# Comparing Deterministic and Stochastic Methods in Geospatial Analysis of Groundwater Fluoride Concentration

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Dataset Description

#### 2.2. Interpolation Methods

#### 2.2.1. Inverse Distance Weighting

_{i}is the known value, n is the total number of known values used in the interpolation process, w

_{i}is the assigned weight to the known value, d

_{i}is the distance between known and predicted values, u is the power parameter where weight decreases as distance increases from the prediction location. In this study, we used the commonly used inverse of the distance raised to the 2nd power [24].

#### 2.2.2. Radial Basis Functions

_{j}) refers to the RBFs, d

_{j}is the distance between the measured value and the predicted value x, f

_{i}(x) is a trend function considered as a member of a basis for the space of polynomials of degree <m, coefficients a

_{i}and b

_{i}are calculated by means of the resolution of the following system of n + m linear equations where n is the number of measured values used in the interpolation of the surface Z(x) [28].

#### 2.2.3. Local Polynomial Interpolation

#### 2.2.4. Kriging Methods

_{i}) and z(x

_{i}+ h) are the sample values at two points separated by the distance interval h [31]. Kriging fits this mathematical function to a defined set of data points, or to all points within a specified radius, to predict the values for the unknown location.

_{0}, Z(x

_{i}) is the measured value at the ith location, ${\lambda}_{i}$ is the weight assigned to the measured value at the ith location and n is the number of measured values. The sum of weights in the above equation is equal to unity, i.e., $\sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathsf{\lambda}}_{\mathrm{i}}=1$.

#### 2.3. Validation of the Interpolation Methods

#### 2.3.1. Leave-One-Out Cross-Validation

#### 2.3.2. Hold-Out Validation

#### 2.3.3. Validation with an Independent Dataset

#### 2.4. Comparison of the Interpolation Methods

_{o}(x

_{i}) and z

_{p}(x

_{i}) are the observed and predicted values at location ‘i’, ‘n’ is the sample size. The smaller the MRE and RMSE values, the better the predictive power of the methods.

_{i}is the observed value and, n is the number of values.

_{i}and y

_{i}are the measured and predicted values, and $\stackrel{-}{X}$ and $\stackrel{-}{Y}$ are the mean of the measured and predicted values.

## 3. Results

#### 3.1. Measured Fluoride Concentration

#### 3.2. Variation Based on Aquifer Type

#### 3.3. Statistical Accuracy of Various Methods

#### 3.4. Correlation and Prediction Error

#### 3.5. Prediction of Contaminated Areas Using Various Methods

#### 3.6. Over- and Under-Estimation of Contaminated Areas

## 4. Discussion

^{2}, depending on the topography and geology, except for areas that were not accessible or there were no existing wells to monitor. Nevertheless, the fluoride concentration patterns were spatially correlated. Even though the r values were low, this did not mean that the values predicted by the interpolation methods were not useful, but rather that they can give a spatial orientation in the high or low values in the region.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**Measured fluoride concentration versus the calculated error in the hold-out validation method.

**Figure 5.**Measured fluoride concentration versus the calculated error in the validation method with an independent dataset.

**Figure 6.**Over- and under-estimation of fluoride concentration obtained using all methods as compared with IDW (note that negative values indicate under-estimation and positive values indicate over-estimation).

**Figure 7.**Residual histogram showing under-estimation and over-estimation in predicted values in the LOOCV method.

**Figure 8.**Residual histogram showing under-estimation and over-estimation in predicted values in the hold-out validation method.

**Figure 9.**Residual histogram showing under-estimation and over-estimation in predicted values in the validation method with an independent dataset.

Aquifer Type | Number of Measured Fluoride Samples | Range (mg/L) | Mean (mg/L) | SD | Number of Samples above 1.5 mg/L of Fluoride |
---|---|---|---|---|---|

Alluvium | 2368 | 0.01–2.77 | 0.47 | 0.38 | 48 |

Banded Gneissic Complex | 380 | 0.05–1.77 | 0.71 | 0.35 | 8 |

Charnockite | 2900 | 0.01–5.00 | 0.62 | 0.48 | 146 |

Gneiss | 6479 | 0.01–5.00 | 0.77 | 0.51 | 563 |

Granite | 307 | 0.01–2.50 | 0.88 | 0.49 | 32 |

Laterite | 38 | 0.05–1.95 | 0.41 | 0.39 | 1 |

Limestone | 67 | 0.05–1.50 | 0.59 | 0.36 | 0 |

Quartzite | 27 | 0.15–1.68 | 0.93 | 0.48 | 5 |

Sandstone | 1013 | 0.01–4.90 | 0.40 | 0.40 | 18 |

Shale | 6 | 0.55–1.35 | 0.92 | 0.29 | 0 |

Total | 13,585 | 0.01–5.00 | 0.66 | 0.49 | 821 |

Measure | IDW | RBF | LPI | OK | Gaussian Kriging | Spherical Kriging | Simple Kriging | UK | EBK |
---|---|---|---|---|---|---|---|---|---|

