Ultra-long period and small-amplitude tsunami generated following the July 2020 Alaska M w 7.8 tsunamigenic earthquake

.


Introduction
Offshore Alaska was the site of a large M w 7.8 earthquake on July 22, 2020 which was followed by a small tsunami with a coastal runup of around half a meter (Fig. 1). The United States Geological Survey (USGS) located the earthquake at 158.554 • W and 55.068 • N at the depth of 28.0 km with an origin time of 06:12:44 (UTC). The earthquake was of dominant thrust mechanism with USGS focal parameters of strike angle: 232 • , dip angle: 20 • and rake angle: 73 • . According to media reports, this offshore earthquake did not make significant damage although the shaking was felt across most of the Alaska Peninsula. A tsunami warning was issued by the US Tsunami Warning System (https://www.tsunami. gov/) following the earthquake; however, the warning was called off after confirming that the tsunami height was minimal based on the tsunami records on offshore Deep-ocean Assessment and Reporting of Tsunamis (DART) devices. The DART systems recorded maximum zeroto-crest amplitudes of less than 1 cm while the nearby coastal tide gauges registered tsunami amplitude of less than 30 cm (Fig. 2).
The July 2020 M w 7.8 tsunamigenic earthquake was generated in the Aleutian subduction zone where the Pacific Plate is subducting beneath the North American Plate at the rate of 5.5-6.0 cm/yr (Li et al., 2016). The Aleutian subduction zone was responsible for the second largest instrumentally-recorded earthquake worldwide: the March 1964 M w 9.2 Alaska earthquake (Plafker et al., 1969) (Fig. 1). Based on the USGS earthquake catalogue, at least 16 M ≧ 7.8 earthquakes were recorded in this zone since 1900, among which the two most notable events are the 1946 (M w 8.6) and 1964 (M w 9.2) events ( Fig. 1, orange circles). The 1946 Aleutian tsunami produced 42 m runup and caused five deaths in the near-field, combined with a runup of 16 m and 159 deaths in the far-field, i.e. Hawaii (Lopez and Okal,. 2006;Okal et al., 2002;Johnson and Satake, 1997). The 1964 Alaska tsunami has been reported to have killed 130 people (Clague et al., 1994) and caused extensive damage. Numerous submarine landslides were reported following the 1964 Alaska earthquake which further intensified the damaging impacts of the event by triggering local landslide tsunamis (Haeussler et al., 2014).
The objective of this research is to understand the sea level characteristics of the July 2020 Alaska tsunami. In particular, there are two unusual signatures in the sea level data of the 2020 tsunami, which motivated this research: (i) the tsunami has a very long period of ~60 min (Fig. 2) which is unusual for a tsunami from an M w 7.8 earthquake; (ii) the tsunami's coastal amplitude is significantly smaller than those normally generated by an M w 7.8 offshore earthquake. For comparison, a similar-magnitude (M w 7.8) and similar-mechanism (thrust) earthquake at the depth of 15.1 km in Kaikoura, New Zealand generated a tsunami with period of 15-20 min and coastal runup of up to 7 m (Power et al., 2017;Heidarzadeh et al., 2019). We note that the focal depth of the Kaikoura event (15.1 km) was shallower than that of the Alaska event (28.0 km). Here, we apply sea level data analysis and numerical modeling of tsunami propagation to study the 2020 Alaska tsunami and to explain the above two unusual characteristics of the tsunami. The innovation of this study is that, for the first time, we apply analytical equations and numerical simulations to explain the unusual tsunami waves generated by an M w 7.8 earthquake.

