# Intertidal Bathymetry Extraction with Multispectral Images: A Logistic Regression Approach

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Study Area and Data Set

#### 2.1. Study Areas

^{2}. The Tagus estuarine surface is characterized by extensive tidal flats, salt marsh vegetation and mudflats, where 40% of the estuarine area is intertidal [2,5,36] (Figure 1). The depths in the Tagus estuary vary between 2 m in the northern part that includes most mudflats, the middle part with average depths of 7 m, and 46 m in the downstream sections at the main navigation channel [5,36]. The Tagus estuary is mesotidal (tidal range between 2 and 4 m) with a maximum amplitude of 3.9 m and has a semidiurnal tide regime [32,36].

^{2}belonging to the Bijagós archipelago area is intertidal zone [38] (Figure 2). Bijagós has a semidiurnal tide regime and is also meso-tidal, with maximum amplitudes of 4.3 m [32,36].

#### 2.2. Satellite Images

#### 2.3. Tide Data

#### 2.4. In Situ Data for Validation

## 3. Methods

#### 3.1. Preprocessing

#### 3.2. Intertidal Zone Pixels’ Selection

#### 3.3. Logistic Regression

**a**and

**k**, where the steepness is correlated with the parameter

**a**. The function will increase if

**a**is positive and will decrease with tidal height (${h}_{i}$), when

**a**is negative. The logistic function superior asymptote is correlated with the

**k**value. The shape of the logistic function is tuned by these parameters, and they are estimated through adjustment with each pixel reflectance. The LowLim in Equation (3), related to the lower asymptote, could be a third parameter to characterize the logistic function. However, this parameter is not related with the shape of the sigmoid and will be neglected in the following analysis. In fact, the correlation between the tide height and the time variable reflectance is described only by the other two parameters. The sigmoid function is shown in Figure 6a for a range of steepness values. According to our experience, the steepness parameter

**a**must be in the range −10 m

^{−1}to −2 m

^{−1}.

**a**and

**k**and predict the surface elevation (${h}_{t}$). The

**a**parameter can be defined through the tide regime and topography, reducing the number of parameters to be estimated. As the function is not linear, the terrain height ${h}_{t}$ and the parameter

**k**are obtained by searching in the bi-dimensional space of the minimum solution of the cost function:

**k**is the maximum reflectance. Hence, the pixel’s height is predicted by searching in the one-dimensional space (local tide range) of the minimum solution of the cost function. The cost function for a range of steepness values is shown in Figure 6b. We observe that the steepness value is not relevant for the predicted height. In fact, the predicted height range is 2.51 to 2.60 m, changing the steepness value from −2 m

^{−1}to −10 m

^{−1}, and the cost function minimum ranges from 0.012 to 0.04 (reflectance).

#### 3.4. Derivation of the Bathymetric Model

#### 3.5. Log-Transformed Ratio Bands Bathymetric Models

## 4. Results

#### 4.1. Pre-Processing

#### 4.2. Intertidal Model Estimation

#### 4.3. Logarithm Ratio Model Estimation

^{th}of August 2018 (3.55 m tide height) and on the 6

^{th}of December 2017 (4.08 m tide height) were used in the Tagus estuary and Bijagós, respectively. The model was calibrated with 507 and 78 depth values in both sites, selected from hydrographic survey in the Tagus estuary [53] and from a nautical chart in Bijagós [54]. Further details concerning the bathymetric model derivation with the ratio-transformed methodology [30,43,51] are presented in the Appendix A (Annex A). We confined the range of the calibration depths between 0 and 10 m (Figure A2) because the selected ratio bands presented best results in shallow waters [30,43,51,52].

#### 4.4. Validation of the Models

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**ANNEX-A—Logarithm ratio model estimation**

^{2}= 0.79). For the Bijagós archipelago, the correlation between the data sets was lower (R

^{2}= 0.54), but the data used in this case for the calibration has not been updated since the 1960’s [54,55], and this could be a large source of error. Perhaps, the sample of depths used in this study for calibration (N = 500 for Tagus estuary and N = 307 for Bijagós) could be considered exaggerated. In fact, some authors defend that few calibration points were needed to calibrate the model, 5 to 10 [51] or 10 control points [43], because the model had only two parameters (${m}_{1}$ and ${m}_{0}$) that required tuning.

**Figure A1.**Linear regression graphics between in situ measurements (depths in meters) and the logarithm ratio. (

**a**) Tagus estuary—13AUG18 S2A L2A image. (

**b**) Bijagós archipelago—25APR18 S2B L2A image. (Red dashed line: linear regression best fit; red rectangle: correlation coefficient (R

^{2}) between data sets and Equation (7) solution).

