Direct energy estimation of the 2011 Japan tsunami using deep-ocean pressure measurements

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117,, doi: /2011jc007635, 2012 Direct energy estimation of the 2011 Japan tsunami using deep-ocean pressure measurements Liujuan Tang, 1,2 Vasily V. Titov, 2 Eddie N. Bernard, 2 Yong Wei, 1,2 Christopher D. Chamberlin, 1,2 Jean C. Newman, 1,2 Harold O. Mofjeld, 2 Diego Arcas, 1,2 Marie C. Eble, 2 Christopher Moore, 2 Burak Uslu, 1,2 Clint Pells, 1,2 Michael Spillane, 1,2 Lindsey Wright, 1,2 and Edison Gica 1,2 Received 26 September 2011; revised 5 June 2012; accepted 18 June 2012; published 3 August [1] We have developed a method to compute the total energy transmitted by tsunami waves, to the case where the earthquake source is unknown, by using deep-ocean pressure measurements and numerical models (tsunami source functions). Based on the first wave recorded at the two closest tsunameters (Deep-Ocean Assessment and Reporting of Tsunamis (DART)), our analysis suggests that the March 11, 2011 Tohoku-Oki tsunami generated off Japan originated from a km long and 100 km wide area, and the total propagated energy is J (with 6% uncertainty). Measurements from 30 tsunameters and 32 coastal tide stations show excellent agreement with the forecasts obtained in real time. Our study indicates that the propagated energy and the source location are the most important source characteristics for predicting tsunami impacts. Interactions of tsunami waves with seafloor topography delay and redirect the energy flux, posing hazards from delayed and amplified waves with long duration. Seafloor topography also gives its spectral imprint to tsunami waves. Travel time forecast errors are path-specific and correlated to the major wave scatterers in the Pacific. Numerical dissipation in the propagation modeling highlights the need of high-resolution inundation models for accurate coastal predictions. On the other hand, it also can be used to account for physical dissipation to achieve efficiency. Our results provide guidelines for the earliest reliable tsunami forecast, warnings of long duration tsunami waves signals and enhancement of the experimental tsunami forecast system. We apply the method to quantify the energy of 15 past tsunamis, independently from earthquake magnitudes. The small tsunami to seismic radiation energy ratios, and their variability ( %), reinforce the importance of using deep-ocean tsunami data, the direct measures of tsunamis, for estimates of tsunami energy and accurate forecasting. Citation: Tang, L., et al. (2012), Direct energy estimation of the 2011 Japan tsunami using deep-ocean pressure measurements, J. Geophys. Res., 117,, doi: /2011jc Introduction [2] The total energy transmitted by tsunami waves is one of the most fundamental macroscopic quantities for interpreting the size of a tsunami, as well as for understanding the physical processes of tsunami propagation and coastal impacts. The objective of this study is to develop a method to estimate the total tsunami propagation energy in real-time. The 2011 Japan tsunami provides a robust test case. 1 Joint Institute for the Study of the Atmosphere and Ocean, University of Washington, Seattle, Washington, USA. 2 NOAA Center for Tsunami Research, Pacific Marine Environmental Laboratory, Seattle, Washington, USA. Corresponding author: L. Tang, NOAA Center for Tsunami Research, Pacific Marine Environmental Laboratory, 7600 Sand Point Way NE, Seattle, WA 98115, USA. (liujuan.tang@noaa.gov) American Geophysical Union. All Rights Reserved /12/2011JC [3] Most recorded historical tsunamis were generated by earthquakes (83%) or earthquake induced landslides (6%) [National Geophysics Data Center, 2011]. Others were caused by volcanic eruptions, landslides and unknown sources. Therefore traditionally seismic data have been the primary data for tsunami studies [Kanamori, 1972; Murty and Loomis, 1980; Kajiura, 1981; Okal, 2003]. Historically, earthquake magnitudes have been used extensively as indicators of tsunami size for tsunami forecasts since the 1940s [Shuto and Fujima, 2009; Satake et al., 2008; Whitmore, 2009]. Seismic waves propagate much faster than tsunami waves providing a time advantage for forecast applications. However, large uncertainties have been seen in the past in determining the earthquake magnitude immediately after an event, such as the moment magnitude (M w ) for the 2004 Sumatra earthquake and the M w for the 2011 Japan earthquake [Ozaki, 2011; Pacific Tsunami Warning Center, Bulletins 001, 002 and 003, December, 2004, basin=pacific; Pacific Tsunami Warning Center, Bulletins 1, 1of28

2 Figure 1. (a) Timeline for assessment of the March 11, 2011 Japan earthquake and tsunami. (b and c) Snapshots of modeled first tsunami wave arrived at the two closest tsunameter stations, and 21401, 30 and 60 min after the earthquake. 2 and 4, 11 March, 2011, archive.php?basin=pacific]. Uncertainties can exist in post event analyses months after an event, such as the M w estimates of the 2011 Japan earthquake [Simons et al., 2011] and the mechanisms of the 2009 Samoa earthquakes [Lay et al., 2010]. Some remain challenges even decades after the events, such as the M = 7.4 and M w = 8.5 for the 1946 Unimak earthquake [Gutenberg and Richter, 1954; López and Okal, 2006], the M = and M w = 9.2 for the 1964 Alaska earthquake [Spaeth and Berkman, 1967; Kanamori, 1977], the M w = 9.5 and 9.2 for the 1960 Chile earthquake [Kanamori, 1977; Fujii and Satake, 2011], etc. Note that a 0.2 increase in earthquake magnitude could correspond to a doubling of the tsunami amplitude [Titov et al., 2011]. In addition, whether or not an earthquake could generate a tsunami, or to what extent, would require tsunami measurements for confirmation. As an example, the 28 March 2005 M w = 8.6 event off the Sumatra coastline did not cause a major tsunami, as one would expect based on the earthquake magnitude alone [Kerr, 2005]. The above events demonstrated that seismic data alone are unable to provide the level of accuracy required for tsunami modeling and forecasts. [4] Previous studies on tsunami energy focused on the details of tsunami generation based on seismic information [Murty and Loomis, 1980; Kajiura, 1981; Okal, 2003; Satake and Tanioka, 2003] and coastal GPS data [Song, 2007]. In this paper, our approach focuses on the characteristics of tsunami propagation (e.g., beyond the stage of tsunami generation), which is also more relevant to the resulting coastal impacts. We estimate the total energy transmitted by the tsunami waves based on inversion of deep-ocean pressure measurements at tsunameter stations (DART). These stations measure the bottom pressure fluctuations directly proportional to the time-varying sea surface elevation induced by the tsunami waves as they propagate past the stations in realtime [González et al., 2005; Meinig et al., 2005; Mofjeld, 2009]. Figure 1b shows the locations of bottom pressure recorders at four stations, which are deployed on the seafloor at water depth greater than 5000 m. Nearby surface buoys relay data and instructions via satellite between the operators and the bottom-mounted units. Presently, a system of 55 tsunameter stations (40 U.S.-, 8 Australian-, 1 Chilean-, 1 China-, 2 Indian-, 1 Indonesian-, 1 Thailand- and 1 Russianowned) are deployed in the Pacific, Atlantic, Indian Oceans, Caribbean Sea, the Gulf of Mexico and South China Sea [Spillane et al., 2008]. The tsunameter network is a crucial component of the tsunami forecast system that is under development and being testing for use by Tsunami Warning Centers in the United States [Titov, 2009]. Tsunameter data have several key features: (1) they are direct measures of tsunami waves; (2) deep-ocean tsunami data are in general the earliest tsunami information available; (3) compared to coastal water level stations [Whitmore, 2003], tsunameter data with high signal-to-noise ratio can be obtained without interference from harbor and local shelf effects (Figure 1b); and (4) the assumption of linear wave dynamics of tsunami propagation in deep ocean allows a tsunami source to be effectively reconstructed based on the best fit to given deep-ocean tsunami data. Tsunameter data have been used 2of28

