Journal of Hydrology 527 (2015) 234–250
Contents lists available at ScienceDirect
Journal of Hydrology
journal homepage: www.elsevier.com/locate/jhydrol
Return period and risk analysis of nonstationary low-flow series
under climate change
Tao Du a, Lihua Xiong a,⇑, Chong-Yu Xu a,b, Christopher J. Gippel c, Shenglian Guo a,d, Pan Liu a
a
State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China
Department of Geosciences, University of Oslo, P.O. Box 1022 Blindern, N-0315 Oslo, Norway
c
Australian Rivers Institute, Griffith University, Nathan, Queensland 4111, Australia
d
Hubei Provincial Collaborative Innovation Center for Water Resources Security, Wuhan University, Wuhan 430072, China
b
a r t i c l e
i n f o
Article history:
Received 2 December 2014
Received in revised form 3 April 2015
Accepted 20 April 2015
Available online 7 May 2015
This manuscript was handled by Andras
Bardossy, Editor-in-Chief, with the
assistance of Peter F. Rasmussen, Associate
Editor
Keywords:
Return period
Risk
Nonstationarity
Low-flow
General Circulation Models (GCMs)
s u m m a r y
Return period and risk of extreme hydrological events are critical considerations in water resources management. The stationarity assumption of extreme events for conducting hydrological frequency analysis
to estimate return period and risk is now problematic due to climate change. Two different interpretations of return period, i.e. the expected waiting time (EWT) and expected number of exceedances
(ENE), have already been proposed in literature to consider nonstationarity in return period and risk analysis by introducing the time-varying moment method into frequency analysis, under the assumption that
the statistical parameters are functions only of time. This paper aimed at improving the characterization
of nonstationary return period and risk under the ENE interpretation by employing meteorological
covariates in the nonstationary frequency analysis. The advantage of the method is that the downscaled
meteorological variables from the General Circulation Models (GCMs) can be used to calculate the nonstationary statistical parameters and exceedance probabilities for future years and thus the corresponding return period and risk. The traditional approach using time as the only covariate under both the EWT
and ENE interpretations was also applied for comparison. Both approaches were applied to annual minimum monthly streamflow series of two stations in the Wei River, China, and gave estimates of nonstationary return period and risk that were significantly different from the stationary case. The
nonstationary return period and risk under the ENE interpretation using meteorological covariates were
found more reasonable and advisable than those of the EWT and ENE cases using time alone as covariate.
It is concluded that return period and risk analysis of nonstationary low-flow series can be helpful to
water resources management during dry seasons exacerbated by climate change.
Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction
Statistical inference is used in hydrological frequency analysis
under the assumption that hydrological events such as flood and
drought are randomly distributed through time. A precondition
for traditional frequency analysis of hydrological variables is the
assumption of stationarity, which means that the controlling environmental factors such as climate and land cover act to generate or
modify the hydrological variable of interest in the same way in the
past, present and future (Gilroy and McCuen, 2012; Katz, 2013;
Benyahya et al., 2014; Charles and Patrick, 2014). However, under
the conditions of climate change, land use change and river
⇑ Corresponding author. Tel.: +86 13871078660; fax: +86 27 68773568.
E-mail addresses: dtgege@126.com (T. Du), xionglh@whu.edu.cn (L. Xiong),
c.y.xu@geo.uio.no (C.-Y. Xu), fluvialsystems@fastmail.net (C.J. Gippel), slguo@whu.
edu.cn (S. Guo), liupan@whu.edu.cn (P. Liu).
http://dx.doi.org/10.1016/j.jhydrol.2015.04.041
0022-1694/Ó 2015 Elsevier B.V. All rights reserved.
regulation, acting individually or together, the assumption of stationarity is suspected, and existing methodology may no longer
be valid (Katz et al., 2002; Xiong and Guo, 2004; Milly et al.,
2005, 2008). To overcome this problem, various approaches have
been developed for conducting nonstationary hydrological frequency analysis (Khaliq et al., 2006). The concept of event return
period (or recurrence interval) and the associated risk of occurrence, with potential consequences of loss of life, social disruption,
economic loss and ecological disturbance, are critical considerations in the management of water resources, especially with regard
to design and operation of hydraulic structures in rivers. Under
nonstationary conditions, estimates of return period and risk are
ambiguous unless the temporally changing environment is explicitly considered in the analysis.
The most common way of handling nonstationarity in hydrological time series is the method of time-varying moment, which
assumes that although the distribution function type of the
T. Du et al. / Journal of Hydrology 527 (2015) 234–250
hydrological variable of interest remains the same, the statistical
parameters are time-varying (Strupczewski et al., 2001; Coles,
2001; Katz et al., 2002; Villarini et al., 2009; Gilroy and McCuen,
2012; Jiang et al., 2015a). With the method of time-varying
moment, it is not difficult to derive a value of a hydrological variable for a return period, with a specific design quantile (Olsen et al.,
1998; Villarini et al., 2009), i.e. T t ¼ 1=pt ¼ 1=ð1 F Z ðzp0 ; ht ÞÞ,
where Tt and pt are the annual return period and exceedance probability, respectively, of the given design quantile zp0 with fitted
annual statistical parameters ht, and FZ is the cumulative distribution function of the hydrological variable of interest. However, for
many planning and design applications, a measure of return period
that varies from one year to the next is impractical. To deal with
this issue, various studies have been carried out for return period
estimation and risk analysis that consider nonstationary conditions
(Wigley, 1988, 2009; Olsen et al., 1998; Parey et al., 2007, 2010;
Cooley, 2013; Salas and Obeysekera, 2013, 2014). Among these
various studies two different interpretations of return period have
already been proposed. The first is that the expected waiting time
(EWT) until the next exceedance event is T years (Wigley, 1988,
2009; Olsen et al., 1998; Cooley, 2013; Salas and Obeysekera,
2013, 2014), and the second is that the expected number of exceedances (ENE) of the event in T years is 1 (Parey et al., 2007, 2010;
Cooley, 2013).
Wigley (1988, 2009) used the EWT interpretation to consider
how nonstationarity can be included in the concepts of return period and risk of extreme events. A normal distribution with a linear
increasing trend in the mean was assumed, and the changes in the
return period and risk were derived by the technique of stochastic
simulation. Building on the work of Wigley (1988), Olsen et al.
(1998) presented a more rigorous mathematical examination of
the effect of nonstationarity on the concepts of return period and
risk, also using the EWT interpretation. Parey et al. (2007, 2010)
introduced the ENE interpretation into the nonstationary framework to derive return levels of air temperature in France. A detailed
review and comparison of the two interpretations of return period
can be found in Cooley (2013). Recently, Salas and Obeysekera
(2013, 2014) extended the geometric distribution to allow for
changing exceedance probabilities over time, considering the cases
of increasing, decreasing and shifting extreme events. Although
these studies vary considerably in their scope and the methodologies employed, they have in common the assumption that the statistical parameters are functions only of time. However, this carries
the unreasonable implication that the identified pattern of nonstationarity in a hydrological time series will continue indefinitely.
Also, while runoff can follow inter-year cyclical patterns, the lack
of a direct physical link between time and runoff means that time
alone is not quite sufficient as an explanatory variable.
The time-varying moment method can be extended to perform
covariate analysis by replacing time with any physical factors that
are known to be causative of the hydrological variable of interest.
Using meteorological variables as covariates could be more effective and have clearer physical meaning for modelling return period
and risk under nonstationary conditions than simply using time as
covariate. This physical covariate analysis approach has previously
been explored for nonstationary frequency analysis of extreme
events (Coles, 2001; Villarini et al., 2010; López and Francés,
2013), but to our knowledge it has not been incorporated with
either the EWT or ENE interpretation of return period in estimating
return period and risk of extreme events under nonstationary
conditions.
