Can the atmospheric boundary layer-based potential evaporation model increase the accuracy of generalized complementary functions?

Evapotranspiration is a vital component of hydrological processes and serves a major role in the hydrological cycle and land surface energy balance (Wang & Dickinson 2012). The complementary principle has been extensively utilized and studied because it can effectively estimate actual evaporation using common meteorological observations and does not require data that is difficult to observe, such as vegetation resistance and soil moisture (Shang et al. 2022). Therefore, the evaluation and improvement of the methods for estimating evapotranspiration based on the complementary principle is of great significance for hydrology science.

The complementary principle was proposed by Bouchet (1963). Several complementary linear models have been developed (Brutsaert & Stricker 1979; Morton 1983; Granger & Gray 1989). Brutsaert & Parlange (1998) used a parameter to represent asymmetry in the complementary relationship (Kahler & Brutsaert 2006; Szilagyi 2007) and developed an asymmetric linear complementary model. Han et al. (2012) proposed a dimensionless form of complementary principle. Inspired by this, Brutsaert (2015) proposed the generalized complementary principle framework and developed a widely used polynomial function (B2015). Later, Han & Tian (2018) revised the boundary conditions of the complementary principle, which served as the foundation for the development of the new sigmoid function (H2018). Gao & Xu (2021) derived the exponential generalized complementary function (G2021), which satisfies the physical boundary conditions and constraints proposed by Brutsaert (2015) and offers a new technique for calculating land surface evapotranspiration. It is important to note that, unlike B2015 and G2021, H2018 is based on the concept of equilibrium evaporation rather than potential evaporation. Some calibration-free equations were proposed by Crago et al. (2016) and Szilagyi et al. (2017) by the rescaling method. The polynomial complementary equation was expanded to a power function by Szilagyi (2022) to increase its adaptability.

The apparent potential evaporation is considered as the evaporation when the surface is assumed to be saturated under actual meteorological conditions (Fisher et al. 2011; Brutsaert 2015). It is calculated using the Penman formula (Penman & Keen 1948) in most studies, and this approach is widely accepted in complementary model studies (Brutsaert 1982, 2015; Liu et al. 2018). On the other hand, potential evaporation (E_po), which is another key variable in complementary models, is defined as the evaporation that occurs in a condition with sufficient water supply which causes changes in both the evaporating surface and the atmosphere surrounding it (Brutsaert 2015). The meteorological elements (such as radiation and temperature) used to determine E_po are typically observed at unsaturated actual conditions, which do not match the original definition of E_po (Szilagyi & Jozsa 2008; Aminzadeh et al. 2016; Yang & Roderick 2019; Liu & Yang 2021; Tu & Yang 2022; Yang et al. 2022; Tu et al. 2023). This causes difficulty to interpret E_po and also the complementary models (Han & Tian 2020). Different understandings and estimations of E_po have been proposed (Slatyer & McIlroy 1961; Monteith 1965; Allen et al. 1998; Szilagyi & Jozsa 2008). The model proposed by Priestly & Taylor (1972) is widely used to estimate E_po, however, due to the difficulty in determining the coefficient α in the model, large uncertainties are brought to the complementary functions by the Priestly and Taylor (P-T) model (Brutsaert & Stricker 1979; Eagleson 2002; Brutsaert 2005; Assouline et al. 2016; Lu et al. 2017). Zhang et al. (2023) found that the sensitivity of α is evidently larger than all the other parameters in the complementary functions. A lot of studies have focused on estimating α (Jury & Tanner 1975; Parlange & Katul 1992; Eichinger et al. 1996; Assouline et al. 2016; Wang et al. 2020; Han et al. 2021; Liu & Yang 2021); however, it was demonstrated that the P-T model is difficult to represent the accurate relationship between the potential evaporation and temperature (Assouline et al. 2016). A calibration-free method for E_po estimation is still required to reduce the uncertainties of the complementary functions. Recently, some studies quantified E_po from the perspective of the Bowen ratio using observations at the open water (Hicks & Hess 1977; McColl & Rigden 2020). Yang & Roderick (2019) considered the independence of surface temperature, net long-wave radiation, and evaporation in saturated surface and (Monteith 1981) proposed the theory of maximum potential evaporation for Bowen ratio and E_po estimation (Tu & Yang 2022; Tu et al. 2023). They also found that the P-T model did not provide an accurate relationship between surface temperature and Bowen ratio.

