# Vadose Zone Modeling in a Small Forested Catchment: Impact of Water Pressure Head Sampling Frequency on 1D-Model Calibration

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Site Description and Monitoring Devices

^{2}has become a completely equipped environmental observatory with permanent sampling and measuring stations that have been in operation since 1986. This watershed is the study site of the Observatoire Hydro-Géochimique de l’Environnement (OHGE, http://ohge.unistra.fr) and belongs to the French and European networks of Critical Zone observatories. With altitudes ranging from 883 to 1146 m (Figure 1B,C), the current climate is mountainous-oceanic with a mean annual rainfall of 1400 mm and an average temperature of approximately 6 °C [38]. The site is well suited for multidisciplinary studies in hydrological, geochemical and forest research [39,40,41,42].

^{®}CR1000 datalogger with a measurement frequency of 10 min, according to other devices monitored in the catchment. All the connecting wires were protected with a 40-mm flexible plastic tube to avoid instrumentation damage. After the in situ installation of these devices, the hole was refilled trying to maintain the natural order of the different layers of soil as close as possible to reduce soil disturbance.

_{21}is the resistance at 21 °C, R

_{s}is the measured resistance, and dT = T

_{s}− 21, T

_{s}is the soil temperature. After preliminary tests conducted in the laboratory, the previous linear calibration relationship proposed by the manufacturer (Equation (1)) appeared to be well suited compared to the proposed nonlinear calibration relationship.

#### 2.2. Vertical Profile and Soil Properties

- The hydraulic retention curve was determined (soil moisture as a function of pressure head);
- The saturated hydraulic conductivity (K
_{sat}) was estimated through 9 constant head permeability tests (Table 1). The measures concerning the layer between −15 cm and −20 cm were discarded due to their abnormally large values and large standard deviation. For the same reason the first measure of each sample was discarded; - The composition of the soil was determined through a laser diffraction particle sizing technique (Figure 2). The soil was mainly composed of sand (70%–80%). A strong gradient of porosity can be observed over the first 15 cm, which is probably related to the richness of organic matter in this soil layer (Figure 2C).

#### 2.3. Numerical Models

#### 2.3.1. Potential and Actual Evapotranspiration

_{BG}in Equation (2)). Previous studies have shown that this approach considers the specificities of the experimental site, especially for the computation of the net radiation in the radiative term (first component of the r.h.s. term in Equation (2)) and the expression of the advective term (second component of the r.h.s. term in Equation (2)). Hence, to remain consistent with previous studies of the Strengbach catchment, the following equation was adopted:

_{BG}is given in mm·d

^{−1}, $\Delta $ is the derivative of saturated vapor pressure versus temperature (mbar·K

^{−1}) expressed as the air temperature T

_{a}(°C) (see Equation (3)), R

_{n}is the net radiation at the canopy surface (cal·cm

^{−2}·d

^{−1}), ${\mathsf{\delta}}_{\mathrm{e}}$ is the saturation vapor pressure deficit (mbar) expressed at the air temperature T

_{a}(°C) and the relative humidity H

_{r}(%) (see Equation (4)), $\mathsf{\gamma}$ is the psychrometric constant (0.66 mbar·K

^{−1}) and U

_{10}is the adjusted wind speed at 10 m height (m·s

^{−1}).

_{BG}; since the data are available at a 10-min time step, net radiation is cumulated every hour and other variables were averaged to obtain hourly PET

_{BG}. Notice that the cumulated values of hourly PET

_{BG}are in concordance with daily or weekly estimations. Between January 2011 and December 2016, the amount of rainfall reached 1228 mm/year, and the amount of PET

_{BG}is 446 mm/year.

^{©}software (https://appgeodb.nancy.inra.fr/biljou/) [50], the AET was calculated as follows:

^{−1}), In is the interception (mm·d

^{−1}) and SEvap is the soil and understored evaporation (mm·d

^{−1}).

#### 2.3.2. Vadose Zone Model

_{r}) [51,52]:

^{−1}], S

_{e}is the effective saturation [-], θ is the volumetric water content [L

^{3}·L

^{−3}], θ

_{s}is the saturated water content [L

^{3}·L

^{−3}], θ

_{r}is the residual water content [L

^{3}·L

^{−3}], h is the pressure head [L], K is the hydraulic conductivity [L·T

^{−1}], z is the vertical distance positive downward [L], t is the time [T], α is a parameter related to the mean pore size [L

^{−1}], n is a parameter reflecting the uniformity of the pore size distribution [-], m is a parameter defined by (m= 1−1/n) [-] [51], K

_{sat}is the saturated hydraulic conductivity [L·T

^{−1}], and L is a parameter related to the tortuosity chosen here equal to 0.5 [-] [52].

