**Research article**
26 Jul 2018

**Research article** | 26 Jul 2018

# Hydrological effects of climate variability and vegetation dynamics on annual fluvial water balance in global large river basins

Jianyu Liu Qiang Zhang Vijay P. Singh Changqing Song Yongqiang Zhang Peng Sun and Xihui Gu

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**Jianyu Liu et al.**Jianyu Liu Qiang Zhang Vijay P. Singh Changqing Song Yongqiang Zhang Peng Sun and Xihui Gu

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^{1}Laboratory of Critical Zone Evolution, School of Earth Sciences, China University of Geosciences, Wuhan 430074, China^{2}Key Laboratory of Environmental Change and Natural Disaster, Ministry of Education, Beijing Normal University, Beijing 100875, China^{3}State Key Laboratory of Earth Surface Processes and Resource Ecology, Beijing Normal University, Beijing 100875, China^{4}Faculty of Geographical Science, Academy of Disaster Reduction and Emergency Management, Beijing Normal University, Beijing 100875, China^{5}Department of Biological and Agricultural Engineering and Zachry Department of Civil Engineering, Texas A&M University, College Station, Texas, USA^{6}CSIRO Land and Water, GPO Box 1700, Canberra ACT 2601, Australia^{7}College of Geography and Tourism, Anhui Normal University, Anhui 241000, China^{8}Department of Atmospheric Science, School of Environmental Studies, China University of Geosciences, Wuhan 430074, China

^{1}Laboratory of Critical Zone Evolution, School of Earth Sciences, China University of Geosciences, Wuhan 430074, China^{2}Key Laboratory of Environmental Change and Natural Disaster, Ministry of Education, Beijing Normal University, Beijing 100875, China^{3}State Key Laboratory of Earth Surface Processes and Resource Ecology, Beijing Normal University, Beijing 100875, China^{4}Faculty of Geographical Science, Academy of Disaster Reduction and Emergency Management, Beijing Normal University, Beijing 100875, China^{5}Department of Biological and Agricultural Engineering and Zachry Department of Civil Engineering, Texas A&M University, College Station, Texas, USA^{6}CSIRO Land and Water, GPO Box 1700, Canberra ACT 2601, Australia^{7}College of Geography and Tourism, Anhui Normal University, Anhui 241000, China^{8}Department of Atmospheric Science, School of Environmental Studies, China University of Geosciences, Wuhan 430074, China

**Correspondence**: Qiang Zhang (zhangq68@bnu.edu.cn)

**Correspondence**: Qiang Zhang (zhangq68@bnu.edu.cn)

Received: 12 Apr 2018 – Discussion started: 23 Apr 2018 – Accepted: 14 Jul 2018 – Published: 26 Jul 2018

The partitioning of precipitation into runoff (*R*) and
evapotranspiration (*E*), governed by the controlling parameter in the Budyko
framework (i.e., *n* parameter in the Choudhury and Yang equation), is
critical to assessing the water balance at global scale. It is widely
acknowledged that the spatial variation in this controlling parameter is
affected by landscape characteristics, but characterizing its temporal
variation remains yet to be done. Considering effective precipitation
(*P*_{e}), the Budyko framework was extended to the annual water
balance analysis. To reflect the mismatch between water supply
(precipitation, *P*) and energy (potential evapotranspiration, *E*_{0}), we
proposed a climate seasonality and asynchrony index (SAI) in terms of both
phase and amplitude mismatch between *P* and *E*_{0}. Considering streamflow
changes in 26 large river basins as a case study, SAI was found to the key
factor explaining 51 % of the annual variance of parameter *n*.
Furthermore, the vegetation dynamics (*M*) remarkably impacted the temporal
variation in *n*, explaining 67 % of the variance. With SAI and *M*, a
semi-empirical formula for parameter *n* was developed at the annual scale to
describe annual runoff (*R*) and evapotranspiration (*E*). The impacts of
climate variability (*P*_{e}, *E*_{0} and SAI) and *M* on *R* and *E*
changes were then quantified. Results showed that *R* and *E* changes were
controlled mainly by the *P*_{e} variations in most river basins over
the globe, while SAI acted as the controlling factor modifying *R* and *E*
changes in the East Asian subtropical monsoon zone. SAI, *M* and *E*_{0} have
larger impacts on *E* than on *R*, whereas *P*_{e} has larger impacts
on *R*.

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Climate variability, vegetation dynamics and water balance are interactive, and this interaction is critical in the evaluation of the impact of climate change and vegetation dynamics on water balance at the basin scale and for the management of water resources (Milly, 1994; Yang et al., 2009; Weiss et al., 2014; Zhang et al., 2016c). The models that can quantify the climate–vegetation–hydrology interactions without calibration using observed evapotranspiration or runoff are particularly needed for hydrological prediction in ungauged basins (Potter et al., 2005). Furthermore, quantifying the influence of climate variability and vegetation dynamics on hydrological variability is critical in differentiating the factors that drive the hydrological cycle in both space and time (Yan et al., 2014; Dagon and Schrag, 2016; Zhang et al., 2016a).

