Abstract

Reference evapotranspiration (ETo) and pan evaporation (Epan) are key parameters in hydrological and meteorological studies. The authors’ objectives were to evaluate the ratio of ETo to Epan (kp) at daily and monthly scales and to predict average ETo in the following years using calibrated kp and observed Epan at the two time scales. Using 50 yr of data obtained at six typical sites in north China, daily and monthly ETo were calculated using the Food and Agriculture Organization estimation method (FAO-56) Penman–Monteith equation, and kp values were determined at the two time scales. Values of kp varied from 0.457 to 0.589 daily and from 0.392 to 0.528 monthly for the six sites. Both daily and monthly kp could be fitted as multilinear functions of longitude, latitude, elevation, and relative humidity. Relatively accurate predictions of daily mean ETo for the subsequent years following the calibration years at all six sites were obtained when the year number L used for calibrating daily mean kp was sufficient (>38). In cases when large deviations occurred between average kp for the L calibration years and the actual kp of the following (L + 1)th year, relatively large prediction errors resulted. For the monthly scale, soil heat flux G fluctuated periodically. When variations of G were included, the calculated monthly ETo values were smaller than the monthly ETo cumulated from daily ETo. Thus, monthly kp values were smaller than daily kp values. Predictions of monthly ETo in 2001 for the six sites were relatively accurate with relative errors ranging from −11.9% to 12.1%. In conclusion, this method is simple and accurate with a small demand for weather data.

1. Introduction

Crop water requirements (CWR) are important parameters in the design of irrigation systems, irrigation scheduling, water resources management, and water cycle research (e.g., Donatelli et al. 2006; Maule et al. 2006; Xu et al. 2006a,b). Direct measures of CWR are difficult because of equipment and funding limitations. CWR can be obtained from estimates of evapotranspiration (ET). The quantity ET is often estimated from measurements of free water evaporation or from calculated reference evapotranspiration (ETo). The quantity ETo is the evapotranspiration from the reference surface, which is a hypothetical grass reference crop with an assumed height of 0.12 m, a fixed surface resistance of 70 s m−1, and an albedo of 0.23, and closely resembles an extensive surface of green, well-watered grass of uniform height, actively growing and completely shading the ground (Allen et al. 1998). The quantity ETo can be considered to be an upper limit of actual ET.

The quantity ETo is useful for characterizing the climatic effects in soil–plant–atmosphere systems. It is important and widely used in agricultural water cycle studies, irrigation schedules, and applied meteorology. The ETo can be obtained by theoretical or empirical equations. Allen et al. (1998) modified the Penman–Monteith (PM) equation using a suite of reference land surface conditions, and the Food and Agriculture Organization adopted this as the standard ETo estimation method (herein denoted FAO-56). FAO-56 has been accepted worldwide as a good ETo estimator compared with other methods (Sumner and Jacobs 2005), both at the daily time step (Allen 1996; Allen et al. 1996, 2000; Liu et al. 1997; Garcia et al. 2004; Temesgen et al. 2005; Alexandris et al. 2006; Cai et al. 2007) and at the monthly scale (Allen et al. 1998; McVicar et al. 2005, 2007). To use FAO-56 requires maximum and minimum air temperature, relative humidity, solar radiation, and wind speed data. Gong et al. (2006) found that in the Changjiang (Yangtze) River basin in China, where ETo is primarily energy limited (average annual precipitation is greater than average annual ETo; Donohue et al. 2007), the calculated ETo was most sensitive to relative humidity, followed by shortwave radiation, air temperature, and wind speed.

