Performance Benchmarking of Tsunami-HySEA model for NTHMP´s Inundation Mapping Activities

Abstract

The Tsunami-HySEA model is used to perform some of the numerical benchmark problems proposed and documented in the “Proceedings and results of the 2011 NTHMP Model Benchmarking Workshop”. The final aim is to obtain the approval for Tsunami-HySEA to be used in NTHMP projects. Therefore, this work contains the numerical results and comparisons for the five benchmarks problems (1, 4, 6, 7, 9) required for such aim. This set of benchmarks considers analytical, laboratory and field data test cases. In particular, the analytical solution of a solitary wave runup on a simple beach, and its laboratory counterpart, two more laboratory test: the runup of a solitary wave on a conically-shape island and the runup onto a complex 3D beach (Monai

Full text

Performance Benchmarking of Tsunami-HySEA Model for NTHMP’s Inundation Mapping
Activities
JORGE MACI
´AS,
1
MANUEL J. CASTRO,
1
SERGIO ORTEGA,
2
CIPRIANO ESCALANTE,
1
and JOSE
´MANUEL GONZA
´LEZ-VIDA
3
Abstract—The Tsunami-HySEA model is used to perform
some of the numerical benchmark problems proposed and docu-
mented in the ‘‘Proceedings and results of the 2011 NTHMP Model
Benchmarking Workshop’’. The final aim is to obtain the approval
for Tsunami-HySEA to be used in projects funded by the National
Tsunami Hazard Mitigation Program (NTHMP). Therefore, this
work contains the numerical results and comparisons for the five
benchmark problems (1, 4, 6, 7, and 9) required for such aim. This
set of benchmarks considers analytical, laboratory, and field data
test cases. In particular, the analytical solution of a solitary wave
runup on a simple beach, and its laboratory counterpart, two more
laboratory tests: the runup of a solitary wave on a conically shaped
island and the runup onto a complex 3D beach (Monai Valley) and,
finally, a field data benchmark based on data from the 1993 Hok-
kaido Nansei-Oki tsunami.
Key words: Numerical modeling, model benchmarking,
tsunami, HySEA model, inundation.
1. Introduction
According to the 2006 Tsunami Warning and
Education Act, all inundation models used in
National Tsunami Hazard Mitigation Program
(NTHMP) projects must meet benchmarking stan-
dards and be approved by the NTHMP Mapping and
Modeling Subcommittee (MMS). To this end, a
workshop was held in 2011 by the MMS, and
participating models whose results were approved for
tsunami inundation modeling were documented in the
‘‘Proceedings and results of the 2011 NTHMP Model
Benchmarking Workshop’’ (NTHMP 2012). Since
then, other models have been subjected to the
benchmark problems used in the workshop, and their
approval and use subsequently requested for NTHMP
projects. For those currently wishing to benchmark
their tsunami inundation models, a first step consists
of completing benchmark problems 1, 4, 6, 7, and 9
in NTHMP (2012). This is the aim of the present
benchmarking study for the case of the Tsunami-
HySEA model. Another preliminary requirement for
achieving MMS approval for tsunami inundation
models is that all models being used by US federal,
state, territory, and commonwealth governments
should be provided to the public as open source. A
freely accessible open source version of Tsunami-
HySEA can be downloaded from the website https://
edanya.uma.es/hysea.
Besides NTHMP (2012) and references therein,
for NTHMP-benchmarked tsunami models, other
authors have performed similar benchmarking efforts
as the one presented here with their particular models,
as is the case of Nicolsky et al. (2011), Apotsos et al.
(2011) or Tolkova (2014). In addition, a model
intercomparison of eight NTHMP models for
benchmarks 4 (laboratory simple beach) and 6 (con-
ical island) can be found in the study by Horrillo et al.
(2015).
2. The Tsunami-HySEA Model
HySEA (Hyperbolic Systems and Efficient Algo-
rithms) software consists of a family of geophysical
1
Departamento de A.M., E. e I.O. y Matema
´tica Aplicada,
Facultad de Ciencias, University of Ma
´laga, Campus de Teatinos,
s/n, 29080 Ma
´laga, Spain. E-mail: jmacias@uma.es
2
Laboratorio de Me
´todos Nume
´ricos, SCAI, University of
Ma
´laga, Campus de Teatinos, s/n, 29080 Ma
´laga, Spain.
3
Departamento de Matema
´tica Aplicada, E.T.S. Telecomu-
nicacio
´n, University of Ma
´laga, Campus de Teatinos, s/n, 29080
Ma
´laga, Spain.
Pure Appl. Geophys. 174 (2017), 3147–3183
2017 The Author(s)
This article is an open access publication
DOI 10.1007/s00024-017-1583-1 Pure and Applied Geophysics
Content courtesy of Springer Nature, terms of use apply. Rights reserved.

codes based on either single-layer, two-layer strati-
fied systems or multilayer shallow-water models.
HySEA codes have been developed by EDANYA
Group (https://edanya.uma.es) from the Universidad
de Ma
´laga (UMA) for more than a decade and they
are in continuous evolution and upgrading. Tsunami-
HySEA is the numerical model specifically designed
for tsunami simulations. It combines robustness,
reliability, and good accuracy in a model based on a
GPU faster than real-time (FTRT) implementation. It
has been thoroughly tested, and in particular has
passed not only all tests by Synolakis et al. (2008),
but also other laboratory tests and proposed bench-
mark problems. Some of them can be found in the
studies by Castro et al. (2005,2006,2012), Gallardo
et al. (2007), de la Asuncio
´n et al. (2013), and
NTHMP (2016).
2.1. Model Equations
Tsunami-HySEA solves the well-known 2D non-
linear one-layer shallow-water system in both spher-
ical and Cartesian coordinates. For the sake of brevity
and simplicity, only the latter system is written:
oh
otþohuðÞ
oxþohvðÞ
oy¼0;
ohuðÞ
otþo
oxhu2þ1
2gh2

þohuvðÞ
oy¼gh oH
oxþSx;
oðhvÞ
otþo
oyhv2þ1
2gh2

þohuvðÞ
ox¼gh oH
oyþSy:
In the previous set of equations, hx;tðÞdenotes
the thickness of the water layer at point x2DR2
at time t, with Dbeing the horizontal projection of
the 3D domain where tsunami takes place. HxðÞis
the depth of the bottom at point xmeasured from a
fixed level of reference. ux;tðÞand vx;tðÞare the
height-averaged velocity in the x- and y-directions,
respectively, and gdenotes gravity. Let us also
define the function gx;tðÞ¼hx;tðÞHðxÞthat
corresponds to the free surface of the fluid.
The terms Sxand Syparameterize the friction
effects and two different laws are considered:
1. The Manning law:
Sx¼ghM2
nukðu;vÞk
h4=3;
Sy¼ghM2
nvkðu;vÞk
h4=3;
where Mn[0 is the manning coefficient.
2. A quadratic law:
Sx¼cfukðu;vÞk;Sy¼cfvkðu;vÞk;
where cf[0 is the friction coefficient. In all the
numerical tests presented in this study the Manning
law is used.
Finally, to perform the BP4 (runup in a simple
beach-experimental) and BP6 (conical island), a
version of the code including dispersion was
used. Dispersive model equations are written as
follows:
Figure 1
Non-scaled sketch of a canonical 1D simple beach with a solitary wave (X
0
=dcot b)
3148 J. Macı
´as et al. Pure Appl. Geophys.
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oh
otþohuðÞ
oxþohvðÞ
oy¼0;
ohuðÞ
otþo
oxhu2þ1
2gh2þ1
2hp

þohuvðÞ
oy¼ðgh þpÞoH
oxþSx;
oðhvÞ
otþo
oyhv2þ1
2gh2þ1
2hp

þohuvðÞ
ox¼ðgh þpÞoH
oyþSy;
ohwðÞ
ot¼p;
hoðhuÞ
oxhu o2ghðÞ
oxþhohvðÞ
oyhv o2ghðÞ
oyþ2hw ¼0:
The dispersive system implemented can be inter-
preted as a generalized Yamazaki model (Yamazaki
et al. 2009) where the term oh
otwis not neglected in the
equation for the vertical velocity. The free divergence
equation has been multiplied by h2to write it with the
conserved variables hu and hv. In addition, due to the
rewriting of the last equation, no special treatment is
required in the presence of wet–dry fronts. The
breaking criteria employed is similar to the criteria
presented by Roeber et al. (2010), based on an ‘‘eddy
viscosity’’ approach.
