Based on the Gaussian Fitting Method to Derive Daily Evapotranspiration from Remotely Sensed Instantaneous Evapotranspiration

School of Water Conservancy, North China University of Water Resources and Electric Power, Zhengzhou 450046, China Henan Key Laboratory of Water Environment Simulation and Treatment, Zhengzhou 450046, China Key Laboratory of Water Cycle & Related Land Surface Processes, Institute of Geographic Sciences and Natural Resources Research, Chinese Academy of Sciences, Beijing 100101, China Department of Civil, Environmental and Geomatics Engineering, Florida Atlantic University, Florida, FL 33431, USA Shenzhen Environmental Monitoring Center, Shenzhen 518049, China


Introduction
Evapotranspiration (ET), which is crucial to the hydrological cycle, is defined as the synthesis process of evaporation and transpiration. It is the link of energy and water exchanges among the biosphere, atmosphere, and hydrosphere [1][2][3][4][5]. In most cases, ET is the largest loss of precipitation, and it is a significant outgoing water flux from the earth's surface. In semiarid areas, the amount of ET almost equals that of precipitation [6]. Hence, accurate estimation of ET is beneficial to improve applications in many fields, such as drought mitigation strategies, irrigation system performance, optimization of irrigation water use, hydrological modeling, and accurate initialization of climate prediction models, and is also very useful for understanding the global climate change, the local to global energy and water cycles, ecosystem processes, and land-atmosphere interaction [7][8][9][10][11][12]. Ground-based observations including Bowen ratio tower, eddy covariance [13], lysimeter, and large aperture scintillometer can provide ET measurements with some advantages. However, regional ET acquired through these methods is time-consuming and labour-intensive because it requires numerous installations and considerable spatial interpolations [8].
Satellite remote sensing makes it possible for acquiring regional ET over various spatial scales, ranging from individual pixels to an entire raster image that may cover a whole river basin [14]. In the last two decades, the recent advances of remote sensing technology together with the requirement for quantifying regional ET have brought about numerous researches in obtaining large-scale ET [8,10,15,16]. Remote sensing can retrieve an instantaneous ET on a regional scale at the time of satellite overpass. However, daily ET estimates are required for water resources monitoring and ecological management purposes [14,17]. Consequently, it is of significance to convert instantaneous ET into daily ET [18,19].
Several instantaneous ET extrapolation methods are proposed and developed to derive daily ET, such as the sine function method [20], the evaporative fraction (EF) method [21], and the reference ET fraction (ETrF) method [22]. Jackson et al. [20] proposed a technique based on the ratio of daily solar radiation to instantaneous solar radiation. Given it was similar to that of solar irradiance throughout the daylight period, they assumed the generic trend for the ET diurnal course could be approximated by a sine function which is named the sine function method. Zhang and Lemeur [23] did researches on the sine function method, and they concluded that the sine function method was preferable to estimate daily ET using remote sensing data. e disadvantage of this method is that it is limited by its empirical nature in applications [14]. e EF, defined as the latent heat flux divided by the latent heat flux plus sensible heat flux (available energy (AE)), is nearly constant during the daytime period [24,25]. Hence, the method can utilize instantaneous EF and continuous measurements of the available energy flux to determine daily ET. Studies [23] showed that the assumption of a constant EF was valid under cloud-free conditions. Sugita and Brutsaert [26] also yielded accurate estimates of daily ET by the EF method. However, some studies found that EF changes with the available energy, surface resistance, and other environmental variables, which caused uncertainties in applying the EF method [23]. Tasumi et al. [27] investigated a method labeled "reference ET fraction," which was defined as the ratio of actual ET to reference ET (ETr) for an alfalfa crop. is method assumes that the instantaneous ETrF is similar to the daily average ETrF. Many studies have been conducted to utilize the ETrF method to derive daily ET, and the results show that the ETrF remains constant during the daytime [28]. e ETrF method, however, seems to perform well under homogeneous surface conditions [29]. e aforementioned instantaneous ET extrapolation methods request numerous variables, and some of the variables may be difficult to attain through remote sensing. For example, the EF method needs an instantaneous EF value and daytime total available energy, the sine function method requests several variables related to geographic location, and the ETrF method demands variables linked with specific crops. To simplify the computation process of daily ET, the study put forward a method of deriving daily ET, which was based on the ET diurnal course and similar to the sine function method. In this paper, we assume that, for clear sky days, the diurnal course of solar radiation and ET can be adequately expressed by the Gaussian fitting curve and then develop the Gaussian fitting approach for calculating daily ET from instantaneous ET. Section 2 includes two subsections: Section 2.1 presents a description on the theory of retrieving instantaneous ET by remote sensing and Section 2.2 introduces the Gaussian fitting method for deriving daily ET. Section 3 describes the datasets and the study area used to assess the Gaussian fitting method. Section 4 shows the results. Section 5 provides discussions on the advantages and limitations of the Gaussian fitting method, and Section 6 summarizes a conclusion of the work.

