Application of particle swarm optimization in optimal placement of tsunami sensors

Abstract

Rapid detection and early warning systems demonstrate crucial significance in tsunami risk reduction measures. So far, several tsunami observation networks have been deployed in tsunamigenic regions to issue effective local response. However, guidance on where to station these sensors are limited. In this article, we address the problem of determining the placement of tsunami sensors with the least possible tsunami detection time. We use the solutions of the 2D nonlinear shallow water equations to compute the wave travel time. The optimization problem is solved by implementing the particle swarm optimization algorithm. We apply our model to a simple test problem with varying depths. We also use our proposed method to determine the placement of sensors for early tsunami detection in Cotabato Trench, Philippines.

Full text

Application of particle swarm
optimization in optimal placement of
tsunami sensors
Angelie Ferrolino
1
, Renier Mendoza
1
, Ikha Magdalena
2
and
Jose Ernie Lope
1
1Institute of Mathematics, University of the Philippines Diliman, Quezon City, Philippines
2Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Bandung, Indonesia
ABSTRACT
Rapid detection and early warning systems demonstrate crucial significance in
tsunami risk reduction measures. So far, several tsunami observation networks have
been deployed in tsunamigenic regions to issue effective local response. However,
guidance on where to station these sensors are limited. In this article, we address the
problem of determining the placement of tsunami sensors with the least possible
tsunami detection time. We use the solutions of the 2D nonlinear shallow water
equations to compute the wave travel time. The optimization problem is solved by
implementing the particle swarm optimization algorithm. We apply our model to
a simple test problem with varying depths. We also use our proposed method to
determine the placement of sensors for early tsunami detection in Cotabato Trench,
Philippines.
Subjects Optimization Theory and Computation, Scientific Computing and Simulation
Keywords Particle swarm optimization, Nonlinear shallow water equations, Tsunami sensors,
Tsunami early warning system, Heuristic algorithm, Finite volume method
INTRODUCTION
While not the most prevalent among all natural disasters, tsunamis rank higher in scale
compared to any others because of its destructive potential. Tsunamis are a series of
ocean waves prompted by the displacement of a large volume of water. They can be
generated by earthquakes, landslides, volcanic eruptions and even meteor impacts,
although they mostly take place in subduction zones caused by underwater earthquakes.
The sudden motion created by the said disturbances gives an enormous shove to the
overlying water, which then travels away from the source.
Tsunamis usually have small initial amplitudes, so they can go unnoticed by sailors.
However, they have far longer wavelengths unlike normal sea waves. Since the rate at
which a wave loses its energy is inversely proportional to its wavelength, tsunami lose little
energy as they propagate. Therefore, out in the depths of the ocean, tsunamis can travel at
high speeds and cross great distances with only little energy loss (Liu, 2009). As they
approach shallower waters, they slow down, and begin to grow in height. Once they crash
ashore, they can cause widespread property damage, destruction of natural resources, and
human injury or death.
Given the devastating potential of tsunamis, it is of great importance to develop
techniques regarding risk assessment and damage mitigation. There are a number of
How to cite this article Ferrolino A, Mendoza R, Magdalena I, Lope JE. 2020. Application of particle swarm optimization in optimal
placement of tsunami sensors. PeerJ Comput. Sci. 6:e333 DOI 10.7717/peerj-cs.333
Submitted 7 August 2020
Accepted 18 November 2020
Published 18 December 2020
Corresponding author
Renier Mendoza,
rmendoza@math.upd.edu.ph
Academic editor
Marcin Woźniak
Additional Information and
Declarations can be found on
page 16
DOI 10.7717/peerj-cs.333
Copyright
2020 Ferrolino et al.
Distributed under
Creative Commons CC-BY 4.0

