HYDROLOGICAL PROCESSES
Hydrol. Process. 29, 1521–1534 (2015)
Published online 1 August 2014 in Wiley Online Library
(wileyonlinelibrary.com) DOI: 10.1002/hyp.10288
Bivariate frequency analysis of nonstationary low-flow series
based on the time-varying copula
Cong Jiang,1 Lihua Xiong,1* Chong-Yu Xu1,2 and Shenglian Guo1,3
1
3
State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China
2
Department of Geosciences, University of Oslo, P.O. Box 1022 Blindern, N-0315 Oslo, Norway
Hubei Provincial Collaborative Innovation Center for Water Resources Security, Wuhan University, Wuhan 430072, China
Abstract:
Many studies have analysed the nonstationarity in single hydrological variables due to changing environments. Yet, few
researches have been done to investigate how the dependence structure between different individual hydrological variables is
affected by changing environments. To investigate how the reservoirs have altered the dependence structure between river flows
at different locations on the Hanjiang River, a time-varying copula model, which takes the nonstationarity in the marginal
distribution and/or the time variation in dependence structure between different hydrological series into consideration, is
presented in this paper to perform a bivariate frequency analysis for the low-flow series from two neighbouring hydrological
gauges. The time-varying moments model with either time or reservoir index as explanatory variables is applied to build the
time-varying marginal distributions of the two low-flow series. It’s found that both marginal distributions are nonstationary, and
the reservoir index yields better performance than the time index in describing the nonstationarities in the marginal distributions.
Then, the copula with the dependence parameter expressed as a function of either time or reservoir index is applied to model the
variable dependence between the two low-flow series. The copula with reservoir index as the explanatory variable of the
dependence parameter has a better fitting performance than the copula with the constant or the time-trend dependence parameter.
Finally, the effect of the time variation in the joint distribution on three different types of joint return periods (i.e. AND, OR and
Kendall) of low flows at two neighbouring hydrological gauges is presented. Copyright © 2014 John Wiley & Sons, Ltd.
KEY WORDS
low flow; reservoir impacts; nonstationarity; dependence structure; time-varying copula; the Hanjiang River
Received 23 January 2014; Accepted 5 July 2014
INTRODUCTION
Due to changing environments (climate change and/or
human activities), the statistical characteristics of
hydrological series in watersheds might be altered, thus
leading to so-called nonstationarity. If nonstationarity in
the hydrological series is not fully taken account of, the
results of the traditional hydrological frequency analysis
based on the stationarity assumption would be invalid in
practice. Milly et al. (2008) raised the questions about
how to cope with the tasks of water management by
means of frequency analysis when the stationarity
assumption of hydrological data series is no longer valid
at the present or in the future. Under such background, the
frequency analysis of nonstationary hydrological series
has now become more and more essential for hydrology
design under changing environments (Vogel et al., 2011;
Gilroy and McCuen, 2012).
*Correspondence to: Lihua Xiong, State Key Laboratory of Water
Resources and Hydropower Engineering Science, Wuhan University,
Wuhan 430072, China.
E-mail: xionglh@whu.edu.cn
Copyright © 2014 John Wiley & Sons, Ltd.
Up to date, most researches of the nonstationary
hydrological frequency analysis are focused on how the
statistical characteristics of one single hydrological
random variable such as annual flood peak or minimum
flow are altered due to changing environments (Rasmussen, 2001; Strupczewski and Kaczmarek, 2001;
Strupczewski et al., 2001a,b; Yue et al., 2002;
Xiong and Guo, 2004; Khaliq et al., 2006; Wong et al.,
2006; Clarke, 2007; Petrow and Merz, 2009; Delgado
et al., 2010; Schmocker-Fackel and Naef, 2010; Silva
et al., 2012). Particularly, Strupczewski et al.
(2001a,2001b) and Strupczewski and Kaczmarek (2001)
presented a nonstationary approach named the timevarying moments model for nonstationary flood frequency analysis. In the time-varying moments model, the
distribution parameters of the series are usually expressed
as functions of time to reflect the nonstationarities of the
series (Khaliq et al., 2006). In addition to the direct usage
of time as the explanatory variable of the distribution
parameters, some other time-varying variables related to
the hydrological series have also been employed. For
example, the distribution parameters of flood, rainfall or
temperature series are usually modelled as functions of
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C. JIANG ET AL.
some climate indices such as Atlantic Multidecadal
Oscillation, North Atlantic Oscillation, Mediterranean
Oscillation and the Western Mediterranean Oscillation
(El Adlouni et al., 2007; Kwon et al., 2008; Villarini
et al., 2010; Villarini et al., 2012; López and Francés,
2013). These covariates may be more effective in
modelling of the nonstationarity of the series than the
time variable, since they can better reflect causal
physical mechanisms between the hydrological series
and the covariates.
As a matter of fact, the changing environments have
altered not only the statistical characteristics of some
single random variables, but also the dependence
(i.e. statistical correlation) structure between different
individual random variables. Thus, in constructing the
time-varying joint probability distribution we need to
know how the joint return period of a pair of two random
hydrological events, for instance, the joint return period
of the discharge events at both the upstream and
downstream stations on the same river larger than the
given respective thresholds, has been impacted by
changing environments. It is therefore necessary to
investigate the evolution of the dependence structure
between different individual random variables, in addition
to the time variation of the marginal distributions for each
individual random variable.
