Mathematical Modeling and Computational Analysis of Underwater Topography with Global Positioning and Echo Sounder Data

Satoshi Iwakami^{1},
Masahiko Tamega^{1},
Masahide Sanada^{1},
Michiaki Mohri^{1},
Yoshitaka Iwakami^{1},
Naoki Okamoto^{1},
Ryousuke Asou^{1},
Shuji Jimbo^{2},
Masaji Watanabe^{3}

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1. Introduction

Recent disastrous heavy rain events and floods caused severe damages including human damages and house damages. Those include 119 fatalities and 213 totally destroyed houses due to 2018 Japan floods (July 2018) [1], 104 fatalities and 3308 totally destroyed houses due to Typhoon 19 (Hagibis, October 2019) and subsequent heavy rain events [2], and 84 fatalities and 1621 totally destroyed houses due to July 2020 heavy rain disaster [3]. As the climate change progresses such disastrous heavy rain events and floods may occur more frequently, and it is important to establish reliable sources of information concerning land water areas such as rivers, reservoirs, and coastal areas.

This study focuses on construction of underwater topography based on data obtained in field measurement. Apparatuses including a RTK-GPS (real time kinematic global positioning system) in VRS (virtual reference station) mode and an echo sounder were used in measurement conducted in Kojima Lake, Okayama Prefecture, Japan. Measurement was conducted on September 28^{th}, 2019, October 4^{th}, 2019, December 25^{th}, 2019, January 6^{th}, 2020, December 26^{th}, 2020, January 27^{th}, 2021, March 17^{th}, 2021, and March 20^{th}, 2021 [4] [5] [6] [7]. Previous studies developed numerical techniques to construct surfaces based on data. Those techniques were applied to data sets obtained in the field measurement for construction of surfaces representing underwater topography. Numerical results show sedimentation during period from January 2020 to January 2021.

2. Application of Numerical Techniques to Data Sets

Numerical techniques developed in previous studies [4] [5] [6] [7] were reapplied to two data sets. One data set, which we call data set 1, consisted of results of measurement conducted on September 28^{th}, 2019, October 4^{th}, 2019, December 25^{th}, 2019, and January 6^{th}, 2020. The other data set, which we call data set 2, consisted of results of measurement conducted on December 26^{th}, 2020, January 27^{th}, 2021, March 17^{th}, 2021, and March 20^{th}, 2021.

The Gauss-Krüger projection transformed latitude components and longitude components of GPS data to xy components of a rectangular coordinate. Combination of those components with vertical components including output results from an echo sounder leads to three dimensional data that lay in an underwater topography. In particular, z component of three dimensional data $\left({x}_{j},{y}_{j},{f}_{j}\right),j=1,2,3,\mathrm{...}$ are given by ${f}_{j}={h}_{j}-{d}_{j}-{z}_{0}-L$, where ${h}_{j}$ is the GPS antenna height, ${d}_{j}$ is the distance between the oscillator of echo sounder and the bottom, ${z}_{0}$ is the geodetic height of the mean sea level, and L is the distance between the antenna and the oscillator. Figure 1 shows three dimensional data of Kojima Lake topographic data.

An underwater topography was represented by a piecewise linear function defined on a triangular mesh. An initial triangular mesh ${T}_{0}$ that contains GPS tracks was set in an xy plane. A sequence of triangular meshes ${T}_{0},{T}_{1},{T}_{2},\mathrm{...}$ were constructed from the initial mesh. A triangular mesh ${T}_{l}(l\ge 1)$ in the sequence was constructed by dividing each element of ${T}_{l-1}$ into four congruent triangles. Figure 2 shows an initial triangular mesh ${T}_{0}$. Figure 2 also shows an approximate outline of Kojima Lake based on data obtained with an online software [8].