Liu & Yang (2021) introduced a physically based model (ABL2021) for estimating potential evaporation (E_po) by integrating the atmospheric boundary layer model with the potential vapor pressure deficit balance. This model offers an advantage by eliminating the requirement for parameter calibration, relying solely on air temperature and specific humidity to estimate the Bowen ratio and surface energy partitioning. ABL2021 is particularly valuable in accurately calculating latent heat flux and the Bowen ratio, providing a robust framework for understanding energy exchanges over water bodies, such as oceans and lakes. Given its solid theoretical foundation, the model has shown high accuracy compared with observed data. This suggests that ABL2021 can provide an important tool for studies of water–atmosphere interactions. Therefore, assessing the accuracy of this new E_po model and its performance in complementary functions is essential to evaluate its potential for broader hydrological and climate-related applications.

The objective of this study is to compare the different approaches to estimating potential evaporation and investigate whether the recently derived potential evaporation model (ABL2021) based on atmosphere boundary layer theory can improve the accuracy for estimating the actual evaporation in the complementary functions. The polynomial function and the exponential function are used to detect the role of the potential evaporation models in the improvement of the accuracy of the generalized complementary functions, and we also explore the mechanisms involved. Next, the accuracy of different complementary functions is also compared and discussed.

Observations of the variables including air temperature (T_a), saturated water vapor pressure difference (VPD), atmospheric pressure (P_a), wind speed at 2 m (W_s), net radiation (NETRAD), latent heat flux (LE), soil heat flux (G), precipitation (P), and sensible heat flux (H) were collected from the 86 sites. Missing values were flagged if the data significantly deviated from the expected range or it was recorded as ‘ − 9999’. If both the adjacent valid data points before and after the missing data were available, a linear interpolation method was applied to estimate the missing values. If such adjacent valid data points were not available, the missing values were excluded from the analysis.

In this study, the widely used polynomial, sigmoid, and exponential complementary functions are examined and compared. Following the generalized complementary theory proposed by Brutsaert (2015), two dimensionless variables which define that are used to establish the polynomial and exponential functions, where apparent potential evaporation (E_pa) refers to the evaporation from a limited saturated surface under the same atmospheric conditions as actual evapotranspiration. In contrast, potential evaporation (E_po) occurs when sufficient water supply leads to changes not only in the evaporating surface but also in the atmospheric conditions interacting with it (Brutsaert 2015).

Since the concept of potential evaporation is used for E_po in B2015 and G2021, whereas equilibrium evaporation is applied in H2018, H2018 is not included in our comparison in this study and we only address H2018 in Section 4.3 for the discussion.

For each complementary function, we compare the estimated and actual values of y, and modify the model parameters until the most accurate estimation of y is obtained. The whale optimization algorithm (WOA) (Mirjalili & Lewis 2016) and the least square method are used to optimize the model parameters based on a matlab program. The optimized parameters in different functions at each site in this study are listed in Table 1.

The bias of E_po estimation by each model at the Taihu Lake sites was determined using the deviation of the fitted straight line from the E_{po_obs} = E_{po_eva} straight line. The ABL2021 model shows the best performance among all the three models at the seven sites in the Taihu Lake, with a bias of 8.76%, while the other two models exhibit a certain degree of overestimation of E_po. The P-T (α = 1.26) shows a relative bias of 34.1% and the relative bias of YR2019 is 26.96%. The ABL2021 model also shows the lowest RMSE of 18.44 mm (Table 2), and the P-T model with α as 1.26 has an RMSE of 33.55 mm which is close to that of YR2019 (29.79 mm). This result illustrates that ABL2021 is more accurate for E_po estimation than the other two models.

The comparison of potential evaporation (E_po) calculations across various sites reveals notable differences in E_po estimation by different models. Furthermore, the performance of complementary evapotranspiration functions largely depends on the precision of E_po estimation. Accordingly, the comparative analysis of different E_po models not only elucidates the variance in complementary models under diverse conditions but also facilitates the optimization of their application. The divergences in the results of E_po models are dominated by their respective energy partitioning mechanisms. Based on the maximum evaporation theory, the YR2019 model considers the equilibrium between surface temperature and radiation. It posits that augmented radiation causes a greater allocation of energy to evaporation, resulting in a maximum evaporation rate (Yang & Roderick 2019) which is larger than the results of other models. Conversely, ABL2021 employs the atmospheric boundary layer theory, utilizing air temperature and humidity to derive potential evapotranspiration, accentuating the role of energy exchange within the boundary layer. In contrast, the P-T model relies on a fixed empirical coefficient, simplifying the calculations and addressing energy partitioning with less precision. Consequently, this model may exhibit poor performance in extreme climates, such as high-temperature conditions (Liu & Yang 2021), thereby it shows the largest range of variations of E_po.

