Han et al. (2012, 2018) proposed a generalized form of the complementary function that expresses E/Epen as a sigmoid function (SGC) of Erad/Epen: 1y=EEpen=11+mxmax-xx-xminnx=EradEpen, where xmax corresponds to the certain maximum value of x under extremely wet environments, and xmin corresponds to the certain minimum value of x under extremely arid environments. In this study, xmax and xmin were set as 1 and 0, respectively, for convenience. The Epen term is defined by Penman's equation (Penman, 1948, 1950), which can be expressed as 2Epen=Δ(Rn-G)Δ+γ+ρcpΔ+γκ2uln⁡z-d0z0mln⁡z-d0z0v(ea*-ea), where Δ (kPa C-1) is the slope of the saturation vapor curve at air temperature; Rn is the net radiation; G is the ground heat flux; γ (kPa C-1) is a psychrometric constant; ρ is the air density; cp is the specific heat; κ= 0.4 is the von Karman constant; u is the wind speed at measurement height; ea* and ea are the saturated and actual vapor pressures of air, respectively; z is the measurement height (Table S1); d0 is the displacement height; z0m and z0v are the roughness lengths for momentum and water vapor, respectively, which are estimated from the canopy height (hc; Table S1), d0=0.67hc, z0m=0.123hc, and z0v=0.1z0m (Monin and Obukhov, 1954; Allen et al., 1998). Erad is the radiation term of Penman evaporation: 3Erad=Δ(Rn-G)Δ+γ. The two parameters m and n of Eq. (1) can be determined by the Priestley–Taylor coefficient α and the asymmetric parameter b (Han and Tian, 2018). 4n=4α1+b-1x0.51-x0.5m=x0.51-x0.5n, where x0.5 is a variable that corresponds to y= 0.5 and equals 0.5+b-1α(1+b-1).

Brutsaert (2015) proposed the polynomial generalized complementary (PGC) function, which describes the relationship between E/Epa and Epo/Epa. We uniformed the independent variable as Erad/Epen to compare the two functions conveniently, and the polynomial function can be expressed as 5y=2-cα2x2-1-2cα3x3-cα4x4, where c is an adjustable parameter. When c= 0, Eq. (5) reduces to 6y=2α2x2-α3x3.

Typically, α has a default value of 1.26 (Priestley and Taylor, 1972). Since some studies have shown that a constant α may cause illogical results and biases in estimating E, it is suggested to specify α for diverse scenarios (Hobbins et al., 2001; Ma et al., 2015a; Sugita et al., 2001; Szilagyi, 2007). According to the complementary principle, under wet conditions, E is close to Epen and the Priestley–Taylor's evaporation (EPT=αErad). Specifically, when E/Epen is larger than a threshold (0.9 is commonly adopted), EPT can be considered to be approximately equal to the observed E; thus, α can be calculated by E/Erad (Kahler and Brutsaert, 2006; Ma et al., 2015a). In this study, α was calculated using this method based on the mean value of E/Erad under wet conditions (E/Epen>0.9). When all the E/Epen values are less than 0.9, α was set as the default value of 1.26. The key parameter b in SGC was calibrated by an optimization algorithm, with the objective function as the minimization of the mean absolute error (MAE) between the estimated E (by Eq. 1) and the observed E. Similarly, the key parameter c in PGC was calibrated by an optimization algorithm, with the objective function as the minimization of the MAE between the estimated E (by Eq. 5) and the observed E. Since we used the optimization algorithm to determine the parameter b in the SGC function, it is a fair choice to use the optimal c value instead of a constant value (c=0) in the PGC function.

To make the model parsimonious, we gave one value for the parameters (α, b and c) at each site for every different timescale. If the parameter was alterable, for example, it was month-dependent, and we would have to calibrate 12 parameters instead of one value for the whole study period. The purpose of this study is to determine the most suitable timescale for the complementary functions, and the variances of the key parameter within a timescale will introduce extra uncertainties. The accuracy will increase when an alterable parameter (that means a higher number of parameters) is used; however, the probability of overfitting risk will increase at the same time. In addition, in comparison to a group of parameters, a general representation of the parameter is more helpful in detecting its overall trend as the change in the timescale.

