Analyzing spatial and temporal variability of annual water-energy balance in nonhumid regions of China using the Budyko hypothesis



[1] On the basis of long time series of climate and discharge in 108 nonhumid catchments in China this study analyzes the spatial and temporal variability of annual water-energy balance using the Budyko hypothesis. For both long-term means and annual values of the water balances in the 108 catchments, Fu's formula derived from the Budyko hypothesis is confirmed. A high correlation and relatively small systematic error between the values of parameter ϖ in Fu's equation optimized from the water balance of individual year and calibrated from the long-term mean water balance show that Fu's equation can be used for predicting the interannual variability of regional water balances. It has been found that besides the annual climate conditions the regional pattern of annual water-energy balance is also closely correlated with the relative infiltration capacity (Ks/ir), relative soil water storage (Smax/E0), and the average slope (tan β). This enables one to estimate the parameter ϖ from catchment characteristics without calibration from the long time series of water balances. An empirical formula for the parameter ϖ in terms of the dimensionless landscape parameters is proposed. Applications of Fu's equation together with the parameter ϖ estimated by this empirical formula have shown that Fu's equation can predict both long-term mean and annual value of actual evapotranspiration accurately and predict both long-term mean and interannual variability of runoff reasonably. This implies that the Fu's equation can be used for predicting the annual water balance in ungauged basins.

1. Introduction

[2] Because of marked variability of the hydrological cycle in time and space, one of the most basic issues in hydrology is to understand the key controlling factors and predict the spatial and temporal variability of the annual water balance [Milly, 1994; Wolock and McCabe, 1999]. Evapotranspiration links with both water and energy balances and plays a key role in the climate-soil-vegetation interactions. The primary controls on the long-term mean annual evapotranspiration (equation image) are precipitation (equation image) and potential evapotranspiration (equation image). Budyko [1948, 1974] proposed a semiempirical expression for the coupled water-energy balances, here defined as the Budyko hypothesis, which is a partition of annual water balance as a function of the relative magnitude of water and energy supply.

1.1. Fu's Equation: Analytical Solutions to the Budyko Hypothesis

[3] Bagrov [1953] made the first attempt to derive the Budyko curve theoretically by imposing an additional condition on the derivatives dE/dP = 1− (E/E0)υ, where υ denotes the effects of catchment characteristics. This condition was further developed by Mezentsev [1955] as dE/dP = [1 − (E/E0)υ]s, from which the integral achieved is E/P = 1/[1 + (P/E0)υ]1/υ when s equals (υ + 1)/υ. This is the same as Choudhury's equation [Choudhury, 1999] proposed consulting the Turc-Pike equation [Turc, 1954; Pike, 1964]. Recall that the expression of dE/dP is empirically proposed, rather than from strict derivations. On the basis of phenomenological considerations, Fu [1981] gave the differential forms of the Budyko hypothesis as equation image = f (E0E, P), when E0 = const; and equation image = f (PE, E0), when P = const. Through dimensional analysis and mathematical reasoning (see Fu [1981] and Zhang et al. [2004] for more details), Fu [1981] finally achieved the analytical solutions (called Fu's equation) to the Budyko hypothesis as

equation image

where ϖ is a constant of integration, and its values range (1, ∞).

1.2. Budyko Hypothesis for Regional Variability

[4] The Budyko hypothesis has been widely applied to investigate the regional variability of annual water balances in the former USSR [Budyko, 1974], in the United States [Milly, 1994; Wolock and McCabe, 1999], and in Australia [Zhang et al., 2004]. However, deviations from the original Budyko curve were observed [Budyko, 1974; Milly, 1994], which means that climate alone is insufficient to account for all the regional variability of mean annual water balance. Besides the mean climate conditions, the regional variability is also related to climate seasonality [Budyko, 1974; Dooge, 1992; Milly, 1994; Wolock and McCabe, 1999; Potter et al., 2005], the spatial average plant available water-holding capacity of the soil [Milly, 1994; Wolock and McCabe, 1999; Zhang et al., 2001; Sankarasubramanian and Vogel, 2002a; Potter et al., 2005], and the variability of rainfall depths and arrival time [Milly, 1994; Potter et al., 2005]. Potter et al. [2005] suggested that infiltration-excess runoff might be an important factor which was ignored in previous studies. Using six parameters to depict basin terrain and climate, Berger and Entekhabi [2001] developed a linear regression relation for mean annual water balance, which was further examined by Sankarasubramanian and Vogel [2002b]. Characterizing and quantifying the interbasin variability of annual water balance and examining its dominant controlling factors remain an ongoing and important problem.

