(1) Overview of the experimental area.

The experiment was carried out in Wanghong Town, Yongning County, Ningxia, spanning 2019–2020. The experimental area is located at 106°11′E longitude and 38°10′N latitude, with an altitude of 1,125 meters. It features a typical continental semi-arid climate, with an average annual rainfall of 198.19 mm and an average annual evaporation of 1,186.73 mm [20]. The soil in the area is sandy loam, and the cultivated layer has an average bulk density of 1.45 g/cm ³. The available nitrogen, phosphorus, and potassium contents, as well as the organic matter content in the cultivated layer of the experimental site, are 85.77 mg/kg, 12.31 mg/kg, 125.84 mg/kg, and 2.3%, respectively. The meteorological parameters of the experimental area are shown in S1 Fig.

(2) Experimental design.

The tested maize variety was Xianyu335, and this experiment was a single-factor experiment focusing on irrigation amount. Based on the irrigation practices of local farmers, a total of nine gradient irrigation amount treatments were set up, with irrigation quotas of 1800 m³/ha (T1), 2160 m³/ha (T2), 2520 m³/ha (T3), 2880 m³/ha (T4), 3240 m³/ha (T5), 3600 m³/ha (T6), 3960 m³/ha (T7), 4320 m³/ha (T8), and 4680 m³/ha (T9). As shown in S1 Table, each treatment had an irrigation frequency of 8 times. The experimental plot was 10 meters long and 5 meters wide, covering an area of 50 square meters. There were three replicates for each plot, resulting in a total of 27 plots. The sowing distance between maize sown plants was 25 cm, and the row spacing was 60 cm. Drip irrigation was employed as the irrigation method. The drip irrigation tape within each plot was arranged in a single straight line, with a dripper spacing of 0.3 m and a dripper flow rate of 2.0 L/h. A water meter was installed at the head of each plot to strictly control the irrigation amount. The total amount of fertilizer applied was 320 kg/ha of nitrogen (N), 120 kg/ha of phosphorus (P₂O₅), and 252 kg/ha of potassium (K₂O). This included a basal application of 30% nitrogen fertilizer and 70% phosphorus-potassium fertilizer. The remaining nutrients were applied in water during the large horn stage, tasseling stage, and filling stage, respectively. Other field operations remained consistent.

(3) Measurement indicators and methods.

1. ① Soil moisture content: Before and after irrigation, the drying and weighing method was used to determine the soil moisture content. The 0–100 cm soil layer was divided into 10 layers at 10 cm intervals, and soil samples were taken using a soil drill. After weighing the fresh soil samples, they were placed in a 105℃oven to dry to a constant weight, and then were weighed again to calculate the soil moisture content. Additional testing has been done after rainfall, which is to determine the soil moisture content once according to the aforementioned method.
2. ② Plant height and biomass: Plant height and biomass were to be measured once at each growth stage of maize, and a total of 5 measurements were taken throughout the entire growth period. Plant height was measured using a tape measure, three maize plants were randomly selected from each plot, and the arithmetic mean of the plant heights from the ground surface to the growth point was calculated. After measuring, the maize plants were dug out and brought back to the laboratory for washing. The oven was adjusted to 105℃ and the plants were withered for 30 minutes, after which they were dried at 80℃ until a constant weight was reached. Then they were weighed using an analytical balance.
3. ③ Leaf Area Index (LAI): It was measured using the LAI-2200 canopy analyzer, during each growth period of maize, measurements were taken under backlight from 16:00–18:00 on clear weather. Three measurements were taken for each plot, and the arithmetic mean was taken.
4. ④ Yield: During the physiological maturity stage, 3 sampling points were randomly selected from each plot, with a 10 row spacing for each sampling point. A representative 20 meter double row was selected from the 10 rows, and plants and ears were counted. The number of ears per hectare was calculated. An ear was collected every 5 ears in each measurement segment, and a total of 20 ears were harvested as samples to determine the number of grains per ear. After all the grains were removed, they were weighed, dried, and then weighed again to calculate the grain moisture content. The average value was taken for each replicate and it was converted into the yield (kg) per hectare.