2.2. Numerical Solution Method
Tsunami-HySEA solves the two-dimensional
shallow-water system using a high-order (second
and third order) path-conservative finite-volume
method. Values of h;hu and hv at each grid cell
represent cell averages of the water depth and
momentum components. The numerical scheme is
conservative for both mass and momentum in flat
bathymetries and, in general, is mass preserving for
arbitrary bathymetries. High order is achieved by a
non-linear total variation diminishing (TVD) recon-
struction operator of the unknowns h;hu;hv and
g¼hH. Then, the reconstruction of His recov-
ered using the reconstruction of hand g. Moreover, in
the reconstruction procedure, the positivity of the
water depth is ensured. Tsunami-HySEA implements
several reconstruction operators: MUSCL (Mono-
tonic Upstream-Centered Scheme for Conservation
Laws, see van Leer 1979) that achieves second order,
the hyperbolic Marquina’s reconstruction (see Mar-
quina 1994) that achieves third order, and a TVD
combination of piecewise parabolic and linear 2D
reconstructions that also achieves third order [see
Gallardo et al. (2011)]. The high-order time
discretization is performed using the second- or
third-order TVD Runge–Kutta method described in
Gottlieb and Shu (1998). At each cell interface,
Tsunami-HySEA uses Godunov’s method based on
the approximation of 1D projected Riemann prob-
lems along the normal direction to each edge. In
particular Tsunami-HySEA implements a PVM-type
(polynomial viscosity matrix) method that uses the
fastest and the slowest wave speeds, similar to HLL
(Harten–Lax–van Leer) method (see Castro and
Ferna
´ndez-Nieto 2012). A general overview of the
derivation of the high-order methods is shown by
Castro et al. 2009. For large computational domains
and in the framework of Tsunami Early Warning
Systems, Tsunami-HySEA also implements a two-
step scheme similar to leap-frog for the deep-water
propagation step and a second-order TVD-weighted
averaged flux (WAF) flux-limiter scheme, described
by de la Asuncio
´n et al. 2013, for close to coast
propagation/inundation step. The combination of
both schemes guaranties the mass conservation in
the complete domain and prevents the generation of
spurious high-frequency oscillations near discontinu-
ities generated by leap-frog type schemes. At the
same time, this numerical scheme reduces computa-
tional times compared with other numerical schemes,
while the amplitude of the first tsunami wave is
preserved.
Figure 2
Water level profiles during runup of the non-breaking wave in the
case H/d=0.019 at time t=55 (d/g)
1/2
for three different
numerical resolutions. Comparison with the analytical solution
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Concerning the wet–dry fronts discretization,
Tsunami-HySEA implements the numerical treat-
ment described by Castro et al. (2005) and Gallardo
et al. (2007) that consists of locally replacing the 1D
Riemann solver used during the propagation step, by
another 1D Riemann solver that takes into account
Figure 3
Maximum runup as a function of time for the three resolutions considered. The black dot showing the analytical maximum runup at t=55 s
Figure 4
Water level profiles during runup of the non-breaking wave in the case H/d=0.019 on the 1:19.85 beach (at times t=35 (d/g)
1/2
,t=40 (d/
g)
1/2
,t=45 (d/g)
1/2
, and t=50 (d/g)
1/2
. Normalized root mean square deviation (NRMSD) and maximum wave amplitude error (ERR) are
computed and shown for each time
3150 J. Macı
´as et al. Pure Appl. Geophys.
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the presence of a dry cell. Moreover, the reconstruc-
tion step is also modified to preserve the positivity of
the water depth. The resulting schemes are well
balanced for the water at rest, that is, they exactly
preserve the water at rest solutions, and are second-
or third-order accurate, depending on the reconstruc-
tion operator and the time stepping method. Finally,
the numerical implementation of Tsunami-HySEA
has been performed on GPU clusters (de la Asuncio
´n
et al. 2011,2013, Castro et al. 2011) and nested-grids
configurations are available (Macı
´as et al.
2013,2014,2015,2016). These facts allow to speed
up the computations, being able to perform complex
simulations, in very large domains, much faster than
real time (Macı
´as et al. 2013,2014,2016).
The dispersive model implements a formal second-
order well-balanced hybrid finite-volume/difference
(FV/FD) numerical scheme. The non-hydrostatic sys-
tem can be split into two parts: one corresponding to the
non-linear shallow-water component in conservative
form and the other corresponding to the non-hydro-
static terms. The hyperbolic part of the system is
discretized using a PVM path-conservative finite-
volume method (Castro and Ferna
´ndez-Nieto 2012
and Pare
´s2006), and the dispersive terms are dis-
cretized with compact finite differences. The resulting
ODE system in time is discretized using a TVD Runge–
Kutta method (Gottlieb and Shu 1998).
3. Benchmark Problem Comparisons
This section contains the Tsunami-HySEA results
for each of the five benchmark problems that are
required by the NTHMP Tsunami Inundation Model
Approval Process (July 2015). The specific version of
Tsunami-HySEA code benchmarked in the present
study is the second order with MUSCL reconstruction
and its second-order dispersive counterpart when
dispersion is required. Detailed descriptions of all
benchmarks, as well as topography data when
required and laboratory or field data for comparison
when applicable, can be found in the repository of
benchmark problems https://gitub.com/rjleveque/
nthmp-benchmark-problems for NTHMP, or in the
NCTR repository http://nctr.pmel.noaa.gov/
Figure 5
Water level profiles during runup of the non-breaking wave in the case H/d=0.019 on the 1:19.85 beach at times t=55 (d/g)
1/2
,t=60 (d/
g)
1/2
, and t=65 (d/g)
1/2
.NRMSD normalized root mean square deviation, MAX maximum amplitude or runup error
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benchmark/. Results from model participating in
original 2011 workshop can be found at NTHMP
(2012). For the sake of completeness, a brief descrip-
tion of each benchmark problem is provided. For BP#1
and BP#4, dealing with analytical solutions or very
simple laboratory 1D configurations, non-dimensional
variables are used everywhere. For problems dealing
with 2D complex laboratory experiments (BP#6 and
BP#7) scaled dimensional problems are solved.
Finally, BP#9 dealing with field data is solved in real-
world not-scaled dimensional variables.
3.1. Benchmark Problem #1: Simple Wave
on a Simple Beach—analytical—CASE H/
d=0.019
In this section, we compare numerical results
for solitary wave shoaling on a plane beach to an
Figure 6
Water level time series at location x/d=9.95 (upper panel) and at location x/d=0.25 (lower panel). Mesh resolution is 800 points
Table 1
Tsunami-HySEA model surface profile errors with respect to the analytical solution for H =0.019 at times t =35:5:65 (d/g)
1/2
. Comparison
with the mean value for NTHMP models in NTHMP (2012)
Model error for case H=0.019
t=35 t=40 t=45 t=50 t=55 t=60 t=65 Mean
RMS
(%)
MAX
(%)
RMS
(%)
MAX
(%)
RMS
(%)
MAX
(%)
RMS
(%)
MAX
(%)
RMS
(%)
MAX
(%)
RMS
(%)
MAX
(%)
RMS
(%)
MAX
(%)
RMS
(%)
MAX
(%)
Tsunami-HySEA model error
1 1 1 0 1 0 0 3 0 1 0 0 2 1 0.85 0.84
Mean error for NTHMP models
22 22 22 12 00 01 53 22
RMS normalized root mean square deviation, MAX maximum amplitude or runup error
3152 J. Macı
´as et al. Pure Appl. Geophys.
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analytic solution based on the shallow-water equa-
tions. The benchmark data for comparison are
obtained from NTHMP (2012) or Synolakis et al.
(2008). In the present case, the model has been run
in non-linear, non-dispersive, and no friction mode
as requested for comparison and verification
purposes. In this problem, the wave of height
His initially centered at distance Lfrom the beach
toe and the shape for the bathymetry consists of an
area of constant depth d, connected to a plane
sloping beach of angle b=arccot(19.85) as
schematically shown in Fig. 1.
