# Mapping Chlorophyll-a Concentrations in the Kaštela Bay and Brač Channel Using Ridge Regression and Sentinel-2 Satellite Images

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Related Work

#### 1.2. Proposed Method

- A method for developing a sophisticated formula for estimation of Chl-a values in coastal waters where we have a limited number of measurements.
- A new regression formula for estimating Chl-a concentrations in coastal waters of study area.

## 2. Materials and Methods

#### 2.1. Study Area

#### 2.2. In Situ Data Set

#### 2.3. Satellite Data

#### 2.4. Methodology

#### 2.4.1. Data Set Construction

#### 2.4.2. Multiple Linear Regression Model

- y is a dependent variable, which, in our case, is a concentration of Chlorophyll-a;
- β
_{0}is a constant, often referred to as bias or intercept; - β
_{i}is a regression coefficient for an independent variable ${X}_{i}$; and - x
_{i(i=1,2,…,n)}are independent variables that are, in our case, the values of Sentinel-2 reflectance bands.

#### 2.4.3. Ridge Regression

- ${\widehat{\beta}}_{n}^{RR}$ is a regression coefficient;
- X is an independent variable that presents a design matrix;
- Y is a response vector;
- k is a tuning parameter; k$\in [0,\infty \rangle $; and
- I
_{p}is identity matrix.

#### 2.4.4. Statistical Evaluation Indices

#### 2.4.5. Data Set Augmentation

#### 2.4.6. Vertical Augmentation

#### 2.4.7. Horizontal Augmentation

_{i}represents each band value (B01-B09, B11-B12, and B8A). Transformation is the application of a mathematical function to each value in a data set, where each set value is replaced by a transformed value, which can generally be written as in (9).

## 3. Results

- Quantitative evaluation was performed on training and testing portions of the data set;
- Qualitative evaluation was performed by inspecting resulting images of Chl-a distribution for certain dates and discussing in the scope of the known events.

- b represents intercept (bias), which is 0.2647;
- β
_{1}, …, β_{12}represents a reflectance value of each band (B01-B09, B11-B12, and B8A) divided by 10,000 and coefficients ${\alpha}_{i}$, ${\gamma}_{i}$, ${\delta}_{i}$, ${\theta}_{i}$, ${\varphi}_{i}$ are listed in Table 6.

- Squared features (${\gamma}_{i}$) for bands B04, B06, B07, and B8A;
- Squared root features (${\delta}_{i}$) for bands B04, B11, and B12;
- Reciprocal transformed features (${\theta}_{i}$) for band B02;
- Logarithmic transformed features (${\varphi}_{i}$) for bands B04, B05, and B8A.

#### 3.1. Comparison with Other Methods for Chl-a Estimation

- Case-2 Regional/Coast Colour (C2RCC) algorithm [13]. We used the C2RCC processor provided in the SNAP processing toolbox from ESA. We collected the Sentinel-2 images from https://scihub.copernicus.eu/ (accessed on 12 October 2021) for several dates of the in situ measurements, processed the whole data set using C2RCC S2-MSI processor, and created a Chl-a map of the area. The processor used default parameters since other algorithms we compared did not use any external data or information. Predicted values of Chl-a concentration were further extracted only on the coordinates of measurement and compared with the in situ measurements.

#### 3.2. Qualitative Validation

## 4. Conclusions and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Sample Availability

## Abbreviations

Chl-a | Chlorophyll-a |

MLR | Multiple Linear Regression |

RRE | Ridge Regression Estimator |

LSE | Least Squares Estimator |

RMSE | Root Mean Square Error |

TSM | Total Suspended Matter |

CDOM | Colored Dissolved Organic Matter |

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**Figure 1.**Map of the geographical location of the study area (Kaštela Bay and The Brač Channel) in the context of Europe and the Republic of Croatia.

**Figure 4.**Distribution of time differences between satellite image and in situ measurement expressed in days.

**Figure 6.**Prediction error plot shows the relationship between in situ and predicted Chlorophyll-a values on train (

**left**) and test (

**right**) data.

**Figure 7.**Map of the predicted Chlorophyll-a and its distribution over the Kaštela Bay and the Brač Channel on 30 September 2017.

**Figure 9.**Map of the predicted Chlorophyll-a and its distribution over Kaštela Bay and the Brač Channel on 2 April 2021—algae blooms occurs.

