Theor Appl Climatol DOI 10.1007/s00704-014-1269-x

ORIGINAL PAPER

Temperature-based approaches for estimating monthly reference evapotranspiration based on MODIS data over North China X. Zheng & Jiaojun Zhu

Received: 24 February 2014 / Accepted: 21 August 2014 # Springer-Verlag Wien 2014

Abstract Reference evapotranspiration (ETo) maps play an important role in distributed hydrological modeling and are particularly useful for regional agricultural and water resource management. In the Three-North Shelter Forest Program, water requirements (i.e., ETo) of different land use types are important preconditions for afforestation program management. The Food and Agriculture Organization Penman– Monteith (FAO-PM) method is the most common method for estimating ETo, but it requires many different types of meteorological data, and few stations with adequate meteorological resources exist in the Three-North regions. In addition, the spatial distribution of ordinary meteorological stations is limited. This study employed two temperature-based ETo methods, Hargreaves and Thornthwaite. The monthly mean, maximum, and minimum air temperatures were estimated using moderate resolution imaging spectroradiometer (MODIS) data. The original coefficients of Hargreaves and mean temperatures of Thornthwaite were modified for regional calibration (with the FAO-PM method as the standard). In

X. Zheng : J. Zhu State Key Laboratory of Forest and Soil Ecology, Institute of Applied Ecology, Chinese Academy of Sciences, Shenyang 110164, People’s Republic of China X. Zheng : J. Zhu Key Laboratory for Management of No-commercial Forests, Liaoning Province Shenyang 110016, People’s Republic of China X. Zheng : J. Zhu Qingyuan Experimental Station of Forest Ecology, Chinese Academy of Sciences, Shenyang 110016, People’s Republic of China X. Zheng : J. Zhu (*) University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China e-mail: [email protected]

the comparison between the original/adjusted Hargreaves and the original/adjusted Thornthwaite methods, the adjusted Hargreaves method was appropriate for estimating ETo in the Three-North regions. The average mean bias error (MBE) was −0.21 mm, the relatively root mean square error (RRMSE) was 13.44 %, the correlation coefficient (R2) was 0.85, and the slope (b) was 1.00 for the monthly ETo. While the MBE was 2.32 mm, the RRMSE was 7.07 %, the R2 was 0.90, and the b was 1.00 for the annual ETo. Therefore, it is possible to estimate monthly and annual ETo values for other parts of the country or the world using adjusted Hargreaves with the estimated air temperature data instead of using the FAO-PM with observed data.

1 Introduction Crop reference evapotranspiration (ETo) is an important variable in agrohydrological systems, as it depends only on climatic parameters and reflects the evapotranspiration of an ideal and well-watered grass surface (Yang et al. 2012; Zheng et al. 2012). The actual evapotranspiration can be derived from ETo by means of proper crop and water stress coefficients (Allen et al. 1998). Therefore, accurate estimation of ETo is very important for various fields of study, including hydrologic water balance, irrigation system design and management, crop yield simulation, water resources planning management, and improving the use of water in agriculture and optimizing lost water (Liang et al. 2010; Zhang et al. 2010; Jahanbani and El-Shafie 2011). Hence, accurately estimating ETo is essential (Meza 2005; Douglas et al. 2009). A number of methods are available for estimating ETo, e.g., Penman (1948), Thornthwaite (1948), Blaney and Criddle (1950), Hargreaves and Samani (1985), and FAO-56 Penman–Monteith (Allen et al. 1998). Among these methods, the FAO-56 Penman–Monteith (FAO-PM) method, which

