**Research article**
24 Jan 2020

**Research article** | 24 Jan 2020

# Inter-annual variability of the global terrestrial water cycle

Dongqin Yin and Michael L. Roderick

^{1,2}

^{1,3}

**Dongqin Yin and Michael L. Roderick**Dongqin Yin and Michael L. Roderick

^{1,2}

^{1,3}

^{1}Research School of Earth Sciences, Australian National University, Canberra, ACT, 2601, Australia^{2}Australian Research Council Centre of Excellence for Climate System Science, Canberra, ACT, 2601, Australia^{3}Australian Research Council Centre of Excellence for Climate Extremes, Canberra, ACT, 2601, Australia

^{1}Research School of Earth Sciences, Australian National University, Canberra, ACT, 2601, Australia^{2}Australian Research Council Centre of Excellence for Climate System Science, Canberra, ACT, 2601, Australia^{3}Australian Research Council Centre of Excellence for Climate Extremes, Canberra, ACT, 2601, Australia

**Correspondence**: Dongqin Yin (dongqin.yin@anu.edu.au, dongqin.yin@cau.edu.cn)

**Correspondence**: Dongqin Yin (dongqin.yin@anu.edu.au, dongqin.yin@cau.edu.cn)

Received: 10 May 2019 – Discussion started: 21 May 2019 – Revised: 20 Nov 2019 – Accepted: 04 Dec 2019 – Published: 24 Jan 2020

Variability of the terrestrial water cycle, i.e. precipitation (*P*),
evapotranspiration (*E*), runoff (*Q*) and water storage change (Δ*S*) is the
key to understanding hydro-climate extremes. However, a comprehensive global
assessment for the partitioning of variability in *P* between *E*, *Q* and Δ*S* is still not available. In this study, we use the recently released global
monthly hydrologic reanalysis product known as the Climate Data Record (CDR)
to conduct an initial investigation of the inter-annual variability of the
global terrestrial water cycle. We first examine global patterns in
partitioning the long-term mean $\stackrel{\mathrm{\u203e}}{P}$ between the various sinks
$\stackrel{\mathrm{\u203e}}{E}$, $\stackrel{\mathrm{\u203e}}{Q}$ and $\stackrel{\mathrm{\u203e}}{\mathrm{\Delta}S}$ and confirm the well-known
patterns with $\stackrel{\mathrm{\u203e}}{P}$ partitioned between $\stackrel{\mathrm{\u203e}}{E}$ and $\stackrel{\mathrm{\u203e}}{Q}$
according to the aridity index. In a new analysis based on the concept of
variability source and sinks we then examine how variability in the
precipitation ${\mathit{\sigma}}_{P}^{\mathrm{2}}$ (the source) is partitioned between the
three variability sinks ${\mathit{\sigma}}_{E}^{\mathrm{2}}$, ${\mathit{\sigma}}_{Q}^{\mathrm{2}}$ and ${\mathit{\sigma}}_{\mathrm{\Delta}S}^{\mathrm{2}}$ along with the three relevant covariance terms, and how
that partitioning varies with the aridity index. We find that the
partitioning of inter-annual variability does not simply follow the mean
state partitioning. Instead we find that ${\mathit{\sigma}}_{P}^{\mathrm{2}}$ is mostly
partitioned between ${\mathit{\sigma}}_{Q}^{\mathrm{2}}$, ${\mathit{\sigma}}_{\mathrm{\Delta}S}^{\mathrm{2}}$ and the
associated covariances with limited partitioning to ${\mathit{\sigma}}_{E}^{\mathrm{2}}$. We
also find that the magnitude of the covariance components can be large and
often negative, indicating that variability in the sinks (e.g. ${\mathit{\sigma}}_{Q}^{\mathrm{2}}$, ${\mathit{\sigma}}_{\mathrm{\Delta}S}^{\mathrm{2}}$) can, and regularly does, exceed
variability in the source (${\mathit{\sigma}}_{P}^{\mathrm{2}}$). Further investigations under
extreme conditions revealed that in extremely dry environments the variance
partitioning is closely related to the water storage capacity. With limited
storage capacity the partitioning of ${\mathit{\sigma}}_{P}^{\mathrm{2}}$ is mostly to ${\mathit{\sigma}}_{E}^{\mathrm{2}}$, but as the storage capacity increases the partitioning of ${\mathit{\sigma}}_{P}^{\mathrm{2}}$ is increasingly shared between ${\mathit{\sigma}}_{E}^{\mathrm{2}}$, ${\mathit{\sigma}}_{\mathrm{\Delta}S}^{\mathrm{2}}$ and the covariance between those variables. In other environments
(i.e. extremely wet and semi-arid–semi-humid) the variance partitioning
proved to be extremely complex and a synthesis has not been developed. We
anticipate that a major scientific effort will be needed to develop a
synthesis of hydrologic variability.

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In describing the terrestrial branch of the water cycle, the precipitation
(*P*) is partitioned into evapotranspiration (*E*), runoff (*Q*) and change in water
storage (Δ*S*). With averages taken over many years, $\stackrel{\mathrm{\u203e}}{\mathrm{\Delta}S}$ is
usually assumed to be zero and it has long been recognized that the
partitioning of the long-term mean annual precipitation ($\stackrel{\mathrm{\u203e}}{P}$) between
$\stackrel{\mathrm{\u203e}}{E}$ and $\stackrel{\mathrm{\u203e}}{Q}$ was jointly determined by the availability of both
water ($\stackrel{\mathrm{\u203e}}{P}$) and energy (represented by the net radiation expressed as
an equivalent depth of water and denoted $\stackrel{\mathrm{\u203e}}{{E}_{\mathrm{o}}}$). Using data from a
large number of watersheds, Budyko (1974) developed an empirical relation
relating the evapotranspiration ratio ($\stackrel{\mathrm{\u203e}}{E}/\stackrel{\mathrm{\u203e}}{P}$) to the aridity
index ($\stackrel{\mathrm{\u203e}}{{E}_{\mathrm{o}}}/\stackrel{\mathrm{\u203e}}{P}$). The resultant empirical relation and other
Budyko-type forms (e.g. Fu, 1981; Choudhury, 1999; Yang et al., 2008;
Roderick and Farquhar, 2011; Sposito, 2017) that partition *P* between *E* and *Q*
have proven to be extremely useful in both understanding and characterizing
the long-term mean annual hydrological conditions in a given region.

However, the long-term mean annual hydrologic fluxes rarely occur in any
given year. Instead, society must (routinely) deal with variability around
the long-term mean. The classic hydro-climate extremes are droughts and
floods but the key point here is that hydrologic variability is expressed on
a full spectrum of time and space scales. To accommodate that perspective,
we need to extend our thinking beyond the long-term mean to ask how the
variability of *P* is partitioned into the variability of *E*, *Q* and Δ*S* (e.g.
Orth and Destouni, 2018).