Like many other places around the world, hydrological processes in China are under the influence of climate change, so the
assumption of stationarity of river flow series is not valid for many
rivers (Zhang et al., 2011). The Wei River, the largest tributary of
235
the Yellow River, is the major source of water supply for the economic hub of Western China – the Guanzhong Plain. The Wei
River basin is one of the most important industrial and agricultural
production zones in China. However, in recent decades the Wei
River basin has suffered a significant decrease in streamflow
(Song et al., 2007; Zuo et al., 2014), which threatens industrial
and agricultural production and socioeconomic development. The
increasing scarcity of water resources under conditions of climate
change is a serious concern not only in the Wei River basin, but
also across the entire nation. There is an urgent need to provide
water resource managers and policy makers with reliable information on the return period and risk characteristics of the low-flow
component of the flow regime of the Wei River under the prevailing nonstationary conditions.
For those reasons, the main goal of this paper is to define and
apply an integrated approach for understanding and quantifying
the differences in calculating the nonstationary return period and
risk of extreme low-flow resulted from using time as the sole
covariate and using meteorological variables as covariates in the
nonstationary frequency analysis of the low-flow series. Note that
the economic development and human activities might have partially contributed in the nonstationarity of extreme hydrological
events as identified in other studies (Xiong et al., 2014; Jiang
et al., 2015b). Thus, in addition to climatic factors, anthropogenic
factors, such as irrigation area, population, and industrial consumption, should be considered as covariates in investigating the
causes of nonstationarity of low-flow events. However, these
anthropogenic factors have not, for the moment, been incorporated
in this paper in investigating the nonstationary exceedance probabilities of low-flow events for future years for the reason that the
values of these anthropogenic factors in the future years are very
difficult to be determined, or the estimated values have too big
uncertainties even compared to the GCMs estimates of climatic
factors. So, in the present study we would like to limit the explanatory factors of future low-flow nonstationarity to just climatic
factors.
When using meteorological covariates, GCMs outputs from the
Coupled Model Intercomparison Project Phase 5 (CMIP5) provide
the statistical parameters of the nonstationary low-flow distribution by substituting downscaled future meteorological variables
into the derived optimal nonstationary model to extend the exceedance probabilities into the future. The EWT interpretation of
return period requires infinite (or as long as possible) future exceedance probabilities (Cooley, 2013) but the GCMs outputs are temporally finite (most of the GCMs just provide continuous
large-scale daily predictors to the year of 2100) (Riahi et al.,
2011; van Vuuren et al., 2011), so the meteorological covariates
cannot be incorporated in estimating return period and risk of
extreme events under nonstationary conditions with the EWT
interpretation. However, the ENE interpretation of return period
requires just finite future exceedance probabilities, which can be
fully provided by the GCMs outputs, and thus the meteorological
covariates are adopted in the nonstationary return period and risk
analysis with the ENE method. Besides, the traditional approach
using time as the only covariate under both the EWT and ENE
interpretations is also applied in this study for the purpose of comparison with the approach using meteorological covariates under
the ENE interpretation. The practical application of this approach
is illustrated using a case study of the Wei River basin.
This paper is organized as follows. The next section describes
the Wei River basin and the available data sets used in the study.
Then the methods for determining the return period (under the
EWT and ENE interpretations) and risk under stationary and nonstationary conditions are described, along with a brief outline of
the methods of nonstationary frequency analysis of low-flow series
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T. Du et al. / Journal of Hydrology 527 (2015) 234–250
using time-varying moment, and statistical downscaling of meteorological variables. The results and discussion of the Wei River case
study follow, and finally the conclusions about the performance of
the proposed approach in the case study and its potential for wider
application are drawn.
The average annual natural discharge of the Wei River is about
10 109 m3, contributing approximately 17% of the discharge of
the Yellow River. The annual discharge of the Wei River at
Linjiacun station during the decade 1991–2000 was 53.9% less than
that during the preceding decade, while at Huaxian station near
the basin outlet it was 50.3% less (Song et al., 2007).
2. Study area and data
2.2. Data
2.1. General characteristics of the study area
The Wei River, the largest tributary of the Yellow River, originates from the Niaoshu Mountain at an elevation of 3485 m above
mean sea level in the Weiyuan county of Gansu province. The Wei
River has a length of 818 km and a drainage area of 134 800 km2,
covering the coordinates of 33°400 –37°260 N, 103°570 –110°270 E in
the southeastern part of the loess plateau (Fig. 1). The Wei River
has two large tributaries, the Jing River and the Beiluo River,
located in the middle and lower reaches of the basin, respectively
(Fig. 1). The Wei River is known regionally as the ‘Mother River’ of
the Guanzhong Plain of the southern part of the loess plateau
because of its key role in the economic development of western
China (Song et al., 2007; Zuo et al., 2014).
The Wei River basin is characterized by semi-arid and
sub-humid continental monsoon climate. Average annual precipitation of the basin is about 540 mm over the period 1954–2009,
but there is a strong decreasing gradient from south to north.
The southern region has a sub-humid climate with annual precipitation ranging from 800 to 1000 mm, whereas the northern region
has a semi-arid climate with annual precipitation ranging from 400
to 700 mm. The annual average temperature over the entire basin
ranges from 6 to 14 °C. The range in the annual potential evapotranspiration is 660–1600 mm, and the basin average annual
actual evapotranspiration from the land surface is about 500 mm.
Four kinds of data were used in this study: observed hydrological data, observed meteorological data, NOAA National Centres for
Environmental Prediction (NCEP) reanalysis data, and GCMs outputs from the CMIP5.
Observed mean daily streamflow data from both Huaxian and
Xianyang gauging stations (Fig. 1) over the period 1954–2009, provided by the Hydrology Bureau of the Yellow River Conservancy
Commission, were the source of the low-flow series (defined here
as the annual minimum monthly streamflow) (Fig. 2a and b).
Huaxian hydrological station, located at 109°460 E, 34°350 N, is about
70 km upstream of the junction of the Wei and Yellow rivers
(Fig. 1). The catchment area upstream of this station is
106 500 km2, or about 80% of the total basin area. Xianyang hydrological station, located at 108°420 E, 34°190 N, is about 120 km
upstream of Huaxian station with a drainage area of 46 480 km2
(Fig. 1).
Temperature and precipitation are two meteorological variables
that are closely related to streamflow and were chosen as covariates for nonstationary frequency analysis of low-flow. Observed
daily average temperature and daily total precipitation series from
22 stations (Fig. 1) for the period 1954–2009 were obtained from
the National Climate Center of the China Meteorological
Administration (source: http://cdc.cma.gov.cn). Considering the
condition of snowpack at the headwater part of the basin may have
Fig. 1. Location, topography, hydro-meteorological stations and river systems of the Wei River basin. The small inset box inside the main China map contains the islands of
the South China Sea.