X. Zheng, J. Zhu

incorporates both energy balance and aerodynamic theory, is considered the most appropriate model to predict ETo and is recommended by the Food and Agriculture Organization of the United Nations (FAO) as the standard for computing ETo from full climate records of meteorological data (Diodato and Bellocchi 2007; Zhang et al. 2010; Raziei and Pereira 2013). However, the main shortcoming of the FAO-PM equation is that it requires a great deal of weather data that are not always available in many locations (Jabloun and Sahli 2008; Gocic and Trajkovic 2010; Sabziparvar and Tabari 2010). This is especially true in developing countries where reliable weather data sets for radiation, relative humidity, and wind speed are limited. Therefore, temperature-based models have been widely applied (Gavilán et al. 2006; Moeletsi et al. 2013; Raziei and Pereira 2013). Temperature-based models include Thornthwaite (1948) and Hargreaves (Hargreaves and Samani 1985). These methods have the advantage of requiring less meteorological data (Chen et al. 2005; Trajkovic 2005). Thornthwaite (1948) explored an empirical ETo model using only mean air temperatures in the east-central USA. Hargreaves and Samani (1985) developed an alternative approach to ETo computation in California, USA, using maximum and minimum air temperature data as the only input variables (Gavilán et al. 2006). These methods were developed for use in specific studies and are most appropriately applied to climates similar to that where they were developed (Bautista et al. 2009; Douglas et al. 2009; Najafi and Tabatabaei 2009). Large errors can be expected when these methods are extrapolated to other climatic areas without recalibrating the constants involved in the formulae (Bautista et al. 2009). Therefore, it becomes necessary to develop procedures for realizing regional and temporal adjustments to Hargreaves and Thornthwaite to obtain the best estimations of ETo (Pereira and Pruitt 2004; Ahmadi and Fooladmand 2008). The Three-North Shelter/Protective Forest Program (TNSFP), covering 42.4 % of China’s territory, is China’s Green Great Wall and the world’s largest ecological program. Based on information from the State Forestry Administration of China, the framework of the TNSFP has been configured over the past 30 years, and its eco-environmental benefits have started to manifest in farmland protection, soil and water conservation, wind reduction, and sand dune fixation in the arid and semiarid Three-North regions (i.e., the western part of Northeast China, the northern region of North China and Northwest China) (Wang and Zhou 2003; Liu et al. 2009). However, several severe ecological problems have been reported in certain regions because TNSFP implementation has not been tailored to local environmental conditions in many areas of the Three-North regions. The major problem in the TNSFP is that protective forests have declined

after a certain number of years following implementation (Zhu et al. 2008). Clearly, discrepancies in the supply and requirements of water represent the key environmental factor for determining the success of afforestation in arid and semi-arid areas (Zheng et al. 2012). Water requirements are usually managed based on ETo estimations of different land use types (Spano et al. 2009). Therefore, it is of significant importance to understand the spatial and temporal variation characteristics of monthly/annual ETo for the establishment of shelter forests in arid and semiarid areas. However, the spatial distribution of meteorological stations is extremely limited in the Three-North regions of China. There are approximately 220 meteorological stations in the Three-North regions (18,495 km2 per station), of which only 40 stations have historical data for calculating ETo with FAOPM (101,725 km2 per station), and the remaining 180 stations cannot satisfy data requirements for FAO-PM. There are thus two challenges for estimating the monthly ETo across northern China: the first is the limited number of meteorological stations (220 stations in total); the other is the limited number of meteorological stations with complete data (for FAO-PM). Remote-sensing techniques may be a sound alternative to provide spatially distributed information because they enable the assessment of environmental conditions in ecosystems (Cristea et al. 2013; Karim et al. 2013). The ETo can therefore be estimated using estimated air temperature data based on remote-sensing data. The purpose of the study was to provide the ETo for all of northern China at high resolution (1 km×1 km). The objective of this study was to calibrate and evaluate two empirical ETo methods (i.e., Thornthwaite and Hargreaves) using estimated temperatures as the input variables for calculating monthly ETo in the semi-arid and arid regions of North China. The estimated temperature data were obtained from remotesensing data and geographic information system (GIS) technology.

2 Materials and methods 2.1 Study area The Three-North regions cover a total area of 4.069× 106 km2 (73° 26′–127°50′ E, 33°30′–50° 12′ N), including 551 counties across 13 provinces and accounting for more than 42.4 % of China’s total territory (Fig. 1). The majority of eastern China is under the influence of the East Asian monsoons, and the inland region has a typical continental climate. Temperature differences across the Three-North regions are equally dramatic. The annual mean temperature decreases from approximately 15 °C in the southern tip to −17 °C in the high

Temperature-based approaches for estimating reference ET

Fig. 1 Study area location, elevation, and location of the meteorological stations. Data from 133 meteorological stations (accounting for 60 %; black circles) were used for the temperature prediction models (i.e., the training data set), and the data from the other 87 stations (accounting for 40 %; white-filled circles) were used for the temperature validation (i.e.,

the test data set). Only 40 stations (a part of the 220 stations, red circles) had complete measured data for FAO-PM, which were used for calibrating the parameters of the Hargreaves and Thornthwaite methods and for verifying the ETo methods (i.e., the original/adjusted Hargreaves and the original/adjusted Thornthwaite)

mountains of the northwestern region, with mean temperatures ranging between 3 and 9 °C in most areas. Precipitation decreases from the east (approximately 750 mm) to the west (less than 50 mm), reflecting the declining influence of moist air from the Pacific Ocean. In topographic terms, there are three predominant geomorphic units in the Three-North regions: high mountains (e.g., Tianshan and Qilian Mountains), basins (e.g., Tarim and Junggar basins), and plains (e.g., Northeast China Plains and North China Plains), with elevations ranging from approximately 155 m below sea level at Ayding Lake to more than 7,000 m above sea level at Kunlun Mountain.

