Management and scheduling of irrigation water requires consideration of evapotranspiration, one of the most important hydrological variables. This study investigates the variations in the daily potential evapotranspiration (ET_{0}), and its aerodynamic (ET_{a}) and radiometric (ET_{r}) components in three areas (western, central and eastern) of the Inner Mongolia Autonomous Region (IMAR) during the growing season (April–September, 2007). In this study, a data-driven approach was followed, and the wavelet transformation analysis method was used to investigate the evapotranspiration characteristics of a relatively large geographic region. The results show that there are close correlations in the variations of ET_{0} with those of ET_{a} and ET_{r}. For the western area of the IMAR, the timing of the largest ET_{a} is 1 month earlier and its wave period is 10 days shorter than those of ET_{0} and ET_{r}. For the central area, the wave period of ET_{a} is 20 days shorter, and the timing of the largest ET_{a} is approximately 1 month earlier than those of ET_{0} and ET_{r}. For the eastern area, there are two large fluctuations in ET_{a}, and they occur 1 month earlier than those of ET_{0}.

## INTRODUCTION

The reference crop evapotranspiration (ET_{0}) is one of the most important hydrological variables, which affects the water and energy exchanges at the land surface/atmosphere interface (Liu *et al.* 2013; Xie *et al.* 2014). To estimate ET_{0} requires a good understanding of the hydrological cycle. Accurate estimates of ET_{0} are critical for scheduling irrigation activities and are also required for a variety of hydro-meteorological applications, e.g. certain hydrological and water-balance models and in calculating the potential/actual evapotranspiration of a region and/or a basin (Blaney & Criddle 1950; Dyck 1983; Hobbins *et al.* 2001a, 2001b; Xu & Li 2003; Xu & Singh 2005; Gong *et al.* 2006). The major factors that affect ET_{0} are elevation, latitude and longitude, meteorological conditions, land use (land cover) and changing climatic conditions at different spatio-temporal scales (Jhajharia *et al.* 2006, 2009a, 2009b).

Many studies have analysed the methods of estimating ET_{0} and their validity in different parts of the world (Jhajharia *et al.* 2004a, 2004b, 2006, 2009a, 2009b). Further, ET_{0} has been evaluated in the context of its spatial and temporal variations. For instance, Thomas (2000) analysed the spatial and temporal characteristics of the ET_{0} trends across China. Jhajharia *et al.* (2012) found that the reduction in ET_{0} in the humid region of northeast India was mainly due to the net radiation and wind speed. In assessing a monthly reference evapotranspiration model, Buttafuoco *et al.* (2010) found that there was spatial uncertainty in the results, which were caused by the uncertainties in the inputs, particularly the temperature, on a regional scale. Using the parametric *t*-test and the non-parametric Mann–Kendall test, Xu *et al.* (2006) analysed the temporal trend of ET_{0} and compared it with the pan evaporation (E_{pan}) trend in the Chang Jiang river basin. They found that for both the ET_{0} and E_{pan} between 1961 and 2000, there were significant decreasing trends. Mo *et al.* (2004) analysed the seasonal and annual variations of ET_{0} between 1984 and 1997 for a mountainous region in the Lu Shi basin, China. Tong *et al.* (2007) analysed the temporal variations of ET_{0} from spring wheat in the last 50 years (1955–2005) in northwest China and tested the ET_{0} trend using the Mann–Kendall test and linear regression analysis. In a study in Iran, Dinpashoh *et al.* (2011) found that the wind speed was the most dominant factor affecting ET_{0} throughout the year except for the winter months. Using the data from 34 meteorological stations between 1957 and 2007 in the Hai He river basin of China, Wang *et al.* (2011) analysed ET_{0} spatial and temporal patterns and trends. Liang *et al.* (2010) analysed the ET_{0} temporal variations between 1961 and 2005 in the Taoer River basin in northeast China.