T. Du et al. / Journal of Hydrology 527 (2015) 234–250
237
Fig. 2. Data analyses of both Huaxian (left panel) and Xianyang (right panel) stations. (a) and (b) are observed low-flow series, (c) and (d) are correlation plots between the
observed low-flow series and annual average temperature Temp, (e) and (f) are correlation plots between the observed low-flow series and annual total precipitation Prep, (g)
and (h) are correlation plots between the observed low-flow series and winter average temperature TempW, and (i) and (j) are correlation plots between the observed lowflow series and winter total precipitation PrepW.
influence on the nonstationarity of the observed streamflow, a
pre-check on the form of the precipitation received at the headwater part was carried out. Result indicated that almost all the precipitation in the three most west meteorological stations at the
headwater part, Yuzhong, Lintao and Minxian, was received in
the form of rainfall, which eliminated the concern about snowpack
influence in this study. Then, the areal average daily series of both
variables (daily average temperature and daily total precipitation)
for the catchments above Huaxian and Xianyang stations were
generated using the Thiessen polygon method (Szolgayova et al.,
2014), and from these the annual average temperature and annual
total precipitation series (denoted by Temp and Prep, respectively)
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T. Du et al. / Journal of Hydrology 527 (2015) 234–250
(Xiong et al., 2014) over the period of 1954–2009 were extracted
for each catchment. As expected, there are linear relations between
the observed low-flow series and these two annual statistics of
meteorological variables (Pearson correlation coefficients of
0.54 and 0.38 for Temp and Prep, respectively for Huaxian, and
0.57 and 0.39 for Xianyang) (Fig. 2c–f). Theoretically, seasonal
statistics of precipitation and temperature might be more suitable
as covariates than annual statistics for modelling the low-flow
events. Considering most of the observed low-flow events occurred
in January and December, winter (December, January and
February) average temperature and winter total precipitation
(denoted by TempW and PrepW, respectively) were also examined.
Correlation plots between the observed low-flow series and
TempW and PrepW (Fig. 2g–j) indicated that seasonal statistics
had lower correlation coefficients comparing with annual statistics
for both stations and thus will not be considered further in this
study.
The NCEP reanalysis daily data and GCMs daily data were
employed to calibrate the statistical downscaling model and derive
future temperature and precipitation scenarios respectively (Wilby
et al., 2002; Wilby and Dawson, 2007). The NCEP reanalysis data
were used to calibrate the multiple linear regression equation
between each predictand (daily average temperature and daily
total precipitation) and NCEP large-scale predictors. Then, future
scenario of the predictand was projected by substituting the
GCMs large-scale predictors into the calibrated multiple linear
regression equation. The 26 candidate predictors of NCEP reanalysis data as described in Wilby and Dawson (2007) for the period of
1954–2009 were obtained from the NOAA Earth System Research
Laboratory (ESRL) (source: http://www.esrl.noaa.gov). The
Representative Concentration Pathways (RCPs) are four greenhouse gas concentration and emissions pathways adopted by the
IPCC for its fifth Assessment Report (AR5), each one is named
according to radiative forcing target level in watts per square
metre for year 2100 (van Vuuren et al., 2011). In this study we used
RCP8.5 scenario, which represents the upper bound of the RCPs,
but future work could extend our analysis by examining the other
RCPs with smaller radiative forcing target levels. RCP8.5 scenario is
characterized by increasing greenhouse gas emissions over time,
representative of scenarios in the literature that leads to comparatively high greenhouse gas concentration levels, and does not
include any specific climate mitigation target (Riahi et al., 2011).
The same 26 predictors of seven different GCMs (CanESM2,
CNRM-CM5,
GFDL-ESM2M,
NorESM1-M,
MIROC-ESM,
MIROC-ESM-CHEM, and CCSM4) under the RCP8.5 scenario for
the period of 2010–2099 inclusive were obtained from the CMIP5
(source: http://cmip-pcmdi.llnl.gov/cmip5).
The NCEP and GCMs data are gridded to different spatial scales,
so data preprocessing was necessary. First, predictors of both data
sets were interpolated to each meteorological station site. For each
meteorological station, the grid it locates in and the eight grids
around it were used for the interpolation of NCEP and GCMs outputs (large-scale predictors) with the Inverse Distance Weighting
method (Bartier and Keller, 1996). Then, areal average series of
every predictor for the catchments above Huaxian and Xianyang
stations were calculated using the Thiessen polygon method.
3. Methodology
In this section, firstly, the exceedance probability of a low-flow
event, which is the key element for determining the return period
and risk, is defined. Then, theories about the return period (under
both EWT and ENE interpretations) and risk of low-flow events
concerning the future exceedance probability under stationary
and nonstationary conditions are described. Finally, in deriving
the future exceedance probability for the determination of the
nonstationary return period and risk of a low-flow event, the
time-varying moment method is employed in the nonstationary
frequency analysis of the observed low-flow series by using time
or meteorological variables as covariates. When using meteorological variables as covariates, the downscaled future meteorological
variables from the GCMs are used to calculate the statistical
parameters and exceedance probabilities for future years. For the
sake of completeness, the methods used in this section are briefly
described in the following subsections.
3.1. Exceedance probability of a low-flow event
The low-flow character of the flow regime is denoted by the
random variable Z. Our interest is on the scarcity of water
resources, so we define the design low-flow quantile zp0 which in
any given year has a probability p0 that the streamflow is lower
than this quantile (Fig. 3a). The probability of a flood event that
is higher than a design flood quantile is usually referred to as the
exceedance probability, or as the exceeding probability (e.g. Salas
and Obeysekera, 2013, 2014). In the case of low-flow (drought)
event, the meaning of exceedance or exceeding is that the drought
severity is exceeded, or the value of the flow statistic is lower than
the design quantile.
Under stationary conditions, the cumulative distribution function of Z is denoted by FZ(z, h), where h is the constant statistical
parameter set. The focus of this study is to analyse the future evolution of return period and hydrological risk of a given design
low-flow decided at a specific year based on the historical observed
data. This specific year is normally defined as the base year or the
initial year and denoted by t = 0. Thus, the given design low-flow at
t = 0 corresponding to an initial return period T0 can be derived by
1
zp0 ¼ F 1
Z ðp0 ; hÞ, where p0 = 1/T0, and F Z is the inverse function of FZ
(similarly hereinafter). The period of years after the initial year is
referred to as ‘‘future’’. In the present study, we adopt Cooley’s
method (Cooley, 2013) where the initial year t = 0 corresponds to
the last observation year. In the stationary case, which means the
controlling environmental factors for future years are the same
as the initial year t = 0. h is constant for every year, and the exceedance probability corresponding to the design quantile zp0 is p0 for
each future year (Fig. 3a), which can be obtained by:
pt ¼ F Z ðzp0 ; hÞ ¼ p0 ;
t ¼ 1; 2; . . . ; 1
ð1Þ
Under nonstationary conditions, the cumulative distribution
function of Z is denoted by FZ(z, ht), where ht varies in accordance
with time or, more directly, with meteorological variables. In nonstationary case, the design low-flow quantile corresponding to the
initial exceedance probability p0 = 1/T0 can be derived from
zp0 ¼ F 1
Z ðp0 ; h0 Þ, where h0 is the statistical parameter set of the initial year t = 0. The statistical parameters are time-varying so the
future exceedance probability corresponding to zp0 is not constant
any more. In this case, the temporal variation in the exceedance
probability corresponding to zp0 can be characterized by the way
the low-flow distribution or, more specifically, the statistical
parameters change through time (Fig. 3b). The exceedance probability for each future year can be obtained by:
pt ¼ F Z ðzp0 ; ht Þ;
t ¼ 1; 2; . . . ; 1
ð2Þ
Generally, ht in Eq. (2) is derived by taking time as the only
covariate in nonstationary modelling of the return period and risk
analysis as described in the introduction (Wigley, 1988, 2009;
Olsen et al., 1998; Parey et al., 2007, 2010; Cooley, 2013; Salas
and Obeysekera, 2013, 2014). In this study, meteorological
covariates, which have clear physical meanings, are considered in
nonstationary modelling. Thus, ht can be derived from the
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T. Du et al. / Journal of Hydrology 527 (2015) 234–250
Fig. 3. Schematic depicting the design low-flow quantile zp0 with (a) constant exceedance probability p0, and (b) time-varying exceedance probabilities pt, t = 1, 2, . . ., 1.
downscaled meteorological covariates of GCMs for future years,
and then be used to more physically estimate the evolution of
the exceedance probabilities pt, return period and risk of zp0 in
the future years.
