# Covariance-Based Selection of Parameters for Particle Filter Data Assimilation in Soil Hydrology

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Particle Filtering

#### 2.2. Correlation Analysis

## 3. Case Studies

## 4. Results

#### 4.1. Case Study #1—Random Boundary Condition

_{sat}were better than K

_{sat}in both methods due to the high influence of these parameters on the water content [21]. However, the estimations of θ

_{sat}by C-GPFM are superior to GPFM at the middle layer.

#### 4.2. Case Study #2—Cyclic Boundary Condition

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Average of parameters posteriors by C-GPFM and GPFM together with the ‘true’ and ‘biased’ values for Case Study #1.

**Figure 5.**Average of parameters posteriors by C-GPFM and GPFM at days #10, #20, #30 and #40 for Case Study #1.

**Figure 6.**Average of state prediction error for the next 20 days at each day of assimilation by C-GPFM and GPFM for Case Study #1.

**Figure 7.**State posterior errors (averaged over time, depth and runs) normalized with respect to the error of the biased (no assimilation) model. Results for the models obtained on days #31, #33, #35, #37 and #39 by C-GPFM and GPFM for Case Study #1.

**Figure 8.**Percentage of the runs in which C-GPFM led to smaller average state errors than GPFM. Results for the models obtained on days #31, #33, #35, #37 and #39 for Case Study #1.

**Figure 9.**Relative absolute error of each parameter (averaged over time) estimated by C-GPFM and GPFM together with the number of times the parameter was selected for adjustment by C-GPFM for Case Study #1. The number of times a parameter was selected for adjustment by C-GPFM is indicated above the corresponding sack.

**Figure 10.**Average of state prediction error for the next 20 days at each day of assimilation by C-GPFM and GPFM for Case Study #2.

Parameter | Description | Depth | ‘True’ | ‘Biased’ |
---|---|---|---|---|

${\theta}_{sat}$ | Saturated water content | 0–20 cm 21–40 cm 41–60 cm | 0.43 0.41 0.43 | 0.48 0.36 0.48 |

$\alpha $ | Air entrance value parameters | 0–20 cm 21–40 cm 41–60 cm | 2.68 2.10 2.68 | 2.1 2.5 2.1 |

${K}_{sat}$ | $\mathrm{Saturated}\text{}\mathrm{hydraulic}\text{}\mathrm{conductivity}\text{}\left. [\mathrm{cm}/\mathrm{day}\right]$ | 0–20 cm 21–40 cm 41–60 cm | 713 230 713 | 613 270 613 |

${\theta}_{res}$ | Residual water content | 0–20 cm 21–40 cm 41–60 cm | 0.045 0.061 0.045 | |

$n$ | Shape parameter | 0–20 cm 21–40 cm 41–60 cm | 0.14 0.10 0.14 |

**Table 2.**The absolute error ratio of the parameter $\alpha $ between the end and the beginning of the drying and wetting periods using GPFM.

Period | 0–20 cm Layer | 21–40 cm Layer | 41–60 cm Layer |
---|---|---|---|

Days 1–10 (wetting) | 2.63 | 1.42 | 0.71 |

Days 11–20 (drying) | 0.36 | 0.24 | 0.77 |

Days 21–30 (wetting) | 1.52 | 11.6 | 2.62 |

Days 31–40 (drying) | 0.78 | 0.72 | 1.39 |

Ratios Multiplication | 1.12 | 2.85 | 1.99 |

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Jamal, A.; Linker, R. Covariance-Based Selection of Parameters for Particle Filter Data Assimilation in Soil Hydrology. *Water* **2022**, *14*, 3606.
https://doi.org/10.3390/w14223606

**AMA Style**

Jamal A, Linker R. Covariance-Based Selection of Parameters for Particle Filter Data Assimilation in Soil Hydrology. *Water*. 2022; 14(22):3606.
https://doi.org/10.3390/w14223606

**Chicago/Turabian Style**

Jamal, Alaa, and Raphael Linker. 2022. "Covariance-Based Selection of Parameters for Particle Filter Data Assimilation in Soil Hydrology" *Water* 14, no. 22: 3606.
https://doi.org/10.3390/w14223606