# Bayesian Calibration and Uncertainty Assessment of HYDRUS-1D Model Using GLUE Algorithm for Simulating Corn Root Zone Salinity under Linear Move Sprinkle Irrigation System

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## Abstract

**:**

_{s}(saturated soil water content) parameters of water flow simulations, dispersivity (λ), and adsorption isotherm coefficient (K

_{d}) parameters of solute transport simulations comparing to the other parameters. A higher level of uncertainty was found for the diffusion coefficient as its corresponding posterior distribution was not considerably changed from its prior distribution. The reason for this phenomenon could be the minor contribution of diffusion to the solute transport process in the soil compared with advection and hydrodynamic dispersion under saline water irrigation conditions. Predictive uncertainty results revealed a lower level of uncertainty in the model outputs for the initial growth stages of corn. The analysis of the predictive uncertainty band also declared that the uncertainty in the model parameters was the predominant source of uncertainty in the model outputs. In addition, the excellent performance of the calibrated model based on 50% quantiles of the posterior distributions of the model parameters was observed in terms of simulating soil water content (SWC) and electrical conductivity of soil water (ECsw) at the corn root zone. The ranges of NRMSE for SWC and ECsw simulations at different soil depths were 0.003 to 0.01 and 0.09 to 0.11, respectively. The results of this study have demonstrated the authenticity of the GLUE algorithm to seek uncertainty aspects and calibration of the HYDRUS-1D model to simulate the soil salinity at the corn root zone at field scale under a linear move irrigation system.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Site Description

^{0.5}. Therefore, the irrigation water chemical characteristics were categorized as C1-S1 in the USSL classification (Table 2). A four-span linear move irrigation system (model 8000, Valmont Corp., Valley, NE, USA) was used to implement irrigation for the aims of the study, and the plots’ sizes of the study were 13.7 m × 27.4 m.

#### 2.2. Data Collection and Management

#### 2.3. HYDRUS-1D Model

#### 2.3.1. Water Flow Modeling

^{3}L

^{−3}), t is time (T), z is vertical space coordinate (L), K(h) is unsaturated hydraulic conductivity function (LT

^{−1}), h is soil pressure head (L), and S(z,t) is sink term (L

^{3}L

^{−3}T

^{−1}) representing the unit volume of water removed by the crop from a unit of volume of soil per time unit.

_{s}is saturated soil volumetric water content (L

^{3}L

^{−3}), θ

_{r}is residual soil volumetric water content (L

^{3}L

^{−3}), and α and n are empirical parameters.

_{s}is saturated hydraulic conductivity (LT

^{−1}), l is an empirical parameter known as tortuosity parameter (dimensionless), S

_{e}is relative soil effective saturation

_{e}= θ − θ

_{r}/θ

_{s}− θ

_{r})

#### 2.3.2. Root Water Uptake

^{−1}), T

_{p}is potential transpiration rate (L

^{3}L

^{−2}.T

^{−1})

_{50}is the pressure head at which water uptake is reduced by 50% and negligible osmotic (salinity) stress exists, and p is an empirical parameter. The π is soil salinity (dS/m), b is the slope of root water uptake reduction, and a is the root water uptake threshold to soil salinity (dS/m).

_{p}), initially, the potential evapotranspiration (ET

_{p}) was calculated by the FAO Penman-Monteith equation [48]. For the next step, the potential evaporation was computed using the following equation [51]:

^{−1}) was estimated by:

#### 2.3.3. Solute Transport

^{−3}), s is solid phase concentration (MM

^{−1}), t is time (T), θ is volumetric soil water content (L

^{3}L

^{−3}), D

_{e}is effective dispersion coefficient (L

^{2}T

^{−1}), q

_{w}is soil water flux (L

^{3}L

^{−2}T

^{−1}), z is vertical coordinates (L).

