Modeling Tsunamis for Improved Hazard Assessment and Detection

This body of work uses state of the art numerical models to assess and reduce tsunami hazard. The first manuscript describes the use of these models to explore nonlinear interaction between tide and tsunami in the context of hazard assessment. Inundation due to several probable maximum tsunamis (PMTs) is considered in the Hudson River Estuary (HRE). Of the sources considered, a submarine mass failure (SMF) poses the most significant tsunami threat in this region and across the entire US East Coast. The next manuscript focuses on the how SMF mechanics effect tsunami generation. In addition to inundation, SMF tsunamis are dangerous because of their short or nonexistent warning times. The final manuscript discusses developments to an algorithm which extend the range of tsunami detection by shore-based HF radar, thereby increasing warning times. Tsunami hazard assessment in the Hudson River Estuary based on dynamic tsunami-tide simulation The first manuscript is part of a tsunami inundation mapping activity carried out along the US East Coast (USEC) since 2010, under the auspice of the National Tsunami Hazard Mitigation program (NTHMP). Two densely built low-lying regions are situated along this coast: Chesapeake Bay and HRE. HRE is the object of this work, with specific focus on assessing tsunami hazard in Manhattan, the Hudson River and East River areas. In the NTHMP work, inundation maps are computed as envelopes of maximum surface elevation along the coast and inland, by simulating the impact of selected PMTs in the Atlantic Ocean margin and basin. At present, such simulations assume a static reference level near shore equal to the local mean high water (MHW) level. Instead we simulate maximum inundation resulting from dynamic interactions between the incident PMTs and a tide, which is calibrated to achieve MHW at its maximum level. To identify conditions leading to maximum tsunami inundation, each PMT is simulated at four different phases of the tide and results are compared to those obtained for a static reference level in the HRE. We conclude that changes in inundation resulting from the inclusion of a dynamic tide in the specific case of the HRE, although of scientific interest, are not significant for tsunami hazard assessment and that the standard approach of specifying a static reference level equal to MHW is conservative. However, in other estuaries with similarly complex bathymetry/topography and stronger tidal currents, a simplified static approach might not be appropriate. Modeling coastal tsunami hazard from submarine mass failures: effect of slide rheology, experimental validation, and case studies off the US East coast We first validate two models simulating tsunami generation by deforming submarine mass failures (SMFs) against laboratory experiments for SMF made of glass beads moving down a steep slope. These are two-layer models, in which the upper layer is water, simulated with the non-hydrostatic 3D (σ-layer) non-hydrostatic model NHWAVE, and the SMF bottom layer is simulated with depth-integrated equations and represented either as a dense Newtonian fluid or a granular medium. Using the dense fluid model, we assess model convergence with grid resolution, and sensitivity of slide motion and generated surface elevations to slide parameters. A more limited validation is conducted for the granular slide model. Both models can accurately simulate time series of surface elevations measured at 4 gages, while providing a good simulation of both the geometry and kinematics of the moving slide material. The viscous slide model, which at present is the only one that can be applied to an arbitrary bottom bathymetry, is then used to simulate the historic Currituck SMF motion, in order to determine relevant viscous slide parameters to simulate SMF tsunamis on the east coast. The same parameters are then applied to simulate tsunami generation from a possible SMF sited near the Hudson River Canyon. Simulations are performed for 3 deforming slides with different dissipation parameters and the rigid slump, and results compared; all SMFs have the same initial volume, location, and geometry. Simulations of tsunami propagation are then done for the tsunamis in two levels of nested grids, using the Boussinesq model FUNWAVE-TVD, and maximum surface elevations computed along a 5 m depth contour off of the coast of New Jersey and New York. At most nearshore locations surface elevations caused by the rigid slump are significantly larger (up to a factor of 2) than those caused by the 3 deforming slides. Hence, the rigid slump provides a conservative estimate of SMF tsunami impact in terms of maximum inundation/runup at the coast, while using a more realistic rheology with some level of SMF deformation, in general, leads to a reduced tsunami impact at the coast. This validates as conservative the tsunami hazard assessment and inundation mapping performed to date as part of NTHMP, on the basis of Currituck SMF proxies simulated as rigid slump. Algorithms for tsunami detection by High Frequency Radar : development and case studies for tsunami impact in British Columbia, Canada To mitigate the tsunami hazard along the shores of Vancouver Island in British Columbia (Canada), Ocean Networks Canada (ONC) has been developing a Tsunami Early Warning System (TEWS), combining instruments (seismometers, pressure sensors) deployed on the sea floor as part of their Neptune Observatory, and a shore-based High-Frequency (HF) radar. This HF radar can remotely sense ocean currents up to a 80 km range, based on the Doppler shift they cause in ocean waves at the radar Bragg frequency. Using this method, however, tsunami detection is limited to shallow water areas where they are sufficiently large due to shoaling and, hence, to the continental shelf. To extend detection range into deep water, thereby increasing warning time, the authors have proposed a new detection algorithm based on spatial correlations of the raw radar signal at two distant locations along the same wave ray. In a previous work, they validated this algorithm for idealized tsunami wave trains propagating over a simple sea floor geometry in a direction normally incident to shore. In the final manuscript, this algorithm is extended and validated for realistic tsunami case studies conducted for seismic sources and using the bathymetry off of Vancouver Island, BC. Tsunami currents computed with a state-of-the-art long wave model are spatially averaged over the HF radar cells aligned along individual wave rays, obtained by solving geometric optic equations. A model simulating the radar backscattered signal in space and time as a function of the simulated tsunami currents is applied for the characteristics of the WERA HF radar deployed by ONC near Tofino, BC. Finally, numerical experiments show that the proposed algorithm works on realistic tsunami data. This is used to develop relevant correlation thresholds for tsunami detection.

to maximum tsunami inundation, each PMT is simulated at four different phases of the tide and results are compared to those obtained for a static reference level in the HRE. We conclude that changes in inundation resulting from the inclusion of a dynamic tide in the specific case of the HRE, although of scientific interest, are not significant for tsunami hazard assessment and that the standard approach of specifying a static reference level equal to MHW is conservative. However, in other estuaries with similarly complex bathymetry/topography and stronger tidal currents, a simplified static approach might not be appropriate.
Modeling coastal tsunami hazard from submarine mass failures: effect of slide rheology, experimental validation, and case studies off the US East coast We first validate two models simulating tsunami generation by deforming submarine mass failures (SMFs) against laboratory experiments for SMF made of glass beads moving down a steep slope. These are two-layer models, in which the upper layer is water, simulated with the non-hydrostatic 3D (σ-layer) non-hydrostatic model NHWAVE, and the SMF bottom layer is simulated with depth-integrated equations and represented either as a dense Newtonian fluid or a granular medium.
Using the dense fluid model, we assess model convergence with grid resolution, and sensitivity of slide motion and generated surface elevations to slide parameters.
A more limited validation is conducted for the granular slide model. Both models can accurately simulate time series of surface elevations measured at 4 gages, while providing a good simulation of both the geometry and kinematics of the moving slide material. The viscous slide model, which at present is the only one that can be applied to an arbitrary bottom bathymetry, is then used to simulate the historic Currituck SMF motion, in order to determine relevant viscous slide parameters to simulate SMF tsunamis on the east coast. The same parameters are then applied to simulate tsunami generation from a possible SMF sited near the Hudson River Canyon.
Simulations are performed for 3 deforming slides with different dissipation parameters and the rigid slump, and results compared; all SMFs have the same initial volume, location, and geometry. Simulations of tsunami propagation are then done for the tsunamis in two levels of nested grids, using the Boussinesq model FUNWAVE-TVD, and maximum surface elevations computed along a 5 m depth contour off of the coast of New Jersey and New York.
At most nearshore locations surface elevations caused by the rigid slump are significantly larger (up to a factor of 2) than those caused by the 3 deforming slides. Hence, the rigid slump provides a conservative estimate of SMF tsunami impact in terms of maximum inundation/runup at the coast, while using a more realistic rheology with some level of SMF deformation, in general, leads to a reduced tsunami impact at the coast. This validates as conservative the tsunami hazard assessment and inundation mapping performed to date as part of NTHMP, on the basis of Currituck SMF proxies simulated as rigid slump.

