# An Effective Kalman Filter-Based Method for Groundwater Pollution Source Identification and Plume Morphology Characterization

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Framework of the Proposed Method

- Step 1: On the basis of the site investigation, the location of possible pollution sources is preliminarily determined and the initial weight and mass-loading rate for each potential pollution source are given based on expert opinion.
- Step 2: Considering the uncertainty of site information, the random hydraulic conductivity field is generated by the LHS (Latin hypercube sampling) technique assuming hydraulic conductivity in a random process.
- Step 3: Groundwater flow and the transport model are constructed and the concentration field library is obtained by Monte Carlo simulation. In the Monte Carlo simulation, each potential pollution source with unit mass-loading rate is calculated at each hydraulic conductivity realization.
- Step 4: According to the weight of the pollution source, the concentration field is randomly selected from the concentration field library and the superposed pollution plume and covariance matrix are generated in combination with the mass-loading rate of the pollution source.
- Step 5: Combined with the existing sampling data, the Kalman filter method is used to update the superposed pollution plume and the covariance matrix.
- Step 6: According to the reduction in the overall uncertainty, new sampling data are selected sequentially using variance reduction with the Kalman filter method.
- Step 7: Without adjusting the weight value of the pollution source, a linear programming model is adopted to identify the source mass-loading rate by using the existing sampling data.
- Step 8: The superposed pollution plume is generated from the concentration field library based on the weight and mass-loading rate values prior to this step.
- Step 9: Combined with the sampling data obtained prior to this step, the Kalman filter method is used to update the superposed pollution plume.
- Step 10: Based on the morphological comparison of pollution plume, the weight value of the pollution source is updated by using the alpha-cut technique.

#### 2.2. Groundwater Contaminant Transport Simulation

_{ij}is the hydraulic conductivity, H is the hydraulic head, W is the volumetric flux per unit volume (positive for inflow and negative for outflow), and x are the Cartesian coordinates.

_{ij}is the dispersion coefficient (a second-order tensor), and R is the source or sink term.

#### 2.3. Stochastic Simulation

#### 2.4. Concentration Field Library

_{j}denotes the concentration field for the jth potential source of the unit pollution mass-loading rate. Afterward, C

_{j}satisfies the following equation.

_{j}denotes the mass-loading rate of jth potential source.

_{j}, m

_{j}), j = 1, …, a), the concentration field corresponding to each hydraulic conductivity realization is generated utilizing the following procedures (Figure 3). First, according to the weight value of each pollution source, randomly select the concentration field from the n concentration fields and assign the zero concentration field to unselected ones. Second, multiply the concentration fields by the mass-loading rate of the pollution sources. Third, superpose the concentration fields under the same hydraulic conductivity field. Lastly, we generate the n concentration field realizations with the given weight and the mass-loading rate.

#### 2.5. Kalman Filter Approach

#### 2.6. Alpha-Cut Technique of a Fuzzy Set

#### 2.7. Linear Optimization Model

## 3. Application of the Proposed Method

#### 3.1. Aquifer Site

^{2}/day.

#### 3.2. Pollution Source Identification Problem

^{3}/day is missed by accident in the site investigation. The potential pollution sources (S1–S5) and the missing pumping well (Q) are shown in Figure 5.

#### 3.3. Erroneous Monitoring Data

^{3}/day pumping rates was introduced to assess the robustness of the proposed approach.

