Ronglin Tang

Zhao-Liang Li
email: lizhaoliang@caas.cn

An improved constant evaporative fraction method for estimating daily evapotranspiration from remotely-sensed instantaneous observations

Keywords: Evapotranspiration, Constant evaporative fraction method, Remote sensing

An improved constant evaporative fraction method for estimating daily evapotranspiration from remotely sensed instantaneous observations

Ronglin Tang, Zhao-liang

Introduction

Evapotranspiration (ET) is a primary mechanism for water and heat transfer between land and the atmosphere. In recent decades, a great number of models with varying degrees of complexity and accuracy (see the ET reviews of [START_REF] Li | A Review of current methodologies for regional evapotranspiration estimation from remotely sensed data[END_REF] and [START_REF] Kalma | Estimating land surface evaporation: a review of methods using remotely sensed surface temperature data[END_REF]) have been developed to estimate regional ET using remote sensing technology. However, these remote sensing models generally provide only instantaneous snapshots of ET at discrete satellite overpass times which are not sufficient to meet the needs of researchers who often require accurate estimates of ET at daily and longer timescales. The most common technique for converting instantaneous ET estimates to daily timescales is based on the assumption of a constant evaporative fraction (EF), defined as the ratio of the latent heat flux (LE) to available energy (R n -G) [START_REF] Brutsaert | Application of self-preservation in the diurnal evolution of the surface energy budget to determine daily evaporation[END_REF][START_REF] Nichols | Evaluation of the evaporative fraction for parameterization of the surface energy balance[END_REF][START_REF] Crago | Conservation and variability of the evaporative fraction during the daytime[END_REF][START_REF] Gómez | Retrieval of evapotranspiration over the Alpilles/ReSeDA experimental site using airborne POLDER sensor and a thermal camera[END_REF][START_REF] Gentine | Analysis of evaporative fraction diurnal behavior[END_REF][START_REF] Sobrino | Application of a simple algorithm to estimate the daily evapotranspiration from NOAA-AVHRR images for the Iberian Peninsula[END_REF][START_REF] Galleguillos | Mapping Daily Evapotranspiration Over a Mediterranean Vineyard Watershed[END_REF]. However, numerous studies [START_REF] Tang | Temporal upscaling of instantaneous evapotranspiration: An intercomparison of four methods using eddy covariance measurements and MODIS data[END_REF][START_REF] Van Niel | Correcting for systematic error in satellite-derived latent heat flux due to assumptions in temporal scaling: Assessment from flux tower observations[END_REF][START_REF] Cammalleri | Applications of a remote sensing-based two-source energy balance algorithm for mapping surface fluxes without in situ air temperature observations[END_REF][START_REF] Delogu | Reconstruction of temporal variations of evapotranspiration using instantaneous estimates at the time of satellite overpass[END_REF][START_REF] Xu | Temporal upscaling and reconstruction of thermal remotely sensed instantaneous evapotranspiration[END_REF]Cha´vez et al., 2008] have reported that daily ET estimates are significantly underestimated by up to 34% [START_REF] Van Niel | Upscaling latent heat flux for thermal remote sensing studies: comparison of alternative approaches and correction of bias[END_REF] when the EF method is applied.

The objectives of this paper are twofold: 1) to develop an improved constant EF method via a theoretical derivation to convert remote-sensing based instantaneous ET to a daily scale and 2) to test the robustness of the improved constant EF method using ground-based ET measurements and four datasets of remote-sensing based instantaneous ET. Section 2 presents the rationale of the conventional and improved constant EF methods. Section 3 provides a brief introduction of the surface meteorological and energy flux measurements and the remote sensing ET datasets. Section 4 describes the results of the evaluation of the conventional and improved constant EF methods.

Finally, the conclusions are presented in Section 5.

Methodology

Conventional constant evaporative fraction method

For the conventional constant evaporative fraction method, assuming that the instantaneous EF equals the daily EF (EF i =EF d ), the daily LE can be derived as follows:

( ) ( ) ()      i d d n d n d ni LE LE EF R G R G RG (1)
where R n is the surface net radiation, W/m 2 , and G is the soil heat flux, W/m 2 . The subscripts i and d indicate the value at the instantaneous and daily time scales, respectively.

Improved constant evaporative fraction method

By rearranging the Penman-Monteith equation, surface evapotranspiration (or latent heat flux) can be mathematically expressed with the following equations [START_REF] Mcnaughton | Predicting effects of vegetation changes on transpiration and evaporation[END_REF][START_REF] Pereira | The Priestley-Taylor parameter and the decoupling factor for estimating reference evapotranspiration[END_REF]

: = ( )      n LE R G (2) * /     (3) 1 1      c a r r (4) * * 1 1      a r r (5) * () ()      p n C VPD r RG (6)
where Δ is the slope of the saturated vapor pressure versus the air temperature curve, kPa/°C; is the psychrometric constant, kPa/°C;  is the decoupling factor and represents the relative contribution of the radiative and aerodynamic terms to the overall ET. Additionally, r * is the critical surface resistance when LE equals equilibrium evapotranspiration, s/m; r a is the aerodynamic resistance, s/m; r c is the surface resistance, s/m;is the air density, kg/m 3 ; C p is the specific heat of the air, J/(kg•°C); and VPD is the vapor pressure deficit of the air, kPa. Given that Eqs. (2-6) are valid and can be applied at both instantaneous and daily time scales [START_REF] Allen | Crop evapotranspiration-Guidelines for computing crop water requirements[END_REF]ASCE-EWRI, 2005], the daily LE can be theoretically linked to instantaneous EF and daily surface available energy with the following equation:

* * () () () ()                           i d i d d n d n i d i i i d i i d nd n i d i d i LE LE R G RG LE RG RG (7)
In the improved constant evaporative fraction method, to achieve a tradeoff between model complexity and accuracy, is assumed to be constant during the day ( d = i ) [START_REF] Goldberg | Quantifying the coupling degree between land surface and the atmospheric boundary layer with the coupled vegetation-atmosphere model HIRVAC[END_REF] and the daily LE can be expressed as follows:

* * * * ( ) = ( ) ( ) () i d i i d i i d improve n d d conv n i d i d d i d LE LE R G LE RG                        (8)
where ()

d improve

LE

is the improved daily LE and ()

d conv

LE

is the daily LE estimated from Eq.

(1) using the conventional constant EF method. For simplicity, r a is calculated by assuming that the atmosphere is under neutral conditions (i.e., no stability correction function is applied). The improved constant EF method is more physically based than the conventional constant EF method and is expected to outperform it. Inputs for the improved method primarily include the instantaneous evaporative fraction, instantaneous and daily air temperature, wind speed, vapor pressure deficit, slope of vapor pressure, and surface available energy. Daily values of air temperature, wind speed, vapor pressure, and surface available energy are computed as the average of multiple (48 half-hourly in this study) measurements from 0:00 h to 24:00 h local time.

The daily slope of the vapor pressure curve is estimated using daily air temperature. Daily VPD is computed as daily saturated vapor pressure (estimated using daily air temperature) minus daily air vapor pressure. Note that a fixed psychrometric constant (0.067 kPa/°C) is used in Eqs. (7-8), primarily because a preliminary analysis suggested that using a variable psychrometric constant (as a function of temperature, humidity, and pressure) does not significantly improve the validation of daily ET compared to using a fixed psychrometric constant.

Test site and data

The Yucheng station (36.8291° N/116.5703° E), which is located on the North China plain, is characterized by sandy loam soil and a subhumid and monsoon climate, with a mean annual temperature and precipitation of 13.1 °C and 528 mm, respectively. Winter wheat is rotated with summer corn at this station. Half-hourly measurements of surface meteorological and energy flux variables, including wind speed, relative humidity, air temperature, atmospheric pressure, surface 4-component radiation (downwelling and upwelling shortwave and longwave radiation), soil heat flux, sensible heat flux and latent heat flux from the eddy covariance (EC) system, which are regularly measured at the Yucheng station, were collected from late April 2009 to late October 2011 for analysis in this study. The specifics of the sensors, the sensor deployment, and the post-processing of turbulent flux measurements can be found in [START_REF] Tang | An inter-comparison of three remote sensing-based energy balance models using Large Aperture Scintillometer measurements over a wheat-corn production region[END_REF][START_REF] Tang | Temporal upscaling of instantaneous evapotranspiration: An intercomparison of four methods using eddy covariance measurements and MODIS data[END_REF] and elsewhere. The daily latent heat flux for the validation was computed as the average of 48 half-hourly EC measurements from 0:00 h to 24:00 h local time.

