Parameter Uncertainty in Water–Salt Balance Modeling of Arid Irrigation Districts

Abstract

Soil salinization poses a major threat to agricultural sustainability in arid regions worldwide, where it is intrinsically linked to irrigated agriculture. In these water-scarce environments, the equilibrium of the water and salt balance is easily disrupted, causing salts to accumulate in the root zone and directly constraining crop growth, thereby creating an urgent need for precise water and salt management strategies. While precise water and salt transport models are essential for prediction and control, their accuracy is often compromised by parameter uncertainty. To address this, we developed a lumped water–salt balance model for the Hetao Irrigation District (HID) in China, integrating farmland and non-farmland areas and vertically structured into root zone, transition layer, and aquifer. A novel calibration approach, combining random sampling with Kernel Density Estimation (KDE), was introduced to identify optimal parameter ranges rather than single values, thereby enhancing model robustness. The model was calibrated and validated using data from the Yichang sub-district. Results showed that the water balance module performed satisfactorily in simulating groundwater depth (R² = 0.79 for calibration, 0.65 for validation). The salt balance module effectively replicated the general trends of soil salinity dynamics, albeit with lower R² values, which reflects the challenges of high spatial variability and data scarcity. This method innovatively addresses the common challenge of parameter uncertainty in the model, narrows the parameter value ranges, enhances model reliability, and incorporates sensitivity analysis (SA) to identify key parameters in the water–salt model. This study not only provides a practical tool for managing water and salt dynamics in HID but also offers a methodological reference for addressing parameter uncertainty in hydrological modeling of other data-scarce regions.

1. Introduction

Soil salinization poses a major constraint to global agricultural sustainable development. According to statistics, the total global area of salinized land reaches 1.1 × 10⁹ hectares, accounting for approximately 10% of the total land area, with these salinized lands predominantly distributed in arid and semi-arid climate zones [1,2]. This is primarily due to scarce precipitation and intense evaporation in these regions, where agricultural production heavily relies on irrigation [3]. Soil salinization leads to significant declines in agricultural productivity, not only threatening the sustainable utilization of land resources, food security, and sustainable agricultural development but also rendering ecosystems more fragile or degraded [4]. In recent years, the frequent occurrence of extreme droughts and rainstorms caused by climate change, compounded by issues such as groundwater over-exploitation, has continued to exacerbate this problem in arid and semi-arid regions [5].

Soil salinization has always been a major challenge for agricultural development in arid and semi-arid regions. Clarifying the patterns and processes of water and salt transport is fundamental for ensuring the construction of ecological irrigation districts [6]. Since the 1960s, researchers both domestically and internationally have conducted extensive studies on the theory and simulation of soil water and salt transport. Methods for studying water and salt transport include field experimental monitoring, water–salt balance models, and numerical simulation [7,8]. Among these, field experimental monitoring uses instruments and equipment for real-time observation of farmland hydrological elements, providing crucial data support for understanding hydrological processes. However, it is costly and difficult to adapt to long-term dynamic research needs [9]. As field experiments are time-consuming and labor-intensive [10], an increasing number of researchers are using models to study soil water and salt transport processes. When it comes to model selection, a critical trade-off exists. Among the available approaches, kinetic models based on Darcy’s law and solute transport equations—such as HYDRUS [11,12], SWAP [13,14], and DRAINMOD—have attracted increasing attention for their ability to describe detailed physical processes. However, these models generally require a large number of soil hydraulic parameters (e.g., hydraulic conductivity, soil water retention curves), which are often difficult to obtain in data-scarce regions. This limits their applicability, and uncertainties in these parameters can significantly affect the reliability of simulation results [15]. With the deepening of theoretical research and the rapid development of computer technology, several models based on the principle of soil water–salt balance have been established, such as ISAREG [16], DRAINMod [17], SaltMod [18], etc., which are widely used in the study of farmland water and salt transport. The SaltMod and SahysMod models were developed by Dutch research institutions [18] based on water–salt balance theory to predict key indicators such as soil salinity, groundwater depth, and mineralization degree. In contrast, the water and salt balance model is based on the fundamental principle of mass conservation, using different conceptual models to depict various relatively independent internal and external hydrological processes [8]. The advantage of water–salt balance models lies in their empirical or conceptual simplification of physical models, resulting in fewer model parameters and avoidance of complex nonlinear relationships. Therefore, they are relatively convenient for model calibration and parameter adjustment, especially suitable for data-scarce regions. To date, a substantial body of research has been established on water–salt balance models in Hetao Irrigation District (HID) [19]. Simulating and predicting regional soil water and salt movement patterns can provide a critical foundation for monitoring and assessing soil salinization in HID. Relevant studies have shown that changes in groundwater depth significantly influence regional evapotranspiration [20], while fluctuations in groundwater levels exhibit a dual impact on the irrigation district: although shallow groundwater can serve as a stable water source for crops, its high mineralization tendency promotes the upward movement of salts to the topsoil through evapotranspiration, thereby exacerbating the risk of soil salinization [21]. Furthermore, water–salt balance models developed based on land use types have revealed the migration patterns of soil salinity from agricultural to non-agricultural areas [22]. Regarding water–salt interactions between cultivated and uncultivated land, groundwater demonstrates significant lateral movement, with cultivated land acting as a notable source of groundwater recharge to wasteland areas [23]. Researchers have used methods such as water balance analysis, water–salt balance calculations, transport equations, and model simulations to deeply investigate the patterns of farmland soil water and salt migration and their impact on agricultural production. These research results provide important theoretical support for scientific irrigation, water–salt balance regulation, and salinization prevention and control.

However, effective prediction and prevention of salinization rely on high-precision water–salt transport models, and the accuracy of model parameters directly determines the scientificity of decision-making. Currently, soil hydraulic parameters are typically initially estimated using pedotransfer functions based on soil texture and bulk density [24], followed by manual trial-and-error adjustments [25]. This approach is not only inefficient but also fails to guarantee parameter reliability. Furthermore, when applying models to investigate soil water–salt dynamics, default values are often used for some intermediate process parameters [26], which significantly compromises model accuracy. In particular, the lack of research on parameter probability distributions in regional water–salt modeling is frequently noted [27], yet most studies resort to alternative methods to circumvent this issue. The limitations of existing models in parameter sensitivity identification and error propagation analysis restrict their practical application in precise early warning of salinization risks. Global optimization algorithms tend to prioritize fitting accuracy but may yield inappropriate parameters, with little discussion in existing research regarding the physical plausibility of these parameter values. Sensitivity analysis (SA) can identify highly sensitive parameters among multiple model inputs [28]. By adjusting these high-sensitivity parameters while simplifying or fixing low-sensitivity ones, the efficiency of model parameterization can be improved [29,30]. Moreover, the occurrence of equifinality—where different parameter combinations produce similar model outcomes—significantly undermines the reliability of model validation. Therefore, conducting sensitivity analysis on model parameters is essential [27].