T ¼ EðXÞ ¼
1
1
x1
X
X
Y
xf ðxÞ ¼
xpx ð1 pt Þ
x¼1
x¼1
ð6Þ
t¼1
3.3. Return period using ENE interpretation
3.2. Return period using EWT interpretation
Under stationary conditions, if X is denoted as the random variable representing the year of the first occurrence of a low-flow that
exceeds (i.e. is lower than) the design quantile, then a low-flow Z
exceeding the design value zp0 for the first time in year X = x,
x = 1, 2, . . ., 1, follows the geometric probability law (Mood et al.,
1974; Salas and Obeysekera, 2013, 2014):
f ðxÞ ¼ PðX ¼ xÞ ¼ ð1 p0 Þx1 p0 ;
x ¼ 1; 2; . . . ; 1
ð3Þ
Noting that Eq. (3) is derived on the assumptions of independence and stationarity, then the expected value of X, i.e. the return
period (expected waiting time interpretation) of the low-flow
exceeding the design quantile zp0 under stationary conditions, is:
T ¼ EðXÞ ¼
1
X
xf ðxÞ ¼ 1=p0
ð4Þ
x¼1
Under nonstationary conditions, the exceedance probability
corresponding to zp0 is no longer constant (Fig. 3b), and then the
geometric probability law considering time-varying exceedance
probabilities pt is (Cooley, 2013; Salas and Obeysekera, 2013,
2014):
Under stationary conditions, if M is denoted as the random variable representing the number of exceedances in T years, then
P
M ¼ Tt¼1 IðZ t < zp0 Þ, where I() is the indicator function. M follows
a binomial distribution (Cooley, 2013):
f ðmÞ ¼ PðM ¼ mÞ ¼
¼ px
x1
Y
ð1 pt Þ;
x ¼ 1; 2; . . . ; 1
EðMÞ ¼
t¼1
The EWT-return period T of the low-flow exceeding the design
quantile zp0 under nonstationary conditions is thus:
m
Tm
pm
0 ð1 p0 Þ
ð7Þ
T
X
p0 ¼ Tp0 ¼ 1
ð8Þ
t¼1
And thus the return period (expected number of exceedances interpretation) of the low-flow exceeding the design quantile zp0 under
stationary conditions is T = 1/p0.
Under nonstationary conditions, the exceedance probability is
not constant and M does not follow a binomial distribution. In this
situation the expected number of exceedances is expressed as
(Cooley, 2013):
EðMÞ ¼
T
T
T
X
X
X
E½IðZ t < zp0 Þ ¼
PðZ t < zp0 Þ ¼
F Z ðzp0 ; ht Þ
T
X
pt
¼
ð5Þ
T
It follows that the expected value of M is 1:
t¼1
f ðxÞ ¼ PðX ¼ xÞ ¼ ð1 p1 Þð1 p2 Þ . . . ð1 px1 Þpx
t¼1
t¼1
ð9Þ
t¼1
The ENE-return period T of the low-flow exceeding the design quantile zp0 under nonstationary conditions can thus be derived by setting Eq. (9) equal to 1 and solving:
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T. Du et al. / Journal of Hydrology 527 (2015) 234–250
T
X
pt ¼ 1
ð10Þ
t¼1
3.4. Hydrological risk
In practical application of hydrological frequency analysis, the
management question is often framed as one of risk, whereby,
for a design life of n years, the hydrological risk R is the probability
of a low-flow event exceeding the design value zp0 before or at year
n. This risk can be derived from the perspective of complement,
which means that there is no exceedance during the design life
of n years. Under the assumption of independence and stationarity,
the probability of the complement is (1 p0)n. Then the hydrological risk under stationary conditions is (Haan, 2002):
R ¼ 1 ð1 p0 Þn
ð11Þ
In recent years, hydrological risk analysis under nonstationary
conditions has become increasingly popular (Rootzén and Katz,
2013; Salas and Obeysekera, 2014; Condon et al., 2015; Serinaldi
and Kilsby, 2015) and provides a different view from return period
and design level for designers by combining the basic information
of design life n and exceedance probabilities pt. As with the stationary case, for a design life of n years, the probability of a low-flow
exceeding the design quantile zp0 before or at year n under the circumstances of time-varying exceedance probabilities pt is:
R ¼ 1 ½ð1 p1 Þð1 p2 Þ . . . ð1 pn Þ ¼ 1 n
Y
ð1 pt Þ
ð12Þ
t¼1
3.5. Nonstationary frequency analysis of low-flow series
The standard way of calculating the nonstationary return
periods under the EWT and ENE interpretations and the risk of a
design quantile zp0 corresponding to an initial return period T0 is
by Eqs. (6), (10) and (12), respectively. An important part of the
procedure is derivation of time-varying exceedance probabilities
pt (Eq. (2)) for future years. This relies on determination of the relationships of the statistical parameters of the low-flow distribution
to the explanatory variables, which is normally referred to as
nonstationary frequency analysis. Several studies have explored
nonstationary return period and risk analysis of extreme events
using only time as covariate in the nonstationary frequency
analysis (e.g. Wigley, 1988, 2009; Olsen et al., 1998; Parey et al.,
2007, 2010; Cooley, 2013; Salas and Obeysekera, 2013, 2014).
Unlike just using time alone as covariate, meteorological variables
have physical meaning and therefore have more convincible
explanatory power to be used as covariates. Thus, in the method presented here, we derive the time-varying exceedance probabilities pt
for future years by employing meteorological covariates in the nonstationary frequency analysis of low-flow events and incorporating
the statistical downscaling of future meteorological variables from
GCMs. As explained before, when meteorological covariates are used
for describing the nonstationary frequency of low-flow events, only
the ENE interpretation can be adopted for the nonstationary return
period and risk analysis, for the current GCMs outputs are provided
only for a finite time period and unable to meeting the indefinite
data requirement by the EWT interpretation (see Eq. (6)).
The nonstationary low-flow series was modelled using the
time-varying moment method, which was built under the
Generalized Additive Models in Location, Scale and Shape
(GAMLSS) framework (Rigby and Stasinopoulos, 2005; Xiong
et al., 2014). Various probability distributions have been suggested
for modelling low-flow events (Matalas, 1963; Eratakulan, 1970;
Smakhtin, 2001; Hewa et al., 2007; Liu et al., 2015). Matalas
(1963) investigated four distributions in modelling low-flow data
of 34 streams and found the Gumbel and Pearson-III distributions
fitted the data well and were more representative than the
Lognormal and Pearson-V distributions. Eratakulan (1970) found
Gamma and Weibull were the first two distributions to be selected
in modelling the low-flow series of 37 stations in the Missouri
River basin. Hewa et al. (2007) introduced the GEV distribution
into the frequency analysis of low-flow data from 97 catchments
of Victoria, Australia. Liu et al. (2015) tested six distributions in
modelling the annual low flows of the Yichang station, China under
nonstationary conditions and found the GEV distribution gave the
best fit. Based on these studies, five two-parameter distributions,
i.e. Gamma (GA), Weibull (WEI), Gumbel (GU), Logistic (LO), and
Lognormal (LOGNO), and two three-parameter distributions, i.e.
Pearson-III (P-III) and GEV, that widely used in modelling
low-flow data were considered as candidates in this study
(Table 1). Considering that the shape parameter j of P-III and
GEV distributions is quite sensitive and difficult to be estimated,
we assumed it to be constant as other studies did (Coles, 2001;
Katz et al., 2002; Gilroy and McCuen, 2012) and nonstationarities
in both the location l and scale r parameters were examined
through monotonic link functions g() (Table 1). The optimal
Table 1
Summary of the distributions used to model the low-flow series in this study.