Figure 7
Comparison of numerically calculated free surface profiles at various dimensionless times for the non-breaking case H/d=0.0185 with the
lab data. Non-dispersive Tsunami-HySEA model
Table 2
Tsunami-HySEA model sea level time series errors with respect to the analytical solution for H =0.019 at x =9.95 and x =0.25.
Comparison with the mean value for NTHMP models in NTHMP (2012), taken from Tables 1–7 b in p. 38
Model error for case H=0.019
x=9.95 x=0.25 Mean
RMS (%) MAX (%) RMS (%) MAX (%) RMS (%) MAX (%)
Tsunami-HySEA 1 1 1 0 0.58 0.68
Mean NTHMP (2012)212121
RMS normalized root mean square deviation, MAX maximum amplitude or runup error
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Figure 8
Comparison of numerically calculated free surface profiles at various dimensionless time for the non-breaking case H/d=0.0185 with the lab
data. Dispersive Tsunami-HySEA model
Table 3
Tsunami-HySEA model surface profile errors with respect to the lab experiment for Case A, H =0.0185 at times t =30:10:70 (d/g)
1/2
. The
values for NTHMP models are taken or computed from data in Table 1–8 a in p. 41 in NTHMP (2012)
Model error for CASE H=0.0185
t=30 t=40 t=50 t=60 t= 70 Mean
RMS
(%)
MAX
(%)
RMS
(%)
MAX
(%)
RMS
(%)
MAX
(%)
RMS
(%)
MAX
(%)
RMS
(%)
MAX
(%)
RMS
(%)
MAX
(%)
NDH 10.35 5.83 6.72 2.27 3.52 9.88 3.13 2.69 9.15 8.44 6.57 5.82
NDN 11 6 9 3 6 13 4 1 33 15 10 8
DH 6.69 3.92 5.35 1.19 4.6 5.12 3.24 1.73 8.63 3.59 5.7 2.1
DN 11 3 8 2 4 3 5 4 12 6 8 3.5
AN114835 75316995
RMS normalized root mean square deviation, MAX maximum amplitude or runup error, NDH Tsunami-HySEA non-dispersive, NDN non-
dispersive models in NTHMP (2012) (Alaska, GeoClaw, and MOST), DH Tsunami-HySEA dispersive, DN dispersive models in NTHMP
(2012) (ATFM, BOSZ, FUNWAVE, NEOWAVE, and SELFE), AN mean of all models in NTHMP (2012)
3154 J. Macı
´as et al. Pure Appl. Geophys.
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3.1.1 Problem Setup
Problem setup is defined by the following items (all
the variables in this BP are non-dimensional and the
computations have been performed in non-dimen-
sional variables):
Friction: no friction (as required).
Parameters:d=1, g=1, and H=0.019 (see
Fig. 1for dand H).
Computational domain: the computational domain
in x spanned from x=-10 to x=70.
Boundary conditions: a non-reflective boundary
condition at the right side of the computational
domain is imposed (beach slope is located to the
left).
Initial condition: the prescribed soliton at time t=0
with the proposed correction for the initial velocity.
These initial data were given by:
gx;0ðÞ¼Hsech2ðcðxX1Þ=dÞ;
where X1¼X0þL, with L¼arccoshðffiffiffiffiffi
20
pÞ=cthe
half-length of the solitary wave, and c¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3H=4d
p
the water wave elevation and
ux;0ðÞ¼
ffiffiffi
g
d
rgðx;0Þ
for the initial velocity (the minus sign meaning
approaching the coast, that in the numerical test is on
the left-hand side).
Grid resolution: the numerical results presented are
for a computational mesh composed of 800 cells,
i.e., Dx=0.1 =d/10. For the convergence analysis
of the maximum runup, two other increased reso-
lutions have been used, Dx=0.05 =d/20 and
Dx=0.025 =d/40 with 1600 and 3200 cells,
respectively.
Figure 9
Comparison of numerically calculated free surface profiles at various dimensionless times for the breaking case H/d=0.3 with the lab data.
Non-dispersive model
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Time stepping: variable time stepping based on a
CFL condition is used.
CFL: CFL number is set to 0.9.
Versions of the code: Tsunami-HySEA third-order
(with Marquina’s reconstruction) and second-order
(with MUSCL reconstruction) models have been
benchmarked using this particular problem. Both
models give nearly identical results.
3.1.2 Tasks to be Performed
To accomplish this benchmark the following four
tasks were suggested:
1. Numerically compute the maximum runup of the
solitary wave.
2. Compare the numerically and analytically com-
puted water level profiles at t=25 (d/g)
1/2
,
t=35 (d/g)
1/2
,t=45 (d/g)
1/2
,t=55 (d/g)
1/2
,
and t=65 (d/g)
1/2
. Note that as we used the
MATLAB scripts and data provided by Juan
Horrillo on behalf of the NTHMP, the numerical
vs analytical comparison is performed at the times
given in the provided data and depicted by the
corresponding MATLAB script that does not
correspond exactly with all the time instants given
in BP1 description. More precisely, they do
correspond to t=35:5:65 (d/g)
1/2
. Therefore,
t=25 (d/g)
1/2
is missing and t=40, 50, and
60 (d/g)
1/2
are shown.
3. Compare the numerically and analytically com-
puted water level dynamics at locations x/
d=0.25 and x/d=9.95 during propagation and
reflection of the wave.
4. Demonstrate scalability of the code.
Figures 2,3,4,5and 6show the plots corre-
sponding to these four tasks.
Figure 10
Comparison of numerically calculated free surface profiles at various dimensionless times for the breaking case H/d=0.3 with the lab data.
Dispersive model
3156 J. Macı
´as et al. Pure Appl. Geophys.
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Figure 11
Maximum runup as a function of time. Upper panel Case A.Lower panel Case C.Red dots mark the maximum runup over time for non-
dispersive model and green dots for the dispersive model
Table 4
Tsunami-HySEA model surface profile errors with respect to the lab experiment for Case C, H =0.30 at times t =15:5:30 (d/g)
1/2
. Non-
dispersive, dispersive model results and the mean of the four models with dispersion in NTHMP (2012) that presented results for this test are
collected in this table. The values for NTHMP models are taken from data in Tables 1–8 b in p. 41 in NTHMP (2012)
Model error for CASE H=0.30
t=15 t=20 t=25 t=30 Mean
RMS (%) MAX (%) RMS (%) MAX (%) RMS (%) MAX (%) RMS (%) MAX (%) RMS (%) MAX (%)
Non-dispersive 22.5 17.33 17.42 52.34 5.17 10.07 2.32 3.09 11.85 20.70
Dispersive 2.25 0.25 3.63 3.84 5.69 11.97 2.28 0.70 3.46 4.18
Mean NTHMP 7 6 9 11 6 10 4 6 6.5 8
RMS normalized root mean square deviation, MAX maximum amplitude or runup error
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3.1.3 Numerical Results
In this section, we present the numerical results
obtained using Tsunami-HySEA for BP1 according
to the tasks to be performed as given in the
benchmark description.
3.1.3.1 Maximum Runup The maximum runup is
reached at t=55 (d/g)
1/2
. In the case of the reference
numerical experiment with Dx=0.1 and 800 cells,
the value for the maximum runup is 0.08724. For the
refined mesh experiments with Dx=0.05 and
Dx=0.025, the computed runups are 0.09102 and
0.9165, respectively. Comparison of the numerical
solutions with the analytical reference is depicted in
Fig. 2showing the convergence of the maximum
runup to the analytical value as mesh size is reduced.
It must be noted that for the analytical solution at
time t=55 (d/g)
1/2
and location x=-1.8 water
surface is located at 0.0909, but this is not the value
of the analytical runup (that must be a value slightly
above 0.92), as can be seen in Fig. 2.
Figure 3depicts the time evolution for the maxi-
mum runup simulated for the three spatial resolutions
considered. The black dot marks the approximate
location of the analytical maximum runup.
3.1.3.2 Water Level at t =35:5:65 (d/g)
1/2
.
