# Surface Water Quality Assessment through Remote Sensing Based on the Box–Cox Transformation and Linear Regression

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

^{2}values achieved for TOC, TDS, and Chl-a water quality models after the band discrimination process were found 0.926, 0.875, and 0.810, respectively, achieving a fair fitting to real water quality data measurements. Finally, a comparison between estimated and measured water quality values not previously used for model development was carried out to validate these models. In this validation process, a good fit of 98% and 93% was obtained for TDS and TOC, respectively, whereas an acceptable fit of 81% was obtained for Chl-a. This study proposes an interesting alternative for ordered and standardized steps applied to generate mathematical models for the estimation of TOC, TDS, and Chl-a based on water quality parameters measured in the field and using satellite images.

## 1. Introduction

^{2}> 85%).

## 2. Materials and Methods

#### 2.1. Study Area

#### 2.2. Methodology

#### 2.2.1. Satellite Imagery Acquisition

#### 2.2.2. Imagery Pre-Processing

#### 2.2.3. Reflectance Data Extraction

#### 2.2.4. Water Quality Monitoring

#### 2.2.5. Box–Cox Transformation of Water Quality Parameters

_{1}power after changing it to a certain amount λ

_{2}(often equal to 0). These transformations could be square roots, logarithms, reciprocals, and/or other common transformations (Table 2) [43]. Hence, the Box–Cox transformation (Equation (3)) is defined as a continuous function that varies as a function of power (λ) [44].

_{1}and λ

_{2}are values that, when substituted in Equation (3), the standard deviation of y’ will be zero.

_{1}values of 2, 1, 0.5, 0.33, 0, −0.5, and 1 shown in Table 2 were investigated to determine which, if any, is most suitable. The software was used to solve for the optimum value of λ

_{1}using maximum likelihood estimation. Once the Box–Cox transformation was performed, the normality of the data was evaluated using the Kolmogorov–Smirnov goodness of fit test.

#### 2.2.6. Multiple Linear Regression

_{0}is the intercept when all the predictors ${x}_{1},{x}_{2},\dots ,{x}_{i}$ are all zero, ${b}_{1}$, ${b}_{2}$, …, ${b}_{i}$ are the linear regression coefficients obtained from the fitted values and $\epsilon $ is a random error corresponding to the n observations that are also assumed to be uncorrelated random variables [46].

#### 2.2.7. Model Performance Evaluation

^{2}) and the root-mean-square error (RMSE). Equation (5) was used to estimate r

^{2}, which is a number between 0 and 1 that measures how well a model estimates an outcome [47]:

#### 2.2.8. Multiple Linear Regression Significance Testing

_{i}, calculated using Equation (8):

#### 2.2.9. Water Quality Model Validation

^{2}were used to estimate the models’ fitness to field water quality measurements.

#### 2.2.10. Water Quality Mapping

## 3. Results and Discussion

#### 3.1. Water Quality from Field Sampling

^{3}. These values are within the reported by Fregoso-López et al. [52] who found a maximum Chl-a concentration of 3.4 mg/m

^{3}and a minimum concentration of 0.3 mg/m

^{3}in the Miguel Hidalgo y Costilla reservoir located in El Fuerte, Sinaloa.

#### 3.2. Box–Cox Transformation

^{2}values greater than 0.85 in all cases. Table 4 shows the results of the Kolmogorov–Smirnov goodness of fit test using the normalized water quality parameters. According to the Kolmogorov one-sample statistic (Dn) values and their respective p-values, the Box–Cox transformation was a good tool to normalize the water quality parameters.

#### 3.3. Multiple Linear Regression Modeling and Discriminant Analysis

^{2}greater than 0.80, considered satisfactory compared with the results of other empirical models used to estimate water quality through remote sensing [55,56].

^{2}value compared to the initial one. This same methodology was carried out for each of the water quality parameters considered in this study and the results of the simplified models are shown in Table 8.

^{2}= 0.875; RMSE = 3.2613) can be considered satisfactory showing a better fit compared to other studies [58]. This could be attributed to the bands used for TDS estimation since low model accuracies have been reported in several studies that have only used B3, B4, and B5 bands (530 to 890 nm) of Landsat 8 [59,60]. According to Zhao et al. [61] (2020), the B3–B5 wavelength range (530–890 nm) can be used to characterize whether the water body contains phytoplankton chlorophyll (560–590 nm), cyanobacteria (620 nm), phycocyanin (650 nm), algae chlorophyll (675 nm), and suspended inorganic matter (810 nm). However, in this study, the discriminant analysis demonstrated that TDS estimation should be carried out using the bands B1, B2, B3, B4, B5, and B6 of Landsat 8. The use of a wider wavelength range could explain the satisfactory fit obtained since higher dissolved content of inorganic and organic substances could be detected, such as the colored dissolved organic matter (CDOM) (420–555 nm). Our results agree with Maliki et al. [62], who successfully predicted the TDS of surface water in Bangladesh using Landsat 8 OLI and multiple linear models (r

^{2}= 0.95).