Four reliable datasets of remote-sensing based instantaneous LE and R n -G at the Yucheng station, which were estimated separately from two typical single-and dual-source energy balance models in our previous studies [START_REF] Tang | An inter-comparison of three remote sensing-based energy balance models using Large Aperture Scintillometer measurements over a wheat-corn production region[END_REF][START_REF] Tang | Temporal upscaling of instantaneous evapotranspiration: An intercomparison of four methods using eddy covariance measurements and MODIS data[END_REF], are extracted to evaluate the performances of the conventional and improved EF methods in practical applications in this study.

Specifically, these datasets are MODIS-based estimates of instantaneous LE and R n -G using the N95 [START_REF] Norman | A two-source approach for estimating soil and vegetation energy fluxes from observations of directional radiometric surface temperature[END_REF] two-source model and the surface energy balance system [SEBS, [START_REF] Su | The Surface Energy Balance System (SEBS) for estimation of turbulent heat fluxes[END_REF] model for 33 clear-sky TERRA overpass times between late April 2009 and late September 2009 and using the N95 [START_REF] Norman | A two-source approach for estimating soil and vegetation energy fluxes from observations of directional radiometric surface temperature[END_REF] In view of the fact that remotely-sensed surface energy balance-based ET models by definition enforce an energy balance among the energy flux components, the widely applied Bowen ratio (BR) correction method and the residual energy (RE) correction method [START_REF] Twine | Correcting Eddy-Covariance Flux Underestimates over a Grassland[END_REF] are introduced in this study to close the energy imbalance between the sum of the eddy covariance (EC) system-measured sensible heat flux, latent heat flux and surface available energy. To determine whether the energy imbalance correction influences the consistency of the model performance, evaluation of the constant EF method and the improved method with the in situ EC measurements is conducted under uncorrected and corrected energy imbalance conditions. October 2011 (one day on which the performance of conventional constant EF method was very significantly deteriorated due to the very small instantaneous R n -G at 10:30 leading to a very large instantaneous EF was excluded from the 404 quality-and completeness-controlled days; see Section of 3.1 of [START_REF] Tang | Temporal upscaling of instantaneous evapotranspiration: An intercomparison of four methods using eddy covariance measurements and MODIS data[END_REF] for a discussion of how these 404 days were screened). Using the paired-t test (null hypothesis: the mean of the estimated daily LE from the improved EF method is equal to that from the conventional EF method; alternative hypothesis: the two statistics are not equal, at the 0.01 level of significance), the daily LE estimated using the improved EF method is shown to be statistically significantly different from that using the conventional EF method under both uncorrected and corrected energy imbalance conditions. Underestimation of the daily LE was generally found in both the conventional and our proposed EF methods.

Results and discussion

However, both the bias and the root mean square error (RMSE), as tabulated in Table 1, decreased by > 31% and 17%-29%, respectively, when our proposed EF method was applied for the half-hourly LE conversion instead of the conventional EF method. The change in the bias demonstrated that the daily LE underestimation often found in the conventional constant EF method was overall significantly smaller for our improved method. It could be concluded that our proposed method outperformed the conventional constant EF method when the acquired instantaneous LE or half-hourly LE corresponding to the daytime overpass times of most polar-orbiting satellites was provided without any bias for conversion to the daily scale, regardless of whether the ground-based EC measurements used for truth validation were uncorrected or corrected for energy imbalance using the BR and RE methods.

In the conversion of four separately acquired datasets of MODIS-based instantaneous LE estimates from the single-source SEBS or dual-source N95 model to the daily scale, our proposed EF method also overall produced lower bias and RMSE values than the conventional constant EF method, as illustrated in Figs. 2-3 and Tables 23. Our proposed method performed much better than the conventional method, with a relatively large RMSE decrease of 42%-54% (see Fig. 2 and Table 2), when the remote-sensing based instantaneous LE and R n -G datasets were from the work of [START_REF] Tang | An inter-comparison of three remote sensing-based energy balance models using Large Aperture Scintillometer measurements over a wheat-corn production region[END_REF]. The positive biases in the SEBS-derived instantaneous LE from Tang et al.

[2011] and N95-derived LE at TERRA overpass times from [START_REF] Tang | Temporal upscaling of instantaneous evapotranspiration: An intercomparison of four methods using eddy covariance measurements and MODIS data[END_REF] contributed in part to the positive biases in daily LE estimated using our proposed EF method. Underestimation of daily LE was still found in the conventional constant EF method. Overall, when applied to convert remote-sensing based instantaneous LE (with a certain degree of bias compared to the ground measurement) acquired at MODIS/TERRA and MODIS/AQUA overpass times to the daily scale, our proposed method also outperformed the conventional constant EF method.

Some analysis of the performances of the two EF methods is performed here. The EF method builds on the assumption of a constant EF and as a result allows the daily ET to be determined directly from the daily surface available energy. The improved method (assuming  d = i ), developed through a theoretical derivation from the physics-based Penman-Monteith equation, considers the non-linear effect of meteorological factors on the ET. The most novel aspect of this study lies in that the factor (

* * d i i d d i d i             
) in the improved method adjusts for the relative difference between the instantaneous evaporative fraction at the time of the overpass and the evaporative fraction during other timeframes. In essence, the improved method relaxes the assumption of a constant evaporative fraction, and  can be interpreted as a correction for the EF method (daily ET in the improved method equals that in the EF method multiplied by the correction factor 

The EF method (Eq. ( 1)) for the conversion of instantaneous LE is a much reduced form of Eq.

(7) (i.e., =1). It is evident that the performance of the conventional constant EF method depends on the magnitude of the  factor. For =1, the conventional constant EF method will produce the estimated daily LE without a bias. For >1, an underestimation in the estimated daily LE will be produced using the conventional constant EF method. For <1, an overestimation in the estimated daily LE will be produced. The underestimation in the estimated daily LE found by a number of authors in previous studies and in this study indicates >1 in most cases. In fact, lots of studies report that the EF generally exhibits on a concave-up shape during the daytime, leading to the underestimation of daily ET when using the EF acquired close to noon as a constant for temporal conversion. The improved constant EF method based on simplifying the calculation of the  factor can evidently improve the estimated daily LE that is often underestimated using the conventional constant EF method, which is expected.

It has to be mentioned that the conversion of instantaneous ET to the daily scale is performed for the same day primarily because of 1) the daily revisit period of the MODIS sensor that provides widely applied land products and 2) the possibly decreased validity of the constancy assumption made in the conversion methods over a longer time scale [START_REF] Alfieri | Effect of the revisit interval and temporal upscaling methods on the accuracy of remotely sensed evapotranspiration estimates[END_REF]. The effectiveness of the improved method during subsequent days is beyond the scope of this paper and is therefore not investigated in the current study. The strengths of the improved method are its more solid physical foundation and the resulting higher accuracy of the estimated daily ET than the conventional EF method. The limitation lies in the requirement of instantaneous and daily ground-based meteorological data (air temperature, VPD, and wind speed), which may be unavailable, as inputs. In practical/operational applications, these inputs, which may come from the reanalysis data estimated using numerical weather prediction models and global climate models, can introduce a certain degree of uncertainties in the estimated daily ET. A sensitivity analysis of the improved EF method using data shown in Fig. 1b shows that a perturbation of 1 °C in air temperature (or a perturbation of 20% in VPD/wind speed or 10% in surface available energy) on average results in a variation of 3.5% (1.8%/1.9% or 0.9%, respectively) in the estimated daily LE.

Conclusions

In this paper, a new constant EF method is proposed through a theoretical derivation to improve the conversion of remote-sensing based instantaneous LE to the daily scale. The improved constant EF method is more physically based and can be applied operationally without too much model complexity and consumption of computation time. It can significantly reduce the underestimation of the daily LE estimated using the conventional constant EF method. The lower bias and RMSE in the conversion results of both ground-based half-hourly LE without any bias and remote-sensing based instantaneous LE with certain degrees of biases in this study demonstrate the great robustness of the improved constant EF method and its superiority over the conventional constant EF method. This improved method can be used instead of the conventional constant EF method to more accurately convert remote-sensing based instantaneous LE estimates to the daily scale.

Fig. 1

 1 Fig. 1 presents the performances of the conventional and improved constant EF methods in converting half-hourly LE measurements at 10:30 (corresponding to the MODIS/TERRA daytime overpass time) and 13:30 (corresponding to the MODIS/AQUA daytime overpass time) local time to the daily scale. These conversions were made over 403 days between late April 2009 and late

Figure 1 .Figure 2 .Figure 3 .

