# 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

- Kool, J.B.; Parker, J.C.; van Genuchten, M.T. Determining Soil Hydraulic Properties from One-step Outflow Experiments by Parameter Estimation: I. Theory and Numerical Studies1. Soil Sci. Soc. Am. J.
**1985**, 49, 1348. [Google Scholar] [CrossRef][Green Version] - Parker, J.C.; Kool, J.B.; van Genuchten, M.T. Determining Soil Hydraulic Properties from One-step Outflow Experiments by Parameter Estimation: II. Experimental Studies1. Soil Sci. Soc. Am. J.
**1985**, 49, 1354. [Google Scholar] [CrossRef] - Van Dam, J.C.; Stricker, J.N.M.; Droogers, P. Inverse method to determine soil hydraulic functions from multistep outflow experiments. Soil Sci. Soc. Am. J.
**1994**, 58, 647–652. [Google Scholar] [CrossRef] - Durner, W.; Schultze, B.; Zurmühl, T. State-of-the-Art in Inverse Modeling of Inflow/Outflow Experiments; University of California: Riverside, CA, USA, 1999. [Google Scholar]
- Puhlmann, H.; Von Wilpert, K.; Lukes, M.; Dröge, W. Multistep outflow experiments to derive a soil hydraulic database for forest soils. Eur. J. Soil Sci.
**2009**, 60, 792–806. [Google Scholar] [CrossRef] - Beydoun, H.; Lehmann, F. Expériences de drainage et estimation de paramètres en milieu poreux non saturé. Comptes Rendus Geosci.
**2006**, 338, 180–187. [Google Scholar] [CrossRef] - Younes, A.; Mara, T.A.; Fajraoui, N.; Lehmann, F.; Belfort, B.; Beydoun, H. Use of Global Sensitivity Analysis to Help Assess Unsaturated Soil Hydraulic Parameters. Vadose Zone J.
**2013**, 12. [Google Scholar] [CrossRef] - Mishra, S.; Parker, J.C. Parameter estimation for coupled unsaturated flow and transport. Water Resour. Res.
**1989**, 25, 385–396. [Google Scholar] [CrossRef] - Younes, A.; Zaouali, J.; Fahs, M.; Slama, F.; Grunberger, O.; Mara, T.A. Bayesian soil parameter estimation: Results of percolation-drainage vs. infiltration laboratory experiments. J. Hydrol.
**2018**, 565, 770–778. [Google Scholar] [CrossRef] - Durner, W.; Iden, S.C. Extended multistep outflow method for the accurate determination of soil hydraulic properties near water saturation. Water Resour. Res.
**2011**, 47. [Google Scholar] [CrossRef][Green Version] - Wendroth, O.; Ehlers, W.; Hopmans, J.W.; Kage, H.; Halbertsma, J.; Wösten, J.H.M. Reevaluation of the evaporation method for determining hydraulic functions in unsaturated soils. Soil Sci. Soc. Am. J.
**1993**, 57, 1436–1443. [Google Scholar] [CrossRef] - Kumar, S.; Sekhar, M.; Reddy, D.V.; Mohan Kumar, M.S. Estimation of soil hydraulic properties and their uncertainty: Comparison between laboratory and field experiment. Hydrol. Process.
**2010**, 24, 3426–3435. [Google Scholar] [CrossRef] - Schelle, H.; Iden, S.C.; Durner, W. Combined transient method for determining soil hydraulic properties in a wide pressure head range. Soil Sci. Soc. Am. J.
**2011**, 75, 1681–1693. [Google Scholar] [CrossRef] - Moreira, P.H.S.; van Genuchten, M.T.; Orlande, H.R.B.; Cotta, R.M. Bayesian estimation of the hydraulic and solute transport properties of a small-scale unsaturated soil column. J. Hydrol. Hydromech.
**2016**, 64, 30–44. [Google Scholar] [CrossRef][Green Version] - Younes, A.; Mara, T.; Fahs, M.; Grunberger, O.; Ackerer, P. Hydraulic and transport parameter assessment using column infiltration experiments. Hydrol. Earth Syst. Sci.
**2017**, 21, 2263–2275. [Google Scholar] [CrossRef][Green Version] - Younes, A.; Mara, T.A.; Voltz, M.; Guellouz, L.; Musa Baalousha, H.; Fahs, M. A new efficient Bayesian parameter inference strategy: Application to flow and pesticide transport through unsaturated porous media. J. Hydrol.
