Temporal Upscaling of Agricultural Evapotranspiration with an Improved Evaporative Fraction Method

Abstract

Evapotranspiration (ET) is a crucial parameter for agricultural management and the hydrologic cycle, and instantaneous satellite images are the primary data source for regional ET. The constant evaporative fraction method (EFO) is a common approach for converting short-time ET (ET_st) to daily ET (ET_day). However, EFO has some limitations due to simple assumptions, including the following: the short-time evaporative fraction (EF_st) equals the daily evaporative fraction (EF_day). This study proposed an improved evaporative fraction method (EFI) through theoretical derivation and data analysis without additional data requirements, enabling the accurate upscaling of ET_st to ET_day. The vapor pressure deficit and available energy were considered in EFI to describe the main effect factor and estimate the deviation between EF_st and EF_day, defining the deviation coefficient and potential deviation between EF_st and EF_day. EFI was tested through four aspects: different agricultural systems, various sites, two growth stages, and different sources of EF_st, comparing estimated ET_day from EFI and measured ET_day. EFI reduced the mean absolute percentage error (MAPE) of ET_day estimation from 23% to 16% when EF_st is derived from measured data compared to EFO. Similarly, the MAPE of ET_day estimation reduced from 38% to 31% when EF_st is derived from a remote sensing model (Surface Energy Balance Algorithm for Land, SEBAL). EFI performs better during the growing period than the fallow season, providing critical information for irrigation practices. Crop type is not a main control factor for the relationship between η (ratio between VPD and R_n-G) and EF_st, and EFI is adaptable to various agricultural systems. The encouraging results of EFI in different scenarios demonstrate its accuracy and robustness. Therefore, EFI is anticipated to upscale EF_st to EF_day, generating a more accurate ET on a regional scale through remote sensing technology.

1. Introduction

Some empirical models were proposed based on satellite images to simulate the daily ET (ET_day). Jackson et al. [1] estimated the ET_day via the temperature difference between canopy and air, which is simple but requires measured data to calibrate. Price [2,3] deduced the relationship between ET_day and the instantaneous surface temperature from the heat flow equation. Based on studies of Seguin and Itier [4], Carlson et al. [5] introduced the average bulk conductance for daily integrated sensible heat flux and correction for nonneutral static stability and established a simplified method for integrating ET_day. These empirical methods provide excellent ET estimation for specific conditions and areas. Unfortunately, these methods have limited applicability in ET_day estimation because some parameters vary for different regions.

Instantaneous ET was calculated more and more accurately with the development of remote sensing technology and theory, and it was extrapolated according to the similarity between the diurnal course of ET and that of other components of the surface energy balance [6]. Jackson et al. [7] assumed the diurnal course of ET follows the diurnal course of radiation for cloud-free days, depicting it with a sine function. However, the practicability of this method was restricted by the topography, weather conditions, and time of instantaneous EF. According to the self-preservation of the reference ET fraction, Allen et al. [8] and Tasumi et al. [9] upscaled the instantaneous ET to ET_day through the ratio of actual ET and alfalfa reference ET. The surface resistance method was another excellent approach for ET_day estimation [10], using the instantaneous surface resistance to replace the daily surface resistance, representing a good performance. However, determining the parameters is a challenge, therefore constraining the application.

The evaporative fraction was defined as the latent heat flux divided by the available energy flux [11]. The constant evaporative fraction during a day was a crucial hypothesis in the constant evaporative fraction method (EFO) [6], which was widely applied for converting the short-time ET to ET_day [12,13,14,15,16,17]. However, some studies [18,19,20,21,22] proved that the ET_day would be underestimated or overestimated when the EFO was adopted. Many efforts [23,24,25,26] have been made to improve the calculation accuracy, considering more circumstance factors. These methods have acquired acceptable results in different regions, but the additional data input might constrain the application.

Based on the deficiencies of previous methods, an improved evaporative fraction method (EFI) is proposed, considering vapor pressure deficit (VPD, expressed by air temperature and surface vapor pressure) and available energy (τ). The VPD and τ are essential parameters in the short-time EF calculation from remote sensing models, so EFI does not require more data than EFO. The objectives of this study were (1) to develop an improved evaporative fraction method without additional data requirements; (2) to test this method in different agricultural systems and for different sites; and (3) to assess the performances of EFI when the short-time EF is obtained from a remote sensing model and ground-measured data.

2. Materials and Methods

2.1. Experiment and Data Collection

2.1.1. Eddy Flux and Micrometeorological Measurements

The field experiment was conducted from 2017 to 2019 in a paddy field (28.45°N, 116.01°E) located in south China where the mean annual temperature is 18.1 °C and annual precipitation is over 1600 mm. The early paddy rice was seeded in late April and harvested in mid-July, followed by transplantation of late paddy rice in late July and late paddy rice harvesting in late October or early November. The paddy rice field flooded during all rice growth periods besides the late tillering and late maturity growth stages (dry condition).

An eddy covariance (EC) system was installed in the center of the paddy field to observe the energy flux, equipped with a fast response 3D sonic anemometer (WindMasterPro, Gill Instruments Inc., Lymington, UK), an open path H₂O gas analyzer (LI-7500A, Li-COR Biosciences Inc., Lincoln, NE, USA), and a data logger (LI-7550, Li-COR Biosciences Inc., Lincoln, NE, USA). The EC system and meteorological sensors were installed at 2.5 m above ground. The meteorological data were collected with a meteorological system, including an air temperature and humidity sensor (HMP155A, Vaisala Instruments Inc., Helsinki, Finland) and a net radiation sensor (NR-lite2, Kipp&Zonen Instruments Inc., Delft, The Netherlands). Meanwhile, soil moisture and temperature sensors (ML2x, Delta-T Devices Inc., Burwell, UK) and a soil heat flux plate (HFP01, Hukseflux Instruments Inc., Delft, The Netherlands) were buried at a depth of 5 cm, and the daily water depth was obtained by rulers.