The Budyko framework was developed to quantify the partitioning of
precipitation into runoff and evapotranspiration (Koster and Suarez, 1999; Xu
et al., 2013) and was widely used to evaluate interactions amongst climate,
catchment characteristics and hydrological cycle (Yang et al., 2009; Cai et
al., 2014; Liu et al., 2017b; Ning et al., 2017). However, the controlling
parameter of the Budyko framework usually needs to be calibrated, based on
observed data. If the controlling parameter can be determined using the
available data, then the Budyko framework can be employed in modeling the
hydrological cycle in ungauged basins (Li et al., 2013). That is why
considerable attention has been devoted to quantifying the relationship
between the controlling parameter and explanatory variables (e.g., Yang et
al., 2009; Abatzoglou and Ficklin, 2017). Most of the relationships were
evaluated at a long-term scale (Abatzoglou and Ficklin, 2017; Gentine et al.,
2012; Li et al., 2013; Xu et al., 2013; Yang et al., 2007, 2009; Zhang et
al., 2016c) due to the steady-state assumption of the Budyko model. However,
hydrological processes, such as water storage, are usually nonstationary due
to climate change and human activities (Greve et al., 2016; Ye et al., 2015).
It should be noted here that the variability of controlling parameters from
year to year may be considerably large in a specific river basin, which can
be significantly affected by variations in vegetation cover and climate
conditions. Hence, it is necessary to develop a model to estimate annual
variations in controlling parameters. In a recent study, Ning et al. (2017)
established an empirical relationship of the controlling parameter at the
annual scale in the Loess Plateau of China. However, the annual values of the
optimized controlling parameter in their study were calibrated with the
*Fu* equation without consideration of the annual water storage
changes (Δ*S*). But Δ*S* was identified as a key factor causing
annual variations in water balance in most river basins, particularly in
river basins of arid regions (e.g., Chen et al., 2013). Therefore, considering
water storage changes, the effective precipitation (*P*_{e}), which is
the difference between precipitation and water storage change (Chen et al.,
2013), was used to extend the Budyko framework to annual-scale water balance
analysis and was used to calibrate *n*.

Climate seasonality (SI, seasonality index) was identified to reflect the non-uniformity in the
intra-annual distribution of water and energy, which plays a role in the
variation in controlling parameter in the Budyko model (Woods, 2003; Ning et
al., 2017; Yang et al., 2012; Abatzoglou and Ficklin, 2017). It is noted
that distributions of water and energy were reflected not only by
differences of seasonal amplitudes of *P* and *E*_{0} but also by the phase
mismatch between *P* and *E*_{0}. In this case, we proposed a climate
seasonality and asynchrony index (SAI) to reflect the seasonality and
asynchrony of water and energy distribution.

Vegetation coverage has also been found to be closely related to the spatial
variation in the controlling parameter (Yang et al., 2009). Li et al. (2013)
and Xu et al. (2013) used vegetation coverage to model the spatial variation in the controlling parameter in the major large basins over the globe at a
long-term scale. However, the effect of climate variability was not
considered, and the impact of vegetation dynamics on the temporal variation in the controlling parameter was not fully investigated. Zhang et al. (2016c) established the relationship of parameter *n* with vegetation changes
over northern China and suggested that the relationship needed to be further
assessed in other river basins across the globe. Also, they confirmed the
impact of climate seasonality on parameter *n* and suggested future studies on
its impacts on *n*. Therefore, this study developed a semi-empirical formula for
parameter *n* with SAI and *M* as predictor variables at the annual scale, using
meteorological and hydrological data from 26 large river basins from around
the globe with a broad range of climate conditions.

Much work has been done to address water balance variations (e.g., Liu et
al., 2017a; Zeng and Cai, 2016; Zhang et al., 2016a, b).
For instance, Zeng and Cai (2016) evaluated the impacts of *P*, *E*_{0} and
Δ*S* on the temporal variation in evapotranspiration for large river
basins. However, little is known about the influence of *M* and SAI on the
hydrological cycle, particularly on their contributions to variations in
runoff and evapotranspiration. The impact of *M* and SAI on the water balance
is critical for water balance modeling. Therefore, based on the developed
semi-empirical formula, this study further assessed the causes of variation in *R* and *E*. The objectives of this study were (1) to propose a climate
SAI to reflect the mismatch of water and
energy; (2) to develop an empirical model for the controlling parameter *n* at
the annual scale using data from 26 large river basins from around the
globe; and (3) to investigate the impact of SAI and other factors on the *R*
and *E* variations.