**2018**, 563, 887–899. [Google Scholar] [CrossRef] - Younes, A.; Zaouali, J.; Kanzari, S.; Lehmann, F.; Fahs, M. Bayesian Simultaneous Estimation of Unsaturated Flow and Solute Transport Parameters from a Laboratory Infiltration Experiment. Water
**2019**, 11, 1660. [Google Scholar] [CrossRef][Green Version] - Mboh, C.M.; Huisman, J.A.; Zimmermann, E.; Vereecken, H. Coupled Hydrogeophysical Inversion of Streaming Potential Signals for Unsaturated Soil Hydraulic Properties. Vadose Zone J.
**2012**, 11. [Google Scholar] [CrossRef] - Younes, A.; Zaouali, J.; Lehmann, F.; Fahs, M. Sensitivity and identifiability of hydraulic and geophysical parameters from streaming potential signals in unsaturated porous media. Hydrol. Earth Syst. Sci.
**2018**, 22, 3561–3574. [Google Scholar] [CrossRef][Green Version] - Pan, F.; Zhu, J.; Ye, M.; Pachepsky, Y.A.; Wu, Y.S. Sensitivity analysis of unsaturated flow and contaminant transport with correlated parameters. J. Hydrol.
**2011**, 397, 238–249. [Google Scholar] [CrossRef] - Brunetti, G.; Šimůnek, J.; Turco, M.; Piro, P. On the use of global sensitivity analysis for the numerical analysis of permeable pavements. Urban Water J.
**2018**, 15, 269–275. [Google Scholar] [CrossRef][Green Version] - van Griensven, A.; Meixner, T.; Grunwald, S.; Bishop, T.; Diluzio, M.; Srinivasan, R. A global sensitivity analysis tool for the parameters of multi-variable catchment models. J. Hydrol.
**2006**, 324, 10–23. [Google Scholar] [CrossRef] - Fajraoui, N.; Ramasomanana, F.; Younes, A.; Mara, T.A.; Ackerer, P.; Guadagnini, A. Use of global sensitivity analysis and polynomial chaos expansion for interpretation of nonreactive transport experiments in laboratory-scale porous media. Water Resour. Res.
**2011**, 47, W02521. [Google Scholar] [CrossRef] - Mualem, Y. A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resour. Res.
**1976**, 12, 513–522. [Google Scholar] [CrossRef][Green Version] - van Genuchten, M.T. A Closed-form Equation for Predicting the Hydraulic Conductivity of Unsaturated Soils. Soil Sci. Soc. Am. J.
**1980**, 44, 892–898. [Google Scholar] [CrossRef][Green Version] - Sudret, B. Global sensitivity analysis using polynomial chaos expansions. Reliab. Eng. Syst. Saf.
**2008**, 964–979. [Google Scholar] [CrossRef] - Homma, T.; Saltelli, A. Importance measures in global sensitivity analysis of nonlinear models. Reliab. Eng. Syst. Saf.
**1996**, 52, 1–17. [Google Scholar] [CrossRef] - Sobol’, I.M. Sensitivity estimates for nonlinear mathematical models. Math. Model. Comput. Exp.
**1993**, 1, 407–414. [Google Scholar] - Sobol′, I.M. Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Math. Comput. Simul.
**2001**, 55, 271–280. [Google Scholar] [CrossRef] - Kaipio, J.; Somersalo, E. Statistical and Computational Inverse Problems; Applied Mathematical Sciences; Springer: New York, NY, USA, 2004; p. 160. [Google Scholar]
- Hastings, W.K. Monte Carlo sampling methods using Markov chains and their applications. Biometrika
**1970**, 57, 97–109. [Google Scholar] [CrossRef] - Metropolis, N.; Rosenbluth, A.W.; Rosenbluth, M.N.; Teller, A.H.; Teller, E. Equation of state calculations by fast computing machines. J. Chem. Phys.
**1953**, 21, 1087–1092. [Google Scholar] [CrossRef][Green Version] - Gallagher, M.; Doherty, J. Parameter estimation and uncertainty analysis for a watershed model. Environ. Model. Softw.