In describing the dependence structure as well as the
joint distribution of multiple hydrological random
variables, the copula function (Nelsen, 2006; Salvadori
et al., 2007) has been widely applied (De Michele and
Salvadori, 2003; Favre et al., 2004; Salvadori and
De Michele, 2004a; Grimaldi and Serinaldi, 2006;
Zhang and Singh, 2006; Zhang and Singh, 2007; Evin
and Favre, 2008; Kao and Govindaraju, 2008; Bárdossy
and Pegram, 2009; Serinaldi et al., 2009; Wang et al.,
2009; Mediero et al., 2010; Wong et al., 2010; GyasiAgyei, 2011; Laux et al., 2011; Salvadori and De
Michele, 2011; Janga and Ganguli, 2012; Requena
et al., 2013). From these researches, however, it is
found that the multivariate hydrological frequency
analysis based on the copula method has rarely
considered the possible time variation of the dependence structure or the joint distribution between the
hydrological random variables.
As one kind of large-scale human activities, reservoirs
have been found to have a profound impact on the flow
regime of the downstream (Batalla et al., 2004; Gao et al.,
2012; Li et al., 2013). Over the past decades, several
large scale reservoirs have been built in the Hanjiang
basin, the largest tributary of the Yangtze River in
China. Considering that the Hanjiang River is the source
area of the middle route of the South-to-North Water
Diversion Project of China, the impacts of these
reservoirs on the flow regime are of a great concern to
Copyright © 2014 John Wiley & Sons, Ltd.
the society. Thus, the major objective of this paper is
to investigate how the reservoirs have altered river
flows at different locations on the same river, and
more importantly, how the reservoirs have altered the
dependence structure between river flows at these
locations, by employing the time-varying copula
model to perform a bivariate frequency analysis of
the low-flow series (annual minimum monthly
streamflow) of two hydrological stations located on
the Hanjiang River.
The remainder of this paper is organized as follows.
First, the study area and data used in this study are
described. Second, the methodology of the timevarying copula model is presented. Third, the results
are presented, including the contents such as the
change-point detection for the two individual low-flow
series as well as their dependence structure, the
modelling of the nonstationary marginal distributions,
the modelling of the variation in the dependence
between the two low-flow series by the copula with
time-varying parameter, and the analysis of the time
variation in the joint return periods. Finally, the
main conclusions together with some discussion are
summarized.
STUDY AREA AND DATA
The Hanjiang River (Figure 1), which controls a
catchment area of 159 000 km2, is the largest tributary
of the Yangtze River in China. In recent decades, many
reservoirs have been built in the Hanjiang basin. The
information of five large-scale reservoirs in this basin
has been listed in Table I. Among the five reservoirs,
the largest one is the Danjiangkou Reservoir, which will
be used as the water source of the middle route of the
South-to-North Water Diversion Project in 2014. Some
studies have revealed that the mean precipitation during
1951–2003 in this basin has no obvious change (Chen
et al., 2006; Zhang et al., 2007), but the flow regime of
the Hanjiang River has been influenced by the regulation
of the reservoirs (Guo et al., 2008; Lu et al., 2009; Ma
et al., 2013).
In the study, the annual minimum monthly runoff
series of the Hanjiang River from both the Ankang
Gauge and Huangzhuang Gauge (denoted by Qa and
Qh, respectively, and shown in Figure 2) during the
period 1954–2011 are used to reveal the impact of the
construction of reservoirs on the low-flow regime of
this river. The Ankang Gauge, located about 30 km
downstream of the Ankang Reservoir, controls a
catchment area of 38 600 km2. The Huangzhuang
Gauge located about 240 km downstream of the
Danjiangkou Reservoir controls a catchment area of
142 056 km2.
Hydrol. Process. 29, 1521–1534 (2015)
NONSTATIONARY BIVARIATE FREQUENCY ANALYSIS BY THE TIME-VARYING COPULA
1523
Figure 1. The map of the Hanjiang River
Table I. Information of the five reservoirs in the Hanjiang Basin
Reservoir
Shiquan
Ankang
Huanglongtan
Danjiangkou
Yahekou
Catchment area (km2)
Total capacity (109 m3)
Completion year
23 400
0.5655
1974
35 700
3.21
1992
10 688
1.165
1978
95 220
20.97
1967
3030
1.316
1960
Figure 2. Two low-flow series and their linear trends
METHODS
Reservoir index calculation
As an indicator to represent the impact of reservoirs on
flow regimes in rivers, a dimensionless reservoir index
has been proposed by López and Francés (2013), which is
defined as
N X
Ai
Vi
RI ¼
(1)
AT
CT
i¼1
where N is the total number of reservoirs upstream of a
hydrological gauge, Ai is the catchment area controlled by
the reservoir i, AT is the total catchment area controlled by
Copyright © 2014 John Wiley & Sons, Ltd.
the hydrological gauge, Vi is the total capacity of the
reservoir i and CT is mean annual runoff at the
hydrological gauge.
In this study, a modification is made to Equation (1).
By assuming that the significance of the effect of each
individual upstream reservoir on the low-flow downstream is not only linked to the ratio of its individual
catchment area to the total upstream catchment area of
the gauging station, i.e. Ai/AT, but also to the ratio of its
individual capacity to the total upstream reservoir
capacity, i.e. Vi/VT, then the CT in Equation (1) is
replaced by the sum of the total capacity VT of
all reservoirs upstream of the hydrological gauge. Thus,
Hydrol. Process. 29, 1521–1534 (2015)
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C. JIANG ET AL.
the reservoir index RI for each hydrological gauge is
modified as follows
N X
Ai
Vi
(2)
RI ¼
AT
VT
i¼1
Framework of the time-varying copula model
In most of current multivariate hydrological frequency
analysis by the copula method, both the statistical
parameters in marginal distributions and the copula
dependence parameter are treated as constant. However,
under changing environments, either the individual
hydrological series or the dependence structure between
the different hydrological series might be nonstationary.