Early research on hydrologic variability focussed on extending the Budyko
curve. In particular, Koster and Suarez (1999) used the Budyko curve to
investigate inter-annual variability in the water cycle. In their framework,
the evapotranspiration standard deviation ratio (defined as the ratio of
standard deviation for *E* to *P*, *σ*_{E}∕*σ*_{P}) was (also) estimated
using the aridity index ($\stackrel{\mathrm{\u203e}}{{E}_{\mathrm{o}}}/\stackrel{\mathrm{\u203e}}{P}$). The classic Koster and
Suarez framework has been widely applied and extended in subsequent
investigations of the variability in both *E* and *Q*, using catchment
observations, reanalysis data and model outputs (e.g. McMahon et al., 2011;
Wang and Alimohammadi, 2012; Sankarasubramanian and Vogel, 2002; Zeng and
Cai, 2015). However, typical applications of the Koster and Suarez framework
have previously been at regional scales and there is still no comprehensive
global assessment for partitioning the variability of *P* into the variability
of *E*, *Q* and Δ*S*. One reason for the lack of a global comprehensive
assessment is the absence of gridded global hydrologic data. Interestingly,
the atmospheric science community have long used a combination of
observations and model outputs to construct gridded global-scale
atmospheric re-analyses and such products have become central to atmospheric
research. Those atmospheric products also contain estimates of some of the
key water cycle variables (e.g. *P*, *E*), such as in the widely used interim
ECMWF Re-Analysis (ERA-Interim; Dee et al. 2011). Though efforts have been
taken to develop land-based products from atmospheric reanalyses, e.g.
ERA-Land (Balsamo et al., 2015) and MERRA-Land (Reichle et al., 2011)
databases, the central aim of atmospheric re-analysis is to
estimate atmospheric variables. That atmospheric-centric aim,
understandably, ignores many of the nuances of soil water infiltration,
vegetation water uptake, runoff generation and many other processes of
central importance in hydrology.

Hydrologists have only recently accepted the challenge of developing their
own re-analysis-type products with perhaps the first serious hydrologic
re-analysis being published as recently as a few years ago (Rodell et al.,
2015). More recently, the Princeton University group has extended this early
work by making available a gridded global terrestrial hydrologic re-analysis
product known as the Climate Data Record (CDR) (Zhang et al., 2018).
Briefly, the CDR was constructed by synthesizing multiple in situ
observations, satellite remote sensing products and land surface model
outputs to provide *gridded* estimates of global land precipitation *P*,
evapotranspiration *E*, runoff *Q* and total water storage change Δ*S*
($\mathrm{0.5}{}^{\circ}\times \mathrm{0.5}{}^{\circ}$, monthly, 1984–2010). In
developing the CDR, the authors adopted local water budget closure as the
fundamental hydrologic principle. That approach presented one important
difficulty. Global observations of Δ*S* start with the GRACE satellite
mission from 2002. Hence before 2002 there is no direct observational
constraint on Δ*S* and the authors made the further assumption that the
mean annual Δ*S* over the full 1984–2010 period was zero at every
grid box. That is incorrect in some regions (e.g. Scanlon et al., 2018) and
represents an observational problem that cannot be overcome. However, our
interest is in the year-to-year variability and for that application, the
assumption of no change in the mean annual Δ*S* over the full 1984–2010
period is unlikely to lead to major problems since we are not looking for
subtle changes over time. With that caveat in mind, the aim of this study is
to use this new 27-year gridded hydrologic re-analysis product to conduct an
initial investigation of the inter-annual variability of the terrestrial
branch of the global water cycle.

The paper is structured as follows. We begin in Sect. 2 by describing the
various climate and hydrologic databases used in this study and also
include a further assessment of the suitability of the CDR database for this
initial variability study. In Sect. 3, we examine relationships between
the mean and variability in the four water cycle variables (*P*, *E*, *Q* and Δ*S*). In Sect. 4, we first relate the variabilities to the classical aridity
index and then use those results to evaluate the theory of Koster and Suarez (1999). Subsequently we examine how the variance of *P* is partitioned into the
variances (and relevant covariances) of *E*, *Q* and Δ*S* and undertake an
initial survey that investigates some of the factors controlling the
variance partitioning. We conclude the paper with a discussion summarizing
what we have learnt about water cycle variability over land by using the CDR
database.

## 2.1 Methods

The water balance is defined by

with *P* the precipitation, *E* the evapotranspiration, *Q* the runoff and Δ*S* the total water storage change in time step *t* (annual in this study). By the
usual variance law, we have

which includes all relevant variances (denoted *σ*^{2}) and covariances
(denoted cov). Equation (2) can be thought of as the hydrologic variance balance
equation.

## 2.2 Hydrologic and climatic data

We use the CDR database (Zhang et al., 2018), which is
a recently released global land hydrologic re-analysis. This product
includes global precipitation *P*, evapotranspiration *E*, runoff *Q* and water
storage change Δ*S* ($\mathrm{0.5}{}^{\circ}\times \mathrm{0.5}{}^{\circ}$,
monthly, 1984–2010). In this study we focus on the inter-annual variability
and the monthly water cycle variables (*P*, *E*, *Q* and Δ*S*) are aggregated to
annual totals. The CDR does not report additional radiation variables and we
use the NASA/GEWEX Surface Radiation Budget (SRB) Release-3.0 (monthly,
1984–2007, $\mathrm{1}{}^{\circ}\times \mathrm{1}{}^{\circ}$) database (Stackhouse et
al., 2011) to calculate *E*_{o} (defined as the net radiation expressed as an
equivalent depth of liquid water, Budyko, 1974). We then calculate the
aridity index ($\stackrel{\mathrm{\u203e}}{{E}_{\mathrm{o}}}/\stackrel{\mathrm{\u203e}}{P}$) using *P* from the CDR and *E*_{o} from
the SRB databases (see Fig. S1a in the Supplement).

In general, we anticipate two important factors, i.e. the water storage
capacity and the presence of ice/snow at the surface, which are most likely
to have influence on the partitioning of hydrologic variability. For the
storage, the active range of the monthly water storage variation was used to
approximate the water storage capacity (*S*_{max}). In more detail, the water
storage *S*(*t*) at each time step *t* (monthly here) was first calculated from the
accumulation of Δ*S*(*t*), i.e. $S\left(t\right)=S(t-\mathrm{1})+\mathrm{\Delta}S\left(t\right)$, where we assumed
zero storage at the beginning of the study period (i.e. *S*(0)=0). With
the resulting time series available, *S*_{max} was estimated as the
difference between the maximum and minimum *S*(*t*) during the study period at
each grid box (see Fig. S1b in the Supplement). The estimated
*S*_{max} shows a large range from 0 to 1000 mm with the majority of values
from 50 to 600 mm (Fig. S1b), which generally agrees with global rooting
depth estimates assuming that water occupies from 10 % to 30 % of the soil
volume at field capacity (Jackson et al., 1996; Wang-Erlandsson et al.,
2016; Yang et al., 2016). To characterize snow/ice cover, and to distinguish
extremely hot and cold regions, we also make use of a gridded global land
air temperature dataset from the Climatic Research Unit (CRU TS4.01
database, monthly, 1901–2016, $\mathrm{0.5}{}^{\circ}\times \mathrm{0.5}{}^{\circ}$)
(Harris et al., 2014) (see Fig. S1c in the Supplement).

