A comparative study of far-field tsunami amplitudes and ocean-wide propagation properties: insight from major trans-Pacific tsunamis of 2010–2015

Mohammad Heidarzadeh,1 Kenji Satake,2 Tomohiro Takagawa,3 Alexander Rabinovich4,5 and Satoshi Kusumoto2 1Department of Civil & Environmental Engineering, Brunel University London, Uxbridge UB8 3PH, UK. E-mail: mohammad.heidarzadeh@brunel.ac.uk 2Earthquake Research Institute, The University of Tokyo, Tokyo 113-0032, Japan 3Port and Airport Research Institute, Yokosuka 239–0826, Japan 4Institute of Ocean Sciences, Sidney, British Columbia, V8L 4B2, Canada 5Shirshov Institute of Oceanology, Russian Academy of Sciences, Moscow 117997, Russia

. (a) Map of the Pacific Ocean and locations of DARTs used in this study. Dashed contours represent maximum simulated tsunami zero-to-crest amplitudes in cm from our Pacific-wide tsunami simulations for the 2015 Illapel event. The black and red numbers close to each DART represent the DART name and maximum observed zero-to-crest tsunami amplitude in cm. The size of the circles is proportional to the observed tsunami amplitudes, while their colour shows the water depth of the DART sensor. Change of text colour between black and white is for increasing the visual clarity of the text. The panel in the top-right corner is the crustal deformation generated by the 2015 Illapel event based on the source model by Heidarzadeh et al. (2016a). (b) Epicentres and magnitudes of past large earthquakes along the Chilean subduction zone.
tating trans-oceanic tsunami that was recorded by a great number of DART stations and by Japanese and Canadian open-ocean cable observatories (Rabinovich & Eblé 2015). The open-ocean records from such a worldwide swarm of great tsunamis are very valuable for studying ocean-wide propagation and deep-ocean properties of tsunamis. Some of the benefits of DART and other open-ocean data are: applications in early tsunami warnings, revealing first negative phases of the tsunami waveforms, and estimation of spectral properties and energy decay of tsunami waves and earthquake source processes (Baba et al. 2004(Baba et al. , 2014Saito et al. 2010;Løvholt et al. 2012;Rabinovich et al. 2013;Allgeyer & Cummins 2014;Heidarzadeh & Satake 2014a;Okal et al. 2014;Watada et al. 2014;Eblé et al. 2015;Zaytsev et al. 2016Zaytsev et al. , 2017. The main purpose of this study is to examine the oceanwide propagation of the 2015 Illapel tsunami through investigating the DART records and comparing them with other recent transoceanic tsunamis. We formulate certain empirical relationships for the durations and amplitudes of initial negative phases of transoceanic tsunamis. Furthermore, the concept of empirical Green's functions is applied to reconstruct the source spectrum of the 2010 Maule tsunami by applying spectral deconvolution using the Illapel tsunami as the empirical Green's function.