Monthly terrestrial water budget data covering a period of 1984–2006 were
collected from 32 large river basins from around the globe (Pan et al.,
2012). The dataset, including *P*, *E*, *R* and Δ*S*, combined data
from multiple sources, such as in situ observations, remote sensing
retrievals, model simulations and global reanalysis products, which were
obtained using assimilation weighted with the estimated error. For more
details on this dataset, reference can be made to Pan et al. (2012). This
dataset, which was deemed to one of the best water budget estimates, has
already been applied to assess the impact of vegetation, topography,
latitude and terrestrial storage on the spatial variability of the
controlling parameter in the Budyko framework and the evapotranspiration
variability over the past several years (Arnell and Gosling, 2013; Li et al.,
2013; Xu et al., 2013; Zeng and Cai, 2016). The dataset has been designed to
explicitly close the water budget. And that the use of data assimilation
might lead to unphysical variability. As a result, Li et al. (2013) found
that more than 20 % of data in six basins among the 32 global basins were
beyond the energy and water limits, and suggested analysis on water–energy
balance using the remaining 26 basins. Following Li et al. (2013), we
evaluated the impact of climate variability and vegetation dynamics on the
spatiotemporal variation in the controlling parameter and the water balance
of the 26 river basins. Detailed information about the characteristics of the
26 basins is given in Table 1. Monthly potential evapotranspiration (*E*_{0})
data from 1901 to 2015 at a spatial resolution of 0.5^{∘} were obtained
from the Climatic Research Unit of the University of East Anglia
(https://crudata.uea.ac.uk/cru/data/hrg/cru_ts_3.24.01/cruts.1701201703.v3.24.01/pet/). A monthly normalized difference vegetation index (NDVI)
covering a period of 1981–2006 was obtained from Global Inventory Modeling
and Mapping Studies (GIMMS) (Buermann, 2002; Li et al., 2013).

## 3.1 The Budyko framework at annual scale

The Budyko framework has been widely used in the assessment of impacts of climate and vegetation variations on the hydrological cycle. There are several analytical equations proposed under the Budyko framework, among which the function deduced by Choudhury (1999) and Yang et al. (2008) has been identified to perform better than other equations (Zhou et al., 2015). The function can be expressed as follows:

where *n* is the controlling parameter of the Choudhury–Yang equation.

The basin stores precipitation first and then releases it as runoff and
evapotranspiration (Biswal, 2016). Affected by water storage changes, *E* is
never equal to the difference between *P* and *R* for a short time
interval. Previous studies have found that storage changes have impacts on
water balance at the annual scale (Donohue et al., 2012). To consider the
influence of variation in water storage, Wang (2012) suggested to use
effective precipitation (*P*_{e}), i.e., ${P}_{\mathrm{e}}=P-\mathrm{\Delta}S$, to
replace precipitation in the water–energy balance. As a result, using the
*P*_{e}, the Choudhury and Yang equation can be extended in
a short timescale:

Parameter *n* controls the shape of the Budyko curve and can be calibrated by
minimizing the mean absolute error (MAE) of runoff (Legates and McCabe, 1999;
Yang et al., 2007). Parameter *n* is a catchment characteristic parameter which
is mainly related to the underlying conditions (i.e., topography and soil),
climate conditions and vegetation cover (Liu et al., 2017a; Yang et al.,
2009; Zhang et al., 2016c). The underlying characteristics are relatively
stable during a short time interval, while climate and vegetation might
undergo considerable variations, which can lead to the change in parameter
*n*. As a result, vegetation dynamics and climate variability were applied to
simulate *n* and assess their impact on runoff and evapotranspiration.

The vegetation coverage (*M*), which is the fraction of land surface covered
with green vegetation in the region, can be calculated as follows (Gutman and
Ignatov, 1998):

where NDVI_{max} and NDVI_{min} represent the dense green vegetation
and bare soil with NDVI_{max}=0.80 and NDVI_{min}=0.05, respectively
(Li et al., 2013; Ning et al., 2017; Yang et al., 2009).

## 3.2 Seasonality and asynchrony of water and energy

The seasonality of *P* and *E*_{0}, which are mainly controlled by solar
radiation, follows a sine distribution (Milly, 1994; Woods, 2003; Berghuijs
and Woods, 2016):

where *t* is the time (months), and *P*(*t*) and *E*_{0} (*t*) are the monthly *P* and *E*_{0}
with the annual mean value of $\stackrel{\mathrm{\u203e}}{P}$ and of $\stackrel{\mathrm{\u203e}}{{E}_{\mathrm{0}}}$, respectively.
The quantities *δ*_{P} and ${\mathit{\delta}}_{{E}_{\mathrm{0}}}$ are
dimensionless seasonal amplitudes, which can be calibrated by minimizing
MAE. The quantity *τ* is the cycle of seasonality, with 6 months in the
tropics and 1 year outside the tropics. The origin of time (*t*=0) was
fixed in April in the previous studies (Milly, 1994; Woods, 2003; Ning et
al., 2017). As a result, if the *δ*_{P} (${\mathit{\delta}}_{{E}_{\mathrm{0}}})$ was
positive, the month with maximum monthly *P* (*E*_{0}) would appear in July,
which corresponds to Northern Hemisphere (e.g., Fig. 1a); while the
Southern Hemisphere would show a January maximum with negative ${\mathit{\delta}}_{P}({\mathit{\delta}}_{{E}_{\mathrm{0}}}$). Considering the difference between seasonal *P* and
*E*_{0}, Wood et al. (2003) defined a climate seasonality index by combining
Eq. (4):

where DI is the dryness index $\left(\frac{\stackrel{\mathrm{\u203e}}{{E}_{\mathrm{0}}}}{\stackrel{\mathrm{\u203e}}{P}}\right)$.