## 2.3 Spatial mask to define study extent

The CDR database provides an estimate of the uncertainty (±1*σ*) for each of the hydrologic variables (*P*, *E*, *Q*, Δ*S*) in each month. We
use those uncertainty estimates to identify and remove regions with high
relative uncertainty in the CDR data. The relative uncertainty is calculated
as the ratio of root mean square of the uncertainty (±1*σ*) to the
mean annual *P*, *E* and *Q* at each grid box following the procedure used by Milly
and Dunne (2002). Note that the long-term mean Δ*S* is zero by
construction in the CDR database, and for that reason we did not use Δ*S* to calculate the relative uncertainty. Grid boxes with a relative
uncertainty (in *P*, *E* and *Q*) of more than 10 % are deemed to have high
relative uncertainty (Milly and Dunne, 2002) and were excluded from the
study extent. The excluded grid boxes were mostly in the Himalayan region,
the Sahara Desert and in Greenland. The final spatial mask is shown in Fig. S2 and this has been applied throughout this study.

## 2.4 Further evaluation of CDR data for variability analysis

In the original work, the CDR database was validated by comparison with
independent observations including (i) mean seasonal cycle of *Q* from 26 large
basins (see Fig. 8 in Zhang et al., 2018), (ii) mean seasonal cycle of
Δ*S* from 12 large basins (Fig. 10 in Zhang et al., 2018), (iii) monthly runoff from 165 medium size basins and a further 862 small basins
(Fig. 14 in Zhang et al., 2018), and (iv) summer *E* from 47 flux towers (Fig. 16
in Zhang et al., 2018). Those evaluations did not directly address
variability in various water cycle elements. With our focus on the
variability we decided to conduct further validations of the CDR database
beyond those described in the original work. In particular, we focussed on
further independent assessments of *E* and we use monthly (as opposed to
summer) observations of *E* from FLUXNET to evaluate the variability in *E*. We
also compare the evapotranspiration *E* in the CDR with two other gridded
global *E* products that were not used to develop the CDR including the
LandFluxEval database ($\mathrm{1}{}^{\circ}\times \mathrm{1}{}^{\circ}$, monthly,
1989–2005) (Mueller et al., 2013) and the Max Planck Institute database
(MPI, $\mathrm{0.5}{}^{\circ}\times \mathrm{0.5}{}^{\circ}$, monthly, 1982–2011) (Jung
et al., 2010). The runoff *Q* in the CDR is further compared with the gridded
European *Q* product E-RUN ($\mathrm{0.5}{}^{\circ}\times \mathrm{0.5}{}^{\circ}$, monthly,
1951–2015) (Gudmundsson and Seneviratne, 2016a).

For the comparison to FLUXNET observations (Baldocchi et al., 2001; Agarwal
et al., 2010) we identified 32 flux tower sites (site locations are shown in
Fig. S3 and details are shown in Table S1) with at least 3 years of
continuous (monthly) measurements using the FluxnetLSM R package (v1.0)
(Ukkola et al., 2017). The monthly totals and annual climatology of *P* and *E*
from CDR generally follow FLUXNET observations, with high correlations and
reasonable root mean square error (Figs. S4–S5, Table S1). Comparison of the
point-based FLUXNET (∼100 m–1 km scale) with the
grid-based CDR (∼50 km scale) is problematic since the CDR
represents an area that is at least 2500 times larger than the area
represented by the individual FLUXNET towers and we anticipate that the CDR
record would be “smoothed” relative to the FLUXNET record. With that in
mind, we chose to compare the ratio of the standard deviation of *E* to *P*
between the CDR and FLUXNET databases and this normalized comparison of the
hydrologic variability proved encouraging (Fig. S6).

The comparison of *E* between the CDR and the LandFluxEval and MPI databases
also proved encouraging. We found that the monthly mean *E* from the CDR
database is slightly underestimated compared with the LandFluxEVAL database
(Fig. S7a) but agrees closely with the MPI database (Fig. S8a). In terms of
variability, the standard deviations of monthly *E* from the CDR are in very
close agreement with the LandFluxEVAL database (Fig. S7c), but there is a
bias and scaling offset for the comparison with the MPI database,
particularly for the grid cells with low standard deviation of *E* (Fig. S8c).
The comparison of runoff *Q* between the E-RUN and CDR databases shows that the
two databases have very similar spatial patterns of both the long-term mean
($\stackrel{\mathrm{\u203e}}{Q}$) and standard deviation (*σ*_{Q}) of the monthly *Q* (Fig. S10). The grid-by-grid comparison results are also encouraging, showing
slight bias of both the long-term mean and standard deviation of monthly *Q* in
the CDR database compared with the E-RUN database (Fig. S11).

We concluded that while the CDR database was unlikely to be perfect, it was nevertheless suitable for an initial exploratory survey of inter-annual variability in the terrestrial branch of the global water cycle.

## 3.1 Mean annual *P*, *E* and *Q* and the Budyko curve

The global pattern of mean annual *P*, *E* and *Q* using the CDR data (1984–2007) is
shown in Fig. 1. The mean annual *P* ($\stackrel{\mathrm{\u203e}}{P}$) is prominent in tropical
regions, southern China, and eastern and western North America (Fig. 1a). The
magnitude of mean annual *E* ($\stackrel{\mathrm{\u203e}}{E}$) more or less follows the pattern of
$\stackrel{\mathrm{\u203e}}{P}$ in the tropics (Fig. 1b) while the mean annual *Q* ($\stackrel{\mathrm{\u203e}}{Q}$) is
particularly prominent in the Amazon, South and Southeast Asia, tropical
parts of western Africa, and in some other coastal regions at higher latitudes
(Fig. 1c).

We relate the grid-box level ratio of $\stackrel{\mathrm{\u203e}}{E}$ to $\stackrel{\mathrm{\u203e}}{P}$ in the CDR database to the classical Budyko (1974) curve using the aridity index ($\stackrel{\mathrm{\u203e}}{{E}_{\mathrm{o}}}/\stackrel{\mathrm{\u203e}}{P}$) (Fig. 2a). As noted previously, in the CDR database, $\stackrel{\mathrm{\u203e}}{\mathrm{\Delta}S}$ is forced to be zero and this enforced steady state (i.e. $\stackrel{\mathrm{\u203e}}{P}=\stackrel{\mathrm{\u203e}}{E}+\stackrel{\mathrm{\u203e}}{Q}$) allowed us to also predict the ratio of $\stackrel{\mathrm{\u203e}}{Q}$ to $\stackrel{\mathrm{\u203e}}{P}$ using the same Budyko curve (Fig. 2b). The Budyko curves follow the overall pattern in the CDR data, which agrees with previous studies showing that the aridity index can be used to predict water availability (e.g. Gudmundsson et al., 2016). However, there is substantial scatter due to, for example, regional variations related to seasonality, water storage change and the landscape characteristics (Milly, 1994a, b; Padrón et al., 2017). With that caveat in mind, the overall patterns are as expected, with $\stackrel{\mathrm{\u203e}}{E}$ following $\stackrel{\mathrm{\u203e}}{P}$ in dry environments ($\stackrel{\mathrm{\u203e}}{{E}_{\mathrm{o}}}/\stackrel{\mathrm{\u203e}}{P}>\mathrm{1.0}$) while $\stackrel{\mathrm{\u203e}}{E}$ follows $\stackrel{\mathrm{\u203e}}{{E}_{\mathrm{o}}}$ in wet environments ($\stackrel{\mathrm{\u203e}}{{E}_{\mathrm{o}}}/\stackrel{\mathrm{\u203e}}{P}\le \mathrm{1.0}$) (Fig. 2).