3 extensively in forecast and hindcast studies for more than 30 past tsunamis [Titov and Tang, 2011; Titov, 2009; Wei et al., 2003, 2008; Tang et al., 2008, 2009; Titov et al., 2005b; Okal and Titov, 2007]. Tsushima et al. [2009, 2011] have used similar approaches for real-time forecasting of near-field tsunamis from cabled ocean bottom pressure data for the 2011 Japan tsunami and other past events. [5] On the afternoon of March 11, 2011 at 14:46:24 local time (05:46:24 UTC), a moment magnitude (M w ) mega-thrust earthquake occurred along the oceanic subduction zone located 130 km east of Sendai, Japan (USGS Global Centroid moment-tensor (CMT) Catalog, The earthquake generated one of the most destructive tsunamis in Japanese history. The toll of 22,941 fatalities and missing and economic losses exceeding $214 billion illustrate the catastrophic regional impact of this tsunami [The 2011 Tohoku Earthquake Tsunami Joint Survey Group, 2011]. Following its generation, the tsunami propagated throughout the Pacific Ocean causing casualties, flooding, and damage throughout the basin. The tragic events surrounding both the 2011 Japan and the 2004 Indian Ocean tsunamis demonstrate the vulnerability of coastal populations irrespective of the level of preparedness, and highlight the need for timely and accurate forecasts and warning [Bernard, 2012; Synolakis et al., 2005; Bernard et al., 2006; The 2011 Tohoku Earthquake Tsunami Joint Survey Group, 2011]. [6] The 2011 Japan tsunami was the first, and the largest, tsunami for which deep-ocean tsunameter data were available in the near field from multiple well-positioned tsunameter stations (Figure 1b). The 2010 Chile tsunami (M w = 8.8) was a large event, but the tsunami arrived at the first tsunameter 3 h after the earthquake (e.g., far-field measurements). During the 2006 Kuril tsunami (M w = 8.3), the nearby tsunameter stations were located to the north of the source area (e.g., one side constrained data set). As to the 2007 Sumatra tsunami (M w = 8.5), data were only available from one tsunameter. Other past tsunamis were relatively small. Therefore, the 2011 Japan tsunami is the first and only event that provides sufficiently high quality tsunameter measurements for a thorough test of the real-time method. [7] Section 2 of this article describes the method. Section 3 presents the assessment of the 2011 Japan tsunami in realtime and forecast results. Section 4 discusses tsunami generation, propagation, energy decay and numerical dissipation in propagation modeling. Section 5 explores the coastal impacts. Section 6 applies the method to quantify propagation energy for fifteen past tsunamis. Concluding remarks are provided in section Method [8] The real-time tsunami forecast scheme utilizes a two-step process: (1) construction of a tsunami propagation scenarios (a tsunami source) via inversion of deep-ocean tsunameter data with pre-computed tsunami source functions and (2) coastal predictions by running high-resolution sitespecific flooding models in real-time [Titov et al., 2005b; Titov, 2009]. [9] Wave dynamics of tsunami propagation in deep-ocean are assumed to be linear [Kânoğlu and Synolakis, 2006; Liu, 2009]. Therefore a propagation scenario (h) can be decomposed into a superposition of a subset of tsunami source functions (h) h ¼ XN i¼1 a i h i ; h is the sea surface elevation, h i is the sea surface elevation of the i-th tsunami source function, a i is the i-th coefficient, N is the total number of tsunami source functions used. A tsunami source function h is a unit propagation scenario that could be generated by an earthquake, a landslide or other type sources. Currently each tsunami source function h in the forecast system represents a unit propagation scenario generated by a M w 7.5 earthquake, which is modeled as the instantaneous rupture of a single rectangular fault plane characterized by pre-defined parameters describing the location, orientation and rupture of the plane [Gusiakov, 1978; Okada, 1985; Gica et al., 2008]. The source sensitivity study by Titov et al. [1999] has established that only a few tsunami source parameters are critical to characterize far-field tsunamis, namely the location and the magnitude. Other parameters have secondary influence and can be predefined for forecast purpose. The Method Of Splitting Tsunami (MOST) model, a suite of finite difference numerical codes based on the longwave approximations [Titov and Synolakis, 1998; Titov and Gonzalez, 1997; Synolakis et al., 2008], are used to computed the tsunami source functions ð1þ h t þ x ½ðh þ dþušþ ½ y ðh þ dþv Š ¼ 0; ð2aþ u t þ u u x þ v u y v t þ u v x þ v v y h ¼ g x C f ujuj h þ d þ fv; h ¼ g y C f vjuj fu; h þ d ð2cþ ð2bþ where d is the undisturbed water depth, u ={u, v}, u and v are the depth-averaged velocities in the x and y directions, respectively, g is the gravity acceleration, f is the Coriolis parameter, C f = gn 2 /(h + d) 1/3, and n is Manning coefficient. In the MOST model, these equations are solved numerically using the splitting method [Titov and Synolakis, 1998]. Dispersion can change the waveshape due to slightly different propagation speeds of waves with different frequencies. This effect can be taken into account without the explicit use of dispersive terms in the governing equations. Shuto [1991] suggested that this process could be simulated by exploiting the numerical dispersion inherent in finite difference algorithms. This method accounts for dispersive effects, but allows the use of no dispersive linear or nonlinear equations for wave propagation modeling. The MOST model uses a numerical dispersion scheme, which the 4 arc min resolution gives a good approximation to the physical dispersion [Burwell et al., 2007]. Currently, the database has 1718 tsunami source function scenarios covering the most active subduction zones in the Pacific, Indian and Atlantic Oceans. [10] In order to exploit the simple linearity assumption of tsunami propagation, and thus to circumvent the large 3of28