In summary, the patterns of soil water and salt transport at the regional scale are complex and highly variable [9,31]. Existing research has predominantly focused on investigating these transport mechanisms and their environmental effects, while studies on the key parameter systems driving these processes and the quantification of their uncertainties remain inadequate. Additionally, few studies have analyzed the distribution characteristics of the parameters. The lack of parameter sensitivity identification and error propagation analysis in current models limits their practical application in precise early warning of salinization risks. Based on previous research, we prioritized the water–salt balance model for parameter uncertainty analysis in this study. This decision was primarily driven by the model’s simple structure and highly lumped parameters, which significantly reduce parameter dimensionality and make quantitative uncertainty analysis computationally more feasible. Therefore, this study establishes an uncertainty parameter water–salt balance model, aiming to clarify the key parameter system in the water–salt model of HID. The model parameters and their value ranges are determined through random sampling and Kernel Density Estimation (KDE) methods to address the uncertainty in model parameter values, providing a precise digital tool for the ecological security and sustainable agricultural development of the irrigation district.

2. Materials and Methods

2.1. Study Area

2.1.1. Study Area Description

As shown in Figure 1, the HID, a typical salinized irrigation area located in the middle and upper reaches of the Yellow River, is the most important agricultural region and an ecologically vulnerable area in Northwest China (40°20′–41°18′ N, 106°20′–109°20′ E). The region belongs to a typical temperate continental climate, with cold winters and mild summers. The average annual precipitation is only 113–220 mm, with over 70% concentrated from July to September, while the multi-year average evaporation is as high as 1035–2346 mm. The groundwater depth in HID is relatively shallow, ranging from 1.5 to 3.0 m [32], and strong evaporation drives the accumulation of soil salinity from deeper layers to the surface, resulting in secondary salinization that severely restricts crop growth. The HID primarily relies on water diverted from the Yellow River for irrigation, with an annual diversion volume of about 4.727 billion m³. The total land area of the irrigation district is 97.65 million mu, of which the cultivated area is 13.7331 million mu, making it an important production base for grain and oil crops.

2.1.2. Hydrogeological Conditions

1. Groundwater Recharge-Runoff-Discharge Characteristics

The dynamics of groundwater in HID are controlled by the water cycle in arid areas, exhibiting typical characteristics of “active vertical movement, weak lateral movement”. Groundwater recharge mainly depends on seepage from the Yellow River diversion irrigation and local lateral runoff. However, influenced by topography and aquifer structure, the scope of lateral recharge is limited. The overall runoff conditions are poor, especially in the plain area where the hydraulic gradient is gentle, and horizontal flow is weak. Discharge is dominated by vertical evapotranspiration, and the lack of horizontal discharge pathways leads to hydraulic stagnation downstream, eventually converging into Wuliangsuhai Lake, forming a closed basin groundwater system dominated by vertical water cycles.

2. Groundwater Dynamic Characteristics

The dynamics of groundwater in HID are driven by irrigation and meteorological factors, showing a significant seasonal cycle, specifically manifested as a “two rises and two falls” intra-annual variation pattern. During the freeze–thaw period, the water level drops to the annual minimum during the freezing stage and rises due to the recharge from meltwater during the thawing stage. During the non-freeze–thaw period, summer irrigation with large amounts of water and autumn irrigation for salt leaching cause two significant rises in the water level, while intense evaporation and autumn irrigation lead to periodic declines in the water level, ultimately forming a regular fluctuation cycle within the year, as shown in Figure 2.

2.2. Farmland-Non-Farmland Water–Salt Balance Model Construction

In the lumped water–salt balance model, the division of vertical balance zones is a key step. Given the necessity of simulating root zone water–salt dynamics and the differences in water–salt processes between different soil layers, this study divides the soil profile vertically into the root zone, a transition layer, and the groundwater aquifer [33,34]. The common root depth of crops in HID is 0.6 m [35]. Accordingly, the root zone is defined as the soil layer within 0.6 m below the surface. This layer is the main area of crop root activity, frequently exchanging water and salt with the atmosphere, plants, and the lower soil layers, and is highly variable. The groundwater aquifer is soil-saturated, and its depth data is an important lower boundary condition for the root zone. Historical field monitoring data show that the average groundwater depth is 1.5–2.5 m, presenting a fluctuating state; therefore, the aquifer is taken as below 1.0 m from the surface. According to other studies [36], the area between the root zone and the aquifer is defined as the transition layer. It is assumed that the water content in this layer remains constant, meaning it does not store or release water; water exchange occurs only between the root zone and the groundwater. The main discharge pathways for groundwater include capillary rise consumed by evaporation, discharge through drainage systems, and groundwater extraction for domestic and industrial use. Recharge mainly comes from canal seepage, field irrigation infiltration, and vertical infiltration recharge from precipitation [35]. On a temporal scale, based on irrigation and seasons in the HID, it is divided into four periods: the growth period (May to August), which is the key period for crop growth; the autumn irrigation period (September to November), for soil drainage and salt leaching after irrigation; the freezing period (December to February of the following year), when soil is frozen and water–salt transport activities weaken; and the thawing period (March to April), when soil thaws, the groundwater level rises, and salts redistribute.

The farmland-non-farmland water–salt balance model is divided into two main modules: the water balance module and the salt balance module. The entire HID is divided into farmland and non-farmland (all landscape areas other than farmland are classified as non-farmland). These two areas are further divided vertically into three layers: the root zone, the transition zone, and the aquifer, establishing an irrigation district farmland–non-farmland root zone soil transition layer–aquifer balance model. The specific steps are as follows: First, based on the groundwater interaction relationship between farmland and non-farmland, the exchange water volume between them is derived using the overall and separate groundwater balance. Second, after determining the water volume, the salt fluxes are generalized based on the salt concentration of each water flux. Based on the salt balance calculation for each body, the salt change process in each layer of farmland and non-farmland is completed, ultimately obtaining the dynamic changes in soil water and groundwater salinity in farmland and non-farmland.