[5] Moreover, the regional water balance is determined by the nonlinear interactions of climate factors, soil properties, and vegetation [Rodriguez-Iturbe and Porporato, 2004; Rodriguez-Iturbe et al., 2006]. Therefore it is desirable to understand the dependence of the Budyko-type formulae on spatial scale. On the basis of field observations and water balances in large river basins, Choudhury [1999] fitted different values of the unique parameter in the empirical Budyko-like formula at field scale (0.07–1.6 km2) and regional scale (1.2–7.0 M km2), and further argued that the parameter was mainly dependent on spatial scale.

1.3. Budyko Hypothesis for Interannual Variability

[6] Building on the Budyko hypothesis, simple frameworks [Schaake, 1990; Dooge, 1992; Dooge et al., 1999; Koster and Suarez, 1999] were suggested for analyzing the interannual variability of water balance induced from long-term variability of atmospheric forcing variables. The effectiveness of this approach was confirmed by using an atmospheric general circulation model [Koster and Suarez, 1999], subsequently examined using observational data [Milly and Dunne, 2002], and further improved by introducing an index of soil moisture storage capacity [Sankarasubramanian and Vogel, 2002a].

[7] If the Budyko hypothesis can be applied for estimating actual evapotranspiration at the annual timescale, this is important for understanding the changes in water cycle especially in the context of decreasing pan evaporation worldwide [Brutsaert and Parlange, 1998; Roderick and Farquhar, 2002]. Another operational hypothesis, in which actual and potential evapotranspiration shows complementary in almost diametrical opposition to the Penman proportional hypothesis [Bouchet, 1963; Brutsaert and Stricker, 1979; Parlange and Katul, 1992], has been invoked for predicting changes in hydrologic cycle [Brutsaert and Parlange, 1998; Szilagyi et al., 2001; Kahler and Brutsaert, 2006]. The Bouchet hypothesis has been examined using annual water balance in a number of catchments [Morton, 1983; Hobbins et al., 2001a, 2001b; Ramirez et al., 2005; Yang et al., 2006]. Most cases indicated that it is valid more qualitatively than quantitatively [Sugita et al., 2001; Brutsaert, 2005]. Regarding the apparent contradiction between the Bouchet and Penman hypotheses, a recent research [Yang et al., 2006] suggested that change in actual evapotranspiration in nonhumid regions is dominated by change in precipitation rather than in potential evapotranspiration, and the complementary relationship between actual and potential evapotranspiration comes about because actual and potential evapotranspiration is correlated via precipitation. In humid regions, change in actual evapotranspiration is controlled by change in potential evapotranspiration rather than precipitation, and this is identical to the Penman hypothesis. Fu's equation can provide a full picture of the evaporation mechanism at the annual timescale. Therefore Fu's equation could be used through top-down analysis for providing an insight into the dynamic interactions among climate, soils, and vegetation and their controls on the annual water balance at the regional scale [Sivapalan, 2003; Farmer et al., 2003].

[8] Using a comprehensive data set available from nonhumid regions of China, this study is aimed to examine Fu's equation from both the long-term water balance in different catchments and annual water balance of individual catchments, to explore both regional and interannual variability in annual water balances and their control factors. In particular, this paper attempts to establish a regional relationship between the parameter (ϖ) in Fu's equation and a limited number of dimensionless landscape characteristics to enable its application to ungauged basins.