4 uncertainties in earthquake sources, here we introduce the concept of a tsunami source. In the context of this paper, a tsunami source refers to the synthetic propagation scenario (h) from a superposition of tsunami source functions (h) that gives a best fit to the observed tsunami amplitude time series (h obs ) in the deep ocean where Z Z E k ¼ e k ds ¼ 0:5r ðh þ dþðu 2 þ v 2 Þds; s Z s Z E p ¼ e p ds ¼ 0:5rg h 2 ds; s s h obs ¼ h þ res t ¼ XN i¼1 a i h i þ res t ; X res 2 t ¼ X ðh obs hþ 2 ¼ X! 2 h obs XN a i h i ; i¼1 i¼1 h obs is the observed sea surface elevation in deep ocean, res t is a residual term between the observation and model. We estimate a i by using the Least Squares method (equation (4)), e.g., minimizing the sum of the squared residuals [Percival et al., 2011]. Figure 2 illustrates h can be obtained in time and space as a linear combination of 6h that gives a best fit to h obs at two tsunameters for the 2011 Japan tsunami. Depending on the length of the observations (the number of tsunameters and inversion time windows) and the tsunami source functions used, a i can vary for the same event. However, as long as the data are sufficient, the propagation time series of h, PN a i h i, should be similar because the residuals are small and the sum of the squared residuals has been minimized. Note by introducing a time shift as another unknown for each tsunami source function in addition to a, the inversion technique can be applied for dynamic sources. [11] In real-time, the inversion starts as soon as the first peak arrived at the first tsunameter station. Better estimates can be achieved when more data become available. However, the inversion result may become inaccurate when unreliable data are used. The reasons are as follows: (1) data recorded at one tsunameter can be highly correlated, since most of the late waves, which may reflect shelf resonance and continentscale standing edge waves [Yamazaki and Cheung, 2011], are from the first one or two incident waves; (2) the late waves may be underestimated due to the numerical dissipation in the shallow water; and (3) data recorded by different tsunameters with similar directionality to the origination area is also highly correlated. Thus data quality and model approximations limit the amount of data that can be used for inversion. The later waves (with relatively low signal-tonoise ratio compared to the first wave) and far-field tsunameter data (which may contain travel time error induced by model approximations) should be excluded for inversion or used with extra caution and additional adjustments. [12] The physical mechanism of tsunami propagation is that energy is transmitted across the ocean by the propagating surface long gravity waves. The best parameter to estimate the size of a tsunami would be the total transmitted energy. The instantaneous tsunami wave energy E at any given time, is the sum of the kinetic energy, E k, and the potential energy E p, and is given as E ¼ E k þ E p ; ð3þ ð4þ ð5þ e k and e p are the surface kinetic and potential energy density. The integration is over the propagation domain. [13] Based on the zero initial velocities assumptions according to the adopted elastic deformation model, we compute the tsunami energy, E T (the total energy transmitted by the tsunami waves), as the difference between the potential energy from the initial deformed body of water and that of the quiescent state in the same basin Z E T E t¼0 ¼ E kt¼0 þ E pt¼0 ¼ 0 þ 0:5rg h 2 0 ds; ð6þ where h 0 is the sea surface elevation at the initial stage (the initial ocean surface deformation). Substituting equation (1) into equation (6), we have the numerical formula E T ¼ 0:5rg X j s j X N i¼1 a i h 0;i;j! 2 ; where j represents j-th wet grid node, s j is a cell area corresponding to the j-th node, h 0,i,j is the initial sea surface displacement from the i-th tsunami source function at j-th node. It should be noted that the undisturbed water depth d has been included in equation (2) while computing h by shallow water equations and subsequently in equation (4) for estimating a i by Least Squares inversion. Thus our energy estimation has taken the water depth into consideration by both model and observation. [14] Energy released from an earthquake and then portions of the earthquake energy transferring into water column are complex dynamic processes at the stage of tsunami generation. However, our goal is not to quantify the energy at the initial stage of tsunami generation. Instead, we try to quantify the amount of energy that propagates outside the source area in the form of surface long gravity waves, which can be well measured by the tsunameter stations. It is also the propagated energy that results in the coastal impacts. Our estimates of the tsunami source (the propagation scenario), and energy, focus on the characteristics of tsunami propagation (equation (3)). They are directly constrained by the deep ocean tsunami data (equation (4)). Regardless the details of earthquake processes for tsunami generation at the initial stage, equation (4) can ensure the propagation scenario gives the best approximation to the tsunami observations, and therefore, the best estimation of the total energy transferred by the tsunami waves (E T ). [15] The tsunami source functions h in the database are computed with a time step of 10 s and a spatial resolution of 4 arc min (approximately 5.7 km at 40 N Latitude in Japan along the E-W direction and 7.4 km along the N-S direction). The outputs, offshore wave height and depth-average velocities of the entire domain, are then compressed and saved every 1 min in time and 16 arc min in space [Tolkova, 2007]. s ð7þ 4of28

5 Figure 2. A synthetic propagation scenario of the two tsunameter inverted source in (a and b) time and (c) space can be obtained by a superposition of (d u) six tsunami source functions that give the best fit to the observations (Figures 2a and 2b). The coefficients, a, were obtained by the Least Squares method. 5of28

6 Table 1. Pre-defined Parameters for Tsunami Source Functions and the Three Tsunami Sources Inverted From the Closest One, Two, and Three Tsunameter Stations for the 2011 Japan Tsunami in Real-Time Name Longitude ( E) Latitude ( N) Tsunami Source Functions a Strike (deg) Dip (deg) Depth (km) First Source (21418) Tsunami Source Coefficients Second Source (21418, 21401) Third Source (21418, 21401, 21413) b b a b a b Tsunami energy E T (J) a All the tsunami source functions have four source parameters, length 100 km, width 50 km, slip 1 m and rake angle 90. The current propagation scenarios do not include inundation and a vertical wall is placed at 20 m water depth [Gica et al., 2008]. The friction term is set to zero. When tsunami waves propagate into shallow water, under the steady state assumption, where there are not any energy losses or inputs, the decrease in transport speed must be compensated by an increase in energy density in order to maintain a constant energy flux. The low spatial resolution and simplified boundary conditions of the propagation model result in inaccurate nearshore dynamics. As a consequence, which will be discussed in section 4.2, the numerical dissipation (due to the low spatial resolution) will cause energy decay in the propagation modeling. [16] Based on consideration of energy conservation, we have developed high-resolution, site-specific inundation forecast models built on the MOST model to simulate the nearshore wave dynamics. Each inundation model contains three telescoping grids with increasing resolution, covering regional, intermediate and nearshore areas (Movies S2 and S3). 1 Once the tsunami source is obtained (as a linear combination of tsunami source functions that gives the best fit to tsunameter data), the linear combination of precomputed time series of offshore wave height and depthaverage velocities are applied as the dynamic boundary conditions for the inundation models. In this way, it not only saves the simulation time of basin wide tsunami propagation, but also provides the validated time series offshore dynamic boundary conditions for the inundation models, which ensure the accuracy of the coastal predictions. Tsunami inundation is a highly nonlinear process; therefore a linear combination would not provide accurate solutions. So the h in equations (2a) (2c) is replaced by h at this stage. Runup and inundation are computed at the coastline. The resolution of the finest grid, typically 1 3 arc s (30 90 m), has to be high enough to resolve the dynamics of a tsunami inside a particular harbor, including influences of major harbor structures such as breakwaters. The grids are derived from the best available bathymetric and topographic data at the time of development, and will be updated as new survey data become available [e.g., Chamberlin, 2007; Carignan et al., 2011]. The inundation forecast models have been validated and optimized for speed and accuracy in order to provide a 4-h event forecasting of amplitude, current and flooding in 1 Auxiliary materials are available in the HTML. doi: / 2011JC minutes of computational time using one single processor [Tang et al., 2009, 2010; Arcas and Uslu, 2010; Righi and Arcas, 2010; Uslu et al., 2010; Wei and Arcas, 2010]. Inundation forecast models have been developed for 57 U.S. coastal communities (44 in the Pacific). We plan to develop a total of 75 forecast models by 2013, with additional models envisioned later for smaller communities. 3. Assessment of Tsunami Energy in Real-Time and Results [17] To illustrate the forecast potential of the method, our real-time assessments of the 2011 Tohoku Japan tsunami, including the tsunami source, energy, and details obtained by inverting data recorded at the first one, then two, and three tsunameter stations are summarized in Table 1 and Figure 1a. The tidal predictions based on harmonic constituents were removed from the raw measurements to obtain the tsunami signals. [18] The first tsunami measurement from the network was available about 30 min after the earthquake at station 21418, located approximately 500 km east of the epicenter (Figure 1b). Data from this station showed a 1.64 m first crest (2.52 m wave height), the largest tsunami wave ever recorded in the deep ocean, and a 22 min peak period, with complex frequency/ period patterns ranging from 5 min to 2 h (Figures 3a 3e). The tsunami energy estimated from the first wave recorded at this single station was J. A second station, owed and maintained by Russian Far Eastern Regional Hydrometeorological Research Institute, located 1000 km northeast of the epicenter, recorded a 0.5 m first crest (0.96 m wave height) with a peak period of 26 min (Figures 3f 3j). The combined inversion of the first waves recorded at the two stations indicated tsunami energy of J, a total of initially displaced water volume of 250 km 3, and a major source area km long and 100 km wide (Figure 4b). The offshore wavelength of the first wave was approximately 360 km (Figure 1c). The northern and eastern boundaries of the tsunami origination area were well constrained by the tsunami arrival time. The western and southern boundaries were estimated from the travel time of the first wave at station Figure 2 illustrates the six tsunami source functions and their coefficients used to construct the propagation scenario, which were obtained by the Least Squares method. Adding measurements from the third station, to the south, confirmed the tsunami energy of J, less than 6% different from the second estimate (Table 1). Hence 6of28