DATA A N D M E T H O D O L O G Y
The data used in this study are DART records from the Pacific Ocean provided by the US National Oceanic and Atmospheric Administration (NOAA). The DART stations are located at water depths of 1.5-5 km (Fig. 1) and the records had sampling intervals of 1 min as they were the Event mode data downloaded from the NOAA website (e.g. Rabinovich & Eblé 2015). The methodology was a combination of: time-series analysis, spectral (Fourier) and wavelet analyses, and numerical simulations. The spectral Fourier analysis reveals the characteristics of the source (i.e. dominating tsunami periods) whereas wavelet analysis demonstrates the temporal changes in these periods. All tsunami records were detided by estimating the tidal signals employing polynomial fitting and then removing them from the original records. The other methods for detiding of DART records are conducting harmonic analysis (e.g. Heidarzadeh & Satake 2013bRabinovich & Eblé 2015) or applying digital filters (e.g. Heidarzadeh & Satake 2014b). It was demonstrated by  that these detiding methods give very similar results for DART records. For the Fourier analysis, we applied the MATLAB command Pwelch which is based on the Welch algorithm (Welch 1967) with Hanning window and 50 per cent of overlaps. The length of the waveform used for Fourier analysis was 3-4 hr which is equivalent to 180-240 data points. For wavelet analysis, we applied the wavelet package by Torrence & Compo (1998) using the Morlet mother function with a wavenumber of 6.0 and a wavelet scale width of 0.10.
For tsunami simulations, we applied the shallow-water numerical model by Satake (1995). Linear simulations with time step of 5.0 s were conducted for a total time of 30 hr. The time step follows the Courant stability condition of the Finite Difference computations (Satake 1995). The inundation on dryland was not permitted because the purpose of the tsunami modelling was far-field tsunami propagation and comparison with observed DART data, not estimation of the tsunami run-up. A 5 arcmin bathymetric grid, resampled 24 M. Heidarzadeh et al.     Fig. 1, was calculated by the analytical formulas of Okada (1985). The source model for the 2010 Maule tsunami was based on that of  which includes average and maximum slip values of 3.8 and 18.8 m, respectively, over a fault plane with 36 subfaults (each 50 km × 50 km). The 2014 Iquique source has a maximum slip of 7 m over a fault plane having 63 subfaults (each 20 km × 20 km) . According to Satake et al. (2013), the source size of the 2011 Tohoku tsunami was 550 km (length) × 200 km (width) (over a fault with 55 subfaults; each 50 km × 50 km or 50 km × 25 km) with average and maximum slip values of 9.5 and 69.1 m, respectively. Table 1 presents a summary of the characteristics of various source models used in this study.  These three effects had not been considered in our numerical model which was based on shallow-water equations.

O B S E RV E D A N D S I M U L AT E D T S U N A M I WAV E F O R M S F O R T H E 2 0 1 5 I L L A P E L T S U N A M I
To include the effects of the indicated factors in the tsunami simulations, we applied the waveform phase-correction method developed by Watada et al. (2014). This method takes into account the fact that tsunami phase velocity is not constant as assumed by the shallow-water theory, but is a function of wave frequency. The method by Watada et al. (2014) transforms the simulated waveforms from the time domain to the frequency domain using the Fourier transform and then replaces the phase part of the tsunami spectrum by the corrected phase spectrum. The inverse Fourier transform enables us to restore the phase-corrected time-series (blue lines in Fig. 2). This method takes into account the ocean-water compressibility and the Earth's elasticity and geopotential perturbations to accurately modify the tsunami arrival time delay, as well as its initial phase reversal. Allgeyer & Cummins (2014) applied a different method by including the effects of ocean-water compressibility and seafloor elastic loading in the tsunami simulations and obtained similar results. In most cases, the corrected waveforms are in good agreement with the observations. The exceptions are a few DARTs in the western part of the Pacific Ocean, in particular, DARTs 52401 and 52402; the reason of this disagreement is not quite clear.

FA R -F I E L D P RO PA G AT I O N A N D T R AV E L T I M E S O F T R A N S -PA C I F I C T S U N A M I S
To better understand the mechanism of the arrival time differences between observations and shallow-water simulations in the farfield, we plotted these delays against the distances from the source ( Fig. 3a), angles from the fault strike (Fig. 3b), and wave amplitudes (Fig. 3c). A linear relationship is observed between the distance from the source and the arrival time delay in good agreement with the results previously reported by Takagawa (2013); Watada et al. (2014) and  for several other trans-Pacific tsunamis. Our data resulted in the following linear relationship between distance from the source in km (L) and the arrival time delay in minutes (T del ) with R 2 = 0.79 for the quality of the fit (Fig. 3a): Theoretically, the arrival time delay should be zero for a distance of zero; therefore, the linear relationship in eq. (1) passes through the origin of the coordinate system. In fact, the 1-2 min delays in the near-field (Fig. 3a) are due to the uncertainties in the travel time measurements. It appears that stations located normal to the fault strike receive longer arrival delays (Fig. 3b). However, the plot of the distance against the angle from fault strike reveals that stations located normal to the fault strike are distal (Fig. 3d). In other words, the observed relationship between the arrival time delay and the angle from the fault in Fig. 3(b) could be due to the distance. According to Figs 3(c)-(e), there is no evident correlation between the arrival time delay and the amplitude or moment magnitude.  Tsunami waveforms from three recent major tsunamis generated offshore of Chile recorded at DART stations. The time axis is shifted to align the waveforms at their first peak amplitude. 9, 13 and 28 min (Fig. 4). The two latter peaks (13 and 28 min) are stronger and appear at most stations. Four stations of 52406, 52402, 51425 and 51407 show a single dominating period at ∼19 min (Fig. 4). Distribution of tsunami energy over frequency and time domains reveals that the aforesaid four peak periods show non-stationary behaviour over time. A combination of Fourier and wavelet results may favour the two periods of ∼13 and ∼28 min as prevailing tsunami periods. Wavelet analysis reveals certain energy at periods >30 min before the arrival times of the first elevation wave (e.g. DARTs 32412, 43412, 46403 and 46408) (Fig. 5). In most wavelet plot, an inverse relationship between the time and wave period is observed (in both observations, Fig. 5a, and simulations, Fig. 5b) which shows shorter-period waves appear in DART stations later than longer-period waves (the dashed-line in Fig. 5a in 43412). This behaviour is also evident from the time-series of the tsunami waveforms in Fig. 2, where in most records the first few waves are longer than the later waves. As compared to the other two large tsunamis off Chile, the dominating period band of the Illapel tsunami (13-28 min) was longer than that from the 2014 Iquique tsunami (14-21 min, after , and shorter than the 2010 Maule tsunami (12-50 min, after Rabinovich et al. 2013). The 2011 Tohoku tsunami showed two dominating periods at 37 and 67 min, according to Heidarzadeh & Satake (2013a).