 123 Figure 1. Comparison of daily LE estimated using the conventional and improved constant EF methods with ground-based EC measurements without an energy imbalance correction and with an energy imbalance correction based on the Bowen ratio (BR) and residual energy (RE) methods.(a), (c), and (e) Converting half-hourly measurements at 10:30 local time to the daily scale. (b), (d), and (f) Converting half-hourly measurements at 13:30 local time to the daily scale.

  two-source model for 55 clear-sky TERRA overpass times and 46 clear-sky AQUA overpass times between late April 2009 and late October 2011. These datasets have been proven to reasonably agree with the ground-based measurements.

	When validated against ground-based large-aperture scintillometer-derived LE, positive biases of
	1 W/m 2 and 35 W/m 2 and RMSEs of 42 W/m 2 and 56 W/m 2 were found in the N95-estimated LE
	and SEBS-estimated LE, respectively (see Fig. 10b in the work of Tang et al. [2011]). Similarly,
	positive and negative biases of 16 W/m 2 and -1 W/m 2 and RMSEs of 46 W/m 2 and 34 W/m 2 were
	found in the N95-estimated LE at TERRA and AQUA overpass times, respectively (see Fig. 8c in
	the work of Tang et al. [2013]). These statistical results fall in between the generally acceptable

upper and lower ranges of bias and RMSE. Different magnitudes of model-estimation bias and error in these three datasets are especially helpful and valuable for testing the robustness/superiority of the improved constant EF method relative to the conventional constant EF method. Please refer to

[START_REF] Tang | An inter-comparison of three remote sensing-based energy balance models using Large Aperture Scintillometer measurements over a wheat-corn production region[END_REF]

and

[START_REF] Tang | Temporal upscaling of instantaneous evapotranspiration: An intercomparison of four methods using eddy covariance measurements and MODIS data[END_REF]

for details on the estimation methods for these instantaneous LE and R n -G values.

Table 1 .

 1 Statistical measures of the performances of conventional and improved constant EF methods in the conversion of half-hourly LE measurements at 10:30 and 13:30 local time to the daily scale.is the mean of the daily EC measurements without correction or with corrections based on the BR and RE methods.

					10:30				13:30
			EF		EF_improved	EF		EF_improved
		(W/m 2 )
			BIAS	RMSE	BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
			(W/m 2 )	(W/m 2 )	(W/m 2 )	(W/m 2 )	(W/m 2 )	(W/m 2 )	(W/m 2 )	(W/m 2 )
	EC	57	-15	22	-6	16	-9	16	-5	12
	BR	69	-15	21	-3	15	-12	17	-7	13
	RE	72	-13	20	0	16	-13	18	-9	15

Table 2 .

 2 Statistical measures of the performances of conventional and improved constant EF methods in the conversion of two datasets (N95-and SEBS-derived instantaneous LE at TERRA overpass times from[START_REF] Tang | An inter-comparison of three remote sensing-based energy balance models using Large Aperture Scintillometer measurements over a wheat-corn production region[END_REF]) to the daily scale.is the mean of the daily EC measurements corrected with the BR and RE methods.

					N95				SEBS
			EF		EF_improved	EF		EF_improved
		(W/m 2 )
			BIAS	RMSE	BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
			(W/m 2 )	(W/m 2 )	(W/m 2 )	(W/m 2 )	(W/m 2 )	(W/m 2 )	(W/m 2 )	(W/m 2 )
	BR	101	-10	25	4	13	-8	24	6	14
	RE	102	-11	24	3	11	-9	23	5	13

Table 3 .

 3 Same as Table2, but for the two datasets of N95-derived instantaneous LE at TERRA and AQUA overpass times from[START_REF] Tang | Temporal upscaling of instantaneous evapotranspiration: An intercomparison of four methods using eddy covariance measurements and MODIS data[END_REF].

		TERRA					AQUA
		EF		EF_improved		EF		EF_improved
	(W/m 2 )	BIAS	RMSE	BIAS	RMSE	(W/m 2 )	BIAS	RMSE	BIAS	RMSE
		(W/m 2 )	(W/m 2 )	(W/m 2 )	(W/m 2 )		(W/m 2 )	(W/m 2 )	(W/m 2 )	(W/m 2 )
	BR 106	-13	29	10	30	122	-11	24	-2	19
	RE 108	-15	30	8	29	124	-14	24	-4	18

Acknowledgments

The staff members at the Yucheng site are acknowledged for their hard work with the setup and maintenance of the ground-based instruments and data collection. Associate Professor Yuanyuan Jia and Professor Chuanrong Li in the Academy of Opto-Electronics, Chinese Academy of Sciences, are thanked for their cooperation in providing the surface measurements. This work was partly supported by the National Natural Science Foundation of China under Grant 41571351 and 41571367 and by the International Science & Technology Cooperation Program of China under Grant 2014DFE10220.


domain: sde sdu

text:
Ronglin Tang

Zhao-Liang Li
email: lizhaoliang@caas.cn

Xiaomin Sun

Yuyun Bi

Temporal upscaling of instantaneous evapotranspiration on clear-sky days using the constant reference evaporative fraction method with fixed or variable surface resistances at two cropland sites

Keywords: evapotranspiration, reference evaporative fraction method, temporal upscaling, fixed or variable resistance

Temporal upscaling of instantaneous evapotranspiration on clear-sky days using the constant reference evaporative fraction method with fixed or variable surface resistances at two cropland sites

Ronglin Tang, Zhao-Liang Li, Xiaomin Sun, Yuyun

Introduction

Land surface evapotranspiration (ET, water in mm equivalent to latent heat flux) is one of the most significant components of the global hydrological cycle, and its estimation plays a significant role in numerical weather forecasting, global climate change modeling, and irrigation water allocation. Although many eddy covariance sites have been established worldwide in the past several decades to measure the long-term transfer of surface water, heat, and momentum across different ecosystems [START_REF] Baldocchi | FLUXNET: A new tool to study the temporal and spatial variability of ecosystem-scale carbon dioxide, water vapor, and energy flux densities[END_REF], these measurements are spatially discrete and of limited spatial representativeness. Remote sensing technology provides the ability to map surface ET over large heterogeneous areas and complements conventional ground-based ET measurements at the tower scale. Because remote sensing models of ET estimation can generally only provide instantaneous values at the clear-sky satellite overpass time [START_REF] Kalma | Estimating land surface evaporation: a review of methods using remotely sensed surface temperature data[END_REF][START_REF] Li | A Review of current methodologies for regional evapotranspiration estimation from remotely sensed data[END_REF][START_REF] Tang | An application of the Ts-VI triangle method with enhanced edges determination for evapotranspiration estimation from MODIS data in arid and semi-arid regions: Implementation and validation[END_REF][START_REF] Tang | Verification of land surface evapotranspiration estimation from remote sensing spatial contextual information[END_REF], several upscaling methods have been developed to convert instantaneous ET data to a daily value so that water consumption by crops can be monitored [START_REF] Shuttleworth | FIFE: the variation in energy partition at surface flux sites[END_REF][START_REF] Trezza | Evapotranspiration using a satellite-based surface energy balance with standardized ground control[END_REF][START_REF] Jackson | Estimation of daily evapotranspiration from one time of day measurements[END_REF][START_REF] Brutsaert | Application of self-preservation in the diurnal evolution of the surface energy budget to determine daily evaporation[END_REF][START_REF] Ryu | On the temporal upscaling of evapotranspiration from instantaneous remote sensing measurements to 8-day mean daily-sums[END_REF][START_REF] Van Niel | Upscaling latent heat flux for thermal remote sensing studies: comparison of alternative approaches and correction of bias[END_REF][START_REF] Delogu | Reconstruction of temporal variations of evapotranspiration using instantaneous estimates at the time of satellite overpass[END_REF].

The constant reference evaporative fraction [EF r , the ratio of actual to reference grass (alfalfa) ET] method, which was proposed by [START_REF] Trezza | Evapotranspiration using a satellite-based surface energy balance with standardized ground control[END_REF] to use a fixed surface resistance of 50 (30) s/m during the daytime and 200 s/m during the nighttime for reference grass (alfalfa) ET estimates and assumes that EF r is relatively constant during the day, is one of the most widely applied schemes for upscaling instantaneous remote sensing estimates of ET. The effectiveness of the constant EF r method in upscaling instantaneous ET data has been investigated and demonstrated by numerous studies that used primarily local (in-situ) metrological observations [START_REF] Colaizzi | Comparison of five models to scale daily evapotranspiration from one-time-of-day measurements[END_REF][START_REF] Chávez | Daily evapotranspiration estimates from extrapolating instantaneous airborne remote sensing ET values[END_REF][START_REF] Allen | Satellite-based energy balance for mapping evapotranspiration with internalized calibration (METRIC)-model[END_REF][START_REF] Liu | A novel method to convert daytime evapotranspiration into daily evapotranspiration based on variable canopy resistance[END_REF]Tang et al., 2013a].