Equations (4)–(5) were applied to represent the mismatch between water and
energy (e.g., Ning et al., 2017). However, the following two issues still
need to be considered: (1) the effect of local climate and catchment
characteristics – the phase of seasonal *P* and *E*_{0} may be not entirely
consistent with that of solar radiation – and (2) the phases between seasonal
*P* and *E*_{0} cannot always be consistent in a specific basin, such as the
Northern Dvina basin (Fig. 1b). The values of *E* for two basins with the
same annual mean *P*, *E*_{0}, *δ*_{P} and ${\mathit{\delta}}_{{E}_{\mathrm{0}}}$ can be
different if the phases of seasonal *P* and *E*_{0} are in mismatch. As a
result, the phase shifts of *P* (*S*_{P}) and ${E}_{\mathrm{0}}\phantom{\rule{0.25em}{0ex}}({S}_{{E}_{\mathrm{0}}}$) should be
considered in the sine function (Berghuijs and Woods, 2016):

As shown in Fig. 2, Eq. (6) with a fitted phase performed much better in
simulating monthly *P* and *E*_{0} than Eq. (4) with a fixed phase, with
*R*^{2} larger than 0.89 for the former but smaller than 0.64 for the latter.

To fully reflect the difference between water and energy, it is necessary to
consider not only the seasonal amplitude difference between *P* and *E*_{0}, but
also the phase difference (i.e., asynchrony) between them (Fig. 1b).
Therefore, an improved climate index describing the difference between water
and energy needs to be developed with the consideration of seasonality and
asynchrony of *P* and *E*_{0}. Based on Eq. (6), we further deduced the
following equations to express the difference between *P* and *E*_{0}:

where $a={\mathit{\delta}}_{P}\mathrm{cos}\frac{\mathrm{2}\mathit{\pi}}{\mathit{\tau}}\frac{{S}_{P}}{\mathrm{12}}-\mathrm{DI}{\mathit{\delta}}_{{E}_{\mathrm{0}}}\mathrm{cos}\frac{\mathrm{2}\mathit{\pi}}{\mathit{\tau}}\frac{{S}_{{E}_{\mathrm{0}}}}{\mathrm{12}}$, $b=-{\mathit{\delta}}_{P}\mathrm{sin}\frac{\mathrm{2}\mathit{\pi}}{\mathit{\tau}}\frac{{S}_{P}}{\mathrm{12}}+\mathrm{DI}{\mathit{\delta}}_{{E}_{\mathrm{0}}}\mathrm{sin}\frac{\mathrm{2}\mathit{\pi}}{\mathit{\tau}}\frac{{S}_{{E}_{\mathrm{0}}}}{\mathrm{12}}$, $\mathit{\phi}=\mathrm{arctan}\phantom{\rule{0.125em}{0ex}}(b/a)$.
Similar to Milly (1994), we defined a SAI to
reflect the mismatch between water and energy in terms of the magnitude and
phase difference between *P* and *E*_{0}:

The SI value calculated by Eq. (5) was an exceptional case for *P* and *E*_{0} in
the same phase shifts. A larger SAI implies a greater difference between *P*
and *E*_{0} in the year. Besides, SAI followed the following three scenarios:
(1) SAI*<*1−DI, given a wet climate with *P*(*t*) *>* *E*_{0}
(*t*) across the whole seasonal cycle (Fig. 3a); (2) SAI*<*DI−1, given
a dry climate with *P*(*t*) *<* *E*_{0} (*t*) across the whole seasonal cycle
(Fig. 3b); (3) $\mathrm{SAI}\ge \mathrm{|}\mathrm{DI}-\mathrm{1}\mathrm{|}$, given that a larger SAI
implies more surplus of *P* for the wet season with *P*(*t*) *>* *E*_{0}
(*t*) (Fig. 3c).

## 3.3 Contributions of SAI and other factors to *R* and *E*

From Eq. (2), using a total differential method, we can redefine the total
differential of *R* and *E* for any timescale by introducing effective
precipitation (*P*_{e}):

The climatic elasticity of evapotranspiration changes to the changes in
precipitation, potential evapotranspiration and *n* can be separately be
expressed as ${\mathit{\epsilon}}_{{P}_{\mathrm{e}}}=\frac{{P}_{\mathrm{e}}}{E}\frac{\partial f}{\partial {P}_{\mathrm{e}}}$,
${\mathit{\epsilon}}_{{E}_{\mathrm{0}}}=\frac{{E}_{\mathrm{0}}}{E}\frac{\partial f}{\partial {E}_{\mathrm{0}}}$,
${\mathit{\epsilon}}_{n}=\frac{n}{E}\frac{\partial f}{\partial n}$.
The climatic elasticity of runoff changes is similar to the climatic
elasticity evapotranspiration changes. The difference operator (d) in
Eqs. (9a) and (9b) refers to the difference of a variable before and after
change points of *R* and *E*, respectively. It is worth noting that Eq. (9)
is derived by the first-order approximation of Taylor expansion. When the
changes in d*P*_{e}, d*E*_{0} and
d*n* are small, the error from approximation can be ignored.
However, due to ignoring the higher orders of the Taylor expansion, the error
will increase as the changes increase (Yang et al., 2014a, b; Zhou et al., 2016).