To evaluate the performance of EFI, the short-time EF (EF_st) from EC systems and the Surface Energy Balance Algorithm for Land (SEBAL) model were used to calculate the daily ET, respectively. We initially collected EC data from all agricultural monitoring systems available on the FLUXNET community (https://fluxnet.org (accessed on 12 November 2024)) and AsiaFlux (http://asiaflux.net (accessed on 14 November 2024)). However, several sites were excluded due to either incomplete records of critical parameters (e.g., net radiation data) or insufficient ancillary information regarding crop types and cultivation schedules. Following rigorous data quality screening and metadata completeness verification, 15 representative sites (Figure 1 and Appendix A) were ultimately selected. The EC systems were installed in agricultural systems, including winter wheat, winter barley, spring barley, soybean, cowpea, sugar beet, rapeseed, mustard, maize, paddy rice, potato fields, and orange orchard, and providers have preprocessed the flux data. The FLUXNET community and AsiaFlux provide 30 min or 60 min flux data, so the short-time data refers to hourly or half-hourly data for EC data analysis and instantaneous data for remote sensing data analysis in this study. In addition, the half-hourly data were aggregated to daily data, and the data processing procedures can be found in Wei et al. [27].

2.1.2. Remote Sensing Data Collection

For the ET estimation using remote sensing technology, many satellites (i.e., MODIS, Landsat, NOAA, Sentinel-2A) have been highly considered by researchers [28,29]. Although some sensors (i.e., WorldView series) can provide higher spatiotemporal resolution, the high price blocks its prevalence. This study adopted Landsat as the input of the remote sensing model, with a relatively high spatial resolution (30 m) and a long record (from 1987), which could be used for long-time monitoring. Quality bands of Landsat images were used to remove the bad pixels with bit arithmetic [30]. The Landsat images and digital elevation model datasets (DEM), which were used to correct the land surface temperature (Equation (D10)), with 30 m spatial resolution, were collected from the United States Geological Survey (https://www.usgs.gov/ (accessed on 28 November 2024)). The European Centre for Medium-Range Weather Forecasts Reanalysis v5 (ERA5) dataset was acquired from medium-range weather forecasts (http://www.ecmwf.int/ (accessed on 28 November 2024)), providing raster meteorological data (air temperature at 2 m, net radiation, and wind speed at 2 m) with hourly temporal resolution and 0.1° spatial resolution. Those datasets were used to calculate the EF_st and ET_day through remote sensing technology. All remote sensing raster data whose spatial resolution was not 30 × 30 m were resampled to 30 × 30 m using cubic interpolation [27].

2.2. Data Preprocessing

The raw data (10 Hz) from EC were processed with EddyPro software (Version 7.0.9), and the 30 min interval initial latent (LE) and sensible (H) heat flux were obtained. The low-quality data flagged by EddyPro was removed, which was influenced by rainfall, instrument malfunction, and other disturbances. Meanwhile, the data with a friction velocity lower than 0.15 m·s⁻¹, which is unreliable [31], was also removed. ReddyProc, an online tool (https://www.bgc-jena.mpg.de/ (accessed on 15 November 2024)), was used for half-hour data gap-filling. The data from January to April 2019 were missing due to equipment failure. The difference between net radiation and soil heat flux (R_n-G) should be equal to turbulent energy (H + LE) according to the energy balance principle [32], and the Bowen ratio (β) was adopted to force energy balance closure [33]. For this method, the residual energy (R_n-G-LE-H) can be partitioned into additional H and LE depending on β. The sum of additional H or LE and original H or LE is the closed H or LE. The specific procedure can be found in Appendix B. The soil heat flux (G) calculation method was as follows:

$$G=G_0+G_{\mathrm{s}}+G_{\mathrm{w}}$$

(1)

$$G_{\mathrm{s}}={\displaystyle \frac{\Delta T_{\mathrm{s}}}{t}}{d}_{\mathrm{s}}({\rho }_{\mathrm{s}}{C}_{\mathrm{s}}+{\rho }_{\mathrm{w}}{C}_{\mathrm{w}}{\theta }_{\mathrm{w}})$$

(2)

$$G_{\mathrm{w}}={\displaystyle \frac{\Delta T_{\mathrm{w}}}{t}}{d}_{\mathrm{w}}{\rho }_{\mathrm{w}}{C}_{\mathrm{w}}$$

(3)

where G₀ is the measured data by soil heat flux plate, W·m⁻²; G_s and G_w represent changes in soil and water heat storage, respectively, W·m⁻²; ΔT_s and ΔT_w is the variation in soil and water temperature, respectively, K; d_s is the depth of layer above soil heat flux sensors, 5 cm; d_w is the water depth, cm; ρ_s is the soil dry bulk density, 1360 kg·m⁻³; ρ_w is water density, 1000 kg·m⁻³; θ_w is the soil water content; and C_s and C_w are the specific heat capacity for soil solid portion (840 J·kg⁻¹·K⁻¹) and water (4190 J·kg⁻¹·K⁻¹).

2.3. Improved Evaporative Fraction Method

The EF_st equals the daily EF (EF_day) according to the constant evaporative fraction method (EFO) [6], so ET_day can be obtained from daily available energy (R_n-G) and EF_st. Unfortunately, EF is not a constant intra-day, which is influenced by many factors [18,23]. Considering the effects of circumstance, the improved evaporative fraction method (EFI) was proposed, including three core procedures. (1) The main control factors of EF were analyzed according to Penman–Monteith. The deviation coefficient (δ) between EF_st and EF_day was defined as the main influence factors of EF. (2) The determination of the potential difference (Ω) between daily EF_day and EF_st. (3) The product of δ and Ω was defined as the deviation between EF_day and EF_st. Then, the EF_day was obtained by finding the sum of the deviation and EF_st. The specific procedure and flowchart of EFI are shown in Figure 2.

