# Use of Global Sensitivity and Data-Worth Analysis for an Efficient Estimation of Soil Hydraulic Properties

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## Abstract

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## 1. Introduction

^{−1}], the saturated hydraulic conductivity; ${\theta}_{r}$ [L

^{3}.L

^{−}

^{3}], the residual water content; and $\alpha $ [L

^{−}

^{1}] and $n$ [−], the MvG shape parameters. A first estimation of the soil parameters is carried out using cumulative outflow measurements. The interest of collecting tensiometric data and the best location of pressure sensors are then investigated by (i) analysis of the pressure sensitivity to the hydraulic soil parameters at different locations using GSA and (ii) performing a Bayesian DWA to quantify the benefit of the pressure head measurements on reducing posterior parameter uncertainty intervals, and thereby, improving the quality of the estimated parameters.

## 2. Materials and Methods

#### 2.1. Laboratory Experiment

#### 2.2. Numerical Model

^{3}.L

^{−3}], the current water content; t [T

^{−1}], the time; $c\left(h\right)$ [L

^{−1}], the specific moisture capacity; ${S}_{s}$ [L

^{−1}], the specific storage; ${\theta}_{s}$ [L

^{3}.L

^{−3}], the saturated water content; $q$ [L.T

^{−1}], the Darcy’s velocity; z [L], the positive downward vertical coordinate and $K\left(h\right)$ [L.T

^{−1}], the hydraulic conductivity. The models of Mualem [24] and van Genuchten [25] were used for water content – pressure head, and conductivity – water content constitutive relationships as follows:

_{e}(-) is the effective saturation.

#### 2.3. Global Sensitivity Analysis

^{th}-dimensional index, ${C}_{\alpha}$ are the polynomial coefficients, and ${\mathsf{\Psi}}_{\alpha}$ are the multivariate orthonormal polynomials of degree $\left|\alpha \right|={\displaystyle {\sum}_{i=1}^{d}{\alpha}_{i}}$. Multivariate orthonormal polynomials are a tensor-product of univariate polynomials [45]. The Legendre polynomials, with non-informative uniform distributions, were used in this work for all parameters. These distributions express the lack of prior information on the parameter values, which renders all plausible values equally likely.

#### 2.4. Bayesian Parameter Inference

^{3}.cm

^{−3}. This value has been accurately obtained by weighing a sample of the saturated soil. The specific storage was fixed at 5E10

^{−9}cm

^{−1}. Therefore, the unknowns vector reduced to $\mathsf{\xi}=\left({K}_{S},\hspace{0.17em}{\theta}_{r},\alpha ,n\right)$. This vector was estimated using a Bayesian inversion to obtain the joint posterior probability distribution functions (jpdfs). These functions were evaluated using the DREAM

_{(ZS)}[46] MCMC sampler, which was largely used in sub-surface hydrology e.g., [9,15,16,46,47,48]. DREAM

_{(ZS)}generates random parameter set sequences which converge asymptotically to the target solution [49]. Statistical measures, such as the mean or the variance, are calculated from the obtained posterior distributions. All parameters were assumed to have uniform distributions over their respective intervals shown in Table 1. As little is known about the model parameters, these large intervals were based on the literature.

_{(ZS)}was used with three parallel chains. We considered that convergence aws reached if the chains were not correlated and the Gelman- Rubin [50] criterion was fulfilled $\left({R}_{stat}\le 1.2\right)$. The last 25% parameter sets (after convergence of the chains) were used for the estimation of the mean parameter values and the corresponding confidence intervals.

## 3. Results and Discussion

#### 3.1. Inversion Using Cumulative Outflow Measurements

#### 3.2. Global Sensitivty Analysis Results

_{3}. This makes sense since the pressure head at ${z}_{3}$ had a reduced sensitivity to the soil parameters because of its vicinity with the lower boundary which had a zero pressure fixed during the whole experiment.

- (i)
- The pressure head prediction was driven by hydraulic parameters and thus, the accuracy of the soil parameters may improve if the pressure data are also considered for calibration.
- (ii)
- If one has to install a single tensiometric sensor for collecting pressure head measurements, it is better to install it near the soil surface (near the top of the column).
- (iii)
- The parameter ${\theta}_{r}$ cannot be estimated from the pressure head measurements regardless to the position of the tensiometric sensor.

_{1}= 30 min which corresponds to the imbibition process, t

_{2}= 60 min which corresponds to the steady-state infiltration and t

_{3}= 80 min which corresponds to the drying process.

_{1}), whereas, the parameter $\alpha $ was much more sensitive during the drying process (t

_{3}).

#### 3.3. Data Worth Analysis Results

- (i)
- The numerical model was used to generate synthetic pressure head data at 18.5 cm using the mean parameter values obtained earlier from the calibration of the cumulative outflow data (Table 2, column 3).
- (ii)
- Then, the simulated pressure values were noised with Gaussian noises of 0.5 cm standard deviation and used as fictive new observations.
- (iii)
- The new fictive pressure observations were supplemented to the real cumulative outflow observations and used for a new Bayesian calibration of the hydraulic parameters.

