# 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

- Simunek, J.; Sejna, M.; Van Genuchten, M.T.; Šimůnek, J.; Šejna, M.; Jacques, D.; Sakai, M. HYDRUS-1D. Simulating the one-Dimensional Movement of Water, Heat, and Multiple Solutes in Variably-Saturated Media. 1998. Version 2. Available online: https://www.pc-progress.com/en/Default.aspx?hydrus-1d (accessed on 1 October 2022).
- Das, N.N.; Mohanty, B.P. Root zone soil moisture assessment using passive microwave remote sensing and vadose zonemodeling. Vadose Zone J.
**2006**, 5, 296–307. [Google Scholar] [CrossRef][Green Version] - Das, N.N.; Mohanty, B.P.; Cosh, M.H.; Jackson, T.J. Modeling and assimilation of root zone soil moisture using remote sensing observations in Walnut Gulch watershed during SMEX04. Remote Sens. Environ.
**2008**, 112, 415–429. [Google Scholar] [CrossRef] - Brandhorst, N.; Erdal, D.; Neuweiler, I. Soil moisture prediction with the ensemble Kalman filter: Handling uncertainty of soil hydraulic parameters. Adv. Water Res.
**2017**, 110, 360–370. [Google Scholar] [CrossRef] - Abbaszadeh, P.; Moradkhani, H.; Yan, H. Enhancing hydrologic data assimilation by evolutionary particle filter and Markov chain Monte Carlo. Adv. Water Res.
**2018**, 111, 192–204. [Google Scholar] [CrossRef] - Bauser, H.H.; Berg, D.; Klein, O.; Roth, K. Inflation method for ensemble Kalman filter in soil hydrology. Hydrol. Earth Syst. Sci.
**2018**, 22, 4921–4934. [Google Scholar] [CrossRef][Green Version] - Berg, D.; Bauser, H.H.; Roth, K. Covariance resampling for particle filter–state and parameter estimation for soil hydrology. Hydrol. Earth Syst. Sci.
**2019**, 23, 1163–1178. [Google Scholar] [CrossRef][Green Version] - Jamal, A.; Linker, R. Inflation method based on confidence intervals for data assimilation in soil hydrology using ensemble Kalman filter. Vadose Zone J.
**2020**, 19, e20000. [Google Scholar] [CrossRef][Green Version] - Reichle, R.H.; McLaughlin, D.B.; Entekhabi, D. Hydrologic data assimilation with the ensemble Kalman filter. Mon. Weather Rev.
**2002**, 130, 103–114. [Google Scholar] [CrossRef] - De Lannoy, G.J.; Reichle, R.H.; Houser, P.R.; Pauwels, V.R.; Verhoest, N.E. Correcting for forecast bias in soil moisture assimilation with the ensemble Kalman filter. Water Resour. Res.
**2007**, 43, 117. [Google Scholar] [CrossRef][Green Version] - DeChant, C.M.; Moradkhani, H. Examining the effectiveness and robustness of sequential data assimilation methods for quantification of uncertainty in hydrologic forecasting. Water Resour. Res.
**2012**, 48, 136. [Google Scholar] [CrossRef] - Yin, S.; Zhu, X. Intelligent particle filter and its application to fault detection of nonlinear system. IEEE Trans. Ind. Electron.
**2015**, 62, 3852–3861. [Google Scholar] [CrossRef] - Jamal, A.; Linker, R. Genetic Operator-Based Particle Filter Combined with Markov Chain Monte Carlo for Data Assimilation in a Crop Growth Model. Agriculture
**2020**, 10, 606. [Google Scholar] [CrossRef] - Moradkhani, H.; Hsu, K.L.; Gupta, H.; Sorooshian, S. Uncertainty assessment of hydrologic model states and parameters: Sequential data assimilation using the particle filter. Water Resour. Res.
**2005**, 41, 480. [Google Scholar] [CrossRef][Green Version] - Moradkhani, H.; DeChant, C.M.; Sorooshian, S. Evolution of ensemble data assimilation for uncertainty quantification using the particle filter-Markov chain Monte Carlo method. Water Resour. Res.
**2012**, 48, 162. [Google Scholar] [CrossRef] - Andrieu, C.; Doucet, A.; Holenstein, R. Particle markov chain monte carlo methods. J. R. Stat. Soc. Ser. B
**2010**, 72, 269–342. [Google Scholar] [CrossRef][Green Version] - Kroes, J.G.; Van Dam, J.C.; Bartholomeus, R.P.; Groenendijk, P.; Heinen, M.; Hendriks, R.F.A.; Van Walsum, P.E.V. SWAP Version 4 (No. 2780). Wageningen Environmental Research. 2017. Available online: https://research.wur.nl/en/publications/swap-version-4 (accessed on 1 October 2022).
- Hamby, D.M. A review of techniques for parameter sensitivity analysis of environmental models. Environ. Monit. Assess.