_{sat}, θ

_{s}, θ

_{r}, n, α) is needed for each layer of the considered domain. The hydraulic retention curves described in Section 2.2 were fitted to estimate the 5 parameters for a 3-layer profile (Table 1). In each layer, data coming from the sampling analyses of the 9 layers were aggregated to correspond to the 3 layers of the discretized domain. A second estimation of the hydrodynamic parameters was obtained from soil granulometry and bulk density through the ROSETTA pedotransfer function [53].

#### 2.4. Numerical Simulations

_{top}= pF 4). The initial pressure head profile is then obtained through the interpolation of the values measured at −42 cm, −71 cm, and −116 cm. Each simulation was subjected to a sufficiently long warm-up period so that no impact of the initial pressure head profile on parameter estimation was observed. An imposed flux boundary condition (BC) was applied to the soil surface, which was combined with a free drainage bottom BC.

#### 2.5. Strategies Investigated for the Model Calibration

_{i}) and simulated values $\left({\widehat{\mathrm{h}}}_{\mathrm{i}}\right)$:

_{inv}is the amount of data (3 times the number of hourly, daily or weekly observation times) used for the inverse problem.

_{p}is the amount of parameters, N

_{inv}is the amount of data used in the inverse problem, and O is the objective function value after minimization. The smaller the criterion value, the better the calibration.

_{sat}, n, and α. Parameters θ

_{s}and θ

_{r}have empirically determined assigned values due to the difficulties in their optimization and to decrease the problems’ degrees of freedom.

## 3. Results

#### 3.1. Parameters

_{sat}values with very large SE, which means that this parameter has little influence on pressure head. Generally, the saturated hydraulic conductivity is known to be a very sensitive parameter [62,63]. Here, the large SE could be related to the lack of dynamics in the pressure head values at depths of 71 cm and 116 cm, as well as to the small pressure variations during 2012. The 2015 optimized values also have quite a large SE value, but a value smaller than the 2012 SE. The optimized K

_{sat}values are very large compared to the values estimated by the ROSETTA pedotransfer function and constant head permeability test (Table 1). Moreover, if previously estimated values are larger for the third layer, optimized values are larger for the first or second layer. This may be due to a higher organic matter content. The “n” optimized values for the first layers are typical of clay-rich soils.

#### 3.2. Simulation Quality Estimation

_{inv}refers to the number of pressure head measurements used in the inverse parameters estimation (N

_{inv}= number of data in Table 4) and N

_{dir}refers to the number of pressure head measurements used for the comparison of direct problem solutions.

_{dir}refers to the number of pressure head measurements used for the comparison of direct problem solutions (N

_{dir}= 3 × 227,371 = 628,113), ${\overline{\mathrm{h}}}_{{\mathrm{L}}_{\mathrm{k}}}$ refers to the mean observed pressure head in layer L

_{k}(k = 1, 2 or 3), and N

_{Lk}refers to the number of values originating from layer L

_{k}. Notice that the RMSE in Table 5 was computed using N

_{dir}instead of N

_{inv}. The desired value for ME is one, whereas the desired values for MBE and MRSE are zero. The modeling efficiency, ME, is a global model performance measure that provides the ratio of deviations regarding the measured values. It compares the square difference between the modelled and measured heads (numerator) with the square difference between the average measured values and the measured values (denominator). The RMSE statistical measure is an average error obtained through a term by term comparison of the square difference between the simulated and observed values. Hence, errors are cumulated using RMSE, whereas the MBE value is a relative measure that distinguishes between under and overestimation. MBE is positive (resp. negative) when the computed values are greater (resp. smaller) on average than the measurements.

#### 3.3. Pressure Head

#### 3.4. Drained Flux

#### 3.5. Water Saturation

_{Lj}refers to the saturation level of the jth layer [-] and Nn

_{Lj}is the number of nodes in layer j.

_{sat}.

## 4. Conclusions

_{sat}, n, α) for each of the three layers. This goal is consistent with what appeared in the literature (see [66,67,68,69]). Estimating the water content at saturation would have required the collection of additional data, which was done in other studies [70,71,72]. Nonetheless, we underline that the present exercise was challenging due to a range of variations in the MP measurements, which were quite extensive compared to the cases generally presented that dealt with (field) data monitoring.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**(

