Optimal placement of tsunami sensors with depth constraint

Abstract

Tsunamis are destructive natural disasters that can cause severe damage to property and the loss of many lives. To mitigate the damage and casualties, tsunami warning systems are implemented in coastal areas, especially in locations with high seismic activity. This study presents a method to identify the placement of near-shore detection sensors by minimizing the tsunami detection time, obtained by solving the two-dimensional shallow water equations (SWE). Several benchmark tests were done to establish the robustness of the SWE model, which is solved using a staggered finite volume method. The optimization problem is solved using particle swarm optimization (PSO). The proposed method is applied to different test problems. As an application, the method is used to find the optimal location of a detection sensor using data from the 2018 Palu tsunami. Our findings show that detection time can
be significantly reduced through the strategic placement of tsunami sensors.

Full text

Optimal placement of tsunami sensors
with depth constraint
Ikha Magdalena
1
, Raynaldi La’lang
1
, Renier Mendoza
2
and
Jose Ernie Lope
2
1Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Bandung, West Java,
Indonesia
2Institute of Mathematics, University of the Philippines Diliman, Quezon City, Philippines
ABSTRACT
Tsunamis are destructive natural disasters that can cause severe damage to property
and the loss of many lives. To mitigate the damage and casualties, tsunami warning
systems are implemented in coastal areas, especially in locations with high seismic
activity. This study presents a method to identify the placement of near-shore
detection sensors by minimizing the tsunami detection time, obtained by solving the
two-dimensional shallow water equations (SWE). Several benchmark tests were
done to establish the robustness of the SWE model, which is solved using a staggered
finite volume method. The optimization problem is solved using particle swarm
optimization (PSO). The proposed method is applied to different test problems. As
an application, the method is used to find the optimal location of a detection sensor
using data from the 2018 Palu tsunami. Our findings show that detection time can
be significantly reduced through the strategic placement of tsunami sensors.
Subjects Optimization Theory and Computation, Scientific Computing and Simulation
Keywords Shallow water equations, Near-shore detection sensors, Particle swarm optimization
INTRODUCTION
Tsunami waves are, in most cases, generated by a seabed deformation caused by some
tectonic plate movement, which leads to a sudden displacement of water that travels from
the ocean to the shore. These waves may also be generated by landslides, submarine
volcanic eruptions, and meteorological disturbances (Joseph, 2011;Okal & Synolakis,
2015). Unlike tidal waves, they characteristically have longer wavelength and amplitudes
that start relatively small but builds up and undergoes shoaling as they reach shallower
areas.
In the past 5 years alone, there have been five major tsunami occurrences: in 2016 in
Kaikoura, New Zealand (Heidarzadeh & Satake, 2017), in 2017 in Greenland (Chao et al.,
2018), in 2018 in Sulawesi, Indonesia (Heidarzadeh, Muhari & Wijanarto, 2018), in
2019 in Sunda Strait, Indonesia (Grilli et al., 2019), and in 2020 in the Aegean Sea
(Triantafyllou et al., 2021). The most devastating was the earthquake-triggered-with-
underwater-landslide tsunami in Sulawesi, which killed more than 1,000 and injured over
600. These regions are among those with the highest tsunami risk due to their proximity to
tectonic plate boundaries. To minimize the damage and casualties of tsunamis in these
high-risk areas, it is necessary to have carefully designed countermeasures, such as tsunami
warning systems.
How to cite this article Magdalena I, La’lang R, Mendoza R, Lope JE. 2021. Optimal placement of tsunami sensors with depth constraint.
PeerJ Comput. Sci. 7:e685 DOI 10.7717/peerj-cs.685
Submitted 12 May 2021
Accepted 1 August 2021
Published 29 September 2021
Corresponding author
Ikha Magdalena,
ikha.magdalena@math.itb.ac.id
Academic editor
Mehmet Cunkas
Additional Information and
Declarations can be found on
page 25
DOI 10.7717/peerj-cs.685
Copyright
2021 Magdalena et al.
Distributed under
Creative Commons CC-BY 4.0