_{l}

^{s}is soil effective diffusion coefficient (L

^{2}T

^{−1}), D

_{lh}is coefficient of hydrodynamic dispersion (L

^{2}T

^{−1}), λ is known as dispersivity [L], θ

_{s}is saturated volumetric water content, and D

_{l}

^{w}is diffusion coefficient in free water (L

^{2}T

^{−1}).

_{d}is known as the distribution coefficient or adsorption isotherm coefficient (L

^{3}M

^{−1}).

#### 2.4. The HYDRUS-1D Model Setup

#### 2.5. Uncertainty Assessment and GLUE Method

- Water flow simulation parameters: [θ
_{r}, θ_{s}, α, n, K_{s}, l] - Solute transport parameters: [λ, D
^{l}_{w}, K_{d}] - Root water uptake = [a, b, h
_{50}, P1]

- 1.
- The prior distributions of parameters were identified based on the HYDRUS-1D model library and existing values in the literature (Table 4). The priors were considered uniformly distributed.

- 2.
- The parameters’ ranges were randomly sampled n times based on the Monte Carlo approach in RStudio 1.41717 environment.
- 3.
- The HYDRUS-1D model was run in the RStudio environment for each parameter set already sampled.
- 4.
- The likelihood values were calculated using inverse error variance as the likelihood function:

_{j}is jth parameter set, O

_{i}is ith observation, y is the model output n is the number of observations.

- 5.
- The threshold for likelihood values for behavioral parameter sets was specified. In this study, 10% of successful parameters after screening operation were used for uncertainty analysis.
- 6.
- The probability of each parameter set was computed using the following equation:

_{j}) is the probability (likelihood weight) of the jth parameter set, n is the number of parameters’ sets, and L(θ

_{j}|O) is the value of the likelihood of the jth parameter set.

- 7.
- The posterior distributions of the parameters and statistics were constructed. The empirical posterior distributions of parameters were achieved by pairs of parameters’ sets (θ
_{j}) and their corresponding probabilities. Then, by using the following equations, the mean and variance of the parameters were calculated:

_{post}and Var

_{post}are the mean and variance of the posterior distribution of the parameters.

- 8.
- For the final step, the simulated values of soil water salinity by the HYDRUS-1D model were sorted based on the corresponding probabilities to create a cumulative distribution function of model outputs (predictive uncertainty). Then 95% confidence intervals for model outputs were retrieved [40].

#### 2.6. Evaluation of the Model Performance

^{2}):

_{i}, S

_{i}, and $\overline{\mathrm{M}}$ are measured value, simulated value, and the average value of measurements. The R2 values close to 1 indicate the good performance model. The NRMSE ranges of <10%, 10–20%, 20–30%, and >30% categorize the model performance as excellent, good, fair, and poor.

## 3. Results and Discussion

#### 3.1. Parameters Uncertainty

_{s}, n, and α as 26, 22, and 8% reduction was obtained in their posterior standard deviation values compared to their priors. Likewise, the posterior SD values of λ and K

_{d}parameters of solute transport parameters were reduced by 16 and 15% compared with their corresponding prior values. Furthermore, the reduction in SD values of posterior distributions root water uptake reduction function for salinity stresses indicated a lower level of uncertainty remaining in these parameters (a and b). Contrastingly, estimations of the rest of the parameters were accomplished with lower confidence (higher uncertainty) as there was no considerable difference between the SD values of their posteriors and priors. The histograms of the parameters’ posterior distributions are presented in Figure 2, Figure 3 and Figure 4. The x-axis of the graphs was considered equal to the prior distribution of the parameters to compare posteriors and priors. Among the water flow simulating parameters, the skewed and peaked posterior distributions have been observed for θ

_{s}, α, and n. It indicates the lower level of uncertainty remained in these parameters after implementing the GLUE algorithm for the HYDRUS-1D model to simulate soil salinity under irrigation with saline water using a linear move irrigation system. The posteriors of θ

_{r}, K

_{s}, and l were slightly changed from their priors as they uniformly covered the upper and lower bounds of the prior distribution. Posterior distributions of solute transport parameters show (Figure 3) that the dispersivity (λ) and adsorption isotherm coefficient (K

_{d}) approximately follow a normal distribution, indicating a reduction in the uncertainty of their estimations.

