Citation:
Linthon, M., S. Koshimura (2019), Study of the bathymetric influence on tsunami propagationnear the coast of Esmeraldas by tsunami simulation and ray tracing analysis, Synopsis of IISEE-GRIPS
master report.

STUDY OF THE BATHYMETRIC INFLUENCE ON
TSUNAMI PROPAGATION NEAR THE COAST OFESMERALDAS BY TSUNAMI SIMULATION AND RAY
TRACING ANALYSIS
Michael Linthon1,2

Supervisor:

Shunichi KOSHIMURA3

ABSTRACTAs a result of the subduction and other geological processes, there is the presence of some bathymetricfeatures which significantly change the gradient of the seabed near Ecuador, such as the oceanic trench
and the submarine Esmeraldas canyon. The main purpose of this study is to give a better understandingof how these alterations in the bathymetry affect the tsunami propagation by conducting ray tracing
analysis and tsunami simulations using different bathymetry data sources of GEBCO, ETOPO, and
INOCAR, for local and distant events along the Pacific and the coast of Ecuador. In addition, we
conducted tsunami propagation simulation using TUNAMI code with a nested grid system of six
domains. We used general and high resolution bathymetric data and analyzed the results in all six
domains. From the results of both analyses, we evaluated the consistency between the directivity of therays and the wave’s propagation for focusing areas such as Esmeraldas. We observed a fasterpropagation of the wave due to the particular bathymetric profile of the canyon in front of Esmeraldas,
which made the first wave arrive at this city; with this result we confirmed the effect of local bathymetry
along the tsunami path. This study can be used as a preliminary evaluation to determine the vulnerabilityof specific locations along the coast regarding the changes in the local bathymetry and topographic
submarine features. Therefore, in the future we expect to conduct further analysis under otherconsiderations not estimated in the present research.
Keywords: Tsunami simulation, ray tracing, bathymetry effect, propagation path, directivity.

1. INTRODUCTION
Ecuador is a country with high seismic risk; its geographical position is part of the "Ring of Fire" inthe Pacific; therefore, it is vulnerable to threats not only from hydro-meteorological nature but also
from the geological environment like earthquakes and tsunamis in most of its territory. Regarding
tsunami studies, one of the most important topics to be discussed, is to know how the tsunami wave
propagates from the source until it reaches the coast. In the case of Ecuador, which is located in front
the subduction zone, there is always a possibility for the occurrence of a tsunami as a result from an
1

Bharatpur Metropolitancity, Bharatpur, Chitwan, Nepal.
IISEE-GRIPS master course student
3
Professor, International Research Institute of Disaster Science (IRIDeS), Tohoku Univ.
2

1earthquake, for this reason, this is a very important issue that should be considered. For a better
understanding of the wave’s propagation, Satake (1988) studied the effects of the bathymetry by theapplication of ray tracing to tsunami propagation, allowing to evaluate the directivity of the wave froma point source model along its propagation path. Ray tracing can also be used to determine focusing
and defocusing areas by analyzing specific points of the coast where the rays concentrate or dissipate
depending on the local bathymetry. In the same way, after the arrival of the first tsunami wave, it
refracts back to the ocean or to other locations in the coast, which can also be estimated by ray tracing
analysis. On the other hand, one of the advantages of this model is that ray paths are displayed in ashort computational time; however, it has a strong sensibility for depth variations that depend on thebathymetry data used for the simulation, and for that reason it is necessary to understand theapplication of the model under different considerations. Since there are no other studies in Ecuador
regarding the application of ray tracing to analyze the wave’s directivity, this research can beconsidered as the first evaluation of this type for local populations along the coast. The current
research focuses on the study of the bathymetry and its influence on the tsunami propagation since it
affects the directivity and the wave direction. By this, focusing and defocusing areas can be
determined as the support of tsunami risk management for vulnerable populations.