## 3.2 Inter-annual variability in *P*, *E*, *Q* and Δ*S*

We use the variance balance equation (Eq. 2) to partition the inter-annual
${\mathit{\sigma}}_{P}^{\mathrm{2}}$ into separate components due to ${\mathit{\sigma}}_{E}^{\mathrm{2}}$, ${\mathit{\sigma}}_{Q}^{\mathrm{2}}$, ${\mathit{\sigma}}_{\mathrm{\Delta}S}^{\mathrm{2}}$ along with the three covariance
components [2cov(*E**Q*), 2cov(*E*,Δ*S*),
2cov(*Q*,Δ*S*)] (Fig. 3). The spatial pattern of ${\mathit{\sigma}}_{P}^{\mathrm{2}}$ (Fig. 3a) is very similar to that of $\stackrel{\mathrm{\u203e}}{P}$ (Fig. 1a), which
implies that the ${\mathit{\sigma}}_{P}^{\mathrm{2}}$ is positively correlated with $\stackrel{\mathrm{\u203e}}{P}$.
In contrast the partitioning of ${\mathit{\sigma}}_{P}^{\mathrm{2}}$ to the various components
is very different from the partitioning of $\stackrel{\mathrm{\u203e}}{P}$ (cf. Figs. 1 and 3).
First we note that while the overall spatial pattern of ${\mathit{\sigma}}_{E}^{\mathrm{2}}$
more or less follows ${\mathit{\sigma}}_{P}^{\mathrm{2}}$, the overall magnitude of ${\mathit{\sigma}}_{E}^{\mathrm{2}}$ is much smaller than ${\mathit{\sigma}}_{P}^{\mathrm{2}}$ and ${\mathit{\sigma}}_{Q}^{\mathrm{2}}$ in
most regions, and in fact ${\mathit{\sigma}}_{E}^{\mathrm{2}}$ is also generally smaller than
${\mathit{\sigma}}_{\mathrm{\Delta}S}^{\mathrm{2}}$. The prominence of ${\mathit{\sigma}}_{\mathrm{\Delta}S}^{\mathrm{2}}$
(compared to ${\mathit{\sigma}}_{E}^{\mathrm{2}}$) surprised us. The three covariance
components [cov(*E*,*Q*), cov(*E*,Δ*S*), cov(*Q*,Δ*S*)] are also
important in some regions. In more detail, the cov(*E*,*Q*) term is
prominent in regions where ${\mathit{\sigma}}_{Q}^{\mathrm{2}}$ is large and is mostly negative
in those regions (Fig. 3e), indicating that years with lower *E* are associated
with higher *Q* and vice versa. There are also a few regions with prominent
positive values for cov(*E**Q*) (e.g. the seasonal hydroclimates of northern
Australia) indicating that in those regions, years with a higher *E* are
associated with higher *Q*. The cov(*E*,Δ*S*) term (Fig. 3f) has a similar
spatial pattern to the cov(*E*,*Q*) term (Fig. 3e) but with a smaller overall
magnitude. Finally, the cov(*Q*,Δ*S*) term shows a more complex spatial
pattern, with both prominent positive and negative values (Fig. 3g) in
regions where ${\mathit{\sigma}}_{Q}^{\mathrm{2}}$ (Fig. 3c) and ${\mathit{\sigma}}_{\mathrm{\Delta}S}^{\mathrm{2}}$ (Fig. 3d) are both large.

These results show that the spatial patterns in variability are not simply a
reflection of patterns in the long-term mean state. On the contrary, we find
that of the three primary variance terms, the overall magnitude of
(inter-annual) ${\mathit{\sigma}}_{E}^{\mathrm{2}}$ is the smallest, implying the least
(inter-annual) variability in *E*. This is very different from the conclusions
based on spatial patterns in the mean *P*, *E* and *Q* (see Sect. 3.1). Further,
while ${\mathit{\sigma}}_{Q}^{\mathrm{2}}$ more or less follows ${\mathit{\sigma}}_{P}^{\mathrm{2}}$ as expected,
we were surprised by the magnitude of ${\mathit{\sigma}}_{\mathrm{\Delta}S}^{\mathrm{2}}$ which, in
general, substantially exceeds the magnitude of ${\mathit{\sigma}}_{E}^{\mathrm{2}}$. Further,
the magnitude of the covariance terms can be important, especially in
regions with high ${\mathit{\sigma}}_{Q}^{\mathrm{2}}$. However, unlike the variances, the
covariance can be both positive and negative and this introduces additional
complexity. For example, with a negative covariance it is possible for the
variance in *Q* (${\mathit{\sigma}}_{Q}^{\mathrm{2}}$) to exceed the variance in *P* (${\mathit{\sigma}}_{P}^{\mathrm{2}}$). To examine that in more detail we calculated the equivalent
frequency distribution for each of the plots in Fig. 3. The results (Fig. S9) further emphasize that in general, ${\mathit{\sigma}}_{E}^{\mathrm{2}}$ is the smallest of
the variances (Fig. S9b). We also note that the frequency distributions for
the covariances (Fig. S9e, f, g) are not symmetrical. In summary, it is clear
that spatial patterns in the inter-annual variability of the water cycle
(Fig. 3) do not simply follow the spatial patterns for the inter-annual mean
(Fig. 1).

## 3.3 Relation between variability and the mean state for *P*, *E*, *Q*

Differences in the spatial patterns of the mean (Fig. 1) and inter-annual
variability (Fig. 3) in the global water cycle led us to further investigate
the relation between the mean and the variability for each separate
component. Here we relate the standard deviation (*σ*_{P}, *σ*_{E}, *σ*_{Q}) instead of the variance to the mean of each water
balance flux (Fig. 4) since the standard deviation has the same physical
units as the mean, making the results more comparable. As inferred
previously, we find *σ*_{P} to be positively correlated with $\stackrel{\mathrm{\u203e}}{P}$
but with substantial scatter (Fig. 4a). The same result more or less holds
for the relation between *σ*_{Q} and $\stackrel{\mathrm{\u203e}}{Q}$ (Fig. 4c). In contrast
the relation between *σ*_{E} and $\stackrel{\mathrm{\u203e}}{E}$ is very different (Fig. 4b). In particular, *σ*_{E} is a small fraction of $\stackrel{\mathrm{\u203e}}{E}$ and this
complements the earlier finding (Fig. 4b) that the inter-annual variability
for *E* is generally smaller than for the other physical variables (*P*, *Q* and
Δ*S*). (The same result was also found using both LandFluxEVAL and MPI
databases; see Fig. S12 in the Supplement.) Importantly, unlike
*P* and *Q*, *E* is constrained by both water and energy availability (Budyko, 1974)
and the limited inter-annual variability in *E* presumably reflects limited
inter-annual variability in the available (radiant) energy (*E*_{o}). This is
something that could be investigated in a future study.