^{l}

_{w}) was not noticeably different from its prior distribution. This could be because the diffusivity contribution to the solute transport procedure for saline water irrigation conditions was trivial compared with advection and hydrodynamic dispersion. In addition, the posterior distribution of the root water uptake threshold (a) was picked and concentrated around the median. However, the other parameters did not significantly change from their priors, indicating a higher level of uncertainty in these parameters after derivation by the GLUE algorithm [23]. This is presumably because of the scale of the study, as our experimental field had plots 13.7 m × 27.4 m dimensions. Studies conducted at the field scale can potentially increase the uncertainty in observational data due to several phenomena, such as uniformity distribution of irrigation applications, preferential flows as a consequence of compaction or shrinkage of dry soil before irrigation events, and soil water redistribution in the soil. On the other hand, the results showed that threshold reduction in root water uptake, which is presented generally in the literature, could be adequately derived for a specific location by the GLUE algorithm. This threshold value could be used as a guideline for leaching requirement specifications and designing irrigation systems. To compare the computed value of root water uptake threshold with the existing values in the literature [54], The value should be divided by 2 because the HYDRUS-1D model uses the values of electrical conductivity of soil water (ECsw). The values in the literature are presented as the electrical conductivity of soil-saturated paste extract (ECe). Our results found ECsw = 3.324 dS/m or ECe = 1.662 dS/m, which was very close to the ones reported by Mass and Hoffman, 1977 (ECe = 1.7 dS/m)—[54]. The relative sensitivity of the parameters can be analyzed by comparing the CV values of the prior and posterior distribution of the parameters. The α, a, and b were the three most sensitive parameters. In contrast, the three l, h

_{50}, and P1 were the least sensitive parameters, respectively, for simulating soil salinity at the corn root zone at the field scale. The scatter plots of likelihood values of behavioral parameters’ sets related to water flow, solute transport, and root water uptake processes are presented in Figure 5 to explore further aspects in identifying the parameters. These plots demonstrate that the single optimum parameter set is identifiable because the parameter set’s corresponding likelihood value was significantly higher in the parameters’ response surface than the majority of the other parameter’s values. The optimum water flow simulation parameters sets are θr = 0.0637, θs = 0.4575, α = 0.0119, Ks = 35.71, n = 1.647, and l = 0.3797. The optimum solute transport and root water uptake parameters are λ = 8.712, D

^{l}

_{w}= 1.35, Kd = 0.83, h

_{50}= −2845, P1 = 2.17, a = 3.63, and b = 5.913. Furthermore, based on Figure 5, ranges with higher probability or lower uncertainty in the parametric surface response of parameters can be identified only for θs and α. The results show that the ranges are: θs = [0.45–0.49], and α = [0.01–0.065].

#### 3.2. Predictive Uncertainty

#### 3.3. The HYDRUS-1D Model Performance

^{3}cm

^{−3}for parameters’ sets based on the 50% and 97.5% quantiles of parameters posteriors and likelihood scatterplots used as calibrated values. The NRMSEs for simulating SWC were from 0.003 to 0.01. The highest accuracy of the model was detected for simulating SWC at 76 cm depth during and out of the growing season. In addition, good adequacy of the model was noticed based on R2 values between simulated and measured SWCs. The calibrated HYDRUS-1D model based on 50% quantiles posteriors resulted in the highest accuracy of the model for simulating SWC compared with the other two series of calibration values.

^{2}values were from 0.13 to 0.60 for both parameters’ sets. The highest adequacy of the model for ECs predictions was observed at 16 cm soil depth for calibrated model based on likelihood values, and the lowest one was noticed for ECsw simulations at 46 cm soil depth for calibrated model based on 97.5% of parameters posterior distributions.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Time series of daily crop evapotranspiration and precipitation (15 May 2016 to 31 December 2016).