World Meteorological Organization, at 2 m above the ground. Although 220 ground metrological stations are available in the Three-North regions (Fig. 1), only 40 have historical data (Tmean, Tmax, Tmin, u, RH, and Rn) for calculating ETo with the FAO-PM method (Table 1). Currently, the remaining 180 stations provide only ordinary data (Tmean, Tmax, Tmin, u, and RH). Meteorological data from January 2001 to December 2009 were used in this study.

2.2 Data collection 2.2.1 Meteorological stations Monthly meteorological data—mean temperature (Tmean), maximum temperature (Tmax), minimum temperature (Tmin), mean wind speed (u), mean relative humidity (RH), and net radiation (Rn)—were obtained from ground meteorological stations and were measured, in line with the standards of the

2.2.2 MODIS data Moderate resolution imaging spectroradiometer (MODIS) data derived from National Aeronautics and Space Administration (NASA) MODIS sensors onboard the Terra satellite were downloaded from the EOS Gateway at https://lpdaac.usgs.gov/. Two different types of products—MOD13A3 (containing the monthly normalized difference vegetation index, NDVI) and MOD11A2 (containing the 8-day nighttime land surface temperature, LST)—were selected for this study. NDVI in MOD13A3 and nighttime LST in MOD11A2 were downloaded in hierarchical data format (HDF) on a sinusoidal projection and were converted to the

X. Zheng, J. Zhu Table 1 Meteorological stations (n=40) with complete data for the FAO-PM method in the ThreeNorth regions

Stations

East longitude (o)

North latitude (o)

Elevation (m)

Mean rainfall (mm)

Climate

Hailaer Fuyu Solun Harbin Aletai

119.45 124.29 121.13 126.46 88.05

49.13 47.48 46.36 45.45 47.44

610 163 500 142 735

321.5 410.1 383.5 492.6 221.5

Semi-arid Semi-humid Semi-arid Semi-humid Arid

Sucheng Yining Wulumuqi Yanqi Tulufan Akesu Keshen Luoqiang Hetian Hamin Ejinaqi Dunhuang Jiuquan Minqin Gangcha Geermu Xining Yuzhong

83.00 81.20 87.39 86.34 89.12 80.14 75.59 88.10 79.56 93.31 101.04 94.41 98.29 103.05 100.08 94.54 101.45 104.09

46.44 43.57 43.47 42.05 42.56 41.10 39.28 39.02 37.08 42.49 41.57 40.09 39.46 38.38 37.20 36.25 36.43 35.52

535 663 935 1,055 35 1,104 1,289 888 1,375 737 941 1,139 1,477 1,368 8,302 2,808 2,295 1,874

283.6 323.3 309.4 78.6 16.0 78.8 70.7 43.4 44.3 49.6 29.8 43.3 87.7 124.8 395.9 42.3 429.3 370.7

Arid Arid Arid Hyperarid Hyperarid Hyperarid Hyperarid Hyperarid Hyperarid Hyperarid Hyperarid Hyperarid Hyperarid Arid Semi-arid Hyperarid Semi-humid Arid

Erenhot Uradbanner Datong Dongsheng Yinchuan Taiyuan Guyuan Yan’an Xifeng Xilinhot Tongliao Changchun Chaoyang Beijing Tianjin Leting Jinghe

111.58 108.31 113.20 109.59 106.13 112.33 106.16 109.30 107.38 116.07 122.16 125.13 120.27 116.28 117.04 118.53 108.58

43.39 41.34 40.06 39.50 38.29 37.47 36.00 36.36 35.44 43.57 43.36 43.54 41.33 39.48 39.05 39.26 34.26

965 1,288 1,067 1,462 1,111 778 1,753 959 1,421 1,003 179 237 170 31 3 11 410

118.0 212.2 347.6 342.0 179.9 425.6 408.3 504.7 538.9 219.6 305.9 533.6 402.9 432.8 502.5 519.5 538.4

Hyperarid Arid Semi-arid Semi-arid Arid Semi-arid Semi-arid Semi-arid Semi-arid Arid Arid Semi-humid Semi-arid Semi-arid Semi-arid Semi-arid Semi-arid

“GeoTIFF” file format on an Albers equal-area conical projection using the MODIS Reprojection Tool (the “MRT” software, downloaded from https://lpdaac.usgs. gov/tools/modis_reprojection_tool), which is based on the nearest neighbor resampling method. For MOD11A2, 46 total nighttime LST 8-day composite images were available for each year. Four consecutive

8-day composites were averaged to create twelve 32-day composite images (except February and December, which were the average of three consecutive 8-day composites), and each 32-day (or 24-day) composite images were used to represent the monthly LST. All image processing procedures and calculations were carried out using the Erdas Imagine 9.2 software.

