Introduction

Atmospheric water demand has been described using potential evaporation (ET p), pan evaporation, and reference crop evapotranspiration (ET o). ET p is the amount of water transpired in a given time by a short green crop, completely shading the ground, of uniform height and with adequate soil water status in the profile1, 2. ET p has been applied as an important parameter for several decades in the field of hydrology, meteorology, agricultural engineering, etc. However, crop conditions in ET p estimation was assumed constant. To avoid ambiguities involved in the definition and interpretation of ET p, ET o was introduced by irrigation engineers and researchers in the late 1970s and early 1980s3, 4. ET o is evapotranspiration rate from a hypothetical grass reference crop with a height of 0.12 m, a fixed surface resistance of 70 sec m−1 and an albedo of 0.23, actively growing, well-watered, and completely shading the ground5. ET o incorporates multi-climatic factors and expresses the evaporative demand of the atmosphere independent of crop type, crop development and management practices. Under the world-widely accepted global warming background6, ET o has become an important agrometeorological parameter for climatological and hydrological studies, as well as for irrigation planning and management7,8,9. ET o has also been incorporated in drought severity and evolution analysis10,11,12. The application of ET o was also related to water use of crops13. ET o has been widely used in different research fields with various objectives because it can be computed from meteorological data.

The methods for estimating ET o (ET p) could be classified as empirical, temperature-based, radiation-based, pan, and combination types. In recent years, several simplified ET o equations were proposed and validated for their applicability14. Of these methods, the temperature-based equations, such as the Thornthwaite1, the Blaney-Criddle15, and the Hargreaves-Samani (1985a)16, 17, were extensively adopted because they mainly use easily-obtained tempereature data. The radiation-based methods were also applied18. The pan evaporation methods were used when the observed pan data were available19,20,21,22. The physically-based combination methods explicitly incorporate physiological and aerodynamic parameters3, 5, 23. The Penman-Monteith (PM) equation was selected as the standard ET o estimation method by the Food and Agriculture Organization (FAO) of the United Nations because it closely approximates ET o at the locations evaluated5, 24, 25. Afterwards the FAO56-PM equation was world widely applied for the validation of the other equations in absence of experimental measurements26.

In recent years, variations of FAO56-PM-based ET o (ET o,s) has been extensively investigated since the FAO56-PM was recommended as a standard ET o estimation method27,28,29,30,31. The ET o,s variations were also analyzed in partial or entire mainland China (EMC) concerning different application objectives32,33,34,35,36,37,38,39,40,41,42,43. Meanwhile, the evaluation of the FAO56-PM method has also been widely conducted by comparing with different ET o estimation methods. The evaluation research mainly focused on answering which method could be an alternative for FAO56-PM either the input data were full, limited or missing9, 44,45,46. It is known that the FAO56-PM equation requires a large data input for estimating ET o, including the geological variables such as elevation and latitude, and the meteorological variables such as minimum air temperature (T min), average air temperature (T ave), maximum temperature (T max), wind speed, relative humidity (RH) and sunshine hour (n). The high data demand of the FAO56-PM method realized its overall high accuracy, but restricted its application in some data-lacking regions. In the regions where the observed long-term meteorological data are difficult to obtain, the FAO56-PM method is not the best choice. To solve this problem, ET o estimation methods with a lower data requirement and a simpler computation process are preferentially applied.

Although ET p and ET o are not equivalent terms, both provide estimates of atmospheric evaporative demand. In the previous studies, there are different understandings about the relationship between ET p and ET o. Several researchers differ the two items strictly32, 47, 48. For instance, FAO56-PM ET o equation is considered a PM ET p equation for specific reference conditions47. For strictly utilization of the items, Allen et al.5 strongly discourage the use of ET p for ET o estimation concerned about the ambiguities in their definitions. A few researchers consider ET o is a kind of ET p 49 or it is reference values of ET p for a uniform grass reference surface50. Noticeably, a lot of researchers look upon ET p and ET o as identical concepts and share similar equations for their estimations51,52,53,54,55. Usually, a climatologist or meteorologist and a hydrologist use the term “potential”, whereas an irrigation scientist uses the term “reference crop”, although the estimation equation could be same. Even some equation-proposers potentially identified the two items, such as Hargreaves and Samani16, 17 adopted “potential” while Hargreaves and Samani16 used the term “reference crop”. Ambiguity between ET o and ET p was expected to be reduced by more extensive definition of ET p as potential crop evapotranspiration or by using one of the ET o definitions56. Take Thornthwaite1 for another example, this method was originally proposed to estimate ET p, but was also applied for estimating ET o in different cases57, 58. Therefore, although there are differences between ET p and ET o, there is close relationship between them and their estimations could be quantitatively linked.