(MATLAB Script and Data from J. Horrillo) The
next two figures show the water level profiles during
the runup of the non-breaking wave in the case H/
d=0.019 on the 1:19.85 beach at times t=35:5:50
(d/g)
1/2
in Fig. 4and times t=55:5:65 (d/g)
1/2
in
Fig. 5. For a quantitative comparison with the ana-
lytical solution, normalized root mean square
deviation (NRMSD) and maximum wave amplitude
error (ERR) are computed and shown for each time.
Table 1presents the values that measure model
surface profile errors with respect to the analytical
solution for H=0.019 at considered times. The error
value for a particular time is rounded towards the
nearest integer. The mean values are computed
exactly, using the exact values for all times. Mean
values for the eight models in NTHMP (2012) report
are presented for comparison (taken from Tables 1–7
a in p. 38).
3.1.3.3 Water Level at Locations x/d =0.25 and x/
d=9.95 Figure 6depicts the comparison of the
water level time series of numerical results at both
locations, x/d=0.25 and x/d=9.95, with the ana-
lytical solution. Table 2collects the values that
measure model sea level time series errors with
respect to the analytical solution for H=0.019 at
locations x=9.95 and x=0.25 . The error value for
each location is rounded towards the nearest integer.
Mean values for Tsunami-HySEA are computed
exactly. Mean values for the eight models in NTHMP
(2012) report are presented for comparison (taken
from Tables 1–7 b in p. 38).
3.1.3.4 Scalability Tsunami-HySEA has the option
of solving dimensionless problems, and this is an
option commonly used. When dimensionless prob-
lems are solved, it makes no sense to perform any test
of scalability as the dimensionless problems to be
solved for the different scaled problems will (if
scaled to unity) always be the same.
3.2. Benchmark Problem #4: Simple Wave
on a Simple Beach—Laboratory
This benchmark is the lab counterpart of BP1
(analytical benchmarking comparison). In this
Figure 12
Scatter plot of non-dimensional maximum runup, R/d, versus non-
dimensional incident wave height, H/d, resulting from a total of
more than 40 experiments conducted by Y. Joseph Zhan. Red dots
indicate the non-dispersive numerical simulations and the green
dots the results for the dispersive model. Numerically they are
slightly different but in the graphic they superimpose
3158 J. Macı
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laboratory test, the 31.73-m-long, 60.96-cm-deep,
and 39.97-cm-wide wave tank located at the Califor-
nia Institute of Technology, Pasadena was used with
water of varying depths. The set of laboratory data
obtained has been extensively used for many code
validations. In this BP4, the datasets for the H/
d=0.0185 non-breaking and H/d=0.30 breaking
solitary waves are used for code validation. The
model has been first run in non-linear, non-dispersive
mode. Then a dispersive version of Tsunami-HySEA
has also been used to assess the influence of
dispersive terms in both, non-breaking and breaking
cases, and in both wave shape evolution and maxi-
mum runup estimation.
Figure 13
Basin geometry and coordinate system. Solid lines represent approximate basin and wavemaker surfaces. Circles along walls and dashed lines
represent wave absorbing material. Red dots represent gage locations for time series comparison. (Figure taken from benchmark description)
Table 5
Laboratory gage positions. See Fig. 13 for graphical location
Gage ID X (m) Y (m) Z (cm) Comment
6 9.36 13.80 31.7 270Transect
9 10.36 13.80 8.2 270Transect
16 12.96 11.22 7.9 180Transect
22 15.56 13.80 8.3 90Transect
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3.2.1 Problem Setup
Problem setup is defined by the following items (all
the variables in this BP are non-dimensional and the
computations have been performed in non-dimen-
sional variables):
•Friction: Manning coefficient was set to 0.03 for
the non-dispersive model and slightly adjusted for
the dispersive model (0.036 for the H/d=0.30 and
0.032 for H/d=0.0185).
•Parameters:d=1, g=1, and H=0.0185 for the
non-breaking Case And H=0.30 for the breaking case.
•Computational domain: the computational domain
in x spanned from x=-10 to x=70.
•Boundary conditions: a non-reflective boundary
condition at the right side of the computational
domain is imposed.
•Initial condition: the prescribed soliton at time
t=0 with the proposed initial velocity. These are
the same conditions as for previous benchmark
problem.
•Grid resolution: the numerical results presented are
for a computational mesh composed of 1600 cells,
i.e., Dx=0.05 =d/20.
•Time stepping: variable time stepping based on a
CFL condition.
•CFL: CFL number is set to 0.9
•Versions of the code: Tsunami-HySEA third-
order (with Marquina’s reconstruction) and sec-
ond-order (with MUSCL reconstruction) non-
dispersive models and second-order (with
MUSCL reconstruction) dispersive model have
been benchmarked using this particular problem.
Both non-dispersive models give nearly identical
results. In this case, dispersion plays an impor-
tant role.
3.2.2 Tasks to be Performed
To accomplish this BP, the four following tasks had
to be performed:
Figure 14
Snapshots at several times showing the wavefront splitting in front of the island for Case B (upper panel) at times t=31, 31.5, and 32 s; and
for Case C (lower panel) at times t=29.5, 30.5, and 31 s. Water elevation in meters
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1. Compare numerically calculated surface profiles
at t/T=30:10:70 for the non-breaking case H/
d=0.0185 with the lab data (Case A).
2. Compare numerically calculated surface profiles
at t/T=15:5:30 for the breaking case H/d=0.3
with the lab data (Case C).
3. Numerically compute maximum runup (Case A
and C).
4. Numerically compute maximum runup R/dvs. H/
d.
3.2.3 Numerical Results
In this section, we present the numerical results
obtained using Tsunami-HySEA for BP4 according
to the tasks to be performed as given in the
benchmark description.
3.2.3.1 Water Level at Times t =30, 40, 50, 60, and
70 (d/g)
1/2
for Case A (H/d =0.0185) Figure 7
shows the numerical results for Task 1 comparing the
computed and measured surface profiles for the low-
amplitude case (A) using the non-dispersive version
of Tsunami-HySEA. Figure 8presents the same
comparison but for the dispersive version of the code.
Table 3gathers the values for the normalized root
mean square deviation (NRMSD) and the maximum
amplitude or runup error (MAX) for this case for both
non-dispersive and dispersive models and compares
them with the mean of the eight models in NTHMP
(2012) performing this benchmark problem. For
comparison purpose, models in NTHMP (2012) have
also been split into dispersive (five of them) and non-
dispersive (three), and the mean values for the errors
are presented in Table 3. Values for NTHMP models
are extracted or computed from data in Tables 1–8 a
in p. 41.
3.2.3.2 Water Level at Times t =10, 15, 20, 25, and
30 (d/g)
1/2
for Case C (H/d =0.3) Figure 9shows
the numerical results for Task 2, comparing the
computed and measured surface profiles for the high-
amplitude case (C) using the non-dispersive version
of Tsunami-HySEA. Figure 10 presents the same
comparison but for the dispersive version of the code.
Table 4gathers the values for the normalized root
Figure 15
Snapshots at several times for the numerical simulation showing the wavefronts collide behind the island. Upper panel for Case B (times
shown t=34, 35 and 35.5 s) and lower panel for Case C (times shown t=32, 33 and 34 s). Water surface elevation in meters
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mean square deviation (NRMSD) and the maximum
amplitude or runup error (MAX) for this case for both
non-dispersive and dispersive models and compares
them with the mean of the four non-dispersive models
in NTHMP (2012) that presented their results for this
test (ATFM, BOSZ, FUNWAVE, and NEOWAVE).
The models in NTHMP (2102) not including disper-
sive terms did not present results for this test (data
extracted from Tables 1–8 b in p. 41).
3.2.3.3 Maximum runup (Case A and C) Figure 11
shows the maximum runup as a function of time for
Case A (upper panel) and Case C (lower panel).
Numerical results for models without and with dis-
persion are presented superimposed in each figure for
comparison. The maximum simulated runup is
marked in the time series. In case (A) the maximum
runup of 0.08066 is reached at time t=56–56.5 s for
the non-dispersive model and at time t=56.5 s for a
height of 0.0802 for the dispersive model. In case
(C) a maximum runup height of 0.5117 is reached at
time t=42–42.5 s for the non-dispersive model and
of 0.512 at t=43.5 s for the dispersive model. These
values are marked with red and green dots (respec-
tively) in Fig. 11.
