Electronic Supplement to
Source Process of the 2011 Tohoku Earthquake Estimated from the Joint Inversion of Teleseismic Body Waves and Geodetic Data Including Seafloor Observation Data: Source Model with Enhanced Reliability by Using Objectively Determined Inversion Settings

by Hisahiko Kubo and Yasumaro Kakehi

Explanation of How the Rupture Propagation in the Dip Direction Affects the Estimated Value of Maximum Slip Duration (T_sd)

Here we consider a question: How does the up-dip rupture propagation affect the estimated value of maximum slip duration (T_sd)?

As shown in Figure 6, the resolution of the semblance analysis in the dip direction is poor. From this reason, if the rupture propagates along the dip direction, the seismic waves produced by this rupture are identified as the ones generated by a point source in the semblance analysis. In that case, the length of S-wave part, which includes the effect of rupture propagation, is interpreted as the duration of slip of a point source, and the value of T_sd is overestimated.

For the 2011 Tohoku earthquake, the seismic arrays were located in the land area (Figure 5), and the rupture during the first 100 s propagated mainly away from the seismic arrays along the up-dip direction (Figure 19). Therefore, the effect of up-dip rupture propagation appears as backward directivity and the estimated value of T_sd is more prolonged compared with the case of forward directivity.

In order to discuss this problem in detail, we assume a rupture model that consists of three subfaults ranging in the dip direction. The three subfaults are ruptured in the up-dip direction and they are designated as F1 (the rupture starting point, proximate to the land), F2 (the subfault located between F1 and F3), and F3 (the shallowest subfault, proximate to the trench). The size of the subfaults is 30km \times 30km. The value of T_sd is 70 s at all three subfaults. The rupture front propagates at a speed of 2.0 km/s. We call this model "Model A".

The duration of the S-wave from this rupture model has a longer duration than T_sd of each subfault. The extended time can be obtained by the sum of the rupture propagation time from F1 to F3 (T1) and the arrival-time difference between seismic waves from F1 and F3 (T2). In model A, T1 is 30.0 s. On the other hand, T2 differs between each array because the directions of back azimuth are different. For example, T2 in Array B is 15.9 s, which is calculated by using the S-wave velocity model of Table 3. Thus the extended time in Array B is about 46 s. The estimated value of T_sd in Array B will be (the value of T_sd at F1) + (the extended time) = 116 s and this value is overestimated compared with the assumed value of T_sd at F1. This situation is schematically illustrated in the Figure S1a.

In this case, the estimated T_sd includes the extended time mentioned above, and the true T_sd is obtained by removing the extended time. However, this correction does not always work. Certainly, it works for the case of Model A, where T_sd has homogeneous distribution, but it does not work for a more general case, where T_sd has heterogeneous distribution. Moreover, we can think of a rupture model, where the total S-wave duration agrees with the value of T_sd at F1 and the estimated T_sd value is equal to the true T_sd value. We will demonstrate such a case.

Let's consider "Model B", where F1 has the longest T_sd (70 + extended time = 116 s), F2 has a middle value of T_sd (93 s), and the F3 has the shortest T_sd (70 s). In this case, the tails of the S-wave parts from the three subfaults reach the array simultaneously and the total duration is equal to the value of T_sd of S1 (Figure S1b). Therefore, the estimated value of T_sd agrees with the true value (the longest value) of T_sd. The correction of the extended time conversely underestimates the true value of T_sd in this case.

In order to make sure that the estimated value of T_sd by the semblance analysis is not overestimated in Model B, we have carried out following semblance analyses against the synthetic waveforms from Model A and Model B. We have added a southern rupture to these two models in order to mimic the southward rupture propagation of the 2011 Tohoku earthquake. The southern rupture starts 100 s after the rupture starts at F1. Figure S2 shows the temporal change of slip in Model A and Model B. The synthetic waveforms are calculated by the discrete wavenumber method (Bouchon, 1981) and the reflection/transmission matrix method (Kennett and Kerry, 1979) using the one-dimensional velocity structure models based on the S-wave velocity model of Table 3. The semblance analysis is conducted at Array B only.

Figure S3 shows the snapshots of the semblance-value distribution on the fault surface for each model. The results for both models show that seismic waves came from the area around the hypocenter for approximately 110 s after the S-wave onset. Thus the estimated value of T_sd by the semblance analysis in Model B agrees closely with the assumed value of T_sd at F1.

Certainly, the rupture propagation in the dip direction can cause overestimation of T_sd because of the poor resolution of the semblance analysis in the dip direction, as shown here. However, it has been also shown that there is a case where the estimated value of T_sd agrees with the true value of T_sd at the subfault where the rupture starts. If we use the corrected (shortened) T_sd for source inversion in that case, we inappropriately exclude the true long T_sd at the rupture-starting subfault. Considering such a case, we have judged that the uncorrected value of T_sd should be used for source inversion.

Figures

Figure S1. Displacements (far-field term) at Array B for (a) Model A and (b) Model B.

Figure S2. Temporal change of slip in (a) Model A and (b) Model B. Slip for every 12 s is shown in each panel. The black star is the hypocenter. The contour interval corresponds to the tick of the color bar.

Figure S3. Snapshots of the semblance-value distribution for (a) Model A and (b) Model B. The semblance values for every 10 s are shown in each panel. 0 s means the S-wave onset time for each array. The black triangles denote the strong ground motion stations used in the semblance analysis. The black star is the hypocenter. The contour interval in each figure corresponds to the tick of its color bar.

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Detailed Information of Teleseismic and Geodetic Stations

Tables

Table S1 [Plain Text Comma-separated Values; 2 KB]. Teleseismic stations used in Kubo and Kakehi (2013)

Table S2 [Plain Text Comma-separated Values; 15 KB]. Terrestrial GPS network stations used in Kubo and Kakehi (2013)

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