China has a total land area of 9,597,000 km2 and is the third largest country in the world. It has a complicated geomorphology which contains different water bodies, glaciers, frozen soils, deserts, basins, mountains, farmland, and forests. The elevations are general lower and lower from the west to the east, shaping a so called “3-level-catena” landform59. The weather stations in China distributed non-uniformly, there are more weather stations in the eastern China, but less in the western regions, especially less on Qinghai-Tibet Plateau. Neither is the distribution of the sites even, nor are the observed climatic elements same for different stations. The total sites available for air temperature and precipitation data are as large as 247460, but when more climatic elements are needed, data from much less number of sites were available, estimated ET o,s values for 200 sites in China by Fan et al.35, while 552 sites are suitable for ET o,s analysis of this research. Not only in China, similar phenomena of difficulty in acquiring long-term and full weather data are also common in other developing countries because of some natural (geographical and climate) and humanity (economic power, knowledge and technology) reasons. Under this condition, for the ET o estimation of China, to date to calibrate a suitable alternative equation which is simpler in computation process using less weather data and has a general good accuracy when compared to the FAO56-PM equation, are still very important for different sub-regions of China and EMC. Although performance of 16 different ET o equations were compared for Xiaotangshan, Changping, Beijing in North China Plain61, a thorough and detail research for selecting a best alternative in EMC has not been conducted.

Based on the reasonable selection of 11 different ET o estimation methods for the calculation of monthly ET o, this research aims to: (1) investigate the spatiotemporal variations and the trends of ET o using climatic data from the selected 552 sites in EMC; (2) compare the performance of the 10 selected ET o estimation methods with the standard FAO56-PM method in different sub-regions of China and EMC for the period 1961–2013; (3) select a best alternative of the FAO56-PM ET o equation, which would be simpler in ET o estimation and use less climatic variables; and (4) calibrate ET o using the alternative equation with the standard FAO56-PM equation.

Data and Methodology

Data

Geographical and weather data from 552 National Meteorological Observatory stations in EMC were collected from the China Meteorological Administration. The data contained both the daily and monthly timescales. The weather data included T max, T min, T ave, U 10 wind speed at 10 m, RH, and n. The data duration was 1961–2013. The elevations of the selected sites covered a large range in EMC (Fig. 1a). To obtain more accurate ET o estimation, the 48 sites reported by Chen et al.62 were used as the radiation correction station. Meteorological station (marked with blue circle) and radiation calibration station (marked with red triangle) and they were set as the centers of the Thiessen polygons to find the other sites which would use same parameters with them for estimating radiation (Fig. 1b). The EMC is divided into seven sub-regions63 considering the differences in topography and climate64 (Fig. 1c). Including the temperate and warm-temperate desert of Northwest China (sub-region I, 61 sites), the temperate grassland of Inner Mongolia (sub-region II, 44 sites), the warm-temperate humid and sub-humid Northeast China (sub-region III, 72 sites), the warm-temperate humid and sub-humid North China (sub-region IV, 104 sites), the subtropical humid Central and South China (sub-region V, 165 sites), the Qinghai-Tibetan Plateau(sub-region VI, 49 sites), and the tropical humid South China (sub-region VII, 57 sites).

Figure 1
figure 1

The DEM, weather station distribution and the sub-region division (I to VII) in China. (ArcGIS 10.2, http://map.baidu.com, Lingling Peng).

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Estimation of ET o using the FAO56-PM method

The FAO56-PM equation for estimating ET o,s is written as bellow (Allen et al.)5:

$$E{T}_{{\rm{O}},{\rm{S}}}=\frac{0.408{\rm{\Delta }}({R}_{{\rm{n}}}-G)+\gamma \frac{900}{{T}_{{\rm{ave}}}+273}{U}_{{\rm{2}}}({e}_{{\rm{s}}}-{e}_{{\rm{a}}})}{{\rm{\Delta }}+\gamma (1+0.34{U}_{{\rm{2}}})}$$
(1)

where G is soil heat flux (MJ m−2 month−1), T ave is mean air temperature at 2 m (°C), \({T}_{ave}=({T}_{{\rm{\max }}}+{T}_{{\rm{\min }}})/2\), U 2 and U 10 are wind speed at 2 and 10 m (m s−1), respectively, U 2 = 0.75 U 10, \({e}_{{\rm{s}}}\) is saturation vapor pressure (kpa), e a is actual vapor pressure (kpa), e s-e a is saturation vapor pressure deficit (kpa), Δ is slope of vapor pressure curve (kpa °C−1), γ is psychrometric constant (kpa °C−1), and R n is net radiation (MJ m−2 month−1). Monthly G is estimated by:

$${G}_{{\rm{K}}}=0.07({T}_{K+1}-{T}_{{\rm{K}}-{\rm{1}}})$$
(2)

where subscripts K + 1, K and K − 1 are order of month, respectively. Annual ET o,s is cumulated from the values of 12 months.

