# Tsunami modeling from the seismic CMT solution considering the dispersive effect: a case of the 2013 Santa Cruz Islands tsunami

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## Abstract

The development of real-time tsunami forecast and rapid tsunami warning systems is crucial in order to mitigate tsunami disasters. The present study shows that tsunami prediction from a seismic centroid moment tensor (CMT) solution would work satisfactorily for the 2013 Santa Cruz Islands earthquake (Mw 8.0) tsunami even though the earthquake source had been modeled as a complicated source characterized by two patches of slip in a past study. We numerically solved the equations for a linear dispersive wave on a spherical coordinate system from the initial tsunami height distribution derived from the CMT solution and a classical scaling law for earthquake faults. The tsunami simulations well explain the observed tsunami arrival times, polarities of initial wave, and maximum amplitudes obtained by deep-ocean pressure measurements. The comparison of the simulation results from dispersive and non-dispersive modeling indicates that the dispersive modeling reproduced the observed waveforms better than the conventional non-dispersive approach. Also, the area affected by a maximum height greater than 0.4 m is decreased by approximately 34% by using dispersion modeling. Those results indicate that the tsunami prediction based on CMT solutions is useful for early warning, and the modeling of dispersion can significantly improve performance.

## Keywords

Seismic CMT solution Dispersion The 2013 Santa Cruz Islands tsunami Numerical simulation## Correspondence/findings

### Introduction

The development of real-time tsunami forecast and rapid tsunami warning systems is an important mission for science technology to mitigate tsunami disasters. The estimations of initial tsunami heights and the propagation effect are essential for tsunami forecasting. In recent years, ocean-bottom observation networks have been installed in deep oceans, and accurate tsunami forecasts have been realized by real-time tsunami data analysis (e.g., Wei et al. 2008; Tang et al. 2009). The real-time data directly estimate the initial tsunami height distribution as the tsunami source. It enables the more reliable prediction of tsunami than from the seismic analysis method where the tsunami source is set from the slip distribution estimated by the seismogram analysis with the assumptions of the subsurface homogeneous/inhomogeneous structure. However, Tsushima et al. (2012) pointed out that the slow tsunami propagation speed fundamentally limits the rapidness of the tsunami source estimation. They reported that it would take more than 20 min to stably estimate the source size even if the ocean-bottom stations are installed very near or inside the source area.

Seismogram analyses can contribute to a rapid tsunami forecast that complement a correct tsunami forecast based on tsunami analysis, because seismic waves propagate much faster (approximately 4,000 m/s) than tsunami (approximately 200 m/s). Gusman and Tanioka (2013) reported that the centroid moment tensor (CMT) solution can be determined by W phase inversion using 5 or 10 min waveform data from the P-wave arrival, and the initial tsunami height distribution from this solution can be used as the tsunami source. They show that the W phase solutions are reliable for use in tsunami modeling. The accuracy of tsunami simulation from a seismically estimated source needs to be understood. Also, it is important to enhance the performance of this method.

Toward these ends, this study focuses on the 2013 Santa Cruz Islands earthquake (Mw 8.0) tsunami propagating in the Coral Sea and Pacific Ocean. By setting an initial tsunami height distribution from simple faulting inferred from the CMT solution of the main shock, we simulate the tsunami propagation from the source to observation points. We investigate how well the simulation from the CMT solution can reproduce observed tsunami waveforms and how tsunami propagation modeling improves the reproduction of the tsunami waveforms.

#### The 2013 Santa Cruz Islands earthquake and tsunami

#### Tsunami data

The National Oceanic and Atmospheric Administration (NOAA) employs the Deep-ocean Assessment and Reporting of Tsunamis (DART) real-time monitoring system to develop effective tsunami forecasting methods and tools (Titov et al. 2005; Tang et al. 2009). The bottom pressure data are continuously recorded at a sampling interval of at least 15 min. When a significant tsunami signal is expected to reach a gauge, 15-s or 1-min sampling data are provided.

#### Numerical simulations

where *φ* is the longitude, *θ* is the colatitude, *R* _{0} is the Earth’s radius, *g* _{0} is the gravity acceleration, *η* is the water height, *h* is the sea depth, and *M* and *N* are the flow rates in the *φ* and *θ* directions, respectively. The right-hand sides of (2) and (3) correspond to dispersive terms. If these terms are equal to zero, these equations represent the linear long-wave equations.

