# The Development of A Rigorous Model for Bathymetric Mapping from Multispectral Satellite-Images

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

^{2}between 0.93 and 0.94, for Granadilla and Corralejo areas with different substrates, i.e., seagrass, sand, and maerl). However, this method needs an optimal atmospheric correction model to achieve effective bathymetry retrieval. Petit et al. [34] combined a mixing model of four seabed albedos, namely sand, corals, algae, and seagrass to the SA model, forming the relaxed abundance sum-to-one constraint least squares on the first spectral derivative (RASC-LSD) model to invert the water depth of coral reef areas. While this model is not suitable for seabeds with water depth exceeding 10 m. Huang et al. [35] solved the QAA model parameters using the Levenberg–Marquardt method and substituted these parameters into the QAA model to retrieve the water depth of Weizhou Island.

## 2. Rigorous Bathymetric Model

#### 2.1. Tidal Fluctuation Impact

_{d}, the geoid, which is the mean sea level, is noted as G

_{i}. To obtain the distance D

_{rw}between the level of sea surface (S

_{s}) and the seafloor (S

_{f}), D

_{t}, D

_{bg}, and D

_{tg}must first be obtained. Therefore, the steps are suggested as follows;

- (1)
- The tide height D
_{t}at a certain time at the tidal station is obtained; - (2)
- The distance D
_{bg}from the geoid to tidal datum is obtained from the tide station; - (3)
- D
_{tg}is calculated through subtracting D_{bg}from D_{t}; - (4)
- The unified geoid step of converting D
_{rw}to D_{w}at the epoch of satellite flight is calculated by subtracting D_{tg}from D_{rw}; - (5)
- Alternatively, the unified to the level of tide height step of converting D
_{w}to D_{rw}at the epoch of satellite flight is calculated as D_{rw}by adding D_{tg}to D_{w}.

#### 2.2. Rigorous Tidal Corrected Bathymetric Model

_{w}, D

_{rw}are the same as above, λ

_{i}is the radiance over the water surface radiance of the satellite image i-band, λ

_{j}is the radiance over the water surface radiance of the satellite image j-band, m

_{1}is a tunable constant to scale the ratio to water depth, n is the fixed constant of all areas, and m

_{0}is the offset of the D

_{w}meter water depth relative to 0 m, n is chosen to assure the logarithm will be positive under any condition and the ratio will produce a linear response with depth, R

_{w}is the reflectance of water.

_{rw}at the epoch of satellite flight is calculated by

_{tg}can be calculated by

_{t}is tidal height at the epoch of satellite flight and can be calculated by a cubic spline interpolation method, i.e., [43].

_{j}),

_{j}(j = 0, 1, …, n) is the node in the interval [x

_{0}, x

_{n}], S(x

_{j}) is the cubic spline interpolation function, D(x

_{j}) (j = 0, 1, …, n) is the corresponding tide height, S(x

_{j}± 0) is the right/left limit of S(x

_{j}) at node x

_{j}, S′(x

_{j}± 0) is the first order right/left derivative of S(x) at node x

_{j}, S″(x

_{j}± 0) is the second order right/left derivative of S(x

_{j}) at node x

_{j}, S″(x

_{0}) is the second order right derivative of S(x

_{0}) at node x

_{0}, D″(x

_{0}) is the second derivative of D(x

_{0}) at node x

_{0}, S″(x

_{n}) is the second derivative of S(x

_{n}) at node x

_{n}, and D″(x

_{n}) is the second derivative of D(x

_{n}) at node x

_{n}.

_{j}) is a cubic polynomial equation on the interval [x

_{j}, x

_{j+1}], S″(x

_{j}) is a linear function on an interval of [x

_{j}, x

_{j+1}], which can be expressed by

_{j}) is the second-order derivative values of S(x), M

_{j}(j = 0, 1, …, n) are the unknown parameters, h

_{j}= x

_{j+1}-x

_{j}(j = 0, 1, …, n − 1).

_{j}) is obtained by integrating Equation (5) twice and integral constant is obtained using S(x

_{j}) = D(x

_{j}), S(x

_{j+1}) = D(x

_{j+1})

_{j}in Equation (6), the derivative function, S′(x) from S(x) has to compute by

_{j−1}, x

_{j}] are expressed by

_{j}in Equation (10) are calculated using the endpoint funtion of Equation (11), i.e.,

- ①
- Building a recurrence equation, i.e.,$$\left\{\begin{array}{l}{\beta}_{1}{=c}_{1}{/b}_{1},\\ {\beta}_{i}{=c}_{i}{/(b}_{i}-{\alpha}_{i}{\beta}_{i-1}),i=2,3,\dots ,n-1\end{array}\right.$$
- ②
- Solve the matix equation $Ly=D$, i.e.,$$\left\{\begin{array}{l}{y}_{1}={D}_{1}/{b}_{1}\\ {y}_{i}=({D}_{i}-{\alpha}_{i}{y}_{i-1})/({b}_{i}-{\alpha}_{i}{\beta}_{i-1}),i=2,3,\dots ,n\end{array}\right.$$
- ③
- Solve the matix equation $UM=y$, i.e.,$$\left\{\begin{array}{l}{x}_{n}={y}_{n},\\ {x}_{i}={y}_{i}-{\beta}_{i}{x}_{i}{}_{+1},i=n-1,n-2,\dots ,2,1\end{array}\right.$$

_{j}(j = 0… n) are computed, and the cubic spline function S(x

_{j}) is obtained.