R n is calculated by:

$${R}_{{\rm{n}}}={R}_{{\rm{ns}}}-{R}_{{\rm{nl}}}$$
(3)
$${R}_{{\rm{ns}}}=(1-\alpha ){R}_{{\rm{s}}}$$
(4)
$${R}_{{\rm{s}}}=[{a}_{{\rm{s}}}+{b}_{{\rm{s}}}(\frac{n}{N})]{R}_{{\rm{a}}}$$
(5)
$${R}_{{\rm{nl}}}=\sigma (\frac{{T}_{\max \,,k}^{4}+{T}_{\min \,,k}^{4}}{2})(0.34-0.14\sqrt{{e}_{{\rm{a}}}})(1.35\frac{{R}_{{\rm{s}}}}{{R}_{{\rm{so}}}}-0.35)$$
(6)

where R ns is net shortwave radiation (MJ m−2 month−1), R nl is net longwave radiation (MJ m−2 month−1), n and N are actual and maximum possible sunshine duration, respectively, R a is the extraterrestrial radiation (MJ m−2 month−1), σ is the Stefan-Boltzmann constant (4.903 × 10−9 MJ K−4 m−2 d−1), α is albedo (α = 0.23), T max,k and T min,k are maximum and minimum absolute temperatures during 24-h, respectively, and R so is clear sky solar radiation (MJ m−2 month−1). The FAO56-PM recommended 0.25 for a s and 0.50 for b s, respectively. For better accuracy, the calibrated values of a s and b s at 48 sites reported by Chen et al.62 were used here (marked with red triangle) for determination of a s and b s values at nearby sites (marked with blue circle in Fig. 1b) using the Thiessen polygon method.

Estimation of ET o using the other 10 selected methods

A preliminary performance comparison of 16 ET o (ET p) methods were conducted (Fig. S1). From the elementary results, ET p equations performed generally worse than ET o equations. Therefore, 10 ET o equations which performed generally well in different regions of the world, i.e., Irmak et al.65, Makkink66, Priestley-Taylor23, Hargreaves-Samani16, Droogers-Allen67, Berti et al.68, Doorenbos-Pruitt4, Wright69 and Valiantzas14, are selected to compare to the FAO56-PM equation. Of which, Valiantzas14 proposed two equations to simplify the FAO56-PM equation. The two Valiantzas14 equations and the Berti et al.68 equation were relatively new, but their performances have not been validated in China. Three Hargreaves-Samani-based equations (HS, MHS_1 and MHS_2) are adopted here because the FAO-56 manual recommended HS as the use of a less demanding method with only data on T ave and extraterrestrial radiation (R a)70. The types, simplified method name, and main equations for estimating ET o of the selected 10 methods (ET o,i, i = 1, 2, …, 10) are given in Table 1. For the Droogers and Allen67 method (simplified as MHS_1), \({S}_{o}=15.392{d}_{{\rm{r}}}({w}_{{\rm{s}}}\,\sin (\phi )\,\sin (\delta )+\,\cos (\phi )\,\cos (\delta )\,\sin ({w}_{{\rm{s}}}))\). For the Berti et al.68 method (simplified as MHS_2), P is precipitation. For the Valiantzas14 method (simplified as Val_1), T dew = [116.91 + 237.3 ln(ea)]/[16.78-ln(ea)]. For the Wright69 method (simplified as KPM), a w = 0.3 + 0.58 exp \([-{(\frac{J-170}{45})}^{2}]\), b w = 0.32 + 0.54 exp \([-{(\frac{J-228}{67})}^{2}]\), where J is Julian day in the year between 1 (1 January) and 365 or 366 (31 December).

Table 1 Equations of ET o,i estimated with the selected 10 methods.
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Performance evaluation of the 10 selected methods

Relative error (RE), standard deviation (θ) and Nash-Sutcliffe efficacy coefficient (NSE)71 are used to assess the performances of monthly ET o,i:

$$RE=\frac{E{T}_{{\rm{o}},{\rm{i}}}-E{T}_{o,s}}{E{T}_{o,s}}$$
(7)
$$\theta =\sqrt{\frac{\sum {({x}_{{\rm{i}}}-\overline{x})}^{2}}{N-1}}$$
(8)
$$NSE=1-\frac{{\sum }_{i=1}^{n}{(E{T}_{o,s}-E{T}_{o,i})}^{2}}{{\sum }_{i=1}^{n}{(E{T}_{o,s}-\overline{E{T}_{o,s}})}^{2}}$$
(9)

where N = 1, 2, …, 636th month. If RE is close to 0, ET o,i is close to ET o,s. The NSE values ranged from −∞ to 1. When NSE is close to 1, the quality of the method for estimating ET o,i is good with high reliability. When NSE is close to 0, ET o,i has an close mean value with ET o,s with an overall reliable estimation, but the errors of the estimation processes are large; when NSE is much less than 0, the estimation is not reliable.

Trend test

The modified nonparametric Mann-Kendall (MMK) test72, which takes into account the effects of auto-correlation in annual time series ET o,L (L = 1, 2, …, n 1, where n 1 = 53 is total year number) based on the standardized Mann-Kendall (MK) method73, 74, is used to test the trend of ET o,L if it is auto-correlated8. The MK test statistic (Z) follows the standard normal distribution with a mean of 0 and variance of 1 under the null hypothesis of no trend in ET o,L. The null hypothesis is rejected if |Z| ≥ Z 1-β/2 at a confidence level of β, where Z 1-β/2 is the (1-β/2)–quantile. If Z is positive (or negative), ET o,L has an upward (or downward) trend. As β = 0.05, if |Z| > 1.96, the trend is significant. The MMK statistic Z * is computed by introducing a correction factor \({n}_{1}^{s}\) to Z to estimate72:

$${Z}^{\ast }=Z/\sqrt{{n}_{1}^{s}},\,{\rm{where}}\,{n}_{1}^{s}=\{\begin{array}{ll}1+\tfrac{2}{{n}_{1}}\sum _{jj=1}^{{n}_{1}-1}({n}_{1}-1){r}_{jj} & for\,jj > 1\\ 1+2\tfrac{{r}_{1}^{{n}_{1}+1}-{n}_{1}{r}_{1}^{2}+({n}_{1}-1){r}_{1}}{{n}_{1}{({r}_{1}-1)}^{2}} & for\,jj=1\end{array}$$
(10)

where r jj is sample autocorrelation coefficient of ET o,L at a lag jj. For denoting significance of a trend, when jj = 0, Z * equals to Z; while as jj > 0, the MMK statistic Z * is utilized.