^{2}value (see Table 3) in comparison with TOC and TDS, likely because Chl-a is a biological parameter showing exponential growth. In addition, Chl-a is more susceptible to seasonal variations related to physical, chemical, and climatic factors [63,64].

^{2}= 0.86 (RMSE = 34.6). Bohn et al. [66] reported r

^{2}= 0.83 estimating Chl-a using Landsat 7 bands B3 and B4. The accuracy of these models was similar to the results of this study (r

^{2}= 0.81, RMSE = 3.1267) (Figure 4c). In the Chl-a model, some bands (such as B2 and B7) appear in the final model generated and do not appear in other studies, such as the one performed by Bohn et al. [66]. This is because these studies estimate Chl-a by calculating predetermined indices such as the normalized difference vegetation index (NDVI), normalized area vegetation index (NAVI), enhanced vegetation index (EVI), and ratio vegetation index (RVI).

^{2}= 0.95) for a linear mixture model used to estimate Chl-a in Lake Balaton, Hungary, using Landsat TM imagery. The accuracy of the water quality models can be improved by removing image interferences. For instance, in this study, the DOS atmospheric correction method was used which assumes that there are dark targets in the image, such as water and dense vegetation. But when the water body is turbid, such as the reservoir in this study, the reflection of water in the near-infrared band is close to 0, which leads to uncertainties of the atmospheric correction over water [68]. Other atmospheric correction methods have been proven to be effective for turbid waters, such as ACOLITE [69,70], ACIX-Aqua [71,72], iCOR [73], POLYMER [74], or MDM [68]. Thus, the performance of these algorithms on the regression models should be investigated in depth in further studies.

#### 3.4. Model Validation

#### 3.5. Spatial and Temporal Distribution of Water Quality Parameters from Optimized Models

^{2}) indicators. A very low RMSE value was obtained when TOC observed and estimated concentrations were contrasted (Figure 6b). This figure also shows a fair estimation for TDS and Chl-a from the optimized models and satellite imagery (Figure 6a,c). The r

^{2}values obtained for TDS and Chl-a were higher than 0.81, which indicated a low variation between the observed and estimated water quality parameters.

## 4. Conclusions

^{2}) values were obtained for the different water quality parameters estimated at the different stages (model development, discrimination, and validation). The obtained models were then used to estimate water quality parameters during periods where field monitoring was not conducted, which represents a crucial tool for decision-making.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Descriptive analysis of the spatial and temporal distribution of water quality parameters (

**a**) TOC, (

**b**) TDS, and (

**c**) Chl-a in the studied reservoir.

**Figure 4.**Estimated and observed values for (

**a**) TOC, (

**b**) TDS, and (

**c**) Chl-a using the simplified models.

**Figure 5.**Validation of TDS (

**a**), TOC (

**b**), and Chl-a (

**c**) with randomly selected data not previously used for model development.

Sensor | Year | Acquisition Date | Path/Row |
---|---|---|---|

Landsat 8 OLI | 2015 | May 4th | 32/43 |

October 27th | |||

2016 | May 22nd | ||

September 11th | |||

2017 | March 6th | ||

September 30th | |||

2018 | February 2nd | ||

October 2nd | |||

2019 | January 17th |

Power | Transformation | Description |
---|---|---|

${\lambda}_{1}=2$ | ${y}^{\prime}={y}^{2}$ | Square |

${\lambda}_{1}=1$ | ${y}^{\prime}=y$ | Untransformed data |

${\lambda}_{1}=0.5$ | ${y}^{\prime}=\sqrt{y}$ | Square root |

${\lambda}_{1}=0.33$ | ${y}^{\prime}=\sqrt[3]{y}$ | Cube root |

${\lambda}_{1}=0{}^{\ast}$ | ${y}^{\prime}=\mathrm{ln}\left(y\right)$ | Logarithm |