Although FAO-56 was relatively accurate for calculating ETo (Gunston and Batchelor 1983; Sumner and Jacobs 2005), the required long-term and continuous meteorological datasets are often lacking. This makes it difficult to determine ETo for regions lacking complete historical climatic data. Therefore, many attempts have been made to estimate ETo using easily obtained and less costly information such as free water evaporation, or pan evaporation (Epan). This is the rainfall-corrected evaporation observed by a special pan under sufficient water supply and is often linearly related with ETo characterized by a linear slope coefficient kp. The relationship is written as

 
formula

In western countries, the class-A pan, 120.7 cm in diameter and 25 cm deep (Allen et al. 1998), is widely used for Epan measurement (Allen 1996; Naoum and Tsanis 2003). However, in many Chinese weather stations, “micropans” of 20-cm diameter and 10-cm height, which are usually filled to a depth of 2 or 3 cm, have been used for a long time (Wang et al. 2006). The ratio of the area for heat transfer to the area for water vapor transfer is much greater for Chinese micropans than for class-A pans, resulting in lower kp values for Chinese micropans compared to class-A pans (McVicar et al. 2007). McVicar et al. (2007) showed how kp varies symmetrically in time (and space) in the Loess Plateau. Rotstayn et al. (2006) developed a physical model for pan evaporation and then Roderick et al. (2007) used this “Penpan” model in Australia to attribute trends in observed Epan given trends in the forcing meteorological data in a water-limited context. Roderick et al. (2007) found that changes in temperature and humidity regimes were generally too small to impact pan evaporation rates and the observed decreases in pan evaporation were mostly due to decreasing wind speed with some regional contributions from decreasing solar irradiance.

Ultimately, our objective is to predict ETo, for example, for local irrigation scheduling; this prediction requires accounting for the relationship between ETo and Epan. Fan et al. (2006) showed that there were clear correlations between Epan and ETo determined by the PM equation based on Loess Plateau climatic data. Further research is needed to determine how broadly applicable the linear relationship between Epan and ETo is. When full climatic data are unavailable, using Epan data can be a good approach to estimate ETo values (Gunston and Batchelor 1983; Allen et al. 1998).

Research based on the PM method has focused on spatial distribution of kp (McVicar et al. 2007), spatial and temporal trends (Gong et al. 2006), the effects of sensitivity coefficients on ETo (Xu et al. 2006a,b), trends of Epan (Roderick et al. 2009a,b), and sampling frequency effects (Hupet and Vanclooster 2001). Also of interest for ETo and Epan measurements and models in the context of a climate change (McVicar et al. 2005), widespread decreases in terrestrial midlatitude near-surface wind speeds in both the Northern (e.g., China: Xu et al. 2006a,b; Jiang et al. 2010; United States: Pryor et al. 2009) and Southern (e.g., Australia: McVicar et al. 2008) Hemispheres need to be accounted for. But there is still a need to compare kp values at daily and monthly scales for less studied regions, to determine whether, and how, different time scales affect kp. It is unknown whether the simple method presented herein can successfully predict ETo. It is necessary to perform such research to determine how to use climatic data more efficiently. Our objectives were 1) to compare kp values at daily and monthly scales, and 2) to predict ETo with the calibrated kp and observed Epan at two time scales.

2. Materials and methods

a. Basic geographical and climatic conditions of the studied weather stations

The six weather stations (Urumqi, Lhasa, Xining, Lanzhou, Huhehaote, and Beijing) were selected from different climatic regions of north China. The data duration for all six stations was from 1955 to 2001. The smallest completeness of daily weather data (including maximum and minimum air temperature, relative humidity, wind speed, and sunshine hour) was 99.3%. The Epan data of the six stations were almost complete (>96.3%). The geographical and average climatic conditions of the six sites are presented in Table 1.

Table 1.

Geographical and mean annual meteorological information of the selected stations.

Geographical and mean annual meteorological information of the selected stations.
Geographical and mean annual meteorological information of the selected stations.

The land area maintained for all weather stations was 25 m by 25 m. The land area was leveled, and the plant cover was mown grass. Air temperature readings were made in a small screened box placed 2 m above the ground surface. The 20-cm-diameter micropan was located on a wooden platform 0.7 m above the ground. The relative humidity was measured with a barothermohygrograph 2.0 m above the ground. The sunlight hours were measured by a Jordan sunshine recorder at a height of 1.6 m above the ground. The wind speed was measured by an anemometer at a height of 10 m above the ground.