#### 3.4. Inversion Using Real Cumulative Outflow and Pressure Measurements

## 4. Conclusions

_{s}and then, the parameter n. The parameter ${\theta}_{r}$ does not affect the pressure at any of the three investigated positions. The results of GSA showed that it is better to collect the tensiometric data at ${z}_{1}=18.5$ cm since the pressure variance is around 10 times more significant than at ${z}_{3}=33$ cm. Further, the analysis of the marginal effects showed that ${K}_{S}$ and $n$ have a strong effect on the pressure head if ${K}_{S}\le 0.6$ cm/min, and $n\le 3$, else their effect on the pressure head is weak. ${K}_{S}$ and $n$ are more sensitive during the imbibition process, whereas, $\alpha $ is much more sensitive during the drying process.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Experimental setup of the laboratory infiltration experiment for soil parameter assessment.

**Figure 4.**Pressure variance curve versus time at 18.5 cm (

**a**) and 33 cm (

**b**) from the soil surface. The shaded areas represent the partial variance of each parameter. The blank region below the variance curve and above the shaded areas represents interactions between parameters.

**Figure 5.**Marginal effects of ${K}_{S}$, ${\theta}_{r}$, $\alpha $ and $n$ on the pressure head at 18.5 cm depth.

**Figure 6.**Histograms of marginal distribution of the hydraulic parameters using real observations of cumulative outflow and pressure head at 18.5 cm.

Parameter | Lower Bound | Upper Bound |
---|---|---|

${K}_{S}$ [cm/min] | 0.10 | 2.0 |

${\theta}_{r}$ [-] | 0.05 | 0.15 |

$\alpha $ [cm^{−1}] | 0.005 | 0.15 |

$n$ [-] | 1.5 | 13. |

**Table 2.**Mean parameter values, confidence intervals (Cis) and size of the CIs when only cumulative outflow observations are used for parameter calibration.

Unit | Mean Value | 99% Confidence Interval | Size of the 99% Confidence Interval | |
---|---|---|---|---|

${K}_{S}$ | (cm/min) | 1.37 | [0.1–2.0] | 1.9 |

${\theta}_{r}$ | - | 0.104 | [0.05–0.15] | 0.1 |

$\alpha $ | (cm^{−1}) | 0.019 | [0.01–0.029] | 0.028 |

$n$ | - | 2.19 | [0.6–3.8] | 3.2 |

**Table 3.**Mean parameter values, CIs and size of the posterior CIs when cumulative outflow observations and fictive pressure data are used for parameter assessment. The fictive pressure data are obtained from noised simulations performed using reference parameter values.

Unit | Reference Value | Mean Value | 99% Confidence Interval | Size of the 99% CI | |
---|---|---|---|---|---|

${K}_{S}$ | (cm/min) | 1.37 | 1.59 | [0.9–2.3] | 1.3 |

${\theta}_{r}$ | - | 0.104 | 0.106 | [0.05–0.15] | 0.1 |

$\alpha $ | (cm^{−1}) | 0.0187 | 0.02 | [0.017–0.025] | 0.007 |

$n$ | - | 2.19 | 2.19 | [1.56–2.64] | 1.08 |

**Table 4.**Mean parameter values, CIs and size of the CIs when cumulative outflow observations and fictive pressure data are used for parameter assessment. The fictive pressure data correspond to noised simulations performed using reference parameter values.

Unit | Reference Value | Mean Value | 99% Confidence Interval | Size of the 99% CI | |
---|---|---|---|---|---|

${K}_{S}$ | (cm/min) | 0.2 | 0.19 | [0.18–0.2] | 0.02 |

${\theta}_{r}$ | - | 0.104 | 0.07 | [0.05–0.15] | 0.1 |

$\alpha $ | (cm^{−1}) | 0.0187 | 0.0132 | [0.012–0.014] | 0.002 |

$n$ | - | 2.19 | 2.64 | [2.4–2.8] | 0.4 |

**Table 5.**Mean parameter values, CIs and size of the CIs when both real cumulative outflow and pressure observations are used for parameter assessment.

Unit | Mean Value | 99% Confidence Interval | Size of the 99% CI | |
---|---|---|---|---|

${K}_{S}$ | (cm/min) | 0.231 | [0.22–0.24] | 0.022 |

${\theta}_{r}$ | - | 0.07 | [0.05–0.15] | 0.1 |

$\alpha $ | (cm^{−1}) | 0.012 | [0.01–0.013] | 0.003 |

$n$ | - | 2.26 | [2–2.5] | 0.5 |

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**MDPI and ACS Style**

Younes, A.; Shao, Q.; Mara, T.A.; Baalousha, H.M.; Fahs, M.
Use of Global Sensitivity and Data-Worth Analysis for an Efficient Estimation of Soil Hydraulic Properties. *Water* **2020**, *12*, 736.
https://doi.org/10.3390/w12030736

**AMA Style**

Younes A, Shao Q, Mara TA, Baalousha HM, Fahs M.
Use of Global Sensitivity and Data-Worth Analysis for an Efficient Estimation of Soil Hydraulic Properties. *Water*. 2020; 12(3):736.
https://doi.org/10.3390/w12030736

**Chicago/Turabian Style**

Younes, Anis, Qian Shao, Thierry Alex Mara, Husam Musa Baalousha, and Marwan Fahs.
2020. "Use of Global Sensitivity and Data-Worth Analysis for an Efficient Estimation of Soil Hydraulic Properties" *Water* 12, no. 3: 736.
https://doi.org/10.3390/w12030736