To allow for such possibility, a general form of the joint
distribution of the hydrological variable pair of (Y t1 ; Y t2) at
any time t should be built by a time-varying copula.
Based on the definition of the copula (Nelsen, 2006;
Salvadori et al., 2007), the time-varying copula can be
expressed as
(3)
H Y 1 ;Y 2 yt1 ; yt2 ¼ C½F 1 yt1 jθt1 ; F 2 yt2 jθt2 jθtc t t t
¼ C u1 ; u2 θc Þ
where C() represents the copula function, F() represents
the cumulative distribution function, θt1 and θt2 are the
time-varying marginal distribution parameters, θtc is the
time-varying copula parameter, and the marginal probabilities ut1 and ut2 in the time-varying copula should be both
uniformly distributed on [0,1].
According to Equation (3), three scenarios for the timevarying copula can be derived, which are: (1) all the
marginal distribution parameters are constant and the
copula parameter is time varying, (2) at least one marginal
distribution parameter is time varying and the copula
parameter is constant and (3) at least one marginal
distribution parameter and also the copula parameter are
both time varying.
The implementation of the time-varying copula model
of Equation (3) includes the two major steps: first, the
determination of the time variation in the marginal
distributions, and second, the modelling of the evolution
of the copula parameter. Figure 3 has outlined the two
main steps of the implementation of time-varying
copula, in which Model 0 corresponds to the copula
model with both stationary marginal distributions and
stationary copula dependence parameter, and Models 1,
2 and 3 correspond to the three scenarios of Equation
(3), respectively.
Time-varying marginal distribution
In this study, five probability distributions, including
four two-parameter distributions, i.e. Gamma, Weibull,
Gumbel and Lognormal (Rigby and Stasinopoulos,
2009), and a three-parameter Pearson type III distribution
(Maidment, 1993), are selected as the candidate
marginal distributions for the two low-flow series. These
probability distributions have been widely employed
in hydrological frequency analysis of low flows (Vogel
and Wilson, 1996; Modarres, 2008; Grandry et al., 2013).
The three parameters (location, scale and shape) of
the marginal distribution are generally denoted by a
Figure 3. The steps of the implementation of the time-varying copula
Copyright © 2014 John Wiley & Sons, Ltd.
Hydrol. Process. 29, 1521–1534 (2015)
NONSTATIONARY BIVARIATE FREQUENCY ANALYSIS BY THE TIME-VARYING COPULA
vector θ = (μ, σ, ν)T corresponding to the symbol used in
Equation (3).
The time-varying marginal distribution for each lowflow series is built by the time-varying moments model.
Rigby and Stasinopoulos (2005) developed a general
class of univariate regression models called Generalized
Addictive Models in Location, Scale and Shape
(GAMLSS), which has been widely applied in frequency
analysis of nonstationary hydrological series (Villarini
et al., 2009b; Villarini et al., 2009a; Villarini et al., 2010;
Villarini et al., 2012; López and Francés, 2013). In this
paper, the time-varying moments model is constructed
based on the GAMLSS framework.
Take the time-varying moment model based on a threeparameter distribution for example. If the observation
of the response variable yt at time t (t = 1, 2, …, n)
follows a distribution with probability density function fY ( yt|μt, σt, νt), and then each distribution parameter can be expressed as a linear function of the
explanatory variables xti (i = 1, 2, …, m) via a link
function as follows
g1 ðμt Þ ¼ α10 þ
m
X
α1i xti
i¼1
g2 ðσt Þ ¼ α20 þ
g3 ðνt Þ ¼ α30 þ
m
X
α2i xti
(4)
i¼1
m
X
α3i xti
i¼1
where g() is the link function, which is determined by the
domain of the statistical parameter, i.e. if the domain of the
distribution parameter θ is θ ∈ R, the link function is g
(θ) = θ, or if θ > 0, g(θ) = ln(θ). αji (j = 1, 2, 3, i = 0, 1, …, m)
are the GAMLSS parameters, which are represented by the
vector α = (α10, …, α1m, α20, …, α2m, α30, …, α3m)T. For a
bivariate frequency analysis, the GAMLSS parameter
vectors in two marginal distributions are defined as α1 and
α2, respectively.
In this study, time and reservoir indexes are separately
introduced as the explanatory variable of the marginal
distribution parameters. Equation (4) gives the general
form of a time-varying marginal distribution, in which all
distribution parameters θ = (μ, σ, ν)T could be time
varying. In practice, only the nonstationarities in the first
two moments are considered, and the shape parameter of
the distribution is often treated as a constant
(Strupczewski et al., 2001a; Renard et al., 2006; El
Adlouni et al., 2007; Gilroy and McCuen, 2012). As done
by Villarini et al. (2009a), a total of four different
marginal distribution models can be considered in the
nonstationarity analysis: (1) all distribution parameters are
constant, (2) only the location parameter is time varying,
Copyright © 2014 John Wiley & Sons, Ltd.
1525
(3) only the scale parameter is time varying and (4) both
the location and scale parameters are time varying. The
final marginal distribution model is selected from the four
models above by comparing the value of the Corrected
Akaike Information Criterion (AICc; Hurvich and Tsai,
1989), which is stricter than the Akaike Information
Criterion (AIC; Akaike, 1974).