The eddy flux data analyzed in this study were obtained from the FLUXNET database (http://fluxnet.fluxdata.org; Baldocchi et al., 2001). Observations from a total of 88 sites around the world were analyzed. Detailed information on these sites is listed in Table S1. These sites were selected from the FLUXNET database because they have observations from a period longer than 5 years. The 88 sites include 11 IGBP (International Geosphere-Biosphere Programme) land cover classes: ENF, evergreen needleleaf forests (27 sites); EBF, evergreen broadleaf forests (8); DBF, deciduous broadleaf forests (13); MF, mixed forests (5); OSH, open shrublands (4); CSH, closed shrublands (1); WSA, woody savannas (3); SAV, savannas (4); GRA, grasslands (15); CRO, croplands (6); and WET, permanent wetlands (2). The climates of the 88 sites range from arid to humid. Among the 88 sites, 11 sites have mean annual precipitation levels lower than 200 mm, 47 sites have precipitation levels between 200–500 mm, and 30 sites have precipitation levels above 500 mm. A total of 11 sites are located in the Southern Hemisphere (i.e., Australia, Brazil, and South Africa), and the others are located in the Northern Hemisphere.

Variables including net radiation, sensible heat flux, latent heat flux, ground heat flux, wind speed, air temperature, air pressure, precipitation, relative humidity, and vapor pressure deficit were acquired from the daily, weekly, and monthly datasets on the FLUXNET website. We analyzed the observations in the growing seasons from April to September for the Northern Hemisphere and from October to March for the Southern Hemisphere. These study periods were selected to avoid the high biases caused by the low level of solar radiation or extremely low evaporation (≈ 0) during the non-growing season. The seasonal and annual data were acquired by averaging the monthly data of the growing seasons. Following Ershadi et al. (2014), the energy-residual-corrected latent heat fluxes were used, which means the residual term in the energy balance is attributed to the latent heat to force the energy balance closure. To investigate the influence of different residual correction methods, the Bowen ratio energy balance method was also adopted in the uncertainty analysis. In the Bowen ratio method, the residual term is attributed to sensible heat and latent heat by preserving the Bowen ratio (Twine et al., 2000). The latent heat, sensible heat, and available energy (Rn – G) are restricted to positive values (Han and Tian, 2018). The energy balance residual (W m-2) and energy balance closure ratio for each site are shown in Table S1.

The Nash–Sutcliffe efficiency (NSE; Legates and McCabe, 1999) is used to evaluate the efficiency of estimating E by the two generalized complementary functions: 7NSE=1-∑(E-Eest)2∑(E-E‾)2, where Eest (W m-2) is the estimated evaporation according to Eqs. (1) or (5), and E¯ is the mean value of E (W m-2).

The relationship between the estimated Eest (site mean values) based on the SGC function (Eq. 1) and the observed E at the 88 sites at multiple timescales is shown in Fig. 1. The regression equations and determination coefficients (R2) were calculated using the site mean results. Each dot in Fig. 1 represents the site mean result averaged by daily (Fig. 1a), weekly (Fig. 1b), monthly (Fig. 1c), and yearly (Fig. 1d) results, and the total observation number is 88 (sites) at each timescale. Most of the results are near the 1:1 line, and all the regression slopes are close to 1 with high R2 (0.95–0.99), which means the sigmoid function exhibits a good performance in estimating E at multiple timescales. The evaluation merits show that the performance varies at each timescale. The NSE values of the SGC functions for each site at different timescales are listed in Table S2. For the 88 sites, nearly half of the sites (40) have the highest NSE at the monthly scale, 12 sites have the highest NSE at the daily scale, 13 sites have the highest NSE at the weekly scale, and 23 sites have the highest NSE at the annual scale. The mean results of NSEH, RH2, and RMSEH (the subscript H corresponds to the sigmoid function proposed in Han and Tian, 2018) of these sites are shown in Table 1. RH2 represents the mean value averaged by the determination coefficients within each site. When the timescale changes from day to month, the mean NSEH increases from 0.33 to 0.55, and RH2 also increases from 0.61 to 0.75 (Table 1). However, they both decrease at the annual scale (NSEH= 0.18 and RH2= 0.61). These results indicate that the SGC function exhibits the highest skill at the monthly scale. We inferred that there is a trade-off between the random error and the number of observations. RMSEH values decrease from 24.56 W m-2 at the daily scale to 7.33 W m-2 at the annual scale, which means that the random error decreases as the timescale increases. At the same time, the fewer observations at the annual scale result in decreased variabilities of x and y, which affect the performance of the SGC function. On the other hand, Morton (1983) did not suggest using the complementary principle for short time intervals (e.g., less than 3 d), mainly considering the lag times associated with heat and water vapor change in the atmosphere, which may provide a possible inference for the weak performance at the daily scale.