The relative contribution (*C*) of *P*_{e}, *E*_{0} and *n* to the *R* and *E* changes can be obtained as follows:

${I}_{{p}_{\mathrm{e}}},{I}_{{E}_{\mathrm{0}}}$ and *I*_{n} denote, respectively, the impacts
of *P*_{e}, *E*_{0} and *n* on *R* or *E*, which can be expressed by
$\frac{\partial E}{\partial {P}_{\mathrm{e}}}\mathrm{d}{P}_{\mathrm{e}}$,
$\frac{\partial E}{\partial {E}_{\mathrm{0}}}\mathrm{d}{E}_{\mathrm{0}}$ and $\frac{\partial E}{\partial n}\mathrm{d}n$. After getting the
contribution of *n* to the *R* and *E* variations, we can further assess the
impacts of *M* and SAI on the variation in *R* and *E*, based on the
semi-empirical model of *n* in terms of *M* and SAI. Following Ning et
al. (2017), using the total differential method, the changes in parameter *n*
can be expressed as follows:

Then, the relative contributions of SAI (*C*_SAI) and *M* (*C*_*M*) to the changes in parameter *n* can be obtained. Combining
with the contribution of *n* to the *R* and *E* changes, the relative contributions
of SAI and *M* to the variations in *R* and *E* can be obtained:

## 4.1 Performance of the proposed SAI in the Budyko framework

Figure 2 shows that Eq. (6) with SAI has a better performance in simulating
*P* and *E*_{0} than Eq. (4) with SI. Here we further assessed the performance
of these two indices, by comparing with the controlling parameter *n* in the
Budyko framework. Parameter *n* for each year was first calibrated by Eq. (2).
The calibrated parameter *n* was called optimized *n*. For the representativeness
of the relation between *n* and other factors, analysis was done at a larger
spatial scale with different climate conditions by combining data from 26
global large basins (Fig. 4).

The correlation coefficient (*r*) between SI and optimized *n* was −0.34
(Fig. 4a). If the asynchrony of seasonal *P* and *E*_{0} was considered in
SI, i.e., SAI, the correlation coefficient increased noticeably with *r* of
−0.51 (Fig. 4b). To further assess the impact of SAI on the fluvial water
balance, we also analyzed the roles of SAI in the Budyko framework. As shown in
Fig. 4e, a larger *n* value was related to a higher evapotranspiration ratio
for a given aridity index, and as SAI increased, the value of controlling
parameter *n* tended to decrease. In other words, catchments with a larger
SAI had a lower evapotranspiration ratio given the same aridity index. This
result is similar to the finding by Zhang et al. (2015), who found that a
larger snow ratio caused a higher runoff index given the same dryness index. In
contrast, this relationship is not distinct for SI (Fig. 4d). In addition,
the SAI can explain 51 % of the annual variance of parameter *n*, while
the SI just explains 22 % (Fig. 4a and b). In short, although SI showed
a significant relationship with *n*, SAI considering both seasonality
and asynchrony of *P* and *E*_{0} was more applicable to represent the
difference between water and energy and performed better in the simulation
of *n* in the Budyko model.

The variation in SAI is also sensitive to climate variability. As shown in
Fig. 6, the climate elasticities of evapotranspiration to precipitation and
parameter *n* increased with SAI, whereas the elasticity of evapotranspiration
to potential evapotranspiration decreased with SAI, which implies that the
variation in evapotranspiration in the catchments with a higher SAI were
more sensitive to the changes in precipitation and parameter *n*, but less
sensitive to the changes in potential evapotranspiration.

## 4.2 A semi-empirical formula for parameter *n*

Previous studies have found that vegetation cover is closely related to the
spatial variation in *n* in different regions (e.g., Li et al., 2013). However,
the new finding in this study is that vegetation dynamics (*M*) also have a
significant impact on the temporal variation in annual values of the parameter
*n* (Figs. 4c and 5c) and evapotranspiration ratio (Fig. 4f). Nevertheless, the
simulation accuracy of *n* can be further improved, particularly at the high
end. As mentioned above, SAI has a significant impact on the variation in
*n*. Therefore, based on the results obtained by Li et al. (2013), it is
possible to develop a more dynamic model to capture the spatiotemporal
variation in parameter *n* and improve the simulation of *n* by incorporating SAI
into the empirical model.