In recent years, wavelet transform analysis has been widely used in the field of water resources and meteorological sciences. For instance, wavelet transform analysis was used to investigate the multi-timescale features of ET_{0} at the Chang Ping station in China (Ren *et al.* 2009). The complexity associated with ET_{0} was solved based on wavelet de-noising and symbolic dynamics by Chen *et al.* (2012). The decomposed inter-decadal and inter-annual components of the rainfall data during the rainy seasons in north China have been analysed using wavelet transformation (Lu 2002). Kişi (2010) investigated the accuracy of a wavelet regression approach for modelling ET_{0}. A synthetic model of wavelets was used to downscale and forecast ET_{0} with support vector machines (Kaheil *et al.* 2008). Wavelet transform analysis in conjunction with power laws was used to perform wet-spell and dry-spell analyses of precipitation from the global climate models (Ashok *et al.* 2011).

In earlier studies, using the Penman–Monteith equation, as recommended by Food and Agriculture Organization of the United Nations (FAO), and meteorological data from the weather stations, ET_{0} and its spatial and temporal variations were investigated (Song *et al.* 2013). However, ET_{0} is affected by both aerodynamic and radiometric variables, as these two variables reflect the combined effects of topography, soil, vegetation and climate. The Inner Mongolia Autonomous Region (IMAR) is predominantly an agricultural region in China, where irrigation activities are highly dependent on the available water resources. Due to variations in the natural geographical and climatic conditions in the IMAR, there is no consistency in the ET_{0} trend. Therefore, an analysis of ET_{0} and its components (ET_{r} and ET_{a}) for different sub-regions are needed in order to understand the ET_{0} variation over one whole year. In view of the uncertainties in ET_{0}, the wavelet transform method has been used in this study to investigate the variations in ET_{0} in the western, central and eastern regions of the IMAR during a growing season.

The main objectives of this study are: (1) to use the wavelet transform method (a data-driven approach) to examine the spatial and temporal patterns of ET_{0}, (2) to investigate the spatial differences of relevant hydrological variables and (3) to improve the understanding of ET_{0} for agricultural water management.

### Study area

^{6}km

^{2}, as shown in Figure 1. The region is 1,700 km wide from the southern to the northern border, and 2,400 km long from the eastern to the western border, and the elevation ranges from 86 m to 3,522 m above mean sea level. It has a typical moderate and continental monsoon climate with small and non-uniform rainfall events. There is a large difference in temperature between summer and winter. Except for Da Xing'an Mountain area which is relatively humid, most areas within the IMAR from west to east are arid, semiarid or sub-humid. In the IMAR, the average annual sunshine is about 2,700 hours. The annual mean air temperature varies between −4 °C and 9.2 °C, and the minimum mean monthly air temperature ranges between −30 °C and −11 °C during January. The maximum mean monthly air temperature ranges between 19 °C and 26 °C during July. The annual mean precipitation ranges between 50 mm and 550 mm. In the IMAR, the precipitation decreases while air temperature increases from the northeast to southwest. The annual precipitation in Oroqen Autonomous County on the eastern end of the IMAR is 486 mm, whereas it is less than 50 mm in Alsa and approximately 38 mm in the Ejina County. Evaporation exceeds 1,200 mm in most of the IMAR and it is more than 3,200 mm in some areas.

Due to the large variations in the climatic and physical conditions, the ET_{0}-distribution in the IMAR is complex. However, from the 135 weather stations within the IMAR (Figure 1), there are three that are representative of the ET_{0} variations. Hence, the data from these three stations have been used for analysis. The selected stations are in: (1) an extreme arid area in Ejina County, (2) a semiarid area in Hohhot and (3) a relative humid area in Erguna Zuo County. They are situated within the western, central and eastern areas of the IMAR, respectively, as shown in Figure 1.

