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Ocean Modelling 114 (2017) 14–32
Contents lists available at ScienceDirect
Ocean Modelling
journal homepage: www.elsevier.com/locate/ocemod
Inter-model analysis of tsunami-induced coastal currents
Patrick J. Lynett a , Kara Gately
b , Rick Wilson
c , Luis Montoya
a , ∗, Diego Arcas d , e , Betul Aytore
f , Yefei Bai g , Jeremy D. Bricker h , 1 , Manuel J. Castro
i , Kwok Fai Cheung
g , C. Gabriel David
j , 2 , Gozde Guney Dogan
f , Cipriano Escalante
i , José Manuel González-Vida
k , Stephan T. Grilli l , Troy W. Heitmann
g , Juan Horrillo
m , Utku Kâno ̆glu
n , Rozita Kian
f , James T. Kirby
o , Wenwen Li p , Jorge Macías i , Dmitry J. Nicolsky
q , Sergio Ortega
r , Alyssa Pampell-Manis m , Yong Sung Park
s , Volker Roeber h , Naeimeh Sharghivand
n , Michael Shelby
l , 3 , Fengyan Shi o , Babak Tehranirad
o , 4 , Elena Tolkova
t , Hong Kie Thio
p , Deniz Velio ̆glu
f , Ahmet Cevdet Yalçıner f , Yoshiki Yamazaki i , Andrey Zaytsev
u , w , Y.J. Zhang
v
a Tsunami Research Center, Sonny Astani Department of Civil & Environmental Engineering, University of Southern California, Los Angeles, CA 90089, USA b NOAA, National Tsunami Warning Center, Palmer, AK 99645, USA c California Geological Survey, Seismic Hazards Mapping Program – Tsunami Projects, Sacramento, CA 95814, USA d NOAA Center for Tsunami Research, 7600 Sand Point Way NE, Seattle, WA 98115, USA e University of Washington, JISAO, 3737 Brooklyn Ave. NE, Seattle, WA 98105, USA f Department of Civil Engineering, Ocean Engineering Research Center, Middle East Technical University, Dumlupinar Boulevard, No:1 Cankaya, Ankara
06800, Turkey g Department of Ocean and Resources Engineering, School of Ocean and Earth Science and Technology, University of Hawaii, Holmes Hall 402, 2540 Dole
Street, Honolulu, HI 96822, USA h International Research Institute of Disaster Science (IRIDeS), Tohoku University, 468-1 E304 AzaAoba, Aramaki, Aoba-ku, Sendai 980-0845, Japan i Facultad de Ciencias, Departamento de Análisis Matemático, University of Málaga, Campus de Teatinos, s/n, Málaga 29080, Spain j Franzius-Institute for Hydraulic, Estuarine and Coastal Engineering, Leibniz University of Hanover, Nienburger Straße 4, Hanover 30167, Germany k E.T.S. Telecomunicación, Departamento de Matemática Aplicada, University of Málaga, Campus de Teatinos, s/n, Málaga 29080, Spain l Department of Ocean Engineering, University of Rhode Island, Narragansett, RI 02882, USA m Tsunami Research Group, Department of Ocean Engineering, Texas A&M University at Galveston, 200 Seawolf Parkway, Galveston, TX 77553, USA n Department of Engineering Sciences, Middle East Technical University, Dumlupinar Boulevard No:1, Cankaya, Ankara 06800, Turkey o Center for Applied Coastal Research, Department of Civil and Environmental Engineering, University of Delaware, Newark, DE 19716, USA p AECOM, 300 S. Grand Ave, Los Angeles, CA 90017, USA q Geophysical Institute, University of Alaska Fairbanks, 903 Koyokuk Drive, Fairbanks, AK 99775-7320, USA r Laboratorio de Métodos Numéricos, SCAI, University of Málaga, Campus de Teatinos, s/n, Málaga 29080, Spain s Division of Civil Engineering, School of Science and Engineering, University of Dundee, Perth Road, Dundee DD1 4HN, United Kingdom
t NorthWest Research Associates, 4118 148th Ave NE Redmond, WA 98052-5164, USA u Special Research Bureau for Automation of Marine Researches, Far Eastern Branch of Russian Academy of Sciences, Gorkiy str. 25, Uzhno-Sakhalinsk
693013, Russia v Virginia Institute of Marine Science, Center for Coastal Resource Management, College of William & Mary, 1375 Greate Road, Gloucester Point, VA
23062-1346, USA w Nizhny Novgorod State Technical University, Nizhny Novgorod 603155, Russia
a r t i c l e i n f o
Article history:
Received 21 September 2016
Revised 22 February 2017
Accepted 5 April 2017
Available online 21 April 2017
a b s t r a c t
To help produce accurate and consistent maritime hazard products, the National Tsunami Hazard Mit-
igation Program organized a benchmarking workshop to evaluate the numerical modeling of tsunami
currents. Thirteen teams of international researchers, using a set of tsunami models currently utilized
for hazard mitigation studies, presented results for a series of benchmarking problems; these results are
summarized in this paper. Comparisons focus on physical situations where the currents are shear and
separation driven, and are thus de-coupled from the incident tsunami waveform. In general, we find that
∗ Corresponding author.