S P E C T R A L A N A LY S I S O F T H E 2 0 1 I L L A P E L T S U N A M I
These results and periods are in good agreement with those found by Zaytsev et al. (2016Zaytsev et al. ( , 2017 based on the analysis of DART records of the 2010, 2011, 2014 and 2015 tsunamis offshore of Mexico.

C O M PA R I S O N O F T H E T S U N A M I WAV E F O R M S F RO M C H I L E A N T R A N S -PA C I F I C T S U N A M I S
Tsunami waveforms from three recent major Chilean tsunamis are shown in Fig. 6 for seven DART stations. The time axis is shifted to align the waveforms at their first peak amplitude. It can be seen that the waveforms from the Illapel tsunami look similar to those from the Maule tsunami, whereas those from the Iquique event are clearly different from the aforesaid two tsunamis (Fig. 6). This is possibly because the epicentres of the Illapel and Maule earthquakes are close to each other (Fig. 6); hence, the resulting tsunamis propagated similar tracks to reach the DART stations. The amplitudes of the Maule tsunami are 2-5 times larger than those of the Illapel (Fig. 6). Since the Illapel earthquake (M w 8.4) was smaller than the Maule (M w 8.8), the former event may serve as an empirical Green's function for the latter earthquake (e.g. Heidarzadeh et al. 2016b).