methods focused on these in literature (Goda, Mori & Yasuda, 2019;Park et al., 2019;
Goda & Risi, 2017). In this research, our goal is to determine the optimal locations of
sensors for the earliest detection of tsunami waves. Through this, we can enable early and
effective response to reduce casualties.
Traditionally, tide gauges, typically placed in a pier, are used for tsunami detection.
They are employed to determine trends in the mean sea level, tidal computation, and
harbor operations and navigation. However, since they are located in areas where
tsunamis still have very high energy, they are frequently destroyed and fail to report the
incoming tsunami and its characteristics. Furthermore, as they are near the coast, tsunami
confirmation may arrive too late for timely evacuation measures which may lead to an
ineffective local response (Barrick, 1979). Hence, we instead consider seafloor-mounted
sensors, particularly the bottom pressure recorders (BPRs). These sensors are capable
of detecting tsunamis, even in open oceans, by measuring variations in hydrostatic bottom
pressure. They use an acoustic coupling to transmit data from the seafloor to the surface
buoys, which then relay via satellite the information to a land-based station (Eble &
Gonzalez, 1990). Further details on these tsunami sensor networks can be found in
Rabinovich & Eblé (2015) and Nagai et al. (2007).
Studies on where to optimally place sensors of tsunami observation networks are
limited. Some considered various technical aspects supported by expert judgments such as
financial limitations (Araki et al., 2008) and legal aspects on geographical boundaries
(Abe & Iwamura, 2013). Attempts to maximize the accuracy of prediction of some tsunami
parameters (e.g., source, amplitude, etc.) were proposed in Mulia, Gusman & Satake
(2017),Meza, Catalán & Tsushima (2020),Mulia et al. (2017),Saunders & Haase (2018).
There are also researches that incorporated the effectiveness of tsunami warnings.
In Omira et al. (2009),Schindelé, Loevenbruck & Hébert (2008), possible locations of
sensors while considering installation constraints were investigated. However, the
researchers did not apply optimization algorithms to solve the problem. In Braddock &
Carmody (2001) and Groen, Botten & Blazek (2010), the optimal location of sensors among
the candidate detection sites were proposed. In their investigations, they did not take
into account bathymetric data since a fixed wave travel speed was used.
Astrakova et al. (2009) developed a problem of placing sensors optimally to detect
tsunami waves as early as possible. However, the algorithm of computation for the
wave travel time (from a source point to a point of interest) is based on the Huygen’s
principle, which is computationally expensive. In Ferrolino, Lope & Mendoza (2020),
wave velocity approximation was based on wave front kinematics. In this article, we
compute for the wave travel time using the 2D nonlinear shallow water equations (SWE).
This approach allows us to more accurately compute the tsunami detection time.
The 2D nonlinear SWE is generally used in the numerical simulation of tsunami
propagation from the open ocean to the coast (Ulutas, 2012). They are accurate for
solving long wave propagation (i.e., waves whose vertical length scale is much greater
than their horizontal length scale), and run-up and inundation problems due to the
introduction of nonlinear terms, which are essential in tsunami modeling. Moreover,
Ferrolino et al. (2020), PeerJ Comput. Sci., DOI 10.7717/peerj-cs.333 2/21