Algorithms for tsunami detection by High Frequency Radar : development and case studies for tsunami impact in British Columbia, Canada
To mitigate the tsunami hazard along the shores of Vancouver Island in British Columbia (Canada), Ocean Networks Canada (ONC) has been developing a Tsunami Early Warning System (TEWS), combining instruments (seismometers, pressure sensors) deployed on the sea floor as part of their Neptune Observatory, and a shore-based High-Frequency (HF) radar. This HF radar can remotely sense ocean currents up to a 80 km range, based on the Doppler shift they cause in ocean waves at the radar Bragg frequency. Using this method, however, tsunami detection is limited to shallow water areas where they are sufficiently large due to shoaling and, hence, to the continental shelf.
To extend detection range into deep water, thereby increasing warning time, the authors have proposed a new detection algorithm based on spatial correlations of the raw radar signal at two distant locations along the same wave ray. In a previous work, they validated this algorithm for idealized tsunami wave trains propagating over a simple sea floor geometry in a direction normally incident to shore. In the final manuscript, this algorithm is extended and validated for realistic tsunami case studies conducted for seismic sources and using the bathymetry off  My wife, Emmy, has been loving, patient and supportive through late nights and changing plans. I am grateful that she reminds me to take care of myself and encourages me to go to the beach once in a while.
Gail Paolino consistently makes even the most convoluted paperwork seem simple. With her in the office, I know that there is always someone looking out for me. It is a relief to know that she is there.      (1)(2)(3)(4) where large tsunamigenic SMF sources are sited [4,2] and parametrized as Currituck SMF proxies [2]. Depth is in meters, in the color scale and bathymetric contours. . . . . . . . . 7 1.3 Regional and near-shore computational grids used in tide, tsunami-only, and tsunami-tide simulations with FUNWAVE-TVD (labeled red boxes correspond to grids defined in Table 2.1). Tide-only simulations are initiated in grid 3b, and then nested into grids 2 and 1. After being initiated in the 1 arcmin grid G4 ( Fig. 1.1), simulations of far-field tsunami sources (CVV, PRT) are carried out in nested grids G3b, G2 and G1.
Simulations of the near-field CRT proxy SMF tsunamis are performed in grids G3a, G2, and G1. All tsunami-tide simulations are initiated in grid G2 and then rerun in grid G1. Color scale is bathymetry (<0) and topography (>0) in meters. . . . . . . . 13 1.4 High-resolution bathymetry/topography in HRE's area of interest, from FEMA's 8 m DEM [33], used to define the finest resolution grid G1's depth matrix. This data set was combined with the 90 m NOAA DEM data to define Grid G2's depth matrix (Table 2.1). Color scale is bathymetry (< 0) and topography (> 0) in meters, referenced to the NAVD88 vertical datum. . . . 16 xii  Table 1.2). The red box marks the footprint of grid G1. Simulated and measured tide time series at the stations are plotted in Figure 1.8, and differences between these are quantified in Table 1  Simulations of the M w 9 PRT seismic source with FUNWAVE-TVD, in grid G4 (truncated at Lon. E. -55; Fig. 1): (a) Initial surface elevation of tsunami source computed in lower red box with Okada's method [37], based on 12 SIFT sub faults [10,38]; the upper red box approximately represents the area of   Table 1). Tsunami generation is performed in grid G1 using NHWAVE with 5 σ-layers in the vertical direction, and results are used as initial condition in FUNWAVE-TVD, also for grid G1. The thick yellow line marks the 5 m depth isobath along which tsunami elevation is computed in Fig. 2  Time series of (a) surface elevation at stations 1-8 ( Fig. 3.2b) in grid G2 (dash lines) and G3 (solid lines); (b,c) spatially-averaged radial velocity U r in radar cells 1-9 aligned along a given wave ray ( Fig. 3.5b): (b) as a function of time; (c) shifted in time by the long wave propagation time t p1 from cell p = 2, ..., 9 to cell 1. 142 3.5 (a) Examples of wave rays computed from an incident direction from west (green; φ t0 = 0) and southwest (red; φ t0 = 45) as a function of bathymetry in grid G3, over the Tofino HF radar sweep area (Fig. 3  We note that, while dynamic simulations predict a slight increase in inundation, this increase may be on the same order as, or even less than sources of uncertainty in the modeling of tsunami sources, such as their initial water elevation, and in bottom friction and bathymetry used in tsunami grids. Nevertheless, results in this paper provide insight into the magnitude and spatial variability of tsunami propagation and impact in the complex inland waterways surrounding New York City, and of their modification by dynamic tidal effects. We conclude that changes 3 in inundation resulting from the inclusion of a dynamic tide in the specific case of the HRE, although of scientific interest, are not significant for tsunami hazard assessment and that the standard approach of specifying a static reference level equal to MHW is conservative. However, in other estuaries with similarly complex bathymetry/topography and stronger tidal currents, a simplified static approach might not be appropriate.

Introduction
Tides and tsunamis are both long waves, whose propagation can accurately be modeled by a long wave theory [1], such as linear Stokes theory in deep water or Saint Venant (a.k.a., Nonlinear Shallow Water equations; NSW) or Boussinesq equations in shallow water, depending on the relative magnitude of nonlinearity and dispersive effects. In deep water, tsunamis are not significantly affected by tides, because both the tidal range is small with respect to depth and tideinduced currents are very weak; hence, tsunami phase speed and shoaling are not significantly affected by the small change in water depth caused by tides. This also applies to shallow coastal water areas that have a simple bathymetry and a fairly straight coastline, as is the case for most of the ocean-exposed US east coast (USEC), from Florida to Massachusetts. There, while tide-induced currents may become larger and tidal range be more significant as compared to the local depth, dynamic tidal effects are still small as compared to those in tsunamis, and tsunami inundation and runup can still be accurately assessed by modeling tsunami propagation over a static high antecedent water level (typically the 10% exceedence tide or the mean high water (MHW) level). This was for instance the approach followed for performing tsunami inundation mapping in Ocean City, MD due to tsunamis caused by submarine mass failures (SMF) along the upper USEC [2].
When assuming a static increase of the mean water level (MWL) in model simulations, both tsunami phase speed and elevation will be affected by the increased depth, yielding larger inundation further onshore. However, in coastal regions where tidal range is large and/or bathymetry is complex (e.g., creating funneling effects), tide-induced currents may become both large and significantly varying in space, leading to both stronger and more dynamic tsunami-tide interactions. In such cases, earlier work (e.g., [3]) indicates that one needs to more carefully and accurately model tide-tsunami interactions to achieve a conservative coastal hazard assessment. This requires, in particular, evaluating whether non-  [13,14,15]. To develop tsunami inundation maps, simulations were performed using the fully nonlinear and dispersive model FUNWAVE-TVD [16,17], by one-way coupling, in a series of coarse to finer nested grids. According to the standard methodology, in these simulations, the reference level in the coastal grids was set to a high tide value (such as the Mean High Water (MHW) level). Hence, potential dynamic interactions between tide-and tsunami-induced flows were neglected.
It should be noted that in this past work Abadie et al. [14] and Tehranirad et al. [15] simulated two flank collapse scenarios for CVV, one with a 450 km 3 volume and geometry similar to Ward and Days [13] original scenario and one, deemed the   [3] found that tsunami waves propagated further on a rising tide in the lower portion of the river; however, upstream the tsunami propagated further at the maximum high tide. Results also showed a potential amplification of tsunami waves directly after high tide. Tolkova [3] concluded that the interaction of the two long waves is completely dependent on the specific environment in which the interaction occurs, which justifies performing site-specific studies. More recently, performing similar studies based on data from a river in Japan, Tolkova et al. [22] showed that the Tohoku 2011 tsunami had caused increased surface elevations in the river by hindering drainage; this translated into increased tsunami inundation during tidal ebb. In the same geographic area, Nakada et al. [23] performed high-resolution simulations of the propagation in Osaka Bay of a large tsunami generated in the Nankai Trough. To quantify tide effects they run many cases in which tsunami propagation started every hour, through two tidal cycles.
They concluded that strong flood or ebb tidal currents modulated tsunami arrival by a few minutes and led to increased elevation in many situations, particularly during strong ebb flows, as compared to a static computation.
As part of the NTHMP USEC work, Tajalli-Bakshs et al. [24] modeled dynamic tsunami-tide interactions in Chesapeake Bay, with particular focus on assessing tsunami hazard in the James River, which is most affected by tidal currents and has the Norfolk Naval facility at its mouth and a nuclear power plant upstream.
They considered the M2 tidal component in the Bay and combined it, for different phases, with the two worst case scenario PMTs identified for this area, i.e., tsunamis generated by an extreme CVV flank collapse and the historical Currituck underwater slide, whose site is located near the mouth of the Bay [25,2] ( Fig. 1.2). While results showed clear nonlinear tsunami-tide interactions, affecting both tsunami elevation and propagation speed in the river, maximum tsunami inundation did not exceed that computed over a static reference level equal to the maximum elevation of the dynamic tide at the river mouth (here, the 10% exceedence maximum tide level).
Earlier studies summarized above all concluded that tsunami-tide interaction effects are largely site-specific. In the Chesapeake Bay, one of the two large estuaries located along the USEC considered in NHTMP work, Tajalli-Bakhsh et al. [24] concluded that this more advanced modeling approach was not necessary for a proper tsunami hazard assessment. Here, following a similar methodology, we simulate the combined effects of tides of various phases on the evolution of tsunami waves in the HRE ( Fig. 1.1 (3) A magnitude 9.0 earthquake in the Puerto Rico Trench [9,10]. The HRE has particularly strong currents (1-2 kts, i.e., nearly twice the speed of currents in Chesapeake Bay) and also has been identified as one of the highest risk areas along the USEC for flooding caused by a tsunami resulting from a submarine mass failure (SMF) occurring in the Hudson River Canyon [4]; this led Grilli et al [2] to define CRT SMF proxy sources in the HRE canyon area ( Fig. 1.2).
Besides being part of the NTHMP work scope of performing conservative tsunami hazard assessment for all the U.S. coastal areas, the HRE is another complex tidal system to assess the importance of nonlinear exchanges of energy between tide and tsunami, similar to the work done by [3] in the Columbia River.
There, Tolkova [3] found that tsunami signals propagating with the low tide were gradually damped out, while those traveling with the high tide were preserved or amplified. This was most apparent at the farthest upstream station for which data for the Tohoku 2011 tsunami were collected. Similar phenomena were observed by Tajalli-Bakhsh et al. [24] for tsunamis propagating up the James River, although as indicated this did not lead to higher inundation than for a static tide level.
If the Hudson River results are consistent with Tolkova's [3] findings, differences between static and dynamic tsunami-tide simulations should be larger at upstream locations when propagating over a high tide.
In the following, to assess dynamic tsunami-tide interactions in the HRE, we perform two sets of simulations. First, for each PMT, we simulate tsunami propagation into the HRE assuming a static tide level equal to the local MHW level.
Then we perform joint tsunami-tide simulations for four phases of tidal forcing achieving a maximum level identical to MHW in the HRE. The methodology for performing combined tsunami-tide simulations, which is similar to that used by Tajalli-Bakhsh [24], is detailed in the next section. We then briefly detail the computational model and present grid set-up. We finally report in detail and compare results of the two sets of simulations. Note that the choice of the MHW for the static and maximum tide levels is consistent with the standard approach in tsunami inundation mapping done for NTHMP. Tajalli-Bakhsh [24] used slightly higher 10% exceedance tide level in Chesapeake Bay, for both static and dynamic simulations, because tsunami hazard was assessed at a nuclear power plant in the James River, which was required to be slightly more conservative.    Table 2.1). Tide-only simulations are initiated in grid 3b, and then nested into grids 2 and 1. After being initiated in the 1 arc-min grid G4 ( Fig. 1.1), simulations of far-field tsunami sources (CVV, PRT) are carried out in nested grids G3b, G2 and G1. Simulations of the near-field CRT proxy SMF tsunamis are performed in grids G3a, G2, and G1. All tsunami-tide simulations are initiated in grid G2 and then rerun in grid G1. Color scale is bathymetry (<0) and topography (>0) in meters.