## 3. Test Field and Data Set

#### 3.1. Test Field

^{2}with a length and a width of 3.06 km × 3.06 km.

#### 3.2. Data Sets and Data Preprocessing

#### 3.2.1. Tidal data and preprocessing

#### 3.2.2. Water Depth (Reference Data) Measurement Using Multibeam Bathymetric System

#### 3.2.3. Satellite Images and Preprocessing

## 4. Results and Discussions

#### 4.1. Bathymetric Retrieval Using our Model

_{j}). The main work in this step is to model S(x) using the second derivative value, i.e., S″(x) = M

_{j}(j = 0, 1, …, n).

_{j}) by Step 1. Then the time of the epoch of satellite flight is substituted into S(x

_{j}) to obtain a tide height of 2.06 m at the epoch of satellite flight (see Figure 6). Then, the tide height D

_{t}= 2.06 m was substituted into Equation (3) to obtain the D

_{tg}, which was equal to −0.215 m.

_{tg}(−0.215 m) were substituted into Equation (1) to calculate the parameters m

_{0}and m

_{1}. Finally, the bathymetric data for the entire test field was retrieved using our model. The result is shown in Figure 7. As seen, the water depths were between −23.10 m and 0 m; the deepest depth reached −23.10 m.

#### 4.2. Accuracy Analysis

#### 4.2.1. “Absolute” Error Analysis of Water Depth

^{2}, accounting for approximately 66% of the entire test field, i.e., the “absolute” error greater than 3.0 m was 3,174,312 m

^{2}, covered approximately 34% of the entire test field. In addition, as observed from Figure 8b, the “absolute” error less than 3.0 m covered 2,375,780 m

^{2}, accounting for approximately 25% of the entire test field, i.e., the “absolute” error greater than 3.0 m was 6,987,820 m

^{2}, covered approximately 75% of the entire test field.

#### 4.2.2. Accuracy Validation Using 4 Check Lines

- (1)
- The mean error of the water depths retrieved by our model was 2.30 m, respectively, while it was 7.71 m by the traditional model, respectively. This means that the mean error of water depth from our model decreased 67% relative to that from the traditional model.
- (2)
- The RMSE relative to “true” water depth by our model was 3.21, while by 7.59 in the traditional model. This means that the RMSE of water depth from our model decreased 56% relative to that from the traditional model.

#### 4.2.3. Accuracy Validation Using Seven Checkpoints

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Principle of tidal fluctuation impact at the epoch of satellite flight, where D

_{s}represents the distance between satellite and sea surface, D

_{bg}represents the distance between the geoid and tidal datum, D

_{tg}represents the distance from the level of sea surface to the geoid; D

_{t}represents the distance from the level of sea surface to tidal datum; D

_{bs}represents the distance between the tidal datum and the sea floor; D

_{w}represents water depth, which is indeed the distance between the geoid and the seafloor; D

_{rw}represents water depth, which is indeed the distance between the level of sea surface and the seafloor.

**Figure 4.**The transects (

**a**) of 4 check lines, water depth data (

**b**) measured by RESON SeaBat 7125 multibeam sounding system, the seven reference points, the, and the four check lines in the test field.

**Figure 9.**a, b, c, and d are the Line a, Line b, Line c, and Line d check line (Ref Figure 4); 1 and 2 are the retrieved water depth error between our model, traditional model, and reference data sets.

ID | Time (Year-Month-Day Hour-Minute-Second) | Time Stamp (s) | Tide Height (m) | ID | Time (Year-Month-Day Hour-Minute-Second) | Time Stamp (s) | Tide Height (m) |
---|---|---|---|---|---|---|---|

1 | 2020-02-22 16:00:00 | 1,582,358,400 | 2.55 | 13 | 2020-02-23 04:00:00 | 1,582,401,600 | 1.68 |

2 | 2020-02-22 17:00:00 | 1,582,362,000 | 2.93 | 14 | 2020-02-23 05:00:00 | 1,582,405,200 | 1.28 |

3 | 2020-02-22 18:00:00 | 1,582,365,600 | 3.32 | 15 | 2020-02-23 06:00:00 | 1,582,408,800 | 1.01 |

4 | 2020-02-22 19:00:00 | 1,582,369,200 | 3.67 | 16 | 2020-02-23 07:00:00 | 1,582,412,400 | 0.89 |

5 | 2020-02-22 20:00:00 | 1,582,372,800 | 3.92 | 17 | 2020-02-23 08:00:00 | 1,582,416,000 | 0.91 |