2. DATAFor this study, it was necessary to prepare the bathymetry data for both ray tracing and tsunami
simulations. Since ray tracing code was set in MATLAB, we required to construct the spatial anddepths matrix, for both far-distant and local events. For far-distant ray tracing simulations, we usedoriginal bathymetric data of 1 arc-min resolution from ETOPO1, that was resampled to 5 arc-min
resolution for simulations with real bathymetry, and also replaced assuming a constant ocean depth of5400 meters for simulations with constant depth. In the same way, for local ray tracing simulations, weused original 15 arc-sec and 1 arc-min grid resolution data from GEBCO and ETOPO1, respectively.

DomainTable 1. Computational domain for the simulation of the TUNAMI model.
Longitude (deg) Latitude (deg) Spatial Resolution Dimension Connecting point
Left

Right

BottomTop

Mts.

Arc-secIF

JF

ISL/JSL/IEL/JEL

1-82.47-78.51

-2.74

2.49

2430

81

181

238

………………

2-80.92-78.69

0.33

2.01

810

37

307

229

72/141/173/216

3-80.28-78.94

0.57

1.63

270

9

553

433

89/34/272/177

4-79.79-79.56

0.95

1.11

90

3

295

193

202/156/299/219

5-79.69

-79.6

0.97

1.06

30

1

358

355

127/20/245/137

6-79.67-79.65

0.98

1.01

10

0.33

187

391

93/42/154/171

On the other hand, for tsunami simulations, we constructed a nested grid system for six
defined areas to compute the TUNAMI model from Tohoku University as shown in Table 1. Thismodel considers the linear propagation for the first three grids, for that reason, we used 15 arc-sec
resolution data from GEBCO for those areas; however, for non-linear propagation in grids 4 to 6, weused finer bathymetry with 10 meters spatial resolution from INOCAR, which was resampledaccording to the considerations of the TUNAMI model for the last three areas.

2

3. METHODOLOGY
3.1. Ray tracing simulation
According to the purpose of the current study, it was necessary to analyze how the bathymetry
influences the way of propagation of the tsunami wave. Since it is known that the velocity of thetsunami is directly related to the water depth √𝑔ℎ, these changes of bathymetry will cause a variationin the propagation path from the source until it reaches the coast; for this reason, ray tracing
methodology was used to analyze the bathymetric effect. Ray tracing is computed as a shortwavelength approximation, which has a strong dependence on the water depth, being sensitive to thechanged inhomogeneity. Considering the earth's sphericity, the governing equations (Satake, 1998) are
as follows:𝑑𝜃⁄𝑑𝑇 = 1⁄𝑛𝑅 ∗ cos 𝜁 ,(1)𝑑𝜑⁄𝑑𝑇 = 1⁄𝑛𝑅 sin 𝜃 ∗ sin 𝜁 ,(2)𝑑𝜁 ⁄𝑑𝑇 = sin 𝜁⁄𝑛2 𝑅 ∗ 𝜕𝑛⁄𝜕𝜃 + cos 𝜁 ⁄𝑛2 𝑅 sin 𝜃 ∗ 𝜕𝑛⁄𝜕𝜑 − 1⁄𝑛𝑅 ∗ sin 𝜁 cot 𝜃 , (3)where φ and θ are the longitude and the colatitude (in radians), respectively at the ray time defined asT, n is referred as the slowness (1⁄√𝑔ℎ ), R corresponds to the earth’s radius defined as 6371 km, andζ is the direction of the ray measured counterclockwise from the south in radians in Eqs. (1) – (3). In
order to solve the above equations, the Runge-Kutta 4th order method was used for the computationand the mid-point method for solving the integrations using interpolated velocities (Gusman et al.,2017).
Likewise, in order to execute the program, it is necessary to input five initial parameters:
the time step and the maximum time of computation both in seconds, the ray azimuth interval indegrees and, XO and Yo that are the point source longitude and latitude, respectively in decimal degrees.
The program only works with positive values of bathymetry; if it is plotted inland or in a region ofnegative values, it does not show any propagation. In such cases, we needed to change the values of
XO and Yo to the nearest point on the coast to see the ray propagation. For this reason, it was importantto check the data, especially if it came from different sourcessince positive values are defined for thebathymetry and negative for topography data.
3.2. Tsunami simulation
In general, tsunami waves are mostly generated after an earthquake occurs, when the deformation of
the seafloor causes a vertical displacement of a large volume of water. From this consideration, we
considered the deformation as an initial condition of a tsunami source, and used the Okada’s (1985)
dislocation model to estimate the deformation of the seafloor by assumed fault parameters.
As previously mentioned, we used the TUNAMI code which was developed and modified
by Tohoku University (Koshimura, 2018). Since for the non-linear simulation, we needed tounderstand the behavior of the wave propagation in shallow waters, for practical use in the numericalmodeling, it was necessary to consider the bottom friction terms, which are related to the interaction of
the bottom. We expressed the shallow water theory for non-linear propagation (Koshimura, 2018) in
Eqs. (4) – (6) with the bottom friction terms for Cartesian coordinate system as follows:𝜕𝜂 𝜕𝑀 𝜕𝑁++=0
𝜕𝑡 𝜕𝑥 𝜕𝑦3(4)𝜕𝑀 𝜕 𝑀2
𝜕 𝑀𝑁
𝜕𝜂 𝑔𝑛2+() + 𝑔𝐷+
𝑀√𝑀2 + 𝑁 2 = 0( )+𝜕𝑡 𝜕𝑥 𝐷
𝜕𝑦 𝐷
𝜕𝑥 𝐷 7/3(5)