Biases may be found when this method is tested using only remote sensing data [START_REF] Cammalleri | Upscaling of evapotranspiration fluxes from instantaneous to daytime scales for thermal remote sensing applications[END_REF].

The key to using the constant EF r method is to accurately estimate the instantaneous (or half-hourly/hourly) and daily ET values for a reference crop using the Penman-Monteith (PM) equation. A fixed surface resistance of 70 s/m was adopted by FAO56 for the reference grass ET estimate from the PM (hereafter referred to as FAO-PM) equation for hourly and daily time steps [START_REF] Allen | Crop evapotranspiration-Guidelines for computing crop water requirements[END_REF]. By comparing the major reference ET equations using weather data from diverse climates across the United States, the Environmental and Water Resources Institute of the American Society of Civil Engineers [ASCE-EWRI, 2005] recommended using a fixed surface resistance of 50 s/m during the daytime and 200 s/m during the nighttime for shorter reference grass ET estimates from the PM (hereafter referred to as ASCE-PM) equation at hourly or shorter computation steps.

Surface resistance not only is a physiological parameter but also has an aerodynamic component. The surface resistance over the course of a day is essentially time-dependent and can be influenced by variations in solar radiation, vapor pressure deficit, and wind speed even for the water-unstressed reference grass/alfalfa surface. [START_REF] Katerji | Modélisation de l'évapotranspiration réelle ETR d'une parcelle de luzerne: rôle d'un coefficient cultural[END_REF] related the surface resistance to aerodynamic resistance and climatological variables through a linear model.

However, the coefficients in their model must be experimentally calibrated before they can be applied to grass and alfalfa. [START_REF] Alves | Modelling surface resistance from climatic variables?[END_REF] reported that the model can be applied only under a limited range of Bowen ratio conditions. [START_REF] Todorovic | Single-layer evapotranspiration model with variable canopy resistance[END_REF] also modeled the surface resistance as a function of aerodynamic resistance and climatic variables, but his model does not require a priori calibration and can be applied without additional constraints. Both surface resistance models have been widely used in the application of the PM equation, and several authors have recommended them for practical use because they provide better estimates of the reference ET than using a fixed value of 70 s/m [START_REF] Pereira | Evapotranspiration: concepts and future trends[END_REF]Rana and Kaerji, 2000;[START_REF] Lecina | Fixed versus variable bulk canopy resistance for reference evapotranspiration estimation using the Penman-Monteith equation under semiarid conditions[END_REF][START_REF] Perez | A simple parameterization of bulk canopy resistance from climatic variables for estimating hourly evapotranspiration[END_REF][START_REF] Liu | A novel method to convert daytime evapotranspiration into daily evapotranspiration based on variable canopy resistance[END_REF].

To the best of our knowledge, almost all recent investigations of the constant EF r upscaling method have used a fixed surface resistance of 50 s/m during the daytime periods to estimate the reference short grass (or 30 s/m for the reference tall alfalfa) ET as recommended by the ASCE-EWRI [START_REF] Colaizzi | Comparison of five models to scale daily evapotranspiration from one-time-of-day measurements[END_REF][START_REF] Allen | Satellite-based energy balance for mapping evapotranspiration with internalized calibration (METRIC)-model[END_REF][START_REF] Chávez | Daily evapotranspiration estimates from extrapolating instantaneous airborne remote sensing ET values[END_REF]Tang et al., 2013a]. No studies have ever attempted to incorporate a variable daytime surface resistance in the upscaling of instantaneous ET data using the constant EF r method. Similar upscaled daily ET results can be expected to be obtained when the daytime variable surface resistance is close to the fixed values of 70 s/m in the FAO-PM equation or 50 s/m in the ASCE-PM equation. Additional studies are necessary to assess the performance of the constant EF r upscaling method under extreme climatic conditions, when the variable surface resistance significantly deviates from the commonly used fixed value.

The objective of this study is to use ground-based measurements to investigate whether using a variable surface resistance in the reference ET estimate from the PM equation can improve the daily (24-h) upscaled ET result of the constant EF r method. Section 2 describes the derivation of the variable surface resistance, the formulations of the three reference ET estimation methods, and the ET-upscaling theory. Section 3 presents the validation sites, the procedures for selecting clear days during each of the three daytime periods, and the method of correcting the energy imbalance in the EC-measured fluxes. In section 4, the daily ETs that are upscaled using the three reference ET estimation methods are compared and validated. A summary and conclusions are provided in Section 5.

Methodology

Derivation of variable surface resistance

Based on the principle of surface energy balance, the difference between the potential ET and actual ET (reference ET in this paper) can be assumed to be caused by an additional sensible heat flux imposed to heat the vegetation surface to provide the extra energy to move saturated air out of vegetation elements. [START_REF] Todorovic | Single-layer evapotranspiration model with variable canopy resistance[END_REF] predicted this additional sensible heat flux as a function of a ''pseudo'' resistance to the additional sensible heat transfer and the difference between the temperature on the potential ET surface and that on the actual ET surface. With the assumption that the ''pseudo'' resistance equals the surface resistance and the temperature difference is approximated as a function of the vapor pressure deficit and air temperature, the surface resistance can be estimated by combining the ET estimated from the PM equation and the additional sensible heat into the surface energy balance equation. A detailed description of the theory and development of the model can be found in [START_REF] Todorovic | Single-layer evapotranspiration model with variable canopy resistance[END_REF]. Here, we focus on the formulations that are used to obtain the surface resistance. In Todorovic's method, the variable surface resistance during the daytime can be solved through the following quadratic equation:

2 ( ) 0 ss cc rr a b c rr    (1)
where r s is the surface resistance, s/m; and r c is the climatological resistance, s/m. The coefficients a, b, and c and the climatological resistance can be derived as follows:

sa cp ee rc E      (2) ( / ) ( / )( )       ca c a s a rr a r r e e (3)
        s a c a e e r b r (4) ( ) s a ee c            , ( 5
)
where Δ is the slope of the saturated vapor pressure versus the air temperature curve, kPa/°C; is the air density, kg/m 3 ; c p is the specific heat of the air, J/(kg•°C); e s -e a is the vapor pressure deficit of the air, kPa; is the psychrometric constant, kPa/°C; E is the surface available energy and equals surface net radiation minus soil heat flux, W/m 2 ; and r a is the aerodynamic resistance, s/m. Eq. ( 1) has only one positive surface resistance solution.

Estimation of Reference ET

Full-form PM equation

Using the solved variable surface resistance from Eq. ( 1) at the hourly (or half-hourly) time scale, the reference ET during the daytime can be derived from the "full-form" PM combination equation:

( ) ( ) / (1 / )            n p s a a TOD PM sa R G c e e r ET rr (6)
where R n is the surface net radiation; G is the soil heat flux; ET TOD-PM is the reference ET estimated using the variable surface resistance that is derived from Todorovic's method.

A short reference crop and a tall reference crop, as proposed by ASCE-EWRI [2005], are adopted to estimate the reference ET for wheat and corn, respectively. When the full-form PM equation is applied to the reference short/tall crop, this equation can be reduced to a simplified form given that r a can be estimated with the procedure presented by [START_REF] Allen | Crop evapotranspiration-Guidelines for computing crop water requirements[END_REF].