they can take into account many types of tsunami generation (Zergani, Aziz &
Viswanathan, 2015;LeVeque, George & Berger, 2011). In solving the 2D nonlinear SWE,
we use the finite volume method in a staggered grid, with a momentum conservative
scheme, as introduced in Pudjaprasetya & Magdalena (2014),Stelling & Duinmeijer
(2003),Magdalena, Rif’atin & Reeve (2020),Magdalena, Iryanto & Reeve (2018),
Andadari & Magdalena (2019),Magdalena, Pudjaprasetya & Wiryanto (2014), and
Magdalena & Rif’atin (2020). This method is very effective and robust because it is well-
balanced, conservative, and capable of handling complex bathymetry and topography.
We apply the Particle Swarm Optimization (PSO) in solving the optimal sensor location
problem. This is a population-based algorithm inspired from the intelligent collective
behavior of animal groups such as flocks of birds. Studies show that these animals are
capable of sharing information among their group, and such ability grants them a survival
advantage (Kennedy & Eberhart, 1995;Shi & Eberhart, 1998). PSO has gained much
attention in solving optimization problems because it is easy to implement,
computationally inexpensive, robust, and quick in convergence.
We first test our proposed approach on benchmark problems before using it to
determine the optimal placement of sensors in the Cotabato trench, located in Mindanao,
Philippines. The Philippines is vulnerable to tsunamis due to the presence of offshore faults
and trenches (Valenzuela et al., 2020). To date, a total of 41 tidal waves classified as
tsunamis have struck the country since 1589 (Bautista et al., 2012). Moreover, compared to
neighboring countries, it has received less attention in tsunami research despite having
regions with high seismic activity (Løvholt et al., 2012).
The remainder of this article is structured as follows: the methods implemented to solve
the problem of optimal sensor placement are discussed in the “Methodology”.“Numerical
Results”obtained on different domains are then presented. Finally, we conclude our
work and provide suggestions for future research.
METHODOLOGY
The tsunami sensor optimal placement problem
Consider the domain with water areas , and the subduction zone P. Let Dbe the part of
the domain where the sensors can be stationed. A sample illustration is shown in Fig. 1.
Our goal is to determine the optimal placement of Lsensors for the earliest detection
of tsunami waves, arising from any source point in P. We denote {p
j
}
Pj=1
as the points
located in the subduction zone P, where p
j
=(x
j
,y
j
). Moreover, let {q
i
}
Li=1
denote the
sensors to be placed in D, where q
i
=(x
i
,y
i
). We also let the configuration Q={q
1
,…,q
L
}of
Lsensors represent a potential solution to the problem of interest.
Assume a source point p
j
∈Pgenerates some perturbation. The wave caused by such
event will move away from the source, arriving at some water point x∈D. We denote
τðpj;xÞ:¼travel time from pjto x:(1)
To calculate the travel time τin (1) from p
j
to an arbitrary point x∈D, we solve the 2D
nonlinear SWE by initiating the wave from p
j
. The time it takes for the wave to reach xwill
Ferrolino et al. (2020), PeerJ Comput. Sci., DOI 10.7717/peerj-cs.333 3/21

be τ(p
j
,x). The numerical solution of the 2D nonlinear SWE is discussed in the next
section. The time it takes for Qto detect a tsunami from p
j
is computed as
tðpj;QÞ¼ min
1iL
sðpj;qiÞ:
Here, we take the minimum travel time over all Lsensors since we want detection by
one sensor to be as early as possible. Next, we need to take into account all possible source
points p
j
to obtain the tsunami registration time T(Q) by configuration Q, that is, we set
TðQÞ¼ max
1jPtðpj;QÞ;
where the maximum of the earlier expression is taken over all source points p
j
. Thus,
we formulate the optimization problem as follows:
Given Lsensors, find Q={q
1
,…,q
L
}that minimizes T(Q) over D, or equivalently,
Q¼arg min
Q2D
TðQÞ:(2)
The minimization problem above is based on the formulation in Astrakova et al. (2009).
The shallow water equations
Shallow water equations are a set of hyperbolic partial differential equations that describe
fluid flow (Sadaka, 2012). They are often used to describe geophysical flows (such as
rivers, lakes, coastal areas, etc.), rainfall runoff, tsunami propagation, or even atmospheric
flows, given the appropriate initial conditions and source terms (Backhaus, 1983;Cea &
Bladé, 2015;Rahman & Mandokhail, 2018;Galewsky, Scott & Polvani, 2004;Fu et al.,
2017;Liu et al., 2008,1995). SWE are derived from the Navier–Stokes equations, with the
main assumption that the horizontal length scale is much greater than the vertical length
scale. The 2D nonlinear shallow water equations are given by
Figure 1 Illustration of a domain W with parts of water D (blue) and subduction zone P (red).
Full-size
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DOI: 10.7717/peerj-cs.333/fig-1
Ferrolino et al. (2020), PeerJ Comput. Sci., DOI 10.7717/peerj-cs.333 4/21

