# Minimizing Errors in the Prediction of Water Levels Using Kriging Technique in Residuals of the Groundwater Model

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Study Area

#### 2.2. Data

#### 2.3. Modeling

- Mapping MODFLOW simulated groundwater heads (model imported from TWDB) into their corresponding coordinates and overlap with observation data to find the MODFLOW estimated values in the observation point.
- Subtracting the observed groundwater head with MODFLOW simulated head and consider as the model residuals.
- Dividing the residuals into two separate datasets, 90 percent of data for fitting kriging methods (calibrating residuals) and 10 percent for validating part (validating residuals), in a random selection.
- Pre-evaluating the calibrating residuals and fit kriging method to generate the estimated residual map for the study domain.
- Comparing the validating residuals with an estimated one to evaluate the accuracy of the kriging method.

#### 2.3.1. Data Preparation

#### 2.3.2. Kriging Method

## 3. Results and Discussion

#### 3.1. Data Investigation

#### 3.2. Model Simulation

#### 3.3. Model Validation

#### 3.4. Comparison to Other Studies

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**Histogram presenting the water level residuals for Edwards–Trinity aquifer (years 1995–2000).

**Figure 6.**Sample variogram and fitted model applied on Edwards aquifer dataset for years 1995 through 2000.

**Figure 7.**Predicted residuals (in meters) after application of kriging on MODFLOW residuals for years 1995 through 2000.

**Figure 8.**Standard Deviation (in meters) for ordinary kriging applied to Edwards–Trinity aquifer for the years 1995 through 2000 (the black dots in the figure represent the observation points used during cross-validation).

Year | Observations ^{1} | MODFLOW (m) | MODFLOW + Kriging (m) | |||||
---|---|---|---|---|---|---|---|---|

Mean a ^{2} | Observed Residuals ^{3} | Standard Error | Mean b ^{4} | Predicted Residuals ^{5} | Error ^{6} | Standard Error | ||

1995 | 727.5 | 764.7 | −37.2 | 3.1 | 728.7 | −36 | 1.2 | 0.7 |

1996 | 665.5 | 706 | −40.5 | 3.3 | 667.1 | −38.9 | 1.6 | 0.4 |

1997 | 692.4 | 725.5 | −33.1 | 2.9 | 693.5 | −32 | 1.1 | 0.4 |

1998 | 682.4 | 716.6 | −34.2 | 2.9 | 682.8 | −33.8 | 0.4 | 0.5 |

1999 | 708.3 | 747.1 | −38.8 | 2.9 | 708.6 | −38.5 | 0.3 | 0.3 |

2000 | 686 | 724.7 | −38.7 | 3.2 | 685.3 | −39.4 | −0.7 | 1.1 |

^{1}Observed average value for water level for the given year (winter season).

^{2}MODFLOW simulated average value for water level for the given year (winter season).

^{3}Difference between observed values and the MODFLOW simulated values (Observations—Mean a).

^{4}Simulated average value for water level for the given year after application of kriging (Mean a + Predicted Residuals).

^{5}Residuals obtained after application of kriging on the observed residuals.

^{6}Difference between predicted and observed residuals for each point.

Year | Min. | 1st Qu. | Median | Mean | 3rd Qu. | Max. | |
---|---|---|---|---|---|---|---|

1995 | Observed Residuals | −141.6 | −53.1 | −32.2 | −37.2 | −20.8 | 55.2 |

Predicted Residuals | −147.1 | −52.1 | −31.9 | −36 | −22.5 | 52.3 | |

Error | −97.3 | −6.7 | −0.1 | 1.2 | 5.2 | 84.5 | |

Standard Error | 0.4 | 0.7 | 0.8 | 0.7 | 0.8 | 0.8 | |

1996 | Observed Residuals | −137.1 | −60.8 | −40.9 | −40.5 | −16.1 | 30.4 |

Predicted Residuals | −132.7 | −56.8 | −43.3 | −38.9 | −14.3 | 33.6 | |

Error | −49.6 | −6.4 | 0.8 | 1.6 | 11.6 | 72.8 | |

Standard Error | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 | |

1997 | Observed Residuals | −135.4 | −51.7 | −32.6 | −33.1 | −11.3 | 62.5 |

Predicted Residuals | −148.1 | −47 | −31.7 | −32 | −9.1 | 45.7 | |

Error | −59 | −6.4 | 0.5 | 1 | 7.7 | 78 | |

Standard Error | 0.2 | 0.4 | 0.4 | 0.4 | 0.4 | 0.5 | |

1998 | Observed Residuals | −131 | −54.3 | −32.3 | −34.2 | −16.9 | 64 |

Predicted Residuals | −145.8 | −53.2 | −33.4 | −33.8 | −19.3 | 43.8 | |

Error | −50 | −8.7 | 0.4 | 0.4 | 7.1 | 61.3 | |

Standard Error | 0.4 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | |

1999 | Observed Residuals | −144.6 | −54 | −31.7 | −38.8 | −18.4 | 52.5 |

Predicted Residuals | −150 | −53.1 | −31.1 | −38.5 | −20.9 | 52.2 | |

Error | −55.2 | −8.2 | 0.4 | 0.3 | 7.7 | 75.9 | |

Standard Error | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | |

2000 | Observed Residuals | −141.1 | −59.3 | −35.7 | −38.7 | −15.8 | 48 |

Predicted Residuals | −139.3 | −56.4 | −39.5 | −39.4 | −19.1 | 36.1 | |

Error | −79.7 | −8.7 | 0 | −0.7 | 8.3 | 66.2 | |

Standard Error | 0.4 | 1.2 | 1.2 | 1.1 | 1.2 | 1.3 |

Year | Observations | MODFLOW (m) | MODFLOW + Kriging (m) | |||||
---|---|---|---|---|---|---|---|---|