E-mail address: [email protected] (L. Montoya). 1 Current address: Department of Hydraulic Engineering, Delft University of Tech-
nology, Netherlands. 2 Current address: Ludwig-Franzius-Institute for Hydraulic, Estuarine and Coastal
Engineering, Leibniz Universität Hannover, Hannover 30167, Germany. 3 Current address: Naval Underwater Warfare Center, Newport, RI 02841, USA. 4 Current address: Moffatt & Nichol, 2185 N California Blvd #500, Walnut Creek,
CA 94596, USA. http://dx.doi.org/10.1016/j.ocemod.2017.04.003
Fig. 1. Summary of numerical and experimental data from BM#1: (a) is a dye visualization from the experiment (modified from L&S); (b) shows the PIV-extracted surface
velocity field from the experiment (modified from L&S); (c) shows a numerical simulation including scalar dye transport to visualize the vortex street; and (d) shows the
experimental data at the two time series locations. In plots (a), (b) and (c), the two time series locations are shown by the numbers.
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3. Simulation result with all dissipation sub-models not included
(e.g. a physically inviscid simulation). The purpose of this test is
to understand the relative importance of numerical vs. physical
(as approximated by the governing equations used in each nu-
merical model) effects on vorticity generation and dissipation
for this class of comparison.
2.2. Benchmark problem #2 (BM#2): tsunami currents in hilo harbor
This benchmark is based on a field dataset from the velocity
data recorded in Hilo Harbor, Hawaii, resulting from the 2011 To-
hoku tsunami. The aim of this benchmark is to understand the
importance of model resolution and numerics on the prediction
of tsunami currents. While modelers will aim to achieve the best
agreement with the measured data, this is not the primary goal of
this exercise. Some of the questions that this benchmark attempts
to address include:
1. What level of accuracy and precision can we expect from
a model with regard to modeling currents on complex
bathymetry?
2. Will a model converge with respect to speed predictions and
model resolution?
3. What is the variation across hydrodynamic different models
(e.g. hydrostatic vs. non-hydrostatic), using the same wave forc-
ing, resolution, and bottom friction (or approximate equivalent
when using different bottom stress models)?
To attempt to most clearly answer these questions, this field
ase will be somewhat idealized, or reduced in complexity, to
ive the modeling results the best chance of an “apples-to-apples”
omparison. For this benchmark, free surface elevation (from tide
tations) and velocity information (from Acoustic Doppler Current
rofilers (ADCPs)) are compared. Data for this benchmark test is
iscussed in detail in Arcos and LeVeque (2015) and Cheung et al.
2013) .
Fig. 2 shows a plot of the bathymetry from Hilo Harbor, Hawaii.
he data is provided in [lat, long] on a 1/3 arcsec grid, taken from
he NOAA National Centers for Environmental Information (NCEI)
sunami DEM database. Note that shown on this figure are also
he simulation “control point” (white dot; upper-most dot in the
gure), the two ADCP locations (black dots) and the tidal station
yellow dot; lower-most dot in the figure). As mentioned above,
his problem has been “reduced” in an attempt to isolate differ-
nces in the employed incident wave forcing. For the bathymetry
ata, this “reduction” manifests as a flattening of the bathymetry
t a depth of 30 m; in the offshore portion of the bathymetry grid,
Fig. 3. Data-scaled modeled mean fluctuation (blue dots) and standard deviation of fluctuation (vertical black lines) for each model for U at location #1 (a), V at location#1
(b), U at location #2 (c), V at location#2 (d). The red-dashed horizontal lines show the standard deviation in the data. (For interpretation of the references to color in this
figure legend, the reader is referred to the web version of this article.)
