pz}}$. The amplitude spectrum is flat until the first corner frequency, and then decreases as a function of $f^{ - 1/2}$. After the second corner frequency, the amplitude spectrum decreases as a function of $f^{ - 3/2}$ in the case of the linear slip-weakening friction law (Fig. 5c in Kaneko et al. 2008). Thus, the amplitude spectrum can be expressed as:

$$ v\left( f \right) = \left\{ {\begin{array}{*{20}c} {D\,{\text{if}} f < f_{\text{ris}} } \\ { D\left( {\frac{{f_{\text{ris}} }}{f}} \right)^{1/2}{\text{if}}\quad f_{{{\text{ris}} }} < f < f_{\text{pz}} } \\ {D\left( {\frac{{f_{\text{ris}} }}{{f_{\text{pz}} }}} \right)^{1/2} \left( {\frac{{f_{\text{pz}} }}{f}} \right)^{3/2}{\text{if}}\quad f > f_{\text{pz}} } \\ \end{array} } \right. $$

(A-1)

As can be seen from Fig. 4, the largest deep/shallow ratio of slip rate $\alpha$ happens when $f \ge f_{\text{pz}}$:

$$ \alpha = \frac{{D^{\text{d}} \sqrt {f_{\text{ris}}^{\text{d}} } f_{\text{pz}}^{\text{d}} }}{{D^{\text{s}} \sqrt {f_{\text{ris}}^{\text{s}} } f_{\text{pz}}^{\text{s}} }}. $$

(A-2)

The formula for second corner frequency $f_{\text{pz}}$ in Mode II is:

$$ f_{\text{pz}} = \frac{{\left( {1 - \upsilon } \right)\Updelta \tau_{\text{s}} v_{\text{R}} }}{{\mu D_{\text{c}} }}A_{\text{II}} (v_{\text{R}} ), $$

(A-3)

where $\nu$ is Poisson’s ratio, $\mu$ is shear modulus, $\Updelta \tau_{\text{s}}$ is strength drop, $v_{\text{R}}$ is the rupture velocity, and $A_{\text{II}}$ is a function of $v_{\text{R}}$ (Equation (5.3.11) in Freund 1990). By combining (A-2) and (A-3) and assuming a uniform Poisson’s ratio and shear modulus, the shallow/deep $D_{\text{c}}$ ratio can be expressed as a function of $\alpha$:

$$ \frac{{D_{\text{c}}^{\text{s}} }}{{D_{\text{c}}^{\text{d}} }} = \alpha \frac{{D^{\text{s}} }}{{D^{\text{d}} }}\left( {\frac{{f_{\text{ris}}^{\text{s}} }}{{f_{\text{ris}}^{\text{d}} }}} \right)^{\frac{1}{2}} \frac{{\Updelta \tau_{\text{s}}^{\text{s}} v_{\text{R}}^{\text{s}} }}{{\Updelta \tau_{\text{s}}^{\text{d}} v_{\text{R}}^{\text{d}} }}\frac{{A_{\text{II}}^{\text{s}} \left( {v_{\text{R}}^{\text{s}} } \right)}}{{A_{\text{II}}^{\text{d}} \left( {v_{\text{R}}^{\text{d}} } \right)}}. $$

(A-4)

The final slip, rise time and rupture velocity can be inferred from slip inversions. The strength drop is our model parameter. For example in our first model, as $\left( {\frac{{f_{\text{ris}}^{\text{s}} }}{{f_{\text{ris}}^{\text{d}} }}} \right)^{\frac{1}{2}} \sim 1$, $\frac{{\Updelta \tau_{\text{s}}^{\text{s}} v_{\text{R}}^{\text{s}} }}{{\Updelta \tau_{\text{s}}^{\text{d}} v_{\text{R}}^{\text{d}} }}\sim 1$, and $\frac{{A_{\text{II}}^{\text{s}} \left( {v_{\text{R}}^{\text{s}} } \right)}}{{A_{\text{II}}^{\text{d}} \left( {v_{\text{R}}^{\text{d}} } \right)}}\sim 1$, we can simplify (A-4) to:

$$ \frac{{D_{\text{c}}^{\text{s}} }}{{D_{\text{c}}^{\text{d}} }} \approx \alpha \left( {\frac{{D^{\text{s}} }}{{D^{\text{d}} }}} \right). $$

(A-5)

As the observed high-frequency slip rate power ratio is at least 10, $\alpha$ has to be larger than $\sqrt {10}$. Given $\frac{{D^{\text{s}} }}{{D^{\text{d}} }}$~2–3 the shallow/deep $D_{\text{c}}$ ratio is larger than 6 to 9. Note that the needed $D_{\text{c}}$ ratio (>10) in our model is larger than this range, as the observed power ratio involves smoothing in time and space. Thus, (A-4) and (A-5) only give a lower bound of the shallow/deep $D_{\text{c}}$ ratio.