2. Data and Methodology

2.1. Study Area and Data Available

[9] Climate and soil control vegetation dynamics, and vegetation exerts important control on the entire water balance and feedback to the atmosphere in nonhumid regions [Rodriguez-Iturbe and Porporato, 2004]. Understanding the water balances in nonhumid regions is a valuable method for understanding the climate-soil-vegetation interactions. In this study, 108 catchments located in the Yellow River basin, the Haihe River basin and several inland basins are selected as the study catchments (see Figure 1). All selected catchments have relatively few human interferences, such as dams and irrigation projects. The land use changes in the study areas during the last two decades were reported to be in range of 2–5% [Liu and Buheaosier, 2000]. From the Yellow River basin, 9 catchments located on the Tibetan Plateau and 54 catchments located on the Loess Plateau are chosen. In addition, 38 catchments are chosen from the Haihe River basin and 7 catchments from several inland river basins in Gansu province of western China. Table 1 summarizes the basic physiographic characteristics of the selected catchments. The drainage areas range from 272 to 94,800 km2, and the dryness index (equation image) varies in the range 1 ∼ 7.

Figure 1.

Study area. Triangles represent hydrological gauges, circles represent meteorological gauges, and the numbers represent the catchment numbers.

Table 1. Basic Characteristics of the 108 Study Catchments
NumberArea, km2Data Length, yearsMean Values, mm/yrequation imageequation imagetan βequation image
equation imageequation imageequation image
Inland River Basins
Tibetan Plateau
Loess Plateau
Haihe River Basin

[10] Monthly discharge data from 1951 to 2000 for the 108 catchments have been provided by the Hydrological Bureau of the Ministry of Water Resources of China. The shortest record length is 6 years, the longest is 46 years, median record length is 29 years, and 88 catchments have more than 20 years discharge data (see Table 1). By ignoring the interannual change of water storage in the catchments, actual evapotranspiration is calculated from the annual water balance and used as the “measured” actual evapotranspiration for analysis. Daily meteorological data of 238 gauges from 1951 to 2000 are obtained from the China Administration of Meteorology, which consists of precipitation, mean, maximum and minimum air temperature, sunshine duration, wind speed, and relative humidity. Daily precipitation data at hydrological stations are also available. Additionally, daily data of solar radiation at 47 meteorological stations (among the 238 stations) are available. The data in the same data period for both discharge and meteorological observation have been chosen for this analysis.

[11] The catchment extent is extracted using digital elevation model (DEM) of 1 km resolution, and resampled to 10 km resolution for calculating the areal average values of hydroclimatic variables. The procedures for calculating catchment average precipitation and potential evapotranspiration are (1) a 10 km gridded data set covering the study area is interpolated from the gauge data (see Yang et al. [2004] for more details), (2) potential evapotranspiration at the daily timescale is estimated in each grid using the Penman equation recommended by Shuttleworth [1993], and (3) the catchment average values are then calculated for each variable. For estimating net radiation (see Appendix A of Yang et al. [2006] for the details), solar radiation is calculated by an empirical equation involving the sunshine duration, in which the parameters (as, bs of equation (A3) of Yang et al. [2006]) are calibrated using the observed solar radiation for each month at the 47 stations. The values of as and bs for each grid are obtained from the nearest station. The first-order estimate of net long-wave radiation is derived from the relative sunshine duration, minimum and maximum near surface air temperature and vapor pressure, as per the method recommended by Allen et al. [1998].

2.2. Description of Catchment Characteristics

[12] In addition to climate, topography, soil and vegetation are main factors affecting the partitioning of rainfall into runoff and evapotranspiration, which are considered in the parameter ϖ in Fu's equation. The average slope tan β of a catchment is used for representing the topography, and is estimated as the average slope of all hillslopes derived from the 1 km DEM. The hydraulic conductivity of soil controls the rainfall infiltration and thus the supply of soil water for evaporation. Considering the rainfall intensity, the relative infiltration capacity [Berger and Entekhabi, 2001] is used for indicating the soil property of infiltration-excess. In this study, the relative infiltration capacity is defined as the ratio of saturated hydraulic conductivity Ks (mm hr−1) to mean precipitation intensity ir (mm hr−1) in 24 hours (M. Sivapalan, personal communication, 2006). The saturated hydraulic conductivity is obtained from the Global Soil Data Task [International Geosphere-Biosphere Programme, 2000] with a 10 km resolution. The mean precipitation intensity is averaged for rainy days of the study period. For representing the vegetation and soil effect on the annual water balance, the plant extractable water capacity recommended by Dunne and Willmott [1996] is employed, which is given as