Fig. 4. Error in component-averaged modeled velocity, as a fraction of the experimental value, for the mean fluctuation (top) and period of oscillation (bottom). The data is
Fig. 5. Data-scaled modeled time-averaged speed squared (blue dots) and the square of the modeled mean speed (smaller green dots) for each model for U at location #1
(a), V at location#1 (b), U at location #2 (c), V at location#2 (d). The red horizontal line in each plot is data line (where, ideally, the blue dots would align), and the green
horizontal dashed line is the square of the experimental mean speed (where, ideally, the green dots would align). (For interpretation of the references to color in this figure
legend, the reader is referred to the web version of this article.)
Fig. 7. Inter-model variability and error for ocean surface elevation measured at the tide gage location. Top plot (a): predictions from all models (thin solid lines), inter-model
mean crest envelope (thick solid line), and inter-model mean trough envelope (thick dashed line). Middle plot (b): comparison of inter-model envelope to measured tide
station data envelope; also shown in the time series from the measured data. Bottom plot (c): mean inter-model error (solid line) and intermodal standard deviation (dashed
line) for the crest envelope.
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Fig. 8. Inter-model variability and error for ocean current speed measured at the ADPC location HA25 (left column) and HA26 (right column). Top row plots (a) and (b): predictions from all models (thin solid lines) and inter-
model mean speed envelope (thick solid line). Middle row plots (c) and (d): Comparison of inter-model envelope to measured ADCP data envelope; also shown in the time series from the measured data. Bottom row plots (e)
and (f): mean inter-model error (blue line) and intermodal standard deviation (green line) for the speed envelope. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of
Fig. 9. Summary of inter-model spatial statistics. Top left (a): inter-model mean of predicted maximum speed in m/s as taken from the 5-m resolution runs. Bottom left
(b): inter-model standard deviation of predicted maximum speed in m/s as taken from the 5-m resolution runs. Right column: inter-model standard deviation of predicted
maximum speed scaled by model-mean maximum speed as taken from the 5-m resolution runs (c), 10-m resolution runs (d), and 20-m resolution runs (e).
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area are near 2 m/s, and represent 50–150% of the model-mean
speed. This inter-model comparison indicates that it is reasonable
to expect large differences between models in areas affected by ed-
dies, and thus velocity predictions from any single model in such
locations must be carefully and conservatively interpreted.
There are two additional important observations from the speed
deviation in Fig. 9 (b). First, in areas not affected by eddies, the
models show excellent convergence with inter-model standard de-
viations less than 0.2 m/s, often representing 10–20% of the mean
speed. Thus, where currents are coupled with the wave (and not
de-coupled in the form of eddies), models converge precisely. Sec-
ond, deviations along the immediate shoreline and in areas of in-
undation are large. While not the focus of this study, this obser-
vation points out that speed predictions for overland flow, among
he models tested here, are highly divergent. As model predictions
or overland flow speed are being increasingly used for structural
oading calculations, additional study is needed to quantify errors
nd variability in inundation velocities.
The properties of a modeled eddy (e.g. tangential speed, ra-
ial speed gradient) are dependent on the magnitude of the shear
n the eddy generation area, or separation area. Numerically, this
hear magnitude may be limited by the numerical resolution, as
elatively coarse resolutions are unable to resolve strong velocity
radients. Thus, there may be a strong connection between the
umerical signature of an eddy and the numerical resolution. Fig.
(c)–(e) show the relative inter-model deviations, scaled by the
nter-model mean speed, for three different resolutions. Clearly, the
ocal speed deviations grow with decreasing grid size. The rea-
Fig. 10. Snapshots of vertical vorticity at the same simulation time for the same numerical model, with 10-m resolution (a) and 5-m resolution (b). Note that the locations
of the ADCP’s are shown by the black dots.
