**5.3 Source rupture inversions for the Rat Islands MW 7.9 earthquake**

We performed the source rupture inversions for the Rat Islands earthquake using the procedure outlined above and the inversion code developed by Kikuchi and Kanamori, provided by Lingling Ye (personal communication) with a subroutine we revised to speed up the calculations of the Green's functions.

#### *5.3.1 Initial depth selection for the rupture model*

For the Rat Islands mainshock, the focal depth published online by ISC is 102.1 km; the centroid depth calculated by the G-CMT group is 104.3 km. The shallowest focal depth for the 23 aftershocks we relocated is 92.3 km. From the consideration that 104.3 km is the centroid depth, the mainshock is a large one with normal faulting, the rupture initial point may be shallower than the centroid depth by tens of kilometers; therefore, we took 92 km as the initial rupture depth. This value is close to that (95 km) used by Ye et al. [3].

#### *5.3.2 Source rupture inversion results using the steep-dip nodal plane*

In the rupture inversion procedure, the nodal plane 2 of the full moment tensor solution obtained using a depth of 105 km, was used as the rupture plane (**Table 1**; strike 308.7°, dip 84.2° and slip -116.5°). The epicenter (51.7028°N; 178.6428°E) used in the inversion was retrieved from the IRIS website. The fault model dimensions are 270 km 210 km, while the size of each sub-fault is 15 km 15 km. The total number of sub-faults is 252.

To perform the source rupture inversion, we needed a rupture propagation velocity (VR). The rupture velocity (*V*R) is often assumed to be a fraction of the shear wave velocity (*β*). For example, Stein and Wysession [26] assumed a formula *V*<sup>R</sup> = 0.7*β.* To obtain a proper rupture velocity, we performed trial inversion tests using a VR from 1.3 km/s to 2.6 km/s with an increment of 0.1 km/s. **Figure 12** shows the variance (the misfit between the observed and the synthetic waveforms) change with rupture velocity when the initial rupture depth was taken 92 km. The minimum variance (0.1570) occurred at a VR = 2.0 km/s.

**Figure 13** shows the source time function and the final slip distribution obtained using an initial depth of 92 km and a VR = 2.0 km/s. The initial point is indicated by a star sign \*. The largest slip (3.52 m) occurred at a depth of about 120 km. The rupture area is about 60 60 km<sup>2</sup> . **Figure 14** shows a waveform comparison between the observed and synthetic seismograms. The fit in each pair between the observed (upper) and synthetic (bottom) traces was generally good, except that at station AAK. This station is in the strike direction of the steep-dip nodal plane.

#### *5.3.3 Source rupture inversion results using the shallow-dip nodal plane*

Based on the well-relocated hypocenter trend, we used the steep-dip plane as the rupture plane. This was the same as that by Twardzik and Ji [5]. However, Ye et al. [3]

#### **Figure 12.** *The variance changes with rupture velocity. The minimum variance occurred at a rupture velocity of 2.0 km/s.*

*Studies on the Source Parameters of the 23 June 2014 Rat Islands, Alaska… DOI: http://dx.doi.org/10.5772/intechopen.104600*

#### **Figure 13.**

*(a) Lower hemispherical projection of the double couple focal mechanism (Table 1). (b) Source time function. (c) Distribution of the slip on the steep-dip nodal plane (Table 1). The star sign \* with "start point" shows the location of the initial rupture. The arrow at a sub-fault shows the direction and the amount of the slip. The maximum slip was about 3.52 m and occurred at a depth of about 120 km. The used rupture velocity VR = 2.0 km/s, at which the variance reached the minimum (Figure 12). The dashed circles show the rupture propagation.*

found that the back-projection images were more straightforwardly reconciled with the shallow-dip plane. They also found their waveform misfits were comparable when the steep-dip plane or the shallow-dip plane was used as the causative plane, and some signals were better fitted using the steep-dip plane. As a result, their preference for the shallow-dip plane was mild. We also performed trial inversion with the key parameters used by Ye et al. [3], VR = 1.5 km/s, and the initial depth = 95 km. **Figure 15** shows the rupture distribution we obtained. The largest slip (3.33 m) occurred at a depth of about 115 km within the largest patch. **Figure 16** shows a waveform comparison between the observed and synthetic seismograms. The fit in each pair between the observed (upper) and synthetic (bottom) traces was also good.