b. The standard reference evapotranspiration equation

The FAO-56 form of the Penman–Monteith equation is

 
formula

where ETo is the reference evapotranspiration (mm day−1), G is soil heat flux (MJ m−2 day−1), T is mean air temperature at 2 m (°C), u2 is wind speed at 2 m (m s−1), es is saturation vapor pressure (kPa), ea is actual vapor pressure (kPa), esea is saturation vapor pressure deficit (kPa), Δ is slope of vapor pressure curve (kPa °C−1), γ is a psychrometric constant (kPa °C−1), and Rn is net radiation (MJ m−2 day−1). The net radiation (Rn) is the difference between the incoming net shortwave radiation (Rns; MJ m−2 day−1) and the outgoing net longwave radiation (Rnl; MJ m−2 day−1), that is, Rn = RnsRnl. The values Rns and Rnl are calculated by

 
formula

and

 
formula

where Rs is solar radiation calculated with the Angstrom formula (Allen et al. 1998):

 
formula

where α is albedo assumed to be 0.23, Ra is extraterrestrial radiation (MJ m−2 day−1), n is actual duration of sunshine (h), N is maximum possible duration of sunshine or daylight hours (h), n/N is relative sunshine duration, as is the regression constant, expressing the fraction of extraterrestrial radiation reaching the earth on overcast days, and as + bs is the fraction of extraterrestrial radiation reaching the earth on clear days (n = N). Prescott originally proposed as = 0.22 and bs = 0.54. We used values reported by Chen et al. (2004) who presented as and bs values for 46 sites in China. The values for our six sites taken from Chen et al. (2004) are presented in Table 2, and their sums are usually lower than the proposed sum of the FAO-56 values (0.75), which is due to high rates of air pollution reducing atmospheric transmittance and in agreement with previous results (McVicar and Jupp 1999).

Table 2.

Parameters reported in Chen et al. (2004).

Parameters reported in Chen et al. (2004).
Parameters reported in Chen et al. (2004).

As the magnitude of daily or 10-day soil heat flux beneath the grass reference surface is relatively small, it is ignored so G is assumed to be zero for a daily scale (Allen et al. 1998). This assumption could be a minor error source of ETo calculation. For the monthly scale, G is calculated by

 
formula

where subscripts i + 1, i, and i − 1 represent the number of month.

The other terms in Eq. (2) are derived from the methods given for daily scale and for monthly scale in Allen et al. (1998, chapters 3 and 4).

c. Pan evaporation method to evaluating ETo

The pan evaporation measurements represent the comprehensive effects of aerodynamic and radiative items such as wind speed, sunlight hours, esea, and net radiation (Rotstayn et al. 2006; Roderick et al. 2007), which are primary drivers of evapotranspiration. Air temperature is also important because it influences some of these drivers (e.g., esea and Rn via RL↓) and other variables that control evapotranspiration.

The coefficient of linear relationship between ETo and Epan was affected by the environmental and meteorological conditions (McVicar et al. 2007). Although albedo may differ between locations, we assumed constant albedo for all of the locations. Assuming constant albedo may be a main environmental effect on kp. For further discussion of environmental and meteorological effects, see McVicar et al. (2007) who performed a sensitivity analysis of kp to spatial interpolation and land surface parameterization. In addition, heat storage in the pan is assumed to be zero (Roderick et al. 2009a,b) to simplify the calculation, which may be an issue when dealing with cloudy days immediately following sunny conditions. With such a small mass of water in a China micropan, this assumption will be less of an issue than for a class-A pan, which holds a greater volume of water. Pan evaporation data were compared with ETo and were further analyzed for variations of kp.

d. Estimation and prediction of average daily ETo for the following years

The approach for predicting average daily ETo is based on Eq. (1). We use calibrated kp values of early years and observed Epan values of the following predicted year to obtain ETo values of the predicted year. With M is the year number for predicting, M = 1, 2, … , nL, n is the total year of the datasets, and L is the number of years used to calibrate the daily mean kp values (kp,cal), then

 
formula

where DN is the total day number of days in L years. The daily mean Epan of the (L + M)th year, , is calculated as

 
formula

where D is the total number of days in the following year, that is, the (L + M)th year. According to Eq. (1), the predicted daily mean ETo of the (L + M)th year, , is

 
formula

As the time period for use of the prediction becomes larger (i.e., temporally extrapolating further away from an observation), there is greater potential for trends in the forcing meteorological data to alter the relationship we have developed for prediction.