Distribution
Probability density function
Gamma
f Z ðzjl; rÞ ¼
2 1Þ z=ðlr2 Þ
zð1=r
1
1=r2
ðlr2 Þ
e
Cð1=r2 Þ
z > 0; l > 0; r > 0
Weibull
Gumbel
Logistic
Lognormal
Pearson-III
GEV
h r i
r1
f Z ðzjl; rÞ ¼ rzlr exp lz
z > 0; l > 0; r > 0
f Z ðzjl; rÞ ¼ r1 exp zrl exp zrl
1 < z < 1; 1 < l < 1; r > 0
2
f Z ðzjl; rÞ ¼ r1 exp zrl f1 þ exp zrl g
1 < z < 1; 1 < l < 1; r > 0
n
o
l2
1ffi 1
exp ½logðzÞ
f Z ðzjl; rÞ ¼ pffiffiffiffi
2r2
2pr z
z > 0; l > 0; r > 0
1 1
h i
zl
zl
1 j2
1
exp lr
f Z ðzjl; r; jÞ ¼ rjljjC1ð1=j2 Þ lr
j þ j2
j þ j2
zl
1
r > 0; j–0; lr
j þ j2 P 0
n ð1=jÞ1
1=j o
f Z ðzjl; r; jÞ ¼ r1 1 þ j zrl
exp 1 þ j zrl
1 < l < 1; r > 0; 1 < j < 1
Moments
Link functions
EðZÞ ¼ l
SDðZÞ ¼ lr
g 1 ðlÞ ¼ lnðlÞ
g 2 ðrÞ ¼ lnðrÞ
EðZÞ ¼ lC r1 þ 1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2ffi
SDðZÞ ¼ l C r þ 1 C r1 þ 1
g 1 ðlÞ ¼ lnðlÞ
g 2 ðrÞ ¼ lnðrÞ
EðZÞ ¼ l þ cr ’ l þ 0:57722r
SDðZÞ ¼ ppffiffi6 r ’ 1:28255r
g 1 ðlÞ ¼ l
g 2 ðrÞ ¼ lnðrÞ
EðZÞ ¼ l
SDðZÞ ¼ ppffiffi3 r ’ 1:81380r
g 1 ðlÞ ¼ l
g 2 ðrÞ ¼ lnðrÞ
1=2 l
e
EðZÞ ¼ wp
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
SDðZÞ ¼ wðw 1Þel
w ¼ expðr2 Þ
EðZÞ ¼ l
Cv ¼ r
Cs ¼ 2j
g 1 ðlÞ ¼ l
g 2 ðrÞ ¼ lnðrÞ
EðZÞ ¼ l rj þ rj g1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
SDðZÞ ¼ r g2 g21 =j
gm ¼ Cð1 mjÞ
g 1 ðlÞ ¼ l
g 2 ðrÞ ¼ lnðrÞ
g 3 ð jÞ ¼ j
g 1 ðlÞ ¼ lnðlÞ
g 2 ðrÞ ¼ lnðrÞ
g 3 ð jÞ ¼ j
T. Du et al. / Journal of Hydrology 527 (2015) 234–250
nonstationary model was selected by penalizing more complex
models in terms of the Akaike Information Criterion (AIC)
(Akaike, 1974) which is calculated as:
AIC ¼ 2 lnðMLÞ þ 2k
ð13Þ
where ML is the maximum likelihood function of models and k is
the number of independently adjusted parameters within the
model, and theoretically 1 < AIC < 1. The model with the smallest AIC value was considered the optimal one.
While the AIC value identifies the optimal model, it is not a
measure of model performance. Goodness-of-fit of the selected
optimal model was assessed qualitatively by the worm (Buuren
and Fredriks, 2001) and the centile curves diagnostic plots, and
quantitatively using the statistics of the Filliben correlation coefficient (denoted by Fr) (Filliben, 1975) and the Kolmogorov–Smirnov
(KS) test (denoted by DKS) (Massey, 1951). Fr and DKS are calculated
by Eqs. (14) and (16), respectively:
Ps
i¼1 ðSðiÞ SÞðBi BÞ
ffi
F r ¼ CorðS; BÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Ps
2 Ps
2
i¼1 ðSðiÞ SÞ
i¼1 ðBi BÞ
ð14Þ
where S(i) are the ordered residuals derived by sorting U1[FZ(zi, hi)],
1 6 i 6 s in ascending order. U1 is the inverse function of the standard normal distribution and s is the length of the observation period (similarly hereinafter). Bi are the standard normal order statistic
medians calculated from U1(bi) where bi are derived by:
8
>
< 1 bs
bi ¼ ði 0:3175Þ=ðs þ 0:365Þ
>
:
0:5ð1=sÞ
ð15Þ
i¼s
Fr have the range of (0, 1] and a Fr bigger than the critical value Fa
indicates that the nonstationary model passes the goodness-of-fit
test.
^ i GðiÞ j
DKS ¼ max1is jG
be overcome by a technique known as downscaling. The statistical
downscaling model (SDSM) provided by Wilby et al. (2002) is a
decision support tool for assessing local climate change impacts
using a robust statistical downscaling technique that combines a
weather generator and multiple linear regression. SDSM has been
widely used in research related to climate change (Wilby and
Dawson, 2013) and is fully described in Wilby et al. (2002) and
Wilby and Dawson (2007).
Theoretically, the downscaling process with SDSM is either
unconditional (as with temperature) or conditional on an event
(as with precipitation amounts). For unconditional process, a direct
multiple linear regression equation between the unconditional pre^ ij on
dictand yUC
and the chosen normalized large-scale predictors u
i
day i is constructed as (Wilby et al., 2003; Wetterhall et al., 2006):
yUC
¼ c0 þ
i
l
X
cj u^ ij þ e
ð17Þ
j¼1
where cj are the estimated regression coefficients deduced by the
least square method, l is the number of chosen predictors and e is
a normally distributed stochastic error term.
For conditional process, a conditional probability of predictand
occurrence xi on day i is directly expressed as a multiple linear
^ ij as:
regression equation of u
xi ¼ g0 þ
l
X
gj u^ ij
ð18Þ
j¼1
i¼1
i ¼ 2; 3; . . . ; s 1
241
ð16Þ
^ i are the empirical cumulative probabilities calculated from
where G
i/(s + 1), and G(i) are the ordered theoretical cumulative probabilities derived by sorting FZ(zi, hi), 1 6 i 6 s in ascending order. DKS
have the range of [0, 1] and a DKS smaller than the critical value
Da indicates that the nonstationary model passes the
goodness-of-fit test.
To summarize, the main steps in deriving time-varying exceedance probabilities pt for future years are:
(i) Nonstationary modelling of the observed low-flow series by
using time alone or meteorological variables as covariates.
(ii) Calculating the design low-flow quantile zp0 corresponding
to an initial return period T0 from the quantile function
zp0 ¼ F 1
Z ðp0 ; h0 Þ, where p0 = 1/T0 and h0 is the fitted statistical parameter set of the initial year t = 0.
(iii) Deriving time-varying exceedance probabilities pt corresponding to zp0 for future years t = 1, 2, . . ., 1 (time as
covariate) or t = 1, 2, . . ., tmax (meteorological covariates)
from Eq. (2), where ht is calculated by extending the optimal
nonstationary model of step (i) into the future under respective case of covariates.