${\lambda}_{1}=-0.5$ | ${y}^{\prime}=\frac{1}{\sqrt{y}}$ | Inverse square root |

${\lambda}_{1}=-1$ | ${y}^{\prime}=\frac{1}{y}$ | Reciprocal |

Parameter | Box–Cox Optimized Mathematical Model | r^{2} |
---|---|---|

TOC | Box–Cox (TOC) = 1 + (TOC^{1.3294} − 1)/(1.3294 × 4.57379^{0.329397}) | 0.96 |

TDS | Box–Cox (TDS) = 1 + (TDS^{4.16779} − 1)/(4.16779 × 97.6453^{3.16779}) | 0.88 |

Chl-a | Box–Cox (Chl-a) = 1 + (Chl-a^{0.333508} − 1)/(0.333508 × 1.43584^{0.666492}) | 0.85 |

Water Quality Normalized Parameter | Kolmogorov-Smirnov Test | |
---|---|---|

Dn Value | p-Value | |

Chl-a | 0.2393 | 0.2544 |

TDS | 0.1644 | 0.7149 |

COT | 0.1554 | 0.7769 |

**Table 5.**Multiple linear regression models for the water quality parameters of the ALM reservoir based on the reflectance values of the Landsat 8 satellite images.

Parameter | Multiple Linear Regression Model | r^{2} | RMSE |
---|---|---|---|

TOC | Box–Cox (TOC) = 9.61963 − 700.238 × B1 + 707.462 × B2 − 39.2047 × B3 − 25.1903 × B4 − 18.2743 × B5 + 216.704 × B6 − 243.629 × B7 | 0.95 | 0.165 |

TDS | Box–Cox (TDS) = 34.849 − 3057.55 × B1 + 4137.63 × B2 − 2526.38 × B3 + 2696.15 × B4 + 1827.6 × B5 − 6080.39 × B6 + 2858.29 × B7 | 0.88 | 3.867 |

Chl-a | Box–Cox (Chl-a) = −38.8501 + 212.068 × B1 + 1213.14 × B2 + 1207.01 × B3 − 2935.1 × B4 + 261.245 × B5 − 2468.64 × B6 + 3907.26 × B7 | 0.87 | 3.430 |

Iteration | Model | Discriminated Bands | r^{2} | RMSE |
---|---|---|---|---|

1 | Box–Cox (TOC) = 9.61963 − 700.238 × B1 + 707.462 × B2 − 39.2047 × B3 − 25.1903 × B4 − 18.2743 × B5 + 216.704 × B6 − 243.629 × B7 | 0 | 0.9608 | 0.1658 |

2 | Box–Cox (TOC) = 9.82457 − 711.379 × B1 + 705.351 × B2 − 48.6016 × B3 − 25.9899 × B5 + 245.128 × B6 − 273.573 × B7 | B4 | 0.9611 | 0.1676 |

3 | Box–Cox (TOC) = 9.03939 − 661.472 × B1 + 667.836 × B2 − 45.8407 × B3 + 147.039 × B6 − 191.869 × B7 | B4, B5 | 0.9423 | 0.1694 |

4 | Box–Cox (TOC) = 8.92542 − 682.488 × B1 + 677.616 × B2 − 38.4808 × B3 + 3.95873 × B6 | B4, B5, B7 | 0.9521 | 0.1829 |

5 | Box–Cox (TOC) = 8.87165 − 688.128 × B1 + 688.322 × B2 − 40.7919 × B3 | B4, B5, B7, B6 | 0.9520 | 0.1835 |

6 | Box–Cox (TOC) = 9.15197 − 620.429 × B1 + 587.138 × B2 | B4, B5, B7, B6, B3 | 0.9350 | 0.2024 |

Iteration | Parameter | Estimate | Standard Error | t-Statistic | p-Value |
---|---|---|---|---|---|