3.2.3.4 Maximum Runup R/d vs. H/d Figure 12
shows the maximum runup, R/d, as a function of H/
dfor the numerical simulations performed without
dispersion (red dots) and including dispersion (green
dots). For the two numerical experiments, with H/
d=0.30 and H/d=0.0185, the computed values for
the maximum runup computed without and with dis-
persion cannot be distinguished in the graphics as the
values only differ slightly. In the same figure a scatter
plot of more than 40 lab experiments conducted by Y.
Joseph Zhan are depicted (Synolakis 1987).
It can be observed that both non-dispersive and
dispersive models perform well in the case of the
non-breaking wave. Nevertheless, this same behavior
does not occur for the breaking wave case. It can be
seen, from Fig. 9, that the non-dispersive model is
Figure 16
Comparison between the computed and measured water level at gauges 6, 9, 16, and 22 for an incident solitary wave in the Case A
(H=0.045). Tsunami-HySEA non-dispersive model
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not able to capture the time evolution of the wave in
this particular case, tending to produce a shock wave
that travels faster than the actual dispersive wave.
Nevertheless, we observe that when the propagation
phase ends and the inundation step takes place, the
non-dispersive model closely reproduces the
observed new wave. Finally, regardless of whether
we are simulating the breaking or non-breaking wave,
if we simply look at the runup time evolution we
observed that both non-dispersive and dispersive
models produce quite close simulated time series
(Fig. 11).
3.3. Benchmark Problem #6: Solitary Wave
on a Conical Island—Laboratory
The goal of this benchmark problem is to compare
computed model results with laboratory
measurements obtained during a physical modeling
experiment conducted at the Coastal and Hydraulic
Laboratory, Engineering Research and Development
Center of the US: Army Corps of Engineers (Briggs
et al. 1995). The laboratory physical model was
constructed as an idealized representation of Babi
Island, in the Flores Sea, Indonesia, to compare with
Babi Island runup measured shortly after the 12
December 1992 Flores Island tsunami (see Fig. 13 for
a schematic picture).
Three cases (A, B, and C) were performed
corresponding to three wavemaker paddle
trajectories.
To accomplish this benchmark, it is suggested that
for
•CASE B: water depth, d=32.0 cm, target
H=0.10, measured H=0.096 (this case was
formerly optional).
Figure 17
Comparison between the computed and measured water level at gauges 6, 9, 16, and 22 for an incident solitary wave in the Case A
(H=0.096). Tsunami-HySEA dispersive model
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•CASE C: water depth, d=32.0 cm, target
H=0.20, measured H=0.181.
To perform the tasks described below in
Sect. 3.3.2.
The Case A, that was formerly mandatory, now is
not included:
•CASE A: water depth, d=32.0 cm, target
H=0.05, measured H=0.045
In any case, we will include the three cases for all
the tasks but for the splitting–colliding item.
3.3.1 Problem Setup
The main features describing the numerical setup of
the problem are:
•Friction: Manning coefficient is set to 0.015 for the
non-dispersive model and to 0.02 for the dispersive
model.
•Computational domain:[-5, 23] 9[0, 28] in
meters.
•Boundary conditions: open boundary conditions.
•Initial condition: the prescribed soliton centered at
x=0 with the proposed correction for the initial
velocity (same expression as in BP1 and BP4, but
extended to two dimensions, with wave elevation
constant and zero velocity in the y-direction).
•Grid resolution: for the non-dispersive model a
spatial grid resolution of 5 cm is used for Case A
and a 2-cm resolution grid for Cases B and C.
Dispersive model uses a 2-cm resolution for the
three cases.
•Time stepping: variable time stepping based on a
CFL condition.
•CFL: 0.9
•Versions of the code: Tsunami-HySEA second
order with MUSCL reconstruction (non-dispersive)
and second-order dispersive with MUSCL recon-
struction codes have been used.
Figure 18
Comparison between the computed and measured water level at gauges 6, 9, 16, and 22 for an incident solitary wave in the Case B
(H=0.096). Tsunami-HySEA non-dispersive model
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3.3.2 Tasks to be Performed
Model simulations must be conducted to address the
following objectives (for cases B and C):
1. Demonstrate that two wavefronts split in front of
the island and collide behind it;
2. Compare computed water level with laboratory
data at gauges 9, 16, and 22 (see Fig. 13 for
graphical location and Table 5for actual
coordinates);
3. Compare computed island runup with laboratory
gage data.
3.3.3 Numerical Results
Note that as we used the MATLAB scripts and data
provided by J. Horrillo (Texas A&M University), we
decided to perform numerical experiments for all the
three cases A, B, and C, and also to present water
level at gauge 6, although not included as mandatory
requirements. For this benchmark, we have used
Tsunami-HySEA non-dispersive and dispersive codes
and have compared shape wave evolution and final
maximum runup.
3.3.3.1 Wave Splitting and Colliding Figure 14
presents snapshots at different times for Case B (in
upper panel) and Case C (in lower panel) showing
how two wavefronts split in front of the island (Task
1). For Case B times t=31, 31.5, and 32 are shown
and for Case C times t=29.5, 30.5, and 31 are
presented.
Figure 15 presents snapshots at different times for
Case B (upper panel) and Case C (lower panel)
showing how two wavefronts, after splitting, collide
Figure 19
Comparison between the computed and measured water level at gauges 6, 9, 16, and 22 for an incident solitary wave in the Case B
(H=0.096). Tsunami-HySEA dispersive model
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behind the island (Task 1). For Case B times t=34,
35, and 35.5 s are shown and for Case C times
t=32, 33, and 34 s are presented.
3.3.3.2 Water Level at Gauges In this section, we
present the comparison of the computed and
measured water level at gauges 6, 9, 16, and 22 for
the cases (A), (B), and (C), respectively. For each
case, first the results for the non-dispersive model
are presented, then the results for the dispersive
code. Figures 16 and 17 show the comparison for
case (A) of the computed with Tsunami-HySEA
non-dispersive and dispersive model and measured
data, respectively. Figures 18 and 19 present the
comparison for case (B) and Figs. 20 and 21 for
case (C). Tables 6,7,and8gather the sea level
time series Tsunami-HySEA non-dispersive and
dispersive models’ error with respect to laboratory
experiment data for CaseA,B,andC,respec-
tively. Comparison with the mean value obtained
for the eight models performing this benchmark in
NTHMP (2012) split into non-dispersive and dis-
persive models is also included.
It can be observed that as we increase the value of
Hmoving from Case A to B and Case C, the
mismatch between the simulated wave and the
measured one increases for the non-dispersive model.
The differences mostly increase in the leading wave.
On the other hand, the dispersive model performs
equally well in all the three cases.
3.3.3.3 Runup Around the Island Figure 22 pre-
sents the runup numerically computed around the
island with the non-dispersive model, compared
against the experimental data for the three cases.
Figure 23 shows the same comparison but for the
dispersive model. The values for the NRMSD and
maximum error runup are computed and shown in the
figures. Table 9gathers the values for these errors for
Tsunami-HySEA (dispersive and non-dispersive) and
Figure 20
Comparison between the computed and measured water level at gauges 6, 9, 16, and 22 for an incident solitary wave in the Case C
(H=0.181). Tsunami-HySEA non-dispersive model
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compared them with the mean of the model in
NTHMP (2012) split in non-dispersive and dispersive
models too.
In this benchmark, the observed behavior of the
simulated maximum runup for non-dispersive and
dispersive models through the three cases considered
is not so easily explained. Now for cases A and B, in
the extremes, both models perform similarly well. In
Case B, the non-dispersive model performs clearly
worse, while the dispersive model performs equally
well.
3.4. Benchmark Problem #7: The Tsunami Runup
onto a Complex Three-Dimensional Model
of the Monai Valley Beach—Laboratory
A laboratory experiment using a large-scale tank
at the central Research Institute for Electric Power
Industry in Abiko, Japan was focused on modeling
the runup of a long wave on a complex beach near the
village of Monai (Liu et al. 2008). The beach in the
laboratory wave tank was a 1:400-scale model of the
bathymetry and topography around a very narrow
gully, where extreme runup was measured. More
information regarding this benchmark can be found in
the study by Synolakis et al. (2008). Figure 24 shows
the computational domain and the bathymetry.