Temperature-based approaches for estimating reference ET

2.3 ETo equations

where Tmean is the average temperature for a given day or period, and I and a are thermal indices, calculated by the following:

2.3.1 FAO-56 Penman–Monteith equation FAO-56 Penman–Monteith (FAO-PM) is considered a standard and the most precise method to estimate ETo. It is expressed as follows (Allen et al. 1998):

ETo ¼

900 U 2ðes−eaÞ T þ 273 Δ þ γ ð1 þ 0:34U 2Þ

0:408ΔðRn−GÞ þ γ

ð1Þ

where Δ is the slope of the saturated vapor pressure curve (kPa °C−1), Rn is the net radiation at crop surface (MJ m−2 day−1), G is the soil heat flux density (MJ m−2 day−1), T is the mean daily air temperature at 2 m height (°C), U2 is the wind speed at 2 m height (m s−1), es is the saturation vapor pressure (kPa), ea is the actual vapor pressure (kPa), es −ea is the saturation vapor pressure deficit (kPa), and γ is the psychometric constant (0.0677 kPa °C−1).

2.3.2 Hargreaves equation The Hargreaves equation was developed by Hargreaves and Samani (1985) using 8 years of daily lysimeter data from Davis and California, and tested in different locations, such as Australia, Haiti, and Bangladesh. Since then, the method has been successfully applied worldwide (Martinez-Cob and Tejero-Juste 2004; Ravazzani et al. 2012; Mendicino and Senatore 2013). The Hargreaves method estimates ETo using only maximum and minimum air temperatures and extraterrestrial radiation data. The equation can be written as follows: ETo ¼ Co  Ra0  ðT max−TminÞ0:5  ðT mean þ 17:8Þ

ð2Þ

I ¼ 12ð0:2T aÞ1:514

ð5Þ

a ¼ 6:75  10−7 I 3 −7:71  10−5 I 2 þ 1:7912  10−2 I þ 0:49239

ð6Þ

where Ta is the climatologically normal annual temperature. 2.4 Estimation of monthly ETo In this study, we used the Hargreaves and Thornthwaite equations to obtain continuous maps of ETo at a 1-km resolution using the following four steps. 2.4.1 Estimation of air temperatures Tmax, Tmean, and Tmin were estimated by means of the combination stepwise regression modeling and spatial interpolation techniques (SRMSIT), as described by Zheng et al. (2013). In the process of SRMSIT, air temperature equations were built using geographical (latitude, LAT; altitude, ALT; and continentality, CON) and remote-sensing variables (nighttime land surface temperature, LST; and normalized difference vegetation index, NDVI).

where Ra’ is the extraterrestrial solar radiation, in millimeters per day; and Co is the conversion parameter (=0.0023). For this study, original Co and locally calibrated values were used.

T y ¼ b0 þ b1LAT þ b2ALT þ b3CON þ b4LST

2.3.3 Thornthwaite equation

where Ty represents the estimated Tmax, Tmean, and Tmin values; and LAT, CON, LST, and NDVI are the independent variables. The spatial interpolation technique, a part of the SRMSIT method, was favorable for solving the unexplained variation, i.e., the residual value (the difference between the observed value and the modeling value), in stepwise regression modeling and for improving the accuracy of air temperature maps (Ninyerola et al. 2000, 2007). We used inverse distance weighting (IDW) to obtain spatial information of the residual values in the ground meteorological stations (Zheng et al. 2013).

Thornthwaite (1948) correlated mean monthly temperatures with ETo values determined from the water balance in valleys of the eastern USA where sufficient moisture was available to maintain active transpiration. The Thornthwaite formula for monthly ETo (mm) is as follows:
 10T mean a ETo ¼ 16 for 0 C ≤ T mean < 26:5 C ð3Þ I ETo ¼ −415:85 þ 32:24T mean−0:43T mean2

for T mean≥26:5 c

ð4Þ

þ b5NDVI

ð7Þ

X. Zheng, J. Zhu

Fig. 2 Coefficient of determination (R2), mean bias error (MBE, °C), and root mean square error (RMSE, °C) results for air temperatures estimated with the SRMSIT method for the period of 2001–2009

The air temperature data from 133 stations, accounting for 60 % of the data (the training data set), were used for monthly air temperature prediction, while air temperature data from 87 other stations, accounting for 40 % (the test data set), were used for model validation.

