Detection of low‐frequency hydro‐acoustic waves as precursor components of destructive tsunamis can enhance the promptness and the accuracy of Tsunami Early Warning Systems (TEWS). We reconstruct the hydro‐acoustic wave field generated by the 2012 Haida Gwaii tsunamigenic earthquake using a 2‐D horizontal numerical model based on the integration over the depth of the compressible fluid wave equation and considering a mild sloped rigid seabed. Spectral analysis of the wave field obtained at different water depths and distances from the source revealed the frequency range of low‐frequency elastic oscillations of sea water. The resulting 2‐D numerical model gave us the opportunity to study the hydro‐acoustic wave propagation in a large‐scale domain with available computers and to support the idea of deep‐sea observatory and data interpretation. The model provides satisfactory results, compared with in situ measurements, in the reproduction of the long‐gravitational waves. Differences between numerical results and field data are probably due to the lack of exact knowledge of sea bottom motion and to the rigid seabed approximation, indicating the need for further study of poro‐elastic bottom effects.

Submarine earthquakes can generate long‐gravitational waves (tsunamis), that propagate at the free surface, and pressure waves (hydro‐acoustic waves), which oscillate between the seabed and the free surface due to the compressibility of sea water. Tsunami waves can travel for long distances and are known for their dramatic effects on coastal areas. Hydro‐acoustic waves travel much faster than the tsunami itself, i.e., at the speed of sound in water; their presence in a pressure record therefore can anticipate the arrival of the tsunami. Several investigations have been carried out [*Nosov*, ; *Stiassnie*, ; *Chierici et al*., ; *Cecioni et al*., ; *Abdolali et al*., ] to study the physical characteristics of hydro‐acoustic waves, clarifying that there exists a relationship between these pressure waves and their tsunamigenic source. Low‐frequency hydro‐acoustic waves generated by seabed motion have been measured during the Tokachi‐Oki 2003 tsunami event [*Nosov et al*., ], by the JAMSTEC (Japan Agency for Marine‐earth Sciences and TEChnology) observatory. *Nosov and Kolesov* [] and *Bolshakova et al*. [] have processed the in situ bottom pressure records in order to estimate amplitude, duration, and velocity of bottom displacement. They also presented the results of a 3‐D finite difference numerical model reproducing the generated hydro‐acoustic waves, and compared the computed signals with the in‐situ observations. More recently, the 28 October 2012 a *M _{w}* = 7.8 earthquake occurred off the West coast of Haida Gwaii archipelago, Canada.

This paper presents numerical modeling of tsunami and hydro‐acoustic waves generated by the 2012 Haida Gwaii earthquake. The earthquake is modeled starting from the USGS data and by using the *Okada* [] formula to reconstruct the residual seabed displacement. The wave field is modeled solving depth‐integrated mild‐slope equation of *Sammarco et al*. [] by means of the Finite Element Method. Model results are compared with field data recorded, during the 2012 Haida Gwaii event, by the Deep‐ocean Assessment and Reporting of Tsunamis (DART^{®}) network (

At 28 October 2012, 03:04 UTC, a powerful *M _{w}* = 7.8 earthquake struck central Moresby Island in the Haida Gwaii archipelago, Canada (Figure ). The earthquake hypocenter was located (52.788°N, 132.101°W) at a depth of 14 km. This was the largest earthquake to hit Canada since 1949, when an 8.1 magnitude quake hit west of the Haida Gwaii Islands (epicenter at 53.62°N, 133.27°W). The 2012 earthquake occurred as an oblique‐thrust faulting on the boundary between the Pacific and North America plates. The Pacific plate actually moves approximately north‐northwest with respect to the North America plate at a rate of about 50 mm/yr. The National Earthquake Information Center (NEIC) reports a strike of 323° and dip of 25° for this earthquake event.

Figure shows the main seismic parameters of the event. Figure a presents the surface projection of the slip distribution as determined by *Lay et al*. []. The slip distribution on the fault plane is shown in Figure b. Contours show the rupture initiation time in seconds with average rupture speed of *V _{r}* = 2.3 km/s (the rupture time ranges between 0.8 and 74.4 s). Figure c shows the residual vertical bottom displacement, calculated by the Okada formula [