The relative error RE is calculated as

 
formula

where , which is the actual daily mean ETo [from Eq. (2)] of the M predicted year.

e. Estimation and prediction of monthly ETo

Although monthly ETo and Epan change periodically, as long as Eq. (1) is statistically effective for monthly scale ETo and Epan data to obtain kp values, we can predict monthly ETo from monthly Epan. The monthly ETo calculated with the FAO-56 equation (, i = 1, 2, … , 12, representing month number) and the monthly Epan cumulated from daily values were divided into 12 groups via months, respectively.

The calibrated monthly kp values are obtained from the slope of Epan versus ETo at the ith month for the early L years when forced through the origin. Predicted monthly ETo in the ith month of the (L + M)th year is

 
formula

where is the cumulative daily Epan of the ith month in the (L + M)th year. Equation (10) is also used for determining REs of the predictions.

3. Results

a. The periodic variation of ETo and Epan

Figure 1 shows the multiyear average monthly variations of ETo and Epan for six sites. The maximum values of ETo and Epan at Urumqi were largest among the six sites. The minimum values of ETo and Epan at Lhasa were largest, while the smallest minimum values of ETo and Epan were at Urumqi. The variations of monthly ETo and Epan were generally similar. The maximum average monthly ETo and Epan of Lhasa, Xining, Lanzhou, and Beijing were around May, which agreed with McVicar et al. (2007), who reported May as having the maximum average monthly Epan on China’s Loess Plateau. For Huhehaote and Urumqi, the maximum ETo and Epan occurred around July, which is likely a response to the differential Asian monsoon.

Fig. 1.

Average monthly (a) ETo and (b) Epan variations from 1955 to 2001 for the selected six stations.

Fig. 1.

Average monthly (a) ETo and (b) Epan variations from 1955 to 2001 for the selected six stations.

b. Comparisons of kp at daily and monthly scales

When we cumulated the calculated daily ETo to 3-day, 10-day, 1-month, and 3-month scales, we obtained almost stable kp values for all six sites at these time scales. The ratios of kp at a larger time scale to kp at daily scale ranged from 1.01 to 1.04 for the six sites. The reasons could be from the assumption that zero heat storage in the water in the pan caused small errors. From our calculations, monthly ETo values calculated using the FAO-56 equation were all smaller than those cumulated from daily ETo.

The linear relations in Eq. (1) between Epan and ETo were fitted at daily and monthly scales. The illustrations of Epan via ETo at the two scales for the six sites are presented in Figs. 2 and 3. There were obvious linear relations between ETo and Epan at both daily and monthly scales for all sites. Mostly, Epan was larger than ETo. The data points showed increasing scatter as Epan values increased. Equation (1) fit the relations of Epan and ETo with a lowest coefficient of determination of 0.740 at the daily scale and 0.626 at the monthly scale. At the daily scale, kp ranged from 0.457 to 0.589; kp at Lanzhou was the largest, while kp at Urumqi was the lowest. At the monthly scale, kp decreased compared to the daily scale and ranged from 0.392 to 0.528. The seasonal kp values reported by Xu et al. (2006a,b) ranged between 0.51 and 0.94 for the Yangtze River basin, and they were generally larger than ours. The seasonal kp of McVicar et al. (2007) reported in their Fig. 11 for central China ranged from 0.4 to 0.7 and had a larger range relative to our values. The ratios of monthly–daily kp were 0.87 for Urumqi, 0.96 for Lhasa, 0.68 for Xining, 0.90 for Lanzhou, 0.97 for Huhehaote, and 0.93 for Beijing. A very low (or high) pan coefficient may be caused by some geological and climatic factors such as elevation, relative humidity, wind speed, air temperature, and so on.

Fig. 2.

Linear relations between Epan and ETo for six stations at daily scale. Total data point number for the regression was 17 167. The slope of the line when forced through the origin was kp.

Fig. 2.

Linear relations between Epan and ETo for six stations at daily scale. Total data point number for the regression was 17 167. The slope of the line when forced through the origin was kp.