Obtaining the Instantaneous ET.
In the study, we adopted the energy balance theory to compute the instantaneous ET. Without considering the energy transported by horizontal advection and consumed by photosynthesis, the energy exchanged between the land surface and the atmosphere can be described by the energy balance equation: where R n , H, LE, and G are the net radiation, sensible heat flux, latent heat flux, and soil heat flux, respectively. Units of the four items are W/m 2 . In equation (1), net radiation, R n , sensible heat flux, H, and soil heat flux, G, can be determined by the following equations, respectively: where S 0 (W/m 2 ) is the downward shortwave radiation, α is the surface albedo, R ld (W/m 2 ) is the downward longwave radiation, σ is the Stefan-Boltzmann constant and the value of it is 5.67 × 10 −8 W·m −2 ·K −4 , ε s is the surface emissivity, T 0 (K) is the aerodynamic temperature and is usually substituted with land surface temperature (LST) in applications [30,31], ρ is the air density, C p (1004 J/(kg·K)) is the specific heat at constant pressure of air, and r a (s/m) is the air aerodynamic resistance and can be calculated by the classical formulae that take into account the stability correction functions for temperature and wind. Readers can refer to the calculation process of r a by Abdelghani et al. [32]. T a (K) is the surface air temperature, and f v is the fractional vegetation cover which was calculated using NDVI images, with NDVI min of the bare soil and NDVI max of dense vegetation. e formula is as follows [33]: 2 Advances in Meteorology where NDVI is the ratio of the differences in reflectivity between the near-infrared (NIR) band and the red (R) band to their sum: where ρ NIR and ρ R are the reflectivities for near-infrared and red bands, respectively. e latent heat flux, LE, can be acquired as the following equation by integrating equations (1)-(4): where variables including α, ε s , LST, and f v can be calculated through remote sensing images, r a is attained by referring to Abdelghani et al. [32], and variables including S 0 , R ld , and T a are acquired from the meteorological stations.