7 Figure 3. (a, f, and k) Time series of observed and modeled wave amplitudes at stations and 21401, and (b, g, and l) Time series of modeled energy flux vector. The top plot shows the entire time series while the bottom plot zooms in to the later waves. (c, h, and m) Wavelet-derived amplitude spectra of the observations show peak periods of 22, 26 and 31 minutes at the three stations respectively. (d, i, and n) The real parts of the spectra are plotted. (e, j, and o) The modeled tsunami waves well reproduce the frequency/ period patterns of the observations. the estimates of tsunami energy were convergent with data from the first two stations. That is to say, measurements from the first two stations were sufficient for determining the energy of this tsunami. The southern boundary was well constrained by the arrival time of the first wave at the third station. [19] The 2-tsunameter inverted source produced accurate predictions of tsunami amplitude time series at 28 other tsunameters throughout the Pacific (Figure 5). Therefore the first wave at the two closest tsunameters was sufficient to tune the major tsunami source parameters for accurate modeling of tsunami propagation in the deep ocean. The Figure 4. Initial ocean surface displacements reconstructed from tsunami sources inverted from the nearest (a) 1-, (b) 2-, and (c) 3-tsunameter stations. The contour lines on the bottom layer correspond to the initial ocean surface displacement in meters. 7of28

8 Figure 5. Forecast and observed tsunami amplitude time series at 30 deep-ocean tsunameters. Locations of the stations were plotted in Figure 8. The observed and forecast maximum crests are listed after hmax= in each figure. Dt, travel time error of the forecast first wave. A time delay of Dt minutes was applied to the forecast time series in each figure. maximum forecast error is at station in Hawaii: the second wave crest of 24 cm is the largest at the station, while the modeled result was only 7 cm. One reason is that is only 60 km offshore. With a high-resolution grid of 2 arc min, the modeled amplitude can be improved to 17 cm. A further nested grid of 18 arc s produces a 19 cm crest. [20] This tsunami source also produced excellent coastal forecasts at 32 coastal sites where high-resolution flooding models and observations were available (Figures 6 8). The forecast accuracy of the maximum crest ( h obs -h forecast /h obs ) at the 32 coastal tide stations is 68%, or a 0.2 m average error. For eight sites with observed amplitude larger than 1.0 m, 8of28

9 Figure 6. Forecast and observed tsunami amplitude time series at 32 coastal tide stations. Forecast time series were computed from high-resolution flooding models in real-time. Locations of the stations are plotted in Figure 7. The observed and forecast maximum crests were listed after hmax= in each figure. Dt, travel time error of the forecast first wave. A time delay of Dt minutes was applied to the forecast time series in each figure. 9of28

10 Figure 7. Filled circles, 32 coastal tide stations in Figure 6. Open circles, forecast sites with no tide gage data available. Stars, earthquake locations. the forecast accuracy is 74%. This is further improved to 87% for four sites with observations greater than 1.5 m. At the Crescent City and Kahului tide stations, which recorded the two largest maximum wave crests of 2.5 and 2.1 m at U.S. West Coast and in Hawaii, the accuracies are 98% and 95% respectively. Higher accuracy was achieved for sites with large amplitudes due to less influence of environmental and numerical noise. Hence the method produces reliable forecast for damaging tsunami impacts. The forecasts of amplitudes, currents and coastal flooding were available approximately 1.75 h after the earthquake: 1.5 h before the tsunami arrived at the first forecast site, Wake Island, and 5 h before the waves reached Hawaii. It should be noted that the forecast was performed in a research environment. Fully implemented operational procedure with automation would reduce the analysis and forecast time to just few minutes after the tsunameter data become available. Data collection processes, including point survey data on inundation and runup for the main Hawaiian Islands and coastal Acoustic Doppler Current Profiler (ADCP) data, are underway. Model comparison with the data will be presented later in a more specific study. [21] The inundation modeling at the most severely impacted Japanese coastlines to the west of the source area also compared well with measurements (Figure 9). The survey data were collected by The 2011 Tohoku Earthquake Tsunami Joint Survey Group [2011]. In areas where the survey data are not available, satellite-based damage estimates conducted by National Geospatial-Intelligence Agency 4 days after the event ( were used. Detailed modeling study of the local tsunami impact is presented in Wei et al. [2012]. The inundation patterns along the Japanese coast and accuracy of predictions at coastal tide stations further validate the deep-ocean measurement derived tsunami source for tsunami forecasting in the near and far fields. [22] To provide guidelines for the earliest reliable forecast, we compare the 1-, 2, and 3-tsunameter inverted sources as in Figures 4, 10 and 11. Figure 4 compares the initial ocean surface deformations, Figure 10 shows the amplitude time series at 30 tsunameter stations, and Figure 11 compares the inundation results in Sendai. Although the shapes of initial deformation are different, all three sources produced similar and good inundation pattern in Sendai (Figure 11). Sources 2 and 3 also produced consistent amplitude time series (Figure 10). The results indicate that the tsunameter constrained tsunami energy is one critical source characteristic. Smaller-scale details, such as the exact contribution of an individual tsunami source function, are of second order importance for the characteristics of the propagating tsunami. As shown in equation (2) and also will be discussed in section 4, the tsunami propagation is strongly affected by the undisturbed water depth d, e.g., seafloor topographic features. Therefore the waves are not very sensitive to details of the initial ocean surface deformation, as long as the energy and the general source area are determined correctly. Figure 10 also shows source 1 (using the first wave at station 21418) produced good forecasts particularly for those stations that have the same (or similar) directionality to the source as the station. These results provide important guidelines for timely and effective forecasts for the system. Reliable regional forecasts may be achieved from a first wave recorded at a single near-field tsunameter. The first wave 10 of 28

11 Figure 8. Energy propagation patterns guided by seafloor topography during the 2011 Japan tsunami. Filled colors show the maximum offshore sea surface elevation in m during 48 h of wave propagation. Contours indicate computed arrival time in hours. White bars represent the maximum wave crests observed at coastal tide stations. The inset shows source geometry of the pre-computed tsunami source functions. 11 of 28

12 Figure 9. Modeling of local impacts in Japan for the 2011 tsunami. Filled colors show the maximum computed sea surface elevation in meters. Black contours show computed tsunami arrival time in minutes. The model setup contains three telescoping grids with increasing resolution of (a) 60, (b) 15 and (c) 2 arc-sec. Modeled flooding extents are indicated as the white areas in Figure 9c. Black lines represent the survey flooding data. Satellite-based damage estimates are indicated in blue lines for areas where survey data are not available. 12 of 28

13 Figure 10. Comparison of modeled tsunami amplitude time series from the 1-, 2- and 3- tsunameter inverted sources at 30 tsunameter stations. A time delay of Dt minutes was applied to the modeled time series in each figure. 13 of 28

14 Figure 11. Japan. Modeled inundation from the (a) 1-, (b) 2- and (c) 3-tsunameter inverted sources at Sendai, recorded at two near-field tsunameters with different directionalities can provide accurate forecasts basin-wide. Adding the third tsunameter has little effect in far-field but it may lead to some improvement in the near-field. 4. Generation, Global Propagation and Energy Decay [23] In this section we shall discuss the generation, propagation and decay of tsunami wave energy and numerical dissipation Generation [24] Figure 12a shows computed tsunami energy time series within the first 3 h in detail. At tsunami origination stage, it takes approximately 2.5 min to generate the first wave, where both kinetic energy E k and potential energy E p account for half of the total initial energy (E T ). At 15 min, the initial energy (E T ) has been converted to wave energy, e.g., kinetic energy is reasonably equal to potential energy. Thereafter E k and E p are conserved with each other. The adopted elastic deformation model assumes instantaneous bottom motion and zero initial velocities here, yet earthquakegenerated tsunamis often involve dynamic bottom motion and nonzero initial velocities. Regardless the ratio of E k to E p at the initial stage, we can expect eventually all the transmittable energy shall convert to ocean wave energy during propagation (e.g., E k = E p ), which then can be well measured by the deep ocean tsunameter stations. Note the travel time to the nearest and most severely inundated Japan coasts ranges from 30 to 60 min (Figure 9). Therefore, our model result suggests it is the wave energy that caused the near-field coastal impacts in Japan Numerical Dissipation [25] Figure 12a shows substantial decay of computed tsunami energy during the period 0.5 to 1 h after tsunami generation, when the first wave propagates into the shallow water near Japan and continues all the way up to Japan coast. Since no friction is considered in the propagation modeling, the decay of energy is due to numerical dissipation nearshore. The nearshore wave dynamics cannot be resolved by the coarse 4 arc min resolution of p the propagation grid. The phase velocity of the wave, C ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi ðg=kþtanhðkdþ, which is approximately gd in the long-wave approximation, decreases in shallow water. The decrease in transport speed requires an increase in energy density in order to maintain a constant energy flux, e.g., an increase in wave amplitude. Meanwhile, the shortened wavelength (l = C/wave period) decreases the number of nodes per wavelength in the computational grid, resulting the increased wave amplitudes being undersampled. For instance with 4 arc min resolution in Japan area, a tsunami wave of 15 min period at water depth of 4000 m would have 31 nodes per wavelength. It then reduces to 4 nodes per wavelength at water depth of 100 m. The numerical dissipation due to the spatial aliasing is amplitude correlated and can be significant for large waves. As an example, the maximum amplitude at Japan coast is 8 m from the 4 arc min ( km) propagation grid, but reached 44 m with a telescoping grid of 15 arc s (360 m) resolution (Figure 9). Therefore the propagation amplitudes should not be used directly for coastal flooding predictions. Note we have applied 14 of 28

