Pacific Marine Environmental Laboratory, National Oceanic and Atmospheric Administration, Seattle, Washington, USA
Joint Institute for the Study of the Atmosphere and Ocean, University of Washington, Seattle, Washington, USA
Corresponding author: H. Zhou, Pacific Marine Environmental Laboratory, National Oceanic and Atmospheric Administration, 7600 Sand Point Way NE, Bldg. 3, Seattle, WA 98115, USA. (firstname.lastname@example.org)
 In this study, we investigate the dispersive effects in the 2009 Samoa tsunami through numerical simulations. The wave propagation is first simulated with a weakly nonlinear and dispersive Boussinesq model and a non-dispersive shallow-water-equations model. Comparison of the numerical results between these models indicates that tsunami propagation is significantly affected by the frequency dispersion east of Tonga Trench. Neglecting dispersive effects results in larger wave heights and speeds. The strong frequency dispersion is primarily attributed to the dramatic variation of water surface elevations generated by the earthquake doublet, and enhanced by the uneven bathymetry in Tonga Trench. Tsunami propagation is also simulated with MOST (“Method of Splitting Tsunamis”), which is based on the shallow water equations but uses numerical dispersion to mimic physical frequency dispersion at operational resolutions. A good agreement is observed between MOST and the Boussinesq model, as well as the field measurements in the leading wave. In the shorter trailing waves, agreement becomes poorer due to the mismatch between numerical and physical dispersions.
and suggested that dispersive effects must be considered in a tsunami if P < 4 , where h is still water depth, d is travel distance, and λ is the length of tsunami source along major propagation axis. In practice, λ may also be taken as the characteristic wavelength of the tsunami source [Løvholt et al., 2012]. In a real ocean, uneven bathymetry may also cause dispersive transformation of long waves [Madsen and Mei, 1969]. Frequency dispersion is comparatively stronger along tsunami propagation directions and in deep water [Dao and Tkalich, 2007]. Neglecting dispersive effects may result in overestimated wave heights [Horrillo et al., 2006; Grilli et al., 2007] and coastal runups [Ioualalen et al., 2007].
 The dispersive propagation of tsunamis can be simulated with the Boussinesq-type models [e.g.,Horrillo et al., 2006; Løvholt et al., 2008, 2012]. The Boussinesq-type models involve the solution of complicated mathematical equations, and require a lot of computational resources. In recent years, parallel computing technology has been implemented into the Boussinesq-type models, making them applicable to real-time operational tsunami simulations [e.g.,Kirby et al., 2009]. Dispersive waves may also be simulated with the long wave models that include non-hydrostatic pressure [e.g.,Stelling and Zijlema, 2003; Yamazaki et al., 2009], or match the physical dispersion with numerical dispersion [e.g., Imamura and Shuto, 1989; Cho et al., 2007].
 At 17:40:10 UTC on 29 September 2009, a moment magnitude (Mw) 8.1 earthquake occurred in the outer trench slope near the northern end of Tonga Trench. The epicenter was at 15.51°S, 172.03°W, approximately 200 km south of the Samoan Islands, and 350 km northeast of Tonga. This event was an outer-rise earthquake caused by the bending of the Pacific Plate before it enters the Kermadec-Tonga Subduction Zone, and was characterized by a prevailing normal fault mechanism. Within two minutes, the first earthquake was followed by two major interplate thrust-faulting slips in the Kermadec-Tonga Subduction Zone with a combined Mw equal to 8.0 [Beavan et al., 2010; Lay et al., 2010]. A tsunami generated by this earthquake doublet caused severe damage to the Samoan Islands and Tonga. A maximum runup of 17.6 m was reported on Tutuila Island of American Samoa; on Tafahi Island of Tonga, a 22.4 m runup was observed [Okal et al., 2010]. At DART (“Deep-ocean Assessment and Reporting of Tsunamis”) station 46407, nearly 7800 km northeast of the epicenter, a leading wave of 4.2 cm wave height was observed to be followed by a train of shorter trailing waves. Some trailing waves have periods shorter than 400 s. By employing the average water depth of 4100 m in the Pacific Ocean, the wavelength-to-depth ratio of these shorter waves is estimated to be below 20, suggesting that frequency dispersion may be important for this event.