In the previous section, we investigated spatial patterns of the mean and
the variability in the global water cycle. In this section, we extend that
by investigating the partitioning of ${\mathit{\sigma}}_{P}^{\mathrm{2}}$ to the three primary
physical terms (${\mathit{\sigma}}_{E}^{\mathrm{2}}$, ${\mathit{\sigma}}_{Q}^{\mathrm{2}}$, ${\mathit{\sigma}}_{\mathrm{\Delta}S}^{\mathrm{2}}$) along with the three relevant covariances. For that, we begin by
comparing the Koster and Suarez (1999) theory against the CDR data and then
investigate how the partitioning of the variance is related to the aridity
index $\stackrel{\mathrm{\u203e}}{{E}_{\mathrm{o}}}/\stackrel{\mathrm{\u203e}}{P}$ (see Fig. S1a in the Supplement).
Following that, we investigate variance partitioning in relation to both our
estimate of the storage capacity *S*_{max} (see Fig. S1b in the Supplement) as well as the mean annual air temperature $\stackrel{\mathrm{\u203e}}{{T}_{\mathrm{a}}}$ (see Fig. S1c in the Supplement) that we use as a surrogate for snow/ice
cover. We finish this section by examining the partitioning of variance at
three selected study sites that represent extremely dry/wet, high/low water
storage capacity and the hot/cold spectrums.

## 4.1 Comparison with the Koster and Suarez (1999) theory

We first evaluate the classical empirical curve of Koster and Suarez (1999)
by relating ratios *σ*_{E}∕*σ*_{P} and *σ*_{Q}∕*σ*_{P} to the aridity index (Fig. 5). The ratio *σ*_{E}∕*σ*_{P} in the CDR database is generally overestimated by the
empirical Koster and Suarez curve, especially in dry environments (e.g.
$\stackrel{\mathrm{\u203e}}{{E}_{\mathrm{o}}}/\stackrel{\mathrm{\u203e}}{P}>\mathrm{3}$) (Fig. 5a). The inference here is
that the Koster and Suarez theory predicts *σ*_{E}∕*σ*_{P} to
approach unity in dry environments while the equivalent value in the CDR
data is occasionally unity but is generally smaller. With *σ*_{E}∕*σ*_{P} generally overestimated by the Koster and Suarez theory
we expect, and find, that *σ*_{Q}∕*σ*_{P} is generally
underestimated by the same theory (Fig. 5b). The same overestimation was
found based on the other two independent databases for *E* (LandFluxEVAL and
MPI) (Fig. S13). This overestimation is discussed further in Sect. 5.

## 4.2 Relating inter-annual variability to aridity

Here we examine how the fraction of the total variance in precipitation accounted for by the three primary variance terms along with the three covariance terms varies with the aridity index ($\stackrel{\mathrm{\u203e}}{{E}_{\mathrm{o}}}/\stackrel{\mathrm{\u203e}}{P}$) (Fig. 6). (Also see Fig. S14 for the spatial maps.) The ratio ${\mathit{\sigma}}_{E}^{\mathrm{2}}/{\mathit{\sigma}}_{P}^{\mathrm{2}}$ is close to zero in extremely wet regions and has an upper limit noted previously (Fig. 5a) that approaches unity in extremely dry regions (Fig. 6a). The ratio ${\mathit{\sigma}}_{Q}^{\mathrm{2}}/{\mathit{\sigma}}_{P}^{\mathrm{2}}$ is close to zero in extremely dry regions but approaches unity in extremely wet regions but with substantial scatter (Fig. 6b). The ratio ${\mathit{\sigma}}_{\mathrm{\Delta}S}^{\mathrm{2}}/{\mathit{\sigma}}_{P}^{\mathrm{2}}$ is close to zero in both extremely dry/wet regions (Fig. 6c) and shows the largest range at an intermediate aridity index ($\stackrel{\mathrm{\u203e}}{{E}_{\mathrm{o}}}/\stackrel{\mathrm{\u203e}}{P}\sim \mathrm{1.0}$).

The covariance ratios are all small in extremely dry (e.g. $\stackrel{\mathrm{\u203e}}{{E}_{\mathrm{o}}}/\stackrel{\mathrm{\u203e}}{P}\ge \mathrm{6.0}$) environments and generally show the largest range in semi-arid and semi-humid environments. The peak magnitudes for the three covariance components consistently occur when $\stackrel{\mathrm{\u203e}}{{E}_{\mathrm{o}}}/\stackrel{\mathrm{\u203e}}{P}$ is close to 1.0, which is the threshold often used to separate wet and dry environments.

## 4.3 Further investigations on the factors controlling partitioning of the variance

Results in the previous section demonstrated that spatial variation in the
partitioning of ${\mathit{\sigma}}_{P}^{\mathrm{2}}$ into ${\mathit{\sigma}}_{E}^{\mathrm{2}}$, ${\mathit{\sigma}}_{Q}^{\mathrm{2}}$,
${\mathit{\sigma}}_{\mathrm{\Delta}S}^{\mathrm{2}}$ and the three covariance components is complex
(Fig. 6). To help further understand inter-annual variability of the
terrestrial water cycle, we conduct further investigations in this section
using two factors likely to have a major influence on the variance
partitioning of ${\mathit{\sigma}}_{P}^{\mathrm{2}}$. The first is the storage capacity
*S*_{max} (see Fig. S1b in the Supplement). The second is the
mean annual air temperature $\stackrel{\mathrm{\u203e}}{{T}_{\mathrm{a}}}$ (see Fig. S1c in the Supplement), which is used here as a surrogate for snow/ice presence.