Seismic-centered tsunami warning systems typically work in three stages (UN-ESCAP,
2009). As the seismic network detects seismic waves, it sends a signal to seismologists in the
warning center. This data is then analyzed to assess whether or not the seismic waves
have the potential to generate a tsunami, including the threat level. The responsible
party then issues the warning to the public and decides on which action to take. While the
exact chain of actions may differ from one country to another, it still follows these general
steps (Murjaya, 2012;Doi, 2003).
An example of a tsunami warning system is the Deep-ocean Assessment and Reporting
of Tsunamis (DART) (Mungov, Eble & Bouchard, 2012;Bernard & Meinig, 2011;Paros
et al., 2011). It comprises three parts: bottom pressure recorders (BPRs), surface buoys,
and satellites. Each BPR is anchored to the seafloor to record barometric pressure and
temperature. The data are then logged to the buoy as the BPRs read the average water level.
Finally, the data are transmitted to the warning center by satellite. Another type of tsunami
warning system are near-shore gauges that are used in alerting high-risk coastal
communities in case of local tsunamis (Satake, 2014;E2S Warning Signals, 2021). These
systems include wave gauges that use ultrasonic waves and buoy and measure offshore sea
levels at water depths of 50 to 200 m (Satake, 2014).
Numerous studies on tsunami warning systems have been undertaken by researchers
from countries located in the Pacific ring of fire. Mulia & Satake (2020) present the
evolution of past to present tsunami observing systems available in Japan. The optimal
placement of sensors in Korea was studied by Lee, Jung & Shin, 2020a,Lee, Jung & Shin
2020b. Meanwhile, the use of the rainfall optimization algorithm in the placement of
sensors in the Cotabato Trench, Philippines was investigated by Ferrolino, Lope &
Mendoza (2020), and an integrated tsunami forecast and warning system, called SIPAT,
has been developed and proven successful in Chile (Catalan et al., 2020).
A warning system’s performance depends on the ability to optimize the function of
each subsystem, and there are many aspects to look at. There have also been numerical
methods developed to reproduce the tsunami generation and propagation (Behrens, 2010;
Cecioni et al., 2014). Wang & Li (2008) introduced a tsunami warning system that utilizes
remote sensing and geographical information systems in monitoring, forecasting,
detection, loss evaluation, and relief management. Mulia et al. (2020) proposed an
enhanced detection system using airborne platforms, and a 3D topology design for
underwater sensor networks was proposed by Lohan, Dube & Agrawal (2020).
Several researchers have proposed methods of optimizing the location of buoys and
gauges. Lee, Jung & Shin (2020a) used the Cornell Multi-grid Coupled Tsunami
(COMCOT) numerical model and a probabilistic approach to gauge the optimal region for
tsunami detection instruments in the eastern sea of Korea. Meza, Catalán and Tsushima
(2019) implemented an inversion algorithm to determine the optimal array configuration
of offshore tsunami sensors for near-field tsunami forecasting based on three tsunami
parameters: arrival time, maximum tsunami amplitude, and forecast skill. Navarrete
et al. (2020) used empirical orthogonal function analysis together with a heuristic
optimization technique to find the optimal locations of a network of tsunameters. Recently,
Wu, Chen & Ghattas (2021) presented a fast and scalable computational framework for
Magdalena et al. (2021), PeerJ Comput. Sci., DOI 10.7717/peerj-cs.685 2/28

finding sensor locations to maximize the expected information gain for a predicted
quantity of interest. These results, however, are not quite general as they are calibrated to
fit specific bathymetric profiles, tsunami characteristics, or detection sensors. Ferrolino
et al. (2020) used two-dimensional shallow water equations (SWE) to compute the travel
time of tsunami waves and particle swarm optimization (PSO) to find the optimal
placement of tsunami sensors. The waves were made to propagate over a wet bed in their
simulations, and no particular constraints were specified. However, in real-life scenarios,
several restrictions must be satisfied in configuring the detection sensors. For example,
near-shore warning systems must be installed in the shallower portion of the sea. To make
the system cost-effective, the gauge must be placed within a prescribed water depth and on
locations where the tsunami can be detected at the earliest possible time.
The objective of this study is to address the limitations in the simulations done by
Ferrolino et al. (2020). First, we use nonlinear SWE with a wet-dry procedure in modeling
the tsunami wave propagation so that the model also applies to a water domain with
islands. Second, we modify the optimization problem that calculates the optimal placement
of sensors by adding a constraint on water depth. This can be used in finding the strategic
locations of near-shore tsunami sensors. For example, in the Philippines, tsunami early
warning systems are installed near the shore (DOST, DOST-PHILVOLCS, DOST-ASTI,
2021).
The numerical solution to the two-dimensional SWE was obtained using the finite
volume method (FVM) on a staggered grid, which has been shown to be robust, accurate,
and inexpensive (Pudjaprasetya & Magdalena, 2014;Magdalena, Rif’atin & Reeve, 2020;
Magdalena, Erwina & Pudjaprasetya, 2015). The arising sensor location problem is a
nonlinear programming because of the constraints. We use a penalty method to transform
the nonlinear programming problem into an unconstrained optimization problem for this
work. Since the resulting cost function is not smooth, we employ PSO, an evolutionary
optimization algorithm that does not require the computation of gradients. It was initially
formulated to simulate social behavior in colonies (Kennedy & Eberhart, 1995), but has
since then been explored to solve problems in broader fields, including product design and
manufacturing (Zhou et al., 2006), smart antenna (Wagih, Elkamchouchi & Lazinica,
2009), and of course tsunami detection (Ferrolino et al., 2020).
Following this introductory section are three parts. The following section discusses the
numerical solution of the SWE using FVM on a staggered grid and the constrained
optimization problem. The third section showcases the robustness and accuracy of the
proposed algorithm to solve the fluid dynamics model numerically. The section also
features our implementation of the PSO algorithm to solve the problem of tsunami
detection, first applied to several test problems and finally to actual data obtained from the
2018 Palu tsunami incident. The final section contains our concluding remarks.
METHODS
Fluid dynamics model
This section discusses the SWE, one of many models in fluid dynamics that can be used to
model fluid motions. Due to their required assumptions, they are especially accurate in
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solving problems involving the ocean (Briganti & Dodd, 2009;Brocchini & Dodd, 2008;Le
Roux, Staniforth & Lin 1998). They can also be used to simulate different tsunami waves,
as done by Geyer & Quirchmayr (2017) and Zergani, Aziz & Viswanathan (2015).
The SWE arise from a special case of the Navier–Stokes equations, which describe the
conservation of mass and momentum. In the case of the SWE, the wavelength of the
disturbance is assumed to be long, relative to the depth of the flow. Due to this assumption,
the horizontal velocity along a vertical column can be assumed to be uniform, hence it is
only a function of the horizontal spatial parameters, namely xand y.
Shallow water equations
The equations in one dimension are obtained by taking a vertical cross-section (in this
case, a y-plane cross-section) of some two-dimensional domain and neglecting flows not
parallel to the cross-section. Assuming uniform fluid density, incompressible-irrotational
flow, and frictionless bed, the set of equations in one dimension is
htþðhuÞx¼0;(1)
utþggxþuux¼0;(2)
where his the total water depth, uis the horizontal velocity, and gis the gravitational
acceleration (throughout this study, the value of gis set to 9.81 m.s
−2
). We also use the
variables dfor water depth and ηfor the free surface elevation, giving the relation h=d+η.
Here, Eq. (1) represents the conservation of mass, while Eq. (2) represents the momentum
balance. Figure 1 illustrates the variables just described.
To extend the model to two dimensions, we will have two equations for momentum
balance in x- and y-direction. We would then need one more variable: vto denote the
horizontal velocity in the y-direction. We also introduce the advection terms vu
y
and uv
x
in
the momentum-conservation-equations. Thus the 2D governing equations read as:
htþðhuÞxþðhvÞy¼0;(3)
utþggxþuuxþvuy¼0;(4)
vtþggyþvvyþuvx¼0:(5)
All the variables used in the two-dimensional SWE are illustrated in Fig. 2.
Figure 1 Illustration of variables used in the 1D SWE. Full-size
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DOI: 10.7717/peerj-cs.685/fig-1
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Finite volume method on a staggered grid
We apply the finite volume method on a staggered grid to solve the SWE numerically. In a
staggered grid arrangement, ηis stored in the cell centre, while vectors uand vare stored in
the cell faces. We use a structured rectangular grid to partition the spatial domain into
intervals of equal size Dxin one dimension, or rectangles of equal size Dx×Dyin two
dimensions; integer indices (x
i
,y
j
) represent the cell centers while half-integer indices
ðxiþ1
2
;yjþ1
2Þrepresent the cell faces. These are visualized in Figs. 3 and 4. We denote by t
n
the discretization of the time dimension.
Note that computing the spatial derivatives of p=hu and q=hv is not as
straightforward, as they are products of two variables stored using two different indices.
We approximate the values using the following upwind scheme to compute hat the cell
faces, based on the direction of the flow:
hiþ1
2¼
hi;uiþ1
2
.0;
hiþ1;uiþ1
20:
((6)
Figure 2 Illustration of variables used in the 2D SWE. Full-size
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DOI: 10.7717/peerj-cs.685/fig-2
Figure 3 Staggered grid for the FVM scheme on the 1D SWE.
Full-size
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DOI: 10.7717/peerj-cs.685/fig-3
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We then define piþ1
2¼hiþ1
2
uiþ1
2
. Now, to compute uux
ðÞ
n
iþ1
2
, we take note of the
following equality:
uux¼p
hux:(7)
This is a modification of the method earlier used by Magdalena, Erwina &
Pudjaprasetya (2015) and is preferred here because the calculation of the advection term is
simpler and less costly. Next, the value of hat half-integer indices and the value of pand u
at integer indices are computed as
hiþ1
2¼1
2hiþhiþ1
ðÞ;(8)
pi¼1
2piþ1
2þpi1
2