2.2.
Obtaining the Daily ET. Jackson et al. put forward the assumption that the diurnal course of ET was similar to that of solar irradiance and could be approximated by a sine function [20,23]. e attempt provided in our study is also an approximation similar to the sine function method, which is named the Gaussian fitting method. According to our observations, we found that not only the diurnal course of net radiation but also the diurnal course of ET during the daytime can be approximated by the Gaussian fitting curve [34]. Hence, we applied the Gaussian fitting curve to fit the diurnal course of ET on several clear sky days. A comparison between ET measurements at EC stations and ET estimates by the Gaussian fitting curve is shown in Figure 1. e points represent the values of net radiation measurements observed every 10 minutes (Figure 1(a)) and ET measurements observed every 30 minutes (Figure 1(b)) throughout the day, respectively. e lines represent the best fits of Gaussian fitting curves to the experimental data. It is obvious that the Gaussian fitting curves closely follow the experimental data during the daytime (from sunrise to sunset).
e Gaussian fitting function can be expressed as follows: To explain the meanings of the variables in equation (8), we display a Gaussian fitting curve to help the description ( Figure 2). Combining equation (8) with Figure 2, the variables in equation (8) are illustrated clearly: y 0 is the offset; x c is the center where y has y max , y c is the maximum value when x � x c , and y max is equal to y c ; (x c , y c ) is the extreme value point; w is the width that equals 2δ, in which δ is the standard deviation of y; and A is the area of the curve. Comparing the diurnal course of latent heat flux with equation (8), we can rewrite equation (8) and attain the similar formula for expressing the diurnal course of ET, as follows: where ET i is the instantaneous ET at the moment of the satellite overpass, ET 0 is the value at sunset or sunrise (usually equals zero), t i is the time of the satellite overpass, and t c is the time when the instantaneous ET arrives at its daily maximum. To acquire the daily ET, we need to calculate the integral of equation (9), which would be a complex process. Fortunately, we notice that the variable A is the area of the curve (equation (9)) and is the approximate integration of instantaneous ET. Hence, the daily ET can be attained as follows: From formula derivation above, we know that the unique advantage of the Gaussian fitting method is that the expression of Gaussian fitting equation has already contained the daily ET. In equation (10), if ASTER satellite data, which pass the western China at about 12 : 15 pm, are used, then t i equals 12.25 (h), and t c is the time when the instantaneous ET arrives at its daily maximum and is usually set to 14.5 (h).
As for w, we adopted an empirical value that was equal to half of the time, during which net radiation is greater than zero (t R n >0 ). e following are the justifications on the determination of w.
To find out the relationship between t R n >0 and w, we utilized one-month measurements of net radiation and ET (except cloudy days) at four EC stations with different underlying surfaces (Table 1). t R n >0 is determined by net radiation measurements and w is calculated by making Gaussian fitting on ET observations. Figure 3 gives the linear fitting results between t R n >0 /2 and w. It reveals that the relationship of t R n >0 /2 and w is close to the 1 : 1 line. As a result, w can be approximated as t R n >0 /2.

Study Area.
Belonging to the middle basin of the second longest inland river of China, Heihe River, the study area is located in Zhangye Oasis, Gansu, China, with the latitude of 38.83°N∼38.93°N and the longitude of 100.32°E∼100.42°E, as shown in Figure 4. Because of being landlocked, it is obviously characterized by an arid climate with a low annual mean rainfall of 124.9 mm, a high potential evaporation of more than 2000 mm, a large temperature difference between day and night, and long hours of sunshine. e study region is covered with maize, orchard, vegetable, woodland, and builtup areas (villages). From May to September, the staple crops in the study region are seed corns, fruit trees, and vegetables.
For retrieving the daily ET, remote sensing data, such as ASTER images and HJ-1 A/B images, together with groundbased measurements including meteorological variables and surface fluxes are utilized.

Remote Sensing Data.
ASTER is a sensor that acquires numerous images in different bands including multispectral visible, near-infrared, and thermal infrared. It is intended to monitor climate, land surface energy balance, and hydrological processes [35]. In the study, the vegetation fraction and the surface emissivity were estimated through ASTER images in the visible and near-infrared wave bands (0.52∼0.86 µm), which had a high spatial resolution of 15 m. As for the variable of LST, the study utilized products provided by the Heihe Plan Science Data Center [36,37], which were retrieved from the thermal infrared wave band (8.125∼11.65 µm) of ASTER images, with a spatial resolution of 90 m. e albedo, which is another imperative variable,    Figure 4) equipped with EC stations and automatic meteorological stations. e 17 elementary sampling plots were divided according to the distribution of crops, shelterbelts, residential areas, roads, and canals, as well as according to soil moisture and irrigation status [39].
Original EC measurements were collected at a sampling frequency of 10 Hz. e processing work on EC data includes spike detection, lag correction of H 2 O/CO 2 relative to the vertical wind component, sonic virtual temperature correction, coordinating rotation using the planar t method, corrections for density uctuation (WPL correction), and frequency response correction. e postprocessing software named "EdiRe" was utilized to make the above corrections (University of Edinburgh; http://www. geos.ed.ac.uk/abs/research/micromet/EdiRe) [40,41]. Besides the corrections made by the EdiRe software, the halfhourly ux data are screened based on four criteria: (1) data are rejected when the sensor is malfunctioning (e.g., when there is a fault diagnostic signal), (2) data are rejected when precipitation occurs within 1 h before and after the collection, (3) incomplete 30 min data are rejected when the missing ratio is larger than 3% in the 30 min raw record, and (4) data are rejected at night when the friction velocity is below 0.1 m/s [13]. e automatic meteorological stations can monitor meteorological variables, such as solar radiation, wind speed, wind direction, air pressure, air temperature, and air humidity, with the sample intervals of 10 minutes and 1 minute. e EC stations can acquire surface uxes including latent heat ux and sensible heat ux with a sample interval of 30 minutes [42].