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on for this appears to be that, as grid size decreases, models
ave a tendency to generate eddies with a larger radial velocity
radient. A larger radial velocity gradient equates to rapid spatial
hanges in fluid speed, and increased sensitivity to the precise path
f the eddy. A troublesome conclusion for modeling follows: in
ddy areas, many models will increasingly diverge with decreas-
ng grid length, at least through the horizontal resolutions tested
ere (down to 5 m). A clear example of how this divergence may
anifest is shown in Fig. 10 . This figure provides vertical vorticity
napshots at the same time from the same numerical model, but
sing two different resolutions. The eddies in the finer resolution
napshot appear smaller with stronger vorticity, in line with the
iscussion above. Also, the large eddy located in the harbor en-
rance takes two different paths in the two simulations. In the 10-
resolution simulation, the eddy passes through the HA25 ADCP
ocation, but in the 5-m simulation the eddy stays to the south
f the ADCP. This is an example of the trouble with in-situ point
easurements of tsunami currents in areas with eddies, and is an-
ther argument for the use of ensemble means for speed predic-
ions, based on many realizations of potential eddies.
An alternative to multi-model or multi-realization ensembles is
o spatially average the output from a single model, i.e. attempt to
verage out the local impact of an eddy. Such an approach would
ave the disadvantage of smoothing peak speeds associated with
n eddy, but might provide numerically convergent results with a
ingle model. Fig. 11 displays an attempt at spatial averaging. Here,
e spatially average across two lengthscales: (1) an “eddy scale”
hich here is estimated to be 100 m based on simulation output
nd (2) ten times the “eddy scale” or 1 km. These averaging areas,
elimited by boxes with sides equals to the selected lengthscales,
re shown in the top plot of Fig. 11 . The box-average maximum
peed for each of the individual models are plotted for both box
izes. For each model, the box-average for the 20-m and 10-m res-
lutions are shown, scaled by the box-average from the 5-m res-
lution result. Thus, if the 10-m (or 20-m) resolution box-average
as numerically converged, it would plot at 1.0 along the vertical
cale. Clearly, using the “eddy scale” as the averaging lengthscale
oes not lead to a set of models that demonstrates convergence at
0-m resolution. The reason for this is that small-scale averages are
ensitive to the precise path of the eddy, which varies among dif-
erent models and among different resolutions for the same model.
hould for a particular model and a particular box location the
ddy exist in the box at one resolution and not another, an espe-
ially poor local convergence will result. On the contrary, averaging
n the large lengthscale, here 1 km, is not sensitive to variations in
ddy’s trajectories, since the larger box encompasses these most
r all trajectories. As a result, we see broad convergence across the
ested models at 10-m resolution. However, averaging model speed
redictions over a 1 km
2 area, while suitable for demonstrating a
orm of individual model convergence, yields output of very lim-
ted use for hazard assessment, where local maximum might gov-
rn hazardous conditions and damage potential.
Following the analysis above, an ensemble mean of maximum
peed is likely to be a robust and informative modeling prod-
ct, with significant and decision-impacting benefits over single
odel, deterministic predictions. The ensemble produced in this
aper used many different models; however it is certainly possi-
le to generate a spectrum of realizations with a single numer-
cal model using a distribution of perturbations to initial condi-
ions, bathymetry, bottom roughness, etc. Some research would be
eeded to specify an appropriate set of perturbations. If an ensem-
le was available, it is not obvious what the most useful way to
resent the statistical information would be. Certainly, means and
eviations could be provided, but this might require expert-level
udgement to use in decision making. An alternative would be a
threshold map”, an example of which is shown in Fig. 12 . This
ap provides the “chance” that any location might experience a
aximum current greater than a set threshold (the threshold is
m/s in Fig. 12 ). The “chance” is based on how many of the mod-
ls in the ensemble predict a speed greater than the threshold. The
dvantage of such a map is that it provides both a speed magni-
ude and confidence together, allowing for discussions of hazard
Fig. 11. The effect of spatial-averaging on model convergence, with the top plot (a) indicating the locations of the “small box” and the “large box” as referenced in the two
lower subplots. The two lower subplots show the box-averaged maximum speeds for each model at 20- and 10-m resolutions in (b) and (c), scaled by the box-averaged
speed from each model’s 5-m resolution simulation; middle subplot for the small box, and lower subplot for the large box. Note that two model results are missing; one
due to the use of different boundary conditions with different resolutions, and the other due to an inability to perform a 5-m resolution simulation.
tsunami hydrodynamic loads of the ASCE 7 standard. J. Struct. Eng. doi: 10.1061/(ASCE)ST.1943-541X.0 0 01499#sthash.YfxKoaeE.dpuf .
ilmen, D.I., Kemec, S., Yalciner, A.C., Düzgün, S., Zaytsev, A., 2015. Development ofa tsunami inundation map in detecting tsunami risk in Gulf of Fethiye, Turkey.