htþðhuÞxþðhvÞy¼0
ðhuÞtþhu2þ1
2gh2

x
þðhuvÞy¼ghzx
ðhvÞtþðhuvÞxþhv2þ1
2gh2

y
¼ghzy
Here, his the local water depth, u,vare the velocities in the xand ydirections, respectively,
g= 9.81 m/s
2
is the gravitational acceleration, and ∇zis the bed profile. An equivalent
form of this system is given by
htþðhuÞxþðhvÞy¼0 (3)
utþuuxþvuyþghx¼0 (4)
vtþuvxþvvyþghy¼0 (5)
derived by setting h=η+zwhere ηis the surface elevation. Note that we can rewrite uu
x
,
vu
y
,uv
x
, and vv
y
in terms of
u
q=hu,
v
q=hv,u,v, and h, that is
uuxþvuy¼1
hððuquÞxuqxuÞþ1
hððvquÞyvqyuÞ(6)
uvxþvvy¼1
hððuqvÞxuqxvÞþ1
hððvqvÞyvqyvÞ:(7)
A compilation of the analytical solutions of the shallow water equations (both 1D
and 2D) for different cases are presented in Delestre et al. (2013). In this article, we solve
the 2D nonlinear SWE numerically by implementing the finite volume method in a
staggered grid with a momentum conservative scheme (Pudjaprasetya & Magdalena, 2014;
Stelling & Duinmeijer, 2003), which will be discussed below. The solution of the SWE
provides us the value of hat any mesh point in the domain in a given time. Hence, we can
determine the time it takes for a disturbance to reach any mesh point in the domain
for the first time (i.e., minimal time), which is the unknown in (1). We will use linear
interpolation to determine wave travel time at any given point, using the three closest mesh
points to this point.
The Finite Volume Method (FVM) is a method in solving partial differential equations
(PDEs) by evaluating the conservative variables across the volume (LeVeque, 2002).
The idea is to divide the domain into parts (see Fig. 2), which we will refer to as
cells/control volumes, then integrate the equation over that volume. We then use Gauss’
Theorem to transform the volume integral into a surface integral. Evaluating this integral
will give us the FVM discretization of a PDE.
Finite Volume Method has been extensively used in various fields such as heat and
mass transfer, gas dynamics and fluid flow problems (Guedri et al., 2009,Lukáčová-
Medvid’ová, Morton & Warnecke, 2002;Demirdžić& Perić, 1990). Its popularity as a
Ferrolino et al. (2020), PeerJ Comput. Sci., DOI 10.7717/peerj-cs.333 5/21

discretization method stems from its high flexibility. The discretization is carried out
directly in the physical domain without the need of transformation between the physical
and computational system. Furthermore, it is easier to implement in unstructured meshes
in comparison to other methods. Another important feature of FVM is that it preserves
conservation, making it quite attractive when modeling problems where flux is of
importance (Moukalled, Mangani & Darwish, 2016).
We will use a staggered grid for the finite volume method. In a staggered grid, the scalar
variables (mass, pressure, density, etc.) are defined at the cell centers, while the velocity or
momentum variables are defined at the center of the cell faces. This is different from a
collocated grid where all variables are stored in the same position (Harlow & Welch, 1965).
Figure 3 shows the 2D staggered grid.
The implementation of FVM on a staggered grid on SWE is described as follows.
Consider a rectangular spatial domain (0, L
x
) × (0, L
y
) with hard wall boundary conditions,
that is, u(0, y,t)=u(L
x
,y,t) = 0 and v(x,0,t)=v(x,L
y
,t) = 0. The domain is meshed with a
grid of M×Ncells. The center of each cell is denoted by x
i,j
, while the center of the bottom
edge, top edge, left edge, and right edge of each cell are denoted by xi;j1
2
,xi;jþ1
2
,xi1
2;j,
and xiþ1
2;j, respectively. Note that the sizes of h,u, and vare M×N,M×(N+ 1), and
(M+1)×N, respectively.
The approximation of the mass Eq. (3) at the cell centered at x
i,j
is
dhi;j
dt þ
hiþ1
2;juiþ1
2;jhi1
2;jui1
2;j
Dxþ
hi;jþ1
2vi;jþ1
2
hi;j1
2
vi;j1
2
Dy¼0
with upwind approximations
Figure 2 Mesh for the 2D finite volume method. Different schemes include (A) cell-vertex scheme;
(B) cell-centered scheme. Full-size
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DOI: 10.7717/peerj-cs.333/fig-2
Ferrolino et al. (2020), PeerJ Comput. Sci., DOI 10.7717/peerj-cs.333 6/21