(1) Model calibration method.

Soil Water Atmosphere Plant (SWAP) model is a mechanistic model developed by Wageningen University in the Netherlands. It was designed to simulate water movement, solute transport, heat transfer, and crop growth processes in soil plant atmosphere continuous systems (SPAC) [21]. According to the sensitivity analysis results of model parameters, field measurements and trial-and-error methods were used to calibrate parameters with strong sensitivity, while model default or reference methods were used to calibrate parameters with low sensitivity. Based on the observed data of LAI, biomass, soil moisture content, and yield from the 2019 irrigation experiment, the model parameters were adjusted and calibrated to achieve global optimization of simulation results for all treatments from T1 to T9. The specific verification method was described in the following text [22]. The established parameter set was validated using the experimental results from 2020.

1. ① accumulated temperature from emergence to flowering (TSUMEA) and Accumulated temperature from flowering to maturity (TSUMAM): they were calculated using temperature data observed at the meteorological station of the experimental base and the biological lower limit temperature of maize.
2. ② initial biomass (TDWI): Sampling was conducted in the field during the two leaf and one heart stage of maize, and the aboveground parts were cut with scissors, dried, and weighed before being converted into initial biomass per hectare based on seedling emergence rate.
3. ③ LAI at emergence (LAIEM): Samples were taken in the field during the two leaf and one heart stage of maize, and all leaves were cut off. The length and width were measured using a ruler, and the leaf area and LAIEM were calculated using the length width coefficient method (full leaf coefficient of 0.75, incomplete leaf coefficient of 0.5).
4. ④ Specific Leaf Area (SLA): During the maize seedling stage, large horn stage, and grain-filling stage, three fully unfolded leaves were taken from each maize plant from top to bottom, and the leaf area was determined using the length-width coefficient method. After the leaves were blanched in an 80 ℃ oven for 30 min, the oven temperature was adjusted to 105 ℃ and the leaves were dried to a constant weight. Then they were weighed with an electronic balance to calculate the SLA of maize in different stages.
5. ⑤ the methods for obtaining parameters of other major crops are shown in S1 Appendix.

(2) Parameter sensitivity analysis.

The Morris sensitivity analysis method was used to analyze the sensitivity of input parameters in the SWAP model. This method was a global sensitivity analysis method proposed by Morris in 1991 [23]. After continuous improvement and refinement, it had significant advantages in the sensitivity analysis of models with multiple parameters, complex structures, and large computational workloads. It could achieve a good balance between computational efficiency and accuracy. The calculation results of this method could simultaneously provide the sensitivity of individual parameters and the interaction between parameters with respect to the output results of the model. Assuming the model had n parameters, namely x₁, x₂, x₃  ...... x_n, the model calculation result could be expressed as y = f(x) = f(x₁, x₂, x₃  ...... x_n). First, the parameters of the model were normalized. The values of each parameter were projected onto the matrix of [0, 1], and the matrix of [0, 1] was discretized into p levels with Δ as the step size. In this way, the model parameters formed an n*p spatial array, which constituted the sampling space for Monte Carlo random values of the model parameters [24]. In a simulation process, it was assumed that the randomly selected parameter samples were , then the parameter sensitivity index was as follows:

(1)

In the formula, represented the parameter sensitivity index when the i-th parameter was disturbed. The larger its value was, the more sensitive the parameter was. J was the parameter set formed after the j-th resampling of the n × p sampling space. I was the parameter number corresponding to the disturbance Δ that occurred after the j-th resampling. m was the number of resampling times during the sensitivity analysis process. The crop parameters of the SWAP model varied greatly among different crop types and varieties, and might even change under different field management and production conditions. Soil was affected by field micro-topography, soil stratification, human activities, and other factors. The values of soil physical properties, hydraulic characteristics, and other parameters often exhibited variability at the field scale. Moreover, crop parameters and soil parameters were important parameters that affected the accuracy of model simulation. Therefore, this sensitivity analysis mainly analyzed two types of parameters: crop parameters and soil parameters. By referring to the literature, the range of values for the main parameters of maize was obtained, and an uncertainty of ±10% above and below the average value was set as the random sampling spatial range. The Morris global sensitivity analysis method was used for qualitative analysis. A total of 59 SWAP model input parameters were selected for sensitivity analysis, including 54 crop parameters and 5 soil parameters. The quantity level p was set to 4, and the number of parameter samples was r = 5900. The sensitivity of leaf area index (LAI), plant height (PH), and yield to these parameters was tested separately.