 In this study, we investigate the dispersive effects of the 2009 Samoa tsunami through numerical simulations. Tsunami propagation in the Pacific Ocean is first simulated with a weakly nonlinear and dispersive Boussinesq model. The model equations are equivalent to those derived by Nwogu , but described in geographic coordinates to consider the earth surface curvature [Zhou et al., 2011]. A high-order finite-difference scheme is applied to solve these equations with a fourth-order accuracy in both time and space. The model has a linear dispersion relation accurate toO(k2h2), where kis wave number. This event is also simulated with a non-dispersive shallow-water-equations (SWE) model. In order to minimize numerical errors, the same high-order numerical scheme is also applied in the SWE model. The dispersive effects are evaluated by comparing the computational results between the two models.
 We further investigate the application of MOST (“Method of Splitting Tsunamis”) to simulate dispersive tsunami propagations. In MOST, the two-dimensional SWE are split into a pair of one-dimensional systems and solved independently through a finite-difference scheme that is accurate to the second order in space and first order in time [Titov and Synolakis, 1998]. Burwell et al.  indicated that the numerical dispersion of MOST can be applied to match the physical frequency dispersion at operational resolutions. In their study, Burwell et al.  demonstrated the effects of numerical dispersion in a series of numerical experiments with regular waves in constant water depth in one dimension. The application of MOST to dispersive waves in a complex water environment is yet to be investigated.
2. Simulations and Discussions
 In this study, we simulate the tsunami propagation in the Pacific basin north of 40°S latitude and east of 140°E longitude. Distinct tsunami waves were observed at five DART stations in this domain, numbered 51425, 51426, 54401, 51406, and 46407. The domain coverage and locations of the DART stations are depicted in Figure 1a. A uniform grid resolution of 2′ is applied to both longitudes and latitudes. The bathymetric and topographic data are derived from the ETOPO 2 two-minute global relief model. Tsunami propagation into Pago Pago Harbor on Tutuila Island is also simulated through three telescoped grids at resolutions of 15″, 6″, and 1.2″, respectively (Figure 1b). Bathymetric and topographic data in the nested grids are derived from the high-resolution digital elevation models developed by the National Geophysical Data Center.
2.1. Tsunami Source
 Immediately after the earthquake on 29 September 2009, NOAA Center for Tsunami Research produced a real-time inundation forecast for Tutuila Island with NOAA's experimental tsunami forecast system (http://nctr.pmel.noaa.gov/samoa20090929/). In this system, tsunami source (initial water surface elevations) is constructed through an inversion algorithm that identifies the best fit between the real-time measurements of DART tsunameters and a linear combination of “unit source functions” in the pre-computed tsunami propagation database [Wei et al., 2008; Tang et al., 2009]. A unit source function includes the time-series of water surface elevations and water velocities in the oceanic basin due to an earthquake in a “unit source”, which has a dimension of 100 × 50 km2, a rake of 90°, and a slip of 1 m in the subduction zone [Gica et al., 2008]. The fault parameters obtained in this approach are primarily to reproduce the initial wave elevations for tsunami simulations, and may not represent the realistic seismological features of an earthquake event.
