# An Automated Approach to Groundwater Quality Monitoring—Geospatial Mapping Based on Combined Application of Gaussian Process Regression and Bayesian Information Criterion

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Site Description and Available Dataset

#### 2.2. Data Preparation and Methodology

#### 2.3. Water Quality Index Calculation Based on PCA and Weighted Factors

#### 2.3.1. PCA Theory

#### 2.3.2. Construction of WQI Based on PCA

#### 2.4. Machine Learning Approach for Geospatial Modelling of WQI with Automatic Kernel Detection

#### 2.4.1. Gaussian Process Regression: General Overview of the Methodology

#### 2.4.2. Hyper-Parameters Selection Using Bayesian Information Criterion

#### 2.4.3. Universal and Ordinary Kriging

#### 2.5. Approach to Geospatial Modelling

#### 2.6. Validation Procedure

#### 2.7. Software

## 3. Results and Discussion

#### 3.1. PCA-Based Weighted Water Quality Index

#### 3.2. Geospatial Modeling

_{3}, Alkalinity, pH mainly contribute to overall variance according to the PCA. We can suggest, that the driving factor of high variability of these characteristics is the aquifer composition itself and basic rocks interacting with waters. The occurrence of trace elements (heavy metals), phosphate, nitrates (PC3, PC4, PC5 loadings) can also significantly contribute to low WQI. Their presence in waters can be explained by various reasons, but more likely, they are associated with the agricultural activity since all of the listed parameters are contained in macro- and micronutrients, and livestock wastes. Additionaly, Quaternary horizons, in general, are highly permeable, so they are exposed to the high risk of contamination by filtrates and pollutants’ further spread.

#### 3.3. PCA-Weighted Approach in WQI Construction

#### 3.4. Automatic Approach to Geospatial Mapping

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

WQI | Water Quality Index |

PCA | Principal Component Analysis |

GPR | Gaussian Process Regression |

BIC | Bayesian Information Criterion |

OK | Ordinary Kriging |

UK | Universal Kriging |

ML | Machine Learning |

FA | Factor Analysis |

SVM | Support Vector Machine |

ANN | Artificial Neural Network |

ABC | Approximate Bayesian Computation |

MLE | Maximum Likelihood Estimation |

RMSE | Root Mean Square Error |

DBSCAN | Density-based spatial clustering of applications with noise |

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**Figure 1.**Location map of the study area. Different colors mark source of collected water samples—wells colored in yellow; rivers colored in pink; and springs colored in blue. Blue lines represent main river streams from open-source 10-m resolution data.

**Figure 2.**Our approach use machine learning methods for weighted WQI calculation (Steps 1 and 2) and geospatial WQI prediction by using Gaussian process regression with automatic kernel search (Steps 3–5).

**Figure 3.**Gaussian Process Regression (red dashed line depicts the predictive mean and orange fill depicts the standard deviation intervals) with noisy measurements (blue dots) of the sigmoid function (solid green line) using Gaussian kernel and constant mean function.

**Figure 4.**The correlation heatmap for chemical parameters in tested freshwater samples. Figure (

**a**) present correlation coefficient) between all measured chemical parameters, while figure (

**b**) present correlation coefficient only for parameters with significant PCA loading. Initial number of water quality parameters for WQI constriction was reduced from twenty-one to fifteen after PCA.

**Figure 5.**The overall distribution of WQI in sample points. (

**A**) The graph presents the number of sample points with observed WQI and the mean value of the WQI. (

**B**) Pie chart of statistical distribution of WQI for tested samples. (

**C**) Distribution of points with estimated WQI across the study area, lower WQI values are corresponding to good groundwater quality, and higher—to poor groundwater quality. (

**D**) Ratio of WQI to spatial coordinates: X—Latitude, Y—Longitude.

**Figure 6.**Geospatial prediction of Water quality index and uncertainty maps based on different techniques: (

**A**)—GPR coupled with BIC; (

**B**)—Ordinary kriging with Gaussian variogram; (

**C**)—Universal kriging, Exponential variogram + linear drift; (

**D**)—Universal kriging, Gaussian variogramm + linear drift.

Principal Components | Comp1 | Comp2 | Comp3 | Comp4 | Comp5 |
---|---|---|---|---|---|