Pure Appl. Geophys. 172, 921–929. doi: 10.10 07/s0 0 024- 014- 0936- 2 . urran, D.R. , 1999. Numerical Methods for Wave Equations in Geophysical Fluid Dy-
namics. Springer-Verlag, New York, Berlin .
lder, J.W. , 1959. The dispersion of marked fluid in turbulent shear flow. J. FluidMech. 5, 544560 .
eorge, D.L. , LeVeque, R.J. , 2006. Finite volume methods and adaptive refinementfor global tsunami propagation and local inundation. Sci. Tsunami Hazards 24,
George, D.L. , 2008. Augmented Riemann solvers for the shallow water equationsover variable topography with steady states and inundation. J. Comput. Phys.
227, 3089–3113 . Gica, E. , Spillane, M. , Titov, V.V. , Chamberlin, C. , Newman, J.C. , 2008. Development
of the forecast propagation database for NOAA’s short-term inundation forecastfor tsunamis (SIFT). In: Proceedings of the NOAA Technical Memorandum (OAR
tsunami hazard along the upper U. S. East Coast: detailed impact around OceanCity, MD. Nat. Hazards 76, 705–746. doi: 10.1007/s11069- 014- 1522- 8 .
Hirt, C.W. , Nichols, B.D. , 1981. Volume Of Fluid (VOF) Method for the Dynamics ofFree Boundaries. J. Comput. Phys. 39, 201–225 .
Horrillo, J., Grilli, S.T., Nicolsky, D., et al., 2015. Performance benchmarking tsunamimodels for NTHMP’s inundation mapping activities. Pure Appl. Geophys. 172,
869. doi: 10.10 07/s0 0 024- 014- 0891- y .
Horrillo, J., Wood, A., Kim, G.-B., Parambath, A., 2013. A simplified 3-D/Navier–Stokes numerical model for landslide tsunami: application to the Gulf of Mex-
ico. J. Geophys. Res. Oceans 118, 6934–6950. doi: 10.10 02/2012JC0 08689 . Imamura, F. , 1989. Tsunami Numerical Simulation With the Staggered Leap-Frog
Scheme (Numerical Code of TUNAMI-N1). School of Civil Engineering, Asian In-stitute of Technology and Disaster Control Research Center, Tohoku University .
Keen, A.S. , Lynett, P.J. , Eskijian, M.L. , Ayca, A. , Wilson, R. , 2017. Monte Carlo–based
approach to estimating fragility curves of floating docks for small craft marinas.J. Waterw. Port Coast. Ocean Eng. 143 (4), 04017004 .
Kim, D.-H., Lynett, P., 2011. Turbulent mixing and scalar transport in shallow andwavy flows. Phys. Fluids 23 (1). doi: 10.1063/1.3531716 .
Kim, D.-H. , Lynett, P. , Socolofsky, S. , 2009. A Depth-integrated model for weakly dis-persive, turbulent, and rotational fluid flows. Ocean Model. 27 (3–4), 198–214 .
University Press . Lloyd, P.M. , Stansby, P.K. , 1997a. Shallow water flow around model conical island of
small slope. I: Submerged. J. Hydraul. Eng. ASCE 123 (12), 1068–1077 . Lloyd, P.M. , Stansby, P.K. , 1997b. Shallow water flow around model conical island of
small slope. II: Submerged. J. Hydraul. Eng. ASCE 123 (12), 1057–1067 . Lynett, P.J. , Borrero, J. , Weiss, R. , Son, S. , Greer, D. , Renteria, W. , 2012. Observations
and modeling of tsunami-induced currents in ports and harbors. Earth Planet.