2.2.1. Farmland-Non-Farmland Water Balance Model

As shown in Figure 3, the schematic diagram of the farmland-non-farmland water balance model simulation consists of six balance bodies: the root zone soil, transition layer, and aquifer for both farmland and non-farmland. It considers multiple soil water and groundwater processes such as rainfall, irrigation infiltration, drainage, and phreatic evaporation, and has the capability to simulate groundwater depth, water fluxes between groundwater bodies, and the groundwater exchange volume between farmland and non-farmland.

• Water Balance Equations for Each Zone

(1) Root Zone Water Balance Equation

The root zone water budget includes three input terms (precipitation infiltration, Yellow River diversion irrigation recharge, and groundwater capillary rise) and three output terms (evapotranspiration, crop root water uptake, and deep percolation), where evapotranspiration consists of soil evaporation and plant transpiration. The irrigation water volume for the farmland and non-farmland areas of the irrigation district is allocated in a certain proportion, approximately 90% for farmland irrigation water diversion and 10% for non-farmland irrigation water diversion [37]. The root zone water balance equation is as follows:

$$∆W_1^j=I^j+P^j-E{T}_a^j-D{P}_I^j-D{P}_P^j+C{R}^j+C{R}_F^j-D{P}_F^j$$

(1)

where the superscript $j=A, NA$ denotes farmland and non-farmland, respectively; $\Delta W_1$ is the change in soil water content in the root zone (mm); $I$ is the irrigation water volume per unit area (mm); $P$ is the precipitation (mm); $E{T}_a$ is the actual evapotranspiration (mm); $D{P}_I$ and $D{P}_P$ are the infiltration amounts of irrigation water and precipitation from the root zone to the transition layer, respectively (mm); $CR$ is the capillary rise water volume from the transition layer to the root zone (mm); $C{R}_F$ is the capillary rise water volume from the transition layer to the root zone during the freezing period (mm); and ${DP}_F$ is the meltwater infiltration volume from the root zone to the transition layer during the thawing period (mm).

(2) Transition Layer Water Balance Equation

Based on the current situation of shallow groundwater depth in HID and long-term monitoring results indicating relatively stable water content in deep soil layers, it is assumed that the transition layer neither stores nor releases water, and water exchange occurs only between the root zone and groundwater. This means the water flux transferred from the transition layer to the root zone equals the water flux transferred from the aquifer to the transition layer. The water balance equation for the transition layer is as follows:

$$\begin{array}{l}∆W_2^j=D{P}_I^j-DP{S}_I^j+D{P}_P^j-DP{S}_P^j-DP{S}_F^j+D{P}_F^j+GC{R}^j-C{R}^j+GC{R}_F^j\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-C{R}_F^j\end{array}$$

(2)

where $\Delta W_2$ represents the change in soil water content in the transition layer (mm); $D{PS}_I$ and $D{PS}_P$ are the infiltration amounts of irrigation water and precipitation from the transition layer to the aquifer, respectively (mm); $GCR$ is the capillary rise water volume from the aquifer to the transition layer (mm); $GC{R}_F$ is the capillary rise water volume from the aquifer to the transition layer during the freezing period (mm); and ${DPS}_F$ is the meltwater infiltration volume from the transition layer to the aquifer during the thawing period (mm).

Assuming the soil water content in the transition layer remains constant across all periods, $\Delta W_2=0$, and the other water fluxes are correspondingly equal, namely: $D{P}_I^j=D{PS}_I^j$, $D{P}_P^j=D{PS}_P^j$, $D{PS}_F^j=D{P}_F^j$, $GC{R}^j=C{R}^j$, $GC{R}_F^j=C{R}_F^j$.

(3) Aquifer Water Balance Equation

Water balance equations for farmland and non-farmland aquifers are as follows:

$$\begin{array}{l}∆W_3^j=-\mu ∆h^j=DP{S}_I^j+DP{S}_P^j+DP{S}_F^j-GC{R}^j-GC{R}_F^j+C{S}^j-D_g^j-GE\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+G_t^j+RL\end{array}$$

(3)

where $\Delta W_3$ is the change in water storage in the aquifer (mm); $\mu$ is the specific yield of the aquifer; $\Delta h$ is the change in groundwater depth (mm); $CS$ is the canal seepage recharge per unit area (mm); $D_g$ is the groundwater drainage per unit area (mm); $GE$ is the groundwater extraction per unit area (mm); $G_t$ is the groundwater exchange flow between farmland and non-farmland per unit area (mm); $RL$ is the lateral groundwater recharge per unit area (mm).

(4) Groundwater Balance Equation

Based on the interactive relationship between farmland and non-farmland groundwater, their changes are always equal: $\Delta h=\Delta h^A=\Delta h^{NA}$. An overall groundwater balance model with a monthly time step is established, and the variables for farmland and non-farmland should satisfy the following:

$$\Delta h=\alpha \Delta h^A+\left(1-\alpha \right)\Delta h^{NA}$$

(4)

$$\alpha ={\displaystyle \frac{A^A}{A^A+A^{NA}}}$$

(5)

$$\alpha G_t^A+\left(1-\alpha \right)G_t^{NA}=0$$

(6)

where $A^A$ and $A^{NA}$ represent the area of farmland and non-farmland, respectively (m²); $\alpha$ is the proportion of farmland area.

(5) Groundwater Depth

Finally, the groundwater depth at the next time step is given by the following equation:

$$h\left(t+1\right)=h\left(t\right)+\Delta h/1000$$

(7)

where $h$ is the groundwater depth (m).

2.

Calculation of Water Fluxes

(1) Irrigation Water Allocation

For the total water diversion volume M, a portion of the irrigation water is directly discharged into drainage ditches as canal abandonment, i.e., surface drainage, and does not participate in the water–salt cycle of the farmland-non-farmland system. This portion is referred to as surface drainage. The surface drainage volume for different months is estimated using the following formula:

$$I^m={\displaystyle \frac{M^m}{A}}\times 1000$$

(8)

$$D{s}^m=\mathrm{m}\mathrm{a}\mathrm{x}\left\{d{g}_a\times \left[I^m+P^m-d{g}_b\times \left(\overline{I^m}+\overline{P^m}\right)\right], 0\right\}$$

(9)

where the superscript m denotes the month; M is the water diversion volume at the canal head (m³); A represents the total area of farmland and non-farmland (m²); $Ds$ is the surface drainage (mm); $d{g}_a$ and $d{g}_b$ are parameters related to surface drainage; $\overline{I^m}$ and $\overline{P^m}$ are the average irrigation water volume per unit area and average precipitation per unit area, respectively (mm).