FAO-PM equation

In the standard FAO-PM equation [START_REF] Allen | Crop evapotranspiration-Guidelines for computing crop water requirements[END_REF], the reference ET is estimated with the simplified PM equation for a hypothetical grass (or alfalfa) with an assumed height of 0.12 m (or 0.5 m), a fixed surface resistance of 70 (or 45) s/m, and an albedo of 0.23 for both the daily and hourly time steps. The FAO-PM equation has the same formulation as Eq. ( 7). The only difference is that the coefficient C d equals 0.34 for the reference short grass and 0.38 for the reference tall alfalfa for a daily cycle in the FAO-PM equation. Both sites are equipped with standard sensors (Table 2) that measure meteorological variables (e.g., air temperature, air pressure, actual vapor pressure, and wind speed/direction) and the soil water and temperature profiles at 30-min intervals. Downward and upward shortwave and longwave radiation are measured with a CNR-1 device at both sites, and the soil heat flux (G) is measured with an HFP-01 soil heat flux plate at a depth of 2 cm at the Yucheng site and 5 cm at the Yingke site. Heat storage above the plate is not accounted for because no more information is available at hand for its correction. Over the entire study period, the multi-day mean surface available energy (R n -G) at the Yingke site is much larger than that at the Yucheng site; the difference can be as much as ~100 W/m 2 at noon (see Figure 1a). In contrast, the multiday mean wind speed (u) at the Yingke site is ~1.3 m/s lower than that at the Yucheng site (Figure 1b). Moreover, the maximum wind speed at the Yucheng site generally occurs at noon, whereas the maximum wind at the Yingke site occurs in the mid-afternoon. The daytime wind speed at the Yingke site is more asymmetric than that at the Yucheng site. The multi-day mean vapor pressure deficit (VPD) at the Yingke site is slightly higher than that at the Yucheng site and has a maximum value of ~1.5 kPa. Over the involved clear-sky days (see Section 3.2 for the selection rules) that are selected for the model assessment, the multi-day mean surface available energy is slightly and moderately (difference < 70 W/m 2 ) larger at the Yingke site and Yucheng site, respectively, than that over the entire study period from mid-morning to mid-afternoon. The multi-day mean surface available energy at the Yingke site is still larger than that at the Yucheng site, but their difference (< 50 W/m 2 ) becomes smaller than that over the entire study period. The multi-day mean wind speed at the Yucheng and Yingke sites over the selected clear-sky days is moderately higher than (difference < 0.6 m/s) and has a similar magnitude to that over the entire study period, respectively, which causes a much larger difference (as high as 1.8 m/s) in wind speed between the Yucheng site and the Yingke site over the selected clear-sky days. Both the Yucheng and the Yingke sites have slightly larger multi-day mean VPD over the selected clearsky days than over the entire study period from the mid-morning to mid-afternoon. A negligible difference is observed in the multi-day mean VPD between the Yucheng and the Yingke sites over the selected clear-sky days. Table 2. Description of the model, manufacturer, and location for the instruments/sensors at the Yucheng [START_REF] Yu | Energy balance simulation of a wheat canopy using the RZ-SHAW (RZWQM-SHAW) model[END_REF][START_REF] Sun | Development of a simple remote sensing evapotranspiration model (Sim-ReSET): algorithm and model test[END_REF] and the Yingke [START_REF] Liu | A comparison of eddy-covariance and large aperture scintillometer measurements with respect to the energy balance closure problem[END_REF] sites involved in this study. T a and e a : air temperature and actual vapor pressure; u: Vapor pressure deficit (kPa) within 180 m and 100 m around the sensor at the Yingke and Yucheng sites [START_REF] Liu | A comparison of eddy-covariance and large aperture scintillometer measurements with respect to the energy balance closure problem[END_REF]Tang et al., 2011a], respectively. To obtain reliable EC-measured H and LE data using the online flux computation and post-field data programs [START_REF] Yu | Overview of ChinaFLUX and evaluation of its eddy covariance measurement[END_REF][START_REF] Liu | A comparison of eddy-covariance and large aperture scintillometer measurements with respect to the energy balance closure problem[END_REF], a series of corrections are performed on the effect of the sonic virtual temperature, the time-lag, the performance of the planar fit coordinate rotation, the density fluctuation, and the frequency response following [START_REF] Webb | Correction of flux measurements for density effects due to heat and water vapour transfer[END_REF] and [START_REF] Burba | A brief practical guide to eddy covariance flux measurements: principles and workflow examples for scientific and industrial applications[END_REF].

Half To recognize the possible time dependencies of the uncertainties and errors, the instantaneous ET values from three daytime periods (10:00-11:00, 12:00-13:00, and 14:00-15:00 local time) between mid-morning and mid-afternoon, which correspond to the overpass times of most of the polar-orbiting satellites, were upscaled. The upscaling of the instantaneous ET data is generally restricted to clear-sky satellite overpass times in practical applications [Li et al., 2013a[Li et al., , 2013b]]. The evaluation at both sites therefore focuses on days that have clear skies at the assumed satellite overpass times. Because the climatological resistance in the denominator on the left-hand side of Eq. ( 1) cannot equal zero, the relative humidity of the atmosphere during the daytime period for a given selected clear or partly cloudy day should be less than 100%, as seen in Eq. ( 2), which leads to a reduction in the available days to 355 days at the Yucheng site and 208 days at the Yingke site. Though some authors have suggested estimating potential surface solar radiation as a factor of extraterrestrial radiation [e.g., [START_REF] Allen | Crop evapotranspiration-Guidelines for computing crop water requirements[END_REF] assigned this factor to be 0.75 at sea level and incorporated the elevation to this factor above this level], the ratio of the observed clear-sky solar radiation to extraterrestrial radiation in previous studies [START_REF] Todorovic | Single-layer evapotranspiration model with variable canopy resistance[END_REF] and at the two sites in this study is shown to vary diurnally and seasonally.

Determining whether a day is clear by comparing the observed surface solar radiation to the potential surface solar radiation determined as a fixed factor of extraterrestrial radiation is likely to miss some days. Here, we briefly review the two-step procedure that was proposed by Tang et al. [2013a] to identify the days that have clear skies at the different time intervals. ,where is the slant-path transmissivity at the solar zenith angle ) on a given partly cloudy day is greater than the minimum of the corresponding   from two adjacent clear days, then that partly cloudy day will be selected for further analysis.

With the constraints provided by these selection criteria, 77 (76) clear-sky and partly cloudy days had clear skies at 10:00-11:00, 68 (73) days were clear at 12:00-13:00, and 100 (80) days were clear at 14:00-15:00 at the Yucheng (Yingke) site (see Table 3). The half-hourly meteorological variables and turbulent heat fluxes on these days were used to evaluate and validate the constant EF r method.

Correction of the energy imbalance in the EC measurements

The energy imbalance in surface EC measurements has been reported in numerous studies [START_REF] Twine | Correcting eddy-covariance flux underestimates over a grassland[END_REF][START_REF] Wilson | Energy balance closure at FLUXNET sites[END_REF][START_REF] Foken | The energy balance closure problem: An overview[END_REF]Tang et al., 2011bTang et al., , 2013b;;[START_REF] Stoy | A data-driven analysis of energy balance closure across FLUXNET research sites: The role of landscape scale heterogeneity[END_REF]. In most cases, the sum of the measured H and LE is less than the measured surface available energy. Although no consensus has been reached on how to best reconcile this energy imbalance, the Bowen ratio (BR) correction method and the residual energy (RE) correction method, which were proposed by [START_REF] Twine | Correcting eddy-covariance flux underestimates over a grassland[END_REF] and have been widely applied in previous years, are adopted in this study to correct the daytime 30-min averaged EC measurements to evaluate the EF r upscaling method under energy-closure conditions. In the BR correction method, the surface available energy is repartitioned into H and LE by conserving the original EC-measured Bowen ratio. In the RE correction method, all of the imbalanced energy goes into the measured LE. When the RE correction method is directly applied to each of the corresponding 30-min averaged LE measurements in a diurnal cycle, the daily LE after correction averaged over the clear-sky or partly cloudy days is shown to be less than that before correction due to the negative available energy during the nighttime, which seems unreliable and is primarily caused by the limitation of the correction method during the nighttime. Moreover, the BR correction method is found to be corrupted if the sum of the original H and LE is close to zero during the nighttime. Therefore, we assume that the ratio of the EC-measured daytime to daily LE before and after the correction is constant to derive the corrected daily LE measurement after applying the BR and RE correction methods to the daytime 30-min averaged EC measurements.

Results and discussion

4.1 Daytime variation in Todorovic's surface resistance and the noon-time evolution of the flux ratio Before evaluating the performance of the constant EF r method in upscaling instantaneous ET, the daytime variation in the half-hourly surface resistance estimated by Todorovic's method is examined in Figure 2. These daytime surface resistances were averaged over the days that had clear skies at 12:00-13:00 local time for the wheat and corn growth stages at the Yucheng and Yingke sites. Figure 2 clearly shows that the daytime surface resistance estimated from Eq. ( 1) exhibits a concave shape with the minimum occurring at noon and the maximum in the early morning or late afternoon. The averaged daytime surface resistance for the wheat (short crop) growth period was higher than that for the corn (tall crop) growth period at both sites. For both the wheat and corn growth periods, the estimated daytime surface resistance at the Yucheng site was between the fixed values recommended by the FAO-PM (70 s/m for short grass and 45 s/m for tall alfalfa) and the ASCE-PM (50 s/m for short grass and 30 s/m for tall alfalfa) equations near noon time (10:00 h -14:00 h) but was higher than them in the early morning and late afternoon. At the Yingke site, the variable surface resistance was higher than the fixed values recommended by the FAO-PM and the ASCE-PM equations.