E M P I R I C A L G R E E N ' S F U N C T I O N S F O R T H E C H I L E A N T S U N A M I S
The concept of empirical tsunami Green's function was formulated and successfully applied to the 2013 Santa Cruz tsunami by Heidarzadeh et al. (2016b). This concept, which is akin to a similar approach in seismology, postulates that for two tsunamis occurring close to each other, the smaller event can be considered as an empirical Green's function to be used to remove the non-source effects from the spectrum of the larger tsunami. According to Heidarzadeh et al. (2016b), spectral deconvolution, conducted by dividing the spectrum of the larger event by that of the smaller event, gives the source spectrum of the larger tsunami. This approach is close to the idea of Rabinovich (1997) where the background spectrum is Here, the epicentres of the Maule and Illapel events are close to each other (Fig. 7), while the energy released by the former earthquake is ∼4 times larger than that of the latter. Therefore, the Illapel tsunami can be used as the empirical Green's function for the Maule event. The ocean-wide DART records of the two events are compared in Fig. 7, while their basin-wide distribution of tsunami amplitudes is shown in Fig. 8. The DART waveforms of the two events look similar, although the amplitudes of the Maule tsunami are 2-5 times larger than those of the Illapel event. The spectral ratios in Fig. 9 (shaded spectra) are the results of the spectral deconvolution and reveal the source spectra of the Maule tsunami, which is in the period range of 13-50 min (Fig. 9). The bottom limit of this period range, that is 13 min, is the period that gives a spectral ratio of ∼1 and the upper limit, that is 62 min, is the period that most of the ratios become either flat or ∼1.0. This tsunami source period band is close to that of 12-50 min previously reported by Rabinovich et al. (2013) and Zaytsev et al. (2016) which was calculated by the direct Fourier analysis of the Maule tsunami waveforms. For comparison, the source periods of other tsunami events worldwide are: 20-90 min for the 2011 Tohoku tsunami (Heidarzadeh & Satake 2014a;Zaytsev et al. 2017) and 2-30 min for the 2017 Bodrum-Kos (Turkey-Greece) tsunami (Heidarzadeh et al. 2017).  . Parameters "a", "b" and "c" are: the maximum amplitude of the initial negative phase, the maximum noise amplitude before the tsunami arrival, and the maximum amplitude of the first frontal crest wave, respectively. The small grey circles in panels (d), (e) and (f) indicate the data available from DART observations for various tsunamis. Fig. 10 shows the maximum amplitude and duration of the observed initial negative phases at DART stations for the four trans-Pacific tsunamis. Here, the duration of the initial phase is defined as the time interval between the beginning of the sea level recession relative to the mean sea level (the initiation of the negative phase) and the time that it reaches the mean sea level again. For the present analysis, the DART data with relatively high signal-to-noise ratios were selected to achieve reliable results (Fig. 10). The durations of the initial phase (D ini ) were 8-29, 20-35, 22-70 and 40-79 min for the Iquique, Illapel, Maule and Tohoku tsunamis, respectively (blue values in Fig. 10). The maximum negative amplitudes of the initial phases (A ini ) were 0.11-0.26, 0.4-0.7, 0.5-2.9 and 1.9-2.5 cm for these tsunami, respectively (red values in Fig. 10), while the amplitude ratios of the negative phases relative to the amplitude of the first elevation wave were 20-40 per cent, 22-41 per cent, 29-61 per cent and 12-67 per cent, respectively (green values in Fig. 10). The amplitudes of the initial phase were significant as compared to the frontal elevation tsunami amplitudes.

C H A R A C T E R I S T I C S O F T H E I N I T I A L N E G AT I V E P H A S E S F O R T R A N S -PA C I F I C T S U N A M I S
Figs 11(a)-(c) shows the durations, the amplitudes, and the amplitude ratios of the initial phases as a function of distances of the DART stations from the source for the four trans-Pacific tsunamis. Neither of them show evident trend with distance. Figs 11(d)-(f) shows their variations with the earthquake moment magnitudes. The durations and amplitudes of the initial negative phases indicate obvious dependence on the earthquake magnitudes, while the amplitude ratios do not. Our results yield the following two equations for the relationship between the properties of the initial phases (i.e. D ini and A ini ) and M w (Figs 11d and e): in which, log is the logarithm to the base 10, D ini is in min, A ini is in cm and the qualities of fit are R 2 = 0.99 and R 2 = 0.95 for eqs (2) and (3), respectively ( Fig. 11 and Table 2). A linear relationship between M w and A ini yields a slightly higher R 2 value (i.e. 0.986), whereas between M w and D ini it is slightly smaller (R 2 = 0.98).
Here, we chose the logarithmic equations for two reasons: first, traditionally the relationship between earthquake magnitude (M w ) and seismic/tsunami wave amplitudes is expressed in logarithmic forms (e.g. Gutenberg & Richter 1954;Abe 1979) and, second, the quality of fit between the linear and logarithmic equations was very similar in our analysis. It should be noted that the M w values are usually reported with one decimal place; therefore, the M w values obtained from eqs (2) and (3) need to be rounded to one decimal place.