Modeling methodology and model grids 1.2.1 Models and modeling methodology
All simulations, both tide and tsunami, alone or combined, are performed using the fully nonlinear and dispersive Boussinesq model FUNWAVE [26,27,9,28], in its most recent Cartesian [16] and spherical [17] implementations referred to as FUNWAVE-TVD (the spherical implementation including Coriolis effects). For the near-field CRT SMF proxy tsunami, the first level grid is initialized with the surface elevation and horizontal velocity computed using the three-dimensional model NHWAVE [32] (see details in [2]) For the dynamic tsunami-tide simulations, we follow the methodology that was first applied by Tajalli-Bakhsh [24] in the Chesapeake Bay estuary, i.e., we: 1. Simulate the propagation of the selected PMTs from their source, in a series of nested grids, to a moderate resolution regional grid (here the 154 m resolution grid G2; Fig. 1 3. Then jointly simulate tide and tsunami, by linearly superimposing incoming tsunami wave elevations and velocities with tidal forcing, along the offshore boundary of a computational grid selected with a depth large enough along its offshore boundary to justify such a linear superposition (here, grid G2).  [33], used to define the finest resolution grid G1's depth matrix. This data set was combined with the 90 m NOAA DEM data to define Grid G2's depth matrix (Table 2.1). Color scale is bathymetry (< 0) and topography (> 0) in meters, referenced to the NAVD88 vertical datum.
4. Finally, simulate effects of tide phase on the three incident tsunamis by considering four different phases when peak tsunami and time-shifted tide signals are superimposed along the boundary of grid G2.

Grid bathymetric data
Besides it's footprint, resolution, and type (spherical or Cartesian), each model grid requires a depth matrix that is developed by interpolating bathymetric and topographic data of resolution commensurate with that of the grid. The key parameters for each model grid are listed in wherever available [33].
In analyzing NOAA's detailed bathymetric data used for coastal hazard assessment in the HRE, we noted a paucity or even a lack of data in the vicinity of Manhattan Island. This region, however, is critical for considering tsunami effects shows the resulting (interpolated) bathymetry and topography for grids G1 and G2. The vertical datum is referenced in all grids to NAVD88. Note that grid G1 is oriented at 18 • clockwise from north ( Fig. 1.3) to allow for a more efficient use of grid points, which significantly reduces the model computational time.

Fresh water discharge
Finally, the fresh water discharge from the Hudson River was estimated at    before. This procedure was finally repeated for grid G1. Based on differences observed between modeled and reference surface elevations at 14 NOAA gages located within grid G1, simulations were repeated with modified bottom friction coefficients in grid G1, to achieve the best possible agreement.

Tide only simulations
OTPS' latest version TPXO8 predicts tidal elevations and currents along the USEC, in a 2 arc-min grid. Considering this is a fairly coarse grid, OTPS' results are more accurate offshore, in deeper water [36]. Accordingly, tide simulations with FUNWAVE were initiated in the larger, coarser resolution, domain G3b, whose boundary is mostly located in fairly deep water. Following Tajalli  After this initial tide simulation, the Manning friction coefficient n was adjusted in grid G1 to improve the agreement between the modeled and known maximum elevations at NOAA's tide gages. By observing discrepancies at these stations (marked in Fig. 1.7a), n was adjusted to 0.015 in the Hudson and East Rivers, north of Battery Point, while a value n = 0.025 was used in the remainder of grid G1; this value was also used in grid G2. The corresponding friction coeffi-  Table 1.2). The red box marks the footprint of grid G1. Simulated and measured tide time series at the stations are plotted in Figure 1.8, and differences between these are quantified in Table 1.2. Color scale is bathymetry (< 0) and topography (> 0) in meters referenced to NAVD88 vertical datum.
tent with the methodology of [2].

Tsunami only simulations
Based on earlier work summarized in the introduction, three PMTs were se-  1.6. The maximum water level elevation with respect to the NAVD88 datum is given at each station for the NOAA high tide (MHW) prediction η p , compared to surface elevation η m modeled with FUNWAVE-TVD in the 154 m resolution grid G2 and 38.5 m resolution grid G1 (note, only 14 stations are located within this grid); the absolute (η m − η p ) and relative differences between these ((η m − η p )/η p ) are listed for each grid.   26 off of the Hudson River canyon [2]. The far-field tsunami sources (PRT, CVV) were specified and their propagation was first modeled in grid G4 (  20 min of propagation, respectively. We see that maximum tsunami elevations are quite directional south-to-north and focus on the upper USEC; this was already pointed out by [9]. After 200 min of propagation, the tsunami is entering the SE corner of grid G3b ( Fig. 1.3).

Dynamic tsunami-tide simulations
Simulations are repeated for the three incident PMTs in combination with the time-varying calibrated tide, which both modulates the reference water level   In deeper water, offshore of the HRE, both tide and tsunamis are long waves of fairly small amplitude, as compared to depth and wavelength, which can thus be linearly combined (i.e., both elevation and current are additive) [1]. For such a superposition to be accurate closer to shore, water depth must be large enough compared to tsunami and tide elevation; here, considering that incident tsunami amplitudes are on the order of 2 m or less ( Fig. 1.13) and tide amplitudes are up to 1 m ( Fig. 1.8), water depth should be on the order of at least 20 m. As indicated before, incident tides computed in grid G3a and tsunamis computed in grid G3b are linearly combined at numerical gages (stations) located along the     period. For the PRT and CRT tsunamis, tsunami-tide synchronization was done for the leading crest ( Fig. 1.13). For the CVV tsunami, a second taller crest arrived just over two hours after the initial crest ( Fig. 1.13); accordingly, besides 4 simulations for the leading crest, 4 additional dynamic simulations were performed for CVV, corresponding to the arrival of this second crest at the 4 phases of the tide. and are strongest near the mouth of the Bay. In this simulation, the strongest currents nearly reach 1.5 m/s (3 knots), which is notably larger (more than twice) than the currents simulated by Tajalli-Bakhsh [24] (and observed) in the wider Chesapeake Bay and even in the James River.
In view of these current patterns, one might anticipate that the second and third phases of dynamic tsunami-tide simulations, in which the largest wave in each incident tsunami reaches the Sandy Hook gage, near the mouth of the HRE, concurrently or 1.5h after high tide, should lead to the maximum amplification of the incident tsunamis, at least, near the mouth of the Bay. Indeed, while for these phases tidal elevations are either maximum or have not yet decreased too much from their highest level, the tsunamis propagating into the Lower Bay will be facing opposite (ebbing) currents that will be increasing or be already quite strong (0.5 to 0.75 m/s in Fig. 1.15c); these opposite currents will continue to strengthen as the tsunamis propagate into New York Harbor and the Hudson and East Rivers (as seen in Fig. 1.15c and d) and should cause the tsunami surface elevation to rise, at least initially. In Fig. 1.15d, while currents are even stronger 3h after high tide, tide surface elevations are starting to become negative and hence it will be harder to achieve higher elevations in the combined results.
Being both long waves, without nonlinear interactions, tide and tsunami should be propagating into the HRE at the same phase speed and their combined level should evolve in a way similar to the individual levels. Nonlinearity, however, will affect these features, first by causing amplitude dispersion effects that   Fig. 1.19 shows that the worst dynamic case scenario is also for high tide, leading to a slightly increased flooding, by up to 0.05 m at the entrance to Lower Bay. Upon entering Lower Bay, this crest interacts with tidal currents that have already been disturbed by more than 2h of tsunami propagation into the bay, and reflection coming back from the upper part of the HRE. The confused currents within the Bay are likely responsible for the mild decrease (by up to -0.05m) in surface elevation seen across the remainder of grid G1. This is confirmed in   Fig. 1.17), slack tide occurs about 1 h after high tide, but in the Hudson River, flooding currents persist for more than two hours after high tide. Hence, leading tsunami crests arriving 1.5 h after high tide (during the initiation of the ebbing current) will still experience a favorable current in the Hudson River and 54 thus will tend to decrease in elevation. Another phenomenon affecting dynamic tsunami-tide simulations is that during lower tide elevation periods, the tsunamis propagate over shallower water areas and hence could end up shoaling somewhat more in some areas than in the static simulations; however, as results of the propagation over a lower tide have shown (i.e., 3 h delay), this in general does not lead to increased maximum flooding in the HRE over the entire simulation. Differences between surface elevation time series for the dynamic tsunami-tide simulations and those for the tsunami propagating over a static MHW level thus mostly result from nonlinear interactions between tide and tsunami currents.

Conclusions
We performed simulations of dynamic tsunami-tide interactions in the Hudson River estuary (HRE) and compared results to the standard tsunami simulations, which are performed over a static MHW tide level. In both cases, the maximum tide level (static or maximum dynamic) was selected as the average maximum tidal  Bay, which is an area especially vulnerable to tsunami inundation. We see that the inundation extent of the dynamic case encompasses that of the static case, except in a very small area for the CVV case.

Acknowledgments
The authors gratefully acknowledge funding for this work, provided by grants #NA14NWS4670041 and #NA15NWS4670029 of the U.S. National Tsunami Haz- ards Mitigation Program (NTHMP). Ongoing and future work will focus on extending the granular slide model, which features more complete and realistic to more accurately simulate tsunami generation from deforming SMFs, in a variety of context and rheology.