### 4.3.1 Relating inter-annual variability to storage capacity

We first relate the partitioning of ${\mathit{\sigma}}_{P}^{\mathrm{2}}$ to water storage
capacity (*S*_{max}) by repeating Fig. 6 but instead we use a logarithmic
scale for the *x* axis and we distinguish *S*_{max} via the background
colour (Fig. 7). To eliminate the possible overlap of grid cells in
the colouring process, all the grid cells over land are further separated
using different latitude ranges (as shown in the four columns of Fig. 7),
i.e. 90–60^{∘} N, 60–30^{∘} N, 30–0^{∘} N and 0–90^{∘} S. We find that *S*_{max} is relatively
high in wet environments ($\stackrel{\mathrm{\u203e}}{{E}_{\mathrm{o}}}/\stackrel{\mathrm{\u203e}}{P}\le \mathrm{1.0}$, Fig. 7a) but shows
no obvious relation to the partitioning of ${\mathit{\sigma}}_{P}^{\mathrm{2}}$. However, in
dry environments ($\stackrel{\mathrm{\u203e}}{{E}_{\mathrm{o}}}/\stackrel{\mathrm{\u203e}}{P}>\mathrm{1.0}$) the ratio ${\mathit{\sigma}}_{E}^{\mathrm{2}}/{\mathit{\sigma}}_{P}^{\mathrm{2}}$ apparently decreases with the increase in
*S*_{max} (Fig. 7a–d). That relation is particularly obvious in extremely
dry environments ($\stackrel{\mathrm{\u203e}}{{E}_{\mathrm{o}}}/\stackrel{\mathrm{\u203e}}{P}\ge \mathrm{6.0}$) at equatorial latitudes
where there is an upper limit of ${\mathit{\sigma}}_{E}^{\mathrm{2}}/{\mathit{\sigma}}_{P}^{\mathrm{2}}$ close to
1.0 when *S*_{max} is small (blue grid cells in Fig. 7c). The interpretation
for those extremely dry environments is that when *S*_{max} is small, ${\mathit{\sigma}}_{P}^{\mathrm{2}}$ is almost completely partitioned into ${\mathit{\sigma}}_{E}^{\mathrm{2}}$ (Fig. 7b, c)
with the other variance and covariance components close to zero. While for
those same extremely dry environments, as *S*_{max} increases, the
partitioning of ${\mathit{\sigma}}_{P}^{\mathrm{2}}$ is shared between ${\mathit{\sigma}}_{E}^{\mathrm{2}}$ and
${\mathit{\sigma}}_{\mathrm{\Delta}S}^{\mathrm{2}}$ and their covariance (Fig. 7c, k, s), while ${\mathit{\sigma}}_{Q}^{\mathrm{2}}$ and its covariance components remain close to zero (Fig. 7g, o, w).
However, at polar latitudes in the Northern Hemisphere (panels in the first
and second columns of Fig. 7) there are variations that could not be easily
associated with variations in *S*_{max}, which led us to further investigate
the role of snow/ice on the variance partitioning in the following section.

### 4.3.2 Relating inter-annual variability to mean air temperature

To understand the potential role of snow/ice in modifying the variance partitioning, we repeat the previous analysis (Fig. 7) but here we use the mean annual air temperature ($\stackrel{\mathrm{\u203e}}{{T}_{\mathrm{a}}}$) to colour the grid cells to (crudely) indicate the presence of snow/ice (Fig. 8). The results are complex and not easy to simply understand. The most important difference revealed by this analysis is in the hydrologic partitioning between cold (first column) and hot (third column) conditions in wet environments ($\stackrel{\mathrm{\u203e}}{{E}_{\mathrm{o}}}/\stackrel{\mathrm{\u203e}}{P}\le \mathrm{0.5}$). In particular, when $\stackrel{\mathrm{\u203e}}{{T}_{\mathrm{a}}}$ is high, ${\mathit{\sigma}}_{P}^{\mathrm{2}}$ is almost completely partitioned into ${\mathit{\sigma}}_{Q}^{\mathrm{2}}$ in wet environments (e.g. $\stackrel{\mathrm{\u203e}}{{E}_{\mathrm{o}}}/\stackrel{\mathrm{\u203e}}{P}\le \mathrm{0.5}$, Fig. 8g). In contrast, when $\stackrel{\mathrm{\u203e}}{{T}_{\mathrm{a}}}$ is low in a wet environment ($\stackrel{\mathrm{\u203e}}{{E}_{\mathrm{o}}}/\stackrel{\mathrm{\u203e}}{P}\le \mathrm{0.5}$ in first column of Fig. 8), there are substantial variations in the hydrologic partitioning. That result reinforces the complexity of variance partitioning in the presence of snow/ice.

## 4.4 Case studies

The previous results (Sect. 4.3) have demonstrated that the partitioning
of ${\mathit{\sigma}}_{P}^{\mathrm{2}}$ is influenced by the water storage capacity
(*S*_{max}) in extremely dry environments ($\stackrel{\mathrm{\u203e}}{{E}_{\mathrm{o}}}/\stackrel{\mathrm{\u203e}}{P}\ge \mathrm{6.0}$) and
that the presence of snow/ice is important (as indicated by mean air
temperature, $\stackrel{\mathrm{\u203e}}{{T}_{\mathrm{a}}}$) in extremely wet environments
($\stackrel{\mathrm{\u203e}}{{E}_{\mathrm{o}}}/\stackrel{\mathrm{\u203e}}{P}\le \mathrm{0.5}$). In this section, we examine, in greater
detail, several sites to gain deeper understanding of the partitioning of
${\mathit{\sigma}}_{P}^{\mathrm{2}}$. For that purpose, we selected three sites based on
extreme values for the three explanatory parameters, i.e.
$\stackrel{\mathrm{\u203e}}{{E}_{\mathrm{o}}}/\stackrel{\mathrm{\u203e}}{P}$ (Fig. S1a), *S*_{max} (Fig. S1b) and $\stackrel{\mathrm{\u203e}}{{T}_{\mathrm{a}}}$
(Fig. S1c). The criteria to select three climate sites are as follows – Site 1: dry ($\stackrel{\mathrm{\u203e}}{{E}_{\mathrm{o}}}/\stackrel{\mathrm{\u203e}}{P}\ge \mathrm{6.0}$) and small *S*_{max} (${S}_{max}\approx \mathrm{0}$), Site 2: dry ($\stackrel{\mathrm{\u203e}}{{E}_{\mathrm{o}}}/\stackrel{\mathrm{\u203e}}{P}\ge \mathrm{6.0}$) and relatively large
*S*_{max} (${S}_{max}\gg \mathrm{0}$) and Site 3: wet ($\stackrel{\mathrm{\u203e}}{{E}_{\mathrm{o}}}/\stackrel{\mathrm{\u203e}}{P}\le \mathrm{0.5}$) and hot ($\stackrel{\mathrm{\u203e}}{{T}_{\mathrm{a}}}>\mathrm{25}$ ^{∘}C). For each of the
three classes, we use a representative grid cell (Fig. 9) to show the
original time series (Fig. 10) and the partitioning of the variability (Fig. 11).

We show the *P*, *E*, *Q* and Δ*S* time series along with the relevant variances
and covariances in Fig. 10. Starting with the two dry sites, at the site
with low storage capacity (Site 1), the time series shows that *E* closely
follows *P*, leaving annual *Q* and Δ*S* close to zero (Fig. 10a). The
variance of *P* (${\mathit{\sigma}}_{P}^{\mathrm{2}}=\mathrm{206.9}$ mm^{2}) is small and almost
completely partitioned into the variance of *E* (${\mathit{\sigma}}_{E}^{\mathrm{2}}=\mathrm{196.9}$ mm^{2}), leaving very limited variance for *Q*, Δ*S* and all three
covariance components (Fig. 10b). At the dry site with larger storage
capacity (Site 2), *E*, *Q* and Δ*S* do not simply follow *P* (Fig. 10c). As a
consequence, the variance of *P* (${\mathit{\sigma}}_{P}^{\mathrm{2}}=\mathrm{2798.0}$ mm^{2}) is
shared between *E* (${\mathit{\sigma}}_{E}^{\mathrm{2}}=\mathrm{1150.2}$ mm^{2}), Δ*S* (${\mathit{\sigma}}_{\mathrm{\Delta}S}^{\mathrm{2}}=\mathrm{800.5}$ mm^{2}) and their covariance component
($\mathrm{2}\mathrm{cov}(E,\mathrm{\Delta}S)=\mathrm{538.4}$ mm^{2}, Fig. 10d). Switching now to
the remaining wet and hot site (Site 3), we note that *Q* closely follows *P*,
with Δ*S* close to zero and *E* showing little inter-annual variation
(Fig. 10e). The variance of *P* (${\mathit{\sigma}}_{P}^{\mathrm{2}}=\mathrm{57}\phantom{\rule{0.125em}{0ex}}\mathrm{374.4}$ mm^{2}) is
relatively large and almost completely partitioned into the variance of *Q*
(${\mathit{\sigma}}_{Q}^{\mathrm{2}}=\mathrm{57}\phantom{\rule{0.125em}{0ex}}\mathrm{296.4}$ mm^{2}), leaving very limited variance for
*E* and Δ*S* and the three covariance components (Fig. 10f). We also
examined numerous other sites with similar extreme conditions as the three
case study sites and found the same basic patterns as reported above.