3.6. Statistical downscaling model (SDSM)
GCMs are a tool for predicting future time series of meteorological variables, thereby extending the time-varying exceedance
probabilities of Eqs. (10) and (12). The coarse spatial resolution
of GCMs data restricts its direct application to local impact studies
(Wilby et al., 2002; Wilby and Dawson, 2007), but this problem can
where gj are the estimated regression coefficients. If xi 6 ri, where
ri is a uniformly distributed random number (0 6 ri 6 1), conditional predictand yCi occurs with the amount of:
yCi ¼ F 1 ½UðZ i Þ
ð19Þ
where F is the empirical distribution function of yCi , U is the normal
cumulative distribution function. Zi is the z-score for day i with the
P
^ ij þ e, where kj are the estimated
expression of Z i ¼ k0 þ lj¼1 kj u
regression coefficients.
Specifically, the downscaling of daily average temperature and
daily total precipitation was carried out according to the unconditional and conditional processes, respectively. A correlation analysis
(Wilby and Dawson, 2007; Hessami et al., 2008) between each predictand and alternative NCEP large-scale predictors indicated that
mean sea level pressure, 500 hPa geopotential height, 500 hPa eastward wind, 850 hPa air temperature, and near-surface air temperature had higher correlations with daily average temperature than
other predictors and thus were selected for the downscaling of daily
average temperature. For daily total precipitation, one additional
predictor, 850 hPa specific humidity, was included. The SDSM models for both daily average temperature and daily total precipitation
were optimized with the respective selected NCEP predictors. Then,
daily average temperature and daily total precipitation for the period of 1954–2009 were simulated by the weather generator in the
SDSM driven by the NCEP reanalysis predictors. The simulation
results were assessed by the Nash–Sutcliffe efficiency (NSE)
between the simulated and observed Temp and Prep. NSE is calculated as (Nash and Sutcliffe, 1970):
Ps
2
ðY obs Y sim
i Þ
NSE ¼ 1 P i¼1 i
2
s
obs
Y mean Þ
i¼1 ðY i
ð20Þ
where Y obs
are the observed meteorological variables, Y sim
are the
i
i
mean
is the mean value of
simulated values corresponding to Y obs
i , Y
Y obs
i . NSE ranges from 1 to 1, with NSE = 1 being the perfect
simulation.
242
T. Du et al. / Journal of Hydrology 527 (2015) 234–250
Normally, the SDSM predictand–predictor relationship, i.e. the
calibrated multiple linear regression equation, is assumed to be
transferrable to future projection period (Wilby et al., 2002;
Wilby and Dawson, 2007, 2013; Wetterhall et al., 2006; Mullan
et al., 2012). Then with the input of the GCMs large-scale predictors, future scenarios of daily average temperature and daily total
precipitation can be derived. Daily average temperature and daily
total precipitation for the period of 2010–2099 generated by the
scenario generator in the SDSM driven by the GCMs predictors
were then used to calculate the statistics of Temp and Prep for
the future years of 2010–2099 for the seven GCMs.
4. Results and discussion
4.1. Nonstationary frequency analysis of low-flow series
The observed low-flow magnitudes from both Huaxian and
Xianyang stations declined through time, with irregular scatter
(Fig. 2a and b). Significant decreasing trends were detected by
the Mann–Kendall test (Mann, 1945; Kendall, 1975; Li et al.,
2014) with the statistics ZMK = 2.71 and ZMK = 3.63 for Huaxian
and Xianyang, respectively, compared to the critical value of
Z1a/2 = 1.96 at a = 0.05. More detailed analysis of the nonstationarity was undertaken as part of modelling the low-flow series
under the GAMLSS framework.
When the low-flow series were modelled using time as covariate, for Huaxian, the AIC values suggested that the Weibull (WEI)
distribution (with logarithmic link functions for both the location
l and scale r parameters) was the optimal distribution, with both
log-transformed parameters modelled as linear functions of time
(Fig. 4a). While for Xianyang, the Gamma (GA) distribution (with
logarithmic link functions for both the location l and scale r
parameters) was the optimal distribution, also with both
log-transformed parameters modelled as linear functions of time
(Fig. 4b). Indeed, the WEI and GA distributions performed very
comparable for both stations (Fig. 4a and b). However, we would
strictly follow the selection criterion of AIC for determining the
optimal nonstationary model for each station.
Fig. 4. Summary of different distributions with different nonstationary models fitted to the observed low-flow series from Huaxian (left panel) and Xianyang (right panel)
stations. Location l and scale r parameters modelled as functions of time in (a) and (b), and meteorological variables Temp and Prep in (c) and (d). The expressions in the red
box is the optimal nonstationary model for respective case of covariate analysis. (For interpretation of the references to colour in this figure legend, the reader is referred to
the web version of this article.)
T. Du et al. / Journal of Hydrology 527 (2015) 234–250
243
Fig. 5. Diagnostic plots for assessing the performance of the optimal nonstationary model using time as covariate: (a) and (b) are worm plots (for a good fit, the data points
should be aligned preferably along the red solid line but within the 95% confidence intervals indicated by the two grey dashed lines); (c) and (d) are centile curves plots (the
blue points are the observed low-flow series, the red line is the 50% centile curve, the dark grey region is the area between the 25% and 75% centile curves and the light grey
region is the area between the 5% and 95% centile curves. Theoretically, the frequency of the observed low-flow events falling within the dark grey region and the light grey
region should be 50% and 90%, respectively). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
For Huaxian, sporadic worm points were without 95% confidence intervals (Fig. 5a), while for Xianyang, all the worm points
were within the 95% confidence intervals (Fig. 5b), indicating
acceptable consistency between the selected model and the
observed low-flow data. The vast majority of the points were
within the 5% and 95% centile curves for both stations
(Fig. 5c and d) indicating that the model captured the variability
of the data. The statistics of the Fr and the DKS (Table 2) also indicated that the selected nonstationary model was an adequate fit
to the low-flow series for respective station.
In fitting the low-flow series to nonstationary models with the
annual average temperature Temp and annual total precipitation
Prep as covariates, the most complex model expressed both link
function-transformed statistical parameters l and r as linear functions of Temp and Prep. This process also included all possible simpler sub-models. The AIC values suggested that the WEI
distribution (with logarithmic link functions for both the location
l and scale r parameters) was the optimal distribution for both
Huaxian and Xianyang stations (Fig. 4c and d). For Huaxian, the
log-transformed l and r were modelled as linear functions of
Temp and Prep, respectively. And for Xianyang, the
log-transformed l was modelled as linear function of Temp, but
r was modelled as a constant. Although the WEI and GA
distributions
performed
comparable
for
both
stations
(Fig. 4c and d), we still strictly followed the selection criterion of
AIC for determining the optimal nonstationary model for each station. Besides, the two seasonal meteorological covariates (winter
average temperature and winter total precipitation) were also
examined for modelling the observed low-flow series. However,
the AIC values of the optimal nonstationary models with the seasonal covariates were 870.9 and 831.7 for Huaxian and Xianyang,
respectively, which were larger than the cases of using the annual
covariates with the AIC values of 863.8 and 817.1 for Huaxian and
Xianyang, respectively (Table 2). This further proved the rationality
of the selection of the annual covariates rather than winter covariates in modelling the nonstationarity of the observed low-flow series in this study.
All the worm points were within the 95% confidence intervals
for both stations, indicating perfect consistency between the
selected model and the observed low-flow data (Fig. 6a and b).
The vast majority of the points were within the 5% and 95% centile
curves (Fig. 6c and d) indicating that the model captured the variability of the data. The statistics of the Fr and the DKS (Table 2) also
indicated that the selected nonstationary model was an adequate
fit to the low-flow series for respective station. In addition, for each
station, the optimal nonstationary model using meteorological
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T. Du et al. / Journal of Hydrology 527 (2015) 234–250
Table 2
Summary of the optimal nonstationary models fitted to the low-flow series of Huaxian and Xianyang stations
of the Wei River using time or meteorological variables as
pffiffiffiffiffiffi
covariates. The critical values of the Filliben correlation coefficient is Fa = 0.978 and the KS test is Da ¼ 1:36= 56 0:182, at a = 0.05 (a Fr bigger than Fa and a DKS smaller than Da
indicate that the nonstationary model passes the goodness-of-fit test).