Introduction 2.1.1 General context
As evidenced by the two recent catastrophic events, of 2004 in the Indian Ocean (IO; e.g., [1,2]) and 2011 in Tohoku (TO), Japan ( [3,4]), extreme tsunamis can devastate the world's increasingly populated coastal areas, causing high fatalities (over 200,000 for the IO tsunami and over 18,000 for the TO tsunami), destroying fragile coastal infrastructures, and imposing enormous loss to the economy of the most impacted countries (e.g., an over $300B loss due to the TO tsunami im-pact in Japan; [5]). Commensurate with their catastrophic impact, the IO and TO events were associated with the 3rd and 5th largest earthquakes ever witnessed in human history (M w 9.3 and 9.1, respectively), and were essentially triggered by the coseismic seafloor deformation induced in the subduction zones (SZ) where their epicenter was located (Andaman SZ and Japan Trench, respectively). As these SZs were in close proximity to the Indonesian and Japanese coastline, respectively, both the tsunami propagation time from the start of each event was short (15-25 min), thus reducing warning times, and the coastal impact was maximized on these countries' coastline as there was no time and space for wave energy spreading to occur; in both cases extreme tsunami runups of over 40 m were measured.
However, for the TO tsunami, despite the large earthquake magnitude, many studies have shown that the largest coastal inundation and runup that impacted the 80 km long Sanriku coast (to the North of the main rupture area), could not be explained by the seismically triggered tsunami alone (e.g., [3] and references herein). Studies based on tsunami waveform inversion (e.g., [6,7,8] ) showed that an additional source of tsunami generation near the trench, to the north of the main rupture, was required to explain these observations. While several seismically related mechanisms were proposed to explain this additional source of wave generation (e.g., [9]), Tappin et al. [4] showed that there is strong evidence Following the methodology detailed in [10], they modeled wave generation from this SMF using the non-hydrostatic three-dimensional (3D) (sigma layer) model NHWAVE [11], in which they specified the SMF law of motion and geometry as bottom boundary conditions, based on earlier numerical and experimental work by Grilli and Watts [12,13], Watts et al. [14], and Enet and Grilli [15,16,17]. Once the waves were generated and slump motion terminated, they continued modeling tsunami propagation towards the coast of Japan with the two-dimensional (2D) fully nonlinear and dispersive Boussinesq long wave model FUNWAVE-TVD, in a series of nested grids of increasingly fine resolution [18,19]. Waves predicted at the locations of nearshore GPS buoys and offshore DART buoys, as well as the modeled coastal inundation and runup, agreed very well with observations, when the SMF was triggered with a 2'30" time delay, consistent with the propagation time of seismic waves from the earthquake epicenter to the assumed SMF location. min delay, 1,600 m deep off the Sissano Spit, whose waves were responsible for the large inundation and runup focused on the spit [35]. This event was the first well documented case supporting the large tsunamigenic and destructive potential of SMF tsunamis, that led to a large number of studies and developments in theoretical/numerical models (e.g., [12,13], [25,14], [36], [37,38], [39,40,41], [42,43], [44], [29], [45]; for details, see the very exhaustive review by Yavari-Ramshe and Ataie-Ashtiani, [46]) and laboratory experiments (e.g., [15,16,17], [13], [28,47], [48], [41], [49], [50]), as well as a reanalysis of past events (e.g., 1946 Unimak) in view of new evidence ( [27]), that had been potential SMF candidates, but whose analysis or modeling had not been fully conclusive [26].
Past volcanic eruptions have also been associated with the generation of large and destructive tsunamis, from pyroclastic flows and/or caldera collapse, such as the 1883 Krakatau [51]

Specific context (USEC)
Although this study has more general goals, its main focus is to improve the modeling of tsunami generation, and coastal inundation and runup, from SMFs lo-  In part due to the lack of specific data on past SMF geometry and parameters at the time and the uncertainty in identifying these and the locations of future tsunamigenic SMFs, Grilli et al. [68] performed Monte Carlo Simulations (MCS) of SMFs triggered by seismicity along the USEC. They defined a series of transects across the coast, initially from New Jersey to Cape Cod, but these were later extended to southern Florida for a total of 91 transects. Thousands of SMFs were simulated along each transect, with random values of seismicity, sediment properties, SMF type (slides and slumps), location/depth, geometry, and excess pore pressure being picked from known probability distributions and/or site-specific field data. Slope stability analyses were performed for each of these and, for each SMF that failed, tsunami generation and runup were estimated based on semiempirical methods [13,14]. Statistics of results were performed, which allowed estimating the 100-and 500-year return period runups along the entire USEC. As indicated before, Currituck is the largest paleo-SMF identified along the western Atlantic Ocean margin and occurred between 24 and 50 ka ago, in a period when sea level was much lower. This event has been extensively studied from the geological and slide triggering points of view (e.g., [20], and references herein).
Tsunami generation from a reconstituted Currituck SMF was first studied by Geist The red box marks the boundary of the 500 m resolution NHWAVE Cartesian grid CT used to simulate the Currituck SMF motion (see Table 1). Depth is in meters, in the color scale and bathymetric contours. then modeled tsunami generation from the four SMF Currituck proxies in areas 1-4 ( Fig. 2.1.2), assuming they behaved as rigid slumps, and computed coastal inundation and runup, to be included in NTHMP inundation maps.
In this work, in view of recent progress made in SMF tsunami models, the modeling of tsunami generation and coastal impact on the USEC using SMF Currituck proxies is revisited, to estimate the impact of slide rheology on tsunami hazard. Earlier work (e.g., [75,13]) indicates that moderate SMF deformation should generally result in a slightly reduced coastal hazard, but overall does not significantly affect tsunami generation. This may not always be the case, however, as local bathymetry, seafloor properties and SMF geometry could play an important role on slide kinematics and tsunami generation. For the purpose of developing the most realistic and accurate inundation maps for NTHMP, it is important to use the most appropriate slide model. Accordingly, in this paper, we first summarize the equations and methods for two recent two-layer SMF models developed and applied by some of the authors. In both models, the upper layer is water modeled with the 3D model NHWAVE and the slide is a depth-integrated layer modeled as: (i) a dense Newtonian fluid [31]; and (ii) a saturated granular medium [76]. Both models are then validated and benchmarked against recent laboratory experiments performed for SMFs made of small glass beads moving down a plane slope. We then apply the dense fluid model (note the granular model is not yet applicable to an arbitrary slope), to the historical Currituck slide and the SMF Currituck proxy in area 1, and compare both slide motion and tsunami generation with those of the same SMFs modeled as rigid slumps. We then draw some conclusions regarding tsunami hazard assessment.
In this work, tsunami generation by SMFs is modeled with the 3D nonhydrostatic model NHWAVE [11], which uses a horizontal Cartesian grid and a boundary fitting σ-coordinate grid in the vertical direction. Once the tsunami is fully generated, the modeling of wave propagation is pursued with the 2D fully nonlinear and dispersive long wave Boussinesq model FUNWAVE-TVD [18,19], in a series of nested grids of increasingly fine resolution (see, e.g., [3,10,62,4], for more details on this approach). It should be pointed out that both NHWAVE and FUN-WAVE are non-hydrostatic, i.e., dispersive, wave models; the inclusion of frequency dispersion in tsunami models has been shown by many authors to be necessary for accurately modeling SMF tsunami generation and propagation, essentially, because of the typically smaller wavelength to depth ratio of SMF tsunami waves (see, e.g., [12,13,25,35,11,81]). Failing to include dispersive/non-hydrostatic effects in the model used to simulate SMF tsunami generation has been shown to cause large errors in the shape and kinematics of initial waves (e.g., [35,11]). Additionally, failing to include dispersion in the model used to propagate the highly dispersive wave trains that are generated will also cause large errors in wave height and steepness (and hence coastal impact), due to a lack of constructive-destructive wave-wave interferences during propagation; the generated wave trains will also typically lack the long oscillatory (dispersive) tail observed experimentally (see the many experimental works listed before), in the field (e.g., [4]), or numerically (e.g., [25,35,65,66,67,10]) for landslide tsunamis, and will be limited to one or two leading waves.
We model tsunami generation by rigid SMFs following the methodology detailed in Grilli et al. [10], in which the SMF law of motion (slump or slide) and geometry are specified as bottom boundary conditions in NHWAVE (see also [13,14,17]); the reader is referred to the references for the details of this method.
We model tsunami generation from deforming SMFs by representing them as a layer of dense fluid [31] or a saturated granular medium layer [76], coupled along the deforming SMF-water interface with NHWAVE, which is used to simulate the resulting wave motion. In both cases, SMF equations of mass and momentum conservation are depth-integrated, similar to those obtained in a long wave generation model, and these include volumetric and bottom friction dissipation terms. In experiments, the actual initial shape of the SMF is modeled, whereas in field case studies, both rigid and deforming slide geometry is modeled as an initial sediment mound with quasi-Gaussian cross-sections and an elliptical footprint over the slope (see, [17,10], for details).
The two deforming SMF models will be validated and benchmarked against laboratory experiments for tsunami generation by slides made of glass beads, moving down a plane slope. However, only the dense fluid slide model will be applied to field case studies, because the other model cannot yet simulate a bathymetry with arbitrary depth h(x, y) (the extension to an arbitrary bathymetry is in progress).
Below, we provide the governing equation for the dense fluid slide model layer, where F = (DU, DV ) with D = h − χ (where h is the fixed bathymetry without slide, and χ denotes the distance from the still water level (swl) to the slide interface; Fig. 2 (i.e., for more dispersive waves), during which velocities can be much more nonuniform over depth than during subsequent tsunami propagation (see, e.g., [36]); (iv) Being fully nonlinear, FUNWAVE-TVD is more accurate than NHWAVE for simulating coastal wave transformations and impact, in particular, regarding wave breaking dissipation and the moving shoreline algorithm; (v) Although both models are heavily and efficiently parallelized, FUNWAVE-TVD is more computationally efficient as it is only 2D, and hence uses grids at least 3 times smaller than the minimum NHWAVE grid with 3 σ-layers that provides a similar approximation of horizontal velocities in the vertical direction; and finally (vi) FUNWAVE-TVD also has a spherical implementation, which allows a more accurate simulation of far-field tsunami propagation. Note, in the present applications, because all simulations will be performed in regional or nearshore grids with small latitudinal and longitudinal ranges, spherical coordinates will not be necessary, although combinations of spherical and Cartesian nested grids have been used in earlier work with FUNWAVE-TVD (see, e.g., [3,10,62,19]).