**Figure 6.**Predictive uncertainty of the HYDRUS-1D mode for simulating soil salinity of corn root zone at (

**a**) 16, (

**b**) 46 and (

**c**) 76 cm depths.

**Figure 7.**The simulated and observed soil water content at (

**a**) 16, (

**b**) 46, and (

**c**) 76 cm soil depths.

**Figure 8.**The simulated and observed electrical conductivity of soil water at (

**a**) 16, (

**b**) 46, and (

**c**) 76 cm soil depths.

Depth (cm) | Soil Texture | Sand (%) | Silt (%) | Clay (%) | Wilting Point (%) | Field Capacity (%) | Saturation (%) | Bulk Density (g.cm^{−2}) |
---|---|---|---|---|---|---|---|---|

0–240 | Silt loam | 18.5 | 55.5 | 26 | 15 | 33 | 45 | 1.38 |

Units | Value | |
---|---|---|

EC | dS/m | 1.2 |

SAR | (meq/L)^{0.5} | 2.64 |

Na^{+} | mg/L | 120 |

Ca^{2+} | mg/L | 58 |

Mg^{2+} | mg/L | 95 |

SO_{4}^{2−} | mg/L | 200 |

PO_{4}^{3−} | mg/L | 6.3 |

NO_{3}^{−} | mg/L | 118 |

K^{+} | mg/L | 6.1 |

Soil Depth (cm) | Soil Water Content (cm^{3}.cm^{−3}) | ECsw (dS/m) |
---|---|---|

0–30 | 0.289 | 2.842 |

30–60 | 0.247 | 2.87 |

60–120 | 0.208 | 5.043 |

120–200 | 0.20 | 5.0 |

Parameters | Units | Range | Mean | SD | CV |
---|---|---|---|---|---|

θ_{r} | - | 0.05–0.08 | 0.065 | 0.0086 | 0.1332 |

θ_{s} | - | 0.3–0.5 | 0.4 | 0.0577 | 0.1443 |

α | 1/cm | 0.001–0.2 | 0.1005 | 0.0574 | 0.5716 |

K_{s} | cm/days | 5–40 | 22.5 | 10.1036 | 0.4490 |

n | - | 1–3 | 2 | 0.5773 | 0.2886 |

l | - | 0.1–1 | 0.55 | 0.2598 | 0.4723 |

λ | cm^{2}/day | 5–30 | 17.5 | 7.2168 | 0.4123 |

D_{l}^{w} | cm^{2}/day | 1–2 | 1.5 | 0.2886 | 0.1924 |

K_{d} | cm^{3}/g | 0.1–1 | 0.55 | 0.2598 | 0.4723 |

a | dS/m | 2–5 | 3.5 | 0.7500 | 0.2142 |

b | % | 4–8 | 4.5 | 1.4433 | 0.3207 |

h_{50} | cm | −5000–−800 | −2900 | 1212.43 | −0.418 |

P1 | - | 1.5–3 | 2.25 | 0.433 | 0.19 |

Parameters | Mean | SD | CV | Quantiles | ||||
---|---|---|---|---|---|---|---|---|

2.5% | 25% | 50% | 75% | 97.5% | ||||

θ_{r} | 0.0642 | 0.00907 | 0.14127 | 0.0505 | 0.0557 | 0.0637 | 0.0716 | 0.0796 |