;(9)
ui¼
ui1
2
;pi.0
uiþ1
2
;pi0:
((10)
These quantities yield the following approximation of ðuuxÞn
iþ1
2
:
uux
ðÞ
iþ1
2¼
piþ1
2
hiþ1
2
uiþ1ui
Dx

;(11)
Figure 4 Staggered grid for the FVM scheme on the 2D SWE.
Full-size
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Thus, we obtain this discretization of the one-dimensional SWE:
hnþ1
ihn
i
Dtþpn
iþ1
2pn
i1
2
Dx¼0;(12)
unþ1
iþ1
2un
iþ1
2
Dtþggnþ1
iþ1gnþ1
i
Dxþuux
ðÞ
n
iþ1
2¼0:(13)
Extending the finite volume method on SWE to two dimensions require us to compute
vuy

n
iþ1
2;jvvy

n
i;jþ1
2and uvx
ðÞ
n
i;jþ1
2
. We only give the details in approximating the first
expression, as the others can be done similarly. This time, we use the equalities
vuy¼q
huy;vvy¼q
hvy;and uvx¼p
hvx:(14)
Analogous to the first advection term, we do an approximation for the following
variables:
hiþ1
2;j¼1
2hi;jþhiþ1;j

;(15)
qiþ1
2;jþ1
2¼1
2qiþ1;jþ1
2þqi;jþ1
2

;(16)
uiþ1
2;jþ1
2¼
uiþ1
2;j;qiþ1
2;jþ1
2
.0
uiþ1
2;jþ1;qiþ1
2;jþ1
2
<0:
((17)
Thus, we have these approximations:
vuy

iþ1
2;j¼
qiþ1
2;j
hiþ1
2;j
uiþ1
2;jþ1
2uiþ1
2;j1
2
Dy
!
;(18)
vvy

i;jþ1
2¼
qi;jþ1
2
hi;jþ1
2
vi;jþ1
2vi;j1
2
Dy
!
;(19)
uvx
ðÞ
i;jþ1
2¼
pi;jþ1
2
hi;jþ1
2
viþ1
2;jþ1
2viþ1
2;j1
2
Dx
!
;(20)
and, finally, we obtain this discretization of the two-dimensional SWE:
hnþ1
i;jhn
i;j
Dtþ
pn
iþ1
2;jpn
i1
2;j
Dxþ
qn
i;jþ1
2qn
i;j1
2
Dy¼0;(21)
unþ1
iþ1
2;jun
iþ1
2;j
Dtþggnþ1
iþ1;jgnþ1
i;j
Dxþuux
ðÞ
n
iþ1
2;jþvuy

n
iþ1
2;j¼0;(22)
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vnþ1
i;jþ1
2vn
i;jþ1
2
Dtþggnþ1
i;jþ1gnþ1
i;j
Dyþvvy