To put the data from the three case study sites into a broader variability context we position the site data onto a backdrop of Fig. 6. As noted previously, at Site 1, the ratio ${\mathit{\sigma}}_{E}^{\mathrm{2}}/{\mathit{\sigma}}_{P}^{\mathrm{2}}$ is very close to unity (Fig. 11a), and under this extreme condition, we have the following approximation:

In contrast, for Site 2 with the same aridity index but higher *S*_{max}, we
have

Finally, at Site 3, we have

## 4.5 Synthesis

The above simple examples demonstrate that aridity $\stackrel{\mathrm{\u203e}}{{E}_{\mathrm{o}}}/\stackrel{\mathrm{\u203e}}{P}$,
storage capacity *S*_{max} and, to a lesser extent, air temperature
$\stackrel{\mathrm{\u203e}}{{T}_{\mathrm{a}}}$, all play some role in the partitioning of ${\mathit{\sigma}}_{P}^{\mathrm{2}}$
to the various components. Our synthesis of the results for the partitioning
of ${\mathit{\sigma}}_{P}^{\mathrm{2}}$ is summarized in Fig. 12. In dry environments with low
storage capacity (${S}_{max}\approx \mathrm{0}$) we have minimal runoff and expect
that ${\mathit{\sigma}}_{P}^{\mathrm{2}}$ is more or less completely partitioned into ${\mathit{\sigma}}_{E}^{\mathrm{2}}$ (Fig. 12a). In those environments, (inter-annual) variations in
storage ${\mathit{\sigma}}_{\mathrm{\Delta}S}^{\mathrm{2}}$ play a limited role in setting the overall
variability. However, in dry environments with larger storage capacity
(${S}_{max}\gg \mathrm{0}$), ${\mathit{\sigma}}_{E}^{\mathrm{2}}$ is only a small fraction of ${\mathit{\sigma}}_{P}^{\mathrm{2}}$ (Fig. 12a), leaving most of the overall variance in ${\mathit{\sigma}}_{P}^{\mathrm{2}}$ to be partitioned to ${\mathit{\sigma}}_{\mathrm{\Delta}S}^{\mathrm{2}}$ and the covariance
between *E* and Δ*S* (Fig. 12c and e). This emphasizes the
hydrological importance of water storage capacity in buffering variations of
the water cycle under dry conditions.

Under extremely wet conditions, the largest difference in variance partitioning is not due to differences in storage capacity but is instead related to differences in mean air temperature. In wet and hot environments, we have maximum runoff and find that ${\mathit{\sigma}}_{P}^{\mathrm{2}}$ is more or less completely partitioned into ${\mathit{\sigma}}_{Q}^{\mathrm{2}}$ (Fig. 12b) while the partitioning to ${\mathit{\sigma}}_{E}^{\mathrm{2}}$ and ${\mathit{\sigma}}_{\mathrm{\Delta}S}^{\mathrm{2}}$ is small. However, in wet and cold environments, the variance partitioning shows great complexity, with ${\mathit{\sigma}}_{P}^{\mathrm{2}}$ being partitioned into all possible components. We suggest that this emphasizes the hydrological importance of thermal processes (melting/freezing) under extremely cold conditions.

However, the most complex patterns to interpret are those for semi-arid to semi-humid environments (i.e. $\stackrel{\mathrm{\u203e}}{{E}_{\mathrm{o}}}/\stackrel{\mathrm{\u203e}}{P}$ ∼ 1.0). Despite a multitude of attempts over an extended period we were unable to develop a simple useful synthesis to summarize the partitioning of variability in those environments. We found that the three covariance terms all play important roles and we also found that simple environmental gradients (e.g. dry/wet, high/low storage capacity, hot/cold) could not easily explain the observed patterns. We anticipate that vegetation-related processes (e.g. phenology, rooting depth, gas exchange characteristics, disturbance) may prove to be important in explaining hydrologic variability in these biologically productive regions that support most of the human population. This result implies that a major scientific effort will be needed to develop a synthesis of the controlling factors for variability of the water cycle in these environments.

Importantly, hydrologists have long been interested in hydrologic
variability, but without readily available databases it has been difficult
to quantify water cycle variability. For example, we are not aware of maps
showing global spatial patterns in variance for any terms of the water
balance (except for *P*). In this study, we describe an initial investigation
of the inter-annual variability of the terrestrial branch in the global
water cycle that uses the recently released global monthly Climate Data
Record (CDR) database for *P*, *E*, *Q* and Δ*S*. The CDR is one of the first
dedicated hydrologic reanalysis databases and includes data for a 27-year
period. Accordingly, we could only examine hydrologic variability over this
relatively short period. Further, we expect future improvements and
modifications as the hydrologic community seeks to further develop and
refine these new reanalysis databases. With those caveats in mind, we
started this analysis by first investigating the partitioning of *P* in the
water cycle in terms of long-term mean and then extended that to the
inter-annual variability using a theoretical variance balance equation (Eq. 2). Despite the initial nature of this investigation we have been able to
establish some useful general principles.

The mean annual *P* is mostly partitioned into mean annual *E* and *Q*, as is well
known, and the results using the CDR were generally consistent with the
earlier Budyko framework (Fig. 2). Having established that, the first
general finding is that the spatial pattern in the partitioning of
inter-annual variability in the water cycle is not simply a reflection of
the spatial pattern in the partitioning of the long-term mean. In
particular, with the variance calculations, the annual anomalies are squared
and hence the storage anomalies do not cancel out like they do when
calculating the mean. With that in mind, we were surprised that the
inter-annual variability of water storage change (${\mathit{\sigma}}_{\mathrm{\Delta}S}^{\mathrm{2}}$)
is typically larger than the inter-annual variability of evapotranspiration
(${\mathit{\sigma}}_{E}^{\mathrm{2}}$) (cf. Fig. 3b and d). The consequence is that ${\mathit{\sigma}}_{\mathrm{\Delta}S}^{\mathrm{2}}$ is more important than ${\mathit{\sigma}}_{E}^{\mathrm{2}}$ for understanding
inter-annual variability of the global water cycle. A second important
generalization is that unlike the variance components which are all
positive, the three covariance components in the theory (Eq. 2) can be both
positive and negative. We report results here showing both large positive
and negative values for the three covariance terms (Fig. 3e, f, g). This was
especially prevalent in biologically productive regions ($\mathrm{0.5}<\stackrel{\mathrm{\u203e}}{{E}_{\mathrm{o}}}/\stackrel{\mathrm{\u203e}}{P}<\mathrm{1.5}$, Fig. 3e, g). When examining the mean
state, we are accustomed to think that *P* sets a limit to *E*, *Q* and Δ*S*, as
per the mass balance (Eq. 1). But the same thinking does not extend to the
variance balance since the covariance terms on the right-hand side of Eq. (2)
can be both large and negative, leading to circumstances where the
variability in the sinks (${\mathit{\sigma}}_{E}^{\mathrm{2}}$, ${\mathit{\sigma}}_{Q}^{\mathrm{2}}$, ${\mathit{\sigma}}_{\mathrm{\Delta}S}^{\mathrm{2}}$) could actually exceed variability in the source (${\mathit{\sigma}}_{P}^{\mathrm{2}}$). These general principles of variance partitioning in the water
cycle above may vary at different timescales (e.g. monthly, daily), and we
expect more details of the variability partitioning across various temporal
scales to be investigated in future studies.