Fig. 3 Linear relationship between monthly Co values and 9-year„ average values of monthly T =ΔT at each station (regression-based calibration)

Temperature-based approaches for estimating reference ET

X. Zheng, J. Zhu

2.4.2 Determination of the parameters of Hargreaves

2.4.3 Adjustment of the Thornthwaite equation

For the empirical Hargreaves coefficient (Co), scientists have proposed a regression-based correction (Vanderlinden et al. 2004; Ravazzani et al. 2012; Mendicino and Senatore 2013). Samani (2000) proposed a formulation, leading to the following equation for Co:

The original Thornthwaite method was adapted by Camargo et al. (1999) to adjust for arid and very humid conditions. For that, the mean temperature (Tmean) was replaced by the effective temperature (Tef), given by the following:

2

−4

Co ¼ 0:00185⋅ΔT −0:0433⋅10 ΔT þ 0:4023

ð9Þ

where Tis the average annual air temperature, and ΔTis the average of Tmax–Tmin. The coefficients a and b were 0.0005 and 0.00159, respectively (R2 =0.90). This type of formulation was also adopted by Lee (2010) for 21 meteorological stations in Korea using recalibrated, but not very different, parameters (a=0.0004, b=0.0013, R2 =0.84). According to the preliminary trial, we used the following equation to calibrate monthly Co:  2 Co ¼ a⋅ T =ΔT Þ þ b⋅T =ΔT þ c þ ΔCo; residual

ð11Þ

ð8Þ

where ΔT is the average of Tmax–Tmin. The recalibration proposed was based on analyses of the annual average of monthly data of the daily temperature range recorded during a 25-year period across 65 stations in the USA, located between 7 and 50° N latitude. Recently, Vanderlinden et al. (2004) derived an alternative equation for correcting Co: . Co ¼ aT ΔT þ b

T ef ¼ βð3T max−T minÞ

ð10Þ

The parameters a, b, and c were computed by minimizing ETo errors between the calibrated methods and the full-data FAO-PM method at the 40 stations based on the average monthly data for the period of 2001–2009 (Mendicino and Senatore 2013). ΔCo,residual represents the residuals of the stepwise regression model fit at each meteorological station, reflecting unexplained variations—that is, other factors, usually more or less local but also those not considered by the model. If ΔCo,residual is spatially interpolated to obtain a surface, the map Co obtained from the stepwise regression model can be corrected. ΔCo,residual values were obtained using IDW based on the complete data sets from the 40 stations. IDW was adopted because IDW outperformed other methods and was the most feasible for residual correction interpolation in a region with limited ground data availability (Duan and Bastiaanssen 2013; Zheng et al. 2013).

where β is named the Camargo parameter (=0.36). However, based on the preliminary trial, Tef derived from the FAO-PM was more related to the mean temperature than the Tef derived from Eq. 11. Therefore, we derived an alternative equation to establish the effective temperatures: T ef ¼ aTmean2 þ bTmean þ c

ð12Þ

The parameters a, b, and c were also computed by minimizing ETo errors between the calibrated methods and the full-data FAO-PM method at the 40 stations based on the average monthly data for the period of 2001–2009. 2.4.4 Validation The 40 stations with complete data for the FAO-PM method were used to validate all of the results for annual/monthly estimated ETo. The performance of the methods (i.e., original Hargreaves, adjusted Hargreaves, original Thornthwaite, and adjusted Thornthwaite methods) was determined by linear regression analyses, always forcing the linear coefficient through the origin (a=0). The slope (b) was used as a measure of accuracy, while the coefficient of determination (R2) was considered a measure of precision. A perfect method should result in b=1 and R2 =1. The performance of the ETo estimates was also evaluated using mean bias error (MBE, mm) and relative root mean square error (RRMSE). ME and RRMSE were calculated by the following equations: 1X ðEToest; i−EToFAOPM; iÞ n i¼1 n

MBE ¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xn ðEToest; i−EToFAOPM; iÞ2 100 i¼1 RRMSE ¼ ⋅ n EToFAOPM; ¯i

ð14Þ

ð15Þ

where EToFAOPM is the ETo of the FAO-PM methods; ET o e s t is ET o of the original Hargreave, original T h or n t h w ai t e , ad j u s t ed H a rgr e av e a nd a dj us t e d

Temperature-based approaches for estimating reference ET

Fig. 4 Spatial and temporal distributions of average calibrated Co values of the Hargreaves method for the study area

X. Zheng, J. Zhu Fig. 5 Coefficient of determination (R2), slope (b), mean bias error (MBE), and relative root mean square error (RRMSE) for ETo estimated by the Hargreaves method with the original (=0.0023) and adjusted Co in the period of 2001–2009

Thornthwaite methods; i is the index of the station number; and n is the number of meteorological stations. MBE describes the direction of the error bias and should ideally be 0. For the RRMSE, the validation is considered to be excellent when the RRMSE is <10 %, good if the RRMSE is between 10 and 20 %, acceptable if the RRMSE is between 20 and 30 %, and poor if >30 % (Campi et al. 2012).