TOC model with all bands | |||||

1 | Constant | 9.61963 | 2.15541 | 4.46302 | 0.0012 |

B1 | −700.238 | 100.765 | −6.94922 | <0.0000 | |

B2 | 707.462 | 109.424 | 6.4653 | 0.0001 | |

B3 | −39.2047 | 53.3205 | −0.735266 | 0.4791 | |

B4 | −25.1903 | 136.359 | −0.184735 | 0.8571 | |

B5 | −18.2743 | 56.7695 | −0.321903 | 0.7542 | |

B6 | 216.704 | 237.583 | 0.912119 | 0.3832 | |

B7 | −243.629 | 244.321 | −0.997167 | 0.3422 | |

TOC model after discriminating B4 | |||||

2 | Constant | 9.82457 | 1.46585 | 6.70228 | <0.0000 |

B1 | −711.379 | 91.8242 | −7.74718 | <0.0000 | |

B2 | 705.351 | 97.8209 | 7.21064 | <0.0000 | |

B3 | −48.6016 | 23.9244 | −2.03147 | 0.0671 | |

B5 | −25.9899 | 40.4164 | −0.643053 | 0.5334 | |

B6 | 245.128 | 185.017 | 1.32489 | 0.2121 | |

B7 | −273.573 | 187.063 | −1.46247 | 0.1716 | |

TOC model after discriminating B4 and B5 | |||||

3 | Constant | 9.03939 | 0.526595 | 17.1657 | <0.0000 |

B1 | −661.472 | 56.2812 | −11.753 | <0.0000 | |

B2 | 667.836 | 80.4374 | 8.30256 | <0.0000 | |

B3 | −45.8407 | 23.0887 | −1.98542 | 0.0704 | |

B6 | 147.039 | 102.673 | 1.43211 | 0.1776 | |

B7 | −191.869 | 134.751 | −1.42388 | 0.18 | |

TOC model after discriminating B4, B5, and B7 | |||||

4 | Constant | 8.92542 | 0.531479 | 16.7936 | <0.0000 |

B1 | −682.488 | 55.898 | −12.2095 | <0.0000 | |

B2 | 677.616 | 83.0813 | 8.15606 | <0.0000 | |

B3 | −38.4808 | 23.4277 | −1.64254 | 0.1244 | |

B6 | 3.95873 | 21.4212 | 0.184804 | 0.8562 | |

TOC model after discriminating B4, B5, B7, and B6 | |||||

5 | Constant | 8.87165 | 0.47848 | 18.5413 | <0.0000 |

B1 | −688.128 | 47.2176 | −14.5735 | <0.0000 | |

B2 | 688.322 | 60.099 | 11.4531 | <0.0000 | |

B3 | −40.7919 | 19.3641 | −2.10657 | 0.0537 | |

TOC model after discriminating B4, B5, B7, B6, and B3 | |||||

6 | Constant | 9.15197 | 0.510097 | 17.9416 | <0.0000 |

B1 | −620.429 | 42.8204 | −14.4891 | <0.0000 | |

B2 | 587.138 | 44.5094 | 13.1913 | <0.0000 |

Parameter | Final Model | Bands Used | r^{2} |
---|---|---|---|

TOC | Box–Cox (TOC) = 9.15197 − 620.429 × B1 + 587.138 × B2 | B1, B2 | 0.9263 |

TDS | Box–Cox (TDS) = 55.7042 − 3387.46 × B1 + 4108.64 × B2 − 2874.84 × B3 + 3514.37 × B4 + 1386.56 × B5 − 3490.39 × B6 | B1, B2, B3, B4, B6 | 0.8753 |

Chl-a | Box–Cox (Cha-a) = −24.4586 + 1204.69 × B2 + 956.358 × B3 − 2506.71 × B4 + 996.356 × B7 | B2, B3, B4, B7 | 0.8100 |

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© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Loaiza, J.G.; Rangel-Peraza, J.G.; Monjardín-Armenta, S.A.; Bustos-Terrones, Y.A.; Bandala, E.R.; Sanhouse-García, A.J.; Rentería-Guevara, S.A.
Surface Water Quality Assessment through Remote Sensing Based on the Box–Cox Transformation and Linear Regression. *Water* **2023**, *15*, 2606.
https://doi.org/10.3390/w15142606

**AMA Style**

Loaiza JG, Rangel-Peraza JG, Monjardín-Armenta SA, Bustos-Terrones YA, Bandala ER, Sanhouse-García AJ, Rentería-Guevara SA.
Surface Water Quality Assessment through Remote Sensing Based on the Box–Cox Transformation and Linear Regression. *Water*. 2023; 15(14):2606.
https://doi.org/10.3390/w15142606

**Chicago/Turabian Style**

Loaiza, Juan G., Jesús Gabriel Rangel-Peraza, Sergio Alberto Monjardín-Armenta, Yaneth A. Bustos-Terrones, Erick R. Bandala, Antonio J. Sanhouse-García, and Sergio A. Rentería-Guevara.
2023. "Surface Water Quality Assessment through Remote Sensing Based on the Box–Cox Transformation and Linear Regression" *Water* 15, no. 14: 2606.
https://doi.org/10.3390/w15142606