Following the phenomenological considerations and the relationships
demonstrated in Fig. 4b and c, the limiting conditions of SAI and *M* were
achieved: (1) if $\mathrm{SAI}\to +\phantom{\rule{0.125em}{0ex}}\mathrm{\infty}$, which indicates that the match
of *P* and *E*_{0} tends to be the worst, and thus *R*→*P*
and *E*→0, i.e., *n*→0; (2) when *M*↑, then *E*↑, which has been demonstrated by previous
studies (i.e., Yang et al., 2009; Li et al., 2013), and thus
*n*↑, which can also be found in Fig. 4c and f. Based on
these limiting conditions, a semi-empirical formula (SEF) for parameter *n* was
obtained as follows:

where *a* and *c* are positive regression coefficients and *b* is negative. Nonlinear
least squares can be used to estimate the values of *a*, *b* and *c*, based on *n*
calibrated from measured data. Then, the final equation was as follows:

As shown in Fig. 5d, the simulated *n* calculated by SEF match well with the
optimized *n* with *R*^{2} of 0.75 and MAE of 0.24. In addition, the Eq. (13) has
also been verified in each catchment among the 26 basins (Table S1 in the Supplement). The
RMSE and MAE for each catchment is relatively small, with mean values of 12.0
and 14.8 mm, respectively. Except for basins 3, 5 and 26, the *R*^{2} values
for simulation of *R* in each catchment are larger than 0.5. These results
indicated that the *M* and SAI as well as the semi-empirical formula can
explain well the variability of the controlling parameter *n*.

In addition to the SEF, linear regression is often applied to simulate *n*.
For example, taking NDVI, latitude and topographic index as explanatory
variables, Xu et al. (2013) applied multiple linear regression to estimate
the spatial variation in *n* for the global large river basins. Considering
the multicollinearity issue, the partial least square regression (PLSR) was
used in this study. As shown in Fig. 5e, the values of Nash–Sutcliffe efficiency (NSE) and mean absolute error (MAE) of the simulated *n* by using PLSR were 0. 65 and 0.27
respectively, which was not as good as the performance of the semi-empirical
formula. Therefore, the SEF was a better choice not only for simulation but
also for explaining the physical meaning.

Cross-validation was used to validate the semi-empirical equation. The
dataset for one basin was used for validation, and the datasets for the
remaining 25 basins were used for calibration. Then the cross-validation
process is repeated 26 times, with each of the 26 basins used once as
validation. Parameter *n* for the validation basin was simulated by the
semi-empirical formula obtained from the other 25 basins. The calibrated
parameters for each basin can be found in Table S2. Subsequently, based on
annual *P*_{e}, *E*_{0} and simulated annual parameter
*n*, simulated annual *R* and *E* were calculated using Eq. (2). The simulated annual
*R* and *E* for each validated basin were combined to compare with the observed *R* and
*E*, respectively (Fig. 7). As shown in Fig. 7a and b, the simulated annual *R* and *E*
that estimated by Budyko model with cross-validation parameter *n* showed a
remarkable agreement with the observed ones, with NSE larger than 0.89
and MAE smaller than 50.52 mm, which is close to the simulation accuracy of
these estimated by Budyko model with simulated parameter *n* by using the
semi-empirical formula (i.e., Eq. 14, Fig. 7c and d). These results indicated
that the semi-empirical formula expressed the spatiotemporal variation in
parameter *n*, and the proposed Eq. (2) with simulated parameter *n* was reliable
for the simulation of annual *R* and *E*.

## 4.3 Contributions of SAI and other factors to *R* and *E* changes

To further assess the impact of SAI on the water balance, here we quantified
the contributions of SAI and other factors, i.e., *P*_{e}, *E*_{0} and
*M*, on the changes in *R* and *E* before and after change point. We used an
ordered clustering test, a Pettitt test method and a change point analysis
method for the “at most one change” (AMOC) method to detect the change points of *R*. To avoid possible
uncertainty within results based on the individual method, the assembled
change points were confirmed with more than one method. If the results for
all the three methods are different, the median change point would be
selected (Liu et al., 2017a). Based on the change points of *R* and the
changes rates of *P*_{e}, *E*_{0}, *M* and SAI before and after change
points (Table S3), the contributions of these four factors to *R* and *E*
were assessed (Figs. 8 and 9; Table S3).

As can be seen from Fig. 8a and c, the *P*_{e} changes controlled
the variation in *R* in most basins, with 18 of the 26 selected basins. The
absolute value of contributions of *P*_{e} changes to *R* changes
ranged from 11 % to 96 %, with the median value at 61 % for the 26
basins (Fig. 8b). In addition to the *P*_{e} changes, the SAI change was also
an important factor for the *R* change with the median absolute contribution
at 16 %. SAI was the dominant factor with the maximum contribution to *R*
changes in six rivers, such as Yangtze, Yellow, Aral, Northern Dvina, Congo
and Mississippi basins. The *E*_{0} changes reduce the *R* in 24 of the 26
basins (Table S4). The *E*_{0} changes had a limited impact on the *R*
changes with the median absolute contribution of 8 %. However, it is the
dominant factor for *R* changes in Parana River basins.