Sci. Lett. 327–328, 68–74 . Lynett, P.J., Borrero, J., Son, S., Wilson, R., Miller, K., 2014. Assessment of the
tsunami-induced current hazard. Geophys. Res. Lett. 41, 2048–2055. doi: 10.1002/2013GL058680 .
slide and associated tsunami: a modelling approach. Mar. Geol. 361, 79–95.
doi: 10.1016/j.margeo.2014.12.006 . Nicolsky, D.J., Suleimani, E., Hansen, R., 2011. Validation and verification of a nu-
merical model for tsunami propagation and runup. Pure Appl. Geophys. 168,1199–1222. doi: 10.10 07/s0 0 024- 010- 0231- 9 .
Okal, E.A. , Fritz, H.M. , Raad, P.E. , Synolakis, C. , Al-Shijbi, Y. , Al-Saifi, M. , 2006. Omanfield survey after the December 2004 Indian Ocean tsunami. Earthq. Spectra 22
(S3), 203–218 . Ozer Sozdinler, C., Yalciner, A.C., Zaytsev, A., et al., 2015. Investigation of hydrody-
namic parameters and the effects of breakwaters during the 2011 Great East
Japan Tsunami in Kamaishi Bay. Pure Appl. Geophy. 172, 3473. doi: 10.1007/s0 0 024- 015- 1051- 8 .
Park, H., Cox, D., Lynett, P., Wiebe, D., Shin, S., 2013. Tsunami inundation model-ing in constructed environments: a physical and numerical comparison of free-
Suppasri, A., Muhari, A., Futami, T., Imamura, F., Shuto, N., 2013. Loss functionsfor small marine vessels based on survey data and numerical simulation of
the 2011 Great East Japan Tsunami. J. Waterw. Port Coast. Ocean Eng. 140 (5),04014018. doi: 10.1061/(ASCE)WW.1943-5460.0 0 0 0244 .
Synolakis, C.E. , 1987. The runup of solitary waves. J. Fluid Mech. 185, 523–545 . Synolakis, C.E. , Bernard, E.N. , Titov, V.V. , Kâno ̆glu, U. , González, F.I. , 2008. Validation
and verification of tsunami numerical models. Pure Appl. Geophys. 165 (11–12),
2197–2228 . itov, V., Kanoglu, U., Synolakis, C., 2016. Development of MOST for real-time
tsunami forecasting. J. Waterw. Port Coast. Ocean Eng. doi: 10.1061/(ASCE)WW.1943-5460.0 0 0 0357 .
okimatsu, K., Ishida, M., Inoue, S., 2016. Tsunami-Induced overturning of buildingsin Onagawa during the 2011 Tohoku Earthquake. Earthq. Spectra 32 (4), 1989–
2007. doi: 10.1193/101815EQS153M .
Tolkova, E., 2014. Land–water boundary treatment for a tsunami model withdimensional splitting. Pure Appl. Geophys. 171 (9), 2289–2314. doi: 10.1007/
ei, G. , Kirby, J.T. , Grilli, S.T. , Subramanya, R. , 1995. A fully nonlinear Boussinesqmodel for surface waves: Part I. Highly nonlinear unsteady waves. J. Fluid Mech.
294, 71–92 .
ei, Y., Bernard, E., Tang, L., Weiss, R., Titov, V., Moore, C., Spillane, M., Hop-kins, M., Kâno ̆glu, U., 2008. Real-time experimental forecast of the Peruvian
tsunami of August 2007 for U.S. coastlines. Geophys. Res. Lett. 35, L04609.doi: 10.1029/2007GL032250 .
amazaki, Y. , Cheung, K.F. , Kowalik, Z. , 2011. Depth-integrated, non-hydrostaticmodel with grid nesting for tsunami generation, propagation, and run-up. Int. J.
Numer. Meth. Fluids 67 (12), 2081–2107 .
ranslated from Russian by Yanenko, N.N. , 1971. In: Holt, M. (Ed.), The Method ofFractional Steps Springer, New York, Berlin, Heidelberg. Translated from Russian
by . eh, H.H.J. , Robertson, I. , Preuss, J. , 2005. Development of Design Guidelines For
Structures That Serve As Tsunami Vertical Evacuation Sites, 4. Washington StateDepartment of Natural Resources, Division of Geology and Earth Resources .
hang, Y. , Baptista, A.M. , 2008. An efficient and robust tsunami model on un-structured grids. Part I: inundation benchmarks. Pure Appl. Geophy. 165 (11),