15 Figure 12. Time series of computed tsunami wave energy in the global domain. The instantaneous energy E was computed from every node for (a) the first 3 h and (b) every 4 nodes for 48 h propagation. Numbers on the top indicate the amount of energy drop (due to numerical dissipation near shore) during that particular time interval along time-axis. For example, the first number, 45, in Figure 12b indicates that 45% of the total energy (E T ) was dropped during the 0 2 h. Black line, a model scenario with improved boundary conditions. Dash-dotted lines, least square fits to the total energy. high-resolution telescoping grids for coastal modeling to improve energy conservation. [26] With tsunami waves propagating in the ocean as well as entering more coastal areas, the total energy continues to decay due to numerical dissipation. Figure 12b shows two additional quick energy drops around and h. Movie S1 indicates these occurred as waves arrive at North and South America. South America is in the direction of the main energy rays (Figure 8); therefore it has more numerical dissipation than North America, even though its travel time is 12 h longer. Note the sub-sampling of the output in every 4 nodes causes the wiggles in the energy time series in Figure 12b, which are absent in Figure 12a when it is computed from every grid point. [27] A numerical experiment was conducted to further explore the numerical dissipation. We replaced the simple coastal reflective boundary conditions of the propagation model with the scheme used for the inundation modeling, while retaining the same vertical wall at 20 m water (E. Tolkova, unpublished presentation, 2012). It allows waves to run down the bathymetric features, if water withdrawal is substantial. Figure 12b shows it can reduce numerical dissipation in the propagation modeling. The numerical dissipation rate obeys the common exponential 15 of 28

16 decay law. As will be discussed in section 4.4, it can be tuned to better simulate the physical dissipation. [28] Although the numerical dissipation is significant in the near field, the 4 arc min grid is still able to produce a valid propagation scenario for coastal forecasts (Figures 5 and 6). The reasons are as follows: [29] 1. The inversion time windows did not include the portion of waves affected by the numerical dissipation. Therefore the total energy estimation was not subjected to the numerical dissipation. [30] 2. The major numerical dissipation (45% in the first 2 h) was mainly caused by the wave propagated westward to Japanese coasts. It did not affect the first wave propagated eastward directly into Pacific Ocean, which contained the highest energy flux and resulted in the major coastal impacts. [31] 3. Compared to the first wave, the energy reflected from Japan coasts was relatively small. As will be discussed in the next section, the maximum energy flux ratio of the reflected wave to the first direct (also the largest) wave is 3% at station [32] 4. The numerical dissipation in the far field provided a reasonable approximation of the physical dissipation and energy trapping in the coastal areas. [33] 5. High resolution nested grids are employed for the coastal forecasts, which improves energy conservation in the coastal areas. [34] In summary, the energy estimation and major energy flux were not affected by the significant, near-field numerical dissipation. The dissipation may cause underestimates of some later wave trains at certain coastal sites. A nested grid approach, e.g., applying a high-resolution grid to cover the coastal area in near field, is being tested to improve the energy conservation for the propagation computation Propagation [35] The propagation of the Japan tsunami has distinct near- and far-field patterns for energy magnitude and directionality (e.g., energy flux). Figures 8 and 9a show the computed maximum offshore sea surface elevation in the global domain and Japan area respectively. Since wave energy is related to the square of wave height, Figures 8 and 9a indicate patterns of the maximum energy density in deep water. Near the source area, larger offshore amplitudes are in the direction perpendicular to the trench. Previous studies on seismic parameters affecting tsunami wave amplitudes by Ben-Menahem and Rosenman [1972] and Okal [1988] have concluded that this is due to the fault orientation, which in most cases coincides with the direction of the ocean trench. In the far field, the ray patterns represent delayed and amplified main signals (Figure 8). The tsunameter data also show the near- and far-field patterns. The first (direct) wave being the largest in near field, yet it has lower amplitude than the main signals in far field (Figure 5). Within the first 2 h, the first wave is significantly greater than the late waves. Then the amplitude ratio of the first to the largest late wave decreases with increasing travel time. When the travel time is longer than 5 h, the ratio has been dropped to less than 1. This indicates energy of the first wave has been redistributed within 5 h, through annulus spreading, multiple refractions, scattering and reflections, and near-field coastal dissipation and trapping. [36] Previous analytic and modeling studies have demonstrated the importance of tsunami wave scattering by seafloor topographic features in the open ocean [Mofjeld et al., 2001; Kowalik et al., 2008]. Mofjeld et al. [2001] have shown the most important factor determining the intensity of scattering and reflection is the depth of a feature compared with the depth of the surrounding area; others include the horizontal extent of the feature, compared with the wavelength, and the incident angle. For a regional depth of 5000 m, features with depth shallower than 1500 m interact significantly with tsunami and those shallower than 400 m can have a major effect on the waves [Mofjeld et al., 2001]. Kowalik et al. [2008] have shown scattering can cause the main signals to be amplified and delayed. In this section, we shall apply the energy flux vector approach introduced by Kowalik and Murty [1993], Kowalik et al. [2008], and Henry and Foreman [2001] to study the later waves. By replacing h with the h and multiplying equation (2a) by rgh, equation (2b) by (h + d)u, and equation (2c) by (h + d)v, adding the resulting equations, we have the energy balance equation e t þ E x x þ E y y ¼ gn2 ðh þ dþ 1=3 u2 þ v 2 juj; where e is the surface energy density e ¼ e k þ e p ¼ 0:5rðhþ dþ u 2 þ v 2 þ 0:5rgh 2 ; E represents the energy flux vector E ¼ E x ; E y ¼ rhþ ð dþ u 2 þ v 2 =2 þ gh fu; vg: ð9þ Since velocity is small in the deep-ocean, by omitting the higher order term in (8), we have e t þ E x x þ E y y ¼ 0: ð8þ ð10þ That is to say, tsunami energy dissipation takes place through bottom friction in the shallow water in coastal areas. For areas in deep-ocean, equation (10) states the change of energy in time is balanced by the fluxes of energy in and out. Kowalik [2008] has illustrated that for the progressive wave, the energy flux vector is perpendicular to the wavefront.the first wave, generated 2.5 min after the earthquake as suggested by the model result, transmits energy mainly in two opposite directions (both perpendicular to the ocean trench): one major portion propagating directly into Pacific (far-field impacts), and the other major portion propagating toward Japan coast (near-field impacts). In near field, energy is dissipated through bottom friction and onshore hydrodynamic processes, some is trapped along Japan coast and shelf, and some is reflected back into Pacific Ocean. A portion of the numerical dissipation can be used to account for the physical dissipation and energy trapping in the coastal areas. [37] Next we use near-field tsunameter data to quantify the energy density ratio of the reflected wave to the direct wave. Figures 3a and 3k show the second largest crests recorded at stations and are 0.28 and 0.18 m respectively. Both of them arrive approximately 2 h after the first peak. The modeled energy flux vector plots (Figures 3b and 3l) and Movie S1 help to identify the source of the wave. It originates 16 of 28