3 Results and discussion 3.1 Spatial distribution of air temperatures We used R2, MBE, and root mean square error (RMSE) to analyze the validation of the final estimated air temperatures with the SRMSIT method for every month in 2001–2009 and found that the SRMSIT method obtained more accurate results in the estimation of monthly Tmean, Tmin, and Tmax in this study (Fig. 2). For example, the average R2 values ranged from 0.90 to 1.00, with an average value of 0.98; the MBE values ranged from −0.70 to 0.25 °C, with an average value of −0.03 °C; and the average RMSE values ranged from 0.12 to 1.41 °C, with an average value of 0.58 °C for monthly Tmean during the period of 2001–2009 (Fig. 2). For the different monthly air temperatures, the SRMSIT method performed variably. The R2 for Tmean was higher than that for Tmin and Tmax during January of 2001 and December of 2009. The average R2 values were 0.98, 0.78, and 0.90; the MBE values were −0.03, 0.13, and 0.90 °C; and the RMSE values were

0.58, 1.34, and 1.19 °C for Tmean, Tmax, and Tmin, respectively (Fig. 2). These results indicated that, compared with mean values, extreme air temperature values are more difficult to predict, which is consistent with the results of Evrendilek et al. (2012) and Boi et al. (2011). Compared with the previous results, Benavides et al. (2007) obtained an RMSE ranging from 1.00 to 2.00 °C using kriging technology for Tmean. Boi et al. (2011) obtained an RMSE between 1.00 and 1.50 °C for Tmin and from 0.50 (winter months) to 1.40 °C (summer months) for Tmax. We obtained more accurate results in the estimation of Tmean, Tmax, and Tmin in this study. The average MBE was −0.03 °C, the RMSE was 0.58 °C, the R2 was 0.98, and b was 1 for the monthly Tmean; the MBE was 0.13 °C, the RMSE was 1.34 °C, the R2 was 0.88, and b was 0.89 for the monthly Tmax; and the MBE was 0.00 °C, the RMSE was 1.19 °C, the R2 was 0.90, and b was 0.91 for the monthly Tmin. Therefore, the SRMSIT method can be used to accurately estimate air temperatures in the different ecosystems of the Three-North regions of China with a limited number of meteorological stations. In addition, the accuracy of spatial air temperature data provides a solid foundation for calculating the monthly and annual ETo at a 1-km resolution.

Fig. 6 Linear relationship between effective temperature (Tef) and the 9-„ year average values of monthly temperatures (Tmean) at each station (regression-based calibration)

Temperature-based approaches for estimating reference ET

X. Zheng, J. Zhu

3.2 Hargreaves method 3.2.1 Calibration of the Hargreaves coefficient As the Hargreaves method is empirical, this process of Co parameter calibration was important. A regression-based calibration (Vanderlinden et al. 2004), which is based on findings that the estimation error is proportional to Tmean but inversely proportional to ΔT (Tmax–Tmin), was attempted, and the following linear relationship was derived between the monthly Co and the 9-year (2001–2009) averaged monthly value of T= ΔT at each station (Fig. 3). In addition to the regression-based calibration, ΔCo,residual values at 40 stations were interpolated to improve the accuracy of the monthly Co using the IDW. Co parameter maps were obtained by combining the regression-based values and the ΔCo, residual maps. However, due to the scarcity of data in the mountainous regions and the fact that the derived linear equation cannot be used in high mountain regions, the average value of Co for each month was chosen as the calibrated Co for the mountainous regions. Finally, monthly Co parameter maps were obtained (Fig. 4). For seasonal changes of Co values, it was clear that a general trend exists among all of the stations in which the Co value increased from winter to spring, i.e., the Co values reached their maximum in April and decreased to their minimum in December. In the geographical distribution of calibrated Co values, they tended to decrease from the eastern

Fig. 7 Coefficient of determination (R2), slope (b), mean bias error (MBE), and relative root mean square error (RRMSE) for ETo estimated by the original Thornthwaite method and the adjusted Thornthwaite method with effective temperatures during the period of 2001–2009

toward the western parts of the study area, which may reflect the different wet and dry conditions in the Three-North regions (Fig. 4). 3.2.2 Evaluation of the calibrated/original Hargreaves equation For the original Hargreaves, the precision of the estimated monthly ETo was not high, with R2 values ranging from 0.00 to 0.61, with an average value of 0.21; b ranging from 0.73 to 1.27, with an average value of 0.91; MBE ranging between −29.49 and 32.97 mm, with an average value of 10.06 mm; and RRMSE ranging between 20.06 and 64.68 %, with an average value of 32.74 % (Fig. 5). When the Hargreaves method was locally calibrated, the accuracy and precision of the estimated ETo for the ThreeNorth regions improved substantially, with R2 ranging from 0.51 to 0.98, with an average value of 0.85; b ranging from 0.87 to 1.15, with an average value of 1.00; MBE ranging from −8.10 to 4.51 mm, with an average value of −0.21 mm; and RRMSE ranging from 5.22 to 45.59 %, with an average value of 13.44 % (Fig. 5). The differences in accuracy between stations were relatively large because the climate varies from station to station. In addition, the validation index revealed that seasonal variations, such as the highest and lowest R 2 values of the adjusted Hargreaves method, likely corresponded to warm and cold months, and the highest and lowest RRMSE values corresponded to warm and cold