Copula with time-varying dependence parameter
In multivariate hydrological frequency analysis, the
Archimedean copulas are widely used for a number of
reasons, such as the ease of construction and the great
variety of copula families (Nelsen, 2006). Recently, a
more flexible multi-parameters copula named extraparameterized Khoudraji-Liebscher’s family has been
applied in hydrological frequency analysis (Salvadori
and De Michele, 2010; Salvadori et al., 2011; De Michele
et al., 2013; Salvadori et al., 2013). In this paper, for
the ease of parameter estimation and avoiding overparameterization, three simple mono-parameter
Archimedean copulas, i.e., Gumbel-Hougaard (GH),
Frank and Clayton copula, which have been used to
build the joint distribution of the low flows (Zhang
et al., 2011; Kuchment and Demidov, 2013), are
selected as the candidates in modelling the timevarying dependence between the two low-flow
series.
Similar to the parameters of the marginal distributions
as expressed in Equation (4), the copula parameter θc can
also be expressed as a linear function of the time-varying
explanatory variables xti (i = 1, 2, …, m) via a proper link
function gc() as follows
m
X
gc θtc ¼ β0 þ
βi xti
(5)
i¼1
where β0, β1, …, βm are the parameters, which are
represented by the vector β = (β0, β1, …, βm)T. The link
function gc() depends on the domain of the copula
parameter, i.e. if θc ∈ R, gc(θc) = θc (for Frank copula), or
if θc > 0, gc(θc) = log(θc) (for GH and Clayton copula).
Similarly, time or reservoir index is separately
introduced as the explanatory variable xti of the copula
parameter to model the evolution process of the
dependence structure between the two low-flow series.
The copula with the dependence parameter expressed as a
function of time t is given as follows:
(6)
gc θ tc ¼ β0 þ β1 t
The presence of the variation with time in the copula
dependence parameter can be determined by comparing
the AICc value of the copula with Equation (6) versus that
of the constant dependence parameter copula.
Hydrol. Process. 29, 1521–1534 (2015)
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C. JIANG ET AL.
In addition, the copula with reservoir index as
explanatory variable is selected by comparing the AICc
values from the totally four scenarios, which are listed
as follows:
(7)
gc θtc ¼ β0
gc θtc ¼ β0 þ β1 Rta
(8)
gc θtc ¼ β0 þ β1 Rth
(9)
gc θtc ¼ β0 þ β1 Rta þ β2 Rth
(10)
The parameters in a time-varying joint distribution
include both the two parameter vectors α1 and α2 in
modelling the time-varying marginal distributions and the
parameter vector β in modelling the time-varying copula
parameter, which are estimated by Inference Function for
Margins (IFM) method (Joe, 1997).
The goodness-of-fit of the time-varying copula model
is examined by testing the goodness-of-fit of both the two
marginal distributions and the copula function. The
goodness-of-fit test for marginal distributions is
performed by the bootstrap Kolmogorov–Smirnov (KS)
test (Wang et al., 2011) at the 5% significance level and
worm plot (van Buuren and Fredriks, 2001), respectively.
The goodness-of-fit test for the copula is performed by
Rosenblatt’s probability integral transform (Rosenblatt,
1952; Genest et al., 2009).
Joint return period under nonstationary framework
Now, there have been three methods used in calculating
the joint return period (JPR) of low-flow events in the
stationary bivariate frequency analysis, i.e. AND method
corresponding to the probability of P(Y1 ≤ y1 ∧ Y2 ≤ y2),
OR method corresponding to the probability of P
(Y1 ≤ y1 ∨ Y2 ≤ y2), and Kendall (KEN) return period
method. Kendall return period is a multivariate return
period first defined by Salvadori and De Michele (2004b),
which has been widely applied in analyzing the joint
return period of floods or droughts (Salvadori et al., 2007;
Salvadori and De Michele, 2010; Salvadori et al., 2011;
De Michele et al., 2013; Gräler et al., 2013; Salvadori
et al., 2013). The JPR of the KEN method for low-flow
events is given by
λ
P½Cðu1 ; u2 jθc Þ ≤ pKEN λ
(11)
¼
PfC½F 1 ðy1 jθ1 Þ; F 2 ðy2 jθ2 Þjθc < pKEN g
λ
¼
K C ð pKEN Þ
T KEN ¼
Copyright © 2014 John Wiley & Sons, Ltd.
where λ is the average interarrival time between low-flow
event occurrences. In this paper, the annual minimum
low-flow series is investigated, so λ should be equal to 1
(i.e. λ = 1). KC() is the Kendall distribution function
(Genest and Rivest, 1993; Barbe et al., 1996; Salvadori
et al., 2007), which is a univariate representation of
multivariate information, and pKEN is a critical probability
level corresponding to the value of KC(pKEN) (Salvadori
et al., 2011; Salvadori et al., 2013).