Validity Veri cation.
To test the validity of the Gaussian tting method, we utilized the Gauss tting curve to simulate the diurnal variation of instantaneous ET observed at EC stations with di erent underlying surfaces. Table 1 gives To make the validity verification more persuasive, we used the whole EC data in June, except that on cloudy, rainy, and data missing days, from four sites (EC1, EC4, EC10, and EC17) to make the Gaussian fitting analysis. Figure 5 gives the Gaussian fitting results on ET measurements on nine days in June 2012 at EC 1. Similarly, fitting analyses for ET measurements on twelve days in June 2012 at EC4, EC10, and EC17 were also carried on from the aspects of the fitting equation, the coefficients of determination (R 2 ), and the root mean square errors (RMSEs), and the results were summarized and displayed in table forms, as shown in Tables 2-4, respectively. For EC1 with the vegetable surface, the R 2 between the simulations by the Gaussian fitting method and the measurements are higher than 0.9 on most days and the RMSEs are lower than 20 W/m 2 on all days. For EC4, EC10, and EC17 with village, maize, and orchard surfaces, respectively, the R 2 are higher than 0.84 and the RMSEs are lower than 20 W/m 2 . Previous studies [16] showed an RMSE of about 20-45 W/m 2 on diurnal ET estimates. erefore, our results are satisfactory. According to the above results, it is obvious that the Gaussian fitting results in June 2012 are quite consistent with the diurnal variation of ET measurements at EC stations, no matter the underlying surface, which means the Gaussian fitting method can describe the diurnal variation of ET accurately and can be used as an approach to simulate daily ET.

Evaluating Daily ET Estimates Derived by the Gaussian
Fitting Method. Four clear sky days, June 24, July 10, and August 11 and 27, 2012, were utilized to calculate the daily ET by the Gaussian fitting method. As mentioned in Section 3.1, remote sensing data (ASTER images and HJ-1 A/B images) with ground-based measurements were combined to retrieve instantaneous ET on the four days, and then, the retrieved instantaneous ET was inputted into the Gaussian fitting method to derive daily ET. e spatially distributed daily ETs on the four days are shown in Figure 6. To evaluate the daily ET estimates by the Gaussian fitting method, analyses in four aspects that are results validation, error analysis, the relationship between daily ET estimates and the land use status, and the comparison between the Gaussian fitting method and the sine function method were performed.

Validating Daily ET Estimates.
Daily ET estimates derived by the Gaussian fitting method for the four days are tested against ET measurements (68 points) at ground-based EC stations (Figure 7). An R 2 of 0.82, an MAE of 0.41 mm, and an RMSE of 0.46 mm were obtained. Almost all the points had absolute errors lower than 1 mm and 69% of the points had absolute errors lower than 0.5 mm. e correlation analysis stated that the retrieved daily ET estimates were quite consistent with the ground-based observations, which means that daily ET estimates by the Gaussian fitting method were close to the 1 : 1 line and had high accuracy.