(3) SWAP-IES Assimilation Simulation System.

Iterative ensemble smoother (IES) algorithm was a continuous data assimilation method that simultaneously utilized all time-series data [25]. Its assimilation idea originated from the optimization method of posterior probability density function sampling. This method estimated the sensitivity information of parameter states using a sample set and iteratively updated parameters using the Gaussian Newton method. It was assumed that the relationship between the input and output of the model could be expressed as:

(2)

In the formula, d_obs was the measured value vector, F was the forward model, m was the unknown parameter vector, and ε was the observation error vector that conformed to a mean of 0 and a covariance of CD = E[εε^T]. Then IES could be implemented in the following steps. Step 1: N_e samples were generated from the prior distribution of parameters to form the initial sample set M₀.

(3)

The superscript in the formula represented the iteration order, and the subscript represented the sample number. Step 2: In the l-th iteration, where l = 1,2,⋯. Given historical observation data at all times, the parameter sample M_l could be updated according to the following equation:

(4)

In the formula, β_l was the parameter for adjusting the iteration step size, CM = △M₀(△M₀)^T/(N_e-1) represented the prior covariance of the parameter, which remained unchanged throughout the entire iteration process. ΔM₀ represented the deviation between matrix M₀ and its mean. G_l was a sensitivity matrix based on set-averaging, which played a role in connecting the input and output changes of the model. Step 3: Step 2 was repeated until the convergence standard or the iteration count set by IES was reached.

The flowchart of the constructed SWAP-IES assimilation yield estimation system as shown in S2 Fig. Using LAI and SW as observation variables of the SWAP-IES, a total of 6 observations were made throughout the entire growth stage. It was assumed that the soil was homogeneous in the horizontal direction, and all crop parameters, except for uncertain parameters, remained unchanged under different water stress treatments. Based on the sensitivity analysis results, five parameters, namely TSUMEA, CVO, SPAN, EFF, and CVL, were selected as the uncertainty parameters to be corrected and dynamically adjusted during the assimilation simulation. During the assimilation process, each parameter generated a prior value set with variances of 0.04, 0.08, 0.1, 0.24, and 0.08, respectively, and the ensemble size was set to 100. The observation variables used for data assimilation, such as LAI and SW, were measured data from field water stress experiments. Gaussian noise was used to perturb the state variables with variances of 0.1 and 0.01, respectively, to generate a sample set of observation values. The assimilation gain was calculated using the sample set, and the uncertainty parameters in the model were adjusted repeatedly until the preset convergence criteria were met.

2.2.1 Simulation of growth indicators.

The SWAP model was calibrated and validated using the results of maize irrigation experiments in 2019 and 2020, respectively. The simulation accuracy of maize LAI, plant height, and biomass during calibration and validation was shown in S4 Fig. Analysis indicated that the SWAP model demonstrated high accuracy in simulating various indicators during the calibration and validation processes. The simulated R² for various growth indicators ranged from 0.94 to 0.98, while the NRMSE ranged from 8.01% to 14.76%. The RMSE for LAI simulation was 0.44 m²/m² and 0.53 m²/m², for plant height simulation it was 12.43 cm and 15.81 cm, and for biomass simulation it was 888.02 kg/ha and 1437.76 kg/ha, respectively. The consistency and accuracy of the simulation were performed well, and the simulation results reached a very high level of accuracy.

