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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 x2DR2

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;tðÞHðxÞthat

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:

Sx¼ghM2

nukðu;vÞk

h4=3;

Sy¼ghM2

nvkðu;vÞk

h4=3;

where Mn[0 is the manning coefﬁcient.

2. A quadratic law:

Sx¼cfukðu;vÞk;Sy¼cfvkðu;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.

Content courtesy of Springer Nature, terms of use apply. Rights reserved.

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

oxhu o2ghðÞ

oxþhohvðÞ

oyhv o2ghðÞ

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

g¼hH. 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

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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.

Content courtesy of Springer Nature, terms of use apply. Rights reserved.

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;0ðÞ¼Hsech2ðcðxX1Þ=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

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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 270Transect

9 10.36 13.80 8.2 270Transect

16 12.96 11.22 7.9 180Transect

22 15.56 13.80 8.3 90Transect

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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.

Content courtesy of Springer Nature, terms of use apply. Rights reserved.

37 km at 139.32E and 42.76N, 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)

Vol. 174, (2017) Performance Benchmarking of Tsunami-HySEA Model 3177

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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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