Error Analyses.
e study made error analyses from two aspects, one is the percent error and the other is the relationship between errors and land use.
According to the error analysis and the numerical analysis theory, the percent error is more scientific and more robust than the absolute error for assessing the accuracy of estimates [43]. erefore, the frequency distribution of the percent errors was conducted. e definition of the percent error is the absolute error divided by the magnitude of the exact value. Its expression is where Δ is the absolute error, δ is the percent error, and L is the exact value. Figure 8 shows the frequency distribution of the percent errors on daily ET estimates, which had a variation from 0% to 18%. An accuracy of the percent errors within 10% was achieved at more than 80% of locations, and the other 20% of locations had the percent errors between 10% and 18%. e results demonstrated that the Gaussian fitting method had high estimation accuracy.
As for the relationship between errors and land uses, we utilized estimations from land covers of vegetable, village, maize, and orchard and their corresponding ground-based observations that were at the site of EC1, EC4, EC10, and EC17 on June 24, July 10, and August 11 and 27, 2012, to calculate and analyze the estimation errors. To avoid abnormal variation, the study used the average values of observations and estimations for four days to analyze, and the results are shown in Table 5. e land cover of village (EC4) had the maximum estimation error, whereas the other three land cover types, the vegetable, the maize, and the orchard, had minor estimation error, which was concordant with the conclusions of Section 4.1 that the Gaussian fitting effects at EC4 were less effective than those at the other three sites. e estimation error analyses indicated that the Gaussian fitting method had higher precision on underlying surfaces covered with vegetation than on bare areas.

Relationship between ET Estimates and Land Use.
Previous studies suggested that the spatial distribution of ET was strongly related to land cover types and that studies on ET estimates at a regional scale always required the incorporation of heterogeneous surface status [44]. Hence, the relationship between ET estimates and land use status can illustrate the rationality of ET estimates in some degree. e land use maps (Figure 4) provided by Heihe Plan Science Data Center [45] were derived from the CASI (aerial remote sensing data of Compact Airborne Spectrographic Imager) and adopted the SVM (support vector machine) as the classification method. Comparing the spatial distribution of daily ET estimates ( Figure 6) with the land use status (Figure 4), it is obvious that ET estimates linked with the land use type well; that is, built-up areas (village) displayed low ET values, and dense vegetation areas (maize and orchard) showed high ET values. 6 Advances in Meteorology       Advances in Meteorology e statistical work was done to make the analyses quantitatively. In the study area, there were four main land use types including maize, vegetable, orchard, and village. For each land use type, average daily ET on the four days was computed (Table 6). Clearly, maize and orchard had the maximum values. Maize had the average daily ET of 6.6 mm, 6.1 mm, 4.2 mm, and 4.6 mm on the four days, respectively; orchard had that of 6.8 mm, 5.9 mm, 4.3 mm, and 5.1 mm on the four days, respectively. Average daily ETfor the vegetable was 5.3 mm and 4.7 mm on June 24 and July 10, respectively, and was around 3.5 mm on the other two days. Village areas     displayed the lowest daily ET, with an average value of around 4 mm on June 24 and July 10 and no more than 2.5 mm on the other two days. e spatial distribution of daily ET was strongly connected with the land cover. In June, July, and August, maize and fruit trees were in the vigorous growth season and with frequent irrigation. As a result, the vegetation cover was dense and the daily ET was high. Compared with maize and orchard, vegetables including pepper, leek, and cauliflower had a sparse vegetation cover during this time, and therefore, their daily ET values were lower than those of maize and orchard. Since villages were covered with buildings and the underlying surfaces were largely bare and solidified, the daily ET values over villages were the lowest.

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Generally, high daily ET always agrees with a dense vegetation cover and low daily ET is usually in accordance with a sparse vegetation cover, which indicates that the daily ET results obtained by the Gaussian fitting method are consistent with the objective knowledge and can well perform the daily ET differences brought about by land use status.

e Comparisons between the Gaussian Fitting Method and the Sine Function Method and the ETrF Method.
Similar to the sine function method, the Gaussian fitting method is also based on the diurnal variation of ET; hence, this section gives the comparisons between the Gaussian fitting method and the sine function method. According to Xu et al. [46], the ETrF method produced the smallest RMSE values during the vegetation growing season; then, the study also compared the Gaussian fitting method to the ETrF method. Applying same data described above, daily ET was estimated by the sine function method and the ETrF method. Figure 9 shows the correlations between measurements and estimates acquired by the sine function method (Figure 9(a)) and those retrieved from the ETrF method (Figure 9(b)), respectively. Figure 9 illustrates that both estimates calculated from the sine function method and the ETrF method have good consistency with the measurements, with an R 2 of 0.8, an MAE of 0.58 mm, and an RMSE of 0.67 mm and with an R 2 of 0.82, an MAE of 0.44 mm, and an RMSE of 0.56 mm, respectively. Compared with the Gaussian fitting method, which has an R 2 of 0.82, an MAE of 0.41 mm, and an RMSE of 0.46 mm, daily ET estimates from the sine function method have higher R 2 , MAE, and RMSE and estimations by the ETrF method have slightly higher MAE and RMSE, which means the Gaussian fitting method is much appropriate than the sine function method and the ETrF method in estimating the daily ET.