Apart from surface drainage, the remaining irrigation diversion volume is allocated to farmland and non-farmland in a 9:1 ratio. A portion of this water directly infiltrates into the groundwater through canal seepage, while the remainder is applied to farmland or non-farmland as irrigation.

$$C{S}^j={\displaystyle \frac{{\eta }_{CS}{M}^j}{A^j}}\times 1000$$

(10)

$$I^j={\displaystyle \frac{{\eta }_{CN}{M}^j}{A^j}}\times 1000$$

(11)

where ${\eta }_{CS}$ is the canal seepage recharge coefficient; ${\eta }_{CN}$ is the canal water use efficiency coefficient; $M^j$ is the actual diverted water volume for farmland or non-farmland after deducting surface drainage (m³).

(2) Field Irrigation Infiltration and Precipitation Infiltration

The infiltration volumes from field irrigation and precipitation are calculated using the following formulas:

$$D{P}_I={\eta }_{I}I$$

(12)

$$D{P}_P={\eta }_{P}P$$

(13)

where ${\eta }_I$ is the irrigation water infiltration recharge coefficient; ${\eta }_P$ is the precipitation infiltration recharge coefficient.

(3) Capillary Rise

Considering the groundwater depth and potential evapotranspiration in HID, the capillary rise during the non-freezing period is calculated with reference to the formula in SALTMOD [38]:

$$CR=F_c{M}_d$$

(14)

$$M_d=\max \left(0, E{T}_P-I-P+D{P}_I+D{P}_P\right)$$

(15)

$$F_c=\left\{\begin{array}{cc}1,& h<0.5h_r\\ 1-(h-0.5h_r)/(h_c-0.5h_r), & 0.5h_r\le h\le h_c\\ 0,& h>h_c\end{array}\right.$$

(16)

where $F_c$ is the capillary rise coefficient; $M_d$ is the soil water deficit in the root zone (mm); $E{T}_P$ is the potential evapotranspiration (mm); $h_r$ is the thickness of the root zone (m); $h_c$ is the lower limit of groundwater depth for capillary rise (m); The capillary rise volume should be greater than 0 and less than the actual evapotranspiration minus the retention of irrigation water and precipitation in the root zone.

$$0\le M_d{T}_a-\left(I-D{P}_I\right)-(P-D{P}_P)$$

(17)

During the freezing period, the groundwater level declines, and a portion of the aquifer’s groundwater transitions to a frozen state in the surface layer. This water volume is related to the groundwater depth h and the negative accumulated temperature ST (°C) [39]:

$$C{R}_F={\epsilon }_F{e}^{-h}\sqrt{ST}$$

(18)

$$W_F=\sum C{R}_F$$

(19)

where ${\epsilon }_F$ is the conversion coefficient for the freezing period; $W_F$ is the total freezing volume for the year (mm).

(4) Meltwater Infiltration Volume

The majority of the groundwater frozen in the root zone during the freezing period will infiltrate back into the aquifer as temperatures rise. This volume of water is allocated proportionally across the thawing period:

$$D{P}_F={\lambda }_F{W}_F\tau \left(m\right)$$

(20)

where ${\lambda }_F$ is the return flow coefficient of frozen water during the thawing period; $\tau \left(m\right)$ is the allocation proportion for different months within the thawing period; $m=3,4$, representing the months.

(5) Groundwater Drainage Volume

With reference to Darcy’s law, the drainage volume of groundwater through the ditch system is determined by the groundwater level and the depth of the drainage ditches. It is calculated as follows [37]:

$$D_g=\left\{\begin{array}{cc}{\gamma }_d\left(h_d-h\right)\times 1000,& h>h_d\\ 0,& h\le h_d\end{array}\right.$$

(21)

where $h_d$ is the depth of the drainage ditch (m); ${\gamma }_d$ is the drainage coefficient (dimensionless).

(6) Other

Based on the average monthly domestic and industrial water usage in HID, the resulting monthly groundwater extraction is considered as $GE=1$ mm. The entire area of the HID is relatively flat, so the lateral groundwater recharge can be assumed as $RL=0$.

The groundwater exchange between farmland and non-farmland is difficult to calculate directly. However, the change in groundwater depth can be directly calculated using the overall groundwater balance equation. Since the groundwater depths in farmland and non-farmland become equal after groundwater transfer, the groundwater exchange can be derived inversely based on the aquifer balance equation for either farmland or non-farmland, as shown below:

$$\begin{array}{l}{G}_t^j=-\mu ∆h^j-(DP{S}_I^j+DP{S}_P^j+DP{S}_F^j-GC{R}^j-GC{R}_F^j+C{S}^j-D_g^j-GE\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+RL)\end{array}$$

(22)

Due to the higher irrigation volume per unit area in farmland, in most cases, the farmland aquifer recharges the non-farmland aquifer, i.e., $G_t^A<0$, $G_t^{NA}>0$.

2.2.2. Farmland and Non-Farmland Salt Balance Model

As shown in Figure 4, this study establishes a root zone–transition layer–aquifer salt balance model encompassing both farmland and non-farmland areas, defining the salt state variables and fluxes for each balance zone. For the root zone, transition layer, and aquifer of both farmland and non-farmland, salt balance equations are formulated separately to derive the salt fluxes between the different balance bodies, thereby quantifying the salt transport processes across soil layers.

• Salt Balance Equation

(1) Root Zone Salt Balance Equation

In the HID, the salt input into the farmland and non-farmland root zones primarily comes from irrigation water, precipitation, and salt transport via upward groundwater movement (capillary rise). The main outputs include leaching and plant salt uptake. On this basis, without considering lateral salt transport in the root zone, the root zone salt balance equation is established as follows:

$$∆S_1^j=S_I^j+S_P^j-S_C^j-S_{DP,I}^j-S_{DP,P}^j-S_{DP,F}^j+S_{CR}^j+S_{CR,F}^j$$

(23)

where $\Delta S_1$ is the change in salt mass in the root zone (g/m²); $S$ represents the salt flux per unit area, with subscripts indicating the corresponding water flow process (including salt input into the root zone and leached salt), consistent with the notations above (g/m²); $S_C$ is the salt uptake by crops/plants per unit area, with values of 4.5 g/m² for crops and 9.0 g/m² for other plants.