**Figure A2.**Bathymetric model for the Tagus estuary achieved through the logarithm ratio band algorithm. S2A L2A 08AUG2018 image.

**Figure A3.**Bathymetric model for the Bijagós archipelago achieved through the logarithm ratio band algorithm. S2A L2A 06DEC2017 image.

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**Figure 1.**Tagus estuary in Lisbon. Red-green-blue (RGB) composite after atmospheric correction of Sentinel-2B imagery on 13 August 2018. The two orange rectangles identify specific intertidal areas at the Tagus estuary (top right—Alcochete and lower left—Seixal and Azinheira).

**Figure 2.**The Bijagós archipelago in Guinea-Bissau. RGB composite after atmospheric correction of Sentinel-2B imagery on 25 April 2018.

**Figure 3.**Tidal height variation: (

**a**) Tide height at Tagus estuary from March to October 2018 and (

**b**) Tide height at Bijagós archipelago from December 2017 to May 2018. The red dots represent the Sentinel-2 images dataset (time—11:21 UTC).

**Figure 4.**Bathymetric model acquired by hydrographic survey in Tagus estuary—Azinheira area (red: 3.1 m above chart datum and green: 7.2 m below chart datum).

**Figure 5.**Guinea-Bissau status of hydrographic surveying in accordance with International Hydrographic Organization (IHO) information (adapted from IHO [55] (p 215)).

**Figure 6.**(

**a**) NIR reflectance time series as a function of the tide height (red circles) and adjusted sigmoid function for a range of steepness parameters (from −2 to −10) and (

**b**) cost function for a range of steepness values.

**Figure 7.**Normalized Difference Water Index (NDWI) temporal variability: (

**a**) Alcochete area in Tagus estuary and (

**b**) Formosa, Maio, and Ponta islands at Bijagós archipelagos.

**Figure 8.**NDWI temporal variability of the three classes: water (blue), intertidal (green), and land (red); (

**a**) Tagus estuary and (

**b**) Bijagós archipelago.

**Figure 9.**(

**a**) Tagus estuary intertidal model estimated using the logistic regression method and 18 S2 images (red: 3.1 m and dark blue: 0.7 m). The two blue rectangles highlight specific intertidal areas at the Tagus estuary (bottom left—Seixal and Azinheira; top right—Alcochete), (

**b**) zoom on the intertidal model estimated for the Tagus estuary—Seixal and Azinheira areas, and (

**c**) zoom on the intertidal model estimated for the Tagus estuary—Alcochete area.

**Figure 10.**(

**a**) Bijagós intertidal model estimated from 19 S2 images (red: 4.2 m and dark blue: 1.3 m). The two yellow rectangles highlight different intertidal areas; (

**b**) zoom on the intertidal model estimated for the Bijagós (top right rectangle in (

**a**)—sandy beaches); and (

**c**) zoom on the intertidal model (bottom left rectangle in (

**a**)—mostly mangroves and mudflats [37,38]).

**Figure 11.**Comparison between derived intertidal model in Tagus estuary and cartographic intertidal area shown in white. Greatest differences are observed along the navigation channel due to intense maritime traffic in the area.

**Table 1.**List of Sentinel-2 A&B images used in this study for intertidal bathymetry extraction. Tide height (m) corresponding to image acquisition time (11:21 UTC). (a) Tagus estuary and (b) Bijagós archipelago.