**A**) Regional map of the Vosges Massif and location of the Strengbach catchment. (

**B**) Contour map of the Strengbach watershed. (

**C**) Topographic map of the Strengbach watershed and location of the studied weathering profile (map from OHGE). (

**D**) Vertical profile in the experimental spruce parcel (blue star in Figure 1C) where sensors have been installed.

**Figure 2.**(

**A**) Undisturbed soil cylinder samplers. (

**B**) Soil composition profile; yellow stars correspond to sampling points and (

**C**) porosity profile obtained through laboratory analysis of the field samples. These values are used for the ROSETTA pedotransfer function. The red lines in (

**C**) represent the interfaces between the three layers of the adopted soil model.

**Figure 3.**Conceptual scheme of the modeling approach. The data frequency changes at different steps during the process. The initial measured data and final simulated data have a 10-min time step. However, throughout the process, data with hourly, daily, and weekly time steps are used. The h

_{mes}(h

_{sim}) is measured (simulated) pressure head, T °C is temperature, rain is rainfall, AET is the calculated actual evapotranspiration, Ksat is the saturated hydraulic conductivity, α and n are parameters for the Mualem-van Genuchten model, DF is the drained flux at −150 cm, SL is the saturation level, MBE is the mean bias error, RMSE is the root mean square error, and ME refers to the modeling efficiency.

**Figure 4.**Pressure head values modeled for the period from May 2012 to September 2016. The model is calibrated using hourly, daily, and weekly data for the period from June to September 2012 (limited by vertical straight lines): pressure head values at depths of 42 cm (

**a**), 71 cm (

**b**) and 116 cm (

**c**).

**Figure 5.**Pressure head values modeled for the period from May 2012 to September 2016. The model is calibrated using hourly, daily, and weekly data for the period from April to December 2015 (limited by vertical straight lines in the picture): pressure head values at depths of 42 cm (

**a**), 71 cm (

**b**), and 116 cm (

**c**).

**Figure 6.**Modeled cumulative (

**a**) and daily (

**b**and

**c**) drained flux at the depth of 150 cm for the period from May 2012 to September 2016. The model is calibrated using hourly, daily, and weekly data for the periods from April to December 2015 (

**b**) and from June to September 2012 (

**c**) (calibration period is limited by vertical straight lines on each picture).

**Figure 7.**Saturation level modeled for the period from May 2012 to September 2016 for a three-layer model. The model is calibrated using hourly, daily and weekly data for the periods from April to December 2015 (

**a**) and from June to September 2012 (

**b**) (calibration period is limited by vertical straight lines on each picture).

**Table 1.**Estimation of hydrodynamic parameters obtained through fitting the hydraulic retention curves and ROSETTA pedotransfer function. The K

_{sat}value of the first parameter set was obtained using the constant head permeability test.

Layer | Retention Curve Fit | ROSETTA | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

K_{sat}(cm/day) | θr (-) | θs (-) | α (1/cm) | n (-) | K_{sat}(cm/day) | θr (-) | θs (-) | α (1/cm) | n (-) | |

1 | 805 | 0.2 | 0.5 | 0.08 | 1.60 | 202 | 0.04 | 0.48 | 0.034 | 1.46 |

2 | 843 | 0.2 | 0.5 | 0.10 | 1.40 | 213 | 0.04 | 0.41 | 0.041 | 1.92 |

3 | 1616 | 0.0 | 0.5 | 0.40 | 1.15 | 262 | 0.04 | 0.42 | 0.041 | 1.96 |

Layer | K_{sat}(cm/day) | θr (-) | θs (-) | α (1/cm) | n (-) |
---|---|---|---|---|---|

1 | 211.989 | 0.044 | 0.510 | 0.014 | 1.067 |

2 | 605.222 | 0.043 | 0.410 | 0.182 | 1.285 |

3 | 3.095 | 0.043 | 0.420 | 0.041 | 1.975 |

**Table 3.**Optimized parameters using hourly, daily, and weekly pressure head data. Data are selected from two periods: from June to September 2012 and from April to December 2015. The adopted values of θ

_{s}and θ

_{r}are shown in Table 1.

Calibration Period | Data Frequency | Layer | K_{sat}(cm/day) | * SE (±) | α (1/cm) | * SE (±) | n (-) | * SE (±) |
---|---|---|---|---|---|---|---|---|