hiþ1
2;j¼
hi;j;if uiþ1
2;j0
hiþ1;j;if uiþ1
2;j,0;
8
<
:
hi;jþ1
2
¼
hi;j;if vi;jþ1
2
0
hi;jþ1;if vi;jþ1
2
,0:
8
<
:
The approximation of the momentum Eq. (4) is implemented at the cell centered at
xiþ1
2;j, and (5) at the cell centered at xi;jþ1
2
. We use the relations (6) and (7) for the
momentum conservative approximation of the advection terms. For positive flow
directions u>0,v> 0, we have
duiþ1
2;j
dt þuqi;j
hiþ1
2;j
uiþ1
2;jui1
2;j
Dx
!
þ
vqi;j1
2
hiþ1
2;j
uiþ1
2;juiþ1
2;j1
Dy
!
þghiþ1;jhi;j
Dx¼0
dvi;jþ1
2
dt þ
uqi1
2;j
hi;jþ1
2
vi;jþ1
2
vi1;jþ1
2
Dx
!
þvqi;j
hi;jþ1
2
vi;jþ1
2
vi;j1
2
Dy
!
þghi;jþ1hi;j
Dy¼0;
where
hiþ1
2;j¼1
2hiþ1;jþhi;j;hi;jþ1
2
¼1
2hi;jþ1þhi;j;
uqi;j¼1
2uqiþ1
2;jþuqi1
2;j;uqiþ1
2;j¼hiþ1
2;juiþ1
2;j;
vqi;j¼1
2vqi;jþ1
2
þvqi;j1
2;vqi;jþ1
2
¼hi;jþ1
2
vi;jþ1
2
;
Figure 3 Illustration of the 2D staggered grid for the finite volume method.
Full-size
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DOI: 10.7717/peerj-cs.333/fig-3
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vqi;j1
2
¼hi;j1
2
vi;j1
2
;vi;j1
2
¼1
2viþ1;j1
2
þvi1;j1
2;
uqi1
2;j¼hi1
2;jui1
2;j;ui1
2;j¼1
2ui1
2;jþ1þui1
2;j1:
Explicit schemes require a careful selection of the time step to fulfill the stability of the
equation. One of the observations of Courant, Friedrichs & Lewy (1967) is that in order for
solutions of a difference equation to converge to the solution of a partial differential
equation, the difference scheme should use all the information contained in the initial data
that influence the solution. This condition is called the CFL condition. The CFL number
(also known as the Courant number) is given by
c¼Dtmaxðx;
yÞ
minðDx;DyÞ
To satisfy the CFL condition, the grid speed, given by Dx
Dtand Dy
Dt, must be at least as large
as the propagation speed in xand y, given by λ
x
and λ
y
, respectively (Toro, 1999;Lax,
2013), that is, 0 ≤c≤1 holds. Thus, for the stability of SWE using FVM scheme on a
staggered grid (Gunawan, 2015), we define the time step as follows:
Dt¼cminðDx;DyÞ
max
i;j1
2hi;juqiþ1
2;jþuqi1
2;jþffiffiffiffiffiffiffiffi
ghi;j
q;1
2hi;jvqi;jþ1
2
þvqi;j1
2þffiffiffiffiffiffiffiffi
ghi;j
q:
The above numerical scheme has been validated through several benchmark tests
in 1D and 2D (e.g., dam breaks, transcritical flows, wave shoaling), where the
obtained numerical results are in very good agreement with the analytical solutions
(Pudjaprasetya & Magdalena, 2014;Magdalena, Iryanto & Reeve, 2018;Magdalena,
Erwina & Pudjaprasetya, 2015;Magdalena & Pudjaprasetya, 2014;Magdalena,
Pudjaprasetya & Wiryanto, 2014).
The particle swarm optimization algorithm
It was illustrated in Ferrolino, Lope & Mendoza (2020) that (2) is neither convex nor
differentiable. Thus, gradient-based and local search algorithms may not be suitable to
solve the problem. This justifies the use of population-based algorithms in determining the
optimal locations.
For this work, we explore the use of the PSO algorithm (Kennedy & Eberhart, 1995;
Shi & Eberhart, 1998) to solve the optimization problem presented in (2). This algorithm
has been used to solve several optimization problems because of its speed and efficacy.
It has been successfully used to solve unconstrained and constrained problems in a variety
of fields such as engineering, communications theory, and operations research (Zhang,
Wang & Ji, 2015). Other applications are in solving problems in clustering analysis
(Chen & Ye, 2004), scheduling (Yu, Yuan & Wang, 2007;Zhang et al., 2014), facility
location (Hajipour & Pasandideh, 2012) and vehicle routing (Belmecheri et al., 2013).
Ferrolino et al. (2020), PeerJ Comput. Sci., DOI 10.7717/peerj-cs.333 8/21