**1994**, 32, 135–154. [Google Scholar] [CrossRef] - Della Peruta, R.; Keller, A.; Schulin, R. Sensitivity analysis, calibration and validation of EPIC for modelling soil phosphorus dynamics in Swiss agro-ecosystems. Environ. Model. Softw.
**2014**, 62, 97–111. [Google Scholar] [CrossRef] - Wu, M.; Ran, Y.; Jansson, P.E.; Chen, P.; Tan, X.; Zhang, W. Global parameters sensitivity analysis of modeling water, energy and carbon exchange of an arid agricultural ecosystem. Agric. For. Meteorol.
**2019**, 271, 295–306. [Google Scholar] - Xu, X.; Sun, C.; Huang, G.; Mohanty, B.P. Global sensitivity analysis and calibration of parameters for a physically-based agro-hydrological model. Environ. Model. Softw.
**2016**, 83, 88–102. [Google Scholar] [CrossRef][Green Version] - De Pue, J.; Rezaei, M.; Van Meirvenne, M.; Cornelis, W.M. The relevance of measuring saturated hydraulic conductivity: Sensitivity analysis and functional evaluation. J. Hydrol.
**2019**, 576, 628–638. [Google Scholar] [CrossRef] - Claverie, M.; Demarez, V.; Duchemin, B.; Hagolle, O.; Ducrot, D.; Marais-Sicre, C.; Dedieu, G. Maize and sunflower biomass estimation in southwest France using high spatial and temporal resolution remote sensing data. Remote Sens. Environ.
**2012**, 124, 844–857. [Google Scholar] [CrossRef] - Linker, R.; Kisekka, I. Concurrent data assimilation and model-based optimization of irrigation scheduling. Agric. Water Manag.
**2022**, 274, 107924. [Google Scholar] [CrossRef] - Zhang, T.; Su, J.; Liu, C.; Chen, W.H. State and parameter estimation of the AquaCrop model for winter wheat using sensitivity informed particle filter. Comput. Electron. Agric.
**2021**, 180, 105909. [Google Scholar] [CrossRef] - Manache, G.; Melching, C.S. Identification of reliable regression-and correlation-based sensitivity measures for importance ranking of water-quality model parameters. Environ. Model. Softw.
**2008**, 23, 549–562. [Google Scholar] [CrossRef] - Shun, H.; Shi, L.; Huang, K.; Zha, Y.; Hu, X.; Ye, H.; Yang, Q. Improvement of sugarcane crop simulation by SWAP-WOFOST model via data assimilation. Field Crops Res.
**2019**, 232, 49–61. [Google Scholar] - Gordon, N.J.; Salmond, D.J.; Smith, A.F. Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE Proc. F (Radar Signal Process.)
**1993**, 140, 107–113. [Google Scholar] [CrossRef][Green Version] - Carpenter, J.; Clifford, P.; Fearnhead, P. Improved particle filter for nonlinear problems. IEE Proc. -Radar Sonar Navig.
**1999**, 146, 2–7. [Google Scholar] [CrossRef] - Kitagawa, G. Monte Carlo filter and smoother for non-Gaussian nonlinear state space models. J. Comput. Graph. Stat.
**1996**, 5, 1–25. [Google Scholar] - Asuero, A.G.; Sayago, A.; González, A.G. The correlation coefficient: An overview. Crit. Rev. Anal. Chem.
**2006**, 36, 41–59. [Google Scholar] [CrossRef] - Rocha, D.; Abbasi, F.; Feyen, J. Sensitivity analysis of soil hydraulic properties on subsurface water flow in furrows. J. Irrig. Drain. Eng.
**2006**, 132, 418–424. [Google Scholar] [CrossRef] - Schaap, M.G.; Leij, F.J.; Van Genuchten, M.T. Rosetta: A computer program for estimating soil hydraulic parameters with hierarchical pedotransfer functions. J. Hydrol.
**2001**, 251, 163–176. [Google Scholar] [CrossRef] - Akoglu, H. User’s guide to correlation coefficients. Turk. J. Emerg. Med.
**2018**, 18, 91–93. [Google Scholar] [CrossRef] [PubMed]

**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 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

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