Overall, the MAE and RMSE for B2015 using the ABL2021 were significantly lower than that of P-T (α = 1.26) and YR2019. With the change in aridity index, the MAE of ABL2021 ranged from 14.2 to 20.4 mm, and the MAE of P-T (α = 1.26) ranged from 19.3 to 32.3 mm. YR2019 showed the largest MAE which ranged from 21.6 to 33.8 mm. The RMSE of ABL2021 varied from 16.8 to 27.3 mm, and the RMSE of P-T (α = 1.26) and YR2019 showed much larger values, which varied from 25.0 to 40.0 mm for P-T (α = 1.26) and from 27.1 to 42.4 mm for YR2019.

In general, B2015 showed better performance at sites with a low aridity index than that with a large aridity index. As the aridity index increased, the RMSE and MAE significantly increased for P-T (α = 1.26) and YR2019. The accuracy of B2015 using the ABL2021 was more stable as the aridity index increased, and the RMSE and MAE did not increase as much as for the YR2019 and P-T (α = 1.26).

The performance of G2021 was generally superior to that of B2015, and the differences in results caused by different E_po models for G2021 were significantly lower than those for B2015. Overall, G2021 also performed worst in alpine deserts and showed the best performance in sites covered by alpine meadows. In comparison with the three E_po models, similar to B2015, G2021 exhibited the best performance when using the ABL2021 to estimate E_po for all landscape types except farmland. For farmland, the best results were also obtained using P-T (α = 1.26). For G2021 with E_po estimated using P-T (α = 1.26), it showed poor applicability in alpine deserts and shrublands, and a better performance in grasslands and alpine meadows. Similar results were also found when using the YR2019 to estimate E_po for G2021. G2021 with ABL2021 exhibited relatively minor variations in its effectiveness as changes in land use types, with poor applicability in alpine deserts but acceptable applicability in grasslands, alpine meadows, mixed forests, broad-leaved forests, coniferous forests, farmland, and permanent wetlands.

In summary, all complementary functions are suitable for alpine meadows, but their performance is poor in alpine deserts and shrublands, which also affects the comparison of different functions in areas with large aridity index. From the comparison of results obtained using various potential evapotranspiration (E_po) models, G2021 is suitable for most underlying surface types. Overall, ABL2021 is more suitable for most land use types as it can reduce the uncertainty in E_po calculations in different complementary functions.

From Figure 8, we can find that when E_po was calculated using P-T with α = 1.26, G2021 showed better performance with evident lower RMSE and MAE than B2015. The differences of RMSE and MAE between H2018 and B2015 were quite limited. Using YR2019, the accuracy of G2021 was also better than that of B2015. The difference between B2015 and G2021 using ABL2021 was quite limited. These results also illustrated that the uncertainties of estimation of actual evaporation induced by different potential evaporation models are larger than different complementary functions.

In general, the parameter d shows low values at vegetation types in wet conditions and larger values at vegetation types in dry conditions, aligning with the previous theoretical analysis of G2021 and demonstrating the model's flexibility.

Therefore, these results indicate that both parameter c in B2015 and parameter d in G2021 are influenced by aridity conditions and vegetation types. The models used to calculate E_po can influence the trends, magnitudes, and the effects of influencing factors on these parameters. Considerable variation in complementary function parameters for the same type of underlying surface is detected. These findings suggest that the uncertainty introduced by the choice of E_po model may also have impacts on the parameter calibration of complementary functions.

E_po estimated by YR2019 shows larger values than the other methods because YR2019 considers that the outgoing long-wave radiation increases when the evaporating surface gets saturated. When the aridity increases, actual evaporation decreases, and the surface temperature increases, thus the outgoing long-wave radiation at the evaporating surface rises, and the net radiation decreases as a consequence. YR2019 adjusts the long-wave radiation and surface temperature to a state with a greater disparity from the current state, and in that state the surface temperature will be lower than the actual temperature and will be larger than the actual net radiation, resulting in a relatively maximum potential evaporation. It can also be noticed that the result estimated by P-T (α = 1.26) is close to that of YR2019.

