# Rupture features of the 2010 Mw 8.8 Chile earthquake extracted from surface waves

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## Abstract

### Keywords

2010 Chile earthquake Rupture velocity Rise time Static stress drop## Background

After the 2010 Chile earthquake, several studies used finite-fault inversion and P-wave back-projection analysis to probe the complex rupture process of the earthquake. Kiser and Ishii (2011) obtained a fast rupture velocity to the north and a slower one to the south, but Delouis et al. (2010) concluded contrarily. Furthermore, the average rupture velocity reported in previous studies varied widely from 2.0 to 3.2 km/s (Delouis et al. 2010; Lay et al. 2010; Kiser and Ishii 2011; Vigny et al. 2011; Wang and Mori 2011). Ruptures derived from different frequency contents also showed various rupture features through P-wave back-projection analysis (e.g., Kiser and Ishii 2011; Wang and Mori 2011).

In previous studies, the rupture characteristics of the 2010 Chile earthquake were mainly inferred using source rupture models inverted from P-waves and GPS data. However, surface waves can also provide evidence for further understanding of the rupture features of large earthquakes (cf. Ben-Menahem 1961; Christensen and Ruff 1986; Zhang and Kanamori 1988; Velasco et al. 1994; Hwang et al. 2001; Ammon et al. 2006). In this study, Rayleigh-wave travel-time delays, caused by the source finiteness of the 2010 Chile earthquake, were used to estimate the fault parameters through rupture directivity analysis (e.g., Velasco et al. 1994; Hwang et al. 2001, 2011; Ammon et al. 2006; Chang et al. 2010; Hwang 2014).

### Data

## Methods

A large earthquake always produces a large rupture and a long source duration, which increase the observed travel time of surface waves (cf. Hwang et al. 2001, 2011; Chang et al. 2010; Hwang 2014). Under the assumption that the rupture process is uniform and has a constant rupture velocity, the increasing travel time depends on the source duration and is half the source duration (e.g., Ben-Menahem 1961; Hwang 2014). Owing to the rupture directivity of the source, we would observe different source durations at various station azimuths. Azimuth-dependent source duration, also called apparent source duration, can be used to derive the fault parameters for large earthquakes through rupture directivity analysis (e.g., Ben-Menahem 1961; Christensen and Ruff 1986; Zhang and Kanamori 1988; Velasco et al. 1994; Hwang et al. 2001, 2011; Hwang 2014). Therefore, the key to analyzing the rupture directivity of a large earthquake by using surface waves is to separate the apparent source duration from the surface-wave travel time observed at different stations.

*T*;

*D*is the epicentral distance; \(\varPhi_{\text{SR}}\) is the station phase after removing the instrumental response; \(\varPhi_{\text{OR}}\) is the initial phase of the source calculated using a known focal mechanism and the velocity structure in the source area (cf. Wang 1981);

*N*is an arbitrary integer for modulating reasonable phase velocities of the long-period part (cf. Chang et al. 2010; Hwang 2014) (Fig. 2b); \(\varPhi_{\text{str}} T\) is the travel time of surface wave traveling purely through the structure; \(t_{\text{s}}\) is the source duration, corresponding to \(T_{\text{ASD}}\) in Eq. (1).

Trampert and Woodhouse (2001) provided a set of spherical harmonic coefficients with periods of 40, 60, 80, 100, and 150 s to reconstruct the global Rayleigh- and Love-wave phase-velocity maps. From these maps, we can optionally calculate the travel time between two points on the Earth’s surface. The calculated travel time (\(t_{\text{cal}}\)) purely propagated along a great-circle path from a point to a point. Therefore, \(t_{\text{cal}}\), independent of the source, was identical to \(\varPhi_{\text{str}} T\) in Eq. (2). Subtracting \(t_{\text{cal}}\) from \(t_{\text{surf}}\) acquired \(\frac{{t_{\text{s}} }}{2}\), half the source duration.