(2) Salt Balance Equation for the Transition Layer

The salt input into the transition zone includes salts leached from the root zone and salts carried by phreatic evaporation. The main output pathways are leaching due to irrigation and precipitation, and upward salt migration to the root zone via capillary water. The balance equation is expressed as follows:

$$\begin{array}{l}∆S_2^j=S_{DP,I}^j+S_{DP,P}^j+S_{DP,F}^j-S_{CR}^j-S_{CR,F}^j-S_{DPS,I}^j-S_{DPS,P}^j-S_{DPS,F}^j+S_{GCR}^j\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} +S_{GCR,F}^j\end{array}$$

(24)

where $\Delta S_2$ is the change in salt mass in the transition layer (g/m²).

(3) Aquifer Salt Balance Equation

Farmland aquifer salt balance equation:

$$∆S_3^j=S_{DPS,I}^j+S_{DPS,P}^j+S_{DPS,F}^j-S_{GCR}^j-S_{GCR,F}^j+S_{CS}^j-S_{Dg}^j-S_{GE}^j-S_{Gt}^j+S_{RL}^j$$

(25)

Non-farmland aquifer salt balance equation:

$$∆S_3^j=S_{DPS,I}^j+S_{DPS,P}^j+S_{DPS,F}^j-S_{GCR}^j-S_{GCR,F}^j+S_{CS}^j-S_{Dg}^j-S_{GE}^j+S_{Gt}^j+S_{RL}^j$$

(26)

where $\Delta S_3$ is the change in salt mass in the aquifer (g/m²).

(4) Soil Salt Content Conversion Formula

The conversion relationship between soil salt content per unit area S (g/m²) and soil salt content per unit mass SC (g/100 g) is as follows:

$$S_i^j=S{C}_i^j{\rho }_i^j{h}_i\times \mathrm{10,000}$$

(27)

where subscript $i=R,T,G$ denotes the root zone, transition layer, and aquifer, respectively; $S$ is the soil salt content per unit area (g/m²); $SC$ is the soil salt content per unit mass (g/100 g); $\rho$ is the soil bulk density (g/m³); $h$ is the thickness of the balance unit (m).

The conversion relationship between the salinity of the soil solution and the soil salt content per unit mass SC is as follows:

$$C_i^j=S{C}_i^j{\rho }_i^j/{\theta }_i^j\times 10$$

(28)

where $C_i$ is the mineralization degree of the soil solution (g/L); $\theta$ is the conversion coefficient.

2.

Calculation of Salt Concentration for Each Water Flux

Except for salt uptake by crops/plants, all other salt transfer processes occur simultaneously with water transfer processes. The salt flux can be expressed as:

$$S_q^j=Q_q^j{C}_q^j$$

(29)

where the subscript q denotes the water flux process; Q is the corresponding water flux (mm); C is the mineralization degree of the corresponding water flux (g/L).

(1) Calculation of Leachate Mineralization Degree in the Root Zone and Transition Layer

The mineralization degree of leachate passing through the root zone and transition layer is related to its initial concentration and the mineralization degree of the soil solution in that layer. It can be expressed as a weighted average of the two, as follows:

$$C_{q1}^j=f_i^j{C}_i^j+(1-f_i^j)C_{q0}^j$$

(30)

where subscripts q₀ and q₁ represent the original flux and the corresponding leachate, respectively. For example: irrigation water $I$ and $D{P}_I$, $D{P}_I$ and $D{PS}_I$; precipitation $P$ and $D{P}_P$, $D{P}_P$ and $D{PS}_P$; meltwater $D{P}_F$ and $D{PS}_F$; $f$ is the leaching coefficient; $C_i$ denotes the mineralization degree of the corresponding soil solution (g/L).

(2) Calculation of Capillary Rise Water Mineralization Degree

Due to the small thickness of the transition layer (0.4 m) and uniform salt distribution, the mineralization degree of its soil solution can be considered as the mineralization degree of capillary rise water entering the root zone [37], i.e.,:

$$C_{CR}^j=C_{CR,F}^j{=C}_T^j$$

(31)

where the subscript T denotes the transition layer.

Given the considerable thickness of the aquifer and the uneven distribution of soil salinity, using the average mineralization degree would introduce significant errors. Therefore, it is necessary to introduce a calibration coefficient as follows:

$$C_{GCR}^j=C_{GCR,F}^j={\xi }^j{C}_G^j$$

(32)

where the subscript G denotes the aquifer; ξ is the calibration coefficient.

2.3. Model Calculation Process and Validation Metrics

The water–salt balance model uses a monthly time step. The calculation flowchart of the farmland-non-farmland water–salt balance model is shown in Figure 5. This study uses the coefficient of determination (R²) to measure the linear correlation between observed and simulated values, and the root mean square error (RMSE) to evaluate model accuracy. The calculation formulas are as follows:

$$R^2=1-{\displaystyle \frac{\sum _{i=1}^n{\left(y_i-y_i^{\prime }\right)}^2}{\sum _{i=1}^n{\left(y_i-\overline{y_i}\right)}^2}}$$

(33)

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}}\sum _{i=1}^n{\left(y_i-y_i^{\prime }\right)}^2$$

(34)

where n represents the number of simulated samples; y_i represents the observed value; y_i’ represents the simulated value; $\overline{y_i}$ is the average value of all samples in the dataset. The closer R² is to 1, the stronger the linear relationship between the observed and simulated values.

2.4. Random Sampling and Kernel Density Estimation

The kernel density estimation (KDE) method is a non-parametric density function estimation method, which is often used in the statistical research of probability distributions [40,41]. Its mathematical formula is as follows:

$$\widehat{f}\left(x\right)={\displaystyle \frac{1}{nh}}\sum _{i=1}^{n}K\left({\displaystyle \frac{x-x_i}{h}}\right)$$

(35)

where $\widehat{f}\left(x\right)$ denotes the probability density function estimated at x, n is the sample size, h represents the bandwidth, and K(⋅) stands for the kernel function.