𝜕𝑁 𝜕 𝑁 2
𝜕 𝑀𝑁
𝜕𝜂 𝑔𝑛2+() + 𝑔𝐷+𝑁√𝑀2 + 𝑁 2 = 0( )+𝜕𝑡 𝜕𝑦 𝐷
𝜕𝑥 𝐷
𝜕𝑦 𝐷 7/3(6)where n is the Manning coefficient, M and N are the discharge fluxes in x and y directions respectively,
and D is the total depth (= 𝜂 + ℎ) which is related to the water level 𝜂 and the water depth ℎ.

4. RESULTS AND DISCUSSION
For far-distant events, two different ray tracing simulations were conducted for each scenario in
Alaska, Nicaragua and Chile, using both artificial bathymetries assuming a 5400 meters constantocean depth and real bathymetry of 5 arc-min resolution data resampled from ETOPO1.

Figure 1. Ray tracing simulation from a point source located in Alaska.
As it can be appreciated in Figure 1, twosimulations were conducted from the same point source in
Alaska. The plot on the left shows the result using the 5400
meters constant depth, of where it indicates that all the west
coast of the United States, Ecuador, Chile, and also the southpart of the Colombia and Peru coasts were affected by thewave directivity. However, according to the plot on the rightusing real bathymetry, we could observe that the mostaffected area was California in North America, in contrary,
Ecuador does not seem to have a significant effect comparedwith the results from the simulations using the artificial
bathymetry.
In the same way, we conducted simulations for
local events under different conditions. At first we considered
two resolution bathymetries, obtaining some spatial variations
regarding the propagation of the rays. Secondly, we evaluatedthe influence of the oceanic trench in the ray propagation. For
this case, we defined two sources at the same latitude but
different longitude considering its position regarding thetrench. Results showed that the trench acts as a barrier
trapping and refracting the waves back to the ocean due to thepoint sources located near the coast and offshore the trench,
Figure 2. Ray tracing simulations
respectively, as shown in Figure 2.
from point sources located near thecoast (top) and offshore the trench(bottom).

4After evaluating how the trench influences the directivity of the wave, we conducted a
deeper analysis for the most important and vulnerable cities along the coast, from 20 point sources
distributed in three areas: north, central and southern Ecuador; considering their position respect to theoceanic trench, we also defined the worst-case scenarios for each area. From the previous results, weobserved a particular trend of the rays focusing on the coast of Esmeraldas in the north area. For the
next step, we did more ray tracing simulations considering the presence of the Esmeraldas canyon,
obtaining the same trend from other point sources located in the north.