D E E P -O C E A N T S U N A M I A M P L I T U D E S F O R T R A N S -PA C I F I C T S U N A M I S
Distribution of deep-ocean tsunami zero-to-crest amplitudes from DART stations throughout the Pacific Ocean are shown in Figs 12-13 for four tsunamis. By excluding stations located in the distances <20 arcdeg (2200 km) from the epicentre, the mean amplitudes of 0.9, 1.7, 6.0 and 15.0 cm were obtained for the Iquique, Illapel, Maule and Tohoku tsunamis, respectively (Figs 12a-d). By taking into account the moment magnitudes of these earthquakes (M w 8.2, 8.4, 8.8 and 9.0, respectively), we can describe the relationship between the earthquake moment magnitude (M w ) and deep-ocean tsunami amplitude ( A tsu ) by an exponential function (Fig. 13). The resulting equation is:   in which, log is the logarithm to the base 10 and A tsu is the deepocean tsunami amplitude in cm recorded at distances <20 arcdeg (2200 km) from the epicentre. The quality of fit is R 2 = 0.99 (Table 2), indicating that the equation fits very well to the measured data.
There is no evident correlation between the deep-ocean amplitudes and distances from the source (Figs 12a-d). To examine the possible effect of source directivity on far-field tsunami amplitudes, we plotted these amplitudes against the angles from the fault strike (Figs 12e-h), but have not also found any correlation between these two parameters. Considering the fact that the effect of directivity on the far-field wave amplitude is well-established, the reason why such an effect cannot be seen here can be attributed to the sparsity of the DARTs over the entire Pacific Ocean and to the irregular bathymetry of the Pacific Basin. Tsunami waves pass through complicated bathymetric features which scatter tsunami energy and waves or trap the waves (Hébert et al. 2001;Heidarzadeh et al. 2016b). 3-D plots of the deep-ocean amplitudes versus distance and angle from the fault strike (Figs 12i-l) also does not show any meaningful correlation between these parameters. Assuming a constant value of M w , we can formulate the relationship between A tsu and A ini : Eq. (5) indicates that the amplitude of the initial negative phase (A ini ) is approximately 25 per cent of that of the amplitude of the first elevation wave (A tsu ), which is in agreement with our observations presented in Section 8. (1) An arrival time difference of up to 16.0 min was detected between the observed and shallow-water simulated waveforms for the 2015 Illapel tsunami; this difference was resolved using a phasecorrection technique. The dominant period band of this tsunami was 13-28 min, which is longer than that of the Iquique tsunami (14-21 min) and shorter than the Maule tsunami (12-50 min).
(2) The waveforms from the 2015 Illapel tsunami looked very similar to those from the 2010 Maule tsunami, although the amplitudes of the 2010 tsunami were approximately 2-5 times larger than of the 2015 tsunami. The Illapel tsunami was used as an empirical Green's function to reconstruct the Maule tsunami and yielded quite reasonable results; the reconstructed source period band of 13-62 min for the Maule event is in a good agreement with the observations.
(3) The duration of the initial negative phases (D ini ) was 8-29, 20-35, 22-70 and 40-79 min for the 2014 Iquique, 2015 Illapel, 2010 Maule and 2011 Tohoku tsunamis, respectively. The maximum negative amplitudes of the initial phases (A ini ) were 0.11-0.26, 0.4-0.7, 0.5-2.9 and 1.9-2.5 cm for the aforesaid tsunamis, respectively, while the amplitude ratios of the negative phases to the first waves were 20-40 per cent, 22-41 per cent, 29-61 per cent and 12-67 per cent for these tsunamis, respectively. Our results yield the relationships between the initial phase parameters (D ini in min and A ini in cm) and earthquake moment magnitudes (M w ) as: M w = 6.129 + 1.629 log(D ini ) and M w = 8.676 + 0.706 log(A ini ).
(4) The mean far-field deep-ocean amplitudes (A tsu ) for the Iquique, Illapel, Maule and Tohoku tsunamis were 0.9, 1.7, 6.0 and 15.0 cm, respectively. No correlation was found between the deep-ocean amplitudes and distance from the source or angle from fault strike. The relationship between A tsu (in cm) and M w can be described as M w = 8.245 + 0.665 log(A tsu ).

A C K N O W L E D G E M E N T S
The DART tsunami records were provided by the U.S. National Oceanic and Atmospheric Administration (NOAA) (https://nctr.p mel.noaa.gov/Dart/). We thank the NOAA team for the maintaining and preparing the DART data, which play the major role in advancing our understanding of the deep-ocean and ocean-wide propagation of tsunamis. Most figures were drafted using the GMT software (Wessel & Smith 1998). This article benefited from constructive comments of two anonymous reviewers. MH was funded by the Brunel University London through the Brunel Research Initiative and Enterprise Fund 2017/18 (BUL BRIEF). ABR was partly supported by the RSF Grant 14-50-00095 and the IORAS Project 0149-2015-0039.