The fundamental difference between P-T, YR2019, and ABL2021 lies in their underlying principles. The Bowen ratio is crucial for understanding the process of surface energy allocation (Andreas & Cash 1996). However, there are significant differences in the relationship between the Bowen ratio and the external environment variables across different models (Assouline et al. 2016). Actually, in the P-T model, the coefficient α is employed to adjust the equilibrium evaporation E_eq (the radiative term in Penman's equation) to the potential evaporation E_{po_PT}. However, the P-T model with α = 1.26 shows the incorrect relationship between surface temperature and Bowen ratio (Yang & Roderick 2019), and the E_po is not sensitive to surface temperature which results in a comparable value of P-T with YR2019 and ABL2021. The P-T model and ABL2021 assume that R_n is not affected by variations in the water supply (Aminzadeh et al. 2016). The YR2019 model, in contrast, argues that E_po declines as water supply increases, and takes into account the effect of evaporation, which varies with the degree of aridity, on net radiation and surface temperature. YR2019 empirically fitted a new Bowen ratio B_o relationship based on long-term global sea surface observations. It highlights the interdependence between surface temperature and radiation, therefore, E_po does not remain constant with changes in water availability. Yang & Roderick (2019) and Tu et al. (2023) argued that net radiation cannot be considered constant due to the changes in surface temperature and net long-wave radiation when the evaporation surface gets saturated and only the net short-wave radiation can be assumed as constant. For the same net solar radiation, drier (wetter) surfaces generally correspond to lower (higher) E and higher (lower) surface temperatures, which directly results in greater (lower) emitted long-wave radiation and therefore lower (greater) net radiation (Tu et al. 2023). This allows YR2019 to dispense with T_s or any related meteorological factors, relying solely on the ultimate external forcing (i.e., net solar radiation), and assuming that under typical conditions, the Bowen ratio B_o of a saturated condition decreases with increasing surface temperature. However, Szilagyi (2022, 2023) argued that when the evaporation surface get saturated, changes in air humidity, cloud coverage, and thickness may affect the net short-wave radiation which may compensate the changes in net long-wave radiation; therefore, the net radiation may still remain constant. More studies are required to further investigate the assumptions of different functions to clarify these above arguments.

Previous studies found that α represents the impacts of the vertical fluxes reaching the surface caused by the growing convective boundary layer. P-T and YR2019 mainly focus on the surface energy balance and do not explicitly consider the interactions between land surface and atmosphere system in the boundary layer. ABL2021 is mainly a gamble between humidity and surface available energy (Liu & Yang 2021), which may be one of the reasons why ABL2021 consistently shows the lowest variations (Figure 14). In ABL2021, the escape velocity g_e into the ABL box increases due to an increase in the surface latent heat flux (LE) and sensible heat flux (H), with their linearly correlated buoyancy fluxes F_B, indicating that dry air above the upper boundary enters the ABL. More dry air leads to a lower relative humidity (RH) in the ABL and an increase in the vapor pressure deficit (VPD), which reduces the buoyancy flux F_B. As the evaporating surface changes from moist to arid, the humidity decreases but the surface available energy is increasing. In the real world, it is hard to grasp and quantify the process of specific energy change for potential evaporation models such as P-T and YR2019. ABL2021 establishes a clear relationship for surface energy allocation (B_o) based on ABL theory, physically linking B_o to various atmospheric states (Liu & Yang 2021). It argues that the state of the lower atmosphere (i.e., air temperature and humidity) determines the surface energy fluxes, which in turn affects atmospheric status (Salvucci & Gentine 2013). Water surface evaporation is dominated by the VPD budget, thus connecting surface heat fluxes with lower atmospheric conditions. Although theoretically both air temperature (T_a) and specific humidity (Q) determine the water surface B_o, in practice, B_o is primarily determined by T_a due to the limited influence of RH (which represents the humidity state of the air and is generally unrelated to water surface T_a).

Opposed to P-T and YR2019, ABL2021 better considers the physical processes of interactions between the land surface and atmosphere system from the perspective of vertical fluxes on VPD changes and surface energy balance. Therefore, it does not contain coefficients, such as α, that need to be calibrated. It shows a potential toward a calibration-free method for E_po estimation and also reduces the uncertainties of the complementary principle.

The parameters in the E_po model may significantly affect its results. Among the three potential evaporation models used in this study, the YR2019 model has fixed parameters, so there is no need to discuss variations of parameters for this model. For the P-T model, the P-T coefficient α is set to 1.26, which is a recommended value based on previous studies (Priestly & Taylor 1972). However, variations in α may affect the E_po calculation. Since the P-T model calculates E_po by multiplying the P-T coefficient α with the radiation component E_rad, increasing or decreasing α will directly scale the E_po results proportionally. Supplementary Figure S1 shows the change of E_po with varying P-T coefficient α, illustrating a linear increase in E_po as α increases in the suggested range of 1.1–1.3 (Szilagyi et al. 2017). This demonstrates that the P-T coefficient α directly influences the estimation of potential evaporation.