Eigenvalues | 6.116 | 2.057 | 1.856 | 1.543 | 1.237 |

Variance (%) | 29.12 | 9.79 | 8.84 | 7.35 | 5.89 |

Cumulative variance (%) | 29.12 | 38.92 | 47.76 | 55.10 | 61.00 |

Parameters loadings | |||||

NH${}_{4}$ | 0.0794 | 0.0041 | 0.5602 | 0.0279 | −0.0603 |

HCO${}_{3}$ | −0.0363 | 0.5385 | 0.0041 | 0.0229 | 0.0137 |

Alkalinity | −0.0364 | 0.5386 | 0.0041 | 0.0228 | 0.0136 |

pH | −0.1731 | 0.3074 | 0.2065 | −0.0889 | −0.1959 |

Hardness of water | 0.2960 | 0.2583 | −0.1245 | −0.0123 | 0.0035 |

Cr | 0.0076 | −0.0764 | −0.0718 | 0.5049 | 0.1270 |

Cu | −0.1188 | 0.0103 | 0.0489 | 0.2093 | 0.4262 |

Fe | −0.0179 | 0.0199 | −0.0408 | 0.6504 | −0.0269 |

Mn | 0.0557 | 0.0913 | 0.1145 | 0.4557 | −0.1452 |

Ni | 0.2217 | −0.1376 | −0.0030 | −0.1010 | −0.0475 |

Zn | −0.0368 | 0.1017 | −0.1915 | 0.0638 | 0.1721 |

SO${}_{4}$ | 0.1987 | −0.0145 | −0.1570 | −0.0894 | 0.3695 |

Cl | 0.5033 | −0.1380 | 0.0726 | 0.0079 | −0.1002 |

NO${}_{3}$ | 0.0666 | −0.1398 | −0.0800 | −0.1048 | 0.5048 |

NO${}_{2}$ | 0.0518 | −0.0645 | 0.1705 | 0.1495 | 0.0442 |

PO${}_{4}$ | 0.0223 | −0.0059 | 0.6047 | −0.0642 | 0.1163 |

Mineralization | 0.3729 | 0.1255 | 0.0228 | −0.0215 | 0.1407 |

Ca | 0.2973 | 0.2457 | −0.1414 | −0.0098 | −0.0152 |

Mg | 0.2552 | 0.2604 | −0.0634 | −0.0169 | 0.0540 |

Na | 0.4440 | −0.0817 | 0.1863 | 0.0101 | −0.0330 |

K | −0.1235 | 0.1455 | 0.2777 | −0.0010 | 0.5150 |

Parameter | Value |
---|---|

Gaussian kernel variance, ${\theta}_{4}$ | 0.0367 |

Gaussian kernel length scale, l | 4.86 |

Periodic kernel variance, ${\theta}_{5}$ | 0.0204 |

Periodic kernel period, T | 5.67 |

Periodic kernel length scale, s | 0.1 |

**Table 3.**Performance evaluation of selected models. Results of cross-validation of the obtained models on 5 different train/test splits.

1 | 2 | 3 | 4 | 5 | Mean | std | ||
---|---|---|---|---|---|---|---|---|

Kriging with BIC | ${R}^{2}$ | 0.729 | 0.487 | 0.609 | 0.641 | 0.702 | 0.637 | 0.098 |

approach | RMSE | 0.060 | 0.072 | 0.071 | 0.062 | 0.059 | 0.065 | 0.0063 |

Ordinary Kriging | ${R}^{2}$ | 0.580 | −0.075 | 0.599 | 0.625 | 0.575 | 0.461 | 0.300 |

Gaussian kernel | RMSE | 0.068 | 0.076 | 0.056 | 0.060 | 0.059 | 0.064 | 0.0085 |

Universal Kriging | ${R}^{2}$ | 0.610 | 0.014 | 0.604 | 0.646 | 0.622 | 0.499 | 0.271 |

Exponential kernel | RMSE | 0.070 | 0.077 | 0.056 | 0.060 | 0.058 | 0.064 | 0.0088 |

Universal Kriging | ${R}^{2}$ | 0.544 | −0.052 | 0.600 | 0.631 | 0.590 | 0.463 | 0.289 |

Gaussian kernel | RMSE | 0.071 | 0.076 | 0.055 | 0.059 | 0.058 | 0.064 | 0.0093 |

Universal Kriging | ${R}^{2}$ | −11.205 | −9.316 | −11.042 | −6.693 | −9.860 | −9.623 | 1.820 |

Polynomial kernel | RMSE | 0.129 | 0.113 | 0.109 | 0.097 | 0.103 | 0.110 | 0.0122 |

Universal Kriging | ${R}^{2}$ | 0.415 | −0.038 | 0.579 | 0.637 | 0.593 | 0.437 | 0.278 |

Periodic kernel | RMSE | 0.080 | 0.076 | 0.057 | 0.059 | 0.058 | 0.066 | 0.0114 |

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**MDPI and ACS Style**

Shadrin, D.; Nikitin, A.; Tregubova, P.; Terekhova, V.; Jana, R.; Matveev, S.; Pukalchik, M. An Automated Approach to Groundwater Quality Monitoring—Geospatial Mapping Based on Combined Application of Gaussian Process Regression and Bayesian Information Criterion. *Water* **2021**, *13*, 400.
https://doi.org/10.3390/w13040400

**AMA Style**

Shadrin D, Nikitin A, Tregubova P, Terekhova V, Jana R, Matveev S, Pukalchik M. An Automated Approach to Groundwater Quality Monitoring—Geospatial Mapping Based on Combined Application of Gaussian Process Regression and Bayesian Information Criterion. *Water*. 2021; 13(4):400.
https://doi.org/10.3390/w13040400

**Chicago/Turabian Style**

Shadrin, Dmitrii, Artyom Nikitin, Polina Tregubova, Vera Terekhova, Raghavendra Jana, Sergey Matveev, and Maria Pukalchik. 2021. "An Automated Approach to Groundwater Quality Monitoring—Geospatial Mapping Based on Combined Application of Gaussian Process Regression and Bayesian Information Criterion" *Water* 13, no. 4: 400.
https://doi.org/10.3390/w13040400