## DATASET AND METHODOLOGY

### Data

_{0}, ET

_{r}and ET

_{a}were calculated using the FAO Penman–Monteith model (Allen

*et al.*1998): where

*ET*is the daily reference crop evapotranspiration rate (mm/d);

_{0}*R*is the net radiation at canopy surface (MJ/m

_{n}^{2}/day);

*G*is the soil heat flux at the soil surface (MJ/m

^{2}/day);

*T*is the mean daily air temperature (°C);

*γ*is the psychometric constant (kPa°/C);

*U*

_{2}is the mean daily wind speed at 2.0 m height (m/s);

*e*is the mean saturation vapor-pressure (kPa);

_{s}*e*is the mean actual vapour-pressure (kPa); (

_{a}*e*−

_{s}*e*) is the saturated vapour pressure deficit (kPa); and Δ is the slope of the saturated vapour pressures temperature curve (kPa°/C).

_{a}*h*is the height above the ground surface. Equation (2) is recommended for a surface with short grass (Allen

*et al.*1998; Gong

*et al.*2006; McVicar

*et al.*2007).

### Wavelet transform theory

_{0}-time or for a finite energy signal, the continuous wavelet transform of the time-series is defined as: where are the wavelet transform coefficients. The characteristics of the ET

_{0}-time-series in frequency

*a*and time domain

*b*at the same time are reflected by . When the frequency resolution of wavelet transforms is low and the time domain is high,

*b*becomes smaller and vice versa. This is why the wavelet transform method is useful for the time-frequency localization analysis that is involved in the analysis of ET

_{0}.

### Mexican hat wavelet

*a*on the

*y*-axis and coefficient

*b*on the

*x-*axis, is used to show the wavelet transform. This plot is able to show the characteristics of the wavelet transform of a time-series. On the same scale as

*a*, the variation of with time shows the characteristics of the time-series. A positive corresponds to ET

_{0}with a high periodicity and vice versa. At the point where is zero, there is abrupt change. If the values of the wavelet transform coefficients are relatively large, this is an indication that there is significant variation in the corresponding time-scale.

*a*at the time-scale

*b*, as follows: The variation of wavelet variance with the scale

*a*can be shown in a wavelet variance plot, which shows the fluctuation of energy distribution with the scale

*a*. The wavelet variance plot can be used to determine the time-scale of main period of the ET

_{0}time-series.

To reduce the interface effect in the wavelet analysis, the most commonly used method is to extend the time-series in both the forward and backward directions. After the wavelet transform, the part of the series being extended can then be abandoned.

Using the Penman–Monteith model and the daily meteorological recorded data between April and September, 2007, ET_{0}, ET_{a} and ET_{r} were calculated for the three selected stations. The ET_{0}, ET_{a} and ET_{r} time-series were extended in both the forward and backward directions, according to Equations (12) and (13). Using the Mexican hat function (MHF), the ET_{0}, ET_{a} and ET_{r} time-series were then analysed at multiple time scales, according to Equations (8) and (9).

## RESULTS AND DISCUSSION

The results of the wavelet analysis and their implications for the three sub-regions of the IMAR (Ejina County in the western area, Hohhot City in the central area and Erguna Zuo County in the eastern area) are discussed in this section. The differences among the three sub-regions are highlighted below.

### Western area – Ejina County

_{0}in Ejina County. For the ET

_{r}and ET

_{a}, the seasonal cycles are similar, in that they increase at the beginning of the growing season and decrease towards the end of the growing season. For both ET

_{0}and ET

_{a}, their peaks are in June, indicating that the highest evapotranspiration is in June. On the other hand, the peak of ET

_{r}is in July. Figures 2(b), 3(b) and 4(b) respectively show the wavelet transforms of the anomaly percentages of ET

_{0}, ET

_{r}and ET

_{a}for the growing season. The strength of the signals is shown by values of wavelet transform coefficient. A positive coefficient means the signal is strong, whereas a negative coefficient means the signal is weak. A coefficient equal to zero means there is abrupt change at this point. From Figures 2(b), 3(b) and 4(b), it can be seen that the isopleths at the upper half of the plots are relatively sparse, which corresponds to the wave period of low frequency on a large scale. For a period of 40–50 days, the ET

_{0}, ET

_{r}and ET

_{a}series show identical trends with two abrupt changes and two transitions (i.e. from low to high and then to low). For both ET

_{0}and ET

_{r}, there are abrupt changes in the middle of May and August, whereas for ET

_{a}there are abrupt changes in early May and early August.