Variation coefficient

The variability of series ET o,L is quantified with a coefficient of variation (C v), calculated with the following equation (Nielsen and Bouma)75:

$${C}_{v}=\frac{\theta }{\overline{E{T}_{{\rm{o}},L}}}\,{\rm{or}}\,{C}_{v}=\frac{\theta }{\overline{E{T}_{{\rm{o}},S}}}$$
(11)

where θ and \(\overline{E{T}_{{\rm{o}},L}}\) are standard deviation and multi-year mean ET o,L series, respectively. Variability levels are classified by C v ≤ 0.1, 0.1 < C v < 1.0 and C v ≥ 1.0 as weak, moderate or strong one, respectively.

Spatial distributions of the climatic variables, ET o,s, ET o,i and the other studied parameters are mapped by the Kriging interpolation method in ArcGIS 10.2 software.

Results

Spatial distribution of climatic variables

The spatial distribution of ET o are closely related to that of the related meteorological elements. Figure 2 illustrates the distribution of multi-year mean T ave, T min, T max, n, U 2 and RH. The distribution of T min, T max, T ave were generally similar, with high values in sub-regions V and VII but lower values in sub-regions II, III and VI. Values of n were higher in sub-regions I, II, III and VI. U 2 values were large in north China especially in sub-regions II and III, and small in sub-region V, generally. Values of RH were higher in the southeast China for sub-regions V and VII. T ave, T max, n, U 2 and RH showed moderate variability, while T min showed strong variability.

Figure 2
figure 2

Spatial distributions of multi-year mean meteorological elements in China. (ArcGIS 10.2, http://map.baidu.com, Lingling Peng).

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Spatial distribution of ET o,s and its trend

The equations and types of the FAO56-PM (for estimating ET o,s) and the other 10 methods (for estimating ET o,i) are shown in Table 1. Detail description of the equations is in the section “Data and methodology”.

Figure 3 shows the spatial distribution of multi-year mean monthly ET o,s and trend test results of annual ET o,s series for each site. The site number for different trends is presented in Table 2. In Fig. 3a, the ET o,s values were higher in the sub-regions I, II, VI and VII than the other 3 sub-regions, ranging from 49 to 108 mm. ET o,s had a moderate spatial variability with a coefficient of variation (C v) being 0.15. In general, ET o,s in western China (high elevations) were larger than in eastern and middle China (low elevations). In Fig. 3b and Table 2, more sites (339) had decrease trends in ET o,s than increase trends (213 sites), and the trends at more sites were insignificant. The sites which had decrease and increase trends occupied 61.4% and 38.6% of the total, respectively. This indicated an overall decrease of ET o,s in China. The common occurrence of insignificance trends was induced by the removing of autocorrelation structures when using the modified nonparametric Mann-Kendall test (MKK) method. It’s reasonable for the trend analysis. The sites with significant decrease (Sig. Dec) trends in ET o,s were mainly located in eastern China, while the sites with significant increase (Sig. Inc) trends in ET o,s were mainly located in middle China (i.e., sub-region VI).

Figure 3
figure 3

Spatial distribution of multi-year mean monthly ET o,s and the annual ET o,s trends over 1961–2013 at the 552 sites in mainland China. (ArcGIS 10.2, http://map.baidu.com, Lingling Peng).

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Table 2 Number of the sites with different trends (tested by the MMK method) for the annual ET o series over the period 1961–2013 in different sub-regions of China.
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The Spatiotemporal variation of ET o,i

The spatial distribution of multi-year mean monthly ET o,i values during 1961–2013 showed remarkable differences between different sub-regions, their variation ranges and the spatial distributions of ET o,i also had different similarity with ET o,s (Fig. 4). All of the 10 methods had different ranges of ET o,i, obtained lower ET o values in the northeastern China (sub-region III), and differed much in spatial distribution when compared with ET o,s. The empirical method for estimating ET o,1 only resembled ET o,s distribution in sub-region III very well, and its ranges were much smaller than ET o,s. ET o,2 and ET o,3 (radiation-based) distributed partly similar with ET o,s. Among the temperature-based methods for estimating ET o (i.e., ET o,4, ET o,5 and ET o,6), ET o,6 resembled the spatial distribution and the value range of ET o,s more. The spatial distribution of ET o,7 and ET o,8 (combination type) were highly similar with that of ET o,s. ET o,9 and ET o,10 (simplified FAO56-PM) had high similarity in spatial distribution with ET o,s, but with much smaller ranges than ET o,s. ET o,i generally had moderate variability (C v < 1.0), of which, C v values of ET o,9 and ET o,10 were the first and the next largest, C v values of ET o,1 to ET o,8 were small.