**2007**, 22, 1000–1020. [Google Scholar] [CrossRef] - Kahl, G.M.; Sidorenko, Y.; Gottesbüren, B. Local and global inverse modelling strategies to estimate parameters for pesticide leaching from lysimeter studies: Inverse modelling to estimate pesticide leaching parameters from lysimeter studies. Pest Manag. Sci.
**2015**, 71, 616–631. [Google Scholar] [CrossRef] [PubMed] - Vrugt, J.A.; Bouten, W. Validity of first-order approximations to describe parameter uncertainty in soil hydrologic models. Soil. Sci. Soc. Am. J.
**2002**, 66, 1740–1751. [Google Scholar] [CrossRef][Green Version] - Hindmarsh, A.C. LSODE and LSODI, two new initial value ordinary differential equation solvers. ACM Signum Newsl.
**1980**, 15, 10–11. [Google Scholar] [CrossRef] - Miller, C.T.; Abhishek, C.; Farthing, M.W. A spatially and temporally adaptive solution of Richards’ equation. Adv. Water Resour.
**2006**, 29, 525–545. [Google Scholar] [CrossRef] - Farthing, M.W.; and Ogden, F.L. Numerical solution of Richards’ equation: A review of advances and challenges. Soil Sci. Soc. Am. J.
**2017**, 81, 1257–1269. [Google Scholar] [CrossRef][Green Version] - Fahs, M.; Younes, A.; Lehmann, F. An easy and efficient combination of the Mixed Finite Element Method and the Method of Lines for the resolution of Richards’ Equation. Environ. Modell. Softw.
**2009**, 24, 1122–1126. [Google Scholar] [CrossRef] - Mara, T.A.; Tarantola, S. Application of global sensitivity analysis of model output to building thermal simulations. Build. Simul.
**2008**, 1, 290–302. [Google Scholar] [CrossRef][Green Version] - Fajraoui, N.; Mara, T.A.; Younes, A.; Bouhlila, R. Reactive transport parameter estimation and global sensitivity analysis using sparse polynomial chaos expansion. Water Air Soil Poll.
**2012**, 223, 4183–4197. [Google Scholar] [CrossRef] - Shao, Q.; Younes, A.; Fahs, M.; Mara, T.A. Bayesian sparse polynomial chaos expansion for global sensitivity analysis. Comput. Methods Appl. Mech. Eng.
**2017**, 318, 474–496. [Google Scholar] [CrossRef][Green Version] - Younes, A.; Delay, F.; Fajraoui, N.; Fahs, M.; Mara, T.A. Global sensitivity analysis and Bayesian parameter inference for solute transport in porous media colonized by biofilms. J. Contam. Hydrol.
**2016**, 191, 1–18. [Google Scholar] [CrossRef] [PubMed] - Mara, T.A.; Belfort, B.; Fontaine, V.; Younes, A. Addressing factors fixing setting from given data: A comparison of different methods. Environ. Modell. Softw.
**2017**, 87, 29–38. [Google Scholar] [CrossRef][Green Version] - Xiu, D.; Karniadakis, G.E. A new stochastic approach to transient heat conduction modeling with uncertainty. Int. J. Heat Mass Tran.
**2003**, 46, 4681–4693. [Google Scholar] [CrossRef] - Laloy, E.; Vrugt, J.A. High-dimensional posterior exploration of hydrologic models using multiple-try DREAM (ZS) and high-performance computing. Water Resour. Res.
**2012**, 48, W01526. [Google Scholar] [CrossRef][Green Version] - Hayek, M.; Younes, A.; Zouali, J.; Fajraoui, N.; Fahs, M. Analytical solution and Bayesian inference for interference pumping tests in fractal dual-porosity media. Computat. Geosci.
**2018**, 22, 413–421. [Google Scholar] [CrossRef] - Linde, N.; Ginsbourger, D.; Irving, J.; Nobile, F.; Doucet, A. On uncertainty quantification in hydrogeology and hydrogeophysics. Adv. Water Resour.
**2017**, 110, 166–181. [Google Scholar] [CrossRef] - Gelman, A.; Carlin, J.B.; Stren, H.S.; Rubin, D.B. Bayesian Data Analysis; Chapmann and Hall: London, UK, 1997. [Google Scholar]
- Gelman, A.; Rubin, D.B. Inference from Iterative Simulation Using Multiple Sequences. Statist. Sci.
**1992**, 7, 457–472. [Google Scholar] [CrossRef]

**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