Temperature-based approaches for estimating reference ET Table 2 Coefficient of determination (R2), mean bias error (MBE, mm/month), and relative root mean square error (RRMSE, %) results for annual ETo estimated by the adjusted Hargreaves method for the period of 2001–2009

R2 b MBE RRMSE (%)

2001

2002

2003

2004

2005

2006

2007

2008

2009

0.93 1.02 17.39 5.73

0.93 1.00 −2.12 5.55

0.91 0.98 −12.58 6.49

0.75 1.00 −0.54 11.75

0.94 1.01 4.38 5.31

0.96 0.99 −15.76 5.77

0.95 0.98 −25.69 6.76

0.88 1.01 14.69 7.61

0.87 1.04 41.15 8.69

months, respectively. It can be concluded that the adjusted Hargreaves method gave better estimates of ETo in warm months. The climate of the Three-North regions is characterized by the synchronization of high temperatures and ample precipitation, leading to the highest MBE values in warm months.

1.10, with an average value of 0.87; MBE ranging from −17.58 to 7.79 mm, with an average value of −2.86 mm; and RRMSE ranging from 21.93 to 151.37 %, with an average value of 51.12 % (Fig. 7). These results prove that the Thornthwaite method with effective temperatures was slightly better than the original Thornthwaite method for the ThreeNorth regions.

3.3 Thornthwaite equation 3.3.1 Adjustment of mean temperature for the Thornthwaite equation The original Thornthwaite method uses mean temperatures higher than 0.00 °C as input data, which is not applicable for a number of stations in the Three-North regions. With the modification of the Thornthwaite method presented by Camargo et al. (1999), the application range of the Thornthwaite is widened by introducing the concept of effective temperature (Tef). In this study, the effective temperature was adjusted by comparing the effective temperature derived from FAO-PM with monthly mean air temperatures (Fig. 6). In the process of the regression-based correction, lower values of the coefficient of determination in warm months and higher values in cold months may indicate that temperature has a greater effect on ETo in cold months than warm months.

3.4 Comparison of ETo with original/adjusted equations in the Three-North regions For weather stations that only have temperature data available, a very common situation in sparsely populated and underdeveloped regions, the best method for determining ETo among the tested formulae (i.e., original Hargreaves, adjusted Hargreaves, original Thornthwaite and adjusted Thornthwaite methods) was the adjusted Hargreaves method (Figs. 5 and 7). This result was different from that of Ahmadi and Fooladmand (2008), who found that the adjusted Thornthwaite method can estimate the accuracy of monthly ETo. The same result was also observed in Tunisia by Jabloun and Sahli (2008) and in México by Bautista et al. (2009). In addition, the process of parameter calibration can improve the estimation practices and thereby the accuracy of the estimations. In this study, better monthly/annual estimations of ETo were obtained using the adjusted Hargreaves method, with an

3.3.2 Evaluation of the calibrated/original Thornthwaite equation When ETo was estimated by the original Thornthwaite method, the estimated ETo was not acceptable, with R2 values ranging from 0 to 0.42, with an average value of 0.04; b ranging from 0.78 to 3.00, with an average value of 1.33; MBE ranging from −50.70 to 11.56 mm, with an average value of −19.79 mm; and RRMSE ranging from 26.91 to 82.52 %, with an average value of 45.16 % (Fig. 7). In addition, the original Thornthwaite method was not applicable in cold months (e.g., January, November, and December) due to the monthly air temperature being less than 0.00 °C (Fig. 7). When the mean air temperature of the original Thornthwaite method was locally calibrated (Fig. 6), the application range of the Thornthwaite was widened in all months in the Three-North regions; however, the estimated ETo was only slightly improved, with R2 values ranging from 0.00 to 0.35, with an average value of 0.08; b ranging from 0.39 to

Fig. 8 Comparison of annual ETo estimates from the adjusted Hargreaves and FAO-PM methods at 40 sites for the period of 2001–2009. The dashed line represents the linear best fit, and the dotted line is the y=x reference

X. Zheng, J. Zhu

average MBE=−0.21 mm, RRMSE=13.44 %, R2 =0.85, and b=1.00 for monthly ETo (Fig. 5), and an average MBE=

2.32 mm, RRMSE=7.07 %, R2 =0.90, and b=1.00 for annual ETo (Table 2 and Fig. 8).