**Figure 10.**Example of excluded boats from Chl-a calculation in the Brač Channel on 30 September 2017.

Station | N | Latitude | Longitude | Minimum | Maximum | Mean | Standard Deviations |
---|---|---|---|---|---|---|---|

S01 | 25 | 43.503411 | 16.208292 | 0.11 | 0.97 | 0.41 | 0.21 |

S02 | 20 | 43.541706 | 16.401844 | 0.07 | 1.36 | 0.62 | 0.36 |

S03 | 42 | 43.53 | 16.453333 | 0.19 | 2.28 | 0.81 | 0.40 |

S04 | 6 | 43.5359 | 16.46885 | 0.19 | 1.79 | 0.75 | 0.49 |

S05 | 7 | 43.53409722 | 16.47113056 | 0.24 | 1.5 | 0.61 | 0.38 |

S06 | 6 | 43.53319444 | 16.48310556 | 0.26 | 4.35 | 1.27 | 1.33 |

S07 | 55 | 43.518333 | 16.381667 | 0.07 | 1.56 | 0.58 | 0.37 |

S08 | 16 | 43.503183 | 16.433797 | 0.26 | 1.13 | 0.55 | 0.19 |

S09 | 15 | 43.488367 | 16.436667 | 0.09 | 1.54 | 0.39 | 0.32 |

S10 | 34 | 43.426719 | 16.393519 | 0.07 | 1.63 | 0.38 | 0.28 |

S11 | 10 | 43.42385 | 16.673533 | 0.10 | 0.59 | 0.31 | 0.16 |

S12 | 11 | 43.432356 | 16.682061 | 0.09 | 0.56 | 0.31 | 0.16 |

S13 | 7 | 43.444792 | 16.690806 | 0.21 | 3.13 | 1.14 | 0.95 |

Band | Spatial Resolution (m) | Central Wavelength (nm) |
---|---|---|

B01 | 60 | 443 |

B02 | 10 | 490 |

B03 | 10 | 560 |

B04 | 10 | 665 |

B05 | 20 | 705 |

B06 | 20 | 740 |

B07 | 20 | 783 |

B08 | 10 | 842 |

B8A | 20 | 865 |

B09 | 60 | 945 |

B10 | 60 | 1375 |

B11 | 20 | 1610 |

B12 | 20 | 2190 |

N = 118 | Minimum | Maximum | Mean | Standard Deviations |
---|---|---|---|---|

B01 | 0.0007 | 6.5535 | 0.6344 | 1.0987 |

B02 | 0.0380 | 6.3595 | 0.6079 | 1.0585 |

B03 | 0.0131 | 6.1288 | 0.5567 | 0.9901 |

B04 | 0.0039 | 6.2861 | 0.4706 | 0.9772 |

B05 | 0.0059 | 6.4309 | 0.5171 | 1.0122 |

B06 | 0.0013 | 6.2887 | 0.5437 | 1.019 |

B07 | 0.0059 | 6.2665 | 0.5608 | 1.0238 |

B08 | 0.0125 | 6.4277 | 0.5714 | 1.0905 |

B09 | 0.0020 | 6.5535 | 0.7290 | 1.4344 |

B11 | 0.0026 | 3.5835 | 0.4883 | 0.7806 |

B12 | 0.0013 | 3.1627 | 0.3980 | 0.6392 |

B8A | 0.0026 | 6.2075 | 0.5770 | 1.0353 |

ChlA | 0.0700 | 2.0500 | 0.5361 | 0.3730 |

Number of Items | Data Set | ${\mathit{R}}^{2}$ | RMSE |
---|---|---|---|

118 (Original) | Train | 0.0163 | 0.3290 |

Test | 0.0046 | 0.4672 | |

328 (Vertically Augmented) | Train | 0.4727 | 0.3031 |

Test | 0.4544 | 0.2190 |

**Table 5.**Analysis of Ridge regression performance on different features, where ${B}_{i}$ represents a value of each Sentinel-2 band (B01-B09, B8A, and B11-12).

Features | Data Set | ${\mathit{R}}^{2}$ | RMSE |
---|---|---|---|

${B}_{i},i=\{1,\dots ,12\}$ | Train | 0.4727 | 0.3031 |

Test | 0.4544 | 0.2190 | |

${B}_{i},{\left({B}_{i}\right)}^{2}i=\{1,\dots ,12\}$ | Train | 0.6147 | 0.2202 |

Test | 0.6008 | 0.3257 | |

${B}_{i},{\left({B}_{i}\right)}^{2},log\left({B}_{i}\right)i=\{1,\dots ,12\}$ | Train | 0.6236 | 0.2689 |

Test | 0.6350 | 0.2512 | |

${B}_{i},{\left({B}_{i}\right)}^{2},log\left({B}_{i}\right),\sqrt{{B}_{i}}i=\{1,\dots ,12\}$ | Train | 0.6277 | 0.2271 |

Test | 0.6266 | 0.2836 | |

${B}_{i},{\left({B}_{i}\right)}^{2},log\left({B}_{i}\right),\sqrt{{B}_{i}},\frac{1}{{B}_{i}},i=\{1,\dots ,12\}$ | Train | 0.6850 | 0.2254 |

Test | 0.6599 | 0.2051 |

${\mathit{\alpha}}_{\mathit{i}}$ | ${\mathit{\gamma}}_{\mathit{i}}$ | ${\mathit{\delta}}_{\mathit{i}}$ | ${\mathit{\theta}}_{\mathit{i}}$ | ${\mathit{\varphi}}_{\mathit{i}}$ |
---|---|---|---|---|