17 Table 2. Wave Crest and Energy Density Ratios Between the First and Honshu Waves at Two Near-Field Tsunameter Stations Crest (cm) Spatial Arrival Time Crest (cm) Arrival Time Crest Ratio h /h Energy Ratio 2 2 h /h The largest crest, h 1 (the first wave) h 34 min 66 1 h 20 min 2.5:1 6.2:1 The second largest crest, h 2 (Honshu wave) 28 2 h 40 min 18 3 h 29 min 1.6:1 2.4:1 Amplitude ratio, h 1 /h Energy density ratio, h /h from a strong oscillation on Honshu coast and shelf at 1 h 56 min after the earthquake. We will refer to it as Honshu wave (HR) hereafter. It is propagated back into Pacific as the second largest wave, characterized by its long peak period, 47 min at and 75 min at (Figure 3). This shows waves originated from different locations, such as the Honshu wave and the first wave directly from the source area (shelf edge), can have substantial different peak periods at the same deep ocean site. At 21413, the Honshu wave has the similar southeast propagation direction as the first wave (Figure 3l). At 21418, the propagation direction is different from the first wave, which is due to the coastal and shelf refraction (Movie S1). Table 2 summarizes the amplitude and energy density flux ratio of the Honshu wave to the first wave at the two stations. The ratio is small, 3% and 8% at and respectively. Compared with the 2 to 1 energy density spatial ratio of the Honshu wave, the first wave shows stronger directionality, with a 6 to 1 spatial ratio between the two stations. At station 21401, it is difficult to distinguish the Honshu wave from the Emperor Seamounts reflection due to their close arrivals (Figures 3f and 3g). [38] Figures 13a 13n and Movie S1 illustrate the dynamics of the tsunami propagation globally and the evolution of wavefronts affected by bathymetric features. The initial sea surface deformation generated two waves (W1 and W2 in Figures 13a and 13b) propagating outwards from the source area. In about half an hour, the first positive wave reached Sendai with a maximum sea surface elevation of 44 m (Figure 9b). The first flood wave reached the Sendai coast at approximate 1 h with maximum computed sea surface elevation ranging from 8 to 44 m (Figure 9c). The shelf and coast near Honshu created multiple reflections and oscillations (Figures 13b 13d). At 1 h 56 min, the Honshu wave propagated back into Pacific as the second largest wave. The Emperor Seamounts reflection E1 arrived at Northern Japan near 6 h after the earthquake, contributing to another group of large waves recorded at Northern Japan tide stations (Figures 13e 13h). Meanwhile, the Koko Guyot and then Mellish Seamount created two sets of circular wavefronts K1 2 and M1 2 by wave scattering (Figures 13f 13h). The Mellish Seamount directed an energy ray away from the main direction of energy propagation toward the Mendocino Escarpment offshore of northern California (Figure 8). At about 7 h, the tsunami arrived at the Hawaiian Island Chains, producing two more circular wavefronts H1 and H2 (Figure 13i). [39] Seamounts and ridges serve as waveguides of tsunami energy [Titov et al., 2005a]. The seamount chains in the Pacific (Emperor-Hawaii Seamounts, Mid-Pacific Mountains, Christmas Ridge, Polynesia Seamounts, Galapagos Islands and Nazca Ridge) provided energy paths for the Japan tsunami to reach Chile s coastline in 20 h with high wave amplitudes (Figures 13k and 13l). The continental shelf trapped these ringing waves for the Pacific Coast of South America, causing the late arrival of the maximum wave (2.5 m crest and 4.6 m wave height) 9 h after the initial tsunami excitation at the tide gage of Arica, Chile. Waves reflected from South America arrived at Hawaii and North America at 37 h (Figure 13m), and reached Japan 46 h after tsunami generation (Figure 13n). Meanwhile, the tsunami propagated into the Indian and Atlantic Oceans (Movie S1). While the direct waves from the source transported most of the energy into the Atlantic Ocean south of Cape Horn, it was the reflected waves from South America that transferred most of the energy into the Indian Ocean south of New Zealand (Figures 13l and 13m). [40] Figure 14 shows some interesting amplitude/energy/ period patterns recorded at two far-field stations offshore Oregon (close to Crescent City) in North America, and offshore Peru in South America. The stations are 370 and 1400 km offshore with water depths of 3266 and 3890 m respectively. Data from the stations exhibit a similar pattern for the delayed and amplified main signal: a sudden deep trough followed by a peak of the same magnitude, with a short period approximately 1/2 and 1/4 1/5 of the two long periods (Figures 14a 14e and 14f 14j). Figure 8 indicates both stations are located at the intersections of multiple energy rays. Station is not only on the ray refracted from Mellish Seamount to Mendocino Escarpment, but also on those reflected by Oregon/Northern California and Canada. Station is on the rays refracted from Christmas Ridge and Galapagos Islands. The approximately simultaneous arrivals of multisource waves may also contribute to the amplified signals (Movie S1). The propagation scenario accurately forecast these delayed and strongly amplified main signals in the far field. [41] Inversion of the deep-ocean tsunami data with pre-computed tsunami source functions requires accurate bathymetry in the models in order to adequately track the travel time and propagation patterns of the tsunami waves. Figure 5 shows the largest travel time error was 12 min at station offshore Chile after more than 20 h propagation time (Figure 5t), while no error was found for the stations to the south of the source area (Figure 5u 5bb). Therefore, the travel time errors are path-specific. Further examination of the error reveals it relates to the effective wave scatterers in the path. Our results indicate: [42] 1. Stations along the path from northwest to southeast across the Pacific Ocean, in which the major Pacific scatterers are located (Movie S1), show the largest travel time error, 5 min for station in Hawaii (0.7 min/h for 7.8 h travel 17 of 28

18 Figure 13. Snapshots of the modeled wave patterns during 48 h propagation. (a and b) The initial ocean surface deformation generated two waves, W1 and W2. (b and c) W1 reached Japan coast and then reflected as wave R1. (d) Wave HR generated by shelf resonance near Honshu propagated 1 h apart from R1. (e) The reflection by Emperor Seamounts created wave front E1. (f) Waves K1 and K2 were created by Koko Guyot. (g) Waves K1 and K2 developed circular fronts. (h) Wave E1 arrived at Northern Japan. Waves M1 and M2 were created by Mellish Seamount. (i) Waves H1 and H2 were created by Hawaiian Islands Chain. (j) Major Waves W1, W2, K1, K2, M1, M2, HR, H1 and H2 were approaching North America. (k) Tsunami propagated in South Pacific. (l) The Easter Fracture Zone refracted the wave front. (m) Waves reflected from South America arrived at North America and Hawaii. Tsunami waves propagated through Indian and Atlantic Oceans. (n) Reflections from South America arrived at Japan 46 h after the earthquake. time), 12 min for station offshore of Chile (0.6 min/h for 20.7 h travel time). [43] 2. Travel time error is negligible ( 1 min, which is the output temporal resolution of the database) for stations with no effective scatterers in the path, such as the stations to the south of the source. Note the first wave that arrived at station northwest of Apia (at 8.5 h with no travel time error, Figure 5z) is the wave segment that passes through the gap 18 of 28