Data and methods
Our data comprises sea level records of the tsunami, ocean bathymetry and the earthquake fault parameters. Sea level records are from tide gauge stations in Sand Point and Dutch Harbor as well as seven DARTs (Fig. 2). The sampling interval for all sea level data is 1 min. Both tide gauge and DART records belong to the National Ocean Service of the US National Oceanic and Atmospheric Administration (NOAA). The initial part of the DART data includes under-sampled seismic waves which are called here as seismic noise (Fig. 2); although they are not useful for tsunami characterization, An et al. (2017) demonstrated that they can be helpful for estimating earthquake parameters. The least-squares method of Grinsted (2008) was employed to estimate the tidal signals and to remove them from the original sea level records. The bathymetry data comes from the General Bathymetric Charts of the Oceans (GEBCO) 2020 digital grid atlas (IOC et al., 2003;Weatherall et al., 2015) which has a spatial resolution of 15 arc-sec. The fault parameters of the July 2020 Alaska earthquake is provided by the USGS (https://earthquake.usgs.gov/earthquakes/eventpage/us7000asvb/fin ite-fault) which is in the form of a finite fault model with 345 subfaults of size 10 km × 10 km. The strike and dip angles of the subfaults were fixed at 232 • and 20 • , respectively, their depths were varied in the range of 6.1-54.0 km and the slip on the subfaults was up to 3.7 m. The USGS source model is obtained through inversion of worldwide seismic observations of the earthquake. Here, we applied a trial-and-error approach to adjust the length, width and slip of the USGS model in order to reproduce tsunami observations. Our method is a combination of sea level data analysis and numerical modeling. Fourier transform and Wavelet (frequency-time) analyses were performed on the sea level data. For Fourier transform, the Welch algorithm (Welch, 1967;Mathworks, 2020) was applied considering half-window overlaps and Hanning windows (Heidarzadeh et al., 2017a). The Fourier analysis was conducted for both tsunami and the background signal (i.e. part of the waveform before tsunami arrival at each sea level station) in order to identify main tsunami energy channels. The lengths of the tsunami and background waveforms used for spectral analyses, were 150-200 min and 120-150 min, respectively. For Wavelet analysis, we used the Wavelet package of Torrence and Compo, (1998) considering the Morlet mother function with a wavenumber of 6 and a scale width of 0.10 .
We employed a well-validated tsunami simulation package called JAGURS (Baba et al., 2015) to simulate tsunami propagation and coastal amplification. We solved Shallow Water Equations on a three-level nested grid system with grid sizes of 180 arc-sec, 60 arc-sec, and 20 arc-sec from the largest domain (far from the coast) to the smallest domain (around the coast), respectively (Fig. 5a). Among a few numerical packages available for tsunami modeling, JAGURS is favored due to its nesting grid capability and its flexibility for both serial and  Johnson and Satake (1997) and Stauder and Bollinger (1966), respectively. Dashed contours give tsunami travel times (TTT) in hours with 0.5 h intervals which are calculated using the TTT program by Geoware, (2011). SZ stands for subduction zone. Names starting with "D" stand for DART stations. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.) parallel computations. We run the tsunami for a total simulation time of 9 h with time step of 1.0 s to satisfy the stability condition of the numerical scheme (Courant et al., 1928;Heidarzadeh et al., 2009). To calculate coseismic crustal deformation due to the earthquake, the analytical dislocation modeling approach of Okada (1985) was applied. An instantaneous coseismic deformation is assumed which implies that the initial sea surface displacement is the same as the crustal deformation. This is the common practice for modeling tsunamis generated by earthquakes because the speed of tsunami wave propagation is approximately hundred times lower than the speed of seismic waves (e. g., Satake, 1987).

Tsunami waveforms and spectral analysis
Deep-ocean and coastal records of the Alaska tsunami measured on DARTs and tide gauges, respectively, are shown in Fig. 2 and their characteristics are listed in Table 1. The deep-ocean zero-to-crest amplitudes are 0.3-1.0 cm with an average of 0.6 cm whereas the coastal amplitudes are 5.7-24.0 cm averaging 14.9 cm ( Table 1). The tsunami amplitudes decay rapidly on DARTs while large-amplitude waves persist for at least 5 h on two coastal records in Sand Point and Dutch Harbor (Fig. 2). Longer duration of tsunami waves on coastal tide gauges is attributed to various coastal and shelf features such as reflections, resonance of the wave inside harbors as well as trapped edge waves (e.g. Saito et al., 2014;Satake, 2014;Satake et al., 2020).
Fourier analysis ( Fig. 3; color spectra) and comparison with background spectra ( Fig. 3; black spectra) revealed that the dominant tsunami period is 51-64 min. It is noted that the tsunami dominant period is influenced by the azimuthal angle of the DART station from the epicenter. We performed Fourier analysis only for DART records because they are mostly free from various coastal and shelf features, and thus carry mainly tsunami source features (e.g. Rabinovich, 1997). Wavelet plots demonstrate the temporal changes of the tsunami dominant periods at three DARTs (Fig. 4). It can be seen that the dominant periods of 51-64 min are the main signal available in the DART waveforms. We note that the strong signal at the period of 2-7 min in the Wavelet plots belong to the under-sampled seismic noise of the earthquake recorded on the DARTs and thus they do not represent the tsunami. The short duration (~2 h) of the energetic tsunami waves on DARTs is also confirmed on Wavelet plots (Fig. 4).