In previous studies, the SGC function was mainly applied at the daily scale. For example, the results of Ma et al. (2015b) in the alpine steppe region showed that the NSE of the sigmoid function is 0.73 at the daily scale, which is equal to our mean value in the grassland (0.73±0.08). The RMSE (11.06 W m-2) is smaller than ours (16.36±1.48 W m-2). The mean NSE of the 20 EC sites from FLUXNET is 0.66 at the daily scale in Han and Tian (2018), approximately 2 times the result in this study, and the RMSE (18.6 ± 0.94 W m-2) is lower than our mean result of 88 sites (24.56 ± 0.95 W m-2).

The SGC function for the five selected sites of different ecosystem types is shown in Fig. 2 to show the performance at multiple timescales (red lines in Fig. 2). These five EC monitoring sites were selected because they have long-term observations (>10 years). The five sites include an evergreen needle forest (CA-TP1, Fig. 2a to d), a deciduous broad forest (US-UMB; Fig. 2e to h), a woody savanna (US-SRM; Figure 2i to l), a cropland (US-Ne2; Fig. 2m to p), and a grassland (US-Wkg; Fig. 2q to t). As observations decrease from the daily to the annual scale, the results converge on the middle part of the sigmoid curves and lie closer to the fitted lines. For some sites, the annual results concentrate on a narrow range with lower annual variabilities (e.g., Fig. 2h, l, and t). Generally, the key parameter (b) of the SGC function at these sites increases from the daily scale to the annual scale, which indicates that the sigmoid curves in the two-dimensional space of Erad/Epen-E/Epen move upwards. A detailed discussion about the variation in the parameters is provided in Sect. 3.4.

The relationship between the estimated Eest (site mean values) based on the PGC function (Eq. 5) and the observed E at the 88 sites at multiple timescales is shown in Fig. 3. The slopes of the regression increase from 0.9 to 1 as the timescale changes from day to month and further increase to 1.01 at the annual scale. The intercept terms decrease from 13.06 W m-2 at the daily scale to 0.01 W m-2 at the monthly scale and further decrease to -0.25 W m-2 at the annual scale. The R2 values increase from 0.83 to 0.99 as the timescale increases. These coefficients of the regression show that the PGC function exhibits the highest skill at the monthly scale. The NSE values of the PGC functions for each site at different timescales are listed in Table S2. For the 88 sites, 42 sites have the highest NSE at the monthly scale, 7 sites have the highest NSE at the daily scale, 14 sites have the highest NSE at the weekly scale, and 25 sites have the highest NSE at the annual scale. The mean values of NSEB, RB2, and RMSEB (the subscript B corresponds to the polynomial function proposed in Brutsaert, 2015) of these sites are shown in Table 1. When the timescale changes from day to month, NSEB increases from 0.19 to 0.50, and RB2 increases from 0.61 to 0.75. They decrease at the annual scale (NSE = 0.25 and RH2= 0.63). Again, these evaluation merits indicate that the PGC function also exhibits the highest skill at the monthly scale, which is the same as for the SGC function.

The PGC function has been applied at multiple timescales in previous studies. Zhang et al. (2017) evaluated the performance of the PGC function in estimating evaporation at four EC flux sites located across Australia, and their results showed that the mean RMSE (24.67 W m-2) and R2 (0.65) are close to our results (RMSE = 26.83 ± 1.16 W m-2 and R2= 0.61) at the daily scale. In Crago and Qualls (2018), the mean RMSE of seven EC sites at the weekly scale was 20.6 W m-2, and the mean R2 was 0.81, which are close to our mean results (RMSE = 19.17 ± 0.95 W m-2 and R2= 0.7).