2012 | hourly | 1 | 1463 | 366 | 0.018 | 0.002 | 1.040 | 0.001 |

2012 | hourly | 2 | 865 | 455 | 0.106 | 0.017 | 2.976 | 0.202 |

2012 | hourly | 3 | 6 | 7 | 0.075 | 0.022 | 2.661 | 1.107 |

2012 | daily | 1 | 168 | 65 | 0.012 | 0.002 | 1.039 | 0.004 |

2012 | daily | 2 | 3321 | 4836 | 0.180 | 0.077 | 2.617 | 0.549 |

2012 | daily | 3 | 1 | 3 | 0.043 | 0.028 | 2.335 | 3.265 |

2012 | weekly | 1 | 1144 | 36220 | 0.003 | 0.001 | 1.031 | 0.007 |

2012 | weekly | 2 | 337 | 436 | 0.213 | 0.060 | 1.465 | 0.114 |

2012 | weekly | 3 | 3 | 7 | 0.041 | 0.022 | 2.631 | 6.437 |

2015 | hourly | 1 | 275 | 9 | 0.0111 | 0.0002 | 1.083 | 0.001 |

2015 | hourly | 2 | 4970 | 392 | 0.1353 | 0.0023 | 1.580 | 0.008 |

2015 | hourly | 3 | 148 | 142 | 0.0307 | 0.0015 | 1.713 | 0.009 |

2015 | daily | 1 | 117 | 83 | 0.0050 | 0.0004 | 1.119 | 0.004 |

2015 | daily | 2 | 2394 | 787 | 0.1193 | 0.0103 | 1.737 | 0.053 |

2015 | daily | 3 | 730 | 665 | 0.0649 | 0.0147 | 1.532 | 0.026 |

2015 | weekly | 1 | 153 | 72 | 0.0105 | 0.0021 | 1.077 | 0.010 |

2015 | weekly | 2 | 4589 | 3016 | 0.1210 | 0.0149 | 1.576 | 0.062 |

2015 | weekly | 3 | 973 | 1405 | 0.0482 | 0.0153 | 1.836 | 0.082 |

Calibration Period | Data Frequency | Number of Data | Objective Function | ^{1} RMSE | ^{2} AIC | ^{3} BIC |
---|---|---|---|---|---|---|

2012 | hourly | 8568 | 1.71 × 10^{7} | 44.7 | 28,299.4 | 28,316.8 |

2012 | daily | 357 | 1.20 × 10^{6} | 57.9 | 1276.6 | 1281.6 |

2012 | weekly | 51 | 1.03 × 10^{5} | 44.9 | 186.5 | 183.8 |

2015 | hourly | 19,659 | 4.00 × 10^{8} | 143 | 84,721.9 | 84,742.5 |

2015 | daily | 822 | 1.75 × 10^{7} | 146 | 3574.8 | 3583.0 |

2015 | weekly | 117 | 2.34 × 10^{6} | 142 | 521.3 | 521.9 |

^{1}Root mean square error;

^{2}Akaike information criterion;

^{3}Bayesian information criterion.

**Table 5.**Statistical measures for direct simulations over the period from 2012 to 2016 using eight different sets of parameters (see Figure 3). All the sets of parameters were obtained through calibration except for the ROSETTA parameters set and laboratory parameters set. The statistical measures are related to the entire profile, and those referring to each layer are provided as supplementary material.

Direct Simulation | ^{1} MBE | ^{2} RMSE | ^{3} ME |
---|---|---|---|

2012 hourly calibration | −13.82 | 201.42 | −0.38 |

2012 daily calibration | −37.83 | 140.80 | 0.33 |

2012 weekly calibration | 94.81 | 903.92 | −26.76 |

2015 hourly calibration | 50.79 | 135.72 | 0.37 |

2015 daily calibration | 28.51 | 130.46 | 0.42 |

2015 weekly calibration | 52.31 | 136.36 | 0.37 |

ROSETTA parameters | 5.83 | 163.05 | 0.10 |

Laboratory parameters | −52.34 | 178.75 | −0.09 |

^{1}Mean bias error

^{2}root mean square error

^{3}model efficiency.

© 2018 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Belfort, B.; Toloni, I.; Ackerer, P.; Cotel, S.; Viville, D.; Lehmann, F. Vadose Zone Modeling in a Small Forested Catchment: Impact of Water Pressure Head Sampling Frequency on 1D-Model Calibration. *Geosciences* **2018**, *8*, 72.
https://doi.org/10.3390/geosciences8020072

**AMA Style**

Belfort B, Toloni I, Ackerer P, Cotel S, Viville D, Lehmann F. Vadose Zone Modeling in a Small Forested Catchment: Impact of Water Pressure Head Sampling Frequency on 1D-Model Calibration. *Geosciences*. 2018; 8(2):72.
https://doi.org/10.3390/geosciences8020072

**Chicago/Turabian Style**

Belfort, Benjamin, Ivan Toloni, Philippe Ackerer, Solenn Cotel, Daniel Viville, and François Lehmann. 2018. "Vadose Zone Modeling in a Small Forested Catchment: Impact of Water Pressure Head Sampling Frequency on 1D-Model Calibration" *Geosciences* 8, no. 2: 72.
https://doi.org/10.3390/geosciences8020072