The simulation accuracy of maize LAI, plant height, and biomass for each treatment during calibration and validation was shown in S3 Table. According to the analysis, the R² of LAI simulation for each treatment ranged from 0.92 to 0.97, the RMSE ranged from 0.35 m²/m² to 0.65 m²/m², and the NRMSE ranged from 9.51% to 14.45%. The R² of plant height simulation for each treatment ranged from 0.93 to 0.99, the RMSE ranged from 9.63 cm to 27.01 cm, and the NRMSE ranged from 5.72% to 18.15%. The R² of biomass simulation for each treatment ranged from 0.93 to 0.99, the RMSE ranged from 636.65 kg/ha to 1986.33 kg/ha, and the NRMSE ranged from 9.51% to 14.45%. It could be seen that the accuracy of simulating maize LAI, plant height, and biomass for each treatment was high to extremely high during model calibration and validation. The established SWAP model parameter set was reliable to some degree for simulating various growth indicators of maize.

2.2.2 Simulation of yield indicators.

The simulation results of maize yield during the calibration and validation of the SWAP model were shown in S5 Fig. During the calibration, the R² for yield simulation was 0.69, the RMSE was 1272.02 kg/ha, and the NRMSE was 15.59%. The absolute error of yield simulation for each treatment ranged from 443.80 kg/ha to 2160.48 kg/ha, and the relative error ranged from 8.71% to 22.23%. During the validation, the R² was 0.53, the RMSE was 18078.9 kg/ha, and the NRMSE was 18.37%. The absolute errors ranged from 2374.1 kg/ha to 27021.8 kg/ha, and the relative errors ranged from 4.38% to 29.21%. The simulation results all achieved a relatively high level of accuracy. However, the RMSE was relatively large, indicating that there were outliers in the simulation results of individual treatments. In the calibration, the simulation errors for T6 and T7 were large, while the simulation accuracy was the highest from T1 to T5. In the validation, the simulation errors from T4 to T6 were relatively large, but the simulation accuracy from T1 to T3 and from T7 to T9 was high. This might have been because the rainfall difference over the two years led to varying degrees of water stress on maize under the same irrigation treatment. Due to the model’s failure to account for the impacts of various stress factors, such as soil salinity, spatial variability of soil properties, weeds, pests, and diseases, on yield, as well as issues related to parameter calibration, including heteroscedasticity and overfitting, along with the model’s assumptions regarding the parameterization of maize growth processes, the model could not achieve a high degree of simulation accuracy for the maize yields of all treatments. Consequently, this led to significant errors in some simulated yield values and a reduction in the overall simulation consistency for the yields of all treatments. To verify the model’s reliability in yield simulation, it was necessary to conduct multi-point and multi-year field experiments to further assess the model’s stability.

2.2.3 Independent validation and confidence interval analysis.

1. (1) Overview of the experimental area and experimental design

The simulation accuracy of the SWAP model for maize yield was further validated using the irrigation experiment conducted in Pingluo County, Ningxia from 2022 to 2023. The experimental area was located in Baofeng Town (E106 ° 43 ′, N39 ° 02 ′), with a temperate continental climate. The average annual rainfall was 177 mm, the average annual evaporation was 1755 mm, the average annual sunshine hours were 3008 h, and the frost-free period was 171 d. The soil in the experimental area was sandy loam, having an average field water holding capacity of 27% and an average bulk density of 1.48 g/cm³. The available nitrogen, phosphorus, and potassium contents, as well as the organic matter content in the cultivated layer of the experimental site, are 74.36 mg/kg, 13.21 mg/kg, 114.28 mg/kg, and 1.9%, respectively. The experiment was a single-factor experiment on irrigation amount, using drip irrigation as the irrigation method. Five gradient irrigation amounts were set up with the conventional irrigation amount of local farmers as the control, and the irrigation quotas were 2400 m³/ha (S1), 2700 m³/ha (S2), 3000 m³/ha (S3), 3300 m³/ha (S4), 3600 m³/ha (S5). The division of experimental plots, fertilization, indicator determination methods, and other field management measures were the same as before. The irrigation amount for each growth period is shown in S4 Table.

1. (2) Validation results and confidence interval analysis.