LOOCV | |||||||||

Predicted mean | 0.67 | 0.67 | 0.67 | 0.67 | 0.67 | 0.67 | 0.67 | 0.67 | 0.67 |

MRE | 0.53 | 0.54 | 0.54 | 0.53 | 0.53 | 0.54 | 0.56 | 0.53 | 0.54 |

RMSE | 0.32 | 0.31 | 0.31 | 0.31 | 0.31 | 0.31 | 0.31 | 0.31 | 0.31 |

CV, predicted (%) | 42 | 37 | 36 | 37 | 38 | 38 | 32 | 37 | 37 |

r, measured vs. predicted | 0.32 | 0.35 | 0.34 | 0.35 | 0.34 | 0.34 | 0.34 | 0.35 | 0.35 |

r, measured vs. error | 0.47 | 0.58 | 0.60 | 0.58 | 0.55 | 0.55 | 0.68 | 0.58 | 0.58 |

Hold-out validation | |||||||||

Predicted mean | 0.67 | 0.67 | 0.68 | 0.67 | 0.68 | 0.68 | 0.65 | 0.67 | 0.68 |

MRE | 0.56 | 0.58 | 0.58 | 0.58 | 0.57 | 0.57 | 0.55 | 0.58 | 0.58 |

RMSE | 0.32 | 0.31 | 0.31 | 0.31 | 0.31 | 0.31 | 0.31 | 0.31 | 0.31 |

CV, predicted (%) | 41 | 36 | 34 | 36 | 37 | 37 | 34 | 36 | 35 |

r, measured vs. predicted | 0.28 | 0.31 | 0.30 | 0.31 | 0.31 | 0.31 | 0.30 | 0.31 | 0.30 |

r, measured vs. error | 0.49 | 0.58 | 0.62 | 0.58 | 0.57 | 0.57 | 0.64 | 0.58 | 0.60 |

Validation with an independent dataset | |||||||||

Predicted mean | 0.86 | 0.86 | 0.87 | 0.86 | 0.86 | 0.85 | 0.82 | 0.86 | 0.86 |

MRE | 0.50 | 0.51 | 0.50 | 0.50 | 0.50 | 0.50 | 0.53 | 0.50 | 0.50 |

RMSE | 1.48 | 1.48 | 1.48 | 1.48 | 1.48 | 1.48 | 1.52 | 1.48 | 1.49 |

CV, predicted (%) | 8 | 9 | 5 | 8 | 11 | 11 | 6 | 8 | 8 |

r, measured vs. predicted | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.07 | 0.00 | 0.02 |

r, measured vs. error | 1.00 | 1.00 | 1.00 | 1.00 | 0.99 | 0.99 | 1.00 | 1.00 | 1.00 |

Method | Fluoride Range and Area in % | |||
---|---|---|---|---|

<0.5 | 0.5 to 1 | 1 to 1.5 | >1.5 | |

Very Low Fluoride | Low Fluoride | Suitable Range | Unsuitable | |

IDW | 28.0 | 61.2 | 10.6 | 0.2 |

RBF | 25.4 | 64.4 | 10.2 | - |

LPI | 26.0 | 63.2 | 10.6 | 0.2 |

OK | 26.5 | 63.6 | 9.9 | - |

Gaussian kriging | 26.6 | 63.2 | 10.2 | - |

Spherical kriging | 26.5 | 63.2 | 10.3 | - |

Simple kriging | 29.6 | 65.9 | 4.5 | - |

UK | 26.5 | 63.6 | 9.9 | - |

EBK | 27.1 | 61.5 | 11.4 | - |

Comparison | Under-Estimated | Equal | Over-Estimated |

IDW minus OK | 11.2 | 77.9 | 10.9 |

IDW minus Gaussian Kriging | 12.9 | 74.4 | 12.7 |

IDW minus Spherical Kriging | 13.2 | 73.9 | 12.9 |

IDW minus Simple Kriging | 11.5 | 63.3 | 25.2 |

IDW minus UK | 11.2 | 77.9 | 10.9 |

IDW minus EBK | 11 | 78.2 | 10.8 |

IDW minus RBF | 7.2 | 85.1 | 7.7 |

IDW minus LPI | 12.3 | 74.8 | 12.9 |

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**MDPI and ACS Style**

Brindha, K.; Taie Semiromi, M.; Boumaiza, L.; Mukherjee, S.
Comparing Deterministic and Stochastic Methods in Geospatial Analysis of Groundwater Fluoride Concentration. *Water* **2023**, *15*, 1707.
https://doi.org/10.3390/w15091707

**AMA Style**

Brindha K, Taie Semiromi M, Boumaiza L, Mukherjee S.
Comparing Deterministic and Stochastic Methods in Geospatial Analysis of Groundwater Fluoride Concentration. *Water*. 2023; 15(9):1707.
https://doi.org/10.3390/w15091707

**Chicago/Turabian Style**

Brindha, K., Majid Taie Semiromi, Lamine Boumaiza, and Subham Mukherjee.
2023. "Comparing Deterministic and Stochastic Methods in Geospatial Analysis of Groundwater Fluoride Concentration" *Water* 15, no. 9: 1707.
https://doi.org/10.3390/w15091707