Figure 4
figure 4

Spatial distribution of multi-year mean monthly ET o,i in China. (ArcGIS 10.2, http://map.baidu.com, Lingling Peng).

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The temporal variations of multi-year mean monthly and annual ET o,i in different sub-regions showed various similarity with that of ET o,s (Figs 5 and 6). In Fig. 5, the variation patterns of monthly ET o,i and ET o,s were general with single peak (valley) around July (January or December), which were also the months that the largest (smallest) differences between ET o,i and ET o,s occurred. The differences between ET o,i and ET o,s curves was the largest for the sub-region I (northwestern China), and was the smallest for the sub-region VI (the Qinghai-Tibetan Plateau). The ET o,s curves were generally in the upper of the 11 curves for different sub-regions. Of the ten curves, ET o,1, ET o,2, ET o,9 and ET o,10 deviated ET o,s much and were not suitable for best alternative of ET o,s. ET o,7 estimated by the FAO24 method had the smallest differences in all of the 7 sub-regions and EMC, followed by ET o,8 estimated by the Wright69 method. The other ET o,i (i = 3, 4, 5, and 6) curves differed but had neither the largest nor the smallest deviations with ET o,s curves. ET o,4 curves for sub-region I and ET o,7 for sub-regions II to VII and EMC were closest to ET o,s curve. Except ET o,7 which was a combination type estimated with a high data-requirement, ET o,6 for sub-regions I, II, IV, VI and EMC were also very close to ET o,s curve. ET o,6 was estimated by the modified Hargreaves-Samani (Berti et al.68), which belonged to the temperature-based type, needed only temperature data, and was simple in computation. In general, both ET o,i and ET o,s curves were regional-, seasonal- and method-specific.

Figure 5
figure 5

Temporal variations of multi-year mean monthly ET o,s and ET o,i in different sub-regions and EMC.

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Figure 6
figure 6

The inter-annual variations of ET o,i and ET o,s in different sub-regions and EMC over 1961–2013.

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In Fig. 6, the annual variations of ET o,i or ET o,s generally had similar temporal variation patterns over 1961–2013 but their values differed a lot. For sub-region I, ET o,i curves ranked with a method order of MHS_1 > KPM > FAO56-PM > HS > FAO24 > MHS_2 > PT > Mak > Val_2 > IRA > Val_1, i.e., ET o,5 > ET o,8 > ET o,s > ET o,4 > ET o,7 > ET o,6 > ET o,3 > ET o,2 > ET o,10 > ET o,1 > ET o,9, while the orders changed for the other sub-regions. Differences between annual ET o,i and ET o,s curves were generally large in sub-regions I and VII, but small in sub-regions III and VI. For sub-regions III, IV and EMC, annual ET o,6 values were much close to ET o,s. For the other sub-regions, annual ET o,7 was also similar to ET o,s. Annual ET o,1 ET o,2, ET o,9 and ET o,10 values were much smaller at most sub-regions and EMC, which was similar to the results of monthly ET o,1 ET o,2, ET o,9 and ET o,10. Also, both annual ET o,i and ET o,s curves were regional-, seasonal-, and method-specific.

Performance comparison of the selected 10 methods for estimating ET o,i

Relative error

Because the estimated ET o,i values were regional-specific, the RE values for ET o,i also showed differences in spatial distributions (Fig. 7). Ranges of RE for ET o,i varied. The range of absolute RE values for ET o,9 was the largest, followed by ET o,10. RE for most of ET o,i covered both negative and positive values, but RE range of ET o,1, ET o,9 and ET o,10 covered only negative values. ET o,7 had the smallest RE range, which reflected that the FAO24 method was more accurate for estimating monthly ET o in EMC. The radiation-based Mak method had smaller RE ranges when compared to the empirical, temperature-based methods and another radiation-based method PT, but it generally underestimated ET o in most of the months and sites and only had local adaptability in sub-region VI, therefore this method shouldn’t be the best alternative for ET o,s in different sub-regions and EMC. Considering the simpler temperature-based ET o type, the MHS_2 method had lower RE than the other temperature-based methods. In general, the spatial distribution of RE for different ET o,i differed at different locations. It revealed the differences in adaptability extents of the applied methods.

Figure 7
figure 7

Spatial distribution of RE values for multi-year mean monthly ET o,i in EMC. (ArcGIS 10.2, http://map.baidu.com, Lingling Peng).

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Generally consistent with the spatial distribution, the temporal variations of RE for monthly ET o,i were also method and sub-region-specific (Fig. 8). The largest (smallest) RE curves generally occurred for ET o,9 (ET o,7) in all of the 7 sub-regions and EMC. The largest negative RE curves were ET o,1, ET o,9 and ET o,10 in all of the 7 sub-regions and EMC, indicating worse performance of the methods IRA, Val_1 And Val_2. Although generally varied with the month, RE values for ET o,4, ET o,6, ET o,7, and ET o,8 ranged between −0.2 to 0.2 in most time of the year for 3, 4, 7 and 7 sub-regions, respectively. The RE values were generally small for EMC when compared to any one of the sub-regions or the methods. In general, in the temperature-based methods, MHS_2 performed the best in most time of the year for most of the sub-regions.