n
i;jþ1
2þuvx
ðÞ
n
i;jþ1
2¼0:(23)
Proposed approach in Tsunami sensor placement
The sensors should be placed in locations where detection time is minimized. We now
discuss the optimization formulation arising from this real-world application. We modify
the minimization problem proposed by Ferrolino et al. (2020),Ferrolino, Lope & Mendoza
(2020) by adding a constraint on the water depth.
Optimization problem
Suppose we have source points R¼fr1;...;rkg, where tsunami waves are generated. Let
sbe an arbitrary point in the spatial domain. We define
sri;s
ðÞ
:¼the time it takes for a wave generated from a source point
rito reach the point s:(24)
To compute sri;sðÞ, we use the two-dimensional SWE presented previously. Suppose
we wish to place lsensors, r¼fs1;...;slg. Since only one sensor is needed to be triggered
for the tsunami to detected, the time it takes for the tsunami to be detected from a source r
i
is given by
tðri;rÞ:¼min
1jlsri;sj

:
Thus, the guaranteed tsunami detection time to reach at least one of the sensors
r¼fs1;...;slgfrom any of the possible sources is given by
TðrÞ¼max
1iktðri;rÞ:(25)
Suppose the sensors can be placed anywhere in the water domain D, then the goal is to
minimize (25). This minimization problem was also used in previous studies (Ferrolino,
Lope & Mendoza, 2020,Ferrolino et al., 2020;Astrakova et al., 2009).
In this work, we introduce a constraint by considering the scenario where the
sensors can only be placed on locations with depth constraint, as in the case of near-shore
tsunami sensors. If we denote by Dthe points in Dthat are within the depth constraint
(see Fig. 5), then the minimization problem we consider is given by
min
rTðrÞ;
subject to r2D:
(26)
The constrained domain Dcan be defined as D={(x,y)∈D:d(x,y)≤d
max
}, where
d
max
is the maximum depth in which a sensor can be placed. The constrained optimization
problem (26) can be reformulated as an unconstrained problem using the penalty method,
that is, by considering the problem
Magdalena et al. (2021), PeerJ Comput. Sci., DOI 10.7717/peerj-cs.685 8/28

min
rTðrÞþlvDnDðrÞ;(27)
where μis the penalty parameter and vDnD(σ) is the characteristic function (or indicator
function) of D\D, whose value is 0 if σ∈Dand 1 otherwise. We set μequal to a
large number to penalize those sensor locations that do not satisfy the depth constraint. In
our simulations, we set μ=10
6
.
It should be noted that this formulation can be easily modified to consider cases when
the tsunami does not originate from a point source (e.g., when the tsunami is due to a
landslide). The minimization problem remains the same but the initial conditions of the
SWE will have to be modified. Likewise, this approach can also be used to determine the
location of deep-ocean tsunami sensors by either setting the depth constraint to a high
value or the value of μto 0.
Particle swarm optimization
As discussed previously, the function τin (24) is computed by solving the two-dimensional
SWE numerically. However, this only gives us the values of τat points located at the cell
centers of the rectangular grid. To calculate the travel time at other locations, we use
bilinear interpolation using the values of τat the surrounding points.
Bilinear interpolation is done by performing linear interpolation in one direction, and
then performing it again in the other direction, on every rectangle [x
i
,x
i+1
]×[y
j
,y
j+1
].
For example, suppose we want to interpolate τusing some function f(x,y) on the unit
square [0, 1] × [0, 1], given its values at the points (0, 0), (1, 0), (0, 1), and (1, 1). Then f
(x,y) is given by
fðx;yÞ¼fð0;0Þð1xÞð1yÞþfð1;0Þxð1yÞþfð0;1Þð1xÞyþfð1;1Þxy;
ðx;yÞ2½0;12:(28)
An illustration of this is provided in Fig. 6.
Figure 5 This is an example of a domain D. The tsunami sources are shown in red and the
set of points that are within the depth constraint is labeled D.
Full-size
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Despite its name, as seen above, the resulting interpolant is not linear; rather, it is the
sum of products of linear functions in xand y, respectively. Hence, the whole function is
piecewise quadratic and may not be differentiable at the cell edges. Furthermore, the
constrained optimization problem is transformed into an unconstrained problem using a
penalty method. The penalty term in (27) includes a characteristic function, which is
binary and not differentiable. In view of this, it is fitting to use optimization methods that
do not rely on gradients. In this work, we explore the use of derivative-free metaheuristic
algorithms in solving the minimization problem.
Metaheuristic algorithms have gained popularity because they only rely on function
evaluations and have the capability of obtaining the global minimizer (Yang, 2014). It was
shown in Ferrolino et al. (2020) how PSO was effective in solving the tsunami location
problem without depth constraint. We adopt the same algorithm for our problem
formulation.
The PSO algorithm starts by randomly generating a swarm of solution candidates
(particles) with their initial velocity vector. Each particle then move around the search-space
according to three parameters: its velocity vector, the vector pointing at its own best-known
position, and the vector pointing at the swarm’s best-known solution. For each iteration,
the step that each particle takes is a linear combination of these three vectors (see Fig. 7),
expecting that the swarm eventually gathers around one point—the optimal solution.
The pseudocode of the algorithm is presented in Algorithm 1. The algorithm starts by
generating swarm
size
particles randomly in the search space D, and each their own initial
Figure 6 The values of the interpolants (red points) are used to calculate the bilinear interpolation
(surface plot) using Equation (28).Full-size
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velocity vector v. They all then follow a series of instructions based on computations
provided in Algorithm 1, where the vectors p(each particle’s best known location) and g
(the entire swarm’s best known location) is updated. This process terminates when either
the maximum number of iterations (max
iter
) is reached, or the change in the objective
value fis less than min
func
. In this study, the function fis the one presented in Eq. (26);
and the value of each hyperparameters are swarm
size
= 100, max
iter
= 100, min
func
=10
−8
,
ω=ϕ
p
=ϕ
g
= 0.5.
More detailed discussions of this method may be found in Kennedy & Eberhart (1995);
Shi & Eberhart (1998),Mezura-Montes & Coello (2011). Our implementation of the
PSO algorithm made use of the built-in command PySwarms, an extensible research
toolkit for particle swarm optimization in Python.
RESULTS
Benchmark test and validation of SWE
In order to get an idea of the numerical model’s robustness and accuracy, some benchmark
tests are performed. The model is considered using different types of initial conditions,
boundary conditions, and bottom profiles. These simulations test the performance of
the SWE.
1D standing wave on a flat bottom
A standing wave, simply put, is a wave that does not travel (horizontally). Rather, it is an
oscillating wave fixed in space. At any given point in space, the peak amplitude of wave
oscillations on a standing wave is constant. This phenomenon can be observed on the
surface of a liquid in a vibrating container.
To produce a standing wave simulation in one dimension, we set up a container of
length L, constant still water depth d; give it initial conditions gðx;0Þ¼cosðpx
LÞ,u(x,0)=0
Figure 7 Illustration of the particle swarm optimization.
Full-size
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and wall boundary conditions u(0, t)=u(L,t) = 0. The result of the simulation on a flat
bottom with length L= 25 m and depth of 4 m, using Dx= 0.2 m, is shown in Fig. 8.
The wave seemingly does not travel horizontally, as a result of interference between
two waves traveling in opposite directions. The water level at any point coscillates in time
with amplitude |η(c, 0)|. The analytical solution to this phenomenon is equivalent to
that of a string of infinite length with some wave number kand angular frequency ω.In
linear and non-dispersive case here, these numbers are just k¼p
Land x¼kffiffiffiffiffi
gd
p, which
gives
gðx;tÞ¼cos p
Lx