The dominant factors of *E* changes were different from those of *R* changes
(Fig. 9). Both the SAI and *M* changes had remarkable impacts on the *E* changes, which were the dominant factors for the *E* changes within
eight and five basins, respectively. Also, the contributions of SAI and *M*
changes to *E* changes were larger than those to *R* changes with the median
absolute contributions of 21 % and 28 %, respectively. Accordingly,
the contribution of *P*_{e} to *E* changes was weaker than that to *R* changes, the median of which dropped from 61 % to 32 %.

In summary, *P*_{e} was the key controlling factor for *R* and *E* in
most river basins. SAI was the dominant factor for both *R* and *E* mainly in
East Asian subtropical monsoon zones because of the monsoon variability (Cook
et al., 2010), such as Yangtze and Yellow River basins. SAI, *M* and *E*_{0} have larger impacts on the *E* changes than *R* changes do, while *P* has
stronger impacts on *R* changes than *E* changes do.

It has been found that both vegetation coverage and climate seasonality have
impacts on water balance (Chen et al., 2013; Li et al., 2013; Zeng and Cai,
2016; Abatzoglou and Ficklin, 2017; Ning et al., 2017; Zhang et al., 2016a).
Li et al. (2013) found that long-term vegetation coverage was closely
related to the spatial variation in the calibrated parameter of the Budyko
model in global river basins. However, vegetation dynamics also influenced
the temporal variation in parameter *n*, but the relationship is yet to be
verified over a larger spatial range (Zhang et al., 2016c; Ning et al.,
2017). Results of this study confirmed that the vegetation dynamics had a
significant impact on both spatial and temporal variations in the
controlling parameter *n* at the global scale.

The seasonality index represents the amplitude difference of seasonal *P* and
*E*_{0}, but does not include the phase difference of seasonal *P* and
*E*_{0}. Investigating the water balance across the Loess Plateau in China,
Ning et al. (2017) found that the seasonal index was closely related to the
controlling parameter. In this study, however, SI showed a worse correlation
with the variation in *n* in the 26 large global river basins than those in
Loess Plateau. All catchments selected by Ning et al. (2017) were in the
monsoon climate zone, where water and energy are strongly coupled, so the
seasonality of *P* and *E*_{0} in most catchments was in the same phase. Hence,
the asynchrony of water and energy was nonexistent and had a limited impact
on the variation in *n*. In contrast, the basins selected in this study covered
a large spatial scale with a wide range of climate types. Most basins had
different phases between seasonal *P* and *E*_{0}, such as the Northern
Dvina, with the phase differences larger than 2 months. The amplitude difference
between seasonal *P* and *E*_{0} cannot adequately represent the difference
between water and energy in the basins with out-of-phase *P* and *E*_{0} (Hickel
and Zhang, 2006). In this case, SAI, considering both amplitude and phase
differences between seasonal of *P* and *E*_{0}, was proposed to reflect
the difference between water and energy. Results showed that the proposed
SAI had a significant impact on *n* and evapotranspiration radio, as well as the
sensitively of evapotranspiration to the variation in precipitation,
potential evapotranspiration and catchment characteristics. SAI can also
be applied to other studies on water–energy balance.

In small-size catchments, interactions amongst climate variability,
vegetation dynamics and water balance are more complex (Li et al., 2013).
Many other factors, such as basins area, latitude, slope gradient, compound
topographic index, and so on (Abatzoglou and Ficklin, 2017; Xu et al., 2013;
Yang et al., 2009), have been identified as playing a role in the spatial
distribution of *n* for small-size catchments. However, in this study, these
factors had few changes at the annual timescale, so they were not
considered in determining the annual variation in *n*. This study demonstrated
that SAI and *M* play an important role in the spatiotemporal variation in *n* in
large river basins, nevertheless, other factors should also be considered in
the simulation of spatial variation in *n* for small-size catchments.

SAI was identified to have a great influence on the changes in *R* and *E*.
In particular, the changes in both *R* and *E* for the two major rivers (i.e., Yangtze
and Yellow River basins) in East Asian monsoon zones is mainly controlled by
SAI. Hoyos and Webster (2007) found that the variation in monsoon systems
has a remarkable effect on the climate seasonal pattern (Hoyos and Webster, 2007).
Using the covariance of *P* and *E*_{0} as an explanatory variable, Zeng and Cai (2016) indicated that the seasonality of *P* and *E*_{0} had a significant
impact on the *E* variation, such as in the Yangtze River basin. Their results are
generally consistent with ours. To assess the impact of ecological
restoration on runoff in the Loess Plateau of China, Liang et al. (2015)
regarded the ecological restoration, i.e., vegetation dynamics, as the cause
of changes in *n*. However, our results showed that SAI also played an
important role in the changes in *n*, particularly for the East Asian
subtropical monsoon zone.