Figure 8
figure 8

The temporal variations of RE values for multi-year mean monthly ET o,i in different sub-regions and EMC.

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The relative error (RE) values of the monthly and annual ET o,i using the selected 10 methods for EMC are presented in Table 3. Values of ET o,1, ET o,2, ET o,9 and ET o,10 underestimated ET o,s in all the 12 months and the whole year, of which, both monthly and annual ET o,9 had the largest deviations, followed by ET o,10. ET o,3 underestimated ET o,s in 8 months (except June, July, August, September) and the whole year. ET o,8 underestimated ET o,s in 6 months in January, February, March, April, November, December but slightly overestimated annual ET o,s. Moreover, the RE values were mostly month-free (i.e., overall larger or smaller than ET o,s in most months of the year) when comparing different estimation methods. However, ET o,4, ET o,5, ET o,6, ET o,7 and ET o,8 overestimated ET o,s in 10, 8, 7, 7 and 6 months of the year, respectively, which resulted to overestimated annual ET o. The RE values for annual ET o,i ranked in an order of ET o,9 > ET o,10 > ET o,1 > ET o,2 > ET o,4 > ET o,3 > ET o,5 > ET o,6 > ET o,8 > ET o,7, corresponding to the method order of Val_1 > Val_2 > IRA > Mak > HS > PT > MHS_1 > MHS_2 > KPM > FAO24. Each method overestimated or underestimated ET o in different months or the whole year when compared to FAO56-PM, but in the temperature type, the MHS_2 method was found to be the closest to FAO56-PM considering. Although the MHS_2 method underestimated the ET 0,s by 24% in January, and 15% and 25% in November and December, but had a very low RE (2%) for the year when compared to the FAO56-PM method.

Table 3 Relative error (RE) values of the 10 selected methods for estimating ET o,i at the monthly and annual timescales for China.
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Standard deviation

The spatial distribution of multiyear mean monthly standard deviation (θ) for ET o,i are illustrated in Fig. S2. All of the ten ET o estimation methods showed larger θ values in the northern China (sub-regions I, II and III), although with different ranges of θ. There was a method order of ranges for θ, i.e., IRA < Val_1 < Mak < PT < MHS_2 < HS < FAO24 < MHS_1 < Val_2 < KPM. In general, a larger ranges of ET o,i corresponded to a larger ranges of θ, the IRA method had a smallest range of θ because it had a smaller range of ET o. In fact, this method largely underestimated ET o,s. The KPM method had a largest range of θ, which indicated the variation scope of ET o,8 values was large. The index θ didn’t reflect the deviations of each ET o,i to ET o,s, it only reflected the deviations of monthly ET o,i to average ET o,i.

The temporal variations of standard deviation (θ) averaged for the 12 months for ET o,i are illustrated in Fig. S3. The standard deviations of the monthly and annual ET o in EMC are presented in Table 4. θ in sub-region I, VI and EMC were generally smaller than the other sub-regions for each month. The MHS_1 had the largest θ for all of the sub-regions and EMC. For EMC, the θ curves ranked in the method order of IRA < Val_1 < Val_2 < HS < Mak < PT < MHS_2 < FAO24 < KPM < MHS_1.

Table 4 Standard deviation values of ET o,i (i = 1 to 10) and ET o,s at the monthly timescale.
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Nash-Sutcliffe efficiency coefficient

The spatial (temporal) distribution of multiyear mean monthly Nash-Sutcliffe efficiency coefficients (NSEs) for ET o,i are illustrated in Figs S4 and S5. In Fig. S4, the ranges of NSE ranked in a method order of FAO24 < Mak < KPM < MHS_2 < PT < HS < MHS_1 < IRA < Val_2 < Val_1. The FAO24 and KPM, as analyzed above, were both combination based ET o methods, although both had smaller NSE ranges, their equations had higher demand of climatic variables. The Mak method had a smaller range of NSE than MHS_2 (between −9.32 and 0.35), it performed better than MHS_2 in sub-regions IV and VI, but it needed addition shortwave radiation (or sunshine hour) when estimating ET o. The NSE of MHS_2 method ranged between −16 and 0.20, it performed well for most sub-regions except VI and VII. From climatic variable demand aspect, the MHS_2 best met a simple equation standard than the other equations, with general good performance. In Fig. S5, the Val_2, Val_1, IRA, HS and Mak were excluded from the ten ET o methods because theire NSE values in each month and each sub-region were generally smaller than 0. Among the left 5 methods, similar to the RE values, the NSE of FAO24 and KPM methods were better, followed by the MHS_2 method, also indicating MHS_2’s better performance for the temporal variations of monthly ET o in the methods with less climatic data demand.

NSE of the monthly and annual ET o,i using the selected 10 methods for EMC are presented in Table 5. Except ET o,7 which was estimated by the combination-based FAO24 method, there were 0, 0, 1, 2, 1, 4, 3, 0 and 0 months fell into the ranges of 0 and 1 for NSE values of ET o,1, ET o,2, ET o,3, ET o,4, ET o,5, ET o,6, ET o,8, ET o,9 and ET o,10, respectively. This indicated a better performance of the MHS_2 method estimated by Berti et al.68 in the non-combination type. For the whole year, only NSEs of the FAO24 and MHS_2 methods were larger than 0, which showed the feasibility of the two methods. But for a best alternative, the combination based FAO24 was not suitable.