cos p
Lffiffiffiffiffi
gd
pt

:(29)
Simulating this at Dx= 0.2 m, Dt= 0.032 s for 20 s gives a mean absolute error of 0.015
m. To simplify things, all simulations below assume hard wall boundary conditions and no
current, unless stated otherwise.
1D wave run-up on a sloping beach
The first nonlinear simulation is wave run-up, which refers to either the phenomenon
or the measure of when an ocean wave reaches a beach or sea dike structure and rises above
still water level. Obviously, as a wave approaches the beach structure, the water depth gets
smaller relative to the amplitude, hence the advection term becomes more significant.
Algorithm 1 Particle swarm optimization.
input:D(search space), f(objective function)
output:g(swarm’s best known location)
hyperparameters: swarm
size
,max
iter
,min
func
,ω,ϕ
p
,ϕg
begin
let swarm generateSwarmðswarmsizeÞ;
for each particle ∈swarm do set its location as the best known position;
let g arg minpfðpÞp:each particle’s best known position
for i 1to max
iter
do
for particle ∈swarm do
let x,v,pas the particle’s position, current velocity, best known position;
let rp;rg randomð0;1Þ;
v xvþrpfpðpxÞþrgfgðgxÞ;
x xþv;
if f(x)<f(p)then p x;
end
g arg minpfðpÞ;
end
end
Magdalena et al. (2021), PeerJ Comput. Sci., DOI 10.7717/peerj-cs.685 12/28

Furthermore, in this simulation the wave interacts with the dry region, something that was
not observed in the previous simulations.
This simulation is based on an experiment by Synolakis (1987) on a solitary wave that
propagates through a single slope. The solitary wave centered at x=x
0
has an initial surface
profile gðx;0Þ¼H
dsech2cxL
4

with c¼ffiffiffiffiffi
3H
4d
q. This way, similar to the step bottom
simulation, the right traveling soliton eventually changes shape, this time as it climbs
the sloping beach. This behavior shown in Fig. 9, where the simulation was done with Dx=
0.1 m and Dt= 0.016 s. Several gauges were placed during the experiment to record the
wave height every 10 s. We then compared these records to the simulation. As seen in
Fig. 9, the model was able to predict the experiment to a pretty high accuracy.
1D dam break over a flat, dry bed
Dam break, as the name suggests, happens when the dam holding a reservoir breaks,
causing a sudden rush of water flowing to the other side. The set up of this simulation is on
aflat bed with 1-m-deep water on one side of the domain (reservoir), and completely dry
on the other side:
hðx;0Þ¼ H;x,x0;
0;x.x0;
(30)
where x
0
represents the location of the barrier. Right after the simulation starts, the
previously contained water will stream to the dry side and start to fill it, while the
water level on the reservoir goes down. The result of the simulation on a 20 m-long,
1 m-deep bed with Dx= 0.1 m is provided in Fig. 10. To test the robustness of the
numerical scheme, we compare it with the analytical solution given by Chanson (2008):
Figure 8 Comparison between numerical and analytical solutions of the 1D standing wave on a flat
bottom, at x = 15 m. Full-size