The concave shape and site-to-site differences in the surface resistance are primarily determined by the daytime evolution of the meteorological variables of surface available energy, vapor pressure deficit, and wind speed at both sites; the sensitivity analysis presented in Figure 3 shows that a high vapor pressure deficit, a low wind speed, a low surface available energy, and a low air temperature can all lead to a high surface resistance and vice versa. Moreover, the sensitivity of the surface resistance to the surface available energy (wind speed, air temperature) decreases when the surface available energy (wind speed, air temperature) increases, and a higher sensitivity can be obtained with a higher vapor pressure deficit. Overall, the ET r,h /ET r,d computed at 12:00-13:00 local time from the three PM equations showed similar seasonal patterns and magnitudes at both sites; higher values occur in the winter and lower values occur in the summer, which reflects the temporal variations in the vapor pressure deficit, the surface available energy, and the disparate percentage of noon-time ET in the daily ET in a yearly cycle. In most cases, the ET r,h /ET r,d ratio varied between ~0.1 and ~0.2 at both sites and was higher at the arid Yingke site than at the sub-humid Yucheng site. The ET r,h /ET r,d using the full-form PM equation was generally higher than that using the ASCE-PM equation. Both were higher than that using the FAO-PM equation.

The discrepancy between the ET r,h /ET r,d ratios estimated using the variable and fixed surface resistances was larger when the ET was small and smaller when the ET was large, which indicates that using these three surface resistances and PM equations in the constant EF r method would yield a larger relative difference in the upscaled daily ET in the winter than that in the summer. However, at both sites, the discrepancy between the ET r,h /ET r,d ratios in the wheat growth stage with short grass as the reference crop was not significantly different from that in the corn growth stage with tall alfalfa as the reference crop.

Evaluation using original flux tower measurements

For a given time period, the paired-t test was used to test the null hypothesis that the mean of one PM model-predicted daily ET equals the mean of another PM model-predicted daily ET against the alternative hypothesis that the two statistics are not equal. After further selecting the days that had two assumed time periods, the paired-t test was separately used for all three PM equations to test the null hypothesis that the mean of the model-predicted ET at one time equals that at another time against the alternative hypothesis that the two statistics are not equal. The level of significance chosen for all tests was 0.05 (95% confidence level).

Evaluation using corrected flux tower measurements

The energy closure ratio, which is defined as the ratio of the sum of the half-hourly ECmeasured H and LE to the surface available energy, for the quality-, completeness-, and relative humidity-controlled days averaged 0.81 and 0.67 at the Yucheng and the Yingke sites, respectively. The paired-t test that was used in Section 4.2 was also applied in this section to evaluate the significance of the statistical results.

Figure 7a-c shows the statistical measures of the performance of the constant EF r upscaling method after using the BR correction method introduced in Section 3.3 to close the energy imbalance of the original EC measurements. At the Yucheng site, different from the results without the correction of surface energy imbalance, using the full-form PM equation and the FAO-PM equation produced the best and the worst results, respectively, from mid-morning to noon, whereas the opposite results were obtained in the mid-afternoon. At the Yingke site, using the three PM equations produced significantly different upscaled daily LEs at a given time from mid-morning to mid-afternoon with no exceptions. Using the FAO-PM equation and the fullform PM equation produced the best and worst results, respectively, from noon to mid-afternoon, whereas the opposite results were obtained in the mid-morning. Moreover, at both sites, using all three PM equations tended to produce progressively smaller upscaled daily LEs from midmorning to mid-afternoon. For a given PM equation, the daily LE upscaled at one time was significantly different from that at another time from mid-morning to mid-afternoon (with an exception at the Yingke site for the full-form PM equation between noon time and midafternoon) and the best model performance tended to occur in the noon time.

Figure 8a-c shows the statistical measures of the performance of the constant EF r upscaling method when using the RE correction method to close the energy imbalance of the original EC measurements. At the Yucheng site, the results were completely the same as those obtained when the BR correction method was applied. At the Yingke site, the results were almost the same as those obtained using the BR correction method. The only difference was that the daily LE upscaled at one time for a given PM equation was significantly different from that at another time from mid-morning to mid-afternoon with no exceptions.

Discussion

The performance of the constant EF r upscaling method in practical applications depends on how close the instantaneous EF r at a given clear-sky satellite overpass time is to the daily average. When the instantaneous EF r is larger (or smaller) than the daily average, the upscaled daily LE will be overestimated (or underestimated). Using the constant EF r method to upscale the LE requires accurate estimates of the reference hourly and daily ET values.

Given the same magnitude of the decrease in the surface resistance for all of the hourly periods during the daytime, the relative increase in the hourly reference ET from mid-morning to mid-afternoon is generally larger than that in the early morning and late afternoon because the reference ET near midday accounts for the largest fraction of the daily average. Due to the combined effect of the reduced nighttime reference ET and the smaller increase in the early morning and late-afternoon reference ET values than the mid-morning to mid-afternoon reference ET values, the relative increase in the hourly reference ET from mid-morning to midafternoon will generally be greater than the relative increase in the daily average, which results in a larger ET r,h /ET r,d and a smaller upscaled daily LE when using the ASCE-PM equation than when using the FAO-PM equation. In contrast, using the greater variable surface resistances derived from Todorovic's method than the fixed values adopted in the ASCE-PM equation produced smaller daytime reference ET estimates and, consequently, smaller daily reference ET estimates. Because the variable surface resistance has a concave shape during the daytime, the relative decrease in the hourly reference ET has a convex shape, and its absolute value is significantly smaller in the midday period than in the early morning and late afternoon. This most likely produces a larger ET r,h /ET r,d in the full-form PM equation than that in the ASCE-PM equation and a smaller upscaled daily LE. The large standard deviation as shown in Figures 678may indicate the instability of the EF r in a diurnal cycle under some circumstances over the wide range of atmospheric and soil moisture conditions at both sites (in other words, using the constant EF r upscaling method can create a large bias in the daily ET).

Progressively smaller upscaled daily ET values from mid-morning to mid-afternoon using the constant EF r method have also been found in previous studies [START_REF] Cammalleri | Upscaling of evapotranspiration fluxes from instantaneous to daytime scales for thermal remote sensing applications[END_REF]Tang et al., 2013a]. When the upscaled daily ET values from the constant EF r method were compared to the EC measurements without an energy imbalance correction, the overestimation (positive bias) during the mid-morning and underestimation (negative bias) during the mid-afternoon in this study and our prior study [Tang et al., 2013a] were not consistent with the results of [START_REF] Cammalleri | Upscaling of evapotranspiration fluxes from instantaneous to daytime scales for thermal remote sensing applications[END_REF], who found a stable underestimation in the upscaled daytime ET from mid-morning to mid-afternoon. Our finding that the noon time overall appeared to be the optimal time for instantaneous ET upscaling has also been revealed by other researchers [START_REF] Colaizzi | Comparison of five models to scale daily evapotranspiration from one-time-of-day measurements[END_REF][START_REF] Delogu | Reconstruction of temporal variations of evapotranspiration using instantaneous estimates at the time of satellite overpass[END_REF]Tang et al., 2013a;[START_REF] Cammalleri | Upscaling of evapotranspiration fluxes from instantaneous to daytime scales for thermal remote sensing applications[END_REF]. Because the ET r,h /ET r,d

does not vary before and after the energy imbalance correction, how the model performance of a given PM equation changes with the energy closure at a given time entirely depends on the relative variation in the corrected hourly and daily EC-measured ET. This may also lead to a change in the same inter-model performances of the three PM equations. Site-to-site and time-totime variations were found in this study in the performances of a given PM equation before and after the energy imbalance was closed, which agreed with [START_REF] Cammalleri | Upscaling of evapotranspiration fluxes from instantaneous to daytime scales for thermal remote sensing applications[END_REF] and Tang et al. [2013a], who found that the closure of EC measurements could have a significant effect on the choice of optimal upscaling technique. Because no studies have ever attempted to incorporate a variable daytime surface resistance in the upscaling of instantaneous ET data using the constant EF r method, our findings that are associated with the best upscaling technique and the optimal upscaling time in the three PM equations in this study cannot be further corroborated by the work of other authors.

Unexpectedly, using variable surface resistances did not produce a much closer upscaled daily ET to using a fixed surface resistance as adopted in the ASCE-PM equation at the subhumid Yucheng site than at the arid Yingke site. This may be because the reference ET estimate is controlled not by the absolute value of the surface resistance but primarily by the ratio of the surface to aerodynamic resistances; both resistances generally have similar diurnal patterns.