*L*), rupture velocity (\(V_{\text{r}}\)), and rise time of dislocation (\(\tau\)). On the basis of the rupture directivity theory for an event with unilateral faulting (cf. Ben-Menahem 1961), \(T_{\text{ASD}}\) can be expressed in the following form:

From Eq. (3), \(T_{\text{ASD}}\) had a linear relationship with \(\cos \varTheta\). When searching for a series of \(\phi_{\text{f}}\), we obtained an optimal \(\phi_{\text{f}}\) under the condition that an optimal linear between \(T_{\text{ASD}}\) and \(\cos \varTheta\) with minimum misfit existed. The misfit was defined as in \({\text{misfit}} = 1 + \gamma\), where \(\gamma\) is cross-correlation coefficient. The slope, \(\frac{L}{C}\) in Eq. (3), was then used to determine the rupture length when giving a known \(C\). Moreover, the intercept, \(\frac{L}{{V_{\text{r}} }} + \tau\), is the average source duration consisting of the rupture time and rise time. In Eq. (3), \(T_{\text{ASD}}\) and \(C\) are functions of period \(T\), so Eq. (3) is used at a specified period. However, when the phase velocity \(C\) across the source area was known for each used period, Eq. (3) can lead to a linear relationship between \(T_{\text{ASD}}\) and \(\frac{\cos \varTheta }{C}\). Consequently, the rupture directivity analysis can be performed by simultaneously using \(T_{\text{ASD}}\) observed from various periods. From the global phase velocity maps of Trampert and Woodhouse (2001), the phase velocities across the source area are 3.97, 4.00, and 4.10 km/s for periods of 60, 80, and 100 s, respectively. An average phase velocity is about 4.02 km/s.

## Results and discussion

### Rupture directivity analysis

Fault parameters estimated from 60-, 80-, and 100-s Rayleigh waves in this study

Fault parameters | Northern segment | Southern segment |
---|---|---|

Rupture length (km) | 313 | 118 |

Source duration (s) | 187 | 100 |

Rupture azimuth (°) | 17 | 171 |

Rise time (s) | 32.3 | |

Rupture velocity (km/s) | 1.67 | 1.18 |

Rupture velocity (km/s) | 2.02 | 1.74 |

^{22}Nm (reported from Global CMT). This is likely because we use long-period signals to estimate the fault parameters, whereas the fault parameters are derived from finite-fault source using short-period ones. Kiser and Ishii (2011) and Wang and Mori (2011) have also reported different rupture features when using various frequency-content signals.

### Rupture velocity and rise time

The rupture velocities were initially estimated at 1.67 km/s for the northern rupture and 1.18 km/s for the southern one from the entire source duration, containing the rupture time and rise time (Eq. (3)). Hence, these two rupture velocities are underestimated because we did not exclude the rise time from the entire source duration for the two ruptures. In Eq. (3), the apparent source time includes three components: rupture time (\(\frac{L}{{V_{\text{r}} }}\)); rise time (\(\tau\)); (\(\frac{L}{C}\cos \varTheta\)), which is created from rupture directivity along the rupture length. However, separating the rise time from the apparent source duration in the time domain is difficult. Here, following Hwang et al. (2011), we first selected the stations approximately normal to the rupture directivity; then, \(\frac{L}{C}\cos \varTheta\) vanished because of \(\varTheta = 90^{^\circ }\), and the apparent source time is \(\frac{L}{{V_{\text{r}} }} + \tau\). This would make the analysis simplified for separating the rise time from the entire source duration. In the time domain, the rupture time and rise time increase the travel time of surface waves, whereas in the frequency domain, the rupture time and rise time all represent sinc functions that produce many nodes in the surface-wave Fourier spectra. Hence, these periods of nodes can be used to indicate rupture time and rise time. In general, the rise time is about 0.10–0.25 times smaller than the rupture time (cf. Geller 1976; Heaton 1990) so that the period of node created from the rupture time is relatively easier to be inspected than that from the rise time. From the Fourier spectra influenced by the sinc functions, the rupture time is a multiple of the period of the node (cf. Chang et al. 2010; Hwang et al. 2001, 2011). Moreover, the longest period among nodes, the first node, directly denotes the rupture time.