Table 5 Nash-Sutcliffe efficiency coefficients of ET o,i in different sub-regions and EMC.
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Therefore, through a comprehensive comparison of spatiotemporal variations from the ten selected methods by relative error, standard deviation and Nash-Sutcliffe efficiency coefficient, the MHS_2 method was preliminary selected as a better one for an alternative of ET o,s with its equation simplicity, least data demand and better performance.

Scatter plots of monthly ET o,i vs. ET o,s

Although the spatiotemporal distribution of multi-year mean ET o,i were analyzed and the performances of all the methods were compared for each sub-region, direct comparison between monthly ET o,i and ET o,s are still necessary, in order that if the required full climatic data for estimating ET o,s are lacking, an relatively accurate alternative method could be selected out from the 10 candidate methods using less weather data. The scatter plots of monthly ET o,i with ET o,s for different sub-regions and EMC are illustrated in Fig. 9. In general, ET o,i deviated more with ET o,s in July, but less for December and January. In all of 7 sub-regions and EMC, ET o,1, ET o,2, ET o,9 and ET o,10 were smaller but ET o,4 and ET o,8 were larger than ET o,s. ET o,3, ET o,5 ET o,6 and ET o,7 were not consistently larger or smaller than ET o,s in different sub-regions. ET o,6 and ET o,7 were close to ET o,s, of which, data points of ET o,7 concentrated to the 1:1 lines the most. Of the 10 ET o,i, ET o,9 deviated the greatest from ET o,s, followed by ET o,10 which showed large scattered distances with the 1:1 lines. From visual comparison, ET o,2, ET o,3, ET o,4, ET o,5, ET o,6, ET o,7 and ET o,8 tended to concentrated to a striating in spite of their deviations from 1:1 line and had good linear correlations with ET o,s in all 7 sub-regions and EMC.

Figure 9
figure 9

Comparisons of ET o,i and ET o,s in each month and sub-region.

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By comprehensive comparisons using RE, standard deviations, NSE and scatter plots, although the two equations proposed by Valiantzas14 are relatively new, both had worse performance than the other methods in different sub-regions and EMC. The two combination type equations Doorenbos-Pruitt4 and Wright69 performed generally well, but had high weather data requirments. The equations proposed by Irmak et al. (2003), Makkink66, Priestley-Taylor23, Hargreaves-Samani16 and Droogers-Allen67 were all simple equations with less data requirements but didn’t perform very well. The Berti et al.68 equation (MHS_2) was a newly proposed temperature-based equation based on modified Hargreaves-Samani. The MHS_2 equation met the least data demand and had general best performance in either the empirical-based, radiation-based, temperature-based or the simplified FAO56-PM equations.

Validation of a best alternative equation for ET o,s

For most sub-regions and EMC, there were good linear correlations between monthly ET o,i and ET o,s. The linear equation is written as:

$$E{T}_{{\rm{o}},{\rm{i}}}=a\,E{T}_{{\rm{o}},{\rm{s}}}+b\,{\rm{or}}\,E{T}_{{\rm{o}},{\rm{s}}}=\frac{E{T}_{o,i}-b}{a}$$
(12)

where a and b are fitted coefficients.

Values of a, b and coefficient of determination (R 2) for various ET o,i and 7 different sub-regions as well as EMC are given in Table 6. R 2 values for ET o,1, ET o,2, ET o,3, ET o,4, ET o,6, ET o,7, ET o,8, ET o,9 and ET o,10 were larger than 0.85 for each sub-region and EMC. Of these, the estimation of ET o,1, ET o,2, ET o,3, ET o,7, ET o,8, ET o,9 and ET o,10 utilized 5, 4, 5, 6, 6, 4 and 4 climatic variables among T min, T ave, T max, RH, U 2, n and P, respectively; whereas ET o,4 and ET o,6 used only 3 (i.e., T min, T ave and T max) with much simpler computation procedures.

Table 6 The fitted a, b and R 2 values for correlating ET o,i with ET o,s in different sub-regions using Equation 12.
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Because temperature data are easier with less cost to observe, and ET o,6 estimated by the MHS_2 method was not only simpler, highly correlated with ET o,s in each month and most sub-regions, but also had generally good similarity in spatiotemporal distribution with ET o,s. Considering both good performance and the correlation with ET o,s, the MHS_2 method was generally good for substituting ET o,s. Therefore, ET o,6 was finally selected as the best alternative for estimating ET o,s in EMC. The calibrated a values were 0.93, 1.00, 1.19, 1.17, 1.15, 1.09, 0.93 and 1.12, and b values were −4.73, −11.1, −11.0, −12.2, 0.28, −16.2, 10.4 and −8.04 for sub-regions I, II, III, IV, V, VI, VII and EMC, respectively.