Mean | Observed Residuals | Standard Error | Mean | Predicted Residuals | Error | Standard Error | ||

1995 | 758.7 | 802.1 | −43.5 | 8.7 | 753.6 | −48.5 | −5.1 | 3 |

1996 | 697.6 | 729 | −31.4 | 14.7 | 689.1 | −39.9 | −8.5 | 5.3 |

1997 | 683 | 707.6 | −24.6 | 5.9 | 677.4 | −30.1 | −5.5 | 3.9 |

1998 | 694.8 | 733 | −38.1 | 10.7 | 694.6 | −38.3 | −0.2 | 8.1 |

1999 | 716.5 | 744.8 | −28.3 | 8.4 | 714 | −30.8 | −2.5 | 3.8 |

2000 | 644.7 | 665.5 | −20.8 | 10 | 646.5 | −19 | 1.8 | 4.6 |

Year | Min. | 1st Qu. | Median | Mean | 3rd Qu. | Max. | |
---|---|---|---|---|---|---|---|

1995 | Observed Residuals | −131.8 | −50 | −33.9 | −43.5 | −24 | −15 |

Predicted Residuals | −132.5 | −55 | −31.8 | −48.5 | −24.8 | −18.9 | |

Error | −7.1 | −1.3 | 0.8 | 5.1 | 8.9 | 29.3 | |

Standard Error | - | - | - | 3 | - | - | |

1996 | Observed Residuals | −138.1 | −55.2 | −30.1 | −31.4 | −1.8 | 61 |

Predicted Residuals | −143.1 | −46.5 | −37.2 | −40 | −5.3 | 40.3 | |

Error | −20.5 | −5.2 | 5 | 8.5 | 20.3 | 52.1 | |

Standard Error | - | - | - | 5.3 | - | - | |

1997 | Observed Residuals | −82.3 | −36.6 | −28.1 | −24.6 | −6.9 | 16.2 |

Predicted Residuals | −80.9 | −44.9 | −31.1 | −30.2 | −7.3 | 16.7 | |

Error | −18.5 | −3.1 | 0 | 5.5 | 20.6 | 38.7 | |

Standard Error | - | - | - | 3.9 | - | - | |

1998 | Observed Residuals | −134.5 | −56.6 | −33.7 | −38.1 | −25.2 | 62 |

Predicted Residuals | −76.3 | −51.7 | −45 | −38.3 | −22.5 | −10.1 | |

Error | −87.3 | −6 | 0.8 | 0.2 | 10.4 | 72.9 | |

Standard Error | - | - | - | 8.1 | - | - | |

1999 | Observed Residuals | −68.1 | −54.8 | −33.6 | −28.3 | −19.3 | 65.5 |

Predicted Residuals | −71 | −52.1 | −39.1 | −30.8 | −22.8 | 32 | |

Error | −38.6 | −2.7 | 0.3 | 2.5 | 9.7 | 35.8 | |

Standard Error | - | - | - | 3.8 | - | - | |

2000 | Observed Residuals | −120.5 | −46.4 | −16.2 | −20.8 | −3.2 | 65.6 |

Predicted Residuals | −71.7 | −45.9 | −20.4 | −19.1 | 4.2 | 41 | |

Error | −48.9 | −4 | 0.5 | −1.7 | 10.2 | 24.7 | |

Standard Error | - | - | - | 4.6 | - | - |

Observation Well | Observed Head (m) | MODFLOW (m) | MODFLOW + Kriging (m) | ||
---|---|---|---|---|---|

Simulated Head | Residuals | Predicted Head | Residuals | ||

1 | 769 | 786.9 | −17.9 | 764.4 | 4.6 |

2 | 745.7 | 760.1 | −14.4 | 741.8 | 3.8 |

3 | 748.3 | 868.8 | −120.5 | 797.2 | −48.9 |

4 | 1059.2 | 993.6 | 65.6 | 1034.6 | 24.7 |

5 | 662.2 | 713.2 | −50.9 | 652.3 | 10 |

6 | 548.9 | 595.9 | −47 | 552.5 | −3.6 |

7 | 586.6 | 608.2 | −21.6 | 574.7 | 12 |

8 | 648.8 | 695 | −46.2 | 647.2 | 1.6 |

9 | 585 | 617.7 | −32.6 | 570.1 | 14.9 |

10 | 616.1 | 663.8 | −47.7 | 618.4 | −2.3 |

11 | 613.4 | 619.5 | −6.2 | 602.2 | 11.1 |

12 | 618.5 | 623.3 | −4.8 | 629.4 | −10.9 |

13 | 623.6 | 620.6 | 3 | 624.1 | −0.5 |

14 | 588.9 | 568.3 | 20.5 | 594 | −5.1 |

15 | 555.1 | 568.4 | −13.3 | 593 | −37.8 |

16 | 346.3 | 344.7 | 1.6 | 347.4 | −1.2 |

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**MDPI and ACS Style**

Asadi, A.; Adhikari, K.
Minimizing Errors in the Prediction of Water Levels Using Kriging Technique in Residuals of the Groundwater Model. *Water* **2022**, *14*, 426.
https://doi.org/10.3390/w14030426

**AMA Style**

Asadi A, Adhikari K.
Minimizing Errors in the Prediction of Water Levels Using Kriging Technique in Residuals of the Groundwater Model. *Water*. 2022; 14(3):426.
https://doi.org/10.3390/w14030426

**Chicago/Turabian Style**

Asadi, Alireza, and Kushal Adhikari.
2022. "Minimizing Errors in the Prediction of Water Levels Using Kriging Technique in Residuals of the Groundwater Model" *Water* 14, no. 3: 426.
https://doi.org/10.3390/w14030426