DOI: 10.7717/peerj-cs.685/fig-8
Magdalena et al. (2021), PeerJ Comput. Sci., DOI 10.7717/peerj-cs.685 13/28

hðx;tÞ¼
H;x,ct;
H
9
2x
ct

2
;ct x,ð2c1Þt;
8
>
<
>
:
(31)
where c¼ffiffiffiffiffiffi
gH
p.AtDx= 0.1 m, Dt= 0.008 s, the numerical scheme gives a mean absolute
error of 0.0034 m.
2D planar surface wave over a parabolic basin
So far the simulations were set on rectangular or trapezoidal shapes. To avoid bias due to
the also rectangular shape of the grids, this simulation will be set on a parabolic basin.
This way, not only do we have a curved bottom profile, all of the perimeter will be
interacting with dry regions. The results of this simulation further confirms the
performance of the wet-dry procedure.
The shape of the basin follows the parabolic function dðx;yÞ¼d01ðx2þy2Þ=L2
ðÞon
an 8,000 m × 8,000 m domain, with L
2
= 8,000 m
2
. The large parabolic shape of the
domain is supposed to mimic that of a lake. We would like to see the motion of the water
surface initialized by gðx;y;0Þ¼2a0d0
Lðx
Ln
2LÞ. The analytical solution of the planar surface
wave obtained by Thacker (1981) is
Figure 9 Comparison between Synolakis’experiment and the numerical solution of the 1D wave
run-up on a sloping beach. Full-size

DOI: 10.7717/peerj-cs.685/fig-9
Magdalena et al. (2021), PeerJ Comput. Sci., DOI 10.7717/peerj-cs.685 14/28

gðx;y;tÞ¼2nd0
Lx
Lcos xty
Lsin xtn
2L;x¼1
Lffiffiffiffiffiffiffiffiffi
2gd0
p:(32)
The parameters used in the simulation were d
0
=1m,ξ= 400 m using Dx=Dy=40m,
and Dt= 4.5 s. As expected, the planar surface in the parabolic basin oscillates with period
T¼2p=x¼2pL=ffiffiffiffiffiffiffi
gd0
p, which can be seen in Fig. 11. Quantitatively, the numerical
solution agrees with the analytical solution with mean absolute error of 0.029 m.
Validation with tsunami-related experiments
Finally, the performance of staggered finite volume scheme for the SWE will be tested on
tsunami-related events. To do this, the model is tested on two experiments: “solitary wave
on a conical island”and “tsunami run-up onto a complex 3D beach.”
Figure 10 Comparison between the analytical and numerical solutions of the 1D dambreak over a
flat, dry bed. Full-size

DOI: 10.7717/peerj-cs.685/fig-10
Magdalena et al. (2021), PeerJ Comput. Sci., DOI 10.7717/peerj-cs.685 15/28

Tsunami run-up onto a conical island
This experiment was done at the Coastal Engineering Research Center, Vicksburg,
Mississippi, as part of a research (Briggs et al., 1995) on a tsunami that struck Babi Island,
Indonesia in 1992. The shape of Babi Island is similar to a truncated cone, as seen in
Fig. 12, and the experiment may serve as a benchmark test for fluid dynamic models in
modeling interaction between waves and such structure. It was conducted on a 25 m ×
30 m basin with a conical island at the center, and waves with an initial solitary wave-like
profile were generated from one side of the tank. The resulting water surface elevations
were measured by 22 gauges. Our numerical simulation was ran with Dx=Dy= 0.2 m.
The results of the simulation were compared with data gathered at 4 of the 22 gauges: G6,
Figure 11 Cross section along y = 0 of the 2D planar surface wave over a parabolic basin.
Full-size

DOI: 10.7717/peerj-cs.685/fig-11
Figure 12 Satellite view of Babi Island (left); conical island experiment setup (right).
Full-size

DOI: 10.7717/peerj-cs.685/fig-12
Magdalena et al. (2021), PeerJ Comput. Sci., DOI 10.7717/peerj-cs.685 16/28

G9, G16, and G22. As seen in Fig. 13, the model was able to replicate the experiment
with great accuracy.
Tsunami run-up onto a complex 3D beach
We then move on to an experiment with a much more complex bathymetry, similar to that
found in nature. The experiment was based on a tsunami that struck Okushiri Island,
Japan, with a really high current, that resulted in an extreme run up at the tip of a very
narrow gulley within a small cove at Monai. It was done at the Central Research Institute
for Electric Power Industry (CRIEPI) in Abiko, Japan on a 1:400 laboratory model of
Monai (Liu, Yeh & Synolakis, 2008) and the setup in shown in Fig. 14. Just like the previous
experiment, a series of waves were generated from the deeper side of the water. The results
of our simulation, with Dx=Dy= 0.025 m, were compared with the recorded height
on the three gauges placed during the experiment. As can be observed in Fig. 15, the model
was able to replicate the experiment with high accuracy.
OPTIMAL PLACEMENT OF DETECTION SENSORS
Having established the robustness of the shallow-water model, we now solve the
optimization problem under different scenarios. This section is divided into three parts.
Figure 13 Comparison between the numerical solution of the tsunami run-up onto aconical island
and data obtained from the laboratory experiment. Full-size

DOI: 10.7717/peerj-cs.685/fig-13
Magdalena et al. (2021), PeerJ Comput. Sci., DOI 10.7717/peerj-cs.685 17/28

First, we investigate the case when there is only one sensor available. In the next part, we
look at how the first case compares with having multiple detection sensors. Finally, we
implement the proposed method using data from the 2018 Palu tsunami incident.
Figure 14 Monai laboratory model (left); contour of Monai laboratory model (right).
Full-size

DOI: 10.7717/peerj-cs.685/fig-14
Figure 15 Comparison between the numerical solution of the tsunami run-up onto a complex
3D beach and data obtained from the laboratory experiment.
Full-size

DOI: 10.7717/peerj-cs.685/fig-15
Magdalena et al. (2021), PeerJ Comput. Sci., DOI 10.7717/peerj-cs.685 18/28