Free surface and bottom pressure data have been collected during this earthquake by the DART network and by Ocean Network Canada. Both networks have instruments located approximately 600 km south the earthquake epicenter. Location of observatories which recorded the waves generated by the 2012 earthquake are shown in Figure , together with bathymetric information and the open‐sea boundary of the numerical domain (dashed line) described below. The DART network has been deployed by the National Oceanic and Atmospheric Administration (NOAA), to support real‐time forecasting of tsunami events. The closest DART instruments that recorded the 2012 earthquake event are: DART46419 (48°45′59″N 129°37′57″W) at 2775 m water depth and 480 km south of the epicenter, and DART46404 (45°51′18″N 128°46′30″W) at 2793 m water depth, 810 km south of the epicenter. Hereinafter, the name of these two stations will shortened, respectively, as *D*_{19} and *D*_{04}. Tsunami warning was issued after recording strong signatures, which began at 3:04 UTC, for a large stretch of the North and Central coast of the Haida Gwaii region and eastward to Hawaii. Later the tsunami alarm was limited, canceled, or downgraded. Ocean Network Canada operates cabled observatories from 100 to 2600 m seaward of southwest coast of Vancouver Island in the northeast Pacific. In this paper, we consider data recorded by three submarine observatories: CORK ODP1027 located 640 km south of epicenter at Cascadia Basin (47°45.7560′N 127°45.5527′W) in 2660 m water depth (hereinafter, *N*_{27}); Bullseye BPR 889, 590 km south of epicenter, at Clayoquot Slope (48°40.2501′N 126°50.8779′W) in 1258 m water depth (*N*_{89}); Barkley Canyon (Upper Slope) 640 km far from epicenter close to the coast (48°25.6379′N 126°10.4851′W), in 392 m water depth (*N _{BC}*). These observatories are equipped with many instruments, including bottom pressure records and seismometers. In the hydro‐acoustic wave frequency band, as introduced later, the bottom pressure component

The wave generation and propagation in weakly compressible (first‐order approximation of fluid density and pressure), inviscid and irrotational fluid is expressed in terms of the fluid velocity potential ^{2} are, respectively, the gradient and the Laplacian in the horizontal plane *x*, *y*, while subscripts denote partial derivatives.

The wave generation is modeled at the bottom boundary, *z* = –*h*(*x*, *y*, *t*), where *h*(*x*, *y*, *t*) = *h _{b}*(

For the first mode (*n* = 0), *β*_{0} is real and is the mode associated with the gravity surface wave, i.e., the tsunami; for the other modes (*n* ≥ 1), namely the hydro‐acoustic modes, *β _{n}* is purely imaginary and they are responsible for acoustic oscillations of the water body.

Equation was named mild‐slope equation in weakly compressible fluid (MSEWC). Superposition of the solutions, *ψ _{n}*, of equation for each mode leads to complete modeling of the fluid potential generated by a fast seabed motion.

In equation , the terms

For incompressible fluid, i.e., in the limit *c _{s}* →

The 2012 earthquake event at Haida Gwaii archipelago, Canada, has been reproduced numerically solving the MSEWC, equation . The model domain is reported in Figure ; it covers an area of about *ζ*_{0} = 1.6 m. Unlike the traditional incompressible tsunami models, which often use the residual vertical displacement of the bottom as the initial free surface displacement, in this depth‐integrated wave model, the transient time‐spatial sea bottom motion is considered for both tsunami and hydro‐acoustic modes. The vertical bottom velocity is assumed to be [*Nosov and Kolesov*, ]:*H*(*t*) is the Heaviside step function and *τ* is the duration of the bottom displacement. Spreading of the rupture front, starting from epicenter to the far edge of earthquake zone, has been reproduced with a velocity of *V _{r}* = 2.3 km/s. In the model, the duration of the seabed displacement is assumed everywhere constant equal to

The modeling of hydro‐acoustic waves has been carried out by solving the MSEWC (6) for *n* ≥ 1 and for a number of frequencies within a finite range. The simulation of a full 3‐D model in the whole area of interest is unreasonably computationally expensive, therefore validation of MSEWC has been carried out by comparison with the solution of the complete 3‐D problem expressed by equation , along vertical cross sections of the computational domain. The orientation of these two sections is shown in Figure and chosen as the line intersecting the earthquake epicenter and each of the two DART stations. Figure shows the water depth, *h _{b}*, and the residual seabed dislocation,

In Figure , the pressure signals at the sea‐bottom both in time (Figures a and c) and frequency (Figures b and d) domains are shown. In each plot, the top plot presents the solution of the 3‐D problem, while the bottom plot presents the MSEWC solution. The two models provide similar results in the time domain, where arrival time and propagation features of the hydro‐acoustic modes show good agreement, even if just the first hydro‐acoustic mode has been reproduced and the limited frequency range *f* = 0.1–0.8 *Hz* has been solved using the MSEWC. These comparisons validate the depth‐integrated equation for real bathymetry and enable us to optimize the model. It has been further verified that the reproduction of frequencies outside of the range *f* = 0.1–0.8 *Hz*, and relative to second and higher hydro‐acoustic modes, increases the computational costs without providing a substantial improvement of the results accuracy.

In order to investigate how the variable bathymetry affects the hydro‐acoustic wave propagation, four positions along the vertical section 2 have been selected (see Figure a): points *A* and *B* are located on the earthquake zone at 1200 and 2900 m water depth; points *C* and *D* are 250 and 500 km far from epicenter at 2950 and 2700 m water depth, respectively. Figure presents the frequency spectra of the numerically reproduced bottom pressure perturbations for these four positions: plots *b*, *c*, *d*, and *e*, respectively, for the positions *A*, *B*, *C*, and *D*.