• In the parameter calibration process of the groundwater balance model, the optimization method based on random sampling and KDE can not only clarify the probability distribution of parameters but also effectively improve the accuracy and reliability of parameter estimation. The specific calibration steps of this method are as follows: Determination of Parameter Value Ranges. First, determine the approximate value range of each key parameter based on existing research and practical experience. For example, the value ranges of parameters such as the irrigation infiltration recharge coefficient, precipitation infiltration recharge coefficient, and canal system water use efficiency coefficient are initially defined through literature and field surveys.
• Generation of Random Parameter Sets. Within the value range of each parameter, use random sampling methods to generate a series of random values, forming a large number of different parameter combinations. These parameter sets cover the possible parameter value ranges, providing diverse inputs for subsequent model simulations. Random sampling methods use uniform distribution and Latin Hypercube Sampling to ensure sufficient coverage of the parameter space.
• Simulation and Model Performance Evaluation. Simulate the groundwater balance model for each parameter set and calculate the coefficient of determination (R²) between the model output and the measured data. The R² value reflects the goodness of fit between the model simulation results and the observed data. A higher R² value indicates that the model’s description of the simulation is more accurate.
• KDE Analysis. Selecting the parameter set with the highest R² is not necessarily the optimal strategy. Although such a set may demonstrate excellent goodness-of-fit, its scarcity might only represent a specific or “overfitted” scenario, lacking general significance. This would lead to unstable KDE analysis results based on these few parameter sets, and the recommended parameter ranges could be excessively narrow, resulting in poor fault tolerance in practical applications. Therefore, we adopted a balanced strategy that considers both goodness-of-fit and the number of parameter sets. We selected three progressively stricter thresholds—R² > 0.6, R² > 0.70, and R² > 0.75—as screening criteria [42]. Through the KDE curve, the optimal value interval for each parameter can be visually identified, i.e., the peak region of the density curve. Based on the KDE analysis results, determine the optimal value range for each parameter. These ranges are based not only on the model’s goodness of fit but also consider the physical rationality of the parameters. The final determined parameter value ranges can serve as recommended parameters for the model and be used for subsequent water–salt balance analysis and prediction.

Through the above steps, combining random sampling, model simulation, and KDE analysis, the parameter calibration process of the groundwater balance model can be systematically optimized. This method not only improves the accuracy of parameter estimation but also provides a statistical basis for the optimal value intervals of parameters, offering scientific support for the simulation and management of groundwater systems.

3. Results

3.1. Data Sources

Given that the Yichang Irrigation District within the HID has a more complete irrigation and drainage system, a typical cropping structure, and relatively comprehensive data such as water diversion and drainage, soil water and salt, the Yichang Irrigation District was selected as the study area to establish the farmland-non-farmland water–salt balance model and verify the model accuracy. This study area is located in the middle and lower reaches of the HID, with a farmland area of 2578.6 km², a non-farmland area of 768.8 km², and a farmland area proportion of 77.0%. The main crops grown are corn, wheat, and sunflowers, and the irrigation method is mainly Yellow River diversion irrigation, with an average annual Yellow River diversion volume of 1.445 billion m³. The Yichang Irrigation District belongs to a typical temperate continental climate, with a multi-year average precipitation of 179.42 mm and an annual average temperature of about 6 °C. Furthermore, the potential evapotranspiration data involved in this study were obtained from the “China 1 km monthly potential evapotranspiration dataset (1901–2023)” released by Peng Shouzhang’s team [43] (National Tibetan Plateau Data Center), with a spatial resolution of 1 km and a time span from January 1901 to December 2023. The groundwater depth in the entire region is relatively shallow, mainly phreatic water, about 1.86 m.

Based on the measured data from the Yichang Irrigation District, this study selected the period from May 2003 to May 2009 as the calibration period for the groundwater balance model, and May 2009 to May 2016 as the validation period. According to the irrigation and soil freezing in HID, it is divided into four periods: the growth period (May to August); the autumn irrigation period (September to November); the freezing period (December to February of the following year); and the thawing period (March to April).

3.2. Water Balance Model Calibration and Validation Results

In this study, the parameter calibration results of the groundwater balance model show that the model exhibits good simulation performance in both the calibration and validation periods. The calibration and validation results of the groundwater balance model are shown in

Table 1. Evaluation metrics for calibration and validation of the water balance model.

Model Phase	Groundwater Depth RMSE/m	Groundwater Depth R2
Calibration	0.19	0.79
Validation	0.24	0.65

. Specifically, the root mean square error (RMSE) for groundwater depth during the calibration and validation periods was 0.19 m and 0.24 m, respectively, and the coefficients of determination (R²) were 0.79 and 0.65, respectively. These metrics reflect the model’s ability to describe the variation process of groundwater depth in the study area. The lower RMSE values indicate smaller deviations between simulated and measured values, while the higher R² values further indicate that the model can reasonably reflect the dynamic changes in groundwater depth.

The main reasons for the differences between the validation period and the calibration period results can be attributed to changes in the canal system efficiency coefficient, irrigation infiltration coefficient, drainage coefficient, etc., over the long term, as well as the impact of adjustments in cropping structure. Improvements in canal water use efficiency and changes in irrigation infiltration directly affect groundwater recharge and field water volume. Fluctuations in the drainage coefficient affect groundwater level dynamics and soil conditions. Adjustments in cropping structure further alter regional evapotranspiration and water resource demand. The interaction and long-term cumulative effects of these factors lead to changes in regional hydrological processes and water resource allocation patterns, resulting in deviations between the validation period and calibration period results.

Based on the frequency of parameter occurrences when the coefficient of determination (R²) was greater than 0.6, 0.7, and 0.75, the probability distributions of the irrigation and precipitation infiltration recharge coefficients for farmland and non-farmland were extracted, as shown in Figure 6, along with the probability distributions of other calibrated parameters, as shown in Figure 7. In the study area, irrigation practices are common in land types other than farmland, such as woodland and grassland. Some non-cultivated areas also receive water supplementation through canal seepage or lateral seepage from farmland irrigation. To more accurately characterize this phenomenon, this study divided the diverted water volume into two parts based on use: farmland irrigation water and non-farmland irrigation water. For setting the irrigation infiltration recharge coefficient, considering the similarity of hydrogeological conditions and irrigation management practices in the study area, the irrigation infiltration recharge coefficients and precipitation infiltration recharge coefficients for farmland and non-farmland during the crop growth period and autumn irrigation period were assigned the same value. For example, the calibrated value for the irrigation infiltration recharge coefficient is 0.34. While previous studies reported a value range of 0.18–0.42 [44], this study narrowed the range to 0.32–0.36 based on maintaining a high R² value. The calibrated value for the precipitation infiltration recharge coefficient is 0.25.