Fig. 9 Spatial distribution of monthly reference evapotranspiration (mm month−1) from 2000 to 2009 over the Three-North regions, China

Temperature-based approaches for estimating reference ET Fig. 10 Spatial distribution of average reference evapotranspiration (mm year−1) from 2000 to 2009 over the Three-North regions, China

The monthly ETo was rather low, and the spatial distributions were fairly homogeneous when using the adjusted Hargreaves method in winter months (January, February, November, and December) (Fig. 9). Monthly ETo values increased from January to July and then decreased from July to December, and the ETo of the summer months (June, July, and August) accounted for more than 60 % of the total annual amount (Fig. 9). Geographically, ETo gradually decreased from south to north in the Three-North regions, and high ETo areas occurred in middle-low regions with plains and basins, and low ETo areas were found in the mountains due to the high altitudes. The average annual ETo estimated by the adjusted Hargreaves method was 817.93 mm, ranging from 300 to 1,300 mm across the region during the period of 2000–2009 in the Three-North regions (Fig. 10). Lower ETo values were distributed in Northeast China, the Tianshan Mountains, and the eastern region of the Tibetan Plateau; higher values were detected in areas from the Tarim Basin to the Qaidam Basin, the western part of Inner Mongolia and Gansu regions, and the eastern part of Xinjiang region. The distribution pattern of annual ETo was similar to that reported by Yin et al. (2008), who interpolated the estimated ETo using the FAO-PM method based on observed station data. The range of annual ETo values was different from that reported by previous researchers, e.g., 500–1,600 mm (Li et al. 2012), or 600– 1,300 mm (Yin et al. 2008). The major reasons for this difference are as follows: first, our ETo data were at a higher spatial resolution (1 km) compared to previous studies. Second, the lack of climate data for locations with no meteorological stations, such as the Tianshan Mountains and Qaidam Basin, leads to less accuracy in those previous studies based on only station data (Yin et al. 2008; Li et al. 2012). Our results were not limited by meteorological stations. Therefore, the final 1-km ETo map determined using the adjusted Hargreaves method with estimated air temperature data not only significantly improved the spatial resolution but also agreed well with the FAO-PM method (Table 2 and Fig. 10).

However, there was a slight difference between the adjusted Hargreaves and FAO-PM methods for estimating ETo (Figs. 5 and 8). The main reasons were as follows: first, ETo was calculated by FAO-PM using data from 40 weather stations for which ea (obtained from relative humidity in this study) was missing. However, Sentelhas et al. (2010) assessed the agreement between ETo estimated with a full data set and with missing ea data, and found that b values were between 1.01 and 1.12 and R2 values were between 0.76 and 0.96. In fact, ETo values estimated with missing ea data were quite different from ETo values estimated with a full data set. Second, the estimated air temperature values were not equal to the observed values. Although we obtained good agreement between the observed air temperature and estimated air temperature data using SRMSIT method (Fig. 2), error (i.e., differences between estimated and observed air temperatures) was unavoidable. Thus, the agreement between the ETo values estimated with the full data set using FAO-PM and with estimated air temperature data using the adjusted Hargreaves method was not perfect. Third, the Hargreaves method was not suitable for some high mountain areas in the Three-North regions due to the much lower temperatures (the original method was developed in arid and semi-arid regions). Although arid and semi-arid regions occupy more than two thirds of the Three-North regions, the alpine climate areas, e.g., the Tianshan Mountains and Hengduan Mountains, represent an important part of this region. Further modifications to the Hargreaves method are needed to facilitate its use in all regions.

4 Conclusions In this study, the Hargreaves and Thornthwaite methods were calibrated using the standard FAO-PM method. The ETo values computed using the original/adjusted Hargreaves and the original/adjusted Thornthwaite methods were compared

X. Zheng, J. Zhu

with those computed using the FAO-PM method at 40 stations with complete data to determine their performance with only temperature data. The original/adjusted Hargreaves methods were better than the original/adjusted Thornthwaite methods, and the adjusted Hargreaves method can provide relatively accurate estimates of monthly/annual ETo at weather stations with only air temperature available in the Three-North regions. In addition, air temperatures (Tmean, Tmax, and Tmin) at a 1-km resolution can be accurately obtained using the SRMSIT method based on MODIS data and GIS technology to compensate for the lack of meteorological station data. The final 1-km monthly/annual ETo data obtained for the ThreeNorth regions using the adjusted Hargreaves method with estimated air temperature data not only significantly improves the spatial resolution but also agrees well with the FAO-PM method. Based on these results, our research provides a convenient approach to estimate ETo at large regional scales. Acknowledgments This research was supported by grants from the National Nature Science Foundation of China (31025007) and “Strategic Priority Research Program - Climate Change: Carbon Budget and Relevant Issues” of the Chinese Academy of Sciences (XDA05060400).

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