#### **Figure 14.**

*Comparison between the 27 observed and synthetic seismograms. For each pair of waveforms, the upper trace is the observed (solid line), and the lower trace is the synthetic (dashed-line), generated with the slip distribution in Figure 13c. Both the observed and synthetic waveforms were filtered with a band-pass filter in the range of 0.01 Hz to 0.1 Hz. The symbols and numbers on the left side of each pair from the top to the bottom indicate the station name,* P*-wave vertical component, station azimuth in degrees, and the ratio between the observed and synthetic maximum amplitudes. As shown, the observed and the synthetic seismograms have a good fit in shape, except at station AAK. The average variance is 0.1570. All the ratios between the maximum amplitudes are close to 1 (the ideal case), except at station AAK. This station is in the strike direction of the steep-dip plane.*

To observe the misfit between the observed and the synthetic seismograms, we found the fit at station AAK (Az 308°) was better in **Figure 16** than that in **Figure 14**. Ye et al. [3] found that the shallow-dip fault plane toward the northwest provides better matches to P waveforms at azimuths from 300° to 340° (their Figure S2) than does the steep-dip fault plane solution (their Figures S3 and S4). This result is exactly the same as that we obtained.

To confirm that the record at AAK does not have a problem, we retrieved the records in the station AAK region, plotted the seismograms, and found the waveform shapes are similar (**Figure 17**). This implies that the recording quality at AAK does not *Studies on the Source Parameters of the 23 June 2014 Rat Islands, Alaska… DOI: http://dx.doi.org/10.5772/intechopen.104600*

#### **Figure 15.**

*(a) Lower hemispherical projection of the double couple focal mechanism (Table 1). (b) Source time function. (c) Distribution of the slip on the shallow-dip nodal plane 1 (Table 1). The star sign \* with "start point" shows the assigned location of the initial rupture. The arrow at a sub-fault shows the direction and the amount of the slip. The obtained maximum slip is about 3.33 m, occurred at about a depth of 115 km. The used rupture velocity VR = 1.5 km/s. The dashed circles show the rupture propagation.*

have a problem, so the better waveform fit at AAK support to select the shallow-dip plane as the rupture plane.

#### *5.3.4 Source rupture inversion results using the simulated plane*

Based on the simulated spatial plane obtained using the well-relocated hypocenters, we found the majority of the hypocenters distributed around a mild dipping plane (**Figure 11**; dip 44.8°). We may assume that the mainshock ruptured on that plane. We performed trial inversions with the values of input parameters, rupture velocity VR = 1.5 km/s, and the initial depth = 95 km. **Figure 18** shows the rupture distribution obtained. **Figure 19** shows a waveform comparison between the observed

#### **Figure 16.**

*Comparison between the 27 observed and synthetic seismograms. For each pair of waveforms, the upper trace is the observed (solid-line); the lower trace is the synthetic (dashed-line), generated with the slip distribution in Figure 15c. The observed waveforms are exactly the same as those in Figure 14. The overall fit between the observed and the synthetic seismograms is also good. The fit at station AAK is better than that, and the average variance (0.1545) is slightly smaller than that, in Figure 14.*

and synthetic seismograms. The observed waveforms are exactly the same as those in **Figure 16**. The fits at stations AAK, KIP, and MIDW are not good; the ratio of the maximum amplitudes at several stations is not close to 1 (the ideal ratio is 1). The average variance (0.2720) is larger than those in **Figures 14** and **16**. The obtained maximum slip is about 3.36 m, which occurred at about a depth of 70 km within a smaller patch. Logically the maximum slip should occur at a depth below the initial depth (95 km), owing to the normal faulting. Since the misfit at several stations is not good, the ratio between the observed and synthetic maximum amplitudes at several stations is far from the ideal value, and the maximum slip occurred at a too shallow depth, the simulated plane is not acceptable to be the rupture plane.

*Studies on the Source Parameters of the 23 June 2014 Rat Islands, Alaska… DOI: http://dx.doi.org/10.5772/intechopen.104600*

#### **Figure 17.**

*Vertical P-wave displacement records in station AAK region (63°N–66°N; 50°W–54°W). It is clear that the shapes of these waveform records are similar, showing that the record at AAK does not have a problem. Along the bottom trace, UCH/BHZ three ruptures are indicated. The first one is small, the second one is larger, and the third is a combination of at least two large ruptures. The station (UCH) distance is 7300 km.*