Fritz et al. simulated the tsunami propagation in this event based on five different fault models, and indicated that only a composite model including both normal and thrust faults might yield an agreeable comparison between the numerical results and measurements at nearby water level stations. This problem was also observed in the tsunami inundation forecast for Tutuila Island. Our initial estimate of the tsunami source came from an inversion solely based on the unit sources with thrust mechanism. This source estimate yielded a good agreement on DART stations 51426 and 54401, but could not explain the records on 51425 with a negative wave before the first peak. Following the preliminary report of the U.S. Geological Survey after the earthquake, three normal-faulting unit sources in the outer-rise zone were added into the tsunami propagation database. The inversion based on both thrust- and normal-faulting unit sources yielded significant improvement of model results. The new source includes three thrust-faulting and two normal-faulting unit sources (Figure 1c). The parameters are provided in Table 1. Due to the bending of the subduction zone, there are overlaps between some unit sources. The overlaps may introduce locally increased slips and short waves. Løvholt et al. applied a low-pass filter to smooth out the short waves in this situation. In the present study, the tsunami source is composed by directly transferring the seafloor displacement to the water surface without any smoothing or filtering. InFigure 1d, we also plot the initial water surface elevations that are employed as the initial conditions for numerical simulations.
Table 1. Source Parameters of the 2009 Samoa Tsunami
Figure 2a plots the maximum water surface elevations computed by the Boussinesq model. Affected by the tsunami source alignment and seafloor bathymetric features, the tsunami radiation pattern shows strong directivity. Propagating to the west, wave amplitudes decrease quickly. High wave energy propagates over a long distance east of Tonga Trench. Comparison between the Boussinesq and the SWE models shows that the SWE model predicts higher wave elevations east of the trench. In Figure 2b, we also plot the relative differences of maximum water surface elevations between the two models. Strong dispersive effects are observed along the major propagation directions east of the trench.
 The strong dispersive effects in this event are primarily attributed to the tsunami source characteristics. The present event was caused by two major earthquakes, which happened nearly simultaneously and were close to each other. The water surface was lifted up to 3.0 m above sea level in the subduction zone west of Tonga Trench. Above the east trench slope, the normal-faulting rupture generated a subsidence of 1.6 m below sea level. Due to the closeness of the two earthquakes, the water surface elevation varies for over 4.5 m in a distance of approximately 15 km, resulting in a large wave steepness. Applyingequation (1) to the present event, we predict significant dispersion appears after the wave travels for nearly 100 km along the main direction of tsunami propagation. Here, we assume the characteristic wavelength is two times of 15 km and employ a water depth of 5000 m. At this distance, the maximum difference between the two models is about 30% as observed in Figure 2b. By comparison, the 2011 Tohoku tsunami was generated by a single thrust-faulting earthquake. Dispersive effects became important after the waves traveled for more than 3800 km in the ocean [Løvholt et al., 2012]. In the present event, dispersive effects are further enhanced by the uneven bathymetry in Tonga Trench. The water depth is over 10000 m on the bottom of the trench, and decreases to around 5000 m on the east brim.
 To investigate the performance of frequency dispersion in more detail, we plot a series of snapshots of the water surface along the 16°10′S latitude in Figure 3. This section is parallel to the eastward propagation axis. Immediately after the simulation begins, the two models deviate from each other in the trench. The SWE model predicts higher and steeper wave forms and faster wave speeds. The Boussinesq model shows that a train of trailing waves are developed following the leading wave. The lengths of the trailing waves measure from 20 to 50 km in the early stage, and increase as waves propagate. These trailing waves are very dispersive and fall beyond the applicable spectrum of the SWE theory.
