The scatter plot of daily ET in Figure 5 showed a significant correlation (p < 0.001) between the PM estimated from nine r _c models with original parameters and the BREB measured values. The R ² values ranged from 0.7 to 0.83, with the PM-KP obtaining the highest correlation and the PM-IS being the lowest. However, the slope of the linear relationships varied significantly, ranging from 0.51 to 1.75 with the PM-MT model being closest to 1. Clearly, four r _c models (the PM-ST, PM-KP, PM-GA, and PM-FA) underestimated daily ET, as indicated by their regression slopes (0.51–0.81) of less than 1 ( Figures 5A, C, F, H ). This was also reflected in their daily mean difference (MBE), ranging from −0.59 to 0.02 mm d⁻¹ ( Figure 6 ), with the PM-FA having the largest value and the PM-ST having the smallest. The PM-GA had the largest underestimation at only half of the true value. An MBE of less than 0 meant underestimation and vice versa. However, the MBE for the FA model was greater than 0, likely because it generally overestimated ET when the daily ET was less than 2 mm d⁻¹ (e.g., sparse vegetation cover) (Li et al., 2015). The index of agreement (d) of these r _c models was greater than 0.85 ( Figure 6 ), with the PM-GA having the highest value and the PM-FA having the lowest. The RMSE values for these four r _c models ranged from 1.07 to 1.47 mm d⁻¹, averaging 1.21 mm d⁻¹ ( Figure 6 ), suggesting a performance order of PM-FA > PM-KP > PM-ST > PM-GA. Notably, the PM-KP performed similarly to the PM-FA, producing values of slope, R ², MBE, RMSE, and d within 0.04, 0.01, 0.08 mm d⁻¹, 0.06 mm d⁻¹, and 0.01 of each other.

Five r _c models, i.e., the PM-CO, PM-JA, PM-IS, PM-MT, and PM-TD, overestimated daily ET, as indicated by their regression slope ( Figures 5B, D, E, G, I ) of larger than 1. The slope of the PM-MT was closest to 1 (1.02), and that of the others were greater than 1.4. Meanwhile, the PM-MT obtained the smallest MBE at 0.32 mm d⁻¹, while the others ranged from 0.98 to 2.05 mm d⁻¹, averaging 1.33 mm d⁻¹ ( Figure 6 ). The PM-CO model had the largest overestimation. The d values of these models were lower than those of the underestimated models, ranging from 0.68 to 0.85 with an average of 0.77. Among these, PM-IS performed the worst and PM-TD performed the best. The RMSE values for the overestimated models ranged from 1.6 to 2.97 mm d⁻¹, averaging 2.52 mm d⁻¹. Performance decreased in the following order: PM-MT > PM-TD > PM-JA > PM-CO > PM-IS ( Figure 6 ).

The accuracy of PM-ST, PM-KP, and PM-FA models without parameterization was acceptable in this study. The PM-KP model performed aligning with the estimation results of Yan et al. (2020) for winter wheat ET in humid regions. However, the RMSE increased by 38.3% compared to their value of 0.81 mm d⁻¹. The FA model did not require parameter correction and needed fewer meteorological factors, making it suitable for widespread application. Its simulation results differed from those of Li et al. (2015) for maize and grape under partial and dense canopy stages, where their RMSE was much high at 7 mm d⁻¹. The ST model, a JA-type model, ranked behind the KP and FA models, requiring LAI values during the calculation process. Having more parameters and complex calculation processes did not improve the accuracy.

