1. Introduction
The limited availability of high-quality water for agricultural purposes is a critical issue in arid and semi-arid regions. However, frequently significant quantities of water with different salinity levels exist in these areas that could be alternative resources for dealing with this problem [
1,
2]. Groundwater, agricultural till water or drainage water, and municipal or industrial wastewater are the main sources of saline waters [
3,
4,
5,
6,
7]. In the last three decades, irrigation with saline water has been extended in semi-arid regions [
8]. Expansion of saline irrigation practices without proper management could increase the risk of degrading soil quality and consequently losing agricultural lands in the long term [
9]. About 77 million hectares of fields have been impacted by salinization worldwide [
10]. Crops yield and yield components reductions as subsequences of irrigation with saline waters have been reported multiple times [
11,
12,
13]. Every year about 1 to 2% of irrigated agricultural lands are reduced due to high soil salinity [
10]. Therefore, appropriate analysis of crop root zone salinity for irrigation with saline waters are critical to explore proper management methods to control and alleviate salinization. Field experiments are usually time and money-consuming, and adapting the results to other locations with different agronomic and irrigation managements is complicated. Hence, using agro-hydrological models could be a reasonable answer to this issue. To date, several agro-hydrological models have been introduced in the literature to simulate the soil water and salinity dynamics under different climate, irrigation, and agronomic conditions. These models are categorized as steady-state and transient based on their developing approach. For instance, the TETrans [
14], SaltMod [
15], SALEACH [
16], and WSBM [
17] are among the models which have been developed so far to simulate soil water content and salinity for different cultivations.
TETrans is a transient model that simulates changes in the solute and water content at the crop root zone through a series of events within a finite discrete soil depth. These events include infiltration of irrigation water, draining soil profile to field capacity, root water uptake through transpiration, and evaporation through the soil surface. The assumptions have been made to pursue the development of this model, which might not be realistic. It has been assumed that each process or event occurs in sequential order within given soil depths. The soil is homogenously distributed over discrete depths. Depletion of soil water content through evapotranspiration does not go below the crop’s threshold to water stress, and dispersion is mostly negligible [
18]. The model uses these sequential components to calculate the salt balance for simulating solute transport at the crop root zone [
14]. Thus, the assumptions are also reflected in solute transport process calculations.
The SaltMod model is another transient model that predicts the salinity of soil water, drainage water, and groundwater. Furthermore, the model computes water table depth and the drainage water quantity in irrigated agricultural fields [
15]. The model computations are mainly based on the water model, salt balance model, and seasonal agronomic practices. The SaltMod has been known to be more reliable for its seasonal rather than daily outputs; because it implements the seasonal water balance for the region of interest, not daily bases [
19]. However, this approach reduces the number of the model’s inputs while neglecting in-season soil water and salinity dynamics, which could substantially affect the model’s predictions.
Shahrokhnia and Wu, 2021, developed the web-based SALEACH model based on a steady-state approach to reproduce crop root zone and drainage water salinity to estimate leaching requirement and irrigation depth [
16]. The models that simulate soil salinity changes based on steady approaches assume that irrigation water flows continuously downward at a constant rate, and the evapotranspiration rate is constant during the growing season. Also, soil-soluble salt concentrations are constant at any point [
20]. However, comparing the assumptions with observational data reveals that these assumptions are unrealistic.
Liu et al., 2022, have introduced the WSBM model that predicts the general trend of soil salinization in the long-term aspect [
17]. The model was developed based on water and salt balance at the crop root zone and groundwater. The model provides preliminary information to explore well-canal irrigation water quality strategies’ effects on the soil. Therefore, the model tends to provide a general perspective rather than accurate simulations to explore the effects of irrigation water management strategies on agricultural lands. However, among the agro-hydrological models, the HYDRUS-1D has been known as a comprehensive model. Only some simplifications have been implemented for the model development. The HYDRUS-1D simulations of soil water flow and solute transport processes are closer to reality than the majority of the agro-hydrological models.
The HYDRUS-1D model is a distinguished numerical model that simulates soil water flow and solute transport [
21], and its reliability has been proved multiple times in the literature. Liu et al., 2022 have illustrated the reliability of the HYDRUS-1D model through calibration and validation process to simulate soil water content and salt movement in 300 cm soil profiles at an irrigated cropland and unirrigated grassland [
22]. Noshadi et al., 2020 have found the HYDRUS-1D model very accurate in simulating soil water content and salt dynamic under wheat cultivation for irrigation with waters with different levels of salinity [
23]. Kanzari et al., 2018 concluded that the HYDRUS-1D model duplicated soil water and salinity dynamics of soil in a semi-arid region [
24]. A study conducted in southwest Queensland, Australia, showed that the HYDRUS-1D model successfully simulates salt leaching in amended profiles [
25]. Ramous et al., 2011 indicated that robust outputs of the HYDRUS-1D model regarding the reduction of maize water and nutrient uptake under osmotic stress were observed in Alvalade and Mitra, Portugal [
26]. Askri et al., 2014, have investigated the interaction effects of waterlogging, salinity, and water shortage on root water uptake of date palms in an oasis by using the HYDRUS-1D model. The model was calibrated and validated using sap flow density, soil hydraulic characteristics, and applied irrigation data. Their results have demonstrated the acceptable performance of the HYDRUS-1D in simulating water uptake of the palm tree [
27]. Skaggs et al., 2006, have studied the performance of the HYDRUS-1D model to imitate the root water uptake and drainage of lysimeter observational data under forage crops (alfalfa and tall wheatgrass) cultivations irrigated with synthetic drainage waters. The researchers have found good agreement between observational data and model simulation for wide ranges of irrigation water salinities (2.5 to 28 dS/m) [
28]. Ali et al., 2021 have inspected the capabilities of the HYDRUS-1D model in quantifying the soil hydraulic conductivity reduction under marginal quality water irrigation. The model outputs were compared with leaching column observational data. The study’s results emphasized that the HYDRUS-1D standard hydraulic conductivity reduction scaling factor needs adjustments. They recommended the non-linear approach as an alternative for determining the hydraulic conductivity scaling factor to enhance model water flow and solute transport outputs [
29]. The performance of the HYDRUS-1D model to simulate water balance and salinity build-up at rice root zone cultivated in micro-lysimeters have been tested by Phogat et al., 2010. The rice crop was irrigated with waters with different salinity levels (0.4 to 10 dS/m). The statistical analysis showed close agreement between model outputs and observational data [
30]. Moreover, this study has shown the reliability of the HYDRUS-1D outputs to be used for calculations of rice water productivity.