Suppose that triangular mesh ${T}_{l}$ consists of m elements ${E}_{1},{E}_{2},\mathrm{...},{E}_{m}$, and nodes $\left({x}_{1},{y}_{1}\right),\left({x}_{2},{y}_{2}\right),\mathrm{...},\left({x}_{n},{y}_{n}\right)$, that elevation of topography ${z}_{i}$ at node $\left({x}_{i},{y}_{i}\right)$ is given for $i=1,2,\mathrm{...},n$, and that an element ${E}_{k}$ contains p data $\left({x}_{j},{y}_{j},{f}_{j}\right),j=1,2,3,\mathrm{...},p$, and that coordinates of vertices of ${E}_{k}$ are $\left({x}_{1},{y}_{1}\right)$, $\left({x}_{2},{y}_{2}\right)$, and $\left({x}_{3},{y}_{3}\right)$. Note that xy coordinates of the first three data are those of the vertices of ${E}_{k}$, and that ${f}_{1},{f}_{2}$ and ${f}_{3}$ are elevations at the vertices $\left({x}_{1},{y}_{1}\right)$, $\left({x}_{2},{y}_{2}\right)$, and $\left({x}_{3},{y}_{3}\right)$, respectively. Consider a linear function $z=ax+by+c$ such that the values of coefficients a, b, and c are those that minimize the square sum

Figure 1. Three dimensional topographic data of Kojima Lake.

Figure 2. Initial mesh. Three dimensional topographic data are also shown.

${\left[{f}_{1}-\left(a{x}_{1}+b{y}_{1}+c\right)\right]}^{2}+\cdots +{\left[{f}_{p}-\left(a{x}_{p}+b{y}_{p}+c\right)\right]}^{2}$. (1)

Once those coefficients are evaluated, value of *f*_{1} is updated, that is, *f*_{1} = *ax*_{1} + *by*_{1} + *c*. With this new value of *f*_{1}, values of coefficients a, b, and c that minimize the square sum (1) are updated and the value of *f*_{2} is updated with equation *f*_{2} = *ax*_{2} + *by*_{2} + *c*. With those new values of *f*_{1} and *f*_{2}, values of coefficients a, b, and c that minimize the square sum (1) are updated, and the value of *f*_{3} is updated with equation *f*_{3} = *ax*_{3} + *by*_{3} + *c*. After those operations are completed
${E}_{k}$, the operations are repeated for the element
${E}_{k+1}$. One cycle of iterations is completed for the triangular mesh when k reaches m, z component or elevation associated with the n nodes,
${z}_{1},{z}_{2},\mathrm{...},{z}_{n}$ are obtained.

Denote by ${Z}_{q}=\left({z}_{1}^{q},{z}_{2}^{q},\mathrm{...},{z}_{n}^{q}\right)$ the n dimensional vector whose components are elevation associated with n nodes after q iterations. The iteration is terminated when the residual becomes less than $\epsilon $, that is

$\Vert {Z}_{q}-{Z}_{q-1}\Vert ={\left[{\left({z}_{1}^{q}-{z}_{1}^{q-1}\right)}^{2}+\cdots +{\left({z}_{n}^{q}-{z}_{n}^{q-1}\right)}^{2}\right]}^{1/2}<\epsilon $.

Values of initial elevation in T_{0} are all set equal to 0, and values of initial elevation for
${T}_{l}$ are obtained from values of final elevation for
${T}_{l-1}$. Figure 3 and Figure 4 show surfaces obtained with
$\epsilon =0.75$. The results shown in Figure 3 and Figure 4 lead to sedimentation during period from January 2020 to January 2021 (Figure 5).

3. Discussion

A triangular mesh is set in a part of region covered the triangular mesh shown by Figureby Figure2 and numerical procedures described in the previous section were repeated. Figure6 shows the initial mesh. Figure7 shows the

Figure 3. Surface over ${T}_{4}$ based on data set 1 with $\epsilon =0.75$, wireframe representation (top), surface with color according to elevation (color).

Figure 4. Surface over ${T}_{4}$ based on data set 2 with $\epsilon =0.75$, wireframe representation (top), surface with color according to elevation (bottom).