One detection sensor
Step bottom
The first simulation was done on a step bathymetry of size 1,000 m × 1,000 m with d=20
m for x< 500 m, and d= 10 m elsewhere. Waves were generated from five evenly
spaced points along the line x= 200 m, as seen in Fig. 16. The feasible region for this
problem is {(x,y): d(x,y)≤10 m}, which is the whole right side of the domain.
The result obtained is pretty intuitive, that is, the optimal location of detection sensor in
this case is right in the middle of the line where the domain changes depth. If located
far up on the y-axis, it will take long for the sensor to detect waves coming from source
points further down (and vice versa). If placed too far to the right, waves from any source
point will take a longer time to reach the sensor.
Sloping bottom
The next simulation was done on a sloping bathymetry of size 10,000 m × 10,000 m, with a
slope of 0.1 and maximum depth of 1,000 m. This time the waves were generated through a
flip-fault motion from six subduction zones in the shape of arcs of the semicircle
xþ5000 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
50002ðy5000Þ2
q, as seen in Fig. 17. The feasible region for this
problem is {(x,y): d(x,y)≤d
max
,x<−5,000}, with d
max
= 900, 800, 700, 600 m. Since in this
case the water depth gradually gets smaller as we approach the shoreline on the right side,
it is interesting to investigate what happens if the depth constraints are varied.
As seen in Fig. 17, not all sensors go to their respective depth constraint. This is expected
because the bottom topography is not flat. As predicted, the sensors obtained with varying
depth constraints lie on the bisector of the semicircle. As the water depth increases, the
location of the sensors move away from the shore and closer to the subduction zones.
Figure 16 Optimal location of one detection sensor on a step bottom.
Full-size

DOI: 10.7717/peerj-cs.685/fig-16
Magdalena et al. (2021), PeerJ Comput. Sci., DOI 10.7717/peerj-cs.685 19/28

However, the sensor cannot get too close to the semicircle to guarantee detection time
from all the possible sources. This can be seen on the obtained locations of sensors for
depths 800 and 900, i.e., the two locations are almost identical. This is possible because the
feasible region for d
max
= 800 m is contained in the feasible region for d
max
= 900 m.
Multiple detection sensors
Step bottom
Using the same setup as the previous one, the PSO algorithm was run to solve the
optimal location of two detection sensors. As seen in Fig. 18, the obtained locations of the
sensors are on the boundary of the feasible region. This is expected because the sensors
must be placed as close as possible to the source points and at the same time must satisfy
the depth constraint. It also makes sense that neither sensor was located right in the
middle, since one sensor can account for waves coming from the upper side while the other
can account for waves coming from the lower side.
Sloping bottom
In this example, the optimal locations of sensors are more spread out to account for
waves coming from different angles. The results are illustrated in Fig. 19. Obviously,
increasing the number of sensors will always decrease the wave detection time. As shown
in Fig. 20, using two sensors instead of one allows the waves to be detected about 37 s
earlier or 68% faster. However, using three or more sensors would be less efficient, as this
only decreases the detection time by 1 s or even less.
Sloping bottom with an island
This simulation starts with the same setup as the sloping bottom simulation, only this time
we introduce a small island in the middle of the domain. In this case, we expect to see
Figure 17 Optimal location of one detection sensor on a sloping bottom with varying depth
constraints. Full-size

DOI: 10.7717/peerj-cs.685/fig-17
Magdalena et al. (2021), PeerJ Comput. Sci., DOI 10.7717/peerj-cs.685 20/28

interesting results on small depth constraints since there are two “shallow”areas here: the
center island and the coastline on the right. It can be seen in Fig. 21 that sensors with depth
constraints of 500, 700 m are not really affected by the island. However for maximum
depth of 300 and 100 m, the pair of sensors are spread apart, with one approaching the
island and the other getting closer to the coastline on the right. Because the domain is
symmetric with respect to the line y= 5,000 m, two solutions are obtained. This is
illustrated in Fig. 21. This configuration could not be tackled in Ferrolino et al. (2020)
Figure 18 Optimal location of two detection sensors on a step bottom.
Full-size

DOI: 10.7717/peerj-cs.685/fig-18
Figure 19 Optimal location of two detection sensors on a sloping bottom with varying depth
constraints. Full-size

DOI: 10.7717/peerj-cs.685/fig-19
Magdalena et al. (2021), PeerJ Comput. Sci., DOI 10.7717/peerj-cs.685 21/28

because the presence of an island in the simulation requires the use of the nonlinear SWE
with wet-dry procedure.
The simulations we have presented so far were formulated so that the optimal
solutions can be verified based on the geometry of the subduction zone and the domain.
The obtained results using PSO yield solutions that agree with the expected geometric
solutions. These tests were done to check that the method works. In a real-world scenario,
the optimal placement of sensors is not trivial because the subduction zones do not
necessarily follow a geometric pattern. Additionally, the bottom topography of oceans is
uneven, making the computation of travel time more complicated. Because we use the 2D
nonlinear SWE in simulating tsunami waves, our proposed method can handle
complex bathymetric profiles and arbitrary tsunami sources. In our following and last
Figure 20 Effects of placing multiple detection sensors on wave detection time (d
max
= 700 m).
Full-size

DOI: 10.7717/peerj-cs.685/fig-20
Figure 21 Two obtained solutions for the optimal placement of two sensors on asloping bottom with
an island subject to varying depth constraints. Full-size