equation image

where droot = min(drmax, dTop), drmax is the maximum depth of the root for each type of vegetation, while the vegetation type is obtained from the USGS Global Land Cover Characteristics Data Base version 2.0 ( with a 1 km resolution; dTop denotes the depth of the topsoil for each soil type, which is derived from a 5 min resolution data set [Food and Agricultural Organization (FAO), 2003]; the moisture contents at field capacity (θf) and wilting point (θw) for each soil type are estimated at matric pressures of −33 and −1500 kPa respectively from the soil water retention as recommended by Dunne and Willmott [1996] using van Genuchten's formula [van Genuchten, 1980]. Other soil water properties are taken from International Geosphere-Biosphere Programme [2000] using the same soil classification as FAO [2003]. The values of Ks, θf, θw, and dTop for the dominant soil type and the dominant type of vegetation are firstly transformed into 10 km gridded data sets, the same resolution as the gridded climatic data set, and then Smax is calculated for each 10 km pixel, and finally Smax and Ks are averaged on each catchment from the 10 km gridded data sets. In this study, Smax is scaled by mean annual potential evapotranspiration in a dimensionless form, i.e., Smax/equation image. The three dimensionless variables for all 108 catchments are given in Table 1.

3. Results and Discussions

3.1. Fu's Curve for Mean Annual Water Balance

[13] On the basis of the long-term mean of annual water balance and climate, Figure 2 presents the water-energy balance for the 108 catchments in two different but equivalent forms of Fu's curves, i.e., E/P vs. E0/P and E/E0 vs. P/E0. Fu's curves with the average values of the parameter ϖ for each region (the Tibetan Plateau, Loess Plateau, Haihe River basin, and inland river basins) are also shown in Figure 2. In Figure 2 (left) the different regional features of Fu's curves stand out more, whereas in Figure 2 (right), these regional differences tend to be hidden. This tends to suggest that actual evapotranspiration is governed more by precipitation rather than potential evapotranspiration in nonhumid regions [Yang et al., 2006].

Figure 2.

Long-term mean values of annual actual evapotranspiration, precipitation, and potential evapotranspiration for the 108 catchments, plotted in two different but equivalent Budyko-type forms (plotted in scattering points) together with Fu's curves with the regional average values of parameter ϖ.

[14] This study calibrates the parameter ϖ in Fu's equation for each catchment in two different methods. From long-term mean water balance and average climate, the unique parameter ϖ can be calculated directly using Fu's equation (see Table 1). It can also be optimized by minimizing the mean absolute error (MAE) [Legates and McCabe, 1999] of the estimated annual evapotranspiration. Figure 3 illustrates the calibrated values versus the optimized values of parameter ϖ for the 108 catchments. A high correlation (r2 = 0.975) and relatively small systematic error (the slope b = 1.07) of the two ϖ values implies that Fu's equation can be used for predicting the interannual variability of regional water balances.

Figure 3.

Comparison of the parameter ϖ optimized from annual water balance with the calibrated one from long-term mean water balance. Each circle denotes one catchment.

3.2. Spatial Variability of Annual Water-Energy Balance

[15] The regional variability of annual water-energy balance has already been noted in Figure 2. This variability may be mainly caused by the differences in annual precipitation and potential evapotranspiration, however, the fact of parameter ϖ differing from catchment to catchment calls more detailed research on examining the regional change in the parameter ϖ of Fu's equation. Figures 4a, 4b, and 4c show the correlations between ϖ and the relative infiltration capacity (Ks/ir), between ϖ and relative soil water storage (Smax/E0), and between ϖ and the average slope (tan β) of the 108 catchments, respectively. The correlation coefficients (r) are −0.60, 0.45, and −0.39, respectively. Applying F test, ϖ was shown to be closely correlated with the three dimensionless factors (at the p > 0.99 significant level).

Figure 4.

Relationships between (a) ϖ and Ks/equation image, (b) ϖ and Smax/equation image, and (c) ϖ and tan β for the 108 catchments. The significance level of 99% is used in the F test for p > 0.99 and n = 108, |r| ≥ 0.21.

[16] Comparing the ϖ values with the catchment areas, it is known that ϖ values for the 108 catchments have no relation (r = −0.025) with catchment sizes (ranging 272–94m800 km2). The spatial scale i.e., catchment size, is not the cause of the change in the ϖ value for the interested catchment size.