Through analysis and comparison with existing research results (

Table 2. Parameters of the groundwater balance model for the Yichang Irrigation District.

Parameter Name	Value	Reference Range
Specific Yield of the Aquifer	0.10	0.09–0.11
Critical Depth for Capillary Rise	2.70	2.58–2.75
Drainage Ditch Depth	2.1	2.05–2.20
Drainage Coefficient	0.023	0.021–0.027
Canal Seepage Groundwater Recharge Coefficient	0.33	0.30–0.37
Canal System Water Efficiency Coefficient	0.67	0.63–0.70
Freezing Period Conversion Coefficient	10.0	9.8–10.4
Recharge Coefficient of Frozen Water	0.90	-
March Groundwater Recharge Proportion	0.40	-
Surface Drainage Parameter a	0.13	0.10–0.16
Surface Drainage Parameter b	0.88	0.86–0.90
Irrigation Water Infiltration Recharge Coefficient-Farmland	0.34	0.32–0.36
Irrigation Water Infiltration Recharge-Non-Farmland	0.34	0.32–0.36
Precipitation Infiltration Recharge Coefficient-Farmland	0.25	0.23–0.27
Precipitation Infiltration Recharge Coefficient-Non-Farmland	0.25	0.23–0.27

), the parameters calibrated in this study are generally reasonable. The comparative results are as follows: In existing studies, the specific yield of the aquifer in HID ranges from 0.02 to 0.10 [8]. The value calibrated in this study is 0.10, with a further narrowed reference range of 0.09–0.11. Within this study area, Mao [45] using the SaltMod model and Sun [35] using a multi-scale water–salt balance model both calibrated the critical capillary rise depth to 3.0 m. The value calibrated in this study is 2.7 m, with a reference range of 2.58–2.75, showing little difference from their results. The drainage ditch depth ranges from 2.05 to 2.20 m, consistent with the actual situation in HID. The drainage coefficient calibrated in this study is 0.023, which is relatively close to the calibrated value of 0.02 in the Yongji irrigation area [35] of HID. This minor difference is likely due to variations in drainage ditch conditions and lining levels across different irrigation areas. Since the canal seepage recharge coefficient and the canal water use efficiency coefficient change inversely, only the canal water use efficiency coefficient result of 0.67 is presented in the text, and its reference range is close to previous research results [46]. Among the calibrated parameters, the freezing period conversion coefficient, frozen water recharge coefficient, and March groundwater recharge proportion are 10.0, 0.9, and 0.4, respectively. The surface drainage parameter ranges are 0.1–0.16 and 0.86–0.90, slightly lower than previous research results. In summary, the results for all calibrated parameters are satisfactory, generally aligning with previous research findings within acceptable fluctuation ranges, demonstrating the rationality of the model.

To quantify the influence of various input parameters on the model output, this study employed the Morris elementary effects method for global SA (Figure 8). The Morris method has been proven to effectively identify the most influential parameters. It requires less computational time and demonstrates clear advantages over local sensitivity analysis [47], particularly since the study revealed that interaction effects play a significant role in hydrological models, indicating parameter correlations and their interdependent importance [48,49].

This method evaluates parameter sensitivity by calculating three key metrics: μ represents the mean of the elementary effects and measures the average strength and direction of the parameter’s influence on the model output (a positive value indicates a positive influence, while a negative value indicates a negative influence); μ* denotes the absolute mean of the elementary effects, which eliminates the possibility of effect cancellation and is used to gauge the pure magnitude of the parameter’s impact on the output, serving as the primary indicator for identifying important parameters; σ signifies the standard deviation of the elementary effects, reflecting the variability in the parameter’s effects—a higher σ value suggests that the parameter’s influence is more dependent on the values of other parameters in the model, indicating significant interactive effects or nonlinear behavior. The parameters Canal System Water Efficiency Coefficient, Drainage Ditch Depth, Drainage Coefficient, and Critical Depth for Capillary Rise all exhibit positive μ values. Among these, Drainage Ditch Depth has the highest μ value, indicating it exerts the strongest positive driving effect on the model output. The parameters Irrigation Water Infiltration Recharge Coefficient-Farmland, Precipitation Infiltration Recharge Coefficient-Farmland, and Recharge Coefficient of Frozen Water all show negative μ values. The Recharge Coefficient of Frozen Water has the largest absolute μ value, demonstrating the strongest negative inhibitory effect on the model output. Furthermore, Drainage Ditch Depth and Recharge Coefficient of Frozen Water are identified as the most influential key parameters affecting the model output, based on their significant μ* values. The σ values for all parameters are relatively similar and generally low. This indicates that the interactions between parameters in the established model are weak, and the model overall demonstrates good parameter independence and additivity.

Figure 9 shows the simulated groundwater depth curves and the observed mean values based on data from 41 monitoring wells during the calibration period (May 2003–May 2009) and the validation period (May 2009–May 2016). The results demonstrate a high degree of consistency between the simulated and observed values. Both the inter-annual fluctuations and seasonal variations show matching trends, verifying the reliable capability of the water balance model to represent the regional groundwater dynamics.

3.3. Calibration and Verification Results of the Salinity Balance Model

Based on the basic data of the Yichang Irrigation District, the main crops grown are corn, wheat, and sunflowers. Therefore, the crop root depth was set at 0.6 m. Historical field monitoring statistics indicate that the average groundwater depth ranges from 1.5 to 2.5 m, showing a fluctuating state. Consequently, the aquifer is considered to start from 1.0 m below the surface, with a transition layer of 0.4 m, below which lies the aquifer thickness. This study uses existing observational data, taking the farmland salinity model as an example, for calibration and validation. Specifically, for the farmland root zone salinity model, the Root Mean Square Error (RMSE) between simulated and measured values is 0.033, and the coefficient of determination (R²) is 0.26. For the non-farmland root zone salinity model, the RMSE is 0.037, and R² is 0.17. These metrics reflect the model’s capability in describing the soil salinity conditions in the study area. Due to the high spatial variability of soil salinity in the farmland root zone, it significantly affects the model calibration and validation results.