3.4.1 Problem Setup
The main items describing the numerical setup of this
problem are:
•Friction: Manning coefficient is set to 0.03
•Computational domain: [0, 5.488] 9[0, 3.402]
(units in meters).
•Boundary conditions: the given initial wave
(Fig. 25) was used to specify the boundary condi-
tion at the left boundary up to time t=22.5 s;
Figure 21
Comparison between the computed and measured water level at gauges 6, 9, 16, and 22 for an incident solitary wave in the Case C
(H=0.181). Tsunami-HySEA dispersive model
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after time t=22.5 s, non-reflective boundary
conditions. Solid wall boundary conditions were
used at the top and bottom boundaries.
•Initial condition: water at rest.
•Grid resolution: a 393 9244-size mesh was used,
with the same resolution (0.014 m) as the bathy-
metry. Table 10 collects grid information.
•Time stepping: variable time stepping based on a
CFL condition.
•CFL: 0.9
•Versions of the code: Tsunami-HySEA second
order with MUSCL reconstruction and WAF
models used for this benchmark. Second-order
model results are presented.
3.4.2 Tasks to be Performed
To accomplish this benchmark, the following tasks
had to be performed:
1. Model the propagation of the incident and reflec-
tive wave according to the benchmark-specified
boundary condition.
2. Compare the numerical and laboratory-measured
water level dynamics at gauges 5, 7, and 9 (in
Fig. 24).
3. Show snapshots of the numerically computed
water level at time synchronous with those of
the video frames; it is recommended that each
Table 6
Sea level time series Tsunami-HySEA model error with respect to laboratory experiment data for Case A (H =0.045). Comparison with the
mean value obtained for the eight models performing this benchmark in NTHMP (2012) separated among non-dispersive (Alaska, Geoclaw,
and MOST) and dispersive models (ATFM, BOSZ, FUNWAVE, NEOWAVE, and SELFE), for a more precise comparison. The values for
NTHMP models are taken from data in Tables 1–9 a in p. 46
Sea level model error for CASE A (H=0.045)
Gauge # 6 Gauge # 9 Gauge # 16 Gauge # 22 Mean
RMS (%) MAX (%) RMS (%) MAX (%) RMS (%) MAX (%) RMS (%) MAX (%) RMS (%) MAX (%)
Tsunami-HySEA 10 3 9 5 9 5 8 10 9 5.6
Mean NTHMP-ND 6 9 7 14 10 10 8 25 8 15
Tsunami-HySEA-D 9 2 9 3 8 2 8 9 8.3 3.9
Mean NTHMP-D 8 7 8 9 9 12 8 12 8 10
Mean All NTHMP 7 8 8 10 9 12 8 18 8 12
RMS normalized root mean square deviation error, MAX maximum runup relative error
Table 7
Sea level time series Tsunami-HySEA model error with respect to laboratory experiment data for Case B (H =0.096). Comparison with the
mean value obtained for the eight models performing this benchmark in NTHMP (2012) separated among non-dispersive (Alaska, Geoclaw,
and MOST) and dispersive models (ATFM, BOSZ, FUNWAVE, NEOWAVE, and SELFE), for a more precise comparison. The values for
NTHMP models are taken from data in Tables 1–9 b in p. 46
Sea level model error for CASE B (H=0.096)
Gauge # 6 Gauge # 9 Gauge # 16 Gauge # 22 Mean
RMS (%) MAX (%) RMS (%) MAX (%) RMS (%) MAX (%) RMS (%) MAX (%) RMS (%) MAX (%)
Tsunami-HySEA 9 1 8 4 10 1 9 10 9 4
Mean NTHMP-ND 8 6 9 7 7 7 9 40 8 15
Tsunami-HySEA-D 8 3 7 5 9 1 6 0 7.6 2.4
Mean NTHMP-D 7 6 8 10 6 7 10 20 8 11
Mean All NTHMP 8 6 8 9 7 7 9 27 8 12
RMS normalized root mean square deviation error, MAX maximum runup relative error
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Table 8
Sea level time series Tsunami-HySEA model error with respect to laboratory experiment data for Case C (H =0.181). Comparison with the
mean value obtained for the eight models performing this benchmark in NTHMP (2012) separated among non-dispersive (Alaska, Geoclaw,
and MOST) and dispersive models (ATFM, BOSZ, FUNWAVE, NEOWAVE, and SELFE), for a more precise comparison. The values for
NTHMP models are taken from data in Tables 1–9 c in p. 46
Sea level model error for CASE C (H=0.181)
Gauge # 6 Gauge # 9 Gauge # 16 Gauge # 22 Mean
RMS (%) MAX (%) RMS (%) MAX (%) RMS (%) MAX (%) RMS (%) MAX (%) RMS (%) MAX (%)
Tsunami-HySEA 8 7 11 2 10 12 7 10 9 8
Mean NTHMP-ND 10 6 11 9 9 3 8 18 9 9
Tsunami-HySEA-D 7 0 10 7 8 6 6 2 7.9 3.9
Mean NTHMP-D 7 3 11 16 7 4 9 12 8 9
Mean All NTHMP 8 5 11 13 8 3 8 15 9 9
RMS normalized root mean square deviation error, MAX maximum runup relative error
Figure 22
Comparison between the computed and measured runup around the island for the three cases. Non-dispersive results
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modeler finds times of the snapshots that best fit
the data.
4. Compute maximum runup in the narrow
valley.
3.4.3 Numerical Result
In this section, we present the numerical results for
BP7 as simulated by Tsunami-HySEA according to
Figure 23
Comparison between the computed and measured runup around the island for the three cases. Tsunami-HySEA dispersive model
Table 9
Runup Tsunami-HySEA model error with respect to laboratory experiment data for all Cases A, B, and C. Comparison with the mean value
obtained for the eight models performing this benchmark in NTHMP (2012) separated among non-dispersive (Alaska, Geoclaw, and MOST)
and dispersive models (ATFM, BOSZ, FUNWAVE, NEOWAVE, and SELFE), for a more precise comparison. The values for NTHMP models
are taken from data in Tables 1–10 in p. 47
Runup model error
CASE A (H=0.045) CASE B (H=0.096) CASE C (H=0.181) Mean
RMS (%) MAX (%) RMS (%) MAX (%) RMS (%) MAX (%) RMS (%) MAX (%)
Tsunami-HySEA 7 0 19 1 5 0 10 0
Mean NTHMP-ND 18 12 21 2 12 5 17 7
Tsunami-HySEA-D 8 0 4 4 6 5 6 3
Mean NTHMP-D 17 4 16 7 10 5 15 5
Mean All NTHMP 18 7 18 5 11 5 16 5
RMS normalized root mean square deviation error, MAX maximum runup relative error
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the tasks to be performed as given in the benchmark
description.
3.4.3.1 Gauge Comparison Figure 26 shows a
comparison of Tsunami-HySEA results with the
laboratory values for the three requested gauges from
t=0tot=30 s. Superimposed is the normalized
root mean square deviation (NRMSD). A mean value
of 7.66% for the NRMSD is obtained for all the three
gauges for the time series simulating the first 30 s.
3.4.3.2 Frame Comparisons In the laboratory
experiment, the evolution of the wave was recorded.
Five frames (Frames 10, 25, 40, 55, and 70) extracted
Figure 24
Computational domain with bathymetry and gauge locations of the scaled model (units in meters)
0 5 10 15 20 25
−0.02
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
0.02
Time (s)
Height (m)
Figure 25
Prescribed input wave for the left boundary condition, defined from t=0tot=22.5 s
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from the video record of the lab experiment with 0.5-
s interval are shown in the left column of Fig. 27.
These frames focus on the narrow gully where the
highest runup is observed. On the right-hand side of
Fig. 27, snapshots of the numerically computed water
level at times t=15, 15.5, 16, 16.5, and 17 in sec-
onds are presented for comparison. A good
agreement of the numerical solution to observations
in time and space is revealed, and it can be observed
how the numerical model is able to capture the rapid
runup/rundown sequence in this particular key
location.
