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Spatial Variations of Aftershock Parameters and their Relation to Geodetic Slip Models for the 2010 Mw8.8 Maule and the 2011 Mw9.0 Tohoku-oki Earthquakes

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Abstract

Recent development in analysis tools and deployments of the geodetic and seismic instruments give an opportunity to investigate aftershock sequences at local scales, which is important for the seismic hazard assessment. In particular, we study the dependencies between aftershock sequences properties and deformational/geological data on a scale of the rupture extension of megathrust earthquakes. For this goal we use, on one hand, published models of inter-, co- and postseismic slip and geological information and, on the other hand, aftershock parameters, obtained by fitting a modified Epidemic Type Aftershock Sequence (ETAS) model. The altered ETAS model takes into account the mainshock rupture extension and it distinguishes between primary and the secondary aftershock triggering involved in the total seismicity rate. We estimate the Spearman correlation coefficients between the spatially distributed aftershock parameters estimated by the modified ETAS model and crustal physical properties for the Maule 2010 Mw8.8 and the Tohoku-oki 2011 Mw9.0 aftershock sequences. We find that: (1) modified ETAS model outperforms the classical one, when the mainshock rupture extension cannot be neglected and represented as a point source; (2) anomalous aftershock parameters occur in the areas of the reactivated fault systems; (3) aftershocks, regardless of their generation, tend to occur in the areas of high coseismic slip gradient, afterslip and interseismic coupling; (4) aftershock seismic moment releases preferentially in regions of large coseismic slip, coseismic slip gradient and interseismically locked areas; (5) b value tends to be smaller in interseismically locked regions.

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Acknowledgements

We are grateful to Rodolfo Console and an anonymous reviewer for their helpful suggestions. This research was funded by the Helmholtz Graduate Research School GeoSim and the University of Potsdam. In addition, Bogdan Enescu was supported by JSPS KAKENHI (Grant 26240004). Furthermore, we would like to acknowledge http://equake-rc.info/SRCMOD/ database for the coseismic slip models, used in the manuscript.

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Authors and Affiliations

  1. GFZ German Research Centre for Geosciences, Potsdam, Germany

    O. Zakharova & S. Hainzl

  2. GEOMAR Helmholtz Centre for Ocean Research Kiel, Kiel, Germany

    D. Lange

  3. Department of Geophysics, Graduate School of Science, Kyoto University, Kyoto, Japan

    B. Enescu

  4. Institute of Statistical Mathematics, Tokyo, Japan

    B. Enescu

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  1. O. Zakharova
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  2. S. Hainzl
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  4. B. Enescu
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Correspondence to O. Zakharova.

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Appendices

Appendix 1: Estimation of ETAS Parameter Uncertainties Using Hessian Matrix

We applied for the estimation of the uncertainty of ETAS parameters a method based on the Taylor expansion of the log-likelihood function \(\mathcal {LL}\) and the second derivatives in the vicinity of its maximum. The second derivative of the \(\mathcal {LL}\) function shows its concavity—the more concave the function the better constrained is the value of the estimated parameter \(\theta\). Moreover, to be able to estimate uncertainties of correlated parameters, it is necessary to take into account changes of all parameters simultaneously. These uncertainties can be evaluated on the basis of the Hessian matrix H (Thacker 1989).

Particularly, the uncertainty estimation includes the following steps:

  1. 1.

    Evaluation of the second-order derivatives of \(\mathcal {LL}\) function and elements of Hessian matrix

    $$\begin{aligned} \mathbf H _{i,j}=\frac{\partial ^2 \mathcal {LL}}{\partial \theta _{i} \partial \theta _{j}}; \end{aligned}$$
    (13)
  2. 2.

    Hessian matrix inversion.

  3. 3.

    Error estimation according to a given confidence interval, assuming that the \(\mathcal {LL}\)-function can be locally approximated by Gaussian distribution.

The described method is applicable for the uncertainty estimation of the ETAS parameters: \(K_0\), K, \(\alpha\), c and p.

Appendix 2: Calculation of Uncertainties Propagation

In this appendix, we present the method of error propagation (Morgan and Henrion 1990) for the uncertainty estimation of the combined aftershock parameters \(D_1\), \(D_2\) and \(TP_2\). Here, the standard deviation \(s_f\) is a function \(f=f(\theta _1, \theta _2, \dots , \theta _m)\) of the m variables \(\theta\)

$$\begin{aligned} s_f = \sqrt{\sum \limits _{j=1}^m \left( \frac{\partial f}{\partial \theta _j} s_{\theta _j}\right) ^2 + 2 \sum \limits _{j=1}^{m} \sum \limits _{k=j+1}^m r_{\theta _j,\theta _k} \left( \frac{\partial f}{\partial \theta _j} s_{\theta _j}\right) \left( \frac{\partial f}{\partial \theta _k} s_{\theta _k}\right) }, \end{aligned}$$
(14)

where \(s_{\theta _j}\) is the standard deviation of the individual variable \(\theta _j\), which is obtained using the Hessian matrix approach (Appendix 1). The correlation coefficient \(r_{\theta _j,\theta _k}\) is calculated based on the estimated values at the M different grid points

$$\begin{aligned} r_{\theta _j,\theta _k} = \frac{\mathrm{cov}(\theta _j,\theta _k)}{\sigma _{\theta _j}\sigma _{\theta _k}} \end{aligned}$$
(15)

with the covariance \(\mathrm{cov}(\theta _j,\theta _k)\) of variables \(\theta _j\) and \(\theta _k\)

$$\begin{aligned} \mathrm{cov}(\theta _j,\theta _k) = \frac{1}{M} \sum \limits _{i=1}^M (\theta _{j_i}-\bar{\theta }_j)(\theta _{k_i}-\bar{\theta }_k). \end{aligned}$$
(16)

and \(\sigma _{\theta _j}\) the standard error with \(\bar{\theta _j}\) being the average value of the variable \(\theta _j\)

$$\begin{aligned} {\sigma _{\theta _j}} = \sqrt{\frac{\sum \nolimits _{i=1}^M {(\theta _{j_i}-\bar{\theta }_j)^2}}{M}}. \end{aligned}$$
(17)

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Zakharova, O., Hainzl, S., Lange, D. et al. Spatial Variations of Aftershock Parameters and their Relation to Geodetic Slip Models for the 2010 Mw8.8 Maule and the 2011 Mw9.0 Tohoku-oki Earthquakes. Pure Appl. Geophys. 174, 77–102 (2017). https://doi.org/10.1007/s00024-016-1408-7

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