## 4. Results and Discussion

#### 4.1. Stochastic Modeling Analysis

#### 4.2. Pollution Source Identification

#### 4.3. Sensitivity Analysis

## 5. Summary and Conclusions

- (1)
- The purpose of the proposed method is to facilitate the remediation strategy of the contaminated sites in an attempt to realize the cost of the pollution source identification and the plume characterization (optimal monitoring network design).
- (2)
- The proposed Kalman filter-based method incorporates multiple techniques such as the concentration field library, the covariance reduction with the Kalman filter, the alpha-cut technique of the fuzzy set, and the linear programming model, which are demonstrated for the pollution source identification and plume characterization.
- (3)
- The performance of this methodology is evaluated on an illustrative groundwater pollution source identification problem and the identified results indicate that the proposed Kalman filter-based optimization model can give satisfactory estimations even when the random hydraulic conductivity field, erroneous monitoring data, prior information shortage of potential pollution sources, and unexpected unknown pumping well are considered.
- (4)
- The results of the sensitivity analysis investigate the effect of various algorithm parameters on convergence. It is concluded that the most important parameter is the setting of α-cuts used at the plume comparison step. The identification results are less sensitive to the setting of initial weights and is not sensitive to the setting of the initial mass-loading rate.
- (5)
- The results from the sensitivity analysis on heterogeneity of hydraulic conductivity field proves that our proposed method would be effective in identifying the true source location and characterizing the pollution plume even under a moderate heterogeneity condition, but the performance may be less satisfied. Additionally, our proposed method is completely ineffective in a highly heterogeneous field (variance = 16.0). It indicates that the proposed method can provide relatively satisfied results for a homogeneous domain or domain with small and moderate heterogeneity, but it cannot validate the transport in the relatively high heterogeneous field.
- (6)
- In this work, our proposed method is designed and assembled for two-dimensional problems and it should be modified to be integrated into three-dimensional problems. In the two-dimensional field, new sampling points are selected to minimize the overall uncertainty of the concentration field. However, for the three-dimensional field, sampling points may exist in different layers and, therefore, exhibit different uncertainties. Whether layered processing is more effective requires further study by comparing it with overall processing. The extension of the alpha-cut technique for comparison of plume in a three-dimensional field is another aspect worthy of further study.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Flow diagram of the pollution source identification using the proposed Kalman filter-based method.

**Figure 9.**The normalized contour map of the pollution plumes: (

**a**) True plume. (

**b**) Superposed plume after the Kalman filter (1 point). (

**c**) Superposed plume after the Kalman filter (2 points). (

**d**) Superposed plume after the Kalman filter (3 points). (

**e**) Superposed plume after the Kalman filter (4 points). (

**f**) Superposed plume after the Kalman filter (6 points).

**Figure 10.**The normalized contour map of the pollution plumes: (

**a**) True plume. (

**b**) Superposed plume after the Kalman filter (14 points) of the moderate heterogeneity case.

Parameters | Unit | Value |
---|---|---|

Grid spacing in x direction | m | 5 |

Grid spacing in y direction | m | 5 |

Saturated thickness | m | 10 |

Top elevation of aquifer | m | 20 |

Bottom elevation of aquifer | m | 10 |

Effective porosity | dimensionless | 0.30 |

Longitudinal dispersivity | m | 4 |

Transverse dispersivity | m | 1.2 |

Effective diffusion coefficient | m^{2}/day | 1.75 × 10^{−4} |

Sampling Number | Source 1 | Source 2 | Source 3 | Source 4 | Source 5 | |||||
---|---|---|---|---|---|---|---|---|---|---|

Mass-Loading Rate (g/day) | Weight | Mass-Loading Rate (g/day) | Weight | Mass-Loading Rate (g/day) | Weight | Mass-Loading Rate (g/day) | Weight | Mass-Loading Rate (g/day) | Weight | |

0 | 200 | 0.7 | 200 | 0.8 | 200 | 0.9 | 200 | 0.6 | 200 | 0.8 |

1 | 1000 | 0.48 | 868.4 | 0.39 | 1000 | 1.0 | 0.01 | 0 | 0.01 | 0 |

2 | 0.01 | 0 | 1000 | 1 | 60.2 | 0 | 0.01 | 0 | 0.01 | 1 |

3 | 0.01 | 0 | 694.72 | 1 | 0.01 | 0 | 0.01 | 0 | 0.01 | 1 |

4 | 0.01 | 0 | 578.52 | 1 | 0.01 | 0 | 0.01 | 0 | 0.01 | 1 |

5 | 0.01 | 0 | 530.64 | 1 | 0.01 | 0 | 0.01 | 0 | 0.01 | 1 |

6 | 0.01 | 0 | 530.64 | 1 | 0.01 | 0 | 0.01 | 0 | 0.01 | 0 |

α-Cuts Setting | Sampling Points Needed | Source 1 | Source 2 | Source 3 | Source 4 | Source 5 | |||||
---|---|---|---|---|---|---|---|---|---|---|---|

Mass-Loading Rate (g/day) | Weight | Mass-Loading Rate (g/day) | Weight | Mass-Loading Rate (g/day) | Weight | Mass-Loading Rate (g/day) | Weight | Mass-Loading Rate (g/day) | Weight | ||