3.3. Interannual Variability of Water-Energy Balance

[17] The Budyko hypothesis is also examined in each catchment using annual data series of water balance and average climate. Similar to Figure 2, Figure 5 plots the annual water balance in the Budyko type curve for one typical catchment located in the Haihe River basin (the catchment number is 76). The results show that the Budyko hypothesis are also valid for interannual variability of water-energy balance in these nonhumid catchments.

Figure 5.

Observed values of annual actual evapotranspiration, precipitation, and potential evapotranspiration for one typical catchment (catchment 76 in the Haihe River basin) plotted in two different but equivalent Budyko-type forms in scattering points and Fu's curve with the calibrated parameter ϖ from long-term mean.

[18] The predictability of Fu's equation with the calibrated ϖ values is further examined. Figure 6 shows the cumulative distribution functions of the mean absolute error (MAE), the square root of the mean square error (RMSE), the coefficient of determination (r2) [Legates and McCabe, 1999], and Nash-Sutcliffe coefficient of efficiency (NSE) [Nash and Sutcliffe, 1970] for the predicted annual evapotranspiration using Fu's equation with the calibrated ϖ values in each catchment. The ranges of MAE, RMSE, r2, and NSE for the 108 catchments are 2.84–67.00 mm, 3.32–84.25 mm, 0.62–1.00, and 0.57–1.00, respectively. In more than 90% of the catchments, the values of MAE and RMSE are less than 39.62 mm and 55.35 mm; and the values of r2 and NSE are larger than 0.82 and 0.76, respectively. In conclusion, Fu's equation can predict the interannual changes in water-energy balance well in the study areas.

Figure 6.

Statistical comparisons of the distributions of criteria for evaluating the estimated annual actual evapotranspiration using Fu's equation with the calibrated parameter from long-term mean of water balance and with the estimated parameter from the empirical equation. The criteria includes the mean absolute error (MAE), the square root of the mean square error (RMSE), the coefficient of determination r2, and the Nash-Sutcliffe coefficient of efficiency (NSE).

3.4. Empirical Formula of Parameter ϖ in Fu's Equation

[19] As shown in Figure 4, the parameter ϖ in Fu's equation is closely correlated with the three dimensionless landscape characteristics. It is possible to estimate the parameter ϖ from catchment characteristics without measured discharge data for applying to ungauged basins. Building on previous investigations, this study selects the above three dimensionless variables as the key descriptors for climate and geomorphology of a catchment to determine ϖ as follows,

equation image

where f1, f2, and f3 are functions to be determined. The overbar denotes the long-term mean values.

[20] On the basis of phenomenological considerations the boundary conditions are achieved

equation image

[21] Considering the correlations shown in Figure 4 and the above boundary conditions, the functional forms of f1, f2, and f3 are selected and thus equation (3) is obtained as

equation image

where the values of coefficients a1 and c1 should be larger than zero, b1 and d1 should be less than zero. Taking logarithm of equation (5), the values of the coefficients a1, b1, c1 and d1 can be estimated by stepwise regression analysis (see Table 2). The final form of equation (5) becomes

equation image
Table 2. Stepwise Regression Results for Determining the Coefficients a1, b1, c1, and d1 in Equation (5) by Increasing Numbers of Variables
VariablesFFα=0.001r2Model Coefficients
b1c1d1ln a1
Ks/equation image66.311.500.385−0.5591.282
Ks/equation image, Smax/equation image46.87.410.471−0.4760.4032.087
Ks/equation image, Smax/equation image, tan β33.35.860.490−0.3680.436−4.4642.158

[22] Comparing with the calibrated values of ϖ, the coefficient of determination r2 and the statistic for F test of the predicted ϖ values of the 108 catchments by equation (6) are 0.491 and 33.3, respectively. Considering the advantage of model interpretation, a linear formula is also derived in this study by stepwise regression analysis, i.e., ϖ = 2.947 − 0.155(Ks/equation image) + 5.882(Smax/equation image) − 2.096 tan β. The coefficient of determination and the statistic for F test of the predicted ϖ values are 0.436 and 26.8, respectively, which is very close to equation (6). It should be also noted that the linear model cannot satisfy the above boundary conditions, and it is mainly for the convenience of model interpretation.