3.4.3.3 Runup in the Valley A maximum simulated
runup height of 0.0891 (compared with the 0.08958
experimentally measured) is reached at time
t=16.3 s at point (5.1559, 1.8896). Figure 28 shows
the frame corresponding to time t=16.3 s, where
the computed maximum runup location is marked
with a red dot.
3.5. Benchmark Problem #9: Okushiri Island
Tsunami—Field
The goal of this benchmark problem is to compare
computed model results with field measurements
gathered after the 12 July 1993 Hokkaido Nansei-Oki
tsunami (also commonly referred to as the Okushiri
tsunami).
3.5.1 Problem setup
The main items describing the setup of the numerical
problem are:
Table 10
Mesh information showing grid resolution, number of cells and
computing time needed for a 200-s simulation
Grid resolution Dx=Dy(m) # of volumes Comput. time [s-
(min)]
393 9244 0.014 95,892 91.54618 (1.52)
Figure 26
Comparison experimental and simulated water level at gauges 5, 7, and 9 from t=0tot=30 s
Figure 27
Comparison of snapshots of the laboratory experiment with the
numerical simulation. aFrame 10—time 15 s, bframe 25—time
15.5 s, cframe 40—time 16 s, dframe 55—time 16.5 s, eframe
70—time 17 s
c
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Figure 28
Maximum simulated runup, reached at time t=16.3 s, at point (5.15592, 1.88961), reaching a maximum height of 0.0891 (versus the 0.08958
experimentally measured)
Figure 29
Computational domain considered (level 0) showing the initial condition for Benchmark problem #9. Nested meshes level 1 and level 2 (the
latter composed of two submeshes) are also depicted
3174 J. Macı
´as et al. Pure Appl. Geophys.
Content courtesy of Springer Nature, terms of use apply. Rights reserved.

•Friction: Manning coefficient 0.03.
•Boundary conditions: non-reflective boundary con-
ditions at open sea, at coastal areas inundation is
computed.
•Computational domain: a nested mesh technique is
used with four levels (i.e., the global mesh with
three levels of refinement, see Figs. 29 and 30).
•Global mesh coverage in lon/lat [138.504,
140.552] 9[41.5017, 43.2984].
– Number of cells: 1152 91011 =1,164,672.
– Resolution: 6.4 arc-sec (&192 m).
•Level 1. Spatial coverage [139.39, 139.664] 9
[41.9963, 42.2702].
– Refinement ratio: 4.
– Number of cells: 616 9616 =379,456.
– Resolution: 1.6 arc-sec (&40 m).
•Level 2. Refinement ratio: 4. Resolution: 0.4 arc-
sec (&12 m).
•Submesh 1: large area around Monai.
•Spatial coverage [139.434, 139,499] 9
[42.0315, 42.0724].
•Number of cells: 584 9368 =214,912.
•Submesh 2: Aonae cape and Hamatsumae
region.
•Spatial coverage [139.411, 139.433] 9
[42.0782, 42.1455].
•Number of cells: 196 9604 =118,384.
•Level 3 (Monai region). Spatial coverage
[139.414, 139.426] 9[42.0947, 42.1033].
•Refinement ratio: 16.
•Number of cells: 1744 91248 =2,216,448.
•Resolution: 0.025 arc-sec (&0.75 m).
•Initial condition: generated by DCRC (Disaster
Control Research Center), Japan. Hipocenter depth
Figure 30
Level 1 nested mesh computational domain containing the two level 2 submeshes, the region to the South including Aonae and Hamatsumae
areas and the coastal region to the West containing Monai area, refined with one level 3 nested mesh around Monai Valley
Vol. 174, (2017) Performance Benchmarking of Tsunami-HySEA Model 3175
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Figure 31
Inundation map of the Aonae Peninsula. This is for t\12.59 min. The color map shows the maximum fluid depth over entire computation.
4-m contours of bathymetry and topography are shown
Figure 32
Zoom on the Aonae Peninsula showing the arrival of the first wave coming from the west at times t=4.75 min and t=5 min. We observe
that this first wave impacts the west coast of the Aonae Peninsula at time close to 5 min after the tsunami generation
3176 J. Macı
´as et al. Pure Appl. Geophys.
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37 km at 139.32E and 42.76N, M
w
7.8 (Taka-
hashi 1996) (Fig. 29, source model DCRC 17a).
•Topobatymetric data: Kansai University.
•CFL: 0.9.
•Version of the code: Tsunami-HySEA WAF.
3.5.2 Tasks to be Performed
To evaluate performance requirements for this
benchmark, the following tasks had to be performed:
1. Compute runup around Aonae.
2. Compute arrival of the first wave to Aonae.
Figure 33
Zoom on the Aonae peninsula showing the first wave arriving from the west at time t=5.25 min and the second wave coming from the east
at time t=9.75 min
Figure 34
Computed and observed water levels at two tide stations located along the west coast of Hokkaido Island, Iwanai in upper panel and Esashi in
lower panel. Observations from Yeh et al. (1996)
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3. Show two waves at Aonae approximately 10 min
apart; the first wave came from the west, the
second wave came from the east.
4. Compute water level at Iwanai and Esashi tide
gauges.
5. Maximum modeled runup distribution around
Okushiri Island.
6. Modeled runup height at Hamatsumae.
7. Modeled runup height at a valley north of Monai.
3.5.3 Numerical Results
In this section, the numerical results obtained with
Tsunami-HySEA for BP9 are presented.
3.5.3.1 Runup Around Aonae Figure 31 shows the
inundation level around Aonae peninsula. The
figure includes 4-m contours of bathymetry and
topography. The contours allow to determine that the
maximum runup height is below 12 m in the eastern
part of the peninsula where the tsunami inundation is
mainly produced by the second wave and where the
topography is flatter producing, despite the lower
runup, a further penetration. The opposite situation
occurs in the western part of the peninsula: a higher
runup ranging from 16 to 20 m, within a narrower
strip, mainly flooded by the first wave arriving from
the west. The southern part of the peninsula is inun-
dated with a runup height of 16 meters and a large
inundated area, suffering both impacts of the western
and eastern tsunami wave.
3.5.3.2 First Wave to Aoane Figure 32 shows the
arrival of the first wave, coming from the west, to the
Figure 35
Computed and observed runup in meters at 19 regions along the coast of Okushiri Island after 1993 Okushiri tsunami. Observations from Kato
and Tsuji (1994)
3178 J. Macı
´as et al. Pure Appl. Geophys.
Content courtesy of Springer Nature, terms of use apply. Rights reserved.

Aonae peninsula at times t=4.75 min and
t=5 min. From this figure, we can conclude the
time of arrival of this first wave to Aoane takes place
at approximately t=5 min. Within 15 s the wave is
close to reaching the western coastline of the penin-
sula and at t=5 min it has already impacted, from
north to south, along all the western seashore.
3.5.3.3 Waves Arriving to Aonae Figure 33 depicts
two snapshots of the arrival of two tsunami waves at
the Aonae peninsula. The first wave arrival, from the
west, is seen at about t=5 min, as is shown in
Fig. 32. The second major wave arrives from the east
at about 9.5 min. Snapshots at time t=5.25 min and
t=9.75 min are presented in Fig. 33.
3.5.3.4 Tide gauges at Iwanai and Esashi Fig-
ure 34 shows the comparison between the computed
and observed water levels at two tide stations located
along the west coast of Hokkaido Island, Iwanai and
Esashi. Besides, the maximum error in the maximum
wave amplitude and the normalized root mean square
deviation (NRMSD) is depicted for both time series.
The errors in the maximum amplitude, although high
(36 and 41%) are analogous to the mean of the
models collected in NTHMP (2012) report (36 and
43%, respectively). No values were given for the
NRMSD there.