DOI: 10.7717/peerj-cs.685/fig-21
Magdalena et al. (2021), PeerJ Comput. Sci., DOI 10.7717/peerj-cs.685 22/28

example, we apply our tsunami sensor detection model to one of the more recent tsunami
incidents: the 2018 Palu tsunami.
Application on real events: 2018 Palu tsunami
On 28 September 2018, Central Sulawesi was struck by a 7.5-Richter-scale earthquake,
which was followed by a tsunami sweeping through the city of Palu.
To simulate the tsunami, we use a gridded bathymetry of Palu Bay provided by
BATNAS Indonesia, with a grid size of 185 m × 185 m. Landslides are generated from two
subduction zones that were proposed by Heidarzadeh, Muhari & Wijanarto (2018) using
backward tsunami ray tracing, see Fig. 22.
Solving the optimization problem using PSO on this simulation without constraints
found the optimal sensor placement at depth d= 595.48 m with coordinates: longitude =
119.81583, latitude = −0.74583. As can be seen in Fig. 23, this location is approximately in
the middle of the region bounded by the two subduction zones. Adding some more
depth constraints, from 450 to 150 m, consistently moved the sensor closer to the shore.
Assuming no constraints, placing the detection sensor according to this simulation
would allow the waves coming from either side to be detected in just about 93 s. This
shaves off up to 240 s compared to the case when the sensor was placed more to the south,
closer to Talise Beach, which would then take either 113 s or 333 s depending on the
source. If a depth constraint of 150 m is applied, the optimally located sensor would still be
able to detect the tsunami in about 106 s, saving up to 227 s.
Figure 22 Filled contour plot of Palu Bay. The possible sources of the tsunami are landslides along
the lines (in red) as proposed by Heidarzadeh, Muhari & Wijanarto (2018).
Full-size

DOI: 10.7717/peerj-cs.685/fig-22
Magdalena et al. (2021), PeerJ Comput. Sci., DOI 10.7717/peerj-cs.685 23/28

CONCLUSION
To conclude, this paper has accomplished two significant results. First, we have derived a
robust and accurate numerical shallow water model (SWE) for simulating water waves in
different situations. The wet-dry procedure that are included in the calculation can
simulate wave propagates over a dry area. Secondly, we have integrated this shallow
water model in conjunction with the meta-heuristic PSO algorithm to solve various
optimization problems of placing tsunami detection sensors, including the 2018 Palu
tsunami incident. The latter findings proved that strategic placement of detection sensors
for tsunami warning systems can drastically improve detection time, giving the responsible
parties more time to evacuate the citizens.
In the previous work by Ferrolino et al., 2020, the tsunami sensors can be placed
anywhere in the water domain. Their results yield the placement of sensors near the
subduction zones for faster detection time. However, subduction zones are usually located
in the deep ocean, making the placement of sensors costly. In the Philippines, tsunami
warning systems are installed near the shores. Because of this depth constraint, we modify
the optimization problem proposed in Ferrolino et al. (2020). We use a penalty method to
solve the constrained minimization problem, which is a generalization of the tsunami
detection problem presented in Ferrolino et al. (2020). The arising minimization problem
is solved using PSO, a robust and easy-to-implement global optimization algorithm.
The benchmark tests for the tsunami sensors location problem are formulated so that
the optimal solution can be inferred geometrically. This way, we can assess if the obtained
Figure 23 Optimal location of one detection sensor, with varying depth constraints, based on 2018
Palu tsunami incident. Full-size

DOI: 10.7717/peerj-cs.685/fig-23
Magdalena et al. (2021), PeerJ Comput. Sci., DOI 10.7717/peerj-cs.685 24/28

solution by PSO is correct. Our numerical results agree with the expected solution.
However, it is still interesting to know what happens if other numerical optimization
algorithms are used. A comparative analysis of recent metaheuristic optimization
algorithms applied to tsunami sensor detection problems merits a separate study.
In our test simulations, the possible tsunami sources are evenly placed along the
subduction zone. However, in reality, some places in the subduction zone may be more
tectonically active than others. This is a limitation of the current study. For future research,
one can use time-series data of earthquakes within the subduction zone and use a
clustering algorithm to identify coordinates along the zone that will most likely be a
tsunami source.
ADDITIONAL INFORMATION AND DECLARATIONS
Funding
This work was supported by Institut Teknologi Bandung. The funders had no role in study
design, data collection and analysis, decision to publish, or preparation of the manuscript.
Grant Disclosures
The following grant information was disclosed by the authors:
Institut Teknologi Bandung.
Competing Interests
The authors declare that they have no competing interests.
Author Contributions
Ikha Magdalena conceived and designed the experiments, performed the experiments,
analyzed the data, performed the computation work, prepared figures and/or tables,
authored or reviewed drafts of the paper, and approved the final draft.
Raynaldi La’lang conceived and designed the experiments, performed the experiments,
analyzed the data, performed the computation work, prepared figures and/or tables,
authored or reviewed drafts of the paper, and approved the final draft.
Renier Mendoza conceived and designed the experiments, performed the experiments,
analyzed the data, performed the computation work, prepared figures and/or tables,
authored or reviewed drafts of the paper, and approved the final draft.
Jose Ernie Lope analyzed the data, authored or reviewed drafts of the paper, and
approved the final draft.
Data Availability
The following information was supplied regarding data availability:
The data used in benchmark tests are available at the NOAA website:
1. Conical Island https://nctr.pmel.noaa.gov/benchmark/Laboratory/Laboratory_
ConicalIsland/index.html
2. Monai Valley https://nctr.pmel.noaa.gov/benchmark/Laboratory/Laboratory_
MonaiValley/
Magdalena et al. (2021), PeerJ Comput. Sci., DOI 10.7717/peerj-cs.685 25/28

The bathymetry data for Palu simulation are available in http://batnas.big.go.id/
pencarian?kategori=Batimetri&keyword=palu/
The Python code used to run the simulations are available at GitHub: https://github.
com/raynaldilalang/SWE-PSO.
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