Numerical simulations and validation
The purpose of numerical modeling was to confirm the coseismic tsunami source of the July 2020 event and to validate it through DART and tide gauge observation. We use the validated source model for further analyses in the next sections. The results of tsunami simulation using the USGS source model is shown in Fig. 5d (blue waveforms) indicating that the simulated waveforms arrive 10-20 min earlier than the observations in some stations. Therefore, it is necessary to further adjust the tsunami source. The main reason for lack of a perfect match between simulations and observation is that the USGS source model is obtained through the inversion of only seismic observations of the earthquake. Therefore, it is natural that such a model does not give a perfect match for tsunami observations although it yields better results for seismic waveforms (e.g. Yokota et al., 2011;Satake and Heidarzadeh, 2017). It has been noted by several authors that a more accurate tsunami source can be achieved by a joint seismic-tsunami inversion (e.g., Gusman et al., 2015).
Inspired by the USGS source model, we considered a rectangular source model with uniform slip to achieve a better match between observations and simulations through a trial-and-error approach. First, the fault length and width are assumed as 110 km and 70 km, respectively. Then, the uniform slip of 1.7 m is calculated based on a depthdependent earth rigidity of 4.3 × 10 10 N/m 2 proposed by Sallarès and Ranero (2019). Similar scaling laws are proposed by An et al. (2018). Other fault parameters such as strike angle (232 • ), dip angle (20 • ), rake angle (73 • ), and depth (28 km) are taken from the USGS model. This model is called as "uniform fault" here. In fact, our uniform fault is a simplified, but improved, version of the USGS fault model which is improved through tsunami observations. The coseismic crustal deformation due to the USGS and our uniform-fault models are given in Fig. 5b and c, respectively.
The tsunami simulations using our uniform fault model produces good agreement between observation and simulation waveforms (Fig. 5d, red waveforms). The simulated waves using the USGS model (Fig. 5d, blue waveforms) arrive earlier and have shorter wave periods whereas those using our uniform fault model match the observation well. Snapshots of tsunami propagation (Fig. 6a) reveal that tsunami arrives in west Canada, Hawaii and west US within 2-5 h. Plot of maximum tsunami amplitudes during the entire tsunami simulations (Fig. 6b) indicates that most of the far-field tsunami amplitudes are directed towards west Canada and west USA. It is long known that tsunami energy travels along normal direction to the fault orientation in the far field (Ben-Menahem and Rosenman, 1972;Okal, 1988;Synolakis and Bernard, 2006). Given the strike angle of 232 • for the 2020 Alaska earthquake (direction NE-SW; Fig. 5b and c), the normal direction to the strike angle is 142 • which is direction NW-SE, towards west Canada and west USA.

The ultra-long period waves of the 2020 Alaska tsunami
The tsunami waveforms and spectra of the July 2020 Alaska event (M w 7.8) are compared with those of the March 2011 Japan event (M w 9.0) at three DART stations (Fig. 7). Despite the much smaller magnitude of the Alaska event compared to the 2011 Japan earthquake, the dominant period of the 2020 Alaska tsunami (51-64 min) is much longer than the dominant period of the 2011 Japan event (20-26 min). Traditionally, it has been thought that the larger the magnitude of the earthquake, the longer its tsunami period would be. However, this is not the case for the 2020 Alaska tsunami. We attribute the ultra-long period of the 2020 Alaska tsunami to the rather shallow water depth of the source region (depth = 100-200 m; Fig. 8b). The dominant period of a tsunami (T) can be approximated using the length of the main coseismic crustal deformation (L) and the water depth (d) at the epicentral area applying the following equation : where g is gravitational acceleration (9.81 m/s 2 ). We note that tsunami energy usually occurs over a period band rather than a single period because of the heterogeneous shape and non-instantaneous occurrence of coseismic crustal deformation. Equation (1) gives the dominant tsunami period using the length of the largest deformation area, rather than the whole period band. For the 2020 Alaska tsunami, the length of the crustal deformation is  (Figs. 5c and 8b). For the 2011 Japan tsunami, the respective values are 80-120 km and 3000-6000 m (Fig. 8a). While the total length of the 2011 event's crustal deformation is > 200 km, we consider only the length of the largest deformation area as we mainly concern about the dominant tsunami period in this study. Using such input parameters, Equation (1) results in dominant period of 17-20 min for the 2011 Japan event (Figs. 8c) and 60-75 min for the 2020 Alaska tsunami (Fig. 8d). These values are consistent with real tsunami dominant periods obtained through Fourier analysis of tsunami observations for the two events (Fig. 7b). Therefore, our analytical study applying Equation (1) explains that the ultra-long period of the 2020 Alaska tsunami is attributed to the generation of the tsunami in extreme shallow water depth of 100-200 m of the continental shelf.