The best alternative MHS_2 could then be widely applied in China for ET o estimation when only temperature data are available. Because there were still deviations in the MHS_2 method, the linear equation correlated for ET o,6 and ET o,s using Equation 12 could be rewritten as follows:

$$E{T}_{{\rm{o}},{\rm{s}}}=A\,E{T}_{{\rm{o}},6}+B$$
(13)

where A and B are numerically equal to 1/a and −b/a, respectively. Equation 13 is also a calibration between ET o,6 and ET o,s.

For easier application of Eq. 13, values A and B for the 552 sites in China were validated. Figure 10 indicates the spatial distribution of A, B, and correlation coefficient (R). Values of A decreased from 1.32 to 0.67 from northwest to southwest and eastern China. B values were the largest in sub-region VI, followed by sub-regions II, III, IV, I, V, and VII, respectively. Values of R ranged between 0.87 and 0.99, were larger than 0.95 in most of China, especially in north China. The general high R values confirmed the applicability of the best alternative MHS_2 method in China after accurate calibration.

Figure 10
figure 10

Spatial distributions of the parameters A, B and R in equation 13 in EMC. (ArcGIS 10.2, http://map.baidu.com, Lingling Peng).

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Discussion

Under the global climate change, decreasing trends in ET o have been observed in different parts of the world32, 76, 77, including China78 and most parts of China, e.g., the Haihe River basin79, the Huang-Huai-Hai Plain80, the northwest China including Xinjiang Uywer Autonomos Region81, southeast China, the Yangtze river basin64, etc. The increasing trends were found at most sites of the Qinghai-Tibetan Plateau34. The trends were also bi-directional in China. This study revealed that annual ET o,s for 61.4% of the study sites had decreasing trends, of them, 9% of the trends were significant. Our research agreed with the former research in the general decreasing trends of ET o,s for EMC, but in the meantime, there were also differences between this research and the previous.. The differences were caused by the changes in the study period, the data source, the ET o,s estimation methods, the site number applied, and the research aims. For example, Wang et al.51 also applied 4189-grid data during 1961–2013 in EMC to estimate ET o and identified the contribution of climatic variables to ET o variability. They revealed that annual ET o decreased with a mean rate of 6.84 mm/decade, and the sites with significant increase trends mainly distributed in the Qinghai-Tibetan Plateau. This research also reported general increasing trends in the same region, i.e. sub-region VI.

The most precise ET o estimation method varied for different regions. The frequently-used methods are the FAO56-PM, HS, and pan measurement etc., these methods have been applied to partial of China or EMC58, 78, 82. Xu et al.82 applied 5 meteorological stations during 1999–2007 in arid-zone of China (i.e., sub-region I, VI of this research) and selected the HS method as the best alternative of ET o,s. This research selected the MHS_2 as a best alternative of ET o,s for different sub-regions and EMC, because it not only had a general high accuracy but also used only temperature data which were easy to observe or collected, even for the sites where the other climatic data were lacking. Moreover, this research also provided the spatial distributions of the calibrated parameters of the MHS_2 method as the best alternative of ET o,s for different sub-regions and EMC, which were very useful for researchers to apply the calibrated MHS_2 method in China.

The MHS_2 method overestimated ET o in the sub-regions V and VII in the high temperature section of EMC (Fig. 7f). RE reached 20% especially in March, April, May and June (Figs 8e,g and 9). Both sub-regions are humid and sub-humid climatic zones of EMC. This reflected the disadvantages of MHS_2 which only applied temperature data for estimating ET o. When temperature is high, ET o,6 obtained with the MHS_2 method could be high but ET o,s may not be as high as it considering also wind speed, relative humidity and sunshine hour. Under the overestimation conditions, the relationship between ET o,6 and ET o,s should be re-calibrated for March, April, May and June. The re-calibrated parameters A, B and R 2 in March, April, May and June for the two sub-regions are presented in Table 7.

Table 7 The re-calibrated parameters A, B and R 2 in March, April, May and June for the V and VII sub-regions using Equation 13.
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Conclusions

Based on monthly climatic data collected from 552 stations during 1961–2013 across different sub-regions of China, a comprehensive comparison between ET o,i (estimated by the IRA, Mak, PT, HS, MHS_1, MHS_2, FAO24, KPM, Val_1 and Val_2 methods) and ET o,s estimated by the FAO56- PM method has been conducted in 7 sub-regions and EMC. 339 and 213 sites had decrease and increase trends in annual ET o,s, indicating a general decrease trend in annual ET o,s. For the spatial distribution, values of multi-year mean monthly ET o,s in western China (high elevations) were larger than in eastern China (low elevations). The step by step comparison of spatiotemporal distribution, RE, standard deviations, NSE and scatter plots between ET o,i and ET o,s either for monthly and annual timescales or different sub-regions and ECM consistently showed the general high accuracy of ET o,6 estimated by the MHS_2 method proposed by Berti et al.68. The MHS_2 method utilized only temperature data, was simple in computation procedure when compared to the other 9 ET o estimation methods, and was highly correlated with ET o,s. It was a best alternative for ET o,s when climatic data were lacking. After accurate validation for the MHS_2 method using equation 13, the calibrated parameters of A and B for each site, sub-region and EMC were obtained. This research is an important contribution to ET o estimation method in China when the high requirements of climatic data could not be met.