(0.2 0.4 0.6 0.8) | 6 | 0.01 | 0 | 530.64 | 1 | 0.01 | 0 | 0.01 | 0 | 0.01 | 0 |

(0.2 0.4 0.6) | 13 | 0.01 | 0 | 578.26 | 1 | 0.01 | 0 | 0.01 | 0 | 0.01 | 0 |

(0.3 0.6 0.78 0.9) | 10 | 0.01 | 0 | 542.36 | 1 | 0.01 | 0 | 0.01 | 0 | 0.01 | 0 |

Initial Weight | Sampling Points | Source 1 | Source 2 | Source 3 | Source 4 | Source 5 | |||||
---|---|---|---|---|---|---|---|---|---|---|---|

Mass-Loading Rate (g/day) | Weight | Mass-Loading Rate (g/day) | Weight | Mass-Loading Rate (g/day) | Weight | Mass-Loading Rate (g/day) | Weight | Mass-Loading Rate (g/day) | Weight | ||

(0.7 0.8 0.9 0.6 0.8) | 6 | 0.01 | 0 | 530.64 | 1 | 0.01 | 0 | 0.01 | 0 | 0.01 | 0 |

(0.7 0.1 0.9 0.6 0.8) | 11 | 0.01 | 0 | 543.28 | 1 | 0.01 | 0 | 0.01 | 0 | 0.01 | 0 |

(0.1 0.1 0.1 0.1 1.0) | 11 | 0.01 | 0 | 551.73 | 1 | 0.01 | 0 | 0.01 | 0 | 0.01 | 0 |

**Table 5.**Sensitivity analysis results of the initial mass-loading rate for the pollution source identification.

Initial Mass-Loading Rate | Sampling Points | Source 1 | Source 2 | Source 3 | Source 4 | Source 5 | |||||
---|---|---|---|---|---|---|---|---|---|---|---|

Mass-Loading Rate (g/day) | Weight | Mass-Loading Rate (g/day) | Weight | Mass-Loading Rate (g/day) | Weight | Mass-Loading Rate (g/day) | Weight | Mass-Loading Rate (g/day) | Weight | ||

(200 200 200 200 200) | 6 | 0.01 | 0 | 530.64 | 1 | 0.01 | 0 | 0.01 | 0 | 0.01 | 0 |

(200 20 200 200 200) | 6 | 0.01 | 0 | 530.64 | 1 | 0.01 | 0 | 0.01 | 0 | 0.01 | 0 |

(195 16 85 16 171) | 6 | 0.01 | 0 | 530.64 | 1 | 0.01 | 0 | 0.01 | 0 | 0.01 | 0 |

Heterogeneity | Sampling Points | Source 1 | Source 2 | Source 3 | Source 4 | Source 5 | |||||
---|---|---|---|---|---|---|---|---|---|---|---|

Mass-Loading Rate (g/day) | Weight | Mass-Loading Rate (g/day) | Weight | Mass-Loading Rate (g/day) | Weight | Mass-Loading Rate (g/day) | Weight | Mass-Loading Rate (g/day) | Weight | ||

small | 6 | 0.01 | 0 | 530.64 | 1 | 0.01 | 0 | 0.01 | 0 | 0.01 | 0 |

moderate | 14 | 0.01 | 0 | 422.32 | 1 | 0.01 | 0 | 0.01 | 0 | 0.01 | 0 |

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## Share and Cite

**MDPI and ACS Style**

Jiang, S.; Fan, J.; Xia, X.; Li, X.; Zhang, R.
An Effective Kalman Filter-Based Method for Groundwater Pollution Source Identification and Plume Morphology Characterization. *Water* **2018**, *10*, 1063.
https://doi.org/10.3390/w10081063

**AMA Style**

Jiang S, Fan J, Xia X, Li X, Zhang R.
An Effective Kalman Filter-Based Method for Groundwater Pollution Source Identification and Plume Morphology Characterization. *Water*. 2018; 10(8):1063.
https://doi.org/10.3390/w10081063

**Chicago/Turabian Style**

Jiang, Simin, Jinhong Fan, Xuemin Xia, Xianwen Li, and Ruicheng Zhang.
2018. "An Effective Kalman Filter-Based Method for Groundwater Pollution Source Identification and Plume Morphology Characterization" *Water* 10, no. 8: 1063.
https://doi.org/10.3390/w10081063