The other six r _c models (CO, JA, IS, GA, MT, and TD) presented great error (RMSE > 1.5 mm d⁻¹) without parameterization ( Figure 6 ). Historically, the MT model was the most widely used in PM equation (r _c =70 s m⁻¹) to estimate grassland ET (Gharsallah et al., 2013). The MT model performed best in regression slope but significantly overestimated in 2022–2023 and underestimated in 2021–2022, resulting in a 4-year total slope of 1.02 ( Figure 5G ). This inconsistency may be due to r _c being assigned a fixed value that did not match the actual situation. The TD model, which did not require parameter correction, showed good results in Liu et al. (2011) and Yan et al. (2020) with RMSE values of 0.79 and 0.85 mm d⁻¹, respectively. They were much lower than those in our study. This discrepancy may be because these studies used only daytime data, whereas using data from the entire day can introduce uncertainty in daily ET. When VPD and R _n-G were less or near zero, and when r _I and r* in the TD model had negative or uncertain values, X had no solution (the solution of the TD method equation, Table 1 ). Thus, the TD model was only applicable when r _I was precise (Perez et al., 2005). In another study, Katerji et al. (2011) compared the KP and TD models over four crops and found that the TD model’s overestimation was attributed to the theoretical limitations, neglecting the effect of aerodynamic resistance. The inability of ST, CO, JA, IS, and GA r _c models to estimate daily ET with original parameters was due to the inadequacy of these parameters, attributed to differences in crop types and significant variations in environmental factors, especially regional climatic water conditions (Forster et al., 2022). Therefore, using these models to estimate ET required parameter calculation.

The scatter plot of the daily ET estimated by the PM equation with seven r _c models (excluding the FA and TD) with calibrated parameters against BREB measurement is shown in Figure 7 . It could be clearly seen that scatter plots of r _c models were closer to the 1:1 line after calibration, indicating a significant correlation between measured values. Compared to using original parameters, seven r _c models (except the FA and TD) performed better after recalibration. d increased from 0.69–0.93 to 0.86–0.93 with the average d value increasing by 13.8%. R ² increased from 0.7–0.83 to 0.78–0.85, with the average R ² increasing by 19%, indicating improved stability and reliability of the r _c models. The PM-KP obtained the highest correlation, while the PM-MT had the lowest.

Five r _c models (PM-CO, PM-JA, PM-IS, PM-GA, and PM-MT) underestimated the daily ET values, with the regression slopes ranging from 0.62 to 0.96, averaging 0.81 ( Figures 7B, D–G ). MBE values ranged from −0.70 to 0.01 mm d⁻¹, averaging −0.26 mm d⁻¹, with the PM-MT having the largest underestimation and the PM-CO having the smallest ( Figure 6 ). The underestimated r _c models produced RMSE values ranging from 1.12 to 1.34 mm d⁻¹, averaging 1.25 mm d⁻¹ ( Figure 6 ), according to their RMSE ranked as PM-CO > IS > GA > JA > MT. The RMSE values were 16.25%–91.64% lower than using the original parameters, averaging 41.16%, indicating that the accuracy of r _c models has been improved.

PM-ST and PM-KP showed a light trend of overestimation with regression slopes of 1.14 and 1, respectively ( Figures 7A, C ). The PM-ST model obtained a better MBE value, while the PM-KP model performed better in terms of RMSE. Their MBEs were 0.00 and 0.29 mm d⁻¹, and the RMSE values were 1.20 and 1.12 mm d⁻¹, respectively ( Figure 6 ). It was worth noting that the RMSE of PM-ST model increased by 3.4%, while the PM-KP model only decreased RMSE by 0.88%, suggesting that the accuracy has not improved.

As described above, the performance of the nine r _c models, based on RMSE values ( Figure 6 ), was ranked as PM-FA > CO > KP > ST > IS > GA > JA > MT > TD. The performance of the first eight models was acceptable. Although parameter recalibration could reduce the RMSE value of the r _c model, they were still higher than that of the PM-FA model. However, this process significantly improved the regression slope of the models. The FA model’s RMSE still performed better than others after parameter correction. Ortega-Farias et al. (2004) and Ortega-Farias et al. (2006) also indicated that the PM-FA model accurately estimated ET for soybean and tomato. However, the regression slope of the PM-FA was the second to last among the eight acceptable models, possibly because it was an empirical method and failed to consider the effect of water stress condition. Unlike with the previous research (Li et al., 2015), the FA model systematically overestimated maize and grapevine ET, with RMSE exceeding 7 mm d⁻¹ during both low and high LAI stages. This overestimation was primarily due to the underestimating canopy resistance, particularly during the sparse canopy stage. While the model accounted physiological control on resistance, it failed to consider the restrictive effects of soil.