Underestimated daily LE values upscaled from the constant EF r method at 12:00-13:00 local time using Todorovic's variable surface resistance in the reference ET estimates at the arid site in this study are also found in the work of [START_REF] Liu | A novel method to convert daytime evapotranspiration into daily evapotranspiration based on variable canopy resistance[END_REF] when the BR correction method is applied to close the energy imbalance of the EC measurements. Both studies produced relative root mean square errors (RMSEs, ratio of the RMSE to the observed mean) with similar magnitudes. However, the constant EF r method using Todorovic's variable surface resistance performs better at the humid Yucheng site than at the arid Yingke site, with lower (relative) absolute BIAS and RMSE values but a higher coefficient of determination (R 2 ). This may be explained by the explanation by [START_REF] Allen | A recommendation on standardized surface resistance for hourly calculation of reference ETo by the FAO56 Penman-Monteith method[END_REF] that the climatic resistance model of Todorovic works reasonably well for rain-fed agriculture but can fail in irrigated areas in semiarid and arid climates. Based on studies by European and American researchers, Allen et al.

[2006] recommended using fixed surface resistance values of 50 s/m for daytime and 200 s/m for nighttime for the reference ET estimates. Nevertheless, several authors have demonstrated that using variable surface resistances is preferential to using a fixed surface resistance of 70 m/s for daytime reference ET estimates, even at semi-arid sites [START_REF] Todorovic | Single-layer evapotranspiration model with variable canopy resistance[END_REF][START_REF] Lecina | Fixed versus variable bulk canopy resistance for reference evapotranspiration estimation using the Penman-Monteith equation under semiarid conditions[END_REF][START_REF] Perez | A simple parameterization of bulk canopy resistance from climatic variables for estimating hourly evapotranspiration[END_REF]. Moreover, using variable surface resistances in this study was indeed able to produce better daily ET estimates in the constant EF r upscaling method in the mid-afternoon compared to using fixed surface resistances at both the Yucheng and Yingke sites when the energy imbalance of the EC measurements was not closed.

The low energy closure ratio (0.67 at the Yingke site) or the surface energy imbalance can be caused by a number of other factors [START_REF] Wilson | Energy balance closure at FLUXNET sites[END_REF][START_REF] Foken | The energy balance closure problem: An overview[END_REF][START_REF] Stoy | A data-driven analysis of energy balance closure across FLUXNET research sites: The role of landscape scale heterogeneity[END_REF],

among which include the measurement errors of surface energy components, the mismatch of source areas between surface flux components, the uncorrection of heat storage for G, the neglect of the heat storage in the canopy, the neglect of photosynthetic and advective energy, and the dispersive fluxes not sampled by the EC system. The reasons for the closure errors in the EC measurements are difficult to realistically identify because so many factors can influence the energy imbalance. These various influencing factors and the applied correction methods can have an impact on the EC-measured ET and the validation results of upscaled daily ET. The error in the validation of upscaled daily ET is therefore difficult to attribute to a certain factor (e.g., poor G representation) or a correction method. Nevertheless, knowledge of the model performances under given conditions at the Yucheng and Yingke sites is still valuable.

The limitation in using the constant EF r upscaling method primarily lies in its requirement of metrological measurements (solar radiation, air temperature, vapor pressure deficit, wind speed) as inputs. However, a great advantage of using this method towards other simpler scaling techniques (e.g., using at-surface solar radiation as the reference variable, as in Cammalleri et al.

[2014] and using surface net radiation as the reference variable, as in [START_REF] Nagler | An empirical algorithm for estimating agricultural and riparian evapotranspiration using MODIS enhanced vegetation index and ground measurements of ET. I. Description of method[END_REF]) is that it can incorporate the effect of advective energy and time-dependent environmental variables on the ET in a diurnal cycle. Moreover, deriving a variable resistance does not require inputs any more than estimating the reference ET from the Penman-Monteith equation when the constant EF r upscaling method is applied. Using variable surface resistance could potentially improve the reference ET estimates and thus the ET upscaling in the constant EFr method [START_REF] Liu | A novel method to convert daytime evapotranspiration into daily evapotranspiration based on variable canopy resistance[END_REF][START_REF] Pereira | Evapotranspiration: concepts and future trends[END_REF][START_REF] Rana | Measurement and estimation of actual evapotranspiration in the field under Mediterranean climate: a review[END_REF]Lecina et al., 2013;[START_REF] Perez | A simple parameterization of bulk canopy resistance from climatic variables for estimating hourly evapotranspiration[END_REF].

Summary and conclusions

In the upscaling of instantaneous ET, both the constant EF r method and the constant global solar radiation ratio method outperform the constant evaporative fraction method, the constant extraterrestrial solar radiation ratio method, and the sinusoidal function method, though there is no agreement on which of the former two methods performs better [START_REF] Xu | Temporal upscaling and reconstruction of thermal remotely sensed instantaneous evapotranspiration[END_REF]Tang et al., 2013;[START_REF] Cammalleri | Upscaling of evapotranspiration fluxes from instantaneous to daytime scales for thermal remote sensing applications[END_REF]. To investigate whether the use of a variable surface resistance in the reference ET estimation from the PM equation, as proposed by [START_REF] Todorovic | Single-layer evapotranspiration model with variable canopy resistance[END_REF], can improve the daily upscaled ET on clear-sky days from the constant EF r method, half-hourly nearsurface meteorological variables and EC-measured LE values were collected as the model inputs and validation data, respectively, at two sites with contrasting climatic conditions, namely, the sub-humid Yucheng station in northern China and the arid Yingke site in northwestern China.

The use of Todorovic's variable surface resistance in this study does not require a priori calibration.

The results before the energy imbalance in the EC measurements was closed are summarized as follows:

i) In the mid-afternoon, using variable surface resistance in the full-form PM equation at both sites produced the best ET upscaling results, and using fixed resistance in the FAO-PM equation produced the worst results.

ii) In the mid-morning to noon, using fixed resistance in the ASCE-PM equation at the Yucheng site produced the best results, and using the FAO-PM equation produced the worst results. At the Yingke site, using the FAO-PM equation and the full-form PM equation produced the best and worst results, respectively.

iii) For all three PM equations, the daily LE upscaled in the mid-morning was not significantly different from that in the noon time, and the model performances were overall statistically better in the mid-morning and noon time than in the afternoon. iv) At both sites, using all three PM equations tended to produce progressively larger upscaled daily LEs from mid-morning to mid-afternoon. For a given time, ET FAO-PM > ET ASCE-PM > ET TOD-PM .

After the energy imbalance was closed using the BR and RE correction methods, some changes were found in the results as follows:

i) In the mid-morning, using the full-form PM equation at both sites produced the best results, and using fixed resistance in the FAO-PM equation produced the worst results.

ii) In the mid-afternoon, using the FAO-PM equation and the full-form PM equation at the Yucheng site produced the best and the worst results, respectively. The opposite results

were obtained at the Yingke site.

iii) In the noon time, using the full-form PM equation at the Yucheng site produced the best results, and using the FAO-PM equation produced the worst results. The opposite results

were obtained at the Yingke site.

iv) For a given PM equation, the daily LE upscaled at one time was significantly different from that at another time from mid-morning to mid-afternoon, and the best model performance tended to occur in the noon time.

v) At both sites, using all three PM equations tended to produce progressively smaller upscaled daily LEs from mid-morning to mid-afternoon.

In summary, using the three PM equations produced significantly different upscaled daily ETs (ET FAO-PM > ET ASCE-PM > ET TOD-PM ) at a given time from mid-morning to mid-afternoon.

Using all three PM equations overall produced the best results at noon at both sites regardless of whether the energy imbalance was closed. When the EC measurements were not corrected for energy imbalance, using variable surface resistance in the full-form PM equation could improve the ET upscaling in the mid-afternoon, but worse results may occur in the mid-morning to noon.

Site-to-site and time-to-time variations were found in the performances of a given PM equation before and after the energy imbalance was closed. At a given site, using the BR correction method yields the same inter-model performances for the three PM equations as using the RE correction method. It has to be mentioned that when detailed meteorological measurements are not available at the ground station, to temporally upscale the remote sensing instantaneous ET in a practical application using the constant EF r method, regional reference ET can be estimated with reanalysis data from numerical weather prediction models and global climate models [START_REF] Tian | Forecasting reference evapotranspiration using retrospective forecast analogs in the southeastern United States[END_REF][START_REF] Ishak | Estimating reference evapotranspiration using numerical weather modelling[END_REF]. When soil heat flux measurement is not available, to close the energy imbalance of EC measurements, semi-empirical relationships as proposed by kustas and Daughtry (1990), [START_REF] Bastiaanssen | A remote sensing surface energy balance algorithm for land (SEBAL). 1. Formulation[END_REF][START_REF] Su | The surface energy balance system (SEBS) for estimation of turbulent heat fluxes[END_REF] can be applied to derive a spatially representative soil heat flux from surface net radiation measurement and remote sensing vegetation index. Note that although the evaluation in this study is only conducted at two sites, it is instructive and can be conclusive to a great extent because these two sites are characterized by contrasting climatic conditions and data involved cover a wide range of soil moisture and vegetation cover conditions. To make a general conclusion, more extensive assessments and validations are required over a wider range of climate, soil moisture, and plant functional type conditions.

  wind speed; WD: wind direction; R s : shortwave radiation; R l : longwave radiation; G: soil heat flux; Pa: atmospheric pressure; T s : soil temperature; SM: soil moisture; H and LE: sensible and latent heat fluxes.