Figure 5. Sedimentation over region over the region covered by the initial mesh shown by Figure 2 during period from January 2020 to January 2021, contour lines $z=0.1$ [m] and $z=0.3$ [m] (top), sedimentation with color according to amount (bottom).

Figure 6. Initial mesh. Three dimensional topographic data are also shown.

sedimentation during period from January 2020 to January 2021.

The area of region covered by the initial mesh shown by Figure 2 is approximately 150,000 m^{2}, and the total sedimentation over the equal to region is approximately 5700.569784 m^{3}. It follows that average increase in elevation of underwater topography over the region is 0.038004 m. The area of region covered by the initial mesh shown by Figure 6 is approximately 25,000 m^{2}, and the total sedimentation over the equal to region is approximately 1508.789762 m^{3}. It follows that average increase in elevation of underwater topography over the region is 0.060352 m.

Major sources of water in Kojima Lake are inflow flow from two rivers Kurashiki River and Sasagase River. Kojima Lake was separated from Kojima Bay by embankment. There are six gates set on the embankment (Figure 2). The water

Figure 7. Sedimentation over the region covered by the initial mesh shown by Figure 6 during period from January 2020 to January 2021, contour lines $z=0.1$ [m] and $z=0.3$ [m] (top), sedimentation with color according to amount (bottom).

level of Kojima Lake is controlled by discharge of water through the gates into Kojima bay during low tide. A possible reason for higher sedimentation over the region shown by Figure 7 is stronger effect of flow generated by the discharge.

Funding

This study was partly supported by a 2020 research grant from the Public Interest Incorporated Foundation Wesco Promotion of Learning Foundation.

References

[1] Ministry of Land, Infrastructure, Transport and Tourism (2018) Summary and Characteristic of Damage in 2018 Japan Floods. (In Japanese)
https://www.mlit.go.jp/river/shinngikai_blog/hazard_risk/dai01kai/dai01kai_siryou2-1.pdf

[2] Cabinet Office, Government of Japan (2020) Situation of Damages and So Forth Concerning 2019 Typhoon 19 as of April 10th, 2020, 9:00. (In Japanese)
http://www.bousai.go.jp/updates/r1typhoon19/pdf/r1typhoon19_45.pdf

[3] Cabinet Office, Government of Japan (2020) Damage Situation and So Forth Concerning July_2020 Heavy Rain Disaster as of January 7th, 2020, 14:00. (In Japanese)
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[4] Iwakami, S., Tamega, M., Jimbo, S. and Watanabe, M. (2019) Numerical Techniques for Underwater Topographic Measurement with GPS and Echo Sounder. International Journal of Information System & Technology, 3, 81-85.
http://ijistech.org/ijistech/index.php/ijistech/article/view/37

[5] Iwakami, S., Tamega, M., Sanada, M., Mohri, M., Iwakami, Y., Jimbo, S. and Watanabe, M. (2020) Study of Underwater Topography Change with Measurement and Analysis. Journal of Physics: Conference Series, 1641, Article ID: 012003.
https://iopscience.iop.org/article/10.1088/1742-6596/1641/1/012003/pdf
https://doi.org/10.1088/1742-6596/1641/1/012003

[6] Iwakami, S., Tamega, M., Sanada, M., Mohri, M., Iwakami, Y., Okamoto, N., Jimbo, S. and Watanabe, M. (2021) Study on Change of Topography in Water Area with Field Measurement. Journal of Geoscience and Environmental Protection, 9, 221-226.
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[7] Iwakami, S., Tamega, M., Sanada, M., Mohri, M., Iwakami, Y., Okamoto, N., Asou, R., Jimbo, S. and Watanabe, M. (2021) Numerical Study of Underwater Topography with Measurement Data. 2nd International Conference on Advanced Information Scientific Development (ICAISD). https://doi.org/10.1088/1742-6596/1641/1/012003

[8] Latitude-Longitude Map.
https://fukuno.jig.jp/app/printmap/latlngmap.html