θ_{s} | 0.442 | 0.04214 | 0.09533 | 0.3417 | 0.412 | 0.4542 | 0.4743 | 0.497 |

α | 0.0614 | 0.05269 | 0.85814 | 0.0062 | 0.018 | 0.0436 | 0.0881 | 0.185 |

K_{s} | 22.97 | 9.653 | 0.42024 | 6.563 | 14.72 | 23.7 | 31.16 | 38.736 |

n | 1.725 | 0.4497 | 0.26069 | 1.1 | 1.337 | 1.69 | 1.992 | 2.702 |

l | 0.547 | 0.2568 | 0.46946 | 0.121 | 0.3341 | 0.5361 | 0.7533 | 0.993 |

λ | 15.42 | 6.023 | 0.39059 | 6.416 | 10.47 | 14.65 | 19.34 | 28.428 |

D_{l}^{w} | 1.53 | 0.2738 | 0.17895 | 1.029 | 1.306 | 1.567 | 1.76 | 1.975 |

K_{d} | 0.5744 | 0.2201 | 0.38318 | 0.129 | 0.4378 | 0.5898 | 0.7277 | 0.964 |

a | 3.324 | 0.3926 | 0.11811 | 2.637 | 2.996 | 3.346 | 3.648 | 3.964 |

b | 5.924 | 1.136 | 0.19176 | 4.122 | 4.938 | 5.993 | 6.82 | 7.873 |

h_{50} | −2846 | 1238 | −0.43499 | −4942 | −3901 | −2777 | −1776 | −852.265 |

P1 | 2.243 | 0.45 | 0.20062 | 1.523 | 1.828 | 2.24 | 2.642 | 2.952 |

SWC | ECsw | ||||||
---|---|---|---|---|---|---|---|

RMSE (cm^{3}.cm^{−3}) | NRMSE | R^{2} | RMSE (dS/m) | NRMSE | R^{2} | ||

Q50% | |||||||

16 cm | 0.003 | 0.01 | 0.84 | 0.30 | 0.11 | 0.41 | |

46 cm | 0.001 | 0.005 | 0.86 | 0.29 | 0.09 | 0.22 | |

76 cm | 0.0006 | 0.003 | 0.72 | 0.42 | 0.09 | 0.50 | |

Q97.5% | |||||||

16 cm | 0.008 | 0.03 | 0.87 | 0.30 | 0.12 | 0.31 | |

46 cm | 0.006 | 0.02 | 0.78 | 0.16 | 0.05 | 0.13 | |

76 cm | 0.003 | 0.01 | 0.47 | 0.46 | 0.1 | 0.58 | |

OptL | |||||||

16 cm | 0.004 | 0.02 | 0.82 | 0.23 | 0.09 | 0.6 | |

46 cm | 0.002 | 0.01 | 0.90 | 0.35 | 0.11 | 0.3 | |

76 cm | 0.0006 | 0.003 | 0.86 | 0.53 | 0.11 | 0.32 |

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## Share and Cite

**MDPI and ACS Style**

Moghbel, F.; Mosaedi, A.; Aguilar, J.; Ghahraman, B.; Ansari, H.; Gonçalves, M.C.
Bayesian Calibration and Uncertainty Assessment of HYDRUS-1D Model Using GLUE Algorithm for Simulating Corn Root Zone Salinity under Linear Move Sprinkle Irrigation System. *Water* **2022**, *14*, 4003.
https://doi.org/10.3390/w14244003

**AMA Style**

Moghbel F, Mosaedi A, Aguilar J, Ghahraman B, Ansari H, Gonçalves MC.
Bayesian Calibration and Uncertainty Assessment of HYDRUS-1D Model Using GLUE Algorithm for Simulating Corn Root Zone Salinity under Linear Move Sprinkle Irrigation System. *Water*. 2022; 14(24):4003.
https://doi.org/10.3390/w14244003

**Chicago/Turabian Style**

Moghbel, Farzam, Abolfazl Mosaedi, Jonathan Aguilar, Bijan Ghahraman, Hossein Ansari, and Maria C. Gonçalves.
2022. "Bayesian Calibration and Uncertainty Assessment of HYDRUS-1D Model Using GLUE Algorithm for Simulating Corn Root Zone Salinity under Linear Move Sprinkle Irrigation System" *Water* 14, no. 24: 4003.
https://doi.org/10.3390/w14244003