Similar to the calculation of the JPR of low-flow events
in the stationary bivariate frequency analysis, the JPRs of
AND, OR and KEN in nonstationary circumstances are
defined as follows
1
(12)
T tAND ¼
PðY 1 ≤ y1 ∧Y 2 ≤ y2 Þ
1
¼ t
C F 1 y1 θ1 Þ; F 2 y2 θt2 Þθtc T tOR ¼
1
PðY 1 ≤ y1 ∨Y 2 ≤ y2 Þ
¼
1
PðY 1 ≤y1 Þ þ PðY 2 ≤ y2 Þ PðY 1 ≤ y1 ∧Y 2 ≤ y2 Þ
¼
1
F 1 y1 θt1 Þ þ F 2 y2 θt2 Þ C F 1 y1 θt1 Þ; F 2 y2 θt2 Þθtc (13)
1
P½Cðut1 ; ut2 jθtc Þ ≤ pKEN 1
(14)
¼ t
P C F 1 y1 jθ1 ; F 2 y2 jθt2 jθtc < pKEN
1
¼ t
K C ðpKEN Þ
T tKEN ¼
RESULTS
Univariate and bivariate change-point analysis
In this section, the nonstationarities in the two individual
low-flow series at Ankang Gauge and Huangzhuang
Gauge, as well as in the dependence structure between
the two low flows are examined. The nonparametric Pettitt
test (Pettitt, 1979) is applied to investigate the presence of
abrupt changes in mean and variance of both the low-flow
series Qa and Qh. According to the results of the Pettitt test,
the variance of Qa presents a significant abrupt change in
1991, while the mean of Qa is stationary. For Qh, the
change point in the mean is found to occur in 1971 and the
change point in the variance is in 1972. Using the method
of detecting the change point in multivariate time series
described by Lavielle and Teyssière (2006), it is found that
Hydrol. Process. 29, 1521–1534 (2015)
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NONSTATIONARY BIVARIATE FREQUENCY ANALYSIS BY THE TIME-VARYING COPULA
a significant change point in the dependence structure
between Qa and Qh took place in 1970. These preliminary
analyses just demonstrate that both the two individual low
flow series and the dependence structure between them are
all nonstationary.
Huangzhuang Gauge Qh are first estimated under
stationary assumption. The results of the five distributions
in fitting the two series have been summarized in Table II.
Under the stationary assumption, the Weibull distribution
and the Gamma distribution have the smallest AICc
values in fitting Qa and Qh, respectively.
Reservoir index
The reservoir indexes for the Ankang Gauge and
Huangzhuang Gauge are denoted by RIa and RIh,
respectively, and calculated by Equation (2). The
Danjiangkou Reservoir was put into operation in November
1967, while the minimum monthly runoff of this year at
Huangzhuang Gauge occurred in January; thus, the
Danjiangkou Reservoir should have its impacts on the low
flow at the Huangzhuang Gauge from 1968 on. As shown in
Figure 4, in 1974 and 1992, RIa had two change points
because of the construction of the Shiquan Reservoir and the
Ankang Reservoir. The capacities of Danjiangkou
Reservoir and the Ankang Reservoir are far larger than
those of the other three reservoirs; hence, the variation in RIh
mainly occurred in 1968 and 1992, the years when the two
reservoirs were put into operation, respectively.
(2) Time-varying marginal distributions with time as
the explanatory variable
The results of time-varying marginal distributions with
time as the explanatory variable of the distribution
parameters are summarized in Table III. In this case,
according to AICc values, the Gamma distribution is the
best model for the low-flow series Qa and the Lognormal
distribution the best model for the low-flow series Qh. The
location parameter of the Gamma distribution for
describing Qa is constant, while the scale parameter has
an increasing temporal trend. As for Qh which follows the
Lognormal distribution, the location parameter has an
increasing temporal trend, while scale parameter is
constant. The results of the model selections mean that
both the two low-flow series are nonstationary.
Modelling the marginal distributions
(3) Time-varying marginal distributions with reservoir
index as the explanatory variable
(1) Stationary marginal distributions
The distribution parameters of the low-flow series at
the Ankang Gauge Qa and the low-flow series at the
Figure 4. The variations in both the reservoir index of Ankang Gauge
(RIa) and the reservoir index at Huangzhuang Gauge (RIh)
Table IV presents the results of the time-varying
marginal distributions with the reservoir index as
explanatory variable of the distribution parameters.
According to AICc values, the Gamma distribution is
selected for fitting both Qa and Qh. Also, it is found
that, in terms of the AICc values, the time-varying
marginal distribution with reservoir index as explanatory variable is better than that with time as the
explanatory variable in modelling the both two
nonstationary low-flow series. This suggests that the
nonstationarity in the low-flow series is more inclined
to be abrupt change rather than trend. Consequently,
the reservoir index is selected as the explanatory
variable to model time-varying marginal distributions
of the low-flow series. The KS test and worm plot
(Figure 5) indicate that the selected models both have a
quite good fitting quality.
Table II. Performance of the five distributions in fitting the two low-flow series under stationary assumption (P-KS is the P value of
bootstrap KS test, and a P-KS bigger than 0.05 means that the distribution passes the goodness-of-fit test at 5% significance level)
PIII
Series
Qa
Qh
Gamma
Weibull
Gumbel
Lognormal
AICc
P-KS
AICc
P-KS
AICc
P-KS
AICc
P-KS
AICc
P-KS
576.11
802.90
0.743
0.915
575.68
801.68
0.970
0.881
574.64
803.47
0.523
0.646
587.30
822.29
0.182
0.2305
579.83
802.44
0.855
0.954
Copyright © 2014 John Wiley & Sons, Ltd.
