The complementary principle has been widely used to estimate evaporation
under different conditions. However, it remains unclear at which timescale
the complementary principle performs best. In this study, evaporation
estimations were conducted at 88 eddy covariance (EC) monitoring sites at
multiple timescales (daily, weekly, monthly, and yearly) by using sigmoid
and polynomial generalized complementary functions. The results indicate
that the generalized complementary functions exhibit the highest skill in
estimating evaporation at the monthly scale. The uncertainty analysis shows
that this conclusion is not affected by ecosystem type or energy balance
closure method. Through comparisons at multiple timescales, we found that
the slight difference between the two generalized complementary functions
only exists when the independent variable (

Terrestrial evaporation (

Recently, the complementary principle, as one of the major types of

The prerequisite of the complementary principle is adequate feedback between
the land surface and the atmosphere, which results in an equilibrium state.
In this situation, the wetness condition of the land surface can be largely
represented by the atmospheric conditions. Therefore, the timescales used
in the complementary principle need to satisfy the adequate feedback
assumption. However, this issue involves the complex processes of
atmospheric horizontal and vertical motion, and these processes are
difficult to explain theoretically. Morton (1983) noted this problem earlier
and suggested that the complementary principle is not suitable for short
timescales (e.g., less than 3 d), mainly because of the potential lag
times associated with the response of energy and water vapor storage to
disturbances in the atmospheric boundary layer. However, there is no solid
evidence or theoretical identification to support this inference. The
original complementary relationship and the AA function are not limited by
applicable timescales. In the derivation of the advanced generalized
complementary functions (SGC of Han and Tian, 2018, and PGC of Brutsaert,
2015), no specific timescale is defined. In practice, the complementary
principle has been widely adopted to estimate

In previous studies, the model validations were mostly completed at the
daily scale (Brutsaert, 2017; Han and Tian, 2018; Wang et al., 2020), and the
datasets of evaporation estimation were often established at the monthly
scale (Ma et al., 2019; Brutsaert et al., 2020). However, each study only
focused on a single timescale. In this study, we assessed the performance of
the complementary functions in evaporation estimation at multiple timescales (daily, weekly, monthly, and yearly). The assessment was carried out
at 88 EC monitoring sites with

This paper is organized as follows: Sect. 1 briefly describes the development of the complementary theory and our motivations to investigate the timescale issue. Section 2 describes the two functions, the parameter calibration method, and the data sources and processing. Section 3 shows and discusses the performance of the complementary functions at multiple timescales, the dependence of the key parameters on timescales, and the uncertainties in the analysis. The conclusions are given in Sect. 4.

Han et al. (2012, 2018) proposed a generalized form of the complementary
function that expresses

Brutsaert (2015) proposed the polynomial generalized complementary (PGC)
function, which describes the relationship between

Typically,

To make the model parsimonious, we gave one value for the parameters
(

The eddy flux data analyzed in this study were obtained from the FLUXNET
database (

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 (

The Nash–Sutcliffe efficiency (NSE; Legates and McCabe, 1999) is used to
evaluate the efficiency of estimating

The relationship between the estimated

The estimated evaporation based on the SGC function (Eq. 1)
vs. the observed site mean evaporation at the daily scale

The evaluation merits (NSE,

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 (

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 (

Plots of

The relationship between the estimated

As in Fig. 1 except for the PGC function (Eq. 5).

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

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

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

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

In addition, we found that the two complementary functions perform
reasonably well at shorter timescales (i.e., day and week), with relatively
high

The key parameters of the two complementary functions (

Plots of the SGC Eq. (1) with

We show the histograms of

Distribution of the key parameter

Distribution of the key parameter

The reduction in

Furthermore, we found that the two key parameters

Relationships between

In this study, the physical meaning of the Priestley–Taylor coefficient

The evaluation merits of the generalized complementary functions may differ
among ecosystem types. However, our results show that such variation
generally does not affect our conclusion that the complementary functions
perform best at the monthly scale. We show the performance of the two
functions at multiple timescales for each ecosystem type in Table S3.
Generally, the SGC function and the PGC function perform best at the monthly
scale in most ecosystem types (9 of 11), with the highest NSE and

In consideration of the substantial discrepancy between the monthly results
and the annual results, we added an analysis at the seasonal scale, which is
between the two time steps. The relationship between the estimated

In addition, we also tested the influence of the different energy balance closure methods. The results based on both the energy residual (ER) closure correction (e.g., Ershadi et al., 2014; Han and Tian, 2018) and the Bowen ratio (BR) closure correction support our conclusion that the generalized complementary functions perform best at the monthly scale (Table S4).

In this study, evaporation estimations were assessed at 88 EC monitoring sites at multiple timescales (daily, weekly, monthly, and yearly) by using two generalized complementary functions (the SGC function and the PGC function). The performances of the complementary functions at multiple timescales were compared, and the variation in the key parameters at different timescales was explored. The main findings are summarized as follows:

The sigmoid and polynomial generalized complementary functions exhibit
higher skill in estimating evaporation at the monthly scale than at the
other evaluated scales. The highest evaluation merits were obtained at this
timescale. The accuracy of the complementary functions highly depends on
the calculation time step. The NSE increases from the daily scale (0.26,
averaged by NSE

The SGC function and the PGC function are approximately identical under
non-humid environments, while the SGC function performs better under super
humid conditions implied by high values of

The key parameter

In this study to determine the most suitable timescale for applying the
complementary principle, the key parameters (

All the data used in this study are from FLUXNET (

The supplement related to this article is available online at:

SH and FT designed the experiments, and LW carried them out. LW developed the model code and performed the simulations. LW prepared the paper with contributions from all co-authors.

The authors declare that they have no conflict of interest.

We are grateful for the financial support from the National Science Foundation
of China (NSFC 51825902, 51579249, 52079147), the Ministry of Science and
Technology of the People's Republic of China (2016YFC0402701), and the State Key
Laboratory of Simulation and Regulation of Water Cycle in River Basin, China
Institute of Water Resources and Hydropower Research (SKL2020ZY06). We thank
the scientists of FLUXNET (

This research has been supported by the National Science Foundation of China (grant nos. NSFC 51825902, 51579249, and 52079147), Ministry of Science and Technology of the People's Republic of China (grant no. 2016YFC0402701) and State Key Laboratory of Simulation and Regulation of Water Cycle in River Basin, China Institute of Water Resources and Hydropower Research (grant no. SKL2020ZY06).

This paper was edited by Marnik Vanclooster and reviewed by three anonymous referees.