Figure 7 shows the Fourier spectra of stations PTCN and RAR. For station PTCN, the first-node period was 60.2 s, which are subtracted from its apparent source duration (94.2 s) to yield a rise time of 34.0 s. Similarly, the rise time was estimated at 30.5 s from station RAR. For an earthquake rupture, the rise is constant and does not vary with station azimuth. Hence, here, we averaged the two rise times as 32.3 s. Since the rise time for the 2010 Chile earthquake was drawn from the Fourier spectral analysis, we recalculated the rupture velocity to be 2.02 km/s for the northern rupture and 1.74 km/s for the southern one after deducting the rise time from their source duration, respectively. On average, the rupture velocity for the 2010 Chile earthquake from this study is still lower than that from source rupture inversion (e.g., Delouis et al. 2010; Lay et al. 2010; Vigny et al. 2011), but it is comparable with the 1992 Nicaragua earthquake (cf. Kikuchi and Kanamori 1995) and the 2011 Tohoku earthquake (cf. Hwang 2014), which were events with bilateral faulting and low rupture velocity. Nevertheless, that the rupture velocity in the northern faulting is faster than that in the southern one is also consistent with the result of Kiser and Ishii (2011) from P-wave back-projection analysis.

As in Fig. 1, several large earthquakes once occurred in the subduction zone, where the Nazca plate subducts under the South American plate. The 2010 Chile earthquake compensated the seismic gap between the 1960 Mw 9.5 Chile earthquake and the 1985 Ms 7.8 earthquake (Ruegg et al. 2009). Previous studies revealed that the 1960 Mw 9.5 Chile earthquake and the 1985 Ms 7.8 earthquake have the average rupture velocity of 2.05 and 2.08 km/s from surface-wave analysis (cf. Furumoto and Nakanishi 1983; Christensen and Ruff 1986). Yin et al. (2016) used compressive sensing (CS) method by using P-waves with frequencies of 0.05–0.5 Hz to derive an average rupture velocity of 1.4 km/s, which also low as also reported by Melgar et al. (2016) in 2.0 km/s. In other words, these earthquakes occurring in the Chile region all show low rupture velocity, as analyzed from low-frequency signals (Fig. 1). In addition, the 2010 Mw 8.8 Chile earthquake exhibited high static stress drop of 50–70 bars (Luttrell et al. 2011; Wang et al. 2015), relatively larger than earthquakes (30 bars) in the subduction zone (cf. Kanamori and Anderson 1975). That is, the 2010 Chile earthquake is an event with a low rupture velocity and a high static stress drop. This is similar to the analysis of the 2011 Tohoku earthquake, which is also a bilateral rupture event (Hwang 2014). Such results imply an inverse relationship between static stress drop and rupture velocity as addressed by Kanamori and Rivera (2004). Several studies also support the inverse relationship (e.g., Tan and Helmberger 2010; Hwang et al. 2012; Causse and Song 2015).

## Conclusions

The rupture directivity analysis by using phase velocities with periods of 60, 80, and 100 s revealed that the 2010 Chile earthquake could be characterized by the following: (1) The earthquake was an asymmetric bilateral faulting event. (2) Two rupture directions were N17°E toward the north and N171°E toward the south, respectively. (3) The rupture length, source duration, and rupture velocity of the northern segment were all larger than those of the southern segment. (4) The rise time of approximately 32.3 s was about 0.17 times of the source duration (187 s), agreeing with the proposal of Geller (1976) and Heaton (1990). As mentioned earlier, the rupture features for the 2010 Chile earthquake from this study were consistent with the locking degree in the source area (Moreno et al. 2012), which indicated the seismogenic structures controlling the earthquake rupture process. Additionally, the low rupture velocity and high static stress drop for the 2010 Chile earthquake seemed to support the inverse relationship between these parameters, as reported by Kanamori and Rivera (2004).

## Notes

### Authors’ contributions

YL drafted the manuscript. RD revised the manuscript and drafted the part of the manuscript. YS analyzed seismic data. CI drew all figures. All authors read and approved the final manuscript.

### Acknowledgements

The author is thankful to the Incorporated Research Institutes for Seismology (IRIS) for providing the seismographs. The author especially thanks the two anonymous reviewers for their suggestions, which greatly improved the manuscript. This study was financially supported by the Ministry of Science and Technology, ROC (Grant No. MOST105-2116-M-019-006).

### Competing interests

The authors declare that they have no competing interests.

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