2.3.1. The Deviation Coefficient Between EF_day and EF_st

From the PM equation [33], EF can be described by a group of meteorological factors as follows.

$$LE={\displaystyle \frac{\Delta (R_{\mathrm{n}}-G)+c_p{\rho }_{\mathrm{a}}\frac{e_{\mathrm{s}}-e_{\mathrm{a}}}{r_{\mathrm{a}}}}{\Delta +\gamma (1+\frac{r_{\mathrm{s}}}{r_{\mathrm{a}}})}}$$

(4)

$$EF={\displaystyle \frac{\Delta }{\Delta +\gamma (1+\frac{r_{\mathrm{s}}}{r_{\mathrm{a}}})}}+{\displaystyle \frac{c_{\mathrm{p}}{\rho }_{\mathrm{a}}}{r_{\mathrm{a}}\Delta +\gamma (r_{\mathrm{a}}+r_{\mathrm{s}})}}\times {\displaystyle \frac{VPD}{R_{\mathrm{n}}-G}}$$

(5)

where Δ represents the slope of the saturation vapor pressure curve, kPa·°C⁻¹; c_p represents the air-specific heat at constant pressure, kPa·°C⁻¹; ρ_a is the air density, kg·m⁻³; e_s represents saturation vapor pressure, kPa; e_a represents the actual vapor pressure, kPa; e_s − e_a represents saturation vapor difference (VPD), kPa; r_a represents the aerodynamic resistance, s·m⁻¹; r_s represents surface resistances, s·m⁻¹; and γ is the psychrometric constant, kPa·°C⁻¹. The intra-day γ, c_p, and ρ_a would not change sharply [34], and so they are regarded as the constants in this study.

Given the EF stability (explained hereinafter) and satellite overpass time, the flux data from daytime (referring to 9:30–14:30 in this study) were analyzed. Meanwhile, the EF_day of a certain location is estimated from the EF_st of the location, so the subsequent variable analysis is conducted on the daytime (9:30–14:30) of the specific location, not the regional scale or a long period.

Many studies [35,36] have proven that r_s varies with VPD, temperature, soil water content, and canopy structures. For a short time, the soil water content and canopy structures will not change sharply. Temperature is always associated with net radiation, and VPD was considered in the η (ratio between VPD). In a brief time frame, solar radiation is the key factor in r_s [37], and the threshold of solar radiation controls the lowest r_s. To be specific, r_s will not decrease continuously when solar radiation reaches the threshold value (relatively high value) [38,39], so r_s shows a stable value during the daytime. The target time of this study is cloud-free time when solar radiation is strong, and r_s might be stable intra-daytime. Meanwhile, many studies [35,36,40,41] have proved that r_s maintains a relatively stable value during the daytime for different crops and meteorological conditions. So, r_s was regarded as a constant in this study.

The r_a can be calculated with crop height and wind speed [34].

$$r_{\mathrm{a}}={\displaystyle \frac{\ln (\frac{z_{\mathrm{m}}-d_0}{z_{\mathrm{om}}})\ln (\frac{z_{\mathrm{h}}-d_0}{z_{\mathrm{oh}}})}{k^2{u}_{\mathrm{z}}}}$$

(6)

$$d_0=0.667h;z_{\mathrm{om}}=0.123h;z_{\mathrm{oh}}=0.1z_{\mathrm{m}}$$

(7)

where z_m is the height of wind measurements; z_h is the height of humidity measurements; d₀ is the zero plant displacement height; z_om is the roughness length governing momentum transfer; z_oh is the roughness length governing the transfer of heat and vapor; k is the von Karman’s constant, 0.41 [34]; u_z is the wind speed at height z; and h is the crop height. z_m and z_h would be the constant if the locations of the sensors were not disturbed by human or non-human factors. Meanwhile, the crop height will not change sharply in one day. So, r_a is just influenced by wind speed during 9:30–14:30 in the same place. Bailey and Davies [42] have illustrated that the ET is insensitive to r_a when the PM equation is applied to estimate the ET.

To achieve a tradeoff between model complexity and accuracy, the effect of wind speed (wind speed and r_a during daytime were analyzed in Appendix C) on r_a was not factored, and r_a was treated as a constant throughout the daytime. Δ can be described by the air temperature, and e_s is a temperature variable. The influence of Δ fluctuation on EF is ignored during the day to simplify the calculation procedure, just considering one temperature variable: e_s.

While we acknowledge that r_a and r_s may vary in longer time scales, our analysis focused specifically on the stable daytime period, which justifies the assumption of constant r_a and r_s for this short time frame. In conclusion, although we recognize that these assumptions might introduce some uncertainties, they were made to simplify the model and reduce its complexity, while still capturing the key dynamics of evapotranspiration during the stable daytime period. So, Equation (5) can be rewritten as follows.

$$EF=d+k\times \eta$$

(8)

$$d={\displaystyle \frac{\Delta }{\Delta +\gamma (1+r_{\mathrm{s}}/r_{\mathrm{a}})}};k={\displaystyle \frac{c_p{\rho }_{\mathrm{a}}}{r_{\mathrm{a}}\Delta +\gamma (r_{\mathrm{a}}+r_{\mathrm{s}})}};\eta ={\displaystyle \frac{VPD}{R_{\mathrm{n}}-G}}$$

(9)

where d and k are the constants of the EF function (Equation (8)); η is the variable of the EF function (Equation (8)). So, EF could be described by VPD and available energy (R_n-G, τ), as suggested in other studies [43,44,45]. For good operability and data accessibility, the deviation coefficient (δ) and potential deviation (Ω) between EF_day and EF_st were used to determine the deviation between EF_day and EF_st, which are explained hereinafter. So, the deviation coefficient (δ) was defined as the η relative deviation between daily and short time.

$$\delta ={\displaystyle \frac{{\eta }_{\mathrm{day}}-{\eta }_{\mathrm{st}}}{{\eta }_{\mathrm{day}}}}$$

(10)

where subscript day and st represent the daily and short-time data (including instantaneous data, hourly data, and half-hourly data), respectively.