Additionally, the performance of the three E_po models in Section 3.1 at the Taihu Lake sites was compared with the results of P-T (α = 1.1), as shown in Supplementary Figure S2. Compared with the P-T model with α set to 1.26, the bias is significantly reduced when α is set to 1.1, with a bias of 17.07%. Furthermore, as shown in Supplementary Table S2, the RMSE of E_po estimation in the P-T model with α = 1.1 is 29.29 mm, which represents an improvement in estimation accuracy compared with P-T (α = 1.26) and even lightly surpasses YR2019. However, it is still larger than the RMSE of ABL2021.

In the ABL2021 model, the key parameter is the product of the atmospheric boundary layer height (h) and the potential virtual temperature gradient (γ_v). In the supplementary materials of Liu & Yang (2021), it is explained that h and γ_v are not independent variables but are mutually constrained. Through a sensitivity analysis of hγ_v, they found that the ABL2021 model shows weak sensitivity to hγ_v. Furthermore, they emphasized that when the value of hγ_v is close to 6–7 K, the error of ABL2021 is minimized compared to observations. This suggests that if hγ_v falls outside this range, the accuracy of the results decreases. Therefore, 6.5 K is set as the value of hγ_v in this study to help to ensure a higher accuracy of the ABL2021 model. Overall, future research should take into account the variability of parameters in the E_po model to further explore improvements in the performance of complementary functions.

The polynomial model B2015 follows a boundary condition in which is considered and there is no formula for a specific E_po and E_pa, and the exponential function model G2021 follows the same boundary condition as B2015. However, the sigmoid function model H2018 (Han & Tian 2018) considers such a condition as is not always maintained, the boundary condition of H2018 considers is ⁠.

In B2015, the parameter c, due to the restricted range (−1,2), resulted in the poor effectiveness in some areas, such as sites covered by grasslands and needleleaf forests. In H2018, although the P-T coefficient α used to calculate does not appear in the input variables, α is still required in the estimation of the parameters m and n in H2018, and the parameter b in the advection-aridity function is also used in the estimation, with b not restricting its range. However, due to the involvement of the parameter α, the accuracy of H2018 is dependent on the value of α to a certain extent. In G2021, the parameter d is delineated as ⁠, which can be flexibly applied to more types of landscapes and different climate conditions.

For B2015, the best simulation of E was achieved when E_po was calculated using ABL2021, which showed better performance than other functions for different vegetation types. The change in its performance with the aridity index was also more stable. Meanwhile, the stability of the accuracy of the P-T equation with α taken as 1.26 was relatively weak compared to YR2023, which implies that the P-T equation did not obviously change in a regular manner with the change of the aridity index. For G2021, the best performance for each vegetation type was also achieved by ABL2021. Generally, comparisons between the three complementary functions show that G2021 showed better performance for actual evaporation estimation regardless of the method used for calculating E_po. The performance of G2021 varies steadily with the variations in aridity. In addition, G2021 is relatively applicable to most vegetation types.

This study investigated the applicability of different evaporation complementary formulas and potential evaporation models only at the monthly scale. The application of complementary functions shows different performances at different time scales. Particularly, the assumptions of different potential evaporation models also need to be examined for different time scales. Comparative studies on different complementary functions and potential evaporation models at daily and other time scales remain to be explored in the future.

There are limited numbers of sites for some vegetation types, such as permanent wetland, alpine meadow, and alpine desert, and such a limited amount of observed data may influence the assessment of the accuracy of complementary functions. The varying number of sites within each aridity index interval may result in a lopsided comparison of different functions. More data collections are required to evaluate the complementary formulas in the future.

In this study, we compared the performance of different potential evapotranspiration models and complementary functions based on 86 flux sites with different landscape types worldwide. We explored the influence of the accuracy of potential evaporation models on the performance of complementary functions. The following conclusions can be drawn:

• (1) The potential evaporation (E_po) estimated by the atmospheric boundary layer model (ABL2021) is closer to the observations than the P-T model and YR2019, and P-T and YR2019 overestimated the observations.

• (2) For the application in B2015 and G2021, ABL2021 shows better performance than YR2019 and P-T. ABL2021 reduces the differences for evaporation estimation of various complementary functions. ABL2021 can improve the accuracy of the complementary functions, and it is suitable for evapotranspiration estimation for most landscape types and different aridity conditions. It suggests that ABL2021 combined with complementary functions shows a potential toward a calibration-free method for evaporation estimations.

• (3) For any E_po models, G2021 performs best among the three complementary functions and it is applicable to most of the landscape types.

The results we obtained in this work may provide valuable reference for the improvement of complementary functions for actual evaporation estimation.

This work was supported by the National Natural Science Foundation of China (grant no. 41661144031).

All relevant data are available from an online repository or repositories.

The authors declare there is no conflict.