## 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 () are precipitation () and potential evapotranspiration (). *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*/*E*_{0})^{υ}, where *υ* denotes the effects of catchment characteristics. This condition was further developed by *Mezentsev* [1955] as *dE*/*dP* = [1 − (*E*/*E*_{0})^{υ}]^{s}, from which the integral achieved is *E*/*P* = 1/[1 + (*P*/*E*_{0})^{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 = *f* (*E*_{0} − *E*, *P*), when *E*_{0} = *const*; and = *f* (*P* − *E*, *E*_{0}), 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

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 km^{2}) and regional scale (1.2–7.0 M km^{2}), 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.