Experimental validation of deforming slide models 2.3.1 Description of laboratory experiments and results
Laboratory experiments of tsunami generation by underwater slides made of glass beads were performed at the Ecole Centrale de Marseille (IRPHE), France, in a precision tank of (useful) length l = 6.27 m and width w = 0.25 m (Fig. 2.3). In each experiment, a Mass W b of beads of density ρ b = 2, 500 kg/m 3 was submerged in fresh water of density ρ w = 1, 000 kg/m 3 , in a reservoir of triangular shape located over a θ = 35 degree slope, fronted by a movable sluice gate. Experiments were started by withdrawing the gate into a bottom cavity within the slope (using a highly repeatable motion controlled by springs, visible in Fig. 2  of each experiment, t = 0, is defined when the gate has just withdrawn into its cavity). In Fig. 2 Fig. 2.4b), which then rebounds, creating two sets of waves moving down and up the tank (Fig. 2.4c). The "onshore" moving waves cause runup on the slope whereas the "offshore" waves reflect on the far end of the tank and propagate back towards the generation area (Fig. 2.4d-f). This behavior is also clearly observed in time series measured at wave gages WG1-WG4 (Fig. 2.5). A detailed analysis of experimental results shows that this and each experiment is highly repeatable, with almost unnoticeable differences between wave gage measurements for two replicates of the same experiment. More details of the experimental set-up and methods, and of results of the 58 experiments will be reported elsewhere.
Here, we benchmark the two models of tsunami generation by deforming slides when the gate has just withdrawn, defined as t = 0. Numerical wave gages are located in the model at the same locations as shown in Fig. 2.3a and simulated time series are compared to experimental data. This is detailed in the next section.

Numerical modeling of laboratory experiments
We model one laboratory experiment of tsunami generation by a deforming slide made of glass beads of density ρ b = 2, 500 kg/m 3 in fresh water of density ρ w = 1, 000 kg/m 3 , with parameters: h = 0.330 m; d b = 4 mm, W b = 2 kg, and no glued beads on the slope (Fig. 2.4), using the two-layer models detailed above, i.e., with either one dense Newtonian fluid sublayer [31] or a saturated granular medium sublayer [76]. To apply the latter model, we simply define a granular material made of spherical grains of density and diameter identical to the glass beads used in experiments. To apply the former model, however, we first need to estimate the density of a dense fluid equivalent to the initial mixture of glass beads and water. Also, because only the diameter and total dry weight of glass beads was measured in experiments, for both models, we need to compute the location of the resting interface between slide and water ( Fig. 2.4a), i.e., the slide initial submergence h s , whose value significantly affects tsunami generation (Grilli and Watts, [13]). Finally, for both models, we need to specify realistic values of the Manning bottom friction coefficient n and the parameter controlling the volumetric energy dissipation. For the dense fluid, the latter is the dynamic viscosity µ and for the granular medium it is the internal Coulomb friction angle ϕ int and dynamic bed friction angle ϕ bed , plus a calibration parameter λ controlling the pore pressure and friction near the bed. As detailed below, results from theoretical and experimental work on granular media will be used to estimate the dense fluid dynamic viscosity and the granular, bed and internal, friction angles will be based on recommended values from earlier experimental work, which will then be slightly adjusted. Finally, for both models, the Manning coefficient will be calibrated for the first generated wave to match observations.
While the maximum regular packing of spheres is φ m = π/(3 √ 2) = 74.0% of the volume occupied by the spheres, studies of randomly packed spheres show that the packing value is typically near (and bounded by) φ = 63.4% [82]. Here, we will assume this value, meaning that for a saturated medium, 1 − φ = 36.  However, as the slide deforms and moves down the slope (Fig. 2.4c-d), the glass bead volume fraction decreases, which based on Eqs. 2.2 causes the equivalent slide density and viscosity to decrease. To account for this, in the dense fluid model, we will be using a lower value of viscosity, µ s = 0.01 kg/(m.s). Additionally, using a lower value of viscosity can help compensate for the nearly free slip of glass beads along the slope and the low bottom friction that results, when glass beads are not glued to the slope. A lower viscosity can also enhance the slide initial acceleration, which as indicated above, may be underestimated in the model Eq.2.1. For this value of viscosity, a Manning coefficient value n = 0.04 was calibrated so that the modeled slide reach the bottom of the slope at the time measured in experiments ( Fig. 2.4).
The selected glass bead experiment is modeled with NHWAVE using the initial slide geometry computed above (T, h s ) and the selected parameters, ρ s , µ s and n, for the dense fluid representing the slide. A sensitivity to grid resolution detailed later shows that results are well converged when using a horizontal resolution with grid size ∆x = ∆y = 0.01 m and 9 σ-layers in the vertical direction. efficiency, this resolution will be used from now on. i.e., ϕ int = 29 deg. and ϕ bed = 6 deg. Then, to improve the agreement of the first generated wave with laboratory measurements, the bed friction was slightly adjusted to: ϕ bed = 9 deg., together with using λ = 0.04 (note that a larger value of λ ∈ [0, 1] will result in a reduced bed Coulomb friction). Finally, the (water on slide or substrate) friction coefficient was set to C f = 0.6. This is a fairly large value, but it allows accounting for sidewall friction in experiments; results will show that with this model, the oscillatory tail in the wave train is still too large as forcing across the width of the tank, which leads to some lateral spreading in the resulting waves. Nonetheless, as indicated the granular model produces surface elevations, which agree reasonably well with experimental results.

Sensitivity of tsunami generation to slide rheology
When modeling SMFs as deforming slides represented by a dense Newtonian fluid, besides density, two main parameters control the slide center of mass motion S(t) down the slope (measured parallel to the slope): the equivalent slide viscosity µ s and Manning coefficient n. When setting these parameters to larger values, this increases the frictional dissipation and resulting forces that are slowing down slide motion in the model. Hence, when µ s or n are increased, we expect that slide  acceleration and velocity will be reduced, once the slide picks up speed and these forces become effective. For the initial part of slide motion down the slope, however, slide velocity is small and the motion should be close to be a purely accelerating motion, S(t) ∼ (1/2)A 2 o (t), with A o the initial slope acceleration down the slope [13,14]; hence one should not observe significant differences in slide motion. Since slide velocity and acceleration are reduced when µ s or n are increased, one would also expect that tsunami generation be reduced. Once the slide is on the flat bottom of the tank, this behavior may be slightly more complex. Results are also as expected regarding the free surface elevations modeled at wave gages, with a reduction in tsunami generation, as µ s or n are increased.

Sensitivity of tsunami generation to slide submergence
Earlier work indicates that besides slide volume/geometry, density, and rheology, one important parameter that affects tsunami generation is the initial slide submergence depth h s [13,14,17], with tsunami wave elevation increasing, the smaller the submergence depth. This is also observed in the full set of laboratory experiments with glass beads, which is not detailed here due to lack of space. In the model, this is verified in Fig. 2 submergence depth the largest leading trough and crest are significantly increased, as compared to the based case, particularly upon generation at gage WG1. For a larger submergence depth, the trend is opposite but by a lesser amount, indicating that nonlinear effects are at play. At gages WG2-WG4, the same trends are preserved but as dispersion distributes the change in the leading wave elevation to the training waves in the oscillatory tail, the effect of submergence depth is less marked than at the first gage.

Case studies off of the US East Coast
In the previous section, on the basis of laboratory experiments, we assessed the accuracy of two models simulating tsunami generation by deforming underwater slides, as well as the sensitivity of the dense fluid model results to parameters such as the equivalent viscosity µ s and the slide-substrate Manning friction coefficient n. In the following we apply the viscous fluid model to the simulation of tsunami generation by SMFs located off of the USEC, on the continental shelf break and slope, and evaluate the effect of slide rheology on coastal tsunami hazard, as compared to earlier simulations in which the SMFs were modeled as rigid slumps [10]. As before, we will limit ourselves to the upper USEC where SMF tsunami hazard was deemed to be higher [68] and consider SMFs with the characteristics of the historical Currituck slide, i.e., Currituck SMF proxies, located in area 1 in Fig. 2.1.2. Before this simulation, however, we revisit the simulation of the historical Currituck slide (which was modeled by Grill et al.     G1) and FUNWAVE-TVD (G1, G2) models to compute tsunami generation by SMFs and propagation to the coast. "Res." is grid resolution and N x and N y indicate the number of grid cells in each direction. and the generated tsunami and coastal impact are compared to those previously obtained by modeling the SMF as a rigid slump.