Particle Swarm Optimization is a population-based algorithm having a group of
individuals (known as particles) moving in a search space to find the best solutions.
The new positions of the particles are determined by their velocities, which are updated
based on their own experience and the experience of neighboring particles. Through this,
PSO can balance exploitation and exploration (He & Wang, 2007).
Algorithm 1 summarizes the steps of PSO, with some alterations proposed in Mezura-
Montes & Coello (2011);Pedersen (2010). We use the MATLAB built-in command
particleswarm in our implementations.
NUMERICAL RESULTS
We now present the results of our numerical simulations. The proposed method was
first tested on a rectangular domain with a semicircular subduction zone, and then later
applied to a real-world problem. Because PSO is heuristic, ten independent runs were
Algorithm 1 Particle swarm optimization algorithm.
Input: npop: swarm size, v: initial particle velocities within the range [−r,r], nvars: number of variables
Output: best particle location
1: Initialization. Create particles at random of size npop within bounds. Also, initialize inertia ωand Nto nbor = max(2,npop×frac), where nbor is the
minimum neighborhood size and frac is the minimum neighbors fraction. Set the stall counter sc =0.
2: Evaluate the cost function ffor all particles. Set best = min (f(p
i
)) where p
i
is the position of particle i. In the latter iterations, p
i
will be the position of
the best objective function that particle ihas found. Let loc be the location such that best =f(loc).
3: For each particle iin the swarm at position x
i
, do the following:
Select a subset Sof Nparticles at random. Determine the best cost function value cfbest(S), and the position xbest(S)ofcfbest(S).
Update the particle’s velocity to v=ωv+w
1
u
1
(p−x)+w
2
u
2
(g−x), where vis the previous velocity, u
1
,u
2
are uniformly (0,1) distributed
random vectors of size nvars, and w
1
,w
2
are the weights for exploitation and exploration, respectively.
Update the particle’s position to x=x+v(implement thee bounds if necessary).
Calculate the cost function F=f(x). Set p=xif F<f(p).
4: Let F= min F(k) out of all kparticles in the swarm. If F<best, set best =Fand loc =x.
5: If the best cost function value is lowered, set val = 0. Otherwise, val =1.
6: Update N.Ifval =0:
Set sc = max(0, sc −1).
Set Nto nbor.
If sc < 2, set ω=2ω.Ifsc > 5, set ω=ω/2.
If val =1:
Set sc =sc +1.
Set N= min(N+nbor,npop).
7: Terminate if the stopping criterion has been satisfied. Else, go back to 3.
Ferrolino et al. (2020), PeerJ Comput. Sci., DOI 10.7717/peerj-cs.333 9/21