 In Figure 4, we present the time series of water surface elevations at six water level stations. The simulated time series are shifted backward for 3 minutes at station 54401, 3.5 minutes at 51406, and 4 minutes at 46407 to better match the measurements. The errors in the computed tsunami arrival times may be due to the inaccurate tsunami source, especially as we neglect the temporal offset between the two earthquakes. Pago Pago Harbor is too close to the epicenter for the frequency dispersion to become effective. The Boussinesq and the SWE models are nearly indistinguishable. Both numerical models predict lower amplitudes of the first peak and trough, similar to the numerical results of Fritz et al. , which employed the composite fault model of Beavan et al. . In the near field, DART stations 51425 and 51426 are close to the epicenter, while station 54401 is off the major passage of waves. Dispersive effects did not play a significant role in the waves passing through these sites. Both models agree well with the measurements. In the far field, the two models predict very different time series at stations 51406 and 46407. The Boussinesq model fits the measurements much better than the SWE model. At station 46407, the Boussinesq model agrees very well with the measurement in the phases of the first four waves. At both stations, the SWE model predicts a trough before the first wave peak. This is not observed in the measurements and numerical results of the Boussinesq model. The leading peaks simulated by the SWE model are more than two times higher than those measured in the field.
2.3. Application of MOST
 Propagation of the 2009 Samoa tsunami is also simulated with MOST. According to Burwell et al. , under the condition where
MOST offers a numerical dispersion that matches the dispersion relation to O(k2h2), where Δs is grid spacing, Δt is time step, and gis gravitational acceleration. For basin-scale simulations, MOST solves the equations in geographic coordinates. Accordingly, the grid spacing Δs in equation (2) can be described as RΔϕ cos θ in the longitudinal direction and RΔθ along the meridian, where R is the radius of the Earth, and ϕ and θ are longitude and latitude, respectively. The current version of MOST requires the same longitudinal resolution at different latitudes in a computational grid. As a result, numerical dispersion in the longitudinal direction becomes weaker at higher latitudes.
 In this study, MOST is run in the same grids as the Boussinesq and the SWE models, except that the resolution is downgraded to 2.5′ in the Pacific basin. At this resolution and a time step of 6 s, the numerical dispersion of MOST roughly matches the physical frequency dispersion at low latitudes, but may become insufficient at high latitudes. The snapshots of water surface predicted by MOST are also plotted in Figure 3. At all moments, MOST agrees well with the Boussinesq model in the leading wave, but shows discrepancies in the shorter trailing waves. Similar results are also observed for the time series at all water level stations, except DART 46407, where MOST predicts slightly higher wave height for the leading wave (Figure 4). The difference between the results of the Boussinesq model and MOST is mainly due to the mismatch between the numerical and physical dispersions. This mismatch is more significant for shorter waves. In general, MOST agrees fairly well with the measurements at all stations, and shows a great improvement over the SWE model.
 In this study, we investigate the dispersive effects in the 2009 Samoa tsunami. By comparing the simulation results between the Boussinesq model and a non-dispersive SWE model, we note significant dispersive effects in the deep water east of Tonga Trench, where the SWE model greatly over-predicts the maximum wave elevations. The strong dispersive effects in this event are primarily attributed to the tsunami source characteristics, where water surface elevations vary for more than 4.5 m within a distance of 15 km. This steep wave profile coincides with the uneven bathymetry in Tonga Trench. The dispersive propagation is also simulated with MOST. MOST has numerical dispersion, which is sensitive to both water depth and grid resolution. The modeling results of MOST compare well in the leading wave with that of the Boussinesq model, as well as the measurements at most water level stations. The mismatch between numerical and frequency dispersions becomes significant in the shorter trailing waves. Nevertheless, for weakly dispersive tsunamis at basin scales, an SWE model with numerical dispersion, such as MOST, may still be an efficient approach to provide tsunami forecast for near-field and distant coasts in real time.
 This study is partially funded by the Pacific Marine Environmental Laboratory (PMEL) and the Joint Institute for the Study of the Atmosphere and Ocean (JISAO) under NOAA Cooperative Agreement NA10OAR4320148, contribution 3827 (PMEL), 2034 (JISAO), and the U.S. Nuclear Regulatory Commission, Office of Nuclear Regulatory Research under Interagency Agreement RES-07-004, project N6401. We are grateful to E. Tolkova at PMEL for helpful discussions on the numerical scheme of MOST.
 The Editor thanks two anonymous reviewers for assisting in the evaluation of this paper.