Currituck slide complex
As mentioned above, we simulate the historical Currituck slide using the deforming fluid-like slide model, in order to derive realistic values for the slide equivalent viscosity µ s and Manning coefficient n, that yield a law of motion of the slide center of mass similar to that of the rigid slumps used earlier. Hence, we focus here on slide kinematics, rather than tsunami generation, which is also computed in the model but will not be detailed here. for the failed area, 128-165 km 3 ), moving over a recreated unfailed bathymetry.
Although the failure clearly occurred in two stages, as inferred from the two existing headwalls (Fig. 2.13a), the initial slump geometry was idealized as a single simulations. With these parameters the slump time of motion were found as, t f = 11.9 min and runout S f = 15.8 km (Fig. 2.13c).
Here, we repeat these simulations for the same unfailed bathymetry and geometric representation of the SMF as in Grilli et al. [10] (Fig. 2.13b), but modeling its kinematics instead as a deforming fluid-like slide of density ρ s =1,900 kg/m 3 , using the NHWAVE model with a dense fluid layer underneath [31]. As shown in  Table 2.1), and 5 σ-layers in the vertical direction. Note that this grid covers a fairly small area, extending more to the SE direction where the deforming slide is moving and tsunami generation occurs, which may not be sufficient to perform actual tsunami hazard assessment, but is sufficient at this stage for the purpose of comparing slide kinematics between rigid and deforming SMFs.
Thus, Fig. 2 Fig. 2.13c, we see that, by contrast, the slump has just covered a pendulum-like rigid motion, while keeping its shape and footprint on the seafloor constant.

Hudson River canyon SMF
We simulate tsunami generation and propagation to the coast by a Currituck  Table 1). Tsunami generation is performed in grid G1 using NHWAVE with 5 σ-layers in the vertical direction, and results are used as initial condition in FUNWAVE-TVD, also for grid G1. The thick yellow line marks the 5 m depth isobath along which tsunami elevation is computed in Fig. 2 Fig. 2.16). Because this is a hypothetical SMF occurring for an existing slope, upon failure, the SMF is initially located below the seafloor (by contrast with the actual Currituck paleoslide whose initial geometry was recreated over the current seafloor). We simulate the SMF motion and related tsunami generation with NHWAVE, assuming either that it behaves as a rigid slump (as in [10]) or as a deforming slide. Once tsunami generation is completed in NHWAVE, simulations are initialized and continued with FUNWAVE-TVD.
In the rigid slump simulations, the SMF moves in the assumed azimuthal direction of motion θ = 136 deg. (from North), and its geometry and kinematics are computed as detailed in [10] and specified as a bottom boundary condition in NHWAVE. During this simulation, as in [10], bottom friction between water and substrate is neglected, which is acceptable considering the large water depth.
In the deforming slide simulations, we use the new NHWAVE two-layer model,  Fig. 2.14). Although the bottom topography is different, we find that the law of motion of the Currituck SMF proxy in Area 1 is very closely that derived for the historical Currituck slide, whether modeled as a rigid slump or a deforming slide, shown in Fig. 2.14 for these parameters (red curves); hence the Hudson Canyon SMF kinematics is not replotted here.  while preserving its shape, the deforming slide has flowed asymmetrically, following the steepest slope on the seafloor, while spreading in all directions ( Fig. 2.17a,b). wave has already formed in the middle of the depression. Fig. 2.18c shows that, at waves generated by the rigid slump are fairly symmetric with respect to the SMF axis of motion (ζ = 0), those generated by the deforming slide are larger on its western side (ζ < 0) than on its eastern side, and the maximum initial free surface depression shown at t = 300 s is also shifted westward, whereas it is on the SMF axis for the rigid slump. As discussed above, this is a result of the asymmetry of the deforming slide thickness on the seafloor. Finally, some critical coastal infrastructures, such as power plants, which have cold water intakes, are also severely impacted by low water conditions occurring during the initial SMF tsunami impact in the form of a large depression wave.
Hence, in Fig. 2.19d, we have similarly plotted the envelope of minimum surface elevation η min computed along the 5 m isobath for all SMF cases considered in Fig. 2.19c. Here we see that consistent with the larger initial depression wave it generates, the rigid slump also causes a larger drawdown at the coast, for most locations, than the deforming slide cases, except to the north (for x > 330 km), likely due to site-specific bathymetric effects.

Conclusions
In this work, we first validated two models simulating tsunami generation by deforming submarine mass failures (SMFs), against laboratory experiments conducted at IRPHE in a small precision wave tank. One case was used, from a large set of highly repeatable laboratory experiments performed for SMF mades of glass beads moving down a steep slope. Both validated models are two-layer models in which the fluid is modeled with the non-hydrostatic 3D (σ-layer) non-hydrostatic model NHWAVE and the SMF layer is depth-integrated and represented either as a dense Newtonian fluid [31] or a granular medium [76]. The latter model is currently limited to a plane slope and does not include dilatancy effects in the granular medium. Both models currently neglect vertical accelerations and hence should be more accurate for moderate to small slopes. At most locations, nearshore tsunami surface elevations caused by the rigid slump were found to be significantly larger (up to a factor of 2) than those caused by the 3 deforming slide cases; for those, surface elevation slightly increased when n decreased. Nearshore minimum surface elevations (tsunami drawdown) were sim-ilarly computed and it was found, again, that the rigid slump caused the largest drawdown in most cases, except along a stretch of the coast of Long Island, likely to site specific focusing bathymetric effects.

Most
In view of these results (see, Fig. 2.19), we conclude that, as expected from earlier work (e.g., [75,13]), the rigid slump provides a conservative estimate of SMF tsunami impact in terms of maximum inundation/runup at the coast and, in most locations, of maximum drawdown; by contrast, using a more realistic rheology with some level of SMF deformation, in general, will reduce tsunami impact at the coast, whether maximum inundation or drawdown. This validates as conservative the tsunami hazard assessment and inundation mapping performed to date as part of NTHMP, on the basis of Currituck SMF proxies simulated as rigid slump.
Clearly the two-layer granular flow NHWAVE model features more complete and realistic physics and thus has greater potential for more accurately modeling SMF tsunami generation, in a variety of context and rheology, once it is extended to arbitrary bottom topography; as indicated, this extension is in progress. Additionally, it is planned to extend the granular layer depth-integrated governing equations to both properly include effects of vertical accelerations, which may be important over steeper slopes, and dilatancy effect. The latter was for instance included in the two-phase model recently proposed by Bouchut et al. (2016) [89], who indicates that when dilation occurs the fluid is sucked into the granular material, the pore pressure decreases and the friction force on the granular phase increases. In the case of contraction (the opposite of dilation), the fluid is expelled from the mixture, the pore pressure increases and the friction force decreases. Including dilatancy will thus allow simulating both volumetric and bed dissipation effects that vary with slide shape and kinematics.

Introduction
Major tsunamis can be enormously destructive and cause large numbers of fatalities along the world's increasingly populated and developed coastlines [1,2]. While the brunt of tsunami impact cannot be easily attenuated, loss of life, however, can be mitigated or even eliminated by providing early warning to coastal populations. Such warnings can be issued based on early detection and assessment of the mechanisms of tsunami generation (e.g., seismicity) as well as detection of the tsunami itself as soon as possible after its generation. The latter is particularly important when the tsunami source is located close to the nearest coastal areas, and thus both energy spreading is low and propagation time is short. This is the case, for instance for co-seismic tsunamis generated in nearshore subduction zones (SZ) (e.g., Japan Trench, Puerto Rico Trench, Cascadia SZ,...), or for submarine mass failures (SMFs), that can be triggered on or near the continental shelf slope by moderate seismic activity [3,4,5]; meteotsunamis, also, may be generated on the continental shelves by fast moving pressure systems (e.g., derechos) [6].
The detection of offshore propagating tsunamis from a nearshore generation area is usually made in deep water, at bottom-mounted pressure sensors (so-called DART buoys), based on which a warning is issued for far-field locations. The detection of onshore propagating tsunamis in shallow water, over the continental shelf, is typically made by bottom pressure sensors and tide gauges that may not survive the impact of large tsunamis; additionally such detection is local (i.e., point-based) 132 and often takes place too late to be used in early warning systems. Hence, with the current detection technology used in tsunami warning systems, there may not be enough time to issue a warning for near-shore seismic or SMF tsunami sources, based on actual tsunami data. When the earthquake is the tsunami triggering mechanism, a warning can be issued based on detecting seismic waves, and from these estimating the earthquake parameters and the likelihood for tsunami generation. For non-seismically induced nearshore SMF tsunamis or for meteotsunamis, a warning can only be issued based on detecting the tsunami at nearshore sensors and, hence, there may not even be enough time to issue it before the tsunami impacts the coast; this is particularly true in the case of a narrow shelf.
The use of shore-based High Frequency (HF) radars to detect incoming tsunami waves was proposed almost 40 years ago by Barrick [7] and, more recently, was supported by numerical simulations (see, e.g., [8], [9], [10], [11]), and by HF radar measurements made during the Tohoku 2011 tsunami in Japan [12,13,14], in Chile [15], and in Hawaii [16]. No realtime tsunami detection algorithms were in place, but an a posteriori analysis of the radar data identified the tsunami current in the measurements. As for other nearshore currents, this works by measuring the Doppler shift tsunami currents induce on the radar signal and from this estimating time series of radial surface currents (i.e., projected on the radar line-of-sight) over a grid of radar cells covering the radar sweep area (typical cell size is one to a few km in each direction, with a range of 10s to 100s of km in the radial direction, depending on radar frequency and power). This dense spatial coverage is another advantage of HF radar detection over standard instrument methods.
Tsunami detection and warning algorithms were proposed in some of these earlier studies, based on both a sufficient magnitude of the tsunami current inferred from the radar Doppler spectrum, combined with identifying its oscillatory nature in space and time. In earlier work based on a 4.5 MHz HF radar (Stradivarius) with a 200 km range, Grilli et al. [17] showed that such algorithms reliably work when tsunami currents are at least U t ∼ 0.15 − 0.20 m/s and thus rise above background noise and currents. Hence, this limits a direct detection of tsunami currents to fairly shallow water and thus nearshore locations, and also means short warning times, unless there is a very wide shelf.
To detect a tsunami in deeper water, beyond the continental shelf, the authors have proposed a new detection algorithm that does not require "inverting" currents, but instead is based on spatial correlations of the raw radar signal at two distant locations along the same wave ray, shifted in time by the tsunami propagation time along the ray. A change in pattern of these correlations indicates the presence of a tsunami, since no other geophysical phenomenon can be responsible.
They validated this algorithm only for idealized tsunami wave trains, propagating over a simple seafloor geometry in a direction normally incident to shore [17].
Here, this algorithm is extended and validated for realistic tsunami case studies conducted for seismic sources and using the bathymetry off of the Pacific Ocean   [20], in this paper, we only detail results for a single M w 9.1 far-field seismic source in the Semidi Subduction zone (SSZ; Fig. 3.1a). Based on time series of tsunami radial currents simulated in each radar cell, we compute the radar signal in the cells using a backscattering model [17], which is applied here for the characteristics of the radar installed in TF (carrier electromagnetic wave (EMW) frequency f EM = 13.5 MHz).