Figure 1 .

 1 Figure 1. Daytime variations of ground-based measurements of (a) surface available energy (R n -G), (b) vapor pressure deficit (VPD) and wind speed (u), averaged from April 1, 2009 to June 30, 2012 or over the selected 125 clear-sky days at the Yucheng (YC) site and from January 1, 2008 to November 30, 2009 or over the selected 93 clear-sky days at the Yingke (YK) site. The thirty-minute averaged sensible heat flux (H) and latent heat flux (LE) are regularly measured by an eddy covariance (EC) system that consists of an open-path CO 2 /H 2 O gas analyzer (model LI-7500, Licor Inc., Lincoln, Nebraska) and a 3-D sonic anemometer/thermometer (model CSAT3, Campbell Scientific Inc., Logan, Utah) at both sites.The EC sensor is elevated to ~4.3 m above surface level (ASL) in late July or early August and is lowered to ~2.9 m ASL in mid-to late-October every year at the Yucheng site but is fixed at ~2.8 m ASL at the Yingke site. At the Yucheng and Yingke sites, the relative height of the EC sensor is ~2.1 m and ~1.8 m above maximum wheat height and ~1.7 m and ~1.0 m above the maximum corn height, respectively. The main contributing area of the EC measurements is

First, days that

  are clear during the entire daytime period are selected by analyzing the observed half-hourly global solar radiation and the corresponding shortwave atmospheric transmissivity (, the ratio of the observed global solar radiation to the extraterrestrial solar radiation). If both values on a given day increase monotonically from sunrise (global solar radiation > 5 W/m 2 ) to midday and then decrease monotonically from midday to sunset (global solar radiation > 5 W/m 2 ), then that day will be classified as a clear-sky day and will be used for further analysis. Moreover, a visual check is made of the selected clear-sky days to remove those that are covered by constant cloud cover during the daytime.Next, days that are partly cloudy during the daytime but have clear skies at the assumed satellite overpass times are selected. Days that do not have clear skies or constant cloud cover over the entire daytime period are classified as partly cloudy days. For each of the three different daytime periods, if the vertical-path shortwave atmospheric transmissivity (

Figure 2 .

 2 Figure 2. Daytime variations of half-hourly surface resistances estimated from the method proposed by[START_REF] Todorovic | Single-layer evapotranspiration model with variable canopy resistance[END_REF] and averaged over the selected days that have clear skies at 12:00-13:00 local time for the wheat and corn growth stages at the Yucheng and Yingke sites.

Figure 3 .Figure 4 .

 34 Figure 3. (a) Contour of Todorovic's variable surface resistance (r s ) with the variations of surface available energy and vapor pressure deficit, when air temperature is at 30 o C and wind speed equals 3.0 m/s. (b) Contour of the r s with the variations of air temperature and wind speed, when surface available energy is at 450 W/m 2 and vapor pressure deficit equals 3.0 kPa.

Figure 5a -Figure 5 .

 5a5 Figure 5a-f compares the daily ET measured by the EC without an energy correction and that (ET FAO-PM , ET ASCE-PM , and ET TOD-PM ) upscaled from the constant EF r method, with the reference ET derived using the FAO-PM equation, the ASCE-PM equation, and the full-form PM equation at the Yucheng and Yingke sites. The corresponding statistical measures of the performance are illustrated in Figure 6a-c. (i) Using the three PM equations produced significantly different upscaled daily LEs at a given time from mid-morning to mid-afternoon, with an exception occurring at the Yingke site in the mid-morning between using the ASCE-PM equation and the full-form PM equation. (ii) At the Yucheng site, using the ASCE-PM equation and the full-form PM equation overall produced the best results from mid-morning to noon and in the mid-afternoon, respectively. Using the FAO-PM equation produced the worst results for all time intervals. At the Yingke site, using the FAO-PM equation and the full-form PM equation produced the best and the worst results, respectively, from mid-morning to noon whereas the opposite results were obtained in the mid-afternoon. (iii) Using all three PM equations tended to produce progressively larger upscaled daily LE from mid-morning to mid-afternoon. For a given time, ET FAO-PM > ET ASCE-PM > ET TOD-PM . (iv) For all three PM equations, the daily LE upscaled in the mid-morning was not significantly different from that in the noon time, and the model performances were overall statistically better in the mid-morning and noon time than those in the mid-afternoon.

Figure 6 .

 6 Figure 6. Statistical measures of the performance of constant reference evaporative fraction upscaling method in converting hourly ET estimated using the FAO-PM, the ASCE-PM, and the full-form PM equations at different clear sky time periods in the daytime to daily value over selected days (10:30 stands for 10:00-11:00 interval, similarly hereafter). The error bar represents one standard deviation of the difference between model-estimated and EC-measured daily ETs. For each of the panels, the left part is for the Yucheng site and the right part is for the Yingke site. The EC-measured LE was not corrected to close the energy imbalance. (a) BIAS, (b) root mean square error (RMSE), (c) coefficient of determination (R 2 ).

Figure 7 .Figure 8 .

 78 Figure 7. Comparison of Same as Figure 6, but for the application of energy closure of the ECmeasured LE using the Bowen ratio correction method.

Table 1 .

 1 Attributes

	Site	Longitude (E)	Latitude (N)	Land cover	Elevation(m)	AP	AT	Data period
	Yucheng	116.5703	36.8291	Crop	28	582	13.1 04/2009-06/2012
	Yingke	100.4103	38.8571	Crop	1519	127	7.2	01/2008-11/2009

of the two sites in China used for the evaluation of the constant reference evaporative fraction upscaling method (AP: Annual mean precipitation, mm; AT: Annual mean temperature, ºC)

Table 3 .

 3 Number

			Time period
	Site	10:00-11:00	12:00-13:00	14:00-15:00
	Yucheng	77	68	100
	Yingke	76	73	80

of the clear-sky and partly cloudy days that had clear skies at three different daytime periods from April 2009 to June 2012 at the Yucheng site and from January 2008 to November 2009 at the Yingke site.

  -hourly averaged atmospheric variables (global solar radiation, air temperature, wind speed, relative humidity, and atmospheric pressure) were collected from April 2009 to June 2012 at the Yucheng site and from January 2008 to November 2009 at the Yingke site. The data were used to drive the reference ET model and the daily (24-h) averaged latent heat flux measured by the EC with and without an energy imbalance correction to validate the upscaled value from the constant EF r method. These data were carefully checked to ensure the quality and completeness of the daily cycle in two steps. First, data spikes and abnormalities in the measured H and LE

	(less than -100 W/m 2 or greater than 700 W/m 2 [Tang et al., 2013; Liu et al., 2011], which
	approximate the lower and upper limits of surface net radiation) were removed, and then days
	with data gaps caused by rainfall events, instrument malfunctions or maintenance, and data
	spikes and abnormalities in the 30-min averaged atmospheric variables or EC-measured H and

LE were excluded. After these data quality and completeness checks, 485 days from April 2009 to June 2012 at the Yucheng site and 272 days from January 2008 to November 2009 at the Yingke site were available for further analysis. The daily measured surface fluxes (surface net 3.2 Selection of days with a clear sky at the assumed "satellite overpass" time

Acknowledgments and Data

The staff members at the Yucheng site are acknowledged for their hard-work on the setup and maintenance of the ground-based instruments and data collection. Associate Professor Yuanyuan Jia and Professor Chuanrong Li in the Academy of Opto-Electronics, Chinese Academy of Sciences, are thanked for their cooperation in providing the surface measurements. EC data at the Yingke site used in this study are downloaded from "Environmental & Ecological Science Data Center for West China, National Natural Science Foundation of China, http://westdc.westgis.ac.cn". This work was partly supported by the National Natural Science Foundation of China under Grant 41571351 and 41571367, and in part by the International Science and Technology Cooperation Program of China under Grant 2014DFE10220.