The PM-CO, IS, and JA r _c models exhibited significant errors before parameterization but achieved satisfactory accuracy afterward ( Figures 5 – 7 ). The PM-CO simulated maize and vineyards more accurately than in Li et al. (2015), with an RMSE 31.3% lower than theirs. Although the PM-CO model had good accuracy, it involved the most complicated calculation process among all r _c models and required the most meteorological parameters. The difficulty of obtaining these data should be considered in practical applications. The successful application of the PM-IS model in this study was consistent with Howell et al. (1997) for wheat, with an RMSE reduction of 18.3%. However, the RMSE for corn and sorghum increased by 29.2% and 57.5%, respectively. This may indicate that the PM-IS model was more suitable for estimating ET of crops with higher plant height and greater canopy temperature differences.

Compared with Zhang et al. (2008) and Li et al. (2015), the accuracy of the PM-JA model has improved, but it was still inferior to other models and ranked last among all acceptable models. Zhang et al. (2008) indicated that the PM-JA model overestimated the vineyard daily ET in an arid desert region of northwest China and was inaccurate after rainfall. The JA model presented uncertain results during the partial canopy stage but simulated ET accurately under the full canopy (Li et al., 2015). This may explain the larger overall RMSE, which also showed that the accuracy of model estimation was not related to the model’s complexity.

For both the PM-KP and ST models, parameterization appeared to be unnecessary. The reduction in RMSE after calibrating the KP model calibration was not negligible, with only a 0.01 difference before and after calibration. This indicated that parameter recalibration did not enhance the KP model’s accuracy, though it did achieve an optimal linear regression slope (value = 1), consistent with Rana et al. (2012). Compared to the simulations of tomato, maize, canola, and tea by Rana et al. (2012); Srivastava et al. (2018); Liu et al. (2012), and Zhang et al. (2022a), the RMSE increased by 43.8%, 66.9%, −3.6%, and 8.9%, respectively, indicating that parameter calibration did not improve accuracy. The PM-ST model’s RMSE was identical to that reported by Xing et al. (2024) for the kiwifruit. However, parameter correction did not improve its accuracy; instead, it slightly increased the RMSE. Nevertheless, the regression slope, d, and MBE values were improved. This may be due to significant annual variations, with 2 years of overestimation and 2 years of underestimation, leading to a slight increase in RMSE.

The PM-GA and MT models still exhibited some significant errors even after parameter calibration, but their results were better than those of the TD model, which did not require correction. In the GA model, the maximum stomatal conductance was set as a constant value. However, in the natural environment, this parameter dynamically fluctuated in response to climatic variations. This discrepancy may contribute to the observed significant underestimation of the GA model (García-Santos et al., 2009; Xu et al., 2021). Similar to the GA model, the MT model also underestimated ET due to its overestimation of r _c and the uncertainty of nighttime r _c. Therefore, these two models should be used with caution.

Although the PM-FA, KP, and ST models can estimate daily ET of winter wheat regardless of whether they were calibrated or not, they each had limitations. Without parameterization, these three models noticeably underestimated daily ET. In this study, calibrating the KP and ST models appeared redundant, as the RMSE of the KP model decreased by only 0.01 mm d⁻¹, while that of the ST model increased by 0.04 mm d⁻¹ after calibration. In comparison, the errors did not improve and even worsened in some cases. Based on the results of Yan et al. (2020), the KP model, originally applied in humid conditions, can still be effectively used under the semiarid conditions of this study. Additionally, the PM-CO, IS, ST, GA, JA, and MT models, similar to previous studies, required parameter calibration before they were used to estimate ET. However, these models required more parameters or meteorological data compared to the KP model. The FA model did not require parameterization, which often demanded extensive soil moisture data, making it useful when parameter calibration was not feasible without measured ET values.