Fig. 3.

As in Fig. 3, but for the monthly scale. Total data point number for the regression was 563.

Fig. 3.

As in Fig. 3, but for the monthly scale. Total data point number for the regression was 563.

There was no single function between kp and the climatic and geographical elements such as longitude, latitude, altitude, and relative humidity. The coefficient kp was affected not by a single geographical or climatic element but by several factors. Two multivariable linear functions obtained for kp and the related influence factors were as follows:

  • for daily scale, 
    formula
  • for monthly scale, 
    formula

where ϕ is longitude, δ is latitude, L is elevation (m), RH is relative humidity (%), the correlation coefficient of the multivariable regression (multir) for daily scale was 0.978, and that for monthly scale was 0.780. Monthly kp was much lower than daily kp at Xining, which decreased the value of multir.

c. Prediction of daily mean ETo

Based on the calibrated daily mean kp of the early L years and the calculated daily mean Epan of the (L + M)th year, the daily mean ETo of the (L + M)th year was predicted (see section 2d). The daily mean is compared with the estimated FAO-56 daily mean ETo of the (L + M)th year (Fig. 4). Estimated and predicted ETo values were similar when ETo was smaller than 3.2 mm day−1. Above 3.2 mm day−1 several points deviated from the 1:1 line. The values of Xining, Lanzhou, Huhehaote, and Beijing were relatively close to the 1:1 line in comparison with Urumqi and Lhasa, especially when daily mean ETo values were relatively small.

Fig. 4.

Comparison of the calculated and predicted daily average ETo of the (L + 1)th year described in section 2d. The solid line is the 1:1 line.

Fig. 4.

Comparison of the calculated and predicted daily average ETo of the (L + 1)th year described in section 2d. The solid line is the 1:1 line.

From Eq. (9), the errors for came from both and . While was obtained from the observed daily Epan values, despite observation errors, it should not be the main error source of . So the main error source for was most likely from . The deviations of with the actual may cause the main errors of . The comparisons of the deviations between and for six sites are presented in Fig. 5. Since was a multiyear mean value, varied steadily but fluctuated. There were larger deviations of relative to for Huhehaote and Urumqi stations, which resulted in relatively large RE for . The largest RE of in comparison with were 23% for Urumqi, 13% for Lhasa, 12% for Xining, 17% for Lanzhou, 19% for Huhehaote, and 14% for Beijing. Larger RE for predictions were found when the deviations of to were larger. The curve fluctuated around for most sites. The deviations of to became smaller when the year number used to determine increased. That was the main reason RE of were low when the year number used to determine was greater than 38. Generally, the curves of were flat and could be treated as a constant value for predicting ETo at the selected six sites.

Fig. 5.

Comparisons of and is the calibrated daily mean kp in total L years, while is the daily mean kp of the (L + 1)th year.

Fig. 5.

Comparisons of and is the calibrated daily mean kp in total L years, while is the daily mean kp of the (L + 1)th year.

For further analyses, we compared the predicted and estimated daily mean ETo values of the latter predicted years using the calibrated of the early L (L = 38, … , 46) years (Fig. 6). Only small differences were found between the predicted and estimated ETo values for all six stations. The RE (%) of the predictions ranged from −4.4 to 11.2 for Urumqi, from 0.6 to 4.8 for Lhasa, from −10.8 to 10.4 for Xining, from −4.1 to 10.9 for Lanzhou, from −3.4 to 8.2 for Huhehaote, and from −5.8 to 5.6 for Beijing. Relatively large differences between calibrated kp and actual kp of the predicted year caused a relatively large RE. The RE decreased when increasing the time of predictions, and if there was a relatively large calibration period then RE decreased. Small differences were found between the predicted and estimated daily mean ETo.

Fig. 6.

The predicted and the calculated daily mean ETo values from 1955 to 2001 at the indicated stations. For example, the calibration periods were 1955–91, 1955–92, … , 1955–2000 with the validation periods 1992, 1993, … , 2001, correspondingly.

Fig. 6.