Figure 3. Ray tracing simulation from a point source in northern Ecuador.
As it can be seen in Figure 3, the rays follow a particular pattern; converging andpropagating from the source following an explicit path arriving to the coast. From these results, we canconfirm that the presence of the Esmeraldas canyon in front of Esmeraldas city considerably
influenced the propagation of thetsunami regarding the wave’s
directivity.
In order to have a better
understanding of this process, we
conducted tsunami simulation using
the TUNAMI code with the six
nested grids, as described in section 2.
After the simulation, we obtained theoutputs each minute for one-hour
propagation, merging the bathymetricand the propagation layers in order tohave a better appreciation of thetsunami wave along the Esmeraldas
canyon. With all the previous
analyses, we observed a consistency
between both results of ray tracing
and TUNAMI model simulations.
From the results of ray tracing
simulations, we appreciated how the
wave's directivity changed its direction along the propagation path, as seen in Figure 3. In this way, we
can define Esmeraldas as a focusing area against a tsunami event generated in the north due to its
location in front of the canyon, which influences the wave to move faster and reach first to Esmeraldas
compared to other points in the north region. From the results of TUNAMI simulation, and as

Figure 4. Snapshot obtained from TUNAMI simulation model
after 22 minutes propagation in Domain 4 (See Table 1).

5

previously discussed, the wave arrived first at Esmeraldas compared to the areas to the side. Due to thebathymetric profile of the canyon, the wave moves faster along the canyon which ends in front of
Esmeraldas as appreciated in Figure 4. This particular trend makes this area more vulnerable comparedto other neighboring populations, being one of the main reasons to continue performing tsunami
studies in that area.

5. CONCLUSIONS
From the results of ray tracing simulations for far distant events, we concluded that there is a clear
difference in the propagation of the wave’s directivity between using artificial and real bathymetries,
since then, we computed other simulations only considering real data. In the same way, for local
simulations, we also observed that there are spatial variations from using bathymetry data at different
resolutions.
From the results of local simulations, considering the presence of the oceanic trench, we
could appreciate how the trench acts as a barrier regarding the position of the source, near the coast or
offshore. From simulations located near the coast, we observed how the rays are trapped and refracted
back to the coast, contrary from an event located offshore the trench, where most of the rays returnedto the deep sea. According to these results, we can infer that an event generated near the coast may
have more influence on coastal populations, compared with a source offshore the trench. Regarding
this last consideration, we computed other simulations along the coast for three different areas,
defining the worst scenarios for most important populations.
After the evaluation of the results from the north area, we observed a particular trend for
Esmeraldas. Simulations from different point sources in the north showed the same tendency, rays
focusing in a specific point in the coast following the shape of the Esmeraldas canyon. In that sense,
we conducted tsunami simulation using the TUNAMI model for linear and non-linear propagation
which considers the deformation of the seafloor. Results showed that there is a consistency between
the ray tracing analysis and tsunami simulation; we observed how the velocity and the wavefront tend
to move faster in the direction of the canyon reaching first to Esmeraldas compared to other areas.
Finally, from the evaluation of all simulations, we defined Esmeraldas as a focusing area from asource located in northern Ecuador; that is due to the presence of the Esmeraldas Canyon, whichcauses the tsunami wave to focus at this point, reaching first to this location in the north.
ACKNOWLEDGEMENTS
This research was conducted during the individual study period of the training course “Seismology,
Earthquake Engineering and Tsunami Disaster Mitigation” by the Building Research Institute, JICA,
and GRIPS.I would like to express my sincere gratitude to my supervisor Dr. Shunichi Koshimura for
his valuable support and experience which helped me to complete my research successfully, and to Dr.
Erick Mas for his guidance and advice during my individual study. I would also like to acknowledge
Dr. Yushiro Fujii for his permanent support from the beginning to the end of the program.
REFERENCES
Gusman, A. R., Satake, K., Shinohara, M., Sakai, S. I., and Tanioka, Y., 2017, Pure and AppliedGeophysics, 174(8), 2925-2943.
Koshimura, S., 2018, Lecture notes on Tsunami Hazard Assessment- Theory of Tsunami Propagation
and Inundation Simulation, IISEE/BRI.
Okada, Y., 1985, Surface deformation due to shear and tensile faults in a half-space, Bulletin of theseismological society of America, 75.4, 1135-1154.
Satake, K., 1988, Effects of bathymetry on tsunami propagation: Application of ray tracing totsunamis, Pure and Applied Geophysics, 126(1), 27-36.

6