As shown in Figures 2(c), 3(c) and 4(c) respectively, the largest variance is on Day 45 (i.e. end of June) for ET_{0}, on Day 40 (i.e. early June) for ET_{a} and on Day 50 (i.e. mid-July or end of June for ET_{r}. However, at the smaller scale, the fluctuations are more complicated and less certain. Figures 2(c), 3(c) and 4(c) also show that during the growing season, the wave period is approximately 45 days for ET_{0}, 40 days for ET_{a} and 50 days for ET_{r}. These results indicate that the changes are dependent on several parameters, such as the roughness of surface and the leaf area index. Hence, to assess these parameters accurately they need to be rescaled in the timeline. Further, the time lag can be attributed to the windy climatic condition in the western area of the IMAR, which has affected ET_{a} after April. Hence, the ET_{a} fluctuations are 1 month earlier than those of ET_{0} and ET_{r}. The ET_{a} results obtained in this study are closely related to ET_{0}, which is consistent with the findings by Huo *et al.* (2004). They found that ET_{a} is dominant in the more arid area of the IMAR (Huo *et al.* 2004).

### Central area – Hohhot city

_{0}, ET

_{a}and ET

_{r}. Figures 5(b), 6(b) and 7(b) show the wavelet transform, and Figures 5(c), 6(c) and 7(c) show the MHF wavelet variance. The seasonal cycles are similar for the three variables. For both ET

_{0}and ET

_{a}, the peaks are in the middle of May, whereas the ET

_{r}peak is in the middle of July. Further, both ET

_{0}and ET

_{a}decrease gradually with time. For a period of 40–50 days, the ET

_{0}and ET

_{r}time-series follow identical trends with two abrupt changes and two transitions. For both ET

_{0}and ET

_{r}, the abrupt changes are at the end of May and end of August, respectively. On the other hand, the abrupt changes in ET

_{a}are at the end of April and during the middle of June. Further, as shown in Figures 5(b), 6(b) and 7(b), the wave periods for ET

_{0}and ET

_{a}are both 10 days, while the wave period for ET

_{a}is 30 days.

The periods and dates corresponding to the largest variations in ET_{0} and ET_{r} are similar to those in Ejina County. Figures 5(c), 6(c) and 7(c) show that during the growing season, the wave periods are approximately 10 days and 45 days for ET_{0}, 10 days, 30 days for ET_{a} and 50 days for ET_{r}. These wave periods are an indication that solar radiation has an effect on the ET_{0} in the Hohhot area, which is consistent with the findings by Huo *et al.* (2004), showing that ET_{r} is beginning to affect ET_{0} in the semi-arid area of the IMAR (Huo *et al.* 2004).

### Eastern area – Erguna Zuo County

_{0}and ET

_{r}, there are significant seasonal fluctuations during the growing season, with peaks in mid-August. On the other hand, for ET

_{a,}there are two peaks in early May and late August. For ET

_{0}and ET

_{r}and for a period of 40–50 days, there are two abrupt changes, occurring in early June and late August and two transitions. As shown in Figures 8(b), 9(b) and 10(b), for ET

_{a}and for a period of 20–25 days

_{,}there are three abrupt changes and three transitions (i.e. from low to high and then to low and to high again).