Tsunami simulations 3.2.1 Numerical models, tsunami source and numerical grids
We extend a HF radar simulator and tsunami detection algorithm proposed earlier by Grillli et al. [17], and apply both to the sweep area of the WERA radar, off of Tofino, BC, based on simulated tsunami currents corresponding to the arrival of a tsunami generated by a M w 9.1 seismic source in the SSZ (Fig. 3.1).
This source was designed by the SAFFR (Science Application for Risk Reduction) group to have the same magnitude as the Tohoku 2011 event and cause maxi- mum impact in northern California [21]. Simulations of tsunami propagation are   performed with FUNWAVE-TVD, a Boussinesq long wave model with extended dispersive properties, which is fully nonlinear in Cartesian grids [18] and weakly nonlinear in spherical grids [19]. The model was efficiently parallelized for use on a shared memory cluster (over 90% scalability is typically achieved), which easily allows using large grids (such as here the G0 grid, which has over 3 million meshes, as detailed below  [27]. The initial surface elevation of the SAFFR seismic source was obtained from Kirby et al. that a nesting ratio with a factor 3-4 reduction in mesh size allowed achieving good accuracy in tsunami simulations, which is the case for grids used here (Table 3.1).
Bathymetric/topographic data for both the 2 arc-min resolution G0 grid and the 0.6 arc-min G1 grid was interpolated from NOAA's 1 arc-min ETOPO-1 data.
Bathymetry for the 270 m and 90 m resolution grids (G2 and G3) was based on the 3 arc-sec data provided for the coast of BC by NOAA's Marine Geology and Geophysics (MGG), wherever available; this higher-resolution data was also used in grids G0 and G1, instead of ETOPO-1 data, in the area overlapping with grid G2. Another MGG 3 arc-sec dataset (the Northwest Pacific data set) was used for areas facing the US coast not covered by the BC bathymetry. Since the MGG BC dataset only included bathymetry, topography for grids G2 and G3 was based on ETOPO-1 data, which clearly is too coarse to accurately simulate coastal tsunami impact in these finer grids; this however is acceptable, since the present work focuses on detecting the tsunami offshore, at a significant distance away from the shoreline.
Except for close to shore, tsunamis are long waves that are well approximated by linear wave theory [28]; hence, tsunami currents and elevations will be on the order of, respectively, where η t h(x, y) (the local depth) is surface elevation at locations (x, y), k t (x, y) =| k t |= 2π/L t is the tsunami wavenumber (with L t 20h the characteristic tsunami wave length), and k t = k t (cos φ t , sin φ t ) is the wavenumber vector, with φ t the angle of the tsunami local direction of propagation with respect to the x axis ( g = 9.81 m/s 2 , is gravitational acceleration). According to Eq. 3.1, assuming no refraction and linear long waves, the local tsunami elevation η t can be predicted based on the initial deep water tsunami elevation η t0 using Green's law, where c 0 = √ gh 0 is the tsunami phase speed in reference depth h 0 . It follows that , and the tsunami current gradually increases as water depth decreases.
As detailed in [17], the proposed HF radar detection algorithm is applied along individual wave rays, which reflect site-specific refraction. These can be computed as a function of bathymetry and the ray's assumed incident direction in deep water φ t0 , independently of tsunami sources, using the geometric optics eikonal equation, which using the long wave celerity (equal to the group velocity) reads, and is solved for φ t (x, y).
The pre-computed wave rays allow identifying radar cells located along specific rays (see details in next section); the tsunami propagation time between each pair of such cells (p, q) is then calculated as, where R(x, y) denotes the radial position of cells in the radar grid and s(R(x, y)) is the curvilinear abscissa along the selected wave ray, with ds = dx cos φ t +dy sin φ t . in grid G0 (a) and G3 (b) for the SAFFR seismic source in the SSZ (Fig. 3.1a).

Tsunami simulation results
As expected, Fig. 3.3a shows energy focusing on northern California and Oregon; however, one can also see significant tsunami elevations near Tofino. This is clearer in Fig. 3.3b, which shows that this tsunami would cause maximum eleva-   (Fig. 3.2), assuming an incident direction from west, i.e., towards east (φ t0 = 0); as expected, wave rays bend based on bathymetry, to become increasingly normal to bathymetric contours, in shallower water close to shore. Figure 3.4b shows time series of tsunami radial currents (i.e., projected in the radar direction U tr = U t · R/R) computed at locations of 9 radar cells numbered 1-9 along a specific wave ray, in increasingly deeper water ( Fig. 3.5b). As expected from Eq. 3.1, while radial velocity is over 0.4 m/s in the shallower water cell (1) in less than 50 m depth, it is less than 0.07 m/s at the deeper cell (9) close to the shelf break, in 500 m depth. The figure also shows that, independently of its magnitude, the pattern of time variation of the tsunami current repeats itself well from station 9 to 1. This is even more apparent in shall see, such time-shifted currents are highly correlated in time, which was one of the main conclusions in the earlier work by [17], based on both idealized tsunami wave train and bathymetry; it thus appears from present results that this key property of tsunami currents for the viability of the proposed detection algorithm is confirmed to apply for realistic tsunami case studies.

Simulations of tsunami detection by HF radar 3.3.1 HF radar detection of tsunami currents
To simulate tsunami detection by HF radar, based on radial currents U tr computed with FUNWAVE-TVD (e.g., Fig. 3.4), we use the HF radar simulator deployed near Tofino, BC, which has a carrier electromagnetic wave (EMW) frequency f EM = 13.5 MHz and a usable maximum range of 85 km. The radar sweep area is outlined in Fig. 3.1a and detailed in Fig. 3.5b; it is covered by radar cells within which the received radar signal is averaged, of length ∆R = 1.5 km in the radial direction and angular opening ∆φ r = 1 degree in the azimuthal direction; the detection sector of the sweep area is 120 degree, implying that cells are 1.48 km wide at a 85 km range and narrower closer to the radar (cell area: ∆S = R ∆R ∆φ r increases with range). The orientation of the radar antennas is such that one side of the sweep area boundary is nearly parallel to the coastline southeast of Tofino ( Fig. 3.5b).
Near-surface ocean currents are inferred from EMW interactions with ocean surface waves, based on the Bragg scattering property that the diffracted radar signal is maximum when it interacts with ocean waves whose wavelength is half the EWM wavelength, Tsunami radial currents ±U tr cause a Doppler effect on surface waves, which causes a shift of the Bragg frequency in the radar signal Doppler spectrum proportional to it, ∆f B = ±U tr /L B . The magnitude of radial currents,Ũ tr (R, t) can thus be inferred (inverted) from this shift, once the radar signal Doppler spectrum is computed; these are currents averaged (overbar) over a radar cell of area ∆S (for a monostatic configuration such as here), centered at R(x, y), and a measuring (or integration) time interval T i (tilde) (here 120 s). To accurately compute the spectrum, the radar cells' spatial dimensions must be sufficiently large to include a statistically meaningful sample of ocean surface waves of various wavelengths, and particularly of length L B . The frequency resolution of the Doppler spectrum near its peak is, ∆f D = 1/T i and that of the inverted current ∆U tr = L B /T i ; hence, to accurately infer surface currents based on a Doppler shift, the measuring time interval must be sufficiently long, typically at least 2 min for a 13.5 MHz (such as used here), yielding ∆U tr = 0.086 m/s. Because of the oscillatory nature of tsunami currents, however, T i cannot be increased too much to improve resolution, as this would gradually reduce the cell-and time-averaged currents, until they have a nearly zero average and become undetectable. As concluded by Grilli et al. [17], the limited resolution of inverted currents combined with their rapidly decreasing magnitude with radar range (and increasing depth; Eq. 3.1) implies that tsunami detection algorithms, such as proposed by Lipa et al. [14], based on "inverting" Doppler spectral shifts would only be reliable nearshore, over the continental shelf, where tsunami currents would be sufficiently larger than background currents (e.g., > 0.15 − 0.20 m/s). By contrast, the new algorithm proposed by Grilli et al. [17], which is tested here on a realistic case study, takes advantage of the high correlation of time-shifted tsunami currents along a wave ray (Fig. 3.4c), which is also observed for the corresponding time-shifted time series of radar signals, to detect tsunami arrival in deeper water by observing a change in the radar signal correlation pattern, and hence does not require tsunami currents to reach large values to be detectable. Applying this algorithm for idealized tsunami wave trains and bathymetry, but in the presence of noise and background current, Grilli et al. [17] showed that the arrival of tsunami currents as low as background values of 0.05-0.1 m/s could be inferred, and thus tsunami detection can take place in deeper water, beyond the continental shelf.