implemented for each scenario that was considered, and we present here the average values
of the results of these runs. Moreover, we set the population to 50 * L, where Lis the
number of sensors, and the number of function evaluations to 10,000.
Semicircle subduction zone with flat bottom
Consider the same domain shown in Fig. 1. Here, Dis the water surface with constant
depth and the rest is dry land. The optimal location of L=1,…, 4 sensors are shown in
Fig. 4. Note that for the case when L= 1, the optimal location of the sensor converged
to the center of the circle. Moreover, as we employ more sensors, they tend to move closer
to the subduction zone, and mimic its shape.
Now, we study how tsunami detection time will be influenced by changes in the number
of sensors. Figure 5 plots the relationship between these two variables. We can see that
detection time decreases as the number of sensors increases. Moreover, we can see from
Table 1 that when we allocate additional sensors from 4, there is no significant difference
Figure 4 Optimal placement of sensors for the domain in Fig. 1.The bottom topography is assumed to
be constant. Different numbers (L) of sensors are presented: (A) L= 1; (B) L= 2; (C) L= 3; (D) L=4.
Full-size
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DOI: 10.7717/peerj-cs.333/fig-4
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compared to other cases. Hence, L= 4 sensors may be adequate in giving us fast time
registration, without introducing too much costs.
Semicircle subduction zone with inverse tangent bottom
In this subsection, we examine how a different bathymetry will affect the optimal location
of tsunami sensors. Specifically, we wish to investigate how the locations will change if
there will be variations in the water depth. For this purpose, we consider an inverse tangent
bottom topography as shown in Fig. 6. The optimal location of L= 1 and 2 sensors
acquired using the inverse tangent bottom in comparison to the optimal locations attained
using a flat bottom are presented in Fig. 7.
Figure 5 Comparison of the tsunami detection time for different number of sensors for the domain
shown in Fig. 1.The bottom topography is assumed to be constant.
Full-size
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DOI: 10.7717/peerj-cs.333/fig-5
Table 1 Improvement in the detection time of tsunami waves due to the increase in number (L)of
sensors for the domain shown in Fig. 1.The bottom topography is assumed to be constant.
LDetection time (s) Improvement in time due to an additional sensor
1 3.84183 –
2 2.51604 1.32579
3 1.46286 1.05318
4 0.88702 0.57584
5 0.87654 0.01048
6 0.87565 0.00089
Ferrolino et al. (2020), PeerJ Comput. Sci., DOI 10.7717/peerj-cs.333 11/21

Observe in Fig. 7A that the sensor moves towards the shallower area. This phenomenon
happens since waves travel slower in shallower areas, caused by the force exerted on them
by the seabed. A similar trend can be observed for the two sensors as shown in Fig. 7B.
Observe that the sensor in the right moves upwards and towards the shallower area.
However, this will make tsunami detection time from the source points near the center
longer. Thus, to accommodate these source points, the sensor on the left move rightwards
and downwards. Note that this will not greatly affect tsunami detection time from the
source points in deeper areas since waves travel much faster there.
Optimal placement of sensors in Cotabato trench
The Cotabato trench is considered as one of the most dangerous major fault zones in the
Philippines since it has high tsunamigenic potential. It is located near a southwestern coast
Figure 6 A bottom topography using the inverse tangent function. The bathymetric profile is shown in (A) while the contour plot is illustrated
in (B). Full-size
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DOI: 10.7717/peerj-cs.333/fig-6
Figure 7 Comparison of the optimal location of L= 1 (A) and L= 2 (B) tsunami sensors using two
different types of water depth: flat bottom topography (black) and inverse tangent bottom
topography (magenta). Full-size
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DOI: 10.7717/peerj-cs.333/fig-7
Ferrolino et al. (2020), PeerJ Comput. Sci., DOI 10.7717/peerj-cs.333 12/21

of Mindanao, Philippines. Some examples of tsunamigenic earthquakes that occured in
this trench are the 1918 Celebes Sea earthquake (M 8.0), the 1976 Moro Gulf earthquake
(M 8.1), and the 2002 Mindanao earthquake (M 7.5) (Stewart & Cohn, 1979;Bautista et al.,
2012).
Consider a portion of the Cotabato Trench as shown in Fig. 8. The top left corner has
latitude 6.3142° and longitude 124.1083°, while the bottom right corner has latitude
5.6158° and longitude 124.8067°. The subduction zone Pis illustrated by the red line,
Figure 8 Top view of a portion of the Cotabato Trench. The red curve is the location of the subduction
zone. Full-size
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DOI: 10.7717/peerj-cs.333/fig-8
Figure 9 The bottom topography of Cotabato Trench. The bathymetric profile is shown in (A) while the contour plot is illustrated in (B).
Full-size
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DOI: 10.7717/peerj-cs.333/fig-9
Ferrolino et al. (2020), PeerJ Comput. Sci., DOI 10.7717/peerj-cs.333 13/21