The simulation and confidence-interval analysis results of the SWAP model for maize experimental yield were presented in S6 Fig. The analysis revealed that the SWAP model had a simulated R² of 0.71, an RMSE of 921 kg/ha, and an NRMSE of 11.62% for maize yield in Pingluo County, indicating good simulation accuracy. Through the confidence-interval analysis of the measured and simulated values of all yield in Yongning County and Pingluo County, it was observed that many yield values fell within the 95% confidence band, and all yield values were included in the 95% prediction band, with an R² of 0.70, an RMSE of 1227.59 kg/ha, and an NRMSE of 14.58%. Overall, the simulation accuracy of the SWAP model for maize yield could meet general needs and had certain applicability. Therefore, the established maize SWAP model parameter set had a certain degree of scientificity and reliability.

2.3.1 Model calibration process.

The calculation results of water stress coefficients (Ws) for each treatment during the model verification were shown in S7 Fig. There were significant differences in the water stress experienced by maize throughout its entire growth period under different irrigation treatments. However, the overall manifestation was that in the early stage, the water stress was relatively mild and the duration was short. It gradually increased after 80 days of emergence and reached the most severe level at 100 days (the average Ws of each treatment ranged from 0.07 to 0.60), and then the water stress gradually decreased. This stage was from the tasseling stage to the milk ripening stage of maize, during which the water consumption was high. Insufficient irrigation could lead to severe water stress. Maize experienced varying degrees of water stress in each treatment from 20 to 70 days after emergence. The average Ws of each treatment decreased from values within the range of 0.99 to 1.0 to values within the range of 0.81 to 0.99 after irrigation on days 38, 48, and 56 after emergence, indicating that water stress was caused by excessive irrigation. On the 31st day after emergence, irrigation alleviated water stress in treatments T1 to T6, with the average Ws increasing from values within the range of 0.83 to 0.95 before irrigation to values within the range of 0.99 to 1.0. However, in treatments T7 to T9, it exacerbated water stress, resulting in a decrease in Ws. On the 69th day after emergence, irrigation alleviated water stress in treatments T1 to T4, with the average Ws increasing from values within the range of 0.90 to 0.96 before irrigation to values within the range of 0.99 to 1.0, but exacerbated water stress in treatments T5 to T9. It could be seen that the maize irrigation scheduling set during the model verification process was unreasonable and needed to be optimized based on water stress conditions.

2.3.2 Model validation process.

The calculation results of water stress coefficients (Ws) for each treatment during model validation were presented in S7 Fig. There were varying degrees of water stress in the early stages of maize growth for treatments T1 to T3. A severe and long-lasting water stress occurred from 70 to 120 days after emergence, with the most intense water stress occurring at 92 days. The Ws values of each sample ranged from 0.18 to 0.47. After irrigation, water stress was alleviated. However, due to the high water consumption from the tasseling stage to the milk ripening stage of maize, severe water stress persisted after irrigation. During the entire growth period of maize, treatments T4 to T9 experienced different degrees of water stress on 10 occasions, and the duration of water stress was closely related to the amount of irrigation. Among them, treatments T4 to T6 had a longer duration of water stress from the tasseling stage to the milk ripening stage of maize, while the duration of water stress episodes in other treatments was relatively shorter. Water stress first emerged on the 32nd day after emergence, and irrigation mitigated the water stress conditions for each treatment. For the 2nd to 6th, as well as the 8th and 10th water stress events, the average Ws before irrigation ranged from 0.98 to 1.0, and it ranged from 0.76 to 0.98 after irrigation, indicating that these water stress events resulted from excessive irrigation. For the 7th water stress event, the pre-irrigation Ws values of treatments T4–T6 ranged from 0.70 to 0.94, and these treatments were relieved to a state without water stress after irrigation. However, the average Ws values of treatments T7–T9 were all 0.99 before irrigation, which was attributed to excessive irrigation. During the 9th water stress event, the average Ws of each treatment ranged from 0.87 to 0.90 before irrigation and increased to 0.99 to 1.0 after irrigation. During model validation, the irrigation scheduling also exhibited some unreasonable issues. Therefore, it was necessary to further optimize the irrigation scheduling based on the water consumption pattern of maize.