2.3.2. The Potential Deviation Between EF_day and EF_st

Potential deviation (Ω) is the possible error when the EF_st is used to express EF_day in the EFO. Ω should theoretically be the maximum EF_day or EF_st. Nevertheless, it is hard to say that the deviation maintains the maximum every moment and EF_day is the target value, so we just use the EF_st to describe the Ω. An adjustment coefficient (t) was introduced to depict the Ω for most conditions in one agricultural system, and Ω was defined as follows when EF_st ≤ 1.

$$\Omega =tE{F}_{\mathrm{st}}$$

(11)

where t represents the adjustment coefficient related to crop type and meteorology, analyzed hereinafter.

2.3.3. Daily EF

Given that the EF_st during the daytime is relatively stable [6] and close to EF_day [46,47], the EF_st was retained in EFI, and the deviation between EF_st and EF_day needed to be determined to obtain a more accurate EF_day through EF_st.

Ω represents the possible maximum deviation between EF_st and EF_day, not the actual deviation between EF_st and EF_day, so the deviation coefficient (δ) was introduced to determine the relative deviation between EF_st and EF_day. The actual deviation between EF_st and EF_day could be described as the product of δ and Ω (Equation (12)).

The EF_day from EFI was used to estimate the daily ET with ground-measured data and a remote sensing model (SEBAL, which needs to input the remote sensing images and conventional meteorological data, and the specific procedure and inputs can be found in Appendix D), respectively.

$$E{F}_{\mathrm{day}}=E{F}_{\mathrm{st}}+\delta \cdot \Omega$$

(12)

$$\lambda E{T}_{\mathrm{day}}=LE=E{F}_{\mathrm{day}}{(R_{\mathrm{n}}-G)}_{\mathrm{day}}$$

(13)

where λ is the latent heat of vaporization, 2.45 MJ·kg⁻¹.

2.4. Validation Methods

To assess the performance of EFI, two sources of short-time EF were adopted: ground-measured data (hourly or half-hourly EF from ground-measured data) and a remote sensing model (instantaneous EF from SEBAL). The averaged ET derived from a 3 × 3 pixel window centered on the flux tower (covering its most flux footprint area) was directly compared with the eddy covariance (EC) measurements to evaluate the consistency between remote sensing estimates and ground-based observations [27].

The root mean squared error (RMSE), mean absolute percentage error (MAPE), determination correlation (R²), and agreement index (AI) [48,49] were adopted in this study to evaluate the EFI performance in different situations.

$$RMSE=\sqrt{{\displaystyle \frac{1}{N}}{(Y_i-f(x_i))}^2}$$

(14)

$$MAPE={\displaystyle \frac{1}{N}}{\displaystyle \sum \left|Y_i-f(x_i)\right|}$$

(15)

$$R^2=1-{\displaystyle \frac{{\displaystyle \sum {(Y_i-f(x_i))}^2}}{{\displaystyle \sum {(Y_i-\overline{Y})}^2}}}$$

(16)

$$AI=1-{\displaystyle \frac{{\displaystyle \sum {\left(Y_i-f(x_i)\right)}^2}}{{\displaystyle \sum {\left(\left|Y_i-\overline{Y}\right|+\left|f(x_i)-\overline{Y}\right|\right)}^2}}}$$

(17)

where Y_i is the measured data, f (x_i) is the estimated value, N is the number of data, and $Y(—)$ is the mean measured value.

3. Results

3.1. Measured Diurnal Evaporative Fraction for Agricultural Systems

The low R_n-G, amplifying the relative sensitivity of evapotranspiration (ET) to rapid variations in soil moisture, VPD, and turbulent mixing, would cause wild fluctuating EF [50,51], and so EF_st from 8:00 to 16:00 (relatively high R_n-G) from EC datasets were analyzed. EF is more affected by the soil water condition (irrigation practice) and available energy (hard to describe quantitatively with limited sites), so the EF variation was displayed according to crop types (Figure 3). Besides rapeseed (Brassica napus subsp. napus), the other crops have higher EF_st during the growth period (GP) than during the fallow season (FS). An adequate water supply (e.g., irrigation and precipitation) and enough available energy resulting in higher evaporation and transpiration during GP might be the reason for higher EF [23,25]. The rapeseed is planted in DE_Kli and US_ARM stations, the crop rotation land, and the FS always concentrates on July to October when the radiation (R_n is 104 W·m⁻²) and precipitation (2.8 mm·d⁻¹) are enough. So, the evaporation condition is favorable for rapeseed fields even during FS. That is why the EF_st is similar during FS and GP. The EF_st of paddy rice fields is higher than that of upland fields because of the ample water supply.

EF is the ratio of LE and R_n-G, and EF_st was more stable and lower at midday than at other times when the R_n-G is typically the largest. With the decrease of R_n-G (denominator of EF), the EF_st becomes sensitive to R_n-G and increases gradually. Due to the data quality limitation, only 80 days of data are available during FS of sugar beet, and the time span is from November to April. So, the EF_st shows a more significant fluctuation during this period. Besides the FS of sugar beet, EF_st between 9:30 and 14:30 is stable for other crops, and the lower R_n-G (denominator of EF) during other times might lead to greater deviation between short-time and daily EF [46,52]. Meanwhile, the overpass time for Landsat is also between 9:30 and 14:30. Every half hour (or hour), EF_st between 9:30 and 14:30 was adopted in the following analysis for a reliable result.