and Dis the water surface above the subduction zone. The corresponding bathymetric
profile of this trench is shown in Fig. 9. We obtained the bathymetric profile of
Cotabato trench from the General Bathymetic Charts of the Oceans (GEBCO),
https://www.gebco.net/.
The placements of L=1,…,6 sensor/s with minimum guaranteed time are illustrated in
Fig. 10. Note here that as we employ more sensors, they become distributed along the
subduction zone. The plot describing the relationship between detection time and the
sensor’s number is shown in Fig. 11, while their corresponding values are presented
in Table 2. Observe that increasing the number of sensors from 6 only shows little
improvement in detection time. So, L= 6 sensors may be adequate in giving us fast time
registration, without introducing additional costs.
Figure 10 Optimal placement of the tsunami sensors in Cotabato Trench. Different numbers (L)of
sensors are presented: (A) L= 1; (B) L= 2; (C) L= 3; (D) L= 4; (E) L= 5; (F) L=6.
Full-size
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DOI: 10.7717/peerj-cs.333/fig-10
Ferrolino et al. (2020), PeerJ Comput. Sci., DOI 10.7717/peerj-cs.333 14/21

CONCLUSION AND RECOMMENDATIONS
The problem of placing sensors optimally is considered to provide the earliest detection of
tsunami waves, which leads to effective warning and response by local communities.
A more accurate calculation of wave travel time is proposed, which is done by solving
the 2D nonlinear shallow water equations. Moreover, the PSO algorithm was implemented
to solve the minimization problem. The proposed model was first tested to simple domains
with varying bathymetries, comprising of a semicircle subduction zone. The calculated
sensor locations for the benchmark problems are geometrically sensible. Afterwards,
the model is applied to calculate the optimal placement of tsunami sensors in
Cotabato Trench. The actual bathymetry profile of this trench is used to acquire a more
Table 2 Improvement in tsunami detection time due to the increase in number (L) of sensors for the
optimal placement of sensors in Cotabato trench.
L Detection time (s) Improvement in time due to an additional sensor
1 224.9557 –
2 90.0057 134.9500
3 69.2824 20.7233
4 57.2515 12.0309
5 36.5071 20.7444
6 24.1509 12.3562
7 20.3382 3.8127
8 18.0073 2.331
Figure 11 Comparison of the tsunami detection time for different number of sensors for the case
when the domain is a portion of the Cotabato trench. Full-size
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DOI: 10.7717/peerj-cs.333/fig-11
Ferrolino et al. (2020), PeerJ Comput. Sci., DOI 10.7717/peerj-cs.333 15/21

realistic estimation of the wave travel time. We note here that our model can be used to any
domain with any form of subduction zone or bathymetry. Moreover, it can consider other
types of tsunami generation by changing the conditions in the 2D nonlinear shallow water
equations.
Numerical results have shown an inverse relationship between the number of
sensors and tsunami detection time. However, employing additional sensors implies more
cost. Imposing constraints in cost and installation (e.g., depth, location) may be done
in future works. Another recommendation is to consider maximizing tsunami warning
efficacy and accuracy of estimation of tsunami parameters as additional objective
functions.
ADDITIONAL INFORMATION AND DECLARATIONS
Funding
This work was funded by the UP System Enhanced Creative Work and Research Grant
(ECWRG-2019-2-11-R) and the Research Grant from 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:
UP System Enhanced Creative Work and Research: ECWRG-2019-2-11-R.
Institut Teknologi Bandung.
Competing Interests
The authors declare that they have no competing interests.
Author Contributions
Angelie Ferrolino 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, analyzed the data, authored
or reviewed drafts of the paper, and approved the final draft.
Ikha Magdalena conceived and designed the experiments, analyzed the data, authored or
reviewed drafts of the paper, and approved the final draft.
Jose Ernie Lope conceived and designed the experiments, 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:
We obtained bathymetric profile data from the Cotabato trench at the General
Bathymetic Charts of the Oceans (GEBCO): https://download.gebco.net/. The top left
corner has latitude 6.3142 degrees and longitude 124.1083 degrees, while the bottom right
corner has latitude 5.6158 degrees and longitude 124.8067 degrees.
Ferrolino et al. (2020), PeerJ Comput. Sci., DOI 10.7717/peerj-cs.333 16/21

We use the MATLAB built-in command particleswarm to solve the arising
minimization problem. The details of the code can be found here: https://www.mathworks.
com/help/gads/particleswarm.html
Supplemental Information
Supplemental information for this article can be found online at http://dx.doi.org/10.7717/
peerj-cs.333#supplemental-information.
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