In order to analyze the seasonal cumulative ET of winter wheat simulated by PM with different r _c models, this study divided the entire growth period of winter wheat into three stages: the seeding period in October and November, the wintering period from December to February of the following year, and the rapid growth period from March to June. The seasonal ET of winter wheat during the seeding, wintering, and rapid growth periods over four growth years, observed using the BERB method, ranged as follows: 50.07 to 98.35 mm with an average of 79.42 mm for the seeding period, 12.56 to 44.2 mm with an average of 27.05 mm for the wintering period, and 251.8 to 409.7 mm with an average of 359.34 mm for the active growing period. Throughout the entire reproductive period, the proportion of total ET was 10.6% to 25.69% during the seeding period, with an average of 17.37%; 2.66% to 9.10% during the wintering period, with an average of 5.91%; and 65.80% to 86.74% during the rapid growth period, with an average of 76.72%.

To compare winter wheat seasonal ET across different growth stages, the differences between BREB observations and PM combined r _c model estimations over 4 years are shown in Figures 8 and 9 . Over the entire growth period, the PM-ST and PM-GA r _c models consistently underestimated seasonal ET. The total ET differences between the two r _c models applied in PM and the BREB measurements varied from −197.12 to 12.27 mm, with an average of −103.59 mm ( Figure 8 ). The MBE and RMSE for the PM-ST and GA models were −152.06 and −115.79 mm, and 157.05 and 121.23 mm, respectively ( Table 2 ). The other seven r _c models (i.e., PM-CO, KP, JA, IS, MT, FA, and TD) overestimated seasonal ET. The differences in ET between these seven r _c models applied in PM and the ET determined by the BREB ranged from −95.33 to 603.52 mm, with an average of 257.65 mm ( Figure 8 ). The MBE values varied from 2.83 to 530.47 mm ( Table 2 ), with the PM-MT model performing the best and the PM-CO model performing the worst. The RMSE values ranged from 29.79 to 535.26 mm ( Table 2 ), based on performance that decreased according to FA > MT > KP > JA > TD > IS > CO.

During the seeding period, the PM-ST, KP, JA GA, MT, and FA models underestimated ET, with MBE ranging from −63.88 to −8.64 mm (averaging −29.19 mm) and RMSE ranging from 18.63 to 67.05 mm (averaging 35.55 mm). The PM-CO, IS, and TD models overestimated ET, with MBE values of 48.64, 84.51, and 79.33 mm, and RMSE values of 52.57, 60.13 and 89.75 mm, respectively ( Table 2 ). The average RMSE values during the seeding period were nearly half of the measured ET. This funding was similar to that of Sun et al. (2007), who suggested that r _c models showed large errors during the sparse canopy stage or when the LAI was less than 2. However, in this study, the errors were even larger.

During the wintering period, when ET was at its lowest, the ST, JA, and MT models underestimated ET, while the other models overestimated it. The MBE values ranged from −3.68 to 135.47 mm, with an average of 44.46 mm, and the RMSE values ranged from 12.98 to 139.04 mm, with an average of 56.29 mm ( Table 2 ). Although these average values were larger than the measured values, the accuracy of the r _c models without parameter calibration did not rely heavily on this stage, as ET during this period accounted for only an average of 5.91% of the total ET, which was consistent with the findings of Gharsallah et al. (2013).

All r _c models recorded their highest RMSE during the rapid growth period, ranging from 57.93 to 375.27 mm, with an average of 186.87 mm. Among these, the FA model performed best and the CO model performed worst, indicating that the rapid growth period was the stage with the least accuracy in estimating ET. Over- or underestimation at this stage did not determine the overall ET estimation trend, as seen with the MT model. The ST, FA, and GA models underestimated ET during this period, with MBE values of −71.31, −21.80, and −123.10 mm, respectively. Among these, the FA model performed the best and the GA performed the worst. The other six r _c models overestimated ET, with MBE values ranging from 54.62 to 371.6 mm ( Table 2 ).