The PGC functions for the five selected sites are also shown in Fig. 2 (green lines). The fitted lines are almost the same as those of the SGC function in most situations when x is not too high. However, they diverge from each other when x becomes larger. Finally, y exceeds 1 when x is larger than 1/α. Generally, the key parameter (c) of the PGC function at these sites decreases from the daily scale to the annual scale, which also indicates that the fitted curves move upwards.

The results from the 88 sites (Figs. 1, 3 and Table 1) show that the performances of the two functions are similar at monthly and annual timescales, while the SGC function performs slightly better than the PGC function at daily and weekly timescales. According to the results in Fig. 2, the two functions with calibrated parameters are approximately identical under non-humid environments, but their difference increases as x(Erad/Epen) increases. We found that the values of α for all sites are greater than 1.0 in our study, which means that the PGC model cannot work properly under the condition of 1/α<Erad/Epen<1.0. At daily and weekly timescales, a substantial number of ecosystems can produce very high Erad/Epen values. Specifically, 63 of the 88 sites have high Erad/Epen values (x>1/α) at the daily scale, and 24 sites have high values at the weekly scale. However, there are only three sites with an x>1/α at the monthly scale, and no site has that value at the yearly scale. For the SGC function, in super-humid conditions, the upper part of the sigmoid curve is nearly flat and closer to the observations (e.g., Fig. 2a, m, and n). However, for the PGC function, theoretically, it cannot be applied when x is over 1/α because the estimated Eest will be higher than Epen, which is illogical. Thus, the sigmoid function performs slightly better at daily and weekly timescales than the polynomial function. However, the difference vanishes at the monthly scale as few high Erad/Epen values occur.

According to the results, the performance of the PGC function is more sensitive to the time step than that of the SGC function. On the one hand, the regression relationship between Eest and the observed E of the 88 sites shows that the performance of the SGC function remains more stable (Fig. 1), while the regression results of the PGC function have higher variation when the timescale changes (Fig. 3). On the other hand, the estimation merits (Table 1) further confirm the sensitivity of the PGC function. From the daily scale to the monthly scale, the increase in NSEH is 0.22, while the increase in NSEB is 0.31; RMSEH decreases by 11.36 W m-2 (46 %), and RMSEB decreases by 13.13 W m-2 (49 %). At the daily scale, quite a few ecosystems (63 of 88 sites) can experience frequent high Erad/Epen (>1/α) values, and the PGC function does not have the ability to simulate E accurately in this situation (Eest>Epen), resulting in lower efficiency. We have carried out an additional analysis that adopts E=Epen for 1/α<Erad/Epen<1.0 in the PGC function, and the resultant NSEB (0.19 vs. 0.19) and RMSEB (26.83 W m-2 vs. 26.68 W m-2) present very similar results. As the timescale increases, the results converge on the middle part of the fitted line, and the number of high x greatly decreases (Fig. 2). Thus, the efficiency of the PGC function obviously increases. This is the reason that the polynomial function acts more sensitively to the time step.

In addition, we found that the two complementary functions perform reasonably well at shorter timescales (i.e., day and week), with relatively high R2 values. Additionally, the estimations of site mean evaporation at shorter timescales are accurate (Figs. 1 and 3), especially for the SGC function. These results suggest that the generalized complementary functions have the ability to estimate evaporation accurately, even at shorter timescales.

The key parameters of the two complementary functions (b of the SGC function and c of the PGC function) vary at multiple timescales (Fig. 2). To explore their changes, the values of 1/b and c at the 88 sites were averaged at each timescale. To take into account the situation in which b is equal to infinity, we used 1/b instead of b in this analysis. Figure 4 shows the change in the two complementary functions with varied parameters at multiple timescales. The averaged 1/b decreases from 0.45 ± 0.05 at the daily scale to 0.24 ± 0.03 at the annual scale (Fig. 4a), and the averaged c decreases from 0.98 ± 0.19 at the daily scale (Fig. 4b) to -0.37 ± 0.22 at the annual scale. The sign of c changes from positive to negative at the monthly scale.