The simulation needs to assume the flow rates and tsunami height at the initial time. The initial flow rates should be zero for all space (Saito 2013). We calculated the initial tsunami height by setting the fault geometry under the assumption of a uniform slip on a single rectangular fault. We selected a fault plane of strike 309°, dip 17°, and rake 61° along subducting plate based on the USGS CMT solution. The length (*L*), width (*W*), and slip amount (*D*) of rectangular fault were inferred from moment magnitude using the scaling law, a relationship of *L* = 2 W and *D*/*L* = 5 × 10^{−5} (e.g., Scholz et al. 1986). The rigidity is assumed 30 GPa to calculate the seismic moment. We assumed the plane to be 119 km in length and 59 km in width and a uniform slip of 5.9 m corresponding to Mw 8.0. The depth of the center of the fault was assumed to be 15 km. Using these parameters, we calculated sea-bottom displacement using a static dislocation model (e.g., Okada 1985) and then the displacement was converted to the initial tsunami height, considering a spatial low-pass filtering effect due to the sea depth of 4 km (e.g., Kajiura 1963; Saito and Furumura 2009). The calculated initial tsunami height is shown in Figure 1. The bathymetry grid was 2 arcminutes (ETOPO2) and time step was 2 s. We assumed total reflection at the coastline. We numerically solved 5-h tsunami using the 2-D linear Boussinesq equation (Equations 1 to 3) with an implicit scheme (Saito et al. 2010) and also using the conventional linear long-wave equations for comparison.

### Result and discussion

#### Tsunami waveform prediction from the CMT solution

The simulation using a well-tuned earthquake source model and nonlinear tsunami propagation simulations would reproduce the tsunami waveform more correctly (Lay et al. 2013). This approach is certainly important for the purpose of precisely investigating the earthquake source process. On the other hand, for the purpose of rapid tsunami warning, the fine-tuning of the slip distribution model would be inappropriate because it would take considerable time to examine the waveforms carefully. Solving the nonlinear tsunami equations also costs computational time. If we consider the linear tsunami equations, however, we can save time by using the database of tsunami Green’s functions. The applicability to a rapid warning for the two-patch event shown above confirms that tsunami prediction using a CMT solution is a powerful candidate for a rapid warning system. Furthermore, in order to avoid a systematic overestimation/underestimation of tsunami height, it would be important to construct an appropriate scaling law to relate the moment magnitude and the tsunami source by analyzing the records of past tsunami events.

#### Improvement of the predictability by dispersive tsunami simulations

Most studies employ the linear long-wave equations for rapid tsunami prediction (e.g., Tsushima et al. 2012; Tang et al. 2009). The linear dispersive equations (Equations 1 to 3) can synthesize the waveforms as rapidly as the linear long-wave equations by using the Green’s function database. Hence, the linear dispersive equations are more appropriate for the tsunami prediction since they model the propagation more precisely across the deep ocean. Figure 2 illustrates an advantage of the dispersive equations in reproducing the observed waveforms. For this case study, the dispersive modeling reproduces observed waveforms better at stations 55012 and 55023 on the Coral Sea bottom than non-dispersive modeling. By introducing dispersive terms, the amplitude of the initial and second phases at station 55012 is significantly improved. At station 55023, the first phase is almost the same between the dispersive and non-dispersive models, while the second and the later phases have different waveforms showing a clear dispersion effect. The root-mean-squared (RMS) residual also decreases from 0.033 and 0.020 m to 0.030 and 0.008 m at stations 55012 and 55023, respectively. Figure 3a,b compares the snapshots calculated based on the dispersive and non-dispersive equations. The difference between them is clearly recognized in the Coral Sea, particularly the region east of station 55012 at the elapsed time of 60 min and the region between stations 55023 and 55012 at the elapsed time of 120 min (shown by arrows in Figure 3a).

### Conclusions

The present study has shown that the tsunami prediction from a CMT solution would work satisfactorily for the 2013 Santa Cruz Islands earthquake tsunami even though the earthquake source had been modeled as a complicated source in a past study. The tsunami simulations well explain the observed tsunami arrival times, maximum amplitudes, and polarities of initial wave observed at the deep ocean-bottom stations. The dispersive modeling significantly improved waveforms in the Coral Sea compared to conventional non-dispersive modeling, which prevents the overestimation of the maximum amplitude. The dispersive modeling decreases the area of maximum amplitude exceeding 0.4 m by 34% from the area calculated based on the non-dispersive modeling. Good performance for a complicated earthquake source confirms that the rapid tsunami calculation using a CMT solution is a powerful candidate for an early tsunami warning system.

## Notes

### Acknowledgements

We thank Dr. Hiroshi Takenaka and two anonymous reviewers for their valuable comments and suggestions. We used tsunami and bathymetry data provided by the National Oceanic and Atmospheric Administration (NOAA). The USGS CMT solution was used for the fault parameter. We are grateful to them for providing the valuable data.

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