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Table III. Results of the marginal distributions with time as the explanatory variable of distribution parameters (t = 1, 2, …, 58)
Distribution parameters
Series
Qa
Qh
Probability
distribution
μ
σ
ν
AICc
P-KS
PIII
Gamma
Weibull
Gumbel
Lognormal
PIII
Gamma
Weibull
Gumbel
Lognormal
106.35
106.38
116.05
119.1
4.633
exp(5.949 + 0.0129t)
exp(5.956 + 0.0126t)
exp(6.271 + 0.00671t)
725.7
5.853 + 0.0134t
exp(1.846 + 0.0189t)
exp(1.829 + 0.0193t)
exp(1.962 0.0200t)
exp(2.761 + 0.0215t)
exp(1.829 + 0.0205t)
0.394
0.419
exp(0.505 + 0.0168t)
exp(5.924 0.0146t)
0.395
0.073
566.00
564.27
564.68
574.84
566.64
790.13
791.12
796.16
819.37
788.50
0.872
0.772
0.672
0.265
0.803
0.745
0.491
0.305
0.016
0.765
0.630
Table IV. Results of the marginal distributions with reservoir index as the explanatory variable of distribution parameters
Distribution parameters
Series
Qa
Qh
Probability
distribution
μ
σ
ν
AICc
P-KS
PIII
Gamma
Weibull
Gumbel
Lognormal
PIII
Gamma
Weibull
Gumbel
Lognormal
106.10
106.06
115.70
94.19
4.633
exp(5.793 + 1.271RIh)
exp(5.78 + 1.301RIh)
exp(5.898 + 1.273RIh)
365.5 + 759.4RIh
5.741 + 1.251RIh
exp(1.606 + 0.900RIa)
exp(1.606 + 0.941RIa)
exp(1.686 0.905RIa)
exp(2.925 + 0.987RIa)
exp(1.601 + 1.020RIa)
0.351
0.343
3.245
exp(4.411 + 1.958RIh)
0.353
0.199
562.25
560.36
562.50
570.56
561.96
776.46
774.25
775.58
788.75
775.29
0.907
0.798
0.721
0.322
0.858
0.911
0.851
0.835
0.334
0.841
0.439
Figure 5. Worm plots in the goodness-of-fit test for the two time-varying marginal distributions with reservoir index as the explanatory variable of the
distribution parameters. (a) is the worm plot for the marginal distribution of the low-flow series at the Ankang Gauge; (b) is for the Huangzhuang Gauge.
The area between the two arc dotted lines corresponds to the 95% confidence interval. As all points fall within the 95% confidence interval, so the two
marginal distributions both have a quite good fitting quality
As shown in Table IV, the scale parameter of the
Gamma distribution for describing Qa relates to the
reservoir index RIa positively, whereas the location
parameter is constant. In the period 1954–1973, when
there was no reservoir constructed upstream the Ankang
Gauge, the coefficient of variation of Qa, denoted by Cv
Copyright © 2014 John Wiley & Sons, Ltd.
(Qa), was 0.201, and then there was a slight increase of
0.019 in 1974 due to the construction of the Shiquan
Reservoir. Further in 1992, the Ankang Reservoir, whose
capacity and catchment area are both significantly larger
than those of the Shiquan Reservoir, was finished, and
thus the coefficient of variation had a sharp increase from
Hydrol. Process. 29, 1521–1534 (2015)
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NONSTATIONARY BIVARIATE FREQUENCY ANALYSIS BY THE TIME-VARYING COPULA
0.220 to 0.482. This change in Cv(Qa) has been
displayed in Figure 6(a). Also, according to Table IV,
the location parameter of the Gamma distribution for
describing Qh is positively related to the reservoir index
RIh, whereas the scale parameter is constant. As shown
in Figure 6(b), the most significant abrupt change of Qh
was in 1968 when the Danjiangkou Reservoir began to
impound, and another visible abrupt change occurred in
1992, because of the construction of the Ankang
Reservoir. Specifically, before 1959 the mean of the
series Qh were 324 m3/s. Due to the regulation of
Danjiangkou Reservoir in 1968, the mean of Qh abruptly
increased to 634 m3/s with an increase of 95.8%. After
1992, the mean of Qh further increased to 664 m3/s. In
general, it is mainly the Ankang Reservoir and
Danjiangkou Reservoir that have led to the
nonstationarities in Qa and Qh, and the effects of the
other three reservoirs can be negligible. It should be
noted that the results of nonstationary analysis for the
series Qa and Qh presented in Figure 6 are roughly
consistent with the results of the Pettitt change-point
analysis for these two low-flow series.
Time-varying copulas
Since the two marginal distributions are found to be
both time varying, we just need to consider the timevarying copulas in the scenarios of Model 2 and Model 3.
(1) Model 2
The results of three copulas, i.e. GH, Frank and
Clayton, all with constant dependence parameter to model
the dependence structure between low-flow series at the
Ankang Gauge Qa and the Huangzhuang Gauge Qh, are
summarized in Table V, where it can be seen that Frank
copula has the smallest AICc value, followed by the GH
and Clayton copula.
Table V. Results of three copulas in modelling the dependence
structure between the two low-flow series in the case of Model 2.
The fourth and fifth columns are the P values of KS test for the
two Rosenblatt’s probability integral transformations Z1 and Z2,
which should be uniformly and independently distributed on
[0,1]. P-Kendall is the P value of the Kendall rank correction test
for Z1 and Z2. In this case, if the P-KS of Z1 and Z2, and P-Kendall
are both bigger than 0.05, Z1 and Z2 will be regarded uniformly
independently distributed on [0,1]
Copula
θc
AICc
GH
1.796 26.84
Frank
4.723 28.96
Clayton 0.808 14.45
P-KS of Z1 P-KS of Z2 P-Kendall
0.798
0.798
0.798
0.966
0.729
0.615
0.877
0.732
0.082
(2) Model 3 with time as the explanatory variable
Table VI summarizes the results of the copulas with a
time-trend dependence parameter in modelling the dependence structure between Qa and Qh. The copula with the
smallest AICc value is Frank copula, followed by GH and
Clayton copula. In terms of AICc values (shown in Table V
and Table VI), the Frank copula with time-trend dependence
parameter is better than that with constant dependence
parameter in modelling the dependence structure between
Qa and Qh. Thus, it can be concluded that the dependence
structure between Qa and Qh is time varying. As the Frank
copula parameter presents a decreasing temporal trend, it
means that the dependence between the two low-flow series
may weaken with time.