3.2. Daily EF from the Improved Evaporative Fraction Method

Every half-hourly and hourly EF during 9:30–14:30 from EC was upscaled to daily EF (00:00–23:59) through the improved evaporative fraction method (EFI) in different agriculture systems. To determine the appropriate adjustment coefficient of Ω (t), MAPE between upscaled EF with EFI and measured daily EF were analyzed, with initial t values 0.1 and 0.01 as the step size. The MAPE shows a similar trend for different agricultural systems, dropping first, then spiking with the increase in t (Figure 4). The MAPE of EFI, with high-value t for a few agricultural systems, might exceed that of EFO. The optimum t (with the smallest MAPE) is between 0.29 and 0.80 for different agricultural systems and concentrates on 0.40–0.60. The suggested values of t for different agricultural systems were given as follows: winter wheat (0.52), winter barley (0.67), spring barley (0.40), soybean (0.34), cowpea (0.48), sugar beet (0.29), rapeseed (0.56), mustard (0.80), maize (0.49), paddy rice (0.57), potato (0.41), and orange (0.47). The optimum t for different agricultural systems was adopted in the following analysis.

The performances of EFI and EFO in EF_day estimation were analyzed through four indexes (Figure 5), with one complete growth period (containing one growth period and one fallow season) as the minimum time unit. Compared to EFO, EFI shows a higher accuracy in EF_day estimation, with lower RMSE (the average RMSE drops from 0.33 to 0.26) and MAPE (the average MAPE drops from 30% to 22%) in different agricultural systems. The upscaled EF_day from EFI has a better consistency with measured EF_day than from EFO, with higher R² and AI. When the EFO has relatively greater deviations, the EFI could diminish the error between upscaled EF_day and measured EF_day more obviously (e.g., winter wheat, winter barley, rapeseed, mustard, and paddy rice). In addition, the results from the two methods when using EF_st from the remote sensing model for different crops were evaluated. Figure 6 shows that the EFI also performed better than EFO when the EF_st from the remote sensing model. Overall, EFI consistently outperforms EFO in EF estimation.

The Wilcoxon signed ranks test [53] was conducted to assess the significant differences between the performances of the two methods (Figure 7) when short-time EF was obtained from ground-measured data. EFI shows significant differences from EFO at the 0.0001 level for the four metrics, demonstrating that EFI outperforms EFO in EF_st upscaling. It should also be noted that upscaled EF_day from EFI and EFO during the growth period (GP) works better than during the fallow season (FS), with a lower error (RMSE and MAPE) and higher consistency (AI and R²). The lower available energy during the FS than the GP, resulting in more fluctuating hourly or half-hourly EF during FS [54], could be one of the reasons why EFI and EFO perform better during GP than during FS.

3.3. Daily ET from Daily Evaporative Fraction

The specific procedures for ET_day estimation with EFI or EFO are as follows. (1) short-time EF calculation (ground-measured or SEBAL); (2) Estimating the δ and Ω; (3) EF_day and ET_day calculation according to Equations (12) and (13).

3.3.1. Daily ET with the EFI When Short-Time EF from Ground-Measured Data

The EF_day was calculated through Equation (12) with every hourly or half-hourly EF (9:30–14:30) from EC data, and ET_day was estimated by Equation (13) with the available energy (R_n-G) from the ground measurements. EFO always underestimates ET_day, and EFI has somewhat corrected this bias. The improvement of ET_day estimation has been particularly prominent for agricultural systems with enough data, including winter wheat, soybean, maize, and paddy rice agricultural systems (Figure 8). The four metrics also demonstrated that EFI performs better than EFO in daily ET estimation (Figure 9), with an average RMSE (0.56 mm·day⁻¹ vs. 0.72 mm·day⁻¹), an average MAPE (16% vs. 23%), an average AI (0.97 vs. 0.94) and an average R² (0.88 vs. 0.79). The EFI also yielded better results during GP than FS, according to MAPE, AI, and R², which coincides with Section 3.2. However, the conclusion is the opposite if the RMSE was used to assess EFI. The higher available energy results in a higher ET during GP, so the RMSE is higher during GP than FS. Therefore, the fact that the RMSE during GP was larger than FS did not prove that EFI is more effective during FS than during GP.

3.3.2. Daily ET from EFI When EF_st from Remote Sensing Model

The SEBAL model, a remote sensing model for energy flux estimation [55], was used to estimate the instantaneous EF, upscaled to daily EF with EFI or EFO. A total of 1181 Landsat scenes from the study area over different periods were analyzed. ET_day was estimated through Equation (13) with the available energy (R_n-G) from ERA5. Compared with measured data from EC systems, daily ET from EFI (solid point) is closer to the 1:1 line than from EFO (hollow point) for most agricultural systems, correcting some anomalous simulated EF_st by SEBAL (Figure 10). However, EFI did not show noticeable better performance in the agricultural systems with limited data, including winter barley, spring barley, cowpea, sugar beet, rapeseed, and mustard. Unreliable results from small samples might be a reason for this [56,57]. Figure 11 reveals that EFI yielded more reliable outcomes than EFO with higher average AI (0.69 vs. 0.56), higher average R² (0.89 vs. 0.84), lower average RMSE (1.27 mm·d⁻¹ vs. 1.57 mm·d⁻¹), and lower average MAPE (31% vs. 38%). The better performance of EFI demonstrated that the improved method could modify the error caused by the hypothesis of EF_day equal to EF_st of EFO to some extent.