Discussion
e Gaussian fitting method applies the Gaussian fitting curve to simulate the diurnal course of ET. Hence, the principle of the method is clear to be understood, and the method is convenient to be utilized. e most obvious advantage of the Gaussian fitting method is that the Gaussian fitting equation has already contained the variable A (the area of curve) which is equal to the daily ET through numerical quadrature. As a result, the acquisition of daily ET is easy. Furthermore, the Gaussian fitting method needs only one time of instantaneous ET retrieved from remote sensing images to estimate daily ET. Compared with the sine function method that is also based on the diurnal course of daytime ET, the Gaussian fitting method requests fewer variables. erefore, it is much simpler to be used.
e Gaussian fitting method also has some uncertainties. e crucial step in the applications of the Gaussian fitting method is determining the variables w and t c . However, w is assumed to be equal to half the length of the time during which net radiation is greater than zero, which is totally dependent on the observations. As for t c , it is highly dependent on the geographic location and its value needs to be adjusted when the study area is different. If there are no ground-based observations for determining variables w and t c , the Gaussian fitting method would lead to inaccurate daily ET estimates. erefore, it is necessary to find alternative methods to overcome these issues in the future.

Conclusions
Remote sensing is a promising tool to retrieve instantaneous ET on a regional scale. However, the daily ET or a longer timescale ET is more significant to monitor and manage the water resource. Hence, it is essential to convert instantaneous ET into daily ET. e study proposes the Gaussian fitting method to derive daily ET from remotely sensed instantaneous ET. Model validation and application were conducted to test the validity and evaluate the accuracy of the Gaussian fitting method. Model validation showed that the Gaussian fitting curve could well describe the diurnal course of ET measurements at EC stations, with high coefficients of determination and low root mean square errors. As a result, the Gaussian fitting method could be taken as an appropriate approach to simulate daily ET. A case study was performed in the middle reaches of Heihe River, China, to derive daily ET from remotely sensed instantaneous ET. e comparison between daily ET estimates derived by the Gaussian fitting method and measurements obtained from EC stations showed an R 2 of 0.82, an MAE of 0.41 mm, and an RMSE of 0.46 mm, which indicated that the simulated daily ET estimates were quite consistent with the ET measurements and had high accuracy. e frequency distribution of the percent errors also displayed high estimation accuracy, with a variation range of 0%∼18%. As for the relationship between daily ET estimates and land cover, the comparison showed that the Gaussian fitting method 10 Advances in Meteorology could well demonstrate daily ET differences caused by land use types. Estimation error analyses stated that the Gaussian fitting method had higher precision on underlying surfaces covered with vegetation than on bare areas. All analyses conclude that the Gaussian fitting method is efficient and feasible to expand instantaneous ET into daily ET.
Data Availability e data used in the manuscript include two types, the remote sensing data and the ground-based data. For remote sensing data, the readers can apply for and obtain ASTER images and HJ-1 A/B images through the official websites of USGS (https://glovis.usgs.gov/) and the Satellite Environment Center, Ministry of Environmental Protection of China (http://www.secmep.cn/). Ground-based data were provided by the HiWATER (Heihe Watershed Allied Telemetry Experimental Research) experiment, which was an ecohydrological and watershed-scale experiment in the Heihe River Basin which is the second longest inland river of China. In the paper, we used EC data and automatic meteorological data. Users can apply for the data from Cold and Arid Regions Science Data Center at Lanzhou (http:// westdc.westgis.ac.cn/).

Disclosure
An earlier and simpler version of a fraction of this paper has been presented as a conference paper in "Geoscience and Remote Sensing Symposium (IGARSS), 2017 IEEE International."

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper. Advances in Meteorology 11