The predicted and the calculated daily mean ETo values from 1955 to 2001 at the indicated stations. For example, the calibration periods were 1955–91, 1955–92, … , 1955–2000 with the validation periods 1992, 1993, … , 2001, correspondingly.

d. Prediction of monthly ETo from 2001

Soil heat flux G was small relative to net radiation Rn (Allen et al. 1998, chapter 3), but G was an important parameter when we calculated monthly ETo using the FAO-56 equation. Figure 7 illustrates the variations of daily mean monthly G for the six sites from January 1999 to December 2000. There were generally similar fluctuations and ranges of monthly G for most of the sites, but monthly G for Lhasa fluctuated less and ranged from −0.46 to 0.44 MJ m−2 day−1. Monthly G at Urumqi fluctuated more than the monthly G at the other sites. The lowest G occurred in October at Urumqi (0.91 MJ m−2 day−1, having an upward direction) and the highest G occurred around April (−0.89 MJ m−2 day−1, having a downward direction) at Urumqi, which followed the obvious fluctuation of air temperature.

Fig. 7.

Daily average values of monthly soil heat flux for the six sites.

Fig. 7.

Daily average values of monthly soil heat flux for the six sites.

A key factor for predicting monthly ETo is to calibrate monthly kp. From the prediction of daily ETo, it was known that using as many years as possible of data for calibration enhanced prediction accuracy. So when calibrating monthly kp values we selected the early 45 yr of data for the six sites. Similar to the daily scale, the monthly average kp (or the calibrated kp) tended to be stable when the length of years for calibration increased. Figure 8 shows the variation of the calibrated monthly kp for the six sites. Monthly kp generally had periodic variations with the maximum occurring in July or August.

Fig. 8.

Variations of the calibrated kp values on a monthly scale. The earliest 45 yr of data (1955–2000) were used for the six sites.

Fig. 8.

Variations of the calibrated kp values on a monthly scale. The earliest 45 yr of data (1955–2000) were used for the six sites.

Figure 9 shows the predicted monthly ETo in 2001 for the six sites. In general, the periodic changes of the predicted monthly ETo were similar to the calculated values. The ETo values from July to September were predicted with a relatively large RE in comparison with the other months.

Fig. 9.

The predicted and calculated monthly ETo (from meteorological data) values in 2001 at the indicated stations.

Fig. 9.

The predicted and calculated monthly ETo (from meteorological data) values in 2001 at the indicated stations.

The RE for the prediction of monthly ETo in 2001 are presented in Table 3. The RE at Lhasa and Beijing were generally low relative to the other sites. The RE values for January, March, October, and November were generally larger than RE of the other months. Similar to the daily scale, low RE values were mainly due to accurately calibrated monthly kp. The coefficient of determination for calibrating monthly kp ranged from 0.830 to 0.898 for Lhasa and Beijing and from 0.683 to 0.879 for the other sites.

Table 3.

RE (%) for the actual and predicted monthly ETo in 2001 for six sites.

RE (%) for the actual and predicted monthly ETo in 2001 for six sites.
RE (%) for the actual and predicted monthly ETo in 2001 for six sites.

Since the prediction procedure is not complicated and it only needs Epan observations and the historical average kp values, this method is promising for simply and accurately predicting daily average ETo and monthly ETo.

4. Conclusions

The relatively good correlation between Epan and ETo makes it possible to use the pan coefficient kp to predict ETo from the observed Epan for selected stations. Values of kp at the daily scale ranging from 0.457 to 0.589 were generally larger than kp at the monthly scale ranging from 0.392 to 0.528 for the six sites. Accurate prediction of daily average ETo for the years following calibration was possible especially when the actual kp of the predicted year was similar to the calibrated kp. The predictions of monthly ETo were relatively accurate. This method for predicting average ETo is operable because of its simplicity.

Acknowledgments

This work was funded by China National Natural Science Foundation (50709028). We thank the anonymous reviewers and Robert Ewing for providing insightful comments that helped us to improve the paper.

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Footnotes

Corresponding author address: Yi Li, College of Water Resources and Architectural Engineering, Northwest Agriculture and Forestry Sci-Tech University, Yangling, Shaanxi, 712100, China. Email: liyikitty@126.com