4. Discussion

4.1. Performance of the Improved Evaporative Method

EFI can upscale the short-time EF to EF_day more accurately and it performs better in ET_day estimation than EFO. Most remote sensing models require VPD and R_n-G for EF_st computation, and EFI typically does not require additional input. Because of this, the EFI can improve the accuracy of ET_day calculation without more data entry, providing a feasible method to upscale the short-time ET to daily ET on a regional scale. According to the sources of daily R_n-G, upscaling methods, and sources of short-time EF, four cases (

Table 1. Brief description of different cases of daily ET estimation.

Case	ETday Symbol	Upscaling Methods	Sources of Daily Rn-G	Sources Short-Time EF
Case Ⅰ	ETOS	EFO	ERA5	SEBAL
Case Ⅱ	ETIS	EFI	ERA5	SEBAL
Case Ⅲ	ETOG	EFO	Ground-measured	EC system
Case Ⅳ	ETIG	EFI	Ground-measured	EC system

) were set to estimate daily ET, compared with measured daily ET from EC systems.

The performance of EFI and EFO for different cases is discrepant (Figure 12). The ET_IG and ET_OG are superior to ET_IS and ET_OS in different cases, respectively, which might result from the more significant uncertainty of estimated short-time EF by SEBAL than measured short-time EF [58]. When EF_st from the EC system (Case III and Case IV), compared with EFO, EFI reduced the average MAPE from 23% to 16% and the average RMSE from 0.72 to 0.56 mm·d⁻¹. As for short-time EF from SEBAL (Case I and Case II), the EFI also performs better than EFO, but the estimated ET_day shows a more obvious difference with measured ET_day than Case III and Case IV, caused by the error of short-time EF estimation. Specifically, SEBAL overestimates or underestimates the EF_st for its model structure limitation in some scenarios. However, EFI does not catch these errors, resulting in the ET_day estimation accuracy not improving compared with EFO for the soybean agricultural system. Nevertheless, the EFI could always improve the daily EF estimation for an accurate short-time EF (Case III and Case IV), with higher R² and AI, lower MAPE, and RMSE. With the advancement of remote sensing technology, the EF_st from remote sensing will be more and more accurate, and EFI is anticipated to deliver more valuable information for agricultural management.

4.2. Comparision with Other Upscaling Methods

Many other methods have been proposed to modify the EFO, including empirical equations [19,23,25] and theoretical derivation [24]. The moisture condition of air indexes was always contained in these methods, which was also considered in EFI. Compared with EFO, these improved methods performed better for different land covers (

Table 2. The information about different improved evaporation methods. Subscript “mea” means measured data; subscript “est” means estimated data; subscript “st” means short-time data; subscript “day” means daily data from 0:00 to 23:59; R_s is solar radiation; RH is the relative humidity; EF is the evaporation fraction; β is the Bowen Ratio; VPD is vapor pressure deficit of the air; A and B are the fitting parameters from measured data and estimated data; R_n is net radiation; G is soil heat flux; T_a is air temperature; r_a is aerodynamic resistance; r_s is surface resistance; r* is critical surface resistance when ET equals equilibrium ET; Ω^* is the decoupling factor when ET equals equilibrium ET.

Providers	Assumptions	Inputs	Core Equations	Accuracy Improvement Compared to EFO
Hoedjes et al. [23]	The linear relationship between Rs, RH and EF; the scaling factor was the ratio of EFmea,st and EFest,st.	Rs, RH, and EFmea,st	$E{F}_{\mathrm{day}}=\left\{\begin{array}{l}E{F}_{\mathrm{mea},\mathrm{st}}\beta >1.5\\ 1.2-(0.0004R_{\mathrm{s},\mathrm{day}}+0.005R{H}_{\mathrm{day}}){\displaystyle \frac{E{F}_{\mathrm{mea},\mathrm{st}}}{E{F}_{\mathrm{est},\mathrm{st}}}}\beta \le 1.5\end{array}\right.$	MAPE: 8% vs. 0.5%
Liu et al. [25]	VPD controls the simulated error between estimated ETday with EFO and measured ETday.	A, B, and VPD	$E{T}_{\mathrm{EF},\mathrm{day}}=E{F}_{\mathrm{st}}{(R_{\mathrm{n}}-G)}_{\mathrm{day}}$$E{T}_{\mathrm{day}}=A\cdot E{T}_{\mathrm{EF},\mathrm{day}}+B\cdot VP{D}_{\mathrm{day}}$	RMSE: 0.24 vs. 0.21 mm·d−1
Liu [19]	The ratio of LE to Rn is a constant.	EFst	${\displaystyle \frac{L{E}_{st}}{R_{n,st}}}={\displaystyle \frac{L{E}_{day}}{R_{day}}}$	MAPE 20% vs. 18%
Tang and Li [24]	The decoupling factor is constant for one day.	Ta, ra, rs, and r *	$E{T}_{\mathrm{day}}=E{F}_{\mathrm{est},\mathrm{st}}{(R_{\mathrm{n}}-G)}_{\mathrm{day}}{\displaystyle \frac{{\Delta }_{\mathrm{day}}}{{\Delta }_{\mathrm{day}}+\gamma }}{\displaystyle \frac{{\Delta }_{\mathrm{st}}+\gamma }{{\Delta }_{\mathrm{st}}}}{\displaystyle \frac{{\Omega }_{\mathrm{st}}^{*}}{{\Omega }_{\mathrm{day}}^{*}}}$	MAPE: 22.8% vs. 11.7%

Note: PR: paddy rice; MA: maize; SB: sugar beet; WW: winter wheat; PO: potato; SBA: spring barley; RA: rapeseed; WBA: winter barley; SO: soybean; OR: orange; CO: cowpea; MU: mustard; HH: half-hourly time steps; HR: hourly time steps; ^a: the field area comes from Google Earth; ^b: the fields surrounded by sufficient paddy fields with same agricultural management; ^c: the distance from the flux tower which more than 80% of contribution to footprint; *: references do not display clearly, but demonstrate study area contribute more than 80% footprint.