Optimal nonstationary model
Huaxian station
Nonstationary Weibulla (t as covariate)
Nonstationary Weibullb (Temp and Prep as covariates)
Xianyang station
Nonstationary Gammac (t as covariate)
Nonstationary Weibulld (Temp as covariate)
Estimated parameters
(standard errors)
AIC values
Filliben correlation
coefficient Fr
KS statistic DKS
la ¼ 7:5266ð0:1652Þ
lb ¼ 0:0229ð0:0066Þ
ra ¼ 0:7664ð0:2600Þ
rb ¼ 0:0145ð0:0082Þ
la ¼ 14:3504ð1:3879Þ
lb ¼ 0:7795ð0:1455Þ
ra ¼ 0:6740ð0:2096Þ
rb ¼ 0:0020ð0:0003Þ
872.4
0.982
0.086
863.8
0.983
0.074
828.5
0.986
0.099
817.1
0.993
0.077
la ¼ 7:1722ð0:1524Þ
lb ¼ 0:0244ð0:0055Þ
ra ¼ 0:7104ð0:1978Þ
rb ¼ 0:0112ð0:0061Þ
la ¼ 15:7336ð1:4568Þ
lb ¼ 0:9804ð0:1553Þ
ra ¼ 0:5528ð0:1054Þ
a
Nonstationarities in both the location ln(lt) = la + lb(t + s) and scale ln(rt) = ra + rb(t + s) parameters of Weibull distribution with time as covariate. s is the length of the
observation period, in this study s = 56.
b
Nonstationarities in both the location ln(lt) = la + lbTempt and scale ln(rt) = ra + rbPrept parameters of Weibull distribution with Temp and Prep as covariates.
c
Nonstationarities in both the location ln(lt) = la + lb(t + s) and scale ln(rt) = ra + rb(t + s) parameters of Gamma distribution with time as covariate. s is the length of the
observation period, in this study s = 56.
d
Nonstationarity in the location ln(lt) = la + lbTempt parameter of Weibull distribution with Temp as covariate, and the constant scale parameter ln(rt) = ra.
Fig. 6. Diagnostic plots for assessing the performance of the optimal nonstationary model using meteorological variables Temp and Prep as covariates: (a) and (b) are worm
plots (for a good fit, the data points should be aligned preferably along the red solid line but within the 95% confidence intervals indicated by the two grey dashed lines); (c)
and (d) are centile curves plots (the blue points are the observed low-flow series, the red line is the 50% centile curve, the dark grey region is the area between the 25% and
75% centile curves and the light grey region is the area between the 5% and 95% centile curves. Theoretically, the frequency of the observed low-flow events falling within the
dark grey region and the light grey region should be 50% and 90%, respectively). (For interpretation of the references to colour in this figure legend, the reader is referred to the
web version of this article.)
T. Du et al. / Journal of Hydrology 527 (2015) 234–250
245
covariates had a smaller AIC value than the optimal model using
time alone as covariate (Fig. 4 and Table 2), confirming the necessity and effectiveness of employing physical covariate analysis in
the nonstationary frequency analysis of low-flow series.
4.2. Statistical downscaling of temperature and precipitation
For the optimal nonstationary model with Temp and Prep as
covariates, the future time series of these two variables allowed
calculation of the time-varying parameters lt and rt for future
years, enabling estimation of the time-varying exceedance probabilities pt into the future. Since for Xianyang station only Temp
was selected in the optimal nonstationary model (Table 2), the
downscaling of Prep for this station will thus not be needed. The
Nash–Sutcliffe efficiency (NSE) between the simulated and
observed Temp and Prep (Fig. 7) suggested an adequate result for
Temp, but the result for Prep was not as good. This was expected,
as it is acknowledged that the downscaling of precipitation is more
problematic than temperature. Other authors have reported similar findings (Wilby and Dawson, 2007; Chen et al., 2012; Yang
et al., 2012). We considered the simulation result acceptable for
this purpose.
The projected annual time series of Temp and Prep for the future
period 2010–2099 for the seven GCMs (Fig. 8) showed strong
increasing trends for Temp of both stations (around 0.0596 °C per
year for the ensemble average for Huaxian, and 0.0573 °C per year
for Xianyang), while the ensemble average value of Prep for
Huaxian was stable at around 470 mm.
4.3. Nonstationary return period and risk using time as covariate
Having determined the optimal nonstationary model using time
alone as covariate with estimated parameters l0 and r0 (Table 2),
the nonstationary return periods T of the design low-flow quantile
zp0 corresponding to the specified initial return period T0 were
computed under both the EWT and ENE interpretations of return
period using Eqs. (6) and (10), respectively. For Huaxian station,
the nonstationary return period T of zp0 under the ENE interpretation was a little bit longer than the case of EWT but they were both
much shorter than the specified T0 (Fig. 9a). For example, when
T0 = 50 years, the values of T under nonstationarity were only
18.6 and 22 years for the EWT and ENE interpretations, respectively. The implication is that for a design low-flow quantile zp0 ,
if stationarity was incorrectly assumed, a low-flow lower than zp0
would be expected to occur about 50 years after the initial year
Fig. 8. Projected meteorological variables from different GCMs for the future period
2010–2099. The ensemble average is the arithmetic average value of all the
individual GCMs: (a) annual average temperature Temp for Huaxian station (°C); (b)
annual total precipitation Prep for Huaxian station (mm); (c) annual average
temperature Temp for Xianyang station (°C).
Fig. 7. Comparisons between simulated and observed meteorological variables for the observation period 1954–2009: (a) annual average temperature Temp for Huaxian
station (°C); (b) annual total precipitation Prep for Huaxian station (mm); (c) annual average temperature Temp for Xianyang station (°C). All variables were simulated by the
weather generator in the SDSM driven by the NCEP reanalysis predictors.
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T. Du et al. / Journal of Hydrology 527 (2015) 234–250
t = 0. Using 50 years as the basis for water resources planning decisions for this exceedance event would be imprudent because nonstationarity of the low-flow series suggests that the value of T
equals to 19 or 22 years might be more appropriate. For
Xianyang station which has relative smaller drainage area, the case
of nonstationary T was very similar to that of Huaxian
(Fig. 9a and b), confirming the necessity of considering nonstationarity in the return period analysis of the Wei River basin.
Given the initial return period T0 and design life n, the hydrological risk R of the design low-flow quantile zp0 for stationary and
nonstationary conditions was calculated using Eqs. (11) and (12),
respectively. Considering the risk results of the two stations were
very similar (Fig. 9c and d), we took the Huaxian result for illustration. Under both stationary and nonstationary cases risk increased
with n, but for any T0, the R for nonstationary conditions was
higher than that for stationary conditions. For example, when
T0 = 50 and n = 40 years, the risks for the stationary and nonstationary conditions were 55.4% and 96.8%, respectively (Fig. 9c).
The implication is that, for a design low-flow quantile zp0 corresponding to T0 = 50, if stationarity was incorrectly assumed, the
probability of low-flow event lower than zp0 occurring before or
at year n = 40 would be 55.4%, whereas in reality, because of nonstationarity, the probability would be 96.8%.