The small coastal amplitude of the July 2020 Alaska tsunami
The coastal amplitudes of the 2020 Alaska tsunami were much smaller than other tsunamis generated by similar-magnitude and similar-mechanism earthquakes. For comparison, the 2016 Kaikoura (New Zealand) and 2012 Haida Gwaii (Canada) tsunamis were both generated by M w 7.8 thrust earthquakes which generated maximum coastal runup of 6 m (Power et al., 2017;Heidarzadeh et al., 2019) and 13 m (Gusman et al., 2016), respectively. We note that the 2020 Alaska earthquake occurred at the depth of 28 km while the depths of the 2016 Kaikoura and 2012 Haida Gwaii events were 15 km and 14 km, respectively.
In general, several factors may influence coastal amplitude of a tsunami including: amplitude of coseismic crustal deformation at the tsunami source (or magnitude of the earthquake), focal depth of the earthquake (FD), water depth at the source area (d 0 ), bathymetric features along the tsunami propagation path that may cause several natural interferences such as reflection of the waves, potential harbor resonance and edge waves (e.g. Raichlen and Lee, 1991;Synolakis, 2003;Yalciner, and Pelinovsky, 2007;Heidarzadeh et al., 2009;Shimozono et al., 2012;Saito et al., 2014). Precise analysis of these effects normally requires numerous numerical simulations which is beyond the scope of this study.
Here, we focus on two of the above-mentioned factors: water depth at the source region and earthquake focal depth. The effect of water depth at the source region (d 0 ) on the coastal amplitude of a tsunami (η) can be explained by Green's law (Sorensen, 2010): where, d is water depth at the coast and η 0 is amplitude of coseismic crustal deformation. By assuming constant water depth at the coast (d), Equation (2) indicates that the deeper the water depth at the source area (d 0 ), the larger the coastal amplitude of a tsunami (η) would be. Regarding earthquake focal depth (FD), it is widely known that deeper earthquakes generate smaller tsunamis and vice versa (e.g. Synolakis, 2003;Satake, 2014).
To quantitatively study the effects of earthquake focal depth (FD) and water depth at the source region (d 0 ) on coastal amplitude of tsunamis, we compare these two parameters for five M w 7.8 thrust events (Fig. 9). These events and the respective references are: 2006 Java (Fujii and Satake, 2006), 2012 Haida Gwaii (Gusman et al., 2016;Leonard and Bednarski, 2014), 2016 Kaikoura (Power et al., 2017;Heidarzadeh and Satake, 2017) and 2016 Ecuador (Heidarzadeh et al., 2017b). Fig. 9 indicates that for similar-size and similar-mechanism earthquakes, coastal tsunami height (or runup) is directly correlated with d 0 (Fig. 9b), but is inversely related to focal depth (FD) (Fig. 9c). Therefore, the small coastal amplitude of the 2020 Alaska tsunami can be explained by the relatively deep focal depth of the earthquake (FD = 28 km) and the extremely shallow water depth around the source region (d 0 = 100-200 m).

Conclusions
The tsunami generated by July 2020 M w 7.8 thrust earthquake was unusual in two ways: (i) the period of the tsunami was very long (51-64 min) and (ii) its coastal amplitude was small (~0.5 m). This study was motivated to explain these peculiar characteristics of the 2020 tsunami. Here, we applied waveform analysis and numerical simulations and achieved the following conclusions: • The deep-ocean zero-to-crest amplitude of the tsunami was 0.3-1.0 cm (average = 0.6 cm) whereas the coastal amplitude recorded on tide gauges was 5.7-24.0 cm (average = 14.9 cm). Tsunami period from this M w 7.8 earthquake was 51-64 min which is much longer than that generated by the 2011 M w 9.0 Japan earthquake (20-26 min). • We proposed a source model for this tsunami comprising length and width of 110 km and 70 km, respectively, with uniform slip of 1.7 m. This source model is located at the water depth of 100-200 m. Our model successfully reproduces the observed tsunami waveforms. • By applying a simple analytical equation for the period of tsunami waves, we showed that the ultra-long period of the 2020 Alaska tsunami (51-64 min) can be reproduced using the water depth around the source region (100-200 m) and the length of the coseismic deformation area (~100 km). • Comparison of the coastal amplitude of the 2020 Alaska event (~0.5 m) with four other similar-size and similar-mechanism earthquakes revealed that the smaller coastal amplitude of this tsunami is due to   the relatively deep focal depth of the earthquake (28 km) and the extremely shallow water around the source region (100-200 m).

Credit author statement
The lead author, Mohammad Heidarzadeh, initiated the research and conceptualized and organized all activities reported in this article. Waveform analyses, spectral analyses and wavelet analyses are conducted by Mohammad Heidarzadeh. The second author, Iyan E. Mulia, conducted numerical simulations of the tsunami.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Fig. 9. a. Epicenters (stars) and focal mechanisms (blue/white circles) of a few M w 7.8 thrust earthquakes worldwide. b. Variations of tsunami runups and water depths at the tsunami sources for events shown in panel "a". Vertical blue bars give the ranges of tsunami runups for each event while pink squares are the average values. c. Same as panel "b" but for earthquake focal depth (FD). (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)