The timing of the largest variations in ET_{0} and ET_{r} in the eastern area is 1 month earlier than those in the western and central areas. From Figures 8(b), 9(b) and 10(b), it can be seen that there are two large fluctuations in ET_{a} (i.e. a 30-day period in early June and a 10-day period in mid-August). Figures 8(c) and 9(c) show that for both ET_{0} and ET_{r}, there is a fluctuation period of approximately 50 days, and Figure 10(c) shows that for ET_{a}, there is a fluctuation period of 25 days during the growing season.

As shown in Figures 8–10, it can be seen that ET_{r} is largest in June and July. These results are consistent with the global trend of radiation. This can be attributed to the large area of the IMAR, with widely varying topography and climatic conditions. It should be noted that the results of ET_{0}, ET_{r} and ET_{a} as presented in this study are for a typical dry year with normal precipitation, in which the relationships among the hydro-meteorological parameters are generally close.

## CONCLUSIONS

A comprehensive evaluation of the daily reference crop evapotranspiration (ET_{0}), aerodynamic (ET_{a}) and radiometric (ET_{r}) in the IMAR has been carried out in this study. The variability of the ET_{0}, ET_{a} and ET_{r} was investigated in three regions of the IMAR during a typical growing season (April–September, 2007). The wavelet transform method (a data-driven approach) was used to analyse the meteorological data from the three selected stations. Such analyses are important to understand the regional differences in ET_{0}, ET_{a} and ET_{r}, information which is useful for irrigation water management.

The results of this study show significant periodicities in the estimated daily ET_{0}, ET_{a} and ET_{r} over the entire growing season. Further, the results show that there are close relationships of ET_{0} with ET_{a} and ET_{r}. The timing of the largest ET_{a} is 1 month earlier than those of the ET_{0} and ET_{r}. In the eastern area, the wave period is 40 days, which is 10 days shorter than that in the western area. The characteristics of the ET_{r} wave are closer to those of the ET_{0} wave than those of the ET_{a} wave. This is an indication that there is a strong influence of ET_{r} on ET_{0}. This may be attributed to the relatively intense solar radiation in the western area during early April. However, the ET_{a} wave is strongly affected by wind. In the central area, the wave period of ET_{a} is 20 days shorter than those of ET_{0} and ET_{r}. Further, for both ET_{0} and ET_{a} within a 10-day period, there are some sporadic small-scale fluctuations, which can be attributed to the short duration rainfall events. For both the western and the central areas, there are two abrupt changes, two transitions and the wave periods for ET_{0}, ET_{a} and ET_{r} are all approximately 50 days. However, for the ET_{0} and ET_{r} in the eastern area, there are three abrupt changes and three transitions. For the ET_{a} in the eastern area, the wave period is approximately 25 days.

The findings from this study can be used to improve the ET_{0}-estimation at both regional and local scales. They can also be used as a technical support for irrigation water management decisions and scheduling. This study has shown that using the wavelet transformation method to analyse ET_{0} enables a more rational understanding of the regional differences in ET_{0}, ET_{a} and ET_{r}.

While this study has used the wavelet transform method to carry out a comprehensive evaluation of ET_{0}, ET_{a} and ET_{r} so as to rationally understand the evapotranspiration characteristics and its variation, there are limitations. First, the modelled results are only based on one meteorological station in each sub-region. Second, only data of a typical year (2007) have been used. Moreover, the results may be limited due to the fundamental assumptions and approaches used in the model. These shortcomings can be overcome by comparing the results of this study with those from other meteorological stations in other regions with similar soil, vegetation, topography, meteorology and climate conditions. Therefore, while further exploration of this approach is needed, this study has provided useful experience and knowledge for the investigation of crop evapotranspiration.

## ACKNOWLEDGEMENTS

We would like to thank the National Natural Science Foundation of China (51369018), 13th Five-Year National Key Development Program (2016YFC0400205-1), the State Key Program of National Natural Science of China (51539005), and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry ([2013]693), China.