${\alpha}_{1}$ = 0.2451 | ${\gamma}_{1}$ = −0.0686 | ${\delta}_{1}$ = 0.0382 | ${\theta}_{1}$ = 0.0001 | ${\varphi}_{1}$ = −0.0079 |

${\alpha}_{2}$ = 0.0747 | ${\gamma}_{2}$ = −0.0057 | ${\delta}_{2}$ = −0.0921 | ${\theta}_{2}$ = −0.1053 | ${\varphi}_{2}$ = −0.3762 |

${\alpha}_{3}$ = −0.0171 | ${\gamma}_{3}$ = −0.0553 | ${\delta}_{3}$ = −0.0442 | ${\theta}_{3}$ = 0.0013 | ${\varphi}_{3}$ = 0.1658 |

${\alpha}_{4}$ = −0.0516 | ${\gamma}_{4}$ = 0.1721 | ${\delta}_{4}$ = −0.1596 | ${\theta}_{4}$ = −0.0001 | ${\varphi}_{4}$ = −0.2753 |

${\alpha}_{5}$ = −0.0321 | ${\gamma}_{5}$ = 0.0774 | ${\delta}_{5}$ = −0.1365 | ${\theta}_{5}$ = −0.0032 | ${\varphi}_{5}$ = −0.1732 |

${\alpha}_{6}$ = −0.0517 | ${\gamma}_{6}$ = −0.2569 | ${\delta}_{6}$ = −0.1073 | ${\theta}_{6}$ = 0.0035 | ${\varphi}_{6}$ = −0.0486 |

${\alpha}_{7}$ = 0.0440 | ${\gamma}_{7}$ = −0.1091 | ${\delta}_{7}$ = −0.0820 | ${\theta}_{7}$ = −0.00003 | ${\varphi}_{7}$ = −0.0969 |

${\alpha}_{8}$ = −0.0128 | ${\gamma}_{8}$ = 0.0763 | ${\delta}_{8}$ = −0.0855 | ${\theta}_{8}$ = −0.0014 | ${\varphi}_{8}$ = 0.0106 |

${\alpha}_{9}$ = 0.0892 | ${\gamma}_{9}$ = 0.0204 | ${\delta}_{9}$ = −0.0072 | ${\theta}_{9}$ = 0.0003 | ${\varphi}_{9}$ = 0.2031 |

${\alpha}_{10}$ = 0.0496 | ${\gamma}_{10}$ = 0.0465 | ${\delta}_{10}$ = −0.2303 | ${\theta}_{10}$ = 0.0003 | ${\varphi}_{10}$ = 0.0025 |

${\alpha}_{11}$ = −0.1300 | ${\gamma}_{11}$ = −0.0366 | ${\delta}_{11}$ = −0.1964 | ${\theta}_{11}$ = −0.0006 | ${\varphi}_{11}$ = −0.1228 |

${\alpha}_{12}$ = 0.2741 | ${\gamma}_{12}$ = 0.1559 | ${\delta}_{12}$ = 0.0312 | ${\theta}_{12}$ = 0.0001 | ${\varphi}_{12}$ = 0.1713 |

**Table 7.**Comparison of statistical evaluation indices of OC3, C2RCC, and our algorithm for predicting Chl-a concentration in respect to in situ measurements.

Algorithm | RMSE | ${\mathit{R}}^{2}$ |
---|---|---|

OC3 (Applied on the whole data set) | 2.9255 | −698.1315 |

C2RCC (Applied on selection of images) | 1.6348 | −6.9908 |

Our algorithm (Applied on test set) | 0.2051 | 0.6599 |

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**MDPI and ACS Style**

Ivanda, A.; Šerić, L.; Bugarić, M.; Braović, M.
Mapping Chlorophyll-a Concentrations in the Kaštela Bay and Brač Channel Using Ridge Regression and Sentinel-2 Satellite Images. *Electronics* **2021**, *10*, 3004.
https://doi.org/10.3390/electronics10233004

**AMA Style**

Ivanda A, Šerić L, Bugarić M, Braović M.
Mapping Chlorophyll-a Concentrations in the Kaštela Bay and Brač Channel Using Ridge Regression and Sentinel-2 Satellite Images. *Electronics*. 2021; 10(23):3004.
https://doi.org/10.3390/electronics10233004

**Chicago/Turabian Style**

Ivanda, Antonia, Ljiljana Šerić, Marin Bugarić, and Maja Braović.
2021. "Mapping Chlorophyll-a Concentrations in the Kaštela Bay and Brač Channel Using Ridge Regression and Sentinel-2 Satellite Images" *Electronics* 10, no. 23: 3004.
https://doi.org/10.3390/electronics10233004