19 Figure 14. Similar delayed and amplified main signals, a sudden deep trough followed by a crest of the same magnitude, recorded at (a) offshore Oregon, North America and (f) offshore Peru, South America. (b and g) Time series of modeled energy flux vector. (c and h) The wavelet-derived amplitude spectra of the observations show the both main signals have a short period approximately 1/2 and 1/4 1/5 of the two long periods. (d and i) The real parts of the spectra are plotted. (e and j) The model forecast from the 4 arc min propagation grid reproduces well the frequency/period patterns of the observations. between the Mid-Pacific Mountains and Marshall Islands Chain (Movie S1). For the 3.7 h travel time from stations to along Aleutian, the added error is only 1 min. We have also noticed for the 2006 Tonga tsunami, which the source was inverted from data recorded at two tsunameters in Hawaii, the models can accurately reproduce the arrival time of the waves reflected from North America that arrived in Hawaii 16 h after the earthquake, as well as the travel time from Hawaii to the Alaska and West Coast stations [Tang et al., 2008]. The lack of effective wave scatterers between Hawaii and North America/Alaska explains it well. [44] 3. Modeling of waves passing through Emperor Seamounts may have 2 3 min travel time error. Figure 5 shows no error for the stations 21401, and to the west of the Emperor Seamounts (Figures 5b 5d), while 2 3 min error for the stations, and 21414, to the east, soon after the first wave propagating over the Emperor Seamounts (Figure 5e and 5f). [45] Therefore wave scatterers can delay not only the main signal but also the first wave due to their shallow water depth. The travel time error is mainly caused by the approximation of major shallow scatterers using the 4 arc min DEM (digital elevation model). Other factors may include the 100 by 50 km spatial resolution of the source area for each tsunami source function, the uncertainty of the oceanic bathymetry, and the MOST model approximations. Since tsunami travel time is important for locating the tsunami origination area, these results provide useful guidelines to select tsunameters with less (or adjusted) travel time error for use in inversion. It would be interesting to test the approach of applying highresolution nested grids covering the major scatterers in the Pacific for the propagation modeling. In addition, we can 19 of 28

20 Figure 15. Decay of amplitude and variance (energy) for the 2011 Japan tsunami at Kahului, Crescent City and Adak tide stations. Black and red solid lines, least square fits to the observations and model results. expect inversion of the tide gage data in Hawaii and some areas in Alaska/West Coast with the models could identify source locations for the 1946 Unimak and 1964 Alaska tsunamis accurately Energy Decay [46] Pacific-wide tsunamis remain dangerous for many hours [Mofjeld et al., 1999, 2000]. Later tsunami waves threaten rescue and recovery operations, especially when they arrive at high tide. Such waves also endanger vessels in shallow water. For example, the 2010 Chile tsunami reflected off distant landmasses, along with large surf and a high tide, caused flooding in Seal Beach and Sunset Beach in Orange County, California 1 day after the earthquake [Wilson et al., 2010]. For these reasons, emergency managers need wave height forecasts to help guide rescue and recovery operations. They also need them to decide when to issue the all-clear. Previous studies by Mofjeld et al. [2000] on forecasting of the later waves were from a statistical approach, based on Van Dorn s [1984, 1987] findings that the energy of later waves tends to decay exponentially with a relatively long e-folding time. Our forecast of the later waves is a deterministic approach since it explicitly gives estimates of amplitude, period, arrival time, and duration of wave groups. As shown in equation (8), tsunami energy dissipation takes place through bottom friction in the shallow water of coastal area. In this section, we use coastal tide gage data and model results to study the decay of tsunami energy. [47] Van Dorn [1984, 1987] has found the time decay of tsunami variance is uniformly exponential for 28 events among eight Pacific tide stations, with the mean decay (e-folding) time (22 h) close to ocean s normal mode transit time (21 h). Three coastal stations, Kahului, Crescent City and Adak, which recorded the largest wave in Hawaii, U.S. West Coast, and Alaska, were used here (Figure 15). The tide gage data were filtered by a low-pass Butterworth filter, with a cut-off of min, to isolate lower-frequency components (including tide), which were then subtracted from the raw data. For each station, 28 days of 1 min filtered data were divided into 12-h segments, beginning 8 days before the earthquake. The variance was computed for each segment and plotted in logarithmic scale to show the decay of variance (energy) with time after earthquake. A least squares fit was applied to the variances for times from the maximum energy density to 5 6 days after the earthquake. Figure 15 shows similar slopes of the fitted lines at three stations. Our result (t 0 = h) confirms Van Dorn s conclusion that tsunami obeys the common decay law et ðþ¼e 0 expð t=t 0 Þ; t max < t < 5 6 days; where t max is the time for the maximum energy density. After 6 days, tsunami signals were too weak to be distinguished from signals caused by tidal and other processes. [48] The three inundation models were rerun for 86 h using the propagation scenario with improved boundary conditions. The same analysis was performed with model time series and results are plotted in red in Figure 15. With t 0 = h, the modeled variances (energy) decay faster than the observations, which may be caused by the numerical dissipation of the propagation model and flood model setup. The 17 h mean decay time of the flood models is consistent 20 of 28

21 with the 17.6 h of the propagation scenario (Figure 12b). We shall expect fine-tuning the numerical dissipation can be used to account for the physical dissipation and trapped coastal energy for the propagation model. Both Crescent City and Kahului tide stations recorded waves with 1-m amplitude around 40 h after the earthquake. Those highlight the need for good simulation of the physical dissipation so the models can better predict the arrival and duration of the major transocean tsunamis. 5. Coastal Impacts [49] The 2011 Japan tsunami was recorded at coastal tide stations globally. Figure 8 shows the maximum crest recorded at 156 coastal tide stations. The incomplete tide gage records from Japan Meteorological Agency near the epicenter show first peak reached more than 8 m at Miyako and Ofunato. High wave crests were also observed, 2.5 and 2.0 m at Crescent City and Port San Luis in California, 2.1 m at Kahului in Hawaii, and 2.5 m at Arica, Chile (4.2, 4.3, 3.9 and 4.6 m peak to trough wave height respectively). [50] The recorded maximum coastal crests and computed offshore energy patterns in Figure 8 reveal three important features for coastal impacts. [51] 1. High coastal amplitudes are in general correlated with high offshore amplitudes. For example, at Japanese and Alaskan tide stations, the recorded coastal amplitude decreases with decreasing offshore amplitude (due to the increasing distance from the source). In North America, high coastal amplitudes were observed at stations in northern California, correlated with the ray of high-energy flux resulting from the distant scattering and refraction from the Mellish Seamount. The further away from this ray, the smaller the recorded coastal amplitudes. Located on the main direction of energy propagation, coastal stations in South America recorded higher amplitudes, compared to those in New Zealand and Australia. These indicate the incident offshore energy is the primary source for coastal impacts. [52] 2. Depending on the shape, orientation and path, local bathymetry can enhance the coastal amplitudes to significantly different extents. For example in Hawaii, the 2.1 m recorded maximum crest at Kahului on North Maui is more than 10 times higher than the 0.19 m at Kaneohe on East Oahu. In California, the 2.03 m crest at Port San Luis is more than triple the 0.66 m at the adjacent Monterey station approximately 160 km to the north. These results point to the importance of high-resolution site-specific flood models, which can take into account the details of local bathymetry, for accurate coastal forecasts. (3) As being discussed in section 4.3, coastal sites located within the energy rays expect delayed and amplified signals (Figure 8). [53] In the following section, we analyze the highresolution flood model results and coastal tide gage data to study the details of coastal impacts in the far field Pacific Islands and Hawaii [54] The tsunami arrived at the first two forecast sites, Wake Island and Midway Islands, 3.5 and 4.5 h after the earthquake. The two tide stations show similar first wave crest of 0.48 and 0.37 m. While the amplitudes decreased at Wake Island afterwards, they were amplified at the Midway Islands for more than an hour, with a recorded maximum of 1.32 m. Waves scattered from nearby Emperor Seamount and Hawaiian Islands chains provided long duration high energy input for the Midway Islands. The Midway Islands consist of two significant pieces of land, Sand Island (1200 acres) and Eastern Island (334 acres), and a small Split Island (6 acres) in between. The maximum land elevation on the three islands is 10.36, 4.32 and 1.34 m above mean high water level respectively (E. Gica, A tsunami forecast model for Midway Island, NOAA OAR Spec. Rep./PMEL Tsunami Forecast Ser., under review, 2012). Spit Island was completely washed over. Eastern and Sand Island were 60% and 20% washed over, respectively ( tsunami.html). Midway residents received approximately 4 h of advanced warning and successfully implemented its tsunami emergency plan. No one was injured and no major damage occurred to the infrastructure. Debris washed onto the airfield, which caused its temporary closure for less than 24 h. Thousands of Bonin petrels were buried alive ( Although the forecast accuracy of the maximum crest is 85%, the Midway model predicted no tsunami flooding. Further investigation shows it was due to the inaccurate topography. With the latest USGS 1-m Lidar topographic data, the updated Midway model reproduced the observed inundation (Gica, report under review, 2012). [55] Located on south coast of Wake Island, the tide station is on the lee side of the incident waves. D. Arcas (A tsunami forecast model for Wake Island, NOAA OAR Spec. Rep./ PMEL Tsunami Forecast Ser., under review, 2012) has shown that Wake Island is particularly protected from tsunami impact. One reason is due to the absence of shallow waters in the vicinity of the island for enhanced shoaling. The first wave being the largest is unique among the 32 coastal stations shown in Figure 6. [56] In Hawaii, the maximum-recorded wave height was at the Kahului tide station with reported inland flooding penetration of 578 m near the harbor area. Analysis of the high-resolution forecast model output allows us to identify that the causes of the flooding at Kahului were mainly due to refraction focusing and local resonances (Movie S2 and Figure 16). Figure 16 shows a peak period near 31 min recorded at the tide station. A similar 31 min period was also observed at Kahului during the 1994 Southern Kuril Islands tsunami and the 2003 Hokkaido tsunami [Tang et al., 2009], which originated from areas close to the source area of the Japan tsunami (Figure 7). This confirmed the conclusion made by Tang et al. [2009] that while coastal tsunami frequency responses are dominated by the local bathymetry, tsunamis originated from areas close to each other tend to excite the same peak period at the same coastal site. That is to say, the bathymetry at the coastal site, at the tsunami origination area, and along the path between the two determine the coastal tsunami frequency responses, while the details at the tsunami origination have little effect. Loomis [1966] has shown that for three past tsunamis, a given station in Hawaii gives its spectral imprint to tsunami incident upon it. Munger and Cheung [2008] have conducted spectral analysis of the computed surface elevation in Hawaii waters for the 2006 Kuril tsunami. They show that the resonant amplification has strong correlation between the length scale of bathymetric features and the frequency content and direction of the tsunami. The forecast peak period can be a 21 of 28