(3) Model 3 with reservoir index as the explanatory
variable
The results of the Model 3 with the reservoir index as
the explanatory variable of the copula parameter are
summarized in Table VII. It can be found that Frank
Figure 6. Results of the modelling of the variation in the marginal distributions of the two low-flow series with time or reservoir index as the explanatory
variable of the distribution parameters. (a) is the quantile plot for the low-flow series at the Ankang Gauge; (b) is for the Huangzhuang Gauge. The model
with time as the explanatory variable displays a smoothing trend variation while the model with reservoir index as the explanatory variable displays
abrupt changes
Copyright © 2014 John Wiley & Sons, Ltd.
Hydrol. Process. 29, 1521–1534 (2015)
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C. JIANG ET AL.
Table VI. Results of three copulas with time as explanatory variable in modelling the dependence structure between the two low-flow
series in the case of Model 3
Copula
θc
AICc
P-KS
of Z1
P-KS
of Z2
P-Kendall
GH
Frank
Clayton
exp(0.772 0.007t)
7.580 0.091t
exp(0.442 0.022t)
27.50
29.41
14.33
0.798
0.798
0.798
0.842
0.763
0.423
0.663
0.542
0.026
Table VII. Results of three copulas with reservoir index as explanatory variable in modelling the dependence structure between the two
low-flow series in the case of Model 3
Copula
θc
AICc
P-KS
of Z1
P-KS
of Z2
P-Kendall
GH
Frank
Clayton
exp(0.970 0.946RIh)
9.096 9.753RIh
exp(0.461 1.681RIh)
30.16
31.10
14.61
0.798
0.798
0.798
0.919
0.545
0.492
0.995
0.773
0.165
copula still has the smallest AICc value, followed by GH
and Clayton copula. The comparison of the AICc values
displayed in Tables VI and VII demonstrates that RIh is
more effective than time in modelling the evolution of the
Frank copula parameter. For this reason, the Frank copula
with RIh as the explanatory variable of the dependence
parameter is determined to model the time-varying
dependence structure between Q a and Q h . The
goodness-of-fit test (Table VII and Figure 7) also
demonstrates that the selected copula model has a
satisfactory fitting performance.
As shown in Figure 8, the major fall in the value of the
dependence parameter of the Frank copula took place in
1968, when the Danjiangkou Reservoir began to
impound, that is roughly consistent with the result of
the change-point detection for the dependence structure
between Qa and Qh. Another visible change point in the
Frank copula parameter was at 1992, the year when
Ankang Reservoir was completed. This reveals that it is
the regulation of the reservoirs that weakens dependence
Figure 8. Evolution of the dependence parameter of the Frank copula with
RIh as the explanatory variable, and also evolutions of both the Kendall’s
tau and Spearman’s rho
Figure 7. Worm plots in the goodness-of-fit test for the copula with reservoir index as the explanatory variable of the dependence parameter. (a) is the
worm plot for the Rosenblatt’s probability integral transformation Z1; (b) is the worm plot for the Rosenblatt’s probability integral transformation Z2. As
all points falls within the 95% confidence interval, so the two probability integral transformations are both uniformly distributed on [0,1]
Copyright © 2014 John Wiley & Sons, Ltd.
Hydrol. Process. 29, 1521–1534 (2015)
NONSTATIONARY BIVARIATE FREQUENCY ANALYSIS BY THE TIME-VARYING COPULA
1531
structure between Qa and Qh, particularly by the
Danjiangkou Reservoir and Ankang Reservoir. Taking
the years 1968 and 1992 as dividing points, each of the
two low-flow series can be divided into three time
segments. As presented in Figure 8, the evolutions in
Kendall’s tau and Spearman’s rho (Nelsen, 2006;
Salvadori et al., 2007) are consistent with the evolution
of the dependence parameter of the Frank copula.
Time variation in the joint return periods
On the basis of the analysis above, the time-varying
joint distribution of the two low flows (Qa,Qh) can be
expressed as the time-varying copula as follows
H Y 1 ;Y 2 Qta ; Qth ¼ C Fr F Ga Qta θta Þ; F Ga Qth θta Þθtc (15)
where FGa() is the Gamma distribution for describing
both Qa and Qh; CFr() stands for the Frank copula. The
time-varying distribution parameters of the two marginal
distributions are given as follows
T
(16)
θta ¼ 106:06; exp 1:606 þ 0:941RI ta
T
θth ¼ exp 5:780 þ 1:301RI th ; 0:343
(17)
The dependence parameter θtc of the Frank copula is
given by
θtc ¼ 9:045-9:644RI th
(18)
Figure 9 presents the evolutions of the JPRs of AND,
OR and KEN under the nonstationary framework
expressed by Equation (15), together with the evolutions
of the Qa and Qh series. It is shown that the three joint
Figure 10. The isolines of design low-flow events with JRP = 50 years for three
different time periods of 1954–1967, 1968–1991 and 1992–2011
return periods are different for the same low-flow pair and
in general TOR < TKEN < TAND.