3.5.3.5 Maximum Runup Around Okushiri Fig-
ure 35 shows a bar plot that compares model runup at
19 regions around Okushiri Island with measured
data. In the NTHMP-provided script, runup error is
Figure 36
Inundation map of the Hamatsumae neighborhood. For t\14 min. The color map shows the maximum fluid depth along the entire
simulation. 4-m contours of bathymetry and topography are shown
Table 11
Tsunami-HySEA model relative error with respect to field
measurement data for runup around Okushiri Island. Comparison
with average error values for models in NTHMP (2102). #OBS
gives the number of observations used to compute the error bars in
Fig. 37
Region Longitude Latitude #
OBS
HySEA
(%)
Mean
NTHMP
(%)
1 139.4292117 42.18818149 3 25 5
2 139.4111857 42.16276287 2 22 8
3 139.4182612 42.13740439 1 66 27
4 139.4280358 42.09301238 1 2 7
5 139.4262450 42.11655479 1 5 6
6 139.4237147 42.10041415 7 2 6
7 139.4289018 42.07663658 1 30 15
8 139.4278534 42.06546152 2 5 10
9 139.4515399 42.04469655 3
a
00
10 139.4565284 42.05169226 5
a
08
11 139.4720138 42.05808988 4 0 2
12 139.5150461 42.21524909 2 0 10
13 139.5545494 42.22698164 6–8 19 14
14 139.4934307 42.06450128 3 32 74
15 139.5474599 42.18744879 1 6 14
16 139.5258982 42.17101221 2 0 11
17 139.5625242 42.21198369 1 9 15
18 139.5190997 42.11305805 3 0 34
19 139.5210766 42.15137635 2 1 19
Mean 12 15
a
When one observed value has been skipped. NTHMP data taken
from Tables 1–11 b in p. 49
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evaluated by a comparison between computed and
measured sets of minimal, maximal, and mean runup
values in unspecified surroundings of prescribed ref-
erence points. We have searched for the set of
observations used for each region to compute the
minimal, maximal, and mean values (these are the
three values required for generating Fig. 35). For
each of these observed values the closer or the two
closer model values were considered for the compu-
tation of the minimum, maximum, and the average in
the given region. When only one measured data were
available in a region, then the three closer model
discretized points were taken (this means a larger
spread in simulated values than in measured data that
can be observed in regions 3, 4,5, 7, 15, and 17). This
procedure was used in all cases, but in the regions
with refined meshes (regions 6, 9, 10, and 11), where
all computed values were used.
In Table 11 the location of the points identifying
these 19 regions are gathered and for each region the
number of observations used to determine minimal,
maximal, and mean values are given in column
#OBS. Finally, this table also presents Tsunami-
HySEA runup error at each location compared with
the mean of models in NTHMP (2012). The main
question that arises when regarding this table is why
there are locations with such a good agreement with
observed data and for other regions the agreement is
so poor. First of all, we are dealing with discrete
observed values taken at locations that we do not
know why or how were chosen. This is a first source
of uncertainty. Second, bathymetry data resolution is
very inhomogeneous: where bathymetry data are
finer, closer model vs observed data comparisons are
obtained. Large errors are associated with low
bathymetry data resolution regions. Finally, numer-
ical resolution also varies, and besides it is finer in
regions with higher resolution bathymetry data and
coarser in regions with low-resolution bathymetry.
Besides, as pointed out in NTHMP (2012), the
Figure 37
Inundation map of the valley north of Monai. 4-m contours of bathymetry and topography are shown
3180 J. Macı
´as et al. Pure Appl. Geophys.
Content courtesy of Springer Nature, terms of use apply. Rights reserved.

accuracy of the seismic source being used and the
accuracy in some of the field observations and tide
gauges may also play an important role to explain the
observed discrepancies. A detailed study trying to
clarify these aspects is needed.
3.5.3.6 Runup Height at Hamatsumae Figure 36
shows the maximum inundation on the Hamatsumae
region computed for [0, 14] min. The color map
shows the maximum fluid depth along the entire
simulation. The figure also depicts 4-m contours of
bathymetry and topography. Maximum runups are
between 8 and 16 meters, with increasing values from
west to east.
3.5.3.7 Runup Height at a Valley North of
Monai Figure 37 shows the maximum inundation at
a valley north of Monai, computed for [0, 4.5] min.
The color map shows the maximum fluid depth along
the entire simulation. The 4-m contours of topogra-
phy allow to determine maximum runups that range
between 8 and 12 m to the south, around 16 to the
north and up to the 31.753 m of maximum computed
runup, very close to the observed value, 31.7 m.
4. Conclusions
The Tsunami-HySEA numerical model is vali-
dated and verified using NOAA standards and criteria
for inundation. The numerical solutions are tested
against analytical predictions (BP1, solitary wave on
a simple beach), laboratory measurements (BP4,
solitary wave on a simple beach; BP6, solitary wave
on a conical island; and BP7, runup on Monai Valley
beach), and against field observations (BP9, Okushiri
island tsunami). In the numerical experiments mod-
eling the propagation and runup of a solitary wave on
a canonical beach, numerical results are clearly below
the established errors by the NTHMP in their 2011
report. For BP1, the mean errors measured are below
1% in all cases. In the case of BP4, several conclu-
sions can be extracted. For the non-breaking case
with H=0.0185 the non-dispersive model produces
accurate wave forms with NRMSD errors, in most
cases, very close to the dispersive model results. For
the breaking wave case with H=0.30 it can be
observed that the shape of the (dispersive) wave
cannot be well captured by the non-dispersive model,
producing large NRMSD errors at the times when the
NLSW model tends to produce a shock. Nevertheless,
the agreement is still high for times when non-steep
profiles are present. Despite this (a dispersive model
is absolutely necessary if we want to accurately
reproduce the time evolution of the wave in the
breaking case) we have observed that measured runup
is accurately reproduced by both models in the two
studied cases. On the other hand, the dispersive ver-
sion of Tsunami-HySEA produces very good results
in both the breaking and non-breaking cases. For
BP6, dealing with the impact of a solitary wave on a
conical island, again non-dispersive and dispersive
Tsunami-HySEA models have been used. Wave
splitting and colliding are clearly observed. Numeri-
cal results are very similar for Case A (A/h=0.045)
and Case B (A/h=0.096) for wave shape. Larger
differences are evident in Case C (A/h=0.181),
where dispersive model performs better for wave
shape, but not for the computed runup. It is note-
worthy that the computed maximum runups for Cases
A and C are very close for both models but they
clearly differ for Case B. Tsunami-HySEA model
figures have been compared with figures in NTHMP
(2012), performing in general better than the mean
when comparing by class of model (dispersive and
non-dispersive). BP7, the laboratory experiment
dealing with the tsunami runup onto a complex 3D
model of the Monai Valley beach, was studied in
detail in (Gallardo et al. 2007). A mean value of
7.66% for the NRMSD is obtained for all the three
gauges for the times series simulating the first 30 s.
The snapshots of the simulation agree well with the
experimental frames and, finally, a maximum simu-
lated runup height of 0.0891 is obtained compared
with the 0.08958 experimentally measured. Com-
parison of BP9 with Okushiri island tsunami
observed data is performed using nested meshes with
two level 2 meshes located one in the South of the
island, covering Aoane and Hamatsumae areas and
the second one to the West containing Monai area.
Finally, one level 3 refined mesh is located covering
the Monai area. Computed runup and arrival times
are in good agreement with observations. Water level
time series at Iwanai and Esashi tide gauges show
Vol. 174, (2017) Performance Benchmarking of Tsunami-HySEA Model 3181
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large NRMSD and large errors in the maximum
amplitude (36 and 41% for ERR) but analogous to the
mean of the models in NTHMP (2012) (36 and 43%
for ERR). For the maximum runup at 19 regions
around Okushiri Island a mean error of 15% is
obtained, the same as the mean of models in NTHMP
(2012), with 10 regions with errors below 10%.
Regions located in areas with refined meshes perform
much better than regions located in coarse mesh
areas.
Acknowledgements
This research has been partially supported by the
Spanish Government Research project SIMURISK
(MTM2015-70490-C2-1-R), the Junta de Andalucı
´a
research project TESELA (P11-RNM7069), and
Universidad de Ma
´laga, Campus de Excelencia
Internacional Andalucı
´a Tech. The GPU and multi-
GPU computations were performed at the Unit of
Numerical Methods (UNM) of the Research Support
Central Services (SCAI) of the University of Malaga.
Open Access This article is distributed under the terms of the
Creative Commons Attribution 4.0 International License (http://
creativecommons.org/licenses/by/4.0/), which permits unrestricted
use, distribution, and reproduction in any medium, provided you
give appropriate credit to the original author(s) and the source,
provide a link to the Creative Commons license, and indicate if
changes were made.
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