HF radar simulator
We simulate tsunami detection by HF radar using the backscattering model (HF radar simulator) of Grilli et al. [17], which accounts for the presence of a time varying surface current in a random sea state. A summary of the main first-order equations of the model is given below; details and second-order equations can be found in reference.
The total surface current over the radar sweep area is assumed to be the sum of: (i) a spatially variable, but nearly stationary at the time scale of radar data acquisition (> O(T i )) residual (mesoscale) current, U 0 (R); and (ii) a spatially and temporally varying current, U t (R, t) induced by the tsunami wave train (e.g., Eq. 3.1), computed here with FUNWAVE-TVD); hence, U (R, t) = U 0 (R) + U t (R, t). The residual current, although stationary, is spatially variable in a way that depends on local and synoptic environmental oceanic conditions; in a specific case such as off of Vancouver Island, this current could be obtained from a regional ocean model, but this will not even be necessary to apply the proposed tsunami detection algorithm. Because the radar signal is simulated over cells of varying sizes ( Fig. 3.5b), the tsunami-induced current computed over the finest FUNWAVE grid G3 is spatially averaged over each radar cell (e.g., Fig. 3.2c) before being used in the radar simulator.
Assuming a small steepness, the surface elevation of random ocean waves is represented by a second-order perturbation expansion, η(R, t) = η 1 (R, t) + η 2 (R, t), with, η 1 (R, t) = =±1 a (K) e i(K.R− Ω(K,R,t)t) dK, (3.5) where the integration is carried out over the wavenumber vectors, K = (K x , K y ) = K(cos θ, sin θ), and wave harmonic amplitudes are given by, with Ψ the directional wave energy density spectrum and Z (K) a complex normal variable (with unit variance and zero mean), independent for each wave harmonic.
The angular frequency of each wave component, Ω(K, R, t), is modulated by the surface current U (R, t). Assuming that the tsunami current is slowly varying in time at the scale of ocean waves, i.e., the tsunami characteristic period, T t T p , the peak spectral wave period, and that waves are in the deep water regime, we have, Ω(K, R, t) t = (Ω g + K.U 0 (R)) t + t 0 K.U t (R, τ )dτ, (3.7) where the integral is a memory term representing the cumulative effects of the tsunami current on the instantaneous wave angular frequency, and, Ω g = √ gK the linear angular wave frequency in deep water. Details of η 2 (R, t) can be found in reference.
Here, we simulate fully developed sea states represented by a Pierson-Moskowitz (PM) directional wave energy density spectrum Ψ(K x , K y ), parametrized as a function of wind velocity at 10 m elevation, U 10 , and with a standard asymmetric angular spreading function, allowing us to model a fraction ξ of wave energy associated with waves propagating in the direction opposite to the dominant wind direction (see details in [17]). For instance, for U 10 = 10 m/s, s = 5, and ξ = 0.1, we find a sea state with significant wave height, H s = 1.71 m, peak spectral wavelength L p = 127.4 m and, assuming deep water, peak period T p = 9.04 s.
In a monostatic radar configuration, radar cells are identified by their range vector R center on the radar (or range R and radar steering angle φ r ). The Bragg vector, K B is defined to point in the radar direction of observation, with K B = (2π/L B ). Up to second-order, the unattenuated backscattered radar signal is denoted by, S(t) = S 1 (t) + S 2 (t), with, where Z again denotes a complex normal variable (with unit variance and zero mean), the factor √ 2 K 2 B ensures consistency with the Doppler spectrum definition, and Ω B is obtained from the wave dispersion relationship in the presence of a current (Eq. 3.7). The expression for the second-order signal S 2 (t) can be found 148 in reference. Accounting for effects of attenuation with range and environmental noise, the radar signal received from each cell is finally modeled as, a geometric attenuation factor function of range R and N the environmental noise, detailed below. F represents the EMW attenuation by the ocean surface, which is computed here with the GRwave model, for the WERA radar frequency [29].
Environmental noise is modeled in each cell as independent complex Gaussian distributions with constant standard deviation σ N , where t indicates that different Gaussian random values with unit standard deviation and zero mean, [G R t (0, 1), G I t (0, 1)], are being generated for each time level t.
Since noise is not affected by range, Eq. 3.9 implies that the radar signal-to-noise ratio (SNR) gradually decreases with range, until the signal becomes undetectable from noise, which sets the effective radar measuring range (here 85 km). Here, we use the same σ N value as in [17], which was based on HF radar experiments done in the Mediterranean sea, for normal temperature and pressure conditions; in future work, σ N will be adjusted using site-specific values of the SNR for the WERA radar deployed offshore of Tofino, once these have been measured.
For an integration time T i , the (non-normalized) radar Doppler spectrum is calculated at time t s (with ∆t s ≤ T i ) as the mean square of the modulus of the Fourier transform of the received radar signal V(t), centered on its mean, over a finite time window [t s − T i /2, t s + T i /2], that is, with f D denoting a set of discrete Doppler frequencies (with ω D = 2πf D ). If the received radar signal is simulated/(recorded) at a constant temporal sampling rate ∆t = T i /N , Eq. 3.11 can be easily computed as a summation from −N/2 to N/2.
Given a directional wave energy density spectrum Ψ(K) and sets of random functions Z (K) (representing random wave phases) and G t (used to simulate noise in each cell), time series of the received signal and corresponding Doppler spectra can be simulated in each radar cell, by applying Eqs. 3.8 to 3.11, in the presence of cell-averaged radial surface currents U r (R, t). More details can be found in [17].

Application of algorithms for tsunami detection to SSZ tsunami
We apply the radar simulator Eqs. 3.5-3.10 to the WERA radar sweep area off of Tofino (Fig. 3.5b), assuming a local wind speed U 10 = 10 m/s, no background current to start with, and using time series of cell-averaged radial tsunami currents U tr (R, t) calculated for the SSZ tsunami from FUNWAVE-TVD's computations (e.g., Fig. 3.4b). We simulate time series of received radar signal V p (t), in each radar cell p, and based on these time-dependent Doppler spectra I(f D , t s ) (Eq. 3.11). This corresponds to cell 5 in Fig. 3.5b, for which Fig. 3.4b shows maximum cellaveraged currents are just above 0.2 m/s; however, the integration time T i = 120 sec used to compute Doppler spectra means the radar signal is based on currents that are also averaged over this time,Ũ tr (R, t), which reduces current magnitude (here, just below 0.2 m/s). Assuming the tsunami is detected immediately upon reaching Station 5, from this location of detection, the tsunami would reach the shore within 20-25 min, which only offers very low warning time. This confirms the conclusion of the idealized study of [17], that TDA1 is only reliable for |Ũ tr |> 0.15−0.20 m/s.
We now evaluate the performance of the Tsunami Detection Algorithm 2 (TDA2) proposed by Grilli et al. [17] on the basis of time-shifted correlations of radar signal. As already pointed out, once shifted by the long wave propagation time ∆t pq between two cells p and q located along the same wave ray, Fig. 3.4c showed that tsunami radial currents appear to be highly correlated. This is verified due to the presence of a highly correlated tsunami current. By contrast these correlations should be flat as a function of time lag in the absence of a tsunami current (and this will hold true in the presence of uncorrelated background currents). This pattern is indeed observed in Figs. 3.7b and 3.7c, for correlations at stations 1 to 9, with and without current, respectively. [Note that, to reduce high frequency noise in correlations, these are computed on the analytical radar signals, which 152 are calculated, for simulated or measured signals, by applying a Fourier transform (FT) to the signal, removing the negative frequencies, and applying an inverse FT; see details in [17].] Correlations averaged over cells 1 to 9 are also plotted in Fig.   3.7 as thick red lines; in Figs. 3.7a and b, these are clearly peaked near the zero time lag in the presence of a tsunami current, but flat in Fig. 3.7c in the absence of a current.
Due to lack of space, we do not show results in the presence of a random background current but these will be shown at the conference, and confirm that currents from both a spatially varying (but nearly stationary at the considered scales) mesoscale current and local effects of environmental conditions (e.g., wind), have no correlation between two cells selected on a wave ray, more particularly It should be finally emphasized that, unlike with TDA1, with TDA2 we do not need to estimate currents by inverting the Doppler spectra and hence that limitations in such inversion in terms of integration time T i T t , resolution ∆U r ∝ 1/T i , and the need for currents to be > 0.15 − 0.20 m/s do not apply. We see for instance that correlations between cell 4.5 and 1 to 9 in Fig. 3.7 all have a similar pattern, whereas the current magnitude is less than 0.05 m/s at station 9, in a 500 m Time correlations, as a function of additional time lag ∆t, between radar cell q = 4.5 (between 4 and 5; 39 km range) and p = 1, ...9 (3 to 72 km range) aligned along the same wave ray (Fig. 3.4b), time-shifted by the long wave propagation time t p4.5 from cell p to 4.5 of: (a) spatially-averaged radial tsunami currents U r (Fig 3.4c); and analytical radar signals simulated with (b) and without (c) tsunami current. Red lines are cell/range averages of each correlation.
Correlations use T C = 600s, centered on tsunami front arrival time at cell 4.5, t = 6, 240s.
depth and reaches nearly 0.3 m/s in station 2, in a 50 m depth. This confirms the conclusions of Grilli et al. [17], but here this is based on a realistic case study.

Conclusion
The detection of tsunamis by HF radars, based on a direct inversion of tsunami currents from the radar signal Doppler spectra (referred to as algorithm TDA1), TDA2 can be easily implemented in a radar system, in a real time tsunami detection mode (rather than simulation mode) for which the radar signal is continuously measured (rather than computed with a radar simulator), processed in all the radar cells (Fig. 3.5b), and time-shifted correlations are dynamically calculated between all pair of cells located along a large number of pre-computed wave rays (Fig. 3.5b). To detect tsunamis from expected (e.g., seismic) or unknown (SMF) tsunami sources, a series of wave rays can be pre-computed for tsunamis incident from a range of potential directions, based on bathymetry, and used in the algorithm. Applying TDA2, the appearance of a peaked correlation between time series of time-shifted radar signals, in pairs of cells located along the same wave ray (for a single pair of cell or averaged over a few cells, from offshore to onshore), will indicate that a tsunami is approaching the radar. In the range of