Although SAI combined with *M* can capture the changes in *n* well (Fig. 5d), the
impact of other factors represented by parameter *n* on the water balance not
only includes SAI and M, but also the human influence, which has been
verified by our previous study (Liu et al., 2017a). As a result, this may
cause uncertainty in our findings. The human influences on *R* and *E* need to be
further investigated.

In this study, a semi-empirical formula was developed to simulate the
spatiotemporal variation in the controlling parameter *n* in the Budyko
model. Influences of climate–vegetation factors on water balance were
evaluated. The Choudhury–Yang equation modified by the effective
precipitation is recommended to calibrate the controlling parameter *n* and
to simulate evapotranspiration (*E*) and runoff (*R*), as well as their variation.

A climate seasonality and asynchrony index, i.e., SAI, is proposed to
reflect the difference between water and energy. Results show that the
optimized *n* has a much higher correlation with SAI than the existing SI,
implying that the phase mismatch between seasonal water and energy should be
considered in the impact assessment of water balance. In general, our
results suggest that the catchments with a larger SAI usually have a larger
evapotranspiration ratio given the same climatic and underlying condition,
and the variation in evapotranspiration tends to be more sensitive to the
changes in precipitation and landscape properties (parameter *n*), whereas it is less
sensitive to the potential evapotranspiration in the catchments with larger
SAI. Furthermore, this study confirms that vegetation dynamics (*M*) also
play an important role in modifying the temporal variation in *n* at the annual
scale. Based on SAI and *M*, a semi-empirical formula for the spatiotemporal
variation in parameter *n* has been developed, and it performs well in the
prediction of annual evapotranspiration and runoff.

Employing the developed semi-empirical formula, the contributions of SAI and
*M*, as well as *P*_{e} and *E*_{0}, to the variation in *E* and *R*
were assessed. Results show that precipitation is the first-order control on
the *R* and *E* changes, and, secondly, SAI was found to control the changes in *R* and *E* in the subtropical monsoon regions of East Asia. SAI, *M* and
*E*_{0} have larger impacts on *E* than on *R*, whereas *P*_{e} has
larger impacts on *R*.

The study assesses the influence of climate variability and vegetation
dynamics on water balance, which highlights the role of climate seasonality
and asynchrony as well as vegetation dynamics in the annual variation in
*n*, and sheds new light on the difference in the contributions of
climate–vegetation factors to the changes in *R* and *E*. This study can be useful
for water–energy modeling, hydrological forecasting and water management.

The monthly potential evapotranspiration data are available free of charge through the Climatic Research Unit, University of East Anglia (UEA, 2017) (https://crudata.uea.ac.uk/cru/data/hrg/cru_ts_3.24.01/). The monthly normalized difference vegetation index (NDVI) data are available at free of charge through the Ecological Forecasting Lab at Ames Research Center, National Aeronautics and Space Administration (NASA, 2014) (https://nex.nasa.gov/nex/projects/1349/).

The supplement related to this article is available online at: https://doi.org/10.5194/hess-22-4047-2018-supplement.

JL and QZ proposed the scientific hypothesis, analyzed the formula, designed the research and wrote the paper. JL, QZ, VPS, CS, YZ and PS contributed to the interpretation of the results and the writing of the paper. YZ reviewed the paper and gave some comments. XG gave some comments and suggestions in the modification of paper.

The authors declare that they have no conflict of interest.

This work is financially supported by the National Science Foundation for
Distinguished Young Scholars of China (grant no. 51425903), Fund for
Creative Research Groups of National Natural Science Foundation of China
(grant no. 41621061), National Natural Science Foundation of China (no.
41771536), Key Project of National Natural Science Foundation of China (grant no. 51190091), National Natural Science Foundation of China under grant no.
41401052, National Key Research and Development Program of China (grant no.
2018YFA0605603) and Fundamental Research Funds for the Central Universities,
China University of Geosciences (Wuhan) (grant nos. CUGCJ1702, CUG180614). We
would like to thank Ming Pan (mpan@princeton.edu) and Dan Li
(danl@princeton.edu) at Princeton University as well as Xianli Xu
(xuxianliww@gmail.com) at the Chinese Academy of Sciences for sharing the basin dataset. Information on the data was provided with great detail in the Data
section and for further messages concerning data please write to zhangq68@bnu.edu.cn. Last but not least, our cordial gratitude
should be extended to the editor and anonymous reviewers for their
professional comments and suggestions, which were greatly helpful for further
quality improvement of our paper.

Edited
by: Pierre Gentine

Reviewed by: two anonymous referees

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*P*

_{e}), the Budyko framework was extended to the annual water balance analysis. To reflect the mismatch between water supply (precipitation,

*P*) and energy (potential evapotranspiration,

*E*

_{0}), a climate seasonality and asynchrony index (SAI) were proposed in terms of both phase and amplitude mismatch between

*P*and

*E*

_{0}.

*P*

_{e}), the Budyko framework was extended to the annual water...