6 | 2020-02-22 21:00:00 | 1,582,376,400 | 4.05 | 18 | 2020-02-23 09:00:00 | 1,582,419,600 | 1.03 |

7 | 2020-02-22 22:00:00 | 1,582,380,000 | 4.03 | 19 | 2020-02-23 10:00:00 | 1,582,423,200 | 1.19 |

8 | 2020-02-22 23:00:00 | 1,582,383,600 | 3.86 | 20 | 2020-02-23 11:00:00 | 1,582,426,800 | 1.35 |

9 | 2020-02-23 00:00:00 | 1,582,387,200 | 3.55 | 21 | 2020-02-23 12:00:00 | 1,582,430,400 | 1.49 |

10 | 2020-02-23 01:00:00 | 1,582,390,800 | 3.14 | 22 | 2020-02-2313:00:00 | 1,582,434,000 | 1.60 |

11 | 2020-02-23 02:00:00 | 1,582,394,400 | 2.66 | 23 | 2020-02-23 14:00:00 | 1,582,437,600 | 1.72 |

12 | 2020-02-23 03:00:00 | 1,582,398,000 | 2.16 | 24 | 2020-02-23 15:00:00 | 1,582,441,200 | 1.91 |

Power Requirements | Transceiver Cable Length | Maximum Band Angle | Data Output |

111/220 VAC, 50/60 Hz Average power 500 W | 25 m standard | 128° (140°) | Depth, side scan, and fragment, 7K data format |

Along-Track Transmit Beam Width | Receive Beam Width of Across-Track | Horizontal Positioning Accuracy (RTK) | Length of Cable from LCU to Processor |

1°(±0.2°) at 400 kHz | 0.54°(±0.03°) at 400 kHz | 2–5 cm | N/A |

Pulse Length | Maximum Ping Rate | Frequency | Work Depth |

From 33–300 μs ec | 50 Hz (±1 Hz) | 400 kHz | 0.2–150 m |

Wave Number | System Depth Limit | Bathymetric Resolution | Data Transmission |

512 EA/ED at 400 kHz | 25 m | 5 cm | Ethernet, 1 Gbit |

Plane Coordinates | Vertical Datum | Projection Mode | 1.5° Band Projection | Scale |
---|---|---|---|---|

2000 national geodetic coordinate system | 1985 National Height Datum | Gauss Kruger projection | Zone 71, central meridian 108.75° | 1:5000 |

Plane Accuracy (95% Confidence) | Sounding Accuracy (95% Confidence) | 100% Seafloor Scanning | System Detection Capability |
---|---|---|---|

2 m | 0.25 m | It has to be done | Characteristics of space objects > 1m^{3} |

0–20 m | 20–30 m |
---|---|

±0.3 m | ±0.4 m |

**Table 6.**The seven chosen reference points, which are marked in Figure 4 with black.

Point | The Water Depth Calculated from the Geoid (m) | Point | The Water Depth Calculated from the Geoid (m) |
---|---|---|---|

1 | 15.89 | 5 | 20.13 |

2 | 14.11 | 6 | 20.03 |

3 | 9.76 | 7 | 21.58 |

4 | 9.99 |

Model | Maximum Absolute Error (m) | Mean Absolute Error (m) | RMSE |
---|---|---|---|

Traditional model | 15.3 | 4.79 | 5.22 |

Our model | 5 | 2.62 | 3.68 |

Error decreasing rate | 54% | 45% | 30% |

Model | Line a | Line b | Line c | Line d | ||||
---|---|---|---|---|---|---|---|---|

Mean Error (m) | RMSE | Mean Error (m) | RMSE | Mean Error (m) | RMSE | Mean Error (m) | RMSE | |

Traditional model | 7.04 | 7.07 | 7.18 | 7.96 | 7.68 | 8.48 | 6.80 | 6.85 |

Our model | 2.24 | 3.45 | 1.62 | 2.28 | 1.47 | 2.04 | 3.88 | 5.07 |

Error decreasing rate | 68% | 51% | 77% | 71% | 80% | 76% | 43% | 26% |

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**MDPI and ACS Style**

Xu, J.; Zhou, G.; Su, S.; Cao, Q.; Tian, Z.
The Development of A Rigorous Model for Bathymetric Mapping from Multispectral Satellite-Images. *Remote Sens.* **2022**, *14*, 2495.
https://doi.org/10.3390/rs14102495

**AMA Style**

Xu J, Zhou G, Su S, Cao Q, Tian Z.
The Development of A Rigorous Model for Bathymetric Mapping from Multispectral Satellite-Images. *Remote Sensing*. 2022; 14(10):2495.
https://doi.org/10.3390/rs14102495

**Chicago/Turabian Style**

Xu, Jiasheng, Guoqing Zhou, Sikai Su, Qiaobo Cao, and Zhou Tian.
2022. "The Development of A Rigorous Model for Bathymetric Mapping from Multispectral Satellite-Images" *Remote Sensing* 14, no. 10: 2495.
https://doi.org/10.3390/rs14102495