Without parameter calibration, the PM-FA, PM-MT, and PM-KP models were acceptable on a seasonal scale, mainly due to their excellent performance during the rapid growth period. They exhibited both over- and underestimation across the three growth periods, resulting in the total ET errors offsetting each other. The FA model performed perfectly in all years ( Figure 8 ) and achieved the best RMSE values ( Table 2 ). The FA model relied on the climatic factors and soil water, which was a function of soil moisture and r _i, and was defined as the water vapor transfer from the soil and plants to the atmosphere (Chen et al., 2022). Due to the inclusion of the moisture content factor, it better reflected the early stage of soil ET. The RMSE of the KP and MT models were acceptable; however, the differences between KP model and BREB for the 2019–2020 and 2022–2023 periods were 112.71 and 99.35 mm, respectively. Additionally, the differences between the MT model and BREB for the 2021–2022 and 2022–2023 periods were −95.33 and 95.11 mm, respectively. These discrepancies seemed unacceptable for practical estimation applications ( Figure 8 ). Therefore, the KP and MT models without calibration should be used with caution in seasonal ET assessments.

It was noteworthy that the MT model was rejected on the uncalibrated daily scale, while the ST was accepted on the daily scale but rejected on the seasonal scale. This suggested that the r _c model, which estimated ET on the daily scale, may not be suitable for estimating seasonal ET, as the same model performed differently at different scales (Howell et al., 1997). This phenomenon may be attributed to variations in climate conditions across years and the differing performance of models at various growth stages of winter wheat. Interestingly, over- and underestimation at different stages can offset each other, improving the overall reproductive cycle results. The MT model performed well on the seasonal scale because it underestimated seeding period daily ET and overestimated it during the rapid growth period ( Figure 8 ), causing the total ET errors to cancel each other out. The ST model significantly underestimated the daily ET across all three growth stages ( Figure 8 and Table 2 ). The accumulation of these large errors resulted in total seasonal ET estimation, indicating that consistent estimation trends across growth stages can lead to larger errors in total ET.

The rejection of the six unacceptable models (i.e., ST, CO, JA, IS, GA, and TD) was primarily due to their poor performance during the rapid growth period ( Figure 8 and Table 2 ). Unlike in previous studies, Liu et al. (2011); Chen et al. (2022); Xing et al. (2024); Li et al. (2015); García-Santos et al. (2009), and Xing et al. (2024), the TD, JA, IS, and CO models significantly overestimated ET during this stage, while the GA and ST models significantly underestimated it. All of the unacceptable models exhibited the worst accuracy during the rapid growth stage, with severe overestimation or underestimation sometimes exceeding twice the ET value at this stage ( Table 2 ). This indicated that the accuracy of seasonal ET estimation primarily depended on this period.

After parameter calibration, six r _c models (i.e., ST, KP, JA, IS, GA, and MT) underestimated the total seasonal ET. The total ET differences between these models and the direct determination method ranged from −235.31 to 74.88 mm, with an average of −68.76 mm. Among them, the ST model obtained the best average value, while the MT model had the worst ( Figure 9 ). The MBE values varied from −179.93 to −1.13 mm ( Table 2 ), with the ST model performing the best and the MT model performing the worst. The positive and negative MBE values on the seasonal scale were entirely consistent with the regression slope. Unlike on the daily scale, there was no instance where the MBE was greater than 0 but the slope was less than 1. The RMSE values ranged from 41.68 to 184.50 mm, with an average of 80.77 mm ( Table 2 ). The models ranked by RMSE were as follows: KP > IS > ST > GA > JA > MT. Only the CO model generally overestimated the seasonal total ET, with differences from the BREB method ranging from −51.84 to 28.11 mm and an average of 2.36 mm ( Figure 9 ). Its MBE value of 2.36 mm was second only to the ST model, and its RMSE of 32.18 mm was the best among all r _c models ( Table 2 ).

Parameter calibration did not improve the accuracy of all r _c models across all growth stages. The RMSE values of the KP, JA, GA, and MT models during the seeding period, JA and MT models during the wintering period, and ST and MT models during the rapid growth period were 2.51%–94.74% higher than those of the uncorrected models. In contrast, the accuracy of the r _c models improved in other situations, with RMSE values decreasing by 9.69% to 91.75%. Except for the MT model, the MBE values for the three growth stages improved, indicating that the degree of overestimation or underestimation was reduced, resulting in more accurate estimation.