As shown in the simulation results in Figure 10, the soil salinity in the farmland root zone of the Yichang Irrigation District ranges from 0.11 to 0.18 g/100 g, exhibiting a fluctuating upward trend. This is consistent with the findings of Wang et al. [23] in HID, where the annual increase in soil salinity was reported as 0.25 g/100 g. Peak salinity values in farmland were observed in July and August (for sunflower) and in May (for corn). Soil salinity decreased significantly after the autumn irrigation period. Non-farmland areas showed a corresponding consistency, with root zone soil salinity accumulating during the growing season and decreasing during the non-growing season. The trends of measured and simulated values are generally aligned, indicating that the simulated results from this salt balance model can effectively reflect the overall inter-annual and intra-annual variations in soil salinity within the region.

4. Discussion

Based on the model calibration and validation results, this study has identified the following primary sources of uncertainty: Firstly, large-scale lumped models face challenges in fully capturing fine-scale variations in actual water–salt dynamics and adequately representing spatial heterogeneity [50]. However, the primary objective of the model in this study is not to predict soil salinity in individual field plots but to assess overall water–salt balance trends and responses to external management measures in the irrigation district [18]. Moreover, relying on mass conservation and various coefficients to describe fluxes makes the model more suitable for investigating the probability distribution of parameters. Secondly, the use of monthly averaged groundwater depth data may lead to lag effects in water level simulations due to the averaging process [51]. Moreover, errors in the water simulation results propagate into the salt transport model. Coupled with the inherent complexity of salt transport and transformation processes, as well as uneven spatiotemporal distribution and scarcity of measured data, these factors collectively increase model uncertainty.

In addition, particular attention should also be paid to the limitations in simulating the freeze–thaw period. Current models employ varied approaches to simulate freeze–thaw cycles. Some models use semi-empirical functions to modify hydraulic conductivity and soil water potential under freezing conditions [52,53], while others couple energy balance equations to explicitly simulate soil freezing and thawing [54]. In this study, due to limited soil thermal property data and the complexity of coupled water-heat-salt transport, the freeze–thaw process is represented in a simplified manner. In the future, as data availability improves, a more physically based representation of these processes will contribute to enhanced simulations.

Regarding parameterization, most parameters in this study were treated as time-invariant during calibration [26]. However, it is recognized that certain factors, such as canal seepage coefficients or irrigation efficiency, may change over time due to variations in management practices or maintenance conditions. Although time-varying parameterization schemes exist in the literature, such as dividing the simulation period into multiple segments based on seasons or management changes, or introducing transfer functions to dynamically update parameters [24], these approaches primarily describe parameter dynamics and do not statistically constrain the parameter space. In contrast, KDE employs smoothing kernel functions for nonparametric estimation of data distributions, effectively capturing local data features. This method not only narrows prediction intervals while preserving high confidence levels but also mitigates errors caused by inappropriate distributional assumptions in parametric prediction models, thereby demonstrating stronger applicability. These advantages have been corroborated in other studies [55,56].

Based on the results of the previous sensitivity analysis, the parameter calibration method employed is relatively reasonable [57]. However, the critical groundwater depth for capillary water rise in this study is slightly higher than previously reported values. This discrepancy may be related to differences in the criteria used to determine the critical depth: when the soil salinity concentration reaching the crop salt tolerance threshold is taken as the basis, a higher critical depth value is often obtained [58]. This study also found that drainage ditch depth exhibits a significant influence on the sensitivity analysis. When the drainage ditch is shallow, the rate of increase in the electrical conductivity of the saturation extract (ECe) in the crop root zone soil accelerates [26]. In contrast, as the ditch depth increases, the growth rate of the ECe value in the root zone soil slows down significantly. Previous research has indicated that when the drainage ditch depth reaches 3 m, the salinity in the root zone soil tends to stabilize [59]. Considering the effectiveness of soil salinity control, this study suggests that the optimal range for drainage ditch depth is between 2.05 and 2.20 m. Excessive deepening of drainage ditches not only increases costs but may also weaken the “dry drainage and salt discharge” effect, thereby hindering salinity regulation [23].

5. Conclusions

Aiming at the parameter uncertainty in the water–salt balance modeling of arid irrigation areas, a parameter calibration method combining random sampling and KDE is proposed and applied to the HID:

• This study successfully developed a farmland-non-farmland water–salt balance model applicable to HID. By conceptually dividing the system into farmland/non-farmland areas and root zone/transition layer/aquifer layers, the model effectively represents regional water–salt transport processes, with particular consideration given to the impact of freeze–thaw cycles. Validation results demonstrate that the model satisfactorily simulates both the dynamic variations in groundwater depth and the overall trends in soil salinity.
• To address parameter uncertainty, this study proposes a parameter calibration strategy that integrates random sampling and KDE. The method begins by randomly generating a large number of parameter combinations. Parameter sets demonstrating higher goodness of fit (R² > 0.6, 0.7, or 0.75) are selected, and KDE is applied to analyze their probability distributions. This approach identifies optimal parameter ranges rather than single optimal values (e.g., irrigation infiltration recharge coefficient: 0.32–0.36; precipitation infiltration recharge coefficient: 0.23–0.27), thereby enhancing the statistical reliability of the parameters and mitigating overfitting. By evaluating parameter sets across multiple fitting thresholds, the strategy avoids the overfitting risks associated with relying solely on the highest R² values, improves model generalizability, and offers a practical and robust methodology for parameter estimation. Furthermore, global sensitivity analysis using the Morris method reveals that drainage ditch depth and recharge coefficient of frozen water are the most influential parameters, exhibiting the strongest positive and negative effects on groundwater depth variation, respectively. Other highly sensitive parameters include the canal system water efficiency coefficient, drainage coefficient, and critical capillary rise depth. The generally low σ values indicate weak interactions among parameters, suggesting that the model structure is relatively independent and that parameter effects are largely additive. The sensitivity analysis results not only validate the rationality of the parameter calibration strategy but also provide important insights for irrigation district management.

Overall, this study effectively quantified and reduced the uncertainty range of parameters by constructing a structurally simplified water–salt balance model and combining the parameter calibration methods of random sampling and KDE, thereby enhancing the applicability and reliability of the model under conditions of scarce data. Sensitivity analysis further clarified the key parameters that affect water and salt dynamics, providing a scientific basis for water and salt management and salinization prevention and control in irrigation districts. Although there is still room for improvement in the simulation accuracy of salt content in the model and the simplification of complex hydrological phenomena such as freeze–thaw processes will bring certain uncertainties, the method system proposed in this paper has provided a valuable reference framework for water–salt modeling and parameter optimization in arid irrigation areas.