We show the histograms of 1/b and c at multiple timescales in Figs. 5 and 6, respectively. At the daily scale, half of the 1/b values are lower than 0.3, and the mean value is 0.45 ± 0.05. At the weekly scale, the peak of the distribution moves left, and almost half of the 1/b values are lower than 0.2, with a mean value of 0.36 ± 0.04. At the monthly scale, the mean value is 0.29 ± 0.04, and the 1/b values continue to decrease. At the annual scale, the mean value decreases to 0.24 ± 0.03, and 61 % of the 1/b values are lower than 0.2. According to Fig. 6, at the daily scale, c follows a normal distribution (p value = 0.17, Kolmogorov–Smirnov test), with a mean value of 0.98 ± 0.21. Nearly 1/3 of the c values are lower than 0. At the weekly scale, the center of the distribution moves left, with a mean value of 0.43 ± 0.24. Half of the c values are lower than 0. At the monthly scale, the mean value is -0.04 ± 0.23, and 58 % of the c values are lower than 0. At the annual scale, the mean value decreases to -0.37 ± 0.25, and 63 % of the c values are lower than 0. These results support our conclusion that 1/b and c decrease as the timescale increases. Generally, the distribution of 1/b and c also moves left within each ecosystem type according to Figs. 5 and 6.

The reduction in 1/b and c indicates that the curves of the complementary functions move upwards as the timescale increases. Under non-humid conditions, the sigmoid function is a concave function, which means 812[fx1+fx2]>fx1+x22, where f is the concave function, and x1 and x2 represent any two values on the x axis. Since most of the results follow the fitted line, the averaged results of the longer time step will move upwards in the two-dimensional space of Erad/Epen-E/Epen, as well the new fitted curve. Although under super humid conditions, the SGC function is a convex function, there are fewer data under this condition as the timescale increases, and the shape of this part is almost unchanged (Fig. 4a). For the PGC function, when x is in the range of 0 to 1/α, most of it is a concave function. For example, in the situation where c is equal to 0, the second derivative is higher than 0 as long as x is lower than 2/3.

Furthermore, we found that the two key parameters b and c present a significant correlation, which provides additional evidence that the two functions can substitute each other in a sense. In other words, the two functions with calibrated parameters substantially provide similar descriptions of the distribution of the results in the state space (x=Erad/Epen, y=E/Epen). They can covert to each other in most situations since the two functions are generally equivalent to the linear asymmetric function when x is neither excessively large nor excessively small. The relationship can be described as follows: 1/b= 0.01c2+0.11c+0.24, with R2 being higher than 0.96 at the monthly scale (Fig. 7). The relationship remains at other timescales with a slight difference in the regression coefficients. At the daily scale, when c is equal to 0, the corresponding b is equal to 4.5, which is the same as that of the theoretical derivation in Brutsaert (2015).

In this study, the physical meaning of the Priestley–Taylor coefficient α, which represents the ratio of EPT (the Priestley–Taylor evaporation) and Erad with the default value of 1.26 (Priestley and Taylor, 1972; Brutsaert and Stricker, 1979), was retained. This fundamental definition of α may result in a smaller range of Erad/Epen in the PGC function. Liu et al. (2016) suggested that αe (the calibrated α with c=0) in the PGC function is only a weak analog of the Priestley–Taylor coefficient, and Brutsaert (2019) directly considered αe to be an adjustable parameter, which can be equal to or smaller than 1. We added the analysis that c is fixed to 0, and α is calibrated as αe. This analysis showed that the two methods provide similar results (mean RMSE = 14.99 W m-2 for αe vs. 16.67 W m-2 for α), and the conclusion of the timescale issue is consistent by adopting either α or αe in the analysis. The optimal αe has a significantly negative linear relationship with the optimal c, and the Pearson correlation coefficient is -0.8. This scenario suggests that calibrating either of the two parameters (αe and c) is equivalent (Han et al., 2012).