Number | Date | Sensor | Tide Height (m) |
---|---|---|---|

(a) | |||

1 | 21 March 2018 | S2A | 0.72 |

2 | 26 March 2018 | S2B | 2.95 |

3 | 05 May 2018 | S2B | 1.40 |

4 | 10 May 2018 | S2A | 2.97 |

5 | 15 May 2018 | S2B | 1.92 |

6 | 19 June 2018 | S2A | 1.68 |

7 | 24 June 2018 | S2B | 3.15 |

8 | 29 July 2018 | S2A | 1.43 |

9 | 03 August 2018 | S2B | 1.58 |

10 | 08 August 2018 | S2A | 3.35 |

11 | 13 August 2018 | S2B | 0.89 |

12 | 18 August 2018 | S2A | 2.19 |

13 | 23 August 2018 | S2B | 2.89 |

14 | 22 September 2018 | S2B | 2.84 |

15 | 27 September 2018 | S2A | 1.17 |

16 | 07 October 2018 | S2A | 3.02 |

17 | 22 October 2018 | S2B | 2.86 |

18 | 27 October 2018 | S2A | 0.99 |

(b) | |||

1 | 01 December 2017 | S2A | 2.19 |

2 | 06 December 2017 | S2B | 4.08 |

3 | 26 December 2017 | S2B | 1.90 |

4 | 10 January 2018 | S2A | 1.39 |

5 | 15 January 2018 | S2B | 2.90 |

6 | 20 January 2018 | S2A | 3.88 |

7 | 25 January 2018 | S2B | 1.36 |

8 | 30 January 2018 | S2A | 3.07 |

9 | 09 February 2018 | S2A | 1.52 |

10 | 19 February 2018 | S2A | 3.97 |

11 | 01 March 2018 | S2A | 3.68 |

12 | 06 March 2018 | S2B | 3.59 |

13 | 21 March 2018 | S2A | 4.01 |

14 | 31 March 2018 | S2A | 4.04 |

15 | 05 April 2018 | S2B | 3.35 |

16 | 15 April 2018 | S2B | 3.75 |

17 | 25 April 2018 | S2B | 1.34 |

18 | 10 May 2018 | S2A | 1.46 |

19 | 20 May 2018 | S2A | 4.16 |

**Table 2.**Number of intertidal candidate pixels for the logistic regression versus application of threshold and saturation index. (a) Tagus estuary and (b) Bijagós archipelago. Red circles: threshold and saturation index selected to generate the intertidal models. Red rectangles: standard deviation of the intertidal model created after the logistic regression and the comparison with the bathymetric survey in the Tagus estuary and depths extracted from a nautical chart in the Bijagós archipelago.

(a) | ||||

Candidate pixels | Standard Deviation (m) | |||

Threshold | sat = 0.2 | sat = 0.3 | ||

0.15 | 1130556 | 0.3463 | 0.3456 | 0.3438 |

1029078 | 0.3413 | 0.3407 | ||

0.17 | 945951 | 0.3468 | 0.3449 | 0.3436 |

0.18 | 873121 | 0.3557 | 0.3536 | 0.3508 |

(b) | ||||

Candidate pixels | Standard Deviation (m) | |||

Threshold | sat = 0.2 | sat = 0.4 | ||

0.10 | 4445416 | 0.7164 | 0.7172 | 0.7179 |

4131866 | 0.7177 | 0.7043 | ||

0.12 | 3895423 | 0.7310 | 0.7157 | 0.7066 |

0.13 | 3664775 | 0.7153 | 0.7155 | 0.7158 |

**Table 3.**Statistical analysis of the differences between the hydrographic survey at Tagus basin (Azinheira) and the satellite-derived bathymetry (Logarithm Ratio and Logistic Regression). N is the number of samples.

Algorithm | Sentinel 2 (Images) | N | Bias (m) | STD (m) | RMSE (m) | Max (m) | Min (m) |
---|---|---|---|---|---|---|---|

Logarithm Ratio | 08AUG18 (3.35 m tide height) | 507 | 1.81 | 1.31 | 2.23 | 3.74 | −6.98 |

Logistic Regression | 18 images (Table 1a) | 508 | −0.51 | 0.34 | 0.61 | 2.18 | −1.20 |

**Table 4.**Statistical analysis of differences between chart depths in the Bijagós archipelago and the satellite-derived bathymetry (Logarithm Ratio and Logistic Regression). N is the number of samples.

Algorithm | Sentinel 2 (Images) | N | Bias (m) | STD (m) | RMSE (m) | Max (m) | Min (m) |
---|---|---|---|---|---|---|---|

Logarithm Ratio | 06DEC2017 (4.08 m tide height) | 78 | 4.84 | 1.26 | 5.00 | −2.21 | −7.54 |

Logistic Regression | 19 images (Table 1b) | 66 | −0.46 | 0.70 | 0.90 | 1.70 | −1.50 |

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**MDPI and ACS Style**

Bué, I.; Catalão, J.; Semedo, Á.
Intertidal Bathymetry Extraction with Multispectral Images: A Logistic Regression Approach. *Remote Sens.* **2020**, *12*, 1311.
https://doi.org/10.3390/rs12081311

**AMA Style**

Bué I, Catalão J, Semedo Á.
Intertidal Bathymetry Extraction with Multispectral Images: A Logistic Regression Approach. *Remote Sensing*. 2020; 12(8):1311.
https://doi.org/10.3390/rs12081311

**Chicago/Turabian Style**

Bué, Isabel, João Catalão, and Álvaro Semedo.
2020. "Intertidal Bathymetry Extraction with Multispectral Images: A Logistic Regression Approach" *Remote Sensing* 12, no. 8: 1311.
https://doi.org/10.3390/rs12081311