Hydro‐acoustic waves oscillate at the resonating frequency, given for each mode by equation , which is therefore inversely proportional to the water depth where these waves are generated. This resonating frequency *f*^{(}^{n}^{)}, namely cutoff frequency [*Tolstoy*; ], identifies at each water depth the lower limit of frequencies associated with propagating modes, whereas *f*^{(}^{n}^{)} to propagate. Figure b shows with the vertical dashed line the cutoff frequency of the first mode at the local water depth (*C*, only frequencies higher than 0.16 Hz cross the shallower area. The solid line in Figure d represents the first acoustic mode for the minimum water depth between generation area and point *C* with 2400 m water depth (*C* (*D* with 1900 m water depth has barricaded propagation of pressure waves with frequencies lower than 0.2 Hz (

In order to correctly reproduce the hydro‐acoustic wave field by means of Finite Element Method in the domain of Figure , a maximum element mesh size of 1 km has been chosen, for a total number of 3,000,000 triangular elements. Solutions were obtained using a high‐speed computer equipped with 12 cores i7 3.20 GHz CPU and 64 GB RAM. The computational time for solving 720 s with a time step of 0.05 s was about 600 h. The results of this large‐scale simulation are shown in Figure in term of free surface elevation, *η*(*x*, *y*, *t*). It can be seen that a complicated perturbation is formed at the free surface and propagates at the sound celerity in water over the entire domain in 12 min after the event. The wave field shows that the hydro‐acoustic waves do not propagate upslope; maximum values of hydro‐acoustic wave amplitude are in deeper water close to the generation area.

Spectrograms of bottom pressure are presented in Figure for three Ocean Neptune Canada observatories and three selected points in front of active fault at different water depths, as shown in Figure . Each spectrogram is normalized by dividing the maximum value among all six points in order to show the ratio of wave amplitudes at different distances and depths. The local characteristic gravitational wave frequency *f _{g}* and first acoustic mode

Numerical simulation of the long–gravitational waves generated by the 2012 Haida Gwaii earthquake has been performed by solving the zero mode of MSEWC (equation ). As for the hydro‐acoustic modes equation is solved by means of Finite Element Method on the numerical domain, which has been discretized in triangular elements with a maximum size of 2.5 km. The computational time is 12 h (on the same computer described in section 3.1) for simulation of 2 h real time. The frequencies in the range *f* = 0–0.03 Hz, with a *df* = 0.002 Hz, have been solved to reconstruct the gravitational wave field. Results of the zero mode simulation are presented in Figures and , in terms of bottom pressure time series at the Ocean Network Canada observatories and at the DART buoys, respectively. In both figures, the red lines show the results of the numerical model, while the gray lines represent the pressure signals recorded at the instruments. The model simulates the tsunami wave magnitude and arrival time properly compared with observation. In Figure , the plots a, b, and c refer, respectively, to the observatories *N _{BC}*,

Reproduction of long‐gravitational waves spreading proportional to square root of the water depth, and low‐frequency hydro‐acoustic waves traveling with the sound celerity in water, have been studied for 2012 Haida Gwaii earthquake in the framework of linear potential theory, using a hyperbolic mild‐slope equation for weakly compressible fluids (MSEWC) [*Sammarco et al*., ]. Here we show a first application of MSEWC to numerically reproduce wave generation and propagation in weakly compressible fluid in a large‐scale domain, overcoming the computational difficulties of three‐dimensional models. The numerical model is validated against a fully 3‐D linear model for the case of varying water depth. The comparison has been done in order to understand the physics of low‐frequency propagation in the far‐field in order to utilize deep‐sea observatories for enhancement of TEWS. At present, the 2‐D numerical model is not able to reproduce the intensity of hydro‐acoustic wave field. However, it shows the importance of deep‐sea observatories where the depth effect does not affect the arriving wave spectrum. It is worth citing that due to the lack of knowledge about spatiotemporal bottom motion in numerical modeling the calculation for hydro‐acoustic waves is still a rough estimate of the exact result. In other words, we have assumed sea bottom deformation spreads uniformly from epicenter (*V _{r}* = 2.3 km∕s) taken place with mean bottom displacement duration (

This work was carried out under the research project FIRB 2008‐FUTURO IN RICERCA (“Design, construction and operation of the Submarine Multidisciplinary Observatory experiment”), funded by the Italian Ministry for University and Scientific Research (MIUR). The authors thank the Ocean Network Canada for the valuable data of bottom pressure and IRIS for seismogram. J.T. Kirby acknowledges the support of the National Tsunami Hazard Mitigation Program, NOAA, grant NA13NWS4670014. The bottom pressure data for this paper are available at ONCs Data Access Center at