22 Figure 16. (a) Observed and forecast time series of wave amplitudes at Kahului tide station. (b) Time series of wavelet-derived amplitude spectra of the observations. Real parts of the spectra are plotted for the (c) observation and (d) forecast results. A time delay of Dt = 6 minutes was applied to the forecast time series in the figures. good indicator for the uncertainty in the forecast maximum amplitude at a particular coastal site. For example, Tang et al. [2009] has shown that the developed Kahului flood model can reproduce well the peak period of, or longer than, 24 min for the past eight tsunamis, while it was less satisfactory in modeling the relatively short peak period of 16 min for the 2006 and 2007 Kuril Islands tsunamis. So less (more) uncertainty in amplitude forecast is expected at Kahului when the forecast peak period is long (short). It may relate to the numerical dissipation for waves of different frequencies. [57] The southwest coast of the Island of Hawaii was severely damaged. Water slammed into the end of the Keauhou bay, destroyed Keauhou Yacht Club, and severely damaged three ocean sports activity offices (S. Bracken, Tsunami cleanup update, online article, Big Island News Center, 2011, Our measured maximum runup reached 4.0 and 3.6 m at Keauhou Bay and Kona respectively. Video footage from an eyewitness (recorded after severe damage from previous waves) show flooding at Kona even 2 h after the tsunami arrived (S. Farmer, video footage, 2011). The Keauhou and Kona flood models are under development. Preliminary result shows the Keauhou model can reproduce the observed inland flooding penetration of 100 m; however the maximum runup is underestimated by 1.4 m. Note the amplitude time series at six coastal stations in Hawaii were well forecast (Figures 6a 6f), including the nearby Kawaihae tide station on the northwest coast of the Big Island (1.0 m maximum recorded crest). The fact that tsunami wave can be strongly amplified at Keauhou Bay was also observed during the 2009 Samoa tsunami. The Samoa tsunami arrived at Hawaii at high tide. While the nearby Kawaihae station recorded only a 0.28 m maximum crest (0.49 m wave height), the Keauhou Parking lot, located at the end of Keauhou Bay, 0.9 m above mean high water level, was flooded with m water (K. Bell, photo data, 2009). This site is currently under investigation as to why it can amplify waves 3 4 times of those at Kawaihae tide station. A 1.14 m maximum crest was recorded at Hilo tide gage, located on the east side of the Big Island (lee side of the incoming waves). No damage was reported for Hilo U.S. West Coast [58] Tide stations in Northern and Central California near the Mendocino Escarpment recorded large waves (Figures 6x 6cc), while much smaller waves were recorded at tide stations in Washington to the north and in southern California to the south (Figures 6r 6w and 6dd 6ff). This correlated with the ray of high-energy flux resulting from the distant scatterings and refractions from the Koko Guyot and Mellish Seamounts and enhanced by Mendocino Escarpment. 22 of 28

23 Figure 17. (a) Observed and forecast time series of wave amplitudes at Crescent City tide station. W, K, M, HR, and H indicate the approximate arrival time of the first waves and feature waves scattered by Koko Guyot, Mellish Seamount, Honshu and Hawaiian Islands Chain. (b) Time series of wavelet-derived amplitude spectra of the observations. Real parts of the spectra are plotted for the (c) observation and (d) forecast results. A time delay of Dt = 9 minutes was applied to the forecast time series in the figures. The further away from this ray, the smaller the recorded coastal amplitudes. [59] At 9 h 15 min, tsunami waves were traveling along Mendocino Escarpment and approaching U.S. West Coast in the sequence of W1, W2, K1, K2, M1, M2, HR, followed by H1, H2 (Figure 13j). The Crescent City tide station recorded three groups of large waves, the fourth, seventh and fourteenth waves of 2.5, 2.2 and 1.9 m crests (4.2, 4.1 and 4.0 m wave height) with complex frequency patterns (Figure 17). Our model results indicate the 4th, 7th and 14th waves occurred at the arrivals of waves K, M and HR, and H respectively (Movies S1 and S3). Kowalik et al. [2008] have identified Koko Guyot and Hess Rise (Mellish Seamount here) as the prime sources for larger energy fluxes at Crescent City delayed by distant scattering during the 2006 Kuril tsunami; and the refraction at Mendocino Escarpment further concentrates the energy flux to Crescent City. The energy flux vectors from the offshore tsunameter clearly demonstrate the delayed and refracted strong energy flux (Figures 14a and 14b). Two major peak periods of 22 and 30 min were recorded at Crescent City tide station (Figure 17). Horrillo et al. [2008] have conducted numerical studies to calculate the natural periods and corresponding natural modes for Crescent City and shelf. They have identified a 21 min period (22 min here, the 1 min difference could due to rounding error), the main peak in the power spectra of the 2006 Kuril tsunami, defined not by the geometry of Crescent City Harbor but the shore line and offshore shelf. They also show a 30 min period was one of the secondary resonance periods for the adjacent shelf. The 22 min peak period (but not the 30 min one) was also observed at the station during the 2006 Tonga tsunami [Tang et al., 2008]. Figure 18 compares the modeled time series of tsunami energy within the Crescent City Harbor during the 2011 Japan and 2006 Kuril tsunamis. The energy level in the harbor reached the maximum when the M waves arrived. The tsunami damaged 30 vessels (10 sunken) inside the Harbor (U.S. Coast Guard, 2011, while the 2006 Kuril tsunami damaged 63 (16 sunken). The total damage is estimated $16 and $22.5 million for the 2011 Japan tsunami and the 2006 Kuril tsunami respectively (M. Hansen, Total damage estimate $38.5M, online article, 2011, triplicate.com/news/local-news/total-damage-estimate385m). The Japan tsunami arrived at Crescent City during low tide (Figure 15d). This, along with effective warning and preparedness, may explain why less damage was reported during this event, even though the energy received by the harbor was 15 times of that of the 2006 Kuril event. 23 of 28

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