Based on the above analysis, we have known that only
the Danjiangkou Reservoir and the Ankang Reservoir
have had the significant impacts on the nonstationarities
in both marginal distributions of the two low-flow series
and their dependence structure. Thus, the whole time
period of 1954–2011 can be divided into three time
segments by the years of 1968 and 1992, when the
Danjiangkou Reservoir and Ankang Reservoir were put
into operation, respectively. Within each time segment
(1954–1967, 1968–1991 and 1992–2011) the joint
distribution of the low flows at the Ankang Gauge and
Huangzhuang Gauge can be treated to be stationary. In
order to investigate how the regulation of the reservoirs
affects the joint return period of the low flows at the two
neighbouring hydrological gauges, the design low-flow
pairs for the given design JRP of 50 years are calculated
according to the joint probability distribution for each of
three time segments. The JRP isolines of the design lowflow pairs are presented in Figure 10. It can be seen that,
for the AND method, from the period of 1954–1967 to
the period of 1968–1991, the JRP isoline moves
horizontally to the right, due to the increase in the mean
of Huangzhuang low flow. Then, from the period of
1968–1991 to the period of 1992–2011, the JRP isoline
moves downward, due to the increase in the coefficient of
variation of Ankang low flow. Meanwhile, with the
decrease of the θc value, the corner of the isoline becomes
smoother because of the weakening in dependence
between the two low-flow series. The JPR isolines of
both OR and KEN methods also present the similar
variation to the JPR isolines of AND.
CONCLUSIONS AND DISCUSSION
Figure 9. Evolutions of the three types of joint return period (AND, OR
and KEN) of the observed low-flow pair (Qa,Qh) together with the
evolutions of observed low flows Qa and Qh
Copyright © 2014 John Wiley & Sons, Ltd.
To investigate how the reservoirs have altered river flows
at different locations of the Hanjiang River as well as the
Hydrol. Process. 29, 1521–1534 (2015)
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C. JIANG ET AL.
dependence structure between the river flows at these
locations, a time-varying copula model is presented in this
paper to model the variable dependence structure between
two nonstationary low-flow series from the Ankang
Gauge and Huangzhuang Gauge, which are both located
at the Hanjiang River. The main conclusions are
presented as follows.
In general, the above findings have highlighted the
importance of considering the time variation of the joint
distribution in the multivariate hydrological frequency
analysis under changing environments. As to using the
time-varying copula in modelling the time variation of the
dependence structure between multiple hydrological
random variables, some comments are made as follows.
1. Both marginal distributions of the low-flow series from
the Ankang Gauge and Huangzhuang Gauge have been
found to be nonstationary; thus, the method of the
marginal distribution parameter estimation based on
stationary assumption is no longer suitable under
nonstationary framework. In modelling the time variation
in the marginal distribution parameters, the reservoir
index performs better than using time as the explanatory
variable. This indicates that the nonstationarities in the
marginal distributions are caused by the construction of
the reservoirs in this basin. It is also found that the
effect of the reservoirs on the two low flows are
different, i.e. for the low-flow series at the Ankang
Gauge, only the coefficient of variation is enlarged,
while for the low-flow series at the Huangzhuang
Gauge, only the mean is enlarged.
2. Both the copula with the dependence parameter
expressed as the function of time and the copula with
the dependence parameter expressed as the function of
the reservoir index have a better performance than the
copula with constant parameter, which means the
dependence structure between the two low-flow series
can be regarded as time varying. Compared with the
time index, the reservoir index has a better performance in modelling the evolution of the dependence
between the two low-flow series at the Ankang Gauge
and the Huangzhuang Gauge, which means that the
construction of reservoirs in this basin is responsible
for the variation of the dependence structure between
the two low-flow series. In terms of the capacity and
catchment area, the Danjiangkou Reservoir plays the
dominant role in leading to the variation in the
dependence structure between the two low-flow
series. The Ankang Reservoir has less degree of
impact on the dependence structure between the two
low-flow series than the Danjiangkou Reservoir, and
the other three reservoirs nearly have no influence on
the dependence structure.
3. Due to the nonstationarities in both the two marginal
distributions of the two low flows at the Ankang
Gauge and the Huangzhuang Gauge and the
dependence structure between these two low flows,
the design low flows corresponding to given design
joint return periods have variations over time, for all
three different JRP calculation methods (AND, OR
and KEN).
1. In this paper, only one explanatory variable, i.e. reservoir
index, is employed to characterize the time variations in
both the marginal distributions and copula parameter in
describing a pair of low-flow events. Since the
hydrological event is the comprehensive outcome of
multi-factors, it should be more persuasive to consider
more explanatory variables that have the physical
relations to the hydrological events of concern.
2. The tail dependence is very important in estimating
the risk of the multivariate hydrological extremes
(Poulin et al., 2007). It’s known that different copulas
have different abilities in characterizing the tail dependence (Nelsen, 2006). For example, the GH copula is
more suitable to model the dependence structure between
the series with upper-tail dependence, while the Clayton
has a stronger ability in characterizing the dependence
structure between the series with lower-tail dependence.
For modelling the time-varying tail dependence between
the hydrological series, the impact of the type of
the copula should be taken into consideration in more
detail in future researches.
Copyright © 2014 John Wiley & Sons, Ltd.
ACKNOWLEDGEMENTS
This research is supported by the National Natural Science
Foundation of China (Grant Nos. 51190094 and 51079098),
which is greatly appreciated. Great thanks are due to the
reviewers for a number of very constructive comments and
suggestions that have helped the substantial improvement of
the manuscript.
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