), such as summer corn and winter wheat croplands in the Northern China Plain (MAPE of LE from 22.8% to 11.7%), paddy fields in the Tai-Lake region of China (RMSE of ET from 0.24 mm·d⁻¹ to 0.21 mm·d⁻¹), and an olive orchard in central Morocco (MAPE of ET from 8% to 0.5%). For the method proposed by Hoedjes et al. [23], the measured EF_st is adopted for dry conditions (Bowen ratio less than 1.5) rather than the estimated EF_st. Meanwhile, the EF_st is also required for wet conditions, and the 1.2, 0.00004, and 0.005 parameters might be somewhat arbitrary. So, this method is unsuitable for areas with scarce observational data. Similarly, Liu et al. [25] applied the measured daily ET to calibrate the estimated ET, generating the parameters A and B. So, the constants A and B might not be applicable for different land covers, requiring new measured data to calibrate. A result was obtained for evaporative flux ratios with R_n [59], and Liu [19] applied it to upscale the instantaneous ET. Although this method does not need the G, reduced inputs, it has a lower accuracy [19]. The G might introduce more uncertainty for ET upscaling; however, ignoring the effect of G also produces large errors in winter [60]. Compared to the first two methods, the method developed by Tang and Li [24] does not need the measured ET or EF as input and performs well. However, this method requires the r_a, r_s, and critical surface resistance when ET equals equilibrium ET (r_s*), which is difficult to quantify precisely through remote sensing, especially for r_s*. The above methods are more accurate than EFO in short-time EF upscaling, but the empirical coefficient is needed, and extra input requirements might hinder the application. So, EFI could be a choice for EF_st upscale when the data are limited.

To validate the effectiveness of EFI, the five different methods were compared to EFI with EF_inst from in situ measurements, including the three evaporation methods (decoupling factor method [24], net radiation method [59], and constant evaporative fraction method) and two other methods (reference evapotranspiration method [8] and sine method [7]). Specific procedures can be found in the corresponding references, and the ET from different methods were named ET(Tang), ET(R_n), ET(EFO), ET(ET₀), and ET(Sine). Figure 13 shows the MAPE between measured ET_day and estimated ET_day from different methods, and the other index can be found in Appendix E. In general, the ETO shows the best performance for all the agricultural systems, with an average MAPE of 15.2% and a stable performance, with a standard deviation of MAPE of 0.04. EFI was developed for regional ET estimation, which is always calculated using remote sensing technology. VPD and R_n-G are needed in most ET remote sensing models, e.g., SEBS [61], METRIC [8], SEBAL [55], and RS-PM [62]. So, the EFI was used to estimate regional ET, which does not require additional input compared to the original methods.

One thing should be noticed: the error of ET(Tang) is more obvious than that of Tang and Li [24]. We carefully reviewed our data processing and calculations and found that the error associated with Tang’s method is indeed larger than EFO in certain cases. This is primarily due to the decoupling factor corresponding to the equilibrium evapotranspiration at the instantaneous scale being much larger than at the daily scale, which leads to the observed larger computed values. This phenomenon can be attributed to the differences in balance evapotranspiration between the instantaneous and daily scales, which causes larger discrepancies in the calculated decoupling factor. The climate conditions at a daily scale likely contribute to this issue, and it is possible that in the work of Tang and Li [24], they had conducted a more detailed review and validation of the base data, which could account for their more consistent results. In contrast, our analysis was based on all available data without conducting such a detailed review, which might explain the differences between our results and those reported by Tang and Li [24]. The method requires the input of r_c and r_a, but they did not provide detailed methods for calculating these resistances. To address this, we used the FAO-56 recommended method to estimate rc and ra, as it does not require additional data. However, it is important to note that the FAO-56 method provides only an approximation, and accurately describing these two complex parameters is challenging. This approximation could introduce additional errors into the calculation process, further contributing to the discrepancies observed in our results compared to Tang and Li [24].

It is true that any of the upscaling methods, including the ones we used in our study, could potentially provide more accurate daily ET estimates if biases were corrected through calibration against in situ measurements. Calibration is indeed an important step for improving the accuracy of any model, particularly in specific environments or conditions. However, the focus of our study was primarily on the development and evaluation of the temporal upscaling method itself rather than performing an extensive calibration of each method against in situ measurements. While calibration can undoubtedly enhance the performance of these methods, the goal here was to demonstrate the effectiveness of EFI under both calibrated and uncalibrated scenarios, as we observed good performance even with a default value of t = 0.5 (Figure 17).

4.3. The Correctness of the Hypothesis of EFI

There are two critical hypotheses for EFI: the deviation coefficient (δ) and potential deviation (Ω). The adjustment coefficient (t) and short-time EF were adopted to depict the Ω, and the optimum t for different agricultural systems was given in Section 3.2, proving the reliability of Ω. The dependability of EFI would be approved if the correctness of δ was demonstrated.

The average half-hour or hourly data (measured by EC systems) were analyzed for various agricultural systems and sites. Figure 14 illustrates a noticeable positive linear connection between η and EF_st for most agricultural systems, demonstrating that the η could describe the EF intra-day when other factors were not considered to a certain extent. Besides this study, the effect of VPD [23,24] and R_n-G [16,50] on EF have been reported. As mentioned above, the results demonstrated the validity of EFI and the rationality of the hypothesis.