To use the HYDRUS-1D model, the proper calibration of the model is vital. Simultaneously the parameters and outputs of the model are subject to uncertainty, and it would be beneficial if they could be quantified. One of the promising approaches for calibration and uncertainty analysis of the models is exploiting Bayesian statistics concepts. The Bayesian statistics concepts use the prior knowledge (prior distribution) about the model parameters and combine this information with observational data (likelihood function) to calculate the uncertainty of the model by achieving posterior distribution of the parameters [
31].
Several algorithms have been developed based on Bayesian statistics concepts to quantify the posterior distributions of parameters. Vrugt et al., 2003, introduced the Shuffled Complex Evolution Metropolis (SCEM-UA) algorithm for uncertainty analysis and optimizing parameters of the hydrological models [
32]. This SCEM-UA was developed to obtain the posterior distribution of the hydrologic model parameters. This algorithm constantly modifies the proposal distribution by combining the strength of the Metropolis algorithm, random search, competitive evolution, and complex shuffling to enhance the trajectory of the algorithm for converging toward the posterior distribution of parameters. Ter Braak, 2006, developed the algorithm known as Differential Evolution Markov Chain (DE-MC), which is the Markov Chain Monte Carlo (MCMC) version of the genetic algorithm Differential Evolution [
33]. DE-MC algorithm runs multiple chains in parallel. The algorithm updates the chains based on the difference between two random parameter vectors, and the Metropolis ratio controls the selection of these vectors. The superiority of the DE-MC algorithm over the conventional MCMC algorithm is the speed of calculation and convergence of the algorithm. Vrugt et al., 2009, introduced the MCMC-based algorithm known as Differential Evolution Adaptive Metropolis (DREAM) [
34]. The DREAM algorithm significantly improved the efficiency of the MCMC simulation. This algorithm runs multiple chains in parallel and adjusts the scale and orientation of the proposal distribution. The DREAM algorithm has been improved multiple times so far, and several versions entitled DREAM (D), DREAM (ZS), and DREAM (Kzs) have been introduced [
35,
36,
37]. However, the complexity of implementing these algorithms for agro-hydrological models would lead the studies to seek alternative algorithms that are less intricate and robust enough to explore uncertainty analysis. One well-known algorithm that fulfills these characteristics is Generalized Likelihood Uncertainty Estimation.
The Generalized Likelihood Uncertainty Estimation (GLUE) is a Bayesian theorem-based algorithm that uses prior information about the parameters and measurement data to estimate uncertainty in model parameters [
38]. Estimating the posterior distribution of the parameters is attained by sampling an enormous number of parameters from the prior distribution by Monte Carlo simulations [
39]. Then, the generated parameter sets’ performances are evaluated by the likelihood function. The uncertainty estimated by GLUE considers all sources of uncertainty, including input, structure, and parameter uncertainties. The calculated value by the likelihood regarding the parameter sets reflects all sources of error with the model [
40]. In the recent decade, the GLUE algorithm has received attention in agro-hydrological studies. Li et al., 2018 have proved that the calibrated DSSAT-CERES based on the GLUE method could accurately reproduce leaf area index, above-ground biomass, grain yield, and above-ground nitrogen of winter wheat in the Beijing area [
41]. Sun et al., 2016 successfully calibrated the parameters of the RZWQM-DSSAT (RZWQM2) model for simulating crop growth and transporting water and nitrogen in a wheat-maize cropping system using the GLUE method with some adjustments in its likelihood function [
42]. Their uncertainty analysis results declared that the parameters related to soil-saturated hydraulic conductivity, nitrogen nitrification and denitrification, and urea hydrolysis were the most effective ones in simulating crop yield components. Sheng et al., 2019 compared two Bayesian theorem-based algorithms of GLUE and DREAM for estimating parameters associated with cultivars in the APSIM-maize model. They found similar performance between GLUE and DREAM algorithms [
43]. Uncertainty assessment of the SWAP model for simulating soil water content (SWC) at a field scale using the GLUE method has been explored by Shafiei et al., 2014 [
40]. Their results revealed that the predictive uncertainty in simulating SWC was low, which proved the good performance of the SWAP model in dry regions at the field scale.
To date, few studies have focused on Bayesian calibration and uncertainty analysis of HYDRUS-1D, explicitly using the GLUE method for simulating soil salt dynamics under sprinkler irrigation systems at field scale. Thus, the main objective of this research was to calibrate and assess the uncertainty of the HYDRUS-1D model for simulating transient conditions of salts in the corn root zone in western Kansas based on the GLUE method.