The ability of the r _c model to estimate seasonal ET depended on the simulation results during the rapid growth period. Six r _c models were accepted, with only the MT model being refused based on the RMSE values. The performance of the r _c models was clearly better than before, with the RMSE value reduced by 68.18%, indicating that the recalibration process significantly improved the accuracy of the r _c models, except for the MT model ( Figure 9 and Table 2 ). This improvement was attributed to better estimation results during the rapid growth period compared to previous results (Li et al., 2014). It should be noted that the underestimation occurred during the seeding stage, and the estimation error during this stage was primarily attributed to significant heterogeneity in water vapor transport within the model (Li et al., 2015). Compared to the daily scale, the accuracy of estimating seasonal ET was higher, consistent with findings by Gharsallah et al. (2013). This may be because the high and low ET estimation canceled each other out over the long-term growth period, making the cumulative seasonal ET close to the measured value (Irmak and Irmak, 2008).

The CO model showed better simulation results in the early stages of growth. This improvement was primarily because the CO followed the resistance law of fluid transfer, coupling the resistance of plants and soil into the overall canopy resistance. It accounted for the combined limiting effects of vegetation and soil on surface water transfer, and providing an accurate estimate of average surface resistance (Li et al., 2015). The FA model ranked second to the CO model. Although it still underestimated seasonal ET as it did on the daily scale, its results were better than those for daily ET estimation.

During the calibration of KP and IS model parameters, VPD had the greatest influence, making it the primary factor affecting the model error. However, when the VPD value was less than 2 kPa, the model error was minimal (Yan et al., 2020). During the seeding and wintering period, the VPD was generally below 2, resulting in better simulation results for these two models.

The GA model was rejected on the daily scale but accepted on the seasonal scale. As a function of R _n and VPD, it performed well during the seeding and wintering periods, and its results for the rapid growth period were also acceptable. Therefore, it can be used to estimate seasonal ET.

The accuracy of the ST model improved during the seeding and the wintering periods. However, its RMSE value increased during the rapid growth period, shifting from underestimation to overestimation. The combined effects of the seeding and the wintering periods improved the overall results, leading to a 64.39% reduction in RMSE.

The ST, JA, and MT models had large errors in estimating ET during the seeding and wintering periods because they all belonged to the JA-type model, which upscaled leaf-level resistance to canopy-level resistance (Lhomme, 2001; Wei et al., 2013). However, during these two growth periods, the LAI value was less than 1, leading to significant underestimation of ET due to the exposed surface. The ST and JA models slightly overestimated ET during the rapid growth period, but the rapid development of winter wheat during this period led to higher ET. The slight overestimation during this period essentially offset the serious underestimation during the seeding and wintering periods, resulting in a slight overestimation in the 4-year daily scale regression, but a more accurate simulation of the seasonal accumulation. Compared with the ET simulated by the JA-type r _c models used by Liu et al. (2020) in the seeding period, wintering period, green period, and maturity period, MBE and RMSE increased.

The MT model consistently underestimated ET across all three growth periods, while the TD model consistently overestimated it, leading to significant overall errors. Therefore, these two models were not suitable for studying winter wheat water consumption in this region at seasonal scale. Notably, the MT method was rejected after calibration due to its significant underestimation of ET during the seeding and rapid growth periods. During the early stages of crop development, the soil surface was nearly bare, and soil evaporation dominated the entire ET process. The assumption of the large leaf model led to significant errors (Wu et al., 2022). The significant overestimation by the TD model may be due to its sensitivity to VPD values, particularly when VPD ranged from 1.5 to 4 kPa. Additionally, research has shown that the TD model cannot be reliably applied at night (Yan et al., 2022).

In summary, after calibration, only the CO model outperformed the FA model, while the other models still performed worse. However, when comparing the calculation processes, it was clear that the FA model was much simpler than the CO model and did not require parameter calibration. The FA model was the most suitable method for seasonal ET estimation unless extremely high accuracy was required. When the FA and CO models lacked the necessary meteorological factor, the IS, GA, JA, ST, and KP models can be used to estimate seasonal ET based on known meteorological data.