The relationship between η and EF_st of the same crop shows a great difference in different places (Figure 14). Three crops (planted in more than four different sites) and three sites (with more than three crops) were chosen to analyze the effect factor on the relationship between η and EF_st (Figure 15). The R2 between η and EF_st shows large fluctuation for the same crop planting in different sites, with 0.80–0.94 for winter wheat, 0.65–0.93 for maize, and 0.74–0.99 for paddy rice. However, the R2 between η and EF_st for different crops in one site shows a relatively stable state, with 0.82–0.87 for BE_Lon, 0.88–0.94 for DE_Kli, and 0.67–0.80 for US_ARM. So, the crop might not be a key factor for the relationship between η and EF_st, but the circumstance factors might be (the circumstance factors would not change sharply during a short time at one location).

4.4. Uncertainty of the ET_day Estimation with EFI

Compared with EFO, EFI upscaled the EF_st to EF_day more accurately, generating a more reliable ET_day. However, some uncertainty about ET_day estimation existed through EFI when the remote sensing technology was applied. Two aspects were analyzed, including sources of EF_st and daily R_n (R_n,day) and the structure of EFI.

4.4.1. Uncertainty About Short-Time EF Calculation and Daily R_n

The ET_day calculated from the measured EF_st was more precise than calculated from the estimated EF_st (SEBAL) in different sites, with an average MAPE of 16% vs. 31%, respectively. So, EF_st calculation and R_n from ERA5 introduced a 15% (31% − 16%) error in ET_day estimation. Nevertheless, the remote sensing approach (e.g., SEBAL) and raster climate dataset (e.g., ERA5) are crucial for regional ET calculation [63,64] at the cost of compromising calculation precision for each point.

The SEBAL, a remote sensing model suited for the homogenous underlying surface, was adopted in this study. However, uncertainties also exist in the structure [47] and some empirical coefficients [65] of SEBAL. The R_n,day and LE_day errors were compared when the remote sensing image and the R_n,day measured data were available at different sites (Figure 16). LE_day estimated error increased along with the R_n,day error, demonstrating that the ERA5 (source of R_n,day) would introduce the mistake when it was used to estimate ET_day. On the other hand, some biases are caused by input data, including polluted pixels and coarse spatial resolution [66,67], even though the quality bands of images were applied to exclude the low-quality pixels.

While potential biases in EF_st may propagate through temporal upscaling processes, this study specifically addresses temporal upscaling methodology rather than intrinsic SEBAL algorithm performance. Regarding ERA-5 reanalysis data, the meteorological variables (surface radiation, air temperature, and wind speed) have been extensively validated in prior global studies showing consistent agreement with ground observations. Given our focus on temporal scaling optimization rather than input data verification, we utilized ERA-5 datasets based on their established reliability documented in the peer-reviewed literature [17,68,69]. While we acknowledge that biases in the SEBAL model’s instantaneous ET estimations and the ERA-5 data could influence the daily ET estimations, our focus was on improving the temporal upscaling process itself. Therefore, we did not analyze the effects of other meteorological factors in the ERA5 dataset and the SEBAL model itself on ET estimates.

4.4.2. Uncertainty About the EFI Structure

It was assumed that EF_day equals EF_st in EFO, which ignores the heterogeneity of EF at different times. EFI considers the main circumstance factors according to P-M and P-T equations, modifying the EFO. However, some uncertainty is contained in the EFI structure.

The deviation coefficient (δ) is conducted from P-M equations in theories. However, EFI did not strictly follow Equation (4), given the model operability and data availability. Besides VPD and (R_n-G), other factors are neglected in the final modified method, which also influences the EF in some instances. According to the qualitative analysis in Section 2.3.1 and results from EFI, simplified processing is acceptable. However, further analysis should be conducted in the following studies to quantify the effects of each circumstance factor on EF.

Potential deviation (Ω) refers to the possible error between EF_day and EF_st, and therefore the adjustment coefficient (t) was introduced to enhance its applicability. The optimum t for the different agricultural systems was given in Section 3.2. However, for different climate conditions or circumstances, the optimum t was not analyzed, for it is challenging to quantify climate condition types and circumstances with the data of this study. EF was influenced by many factors, especially the soil water content [52,70]. Irrigation practice during GP maintained the moist soil, leading to a stable EF_day. However, EF_day fluctuated sharply during the FS (Figure 3) brought on by rain or other factors, leading to inaccurate potential error estimation. Future research that takes rainfall information into account might help diminish the error. To improve regional ET_day estimation accuracy, the interaction of uncertainty between EF_st and EFI should be explored in the future.

However, it is important to note that the purpose of this study was to develop and validate EFI at a localized scale using the available crop-specific data. While we understand that applying this method on a larger scale would require broader crop-type datasets, we believe the methodology can still provide valuable insights at the regional level where crop-type information is accessible. Moreover, when a specific value for t is not set, we assume a general value of 0.5 for all vegetation types. This approach still yields results that are superior to the original method. The specific results can be seen in Figure 17. Future research could aim to expand the applicability of this method by integrating more generalized crop classifications or by using remotely sensed data that can infer crop types at a larger scale. Additionally, efforts to create more comprehensive crop-type datasets would be beneficial for scaling up this approach.

5. Conclusions

This study developed an improved evaporative fraction method (EFI) to upscale EF_st to EF_day, considering saturation vapor difference and available energy without additional data requirements compared to the constant evaporative fraction method (EFO). EFI was tested for different agricultural systems, various sites, growth stages, and different sources of EF_st, respectively. The optimum adjustment coefficient was given for different agricultural systems. The comparison between estimated ET_day and measured ET_day (from EC) demonstrates that the accuracy of EFI is superior to EFO, yielding a lower error (RMSE and MAPE) and a better consistency (AI and R²). Meanwhile, the crop might not be the main controlling factor for the relationship between η and EF_st, but meteorological conditions. Since EFI requires no extra data, it is expected to be applied in converting EF_st to EF_day at a large scale, generating a more accurate ET_day.