Our initial attempt to develop deeper understanding of variance partitioning
was based on a series of case studies located in extreme environments
(wet/dry vs. hot/cold vs. high/low water storage capacity). The results
offered some further insights about hydrologic variability. For example,
under extremely dry (water-limited) environments, with limited storage
capacity (*S*_{max}), we found that *E* follows *P* and ${\mathit{\sigma}}_{E}^{\mathrm{2}}$ follows
${\mathit{\sigma}}_{P}^{\mathrm{2}}$, with ${\mathit{\sigma}}_{Q}^{\mathrm{2}}$ and ${\mathit{\sigma}}_{\mathrm{\Delta}S}^{\mathrm{2}}$ both
approaching zero. However, as *S*_{max} increases, the partitioning of
${\mathit{\sigma}}_{P}^{\mathrm{2}}$ progressively shifts to a balance between ${\mathit{\sigma}}_{E}^{\mathrm{2}}$, ${\mathit{\sigma}}_{\mathrm{\Delta}S}^{\mathrm{2}}$ and cov(*E*,Δ*S*) (Figs. 10–12). This
result explains the overestimation of *σ*_{E}∕*σ*_{P} by the
empirical theory of Koster and Suarez (1999) which implicitly assumed no
inter-annual change in storage. The Koster and Suarez empirical theory is
perhaps better described as an upper limit that is based on minimal storage
capacity, and that any increase in storage capacity would promote the
partitioning of ${\mathit{\sigma}}_{P}^{\mathrm{2}}$ to ${\mathit{\sigma}}_{\mathrm{\Delta}S}^{\mathrm{2}}$, particularly
under dry conditions (Figs. 10–12).

In extremely wet/hot environments (i.e. no snow/ice presence) we found ${\mathit{\sigma}}_{P}^{\mathrm{2}}$ to be mostly partitioned to ${\mathit{\sigma}}_{Q}^{\mathrm{2}}$ (with both ${\mathit{\sigma}}_{E}^{\mathrm{2}}$ and ${\mathit{\sigma}}_{\mathrm{\Delta}S}^{\mathrm{2}}$ approaching zero, Fig. 10). In contrast, in extremely wet/cold environments, the partitioning of ${\mathit{\sigma}}_{P}^{\mathrm{2}}$ was highly (spatially) variable, presumably because of spatial variability in the all-important thermal processes (freeze/melt).

The most complex results were found in mesic biologically productive
environments ($\mathrm{0.5}<\stackrel{\mathrm{\u203e}}{{E}_{\mathrm{o}}}/\stackrel{\mathrm{\u203e}}{P}<\mathrm{1.5}$), where all
three covariance terms (Eq. 2) were found to be relatively large and
therefore they all played critical roles in the overall partitioning of
variability (Fig. 6). As noted above, in many of these regions, the
(absolute) magnitudes of the covariances were actually larger than the
variances of the water balance components *E*, *Q* and Δ*S* (e.g. Fig. 3).
That result demonstrates that deeper understanding of the process-level
interactions that are embedded within each of the three covariance terms
(e.g. the role of seasonal vegetation variation) will be needed to develop
process-based understanding of variability in the water cycle in these
biologically productive regions ($\mathrm{0.5}<\stackrel{\mathrm{\u203e}}{{E}_{\mathrm{o}}}/\stackrel{\mathrm{\u203e}}{P}<\mathrm{1.5}$).

The syntheses of the long-term mean water cycle originated in 1970s (Budyko, 1974), and it took several decades for those general principles to become widely adopted in the hydrologic community. The hydrologic data needed to understand hydrologic variability are only now becoming available. With those data we can begin to develop a process-based understanding of hydrologic variability that can be used for a variety of purposes; e.g. deeper understanding of hydro-climatic behaviour, hydrologic risk analysis, climate change assessments and hydrologic sensitivity studies are just a few applications that spring to mind. The initial results presented here show that a major intellectual effort will be needed to develop a general understanding of hydrologic variability.

The global terrestrial water budget used in this study can be accessed at http://stream.princeton.edu:8080/opendap/MEaSUREs/WC_MULTISOURCES_WB_050/ (Zhang et al., 2018). The NASA/GEWEX Surface Radiation Budget (SRB) is available at https://eosweb.larc.nasa.gov/project/srb/srb_table (last access: 10 August 2018; Stackhouse et al., 2011). The global land air temperature dataset from the Climatic Research Unit (CRU TS4.01 database) can be downloaded from http://data.ceda.ac.uk/badc/cru/data/cru_ts/cru_ts_4.01 (Harris et al., 2014). The FLUXNET data are available at https://fluxnet.fluxdata.org/ (last access: 8 March 2018). The LandFluxEval, MPI and E-RUN databases used for further validation are published by Mueller et al. (2013), Jung et al. (2010) and Gudmundsson and Seneviratne (2016b), respectively.

The supplement related to this article is available online at: https://doi.org/10.5194/hess-24-381-2020-supplement.

DY and MLR designed the study and are both responsible for the integrity of the manuscript. DY performed the calculations and analyses, and prepared the original manuscript, and MLR contributed to the interpretation, discussion and writing of the manuscript.

The authors declare that they have no conflict of interest.

We thank Anna Ukkola for help in accessing the FLUXNET database. We thank the reviewers (including René Orth and two anonymous reviewers) for helpful comments that improved the manuscript.

This research has been supported by the Australian Research Council (grant nos. CE11E0098 and CE170100023) and the National Natural Science Foundation of China (grant no. 51609122).

This paper was edited by Anke Hildebrandt and reviewed by Rene Orth and two anonymous referees.

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- Abstract
- Introduction
- Methods and data
- Mean and variability of water cycle components
- Relating the variability of water cycle components to aridity
- Discussion and conclusions
- Data availability
- Author contributions
- Competing interests
- Acknowledgements
- Financial support
- Review statement
- References
- Supplement

- Abstract
- Introduction
- Methods and data
- Mean and variability of water cycle components
- Relating the variability of water cycle components to aridity
- Discussion and conclusions
- Data availability
- Author contributions
- Competing interests
- Acknowledgements
- Financial support
- Review statement
- References
- Supplement