[23] This study applies Fu's equation together with the parameter ϖ estimated by equation (6) to predict the mean annual values of actual evapotranspiration and runoff for the 108 catchments. Indicated by the results in Figure 7, the predicted mean annual evapotranspiration in the 108 catchments can explain 95.2% (94.7% for a regression through the origin [Snedecor and Cochran, 1980]) of the variance of the observed values, and the predicted mean annual runoff can explain 62% (57.2% for a regression through the origin) of the observed values (where the runoff coefficient is 0.112 ± 0.057).

Figure 7.

Predicted values of mean annual evapotranspiration using Fu's equation with the estimated parameter ω from the empirical formula plotted versus observed values and predicted mean annual runoff plotted versus observed values. The 1:1 line is plotted for comparison; b denotes the slope of linear regression, and a denotes the intercept.

[24] It also applies to predict the annual values of actual evapotranspiration for the 108 catchments at the same data period. Figure 8 displays the time series of the predicted annual values of actual evapotranspiration compared with the measured values in four typical catchments from the four regions, respectively. Generally, the accuracy declined slightly comparing with the predictions using the calibrated ϖ values (also see Figure 6), whereas only in 5 of the 108 catchments, the values of NSE are less than 0.6. Figure 9 compares the observed interannual variability of runoff σQ (defined as the standard deviations of annual values of runoff, following Koster and Suarez [1999]) with the values predicted by Fu's equation using the estimated ϖ values by equation (6). The correlation coefficient r is 0.824, the gradient b is 0.72, and the intercept a is 9.0 mm. All results show that the parameter ϖ can be estimated from regional characteristics by an empirical formula i.e., equation (6) without calibration, and Fu's equation is reliable and robust for predicting annual evapotranspiration in different regions.

Figure 8.

Comparisons of annual values of actual evapotranspiration between the estimation by Fu's equation with the empirical formula of parameter ϖ (solid line) and the estimation by the water balance (circles) in four typical catchments.

Figure 9.

Comparison of observed interannual variability of runoff σQ (as defined by Koster and Suarez [1999]) with the predicted values by Fu's equation with the empirical formula of parameter ϖ.

4. Conclusions

[25] Through analyzing annual water balances in 108 nonhumid catchments of China, the spatial and temporal variability in annual water-energy balances have been examined. For both long-term water balances in the 108 catchments and annual water balances of individual catchments, Fu's formula derived from the Budyko hypothesis is confirmed. The Budyko-type curves plotted for the 108 catchments have shown significant regional patterns. By examining the spatial changes in the calibrated parameter ϖ in Fu's equation from the long-term water balance in the 108 catchments, it is understood that in addition to the mean climate conditions, the regional feature of water-energy balance is also closely correlated to the relative infiltration capacity (Ks/equation image), relative soil water storage (Smax/E0), and the average slope (tan β), but has nearly no correlation (r = −0.025) with the spatial scale (i.e., the catchment area). It is found that the optimized values of Fu's parameter ϖ from annual values of water balance in each of the 108 study catchments have a high correlation and relatively small systematic error comparing with the calibrated ϖ values from long-term mean annual water balance in the same study periods. This implies that Fu's equation can be used for predicting the interannual variability of regional water balances. Furthermore, indicated by the results, Fu's equation can predict accurately both long-term mean and interannual actual evapotranspiration. Through a stepwise regression analysis, an empirical formula of the parameter ϖ has been derived, and proved to be able to predict annual actual evapotranspiration accurately, as well as predict the mean annual and interannual variability of runoff reasonably. This is especially useful for predictions in ungauged basins.


[26] This research was partially sponsored by the Core Research for Evolutional Science and Technology (CREST) program of the Japan Science and Technology Agency (JST) and was partially supported by the National 973 Project of China (2006CB403405) and the National Natural Science Foundation of China (50679029). The authors would like to express their appreciation to Murugesu Sivapalan, Marc Parlange, Michiaki Sugita, Steve Melching, and two anonymous reviewers, whose comments and suggestions led to significant improvements in the submitted manuscript as well as raising our interest in the spatial and temporal variability of annual water balances. We would also like to thank Baopu Fu for his insightful suggestions.