4.4. Nonstationary return period and risk using meteorological
covariates
Having determined the optimal nonstationary model using
meteorological covariates with estimated parameters l0 and r0
(Table 2) and downscaled future Temp and Prep (Fig. 8), the nonstationary return period T of the design low-flow quantile zp0 corresponding to the specified initial return period T0 was computed
under the ENE interpretation of return period using Eq. (10). For
Huaxian station, the comparison of T and T0 under the ENE interpretation with meteorological covariates (Fig. 10a) was quite different to those under the EWT and ENE interpretations with time
covariate (Fig. 9a). In the case of the ENE interpretation with meteorological covariates, for T0 < 20, the nonstationary T was slightly
longer than the stationary T0, while for T 0 P 20, the nonstationary
T was shorter than the stationary T0 (Fig. 10a), but to a lesser
degree than the case of using time covariate (Fig. 9a). For example,
when T0 = 10 years, the value of T under the correct assumption of
nonstationarity was 12 years, while when T0 = 50 years, the value
of T under the correct assumption of nonstationarity was 33 years.
This characteristic of the nonstationary T can be more clearly
reflected from Xianyang station (Fig. 10b) where the values of T
corresponding to T0 = 10 and T0 = 50 were 20 and 39 years, respectively. In this case, the implications for water resources planning
decisions based on an incorrect assumption of stationarity would
depend on the magnitude of T0.
Given the initial return period T0 and design life n, the hydrological risk R of the design low-flow quantile zp0 for stationary and nonstationary conditions was calculated using Eqs. (11) and (12),
respectively (Fig. 10c and d). The nonstationary risk results
(Fig. 10c and d) were quite similar to those for the nonstationary
return periods (Fig. 10a and b). For Huaxian station, for n < 20, the
R for nonstationary conditions was slightly lower than the R for stationary conditions, while for n P 20, the R for nonstationary
Fig. 9. Nonstationary return period T and hydrological risk R of the Wei River design low-flow quantile zp0 (corresponding to the initial return period T0) under the EWT and
ENE interpretations, where the future exceedance probabilities pt of Eqs. (6), (10) and (12) are derived from the optimal nonstationary model with time as covariate
(Fig. 4a and b). (a) and (b) are relations of the nonstationary return period T and the initial return period T0; (c) and (d) are nonstationary hydrological risks R as a function of
design life n for zp0 with different initial return periods T0 (the dashed lines are the risk under stationary conditions and the solid lines are the risk under nonstationary
conditions).
T. Du et al. / Journal of Hydrology 527 (2015) 234–250
247
Fig. 10. Nonstationary return period T and hydrological risk R of the Wei River design low-flow quantile zp0 (corresponding to the initial return period T0) under the ENE
interpretation, where the future exceedance probabilities pt of Eqs. (10) and (12) are derived by substituting the downscaled Temp and Prep of 2010–2099 (Fig. 8) into the
optimal nonstationary model with meteorological covariates (Fig. 4c and d). (a) and (b) are relations of the nonstationary return period T and the initial return period T0; (c)
and (d) are nonstationary hydrological risks R as a function of design life n for zp0 with different initial return periods T0 (the dashed lines are the risk under stationary
conditions and the solid lines are the risk under nonstationary conditions). Because of the inverse solving of Eq. (10) and use of meteorological covariates, the nonstationary T
and R are not as smooth as the case of Fig. 9.
conditions was higher than the R for stationary conditions (Fig. 10c).
For example, when T0 = 50 and n = 10 years, the risks for the stationary and nonstationary conditions were 18.3% and 14.1%, respectively. The implication is that, for a design low-flow quantile zp0
corresponding to T0 = 50, if stationarity was incorrectly assumed,
the risk of a low-flow event lower than zp0 occurring before or at
year n = 10 would be 18.3%, whereas in reality, because of nonstationarity, the probability of this event was 14.1%. In contrast, when
T0 = 50 and n = 40 years, the nonstationary risk probability was
79.0%, which was much higher than the stationary case of 55.4%.
This characteristic of the nonstationary R can be more clearly
reflected from Xianyang station (Fig. 10d) where the values of R corresponding to T0 = 50, n = 10 and T0 = 50, n = 40 were 5.8% and
68.4%, respectively. In this case, the implications for risk associated
with water resources planning based on an incorrect assumption of
stationarity would depend on the chosen design life n.
4.5. Discussion
The nonstationary return periods and risks of low-flow events
using either time or meteorological variables as covariates were
clearly different from those where the incorrect assumption of stationarity was applied (Figs. 9 and 10). This result demonstrates the
importance of considering nonstationarity when estimating return
period and hydrological risk. There were also large differences
between the results for the two kinds of covariate under the nonstationary framework.
When using time alone as covariate, under both the EWT and
ENE interpretations, both the nonstationary return period and
hydrological risk values suggest that, in the future, the occurrence
of low-flow events will be a more serious problem compared with
that suggested by analysis based on the assumption of stationarity (Fig. 9). While when using temperature and precipitation as
covariates in the nonstationary model under the ENE interpretation, the comparison of return period and risk of low-flow events
with the stationary model depends on the magnitude of the initial
return period and the length of the design life (Fig. 10). Using
time as covariate, the fitted moments of the observed low-flow
series, i.e. the mean E(Z) and standard deviation SD(Z), derived
from the fitted statistical parameters l and r according to the
relations in Table 1 monotonously decrease with time for both
stations (Fig. 11 black lines). However, there has been a noticeable upward movement in the annual minimum monthly streamflow since the mid-1990s (Fig. 2a and b), which contradicts the
patterns of sustained decrease in E(Z) and SD(Z) over time
(Fig. 11 black lines). Therefore, simply using time alone as covariate and assuming the statistical parameters monotonously
change indefinitely is inappropriate. In contrast, the nonstationary model using temperature and precipitation as covariates provides better model performance (Figs. 4–6 and Table 2) and more
reasonable statistical parameters and fitted moments (Fig. 11 red
lines). Overall, the analysis suggests that the nonstationary return
period and risk of a low-flow event derived by a model that
includes meteorological variables would produce more reliable
information than time covariate to assist decision making in the
management of water resources during naturally dry periods that
are being progressively exacerbated over time by the effects of
climate change.
248
T. Du et al. / Journal of Hydrology 527 (2015) 234–250
Fig. 11. Results of the fitted moments, i.e. the mean E(Z) and standard deviation SD(Z) (Table 1), of the observed low-flow series from (a) Huaxian and (b) Xianyang stations.
For each station, E(Z) (solid lines) and SD(Z) (dotted lines) are derived from the fitted statistical parameters l and r of two nonstationary models with different covariates, one
(black lines) is the optimal nonstationary model using time as covariate (Fig. 4a and b), and the other (red lines) is the optimal nonstationary model using meteorological
covariates (Fig. 4c and d). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
5. Conclusions
Two interpretations of return period, i.e. the expected waiting
time (EWT) and expected number of exceedances (ENE), were considered to explore the nonstationary return period and risk of the
annual minimum monthly streamflow in the Wei River, China.
Under the ENE interpretation, meteorological variables were
employed in the nonstationary frequency analysis and downscaled
future meteorological variables from the CMIP5 GCMs outputs
were used for deriving the future statistical parameters of the
low-flow distribution and the exceedance probabilities for future
years. This approach was compared with the case of using time
as covariate in the frequency analysis under both the EWT and
ENE interpretations. A number of conclusions that have implications for assessing risk in water resources management in the situation of nonstationary river flow data were drawn as follows.
(1) The annual minimum monthly streamflow series of both
Huaxian and Xianyang stations in the Wei River from 1954
to 2009 exhibited a strong nonstationarity. When using time
as covariate, the Weibull and Gamma distributions (both
with logarithmic link functions for the two model parameters) were the best distributions for modelling the observed
low-flow series of Huaxian and Xianyang, respectively.
Significant nonstationarities were detected in both model
parameters and the optimal nonstationary model expressed
both log-transformed parameters as linear functions of time.
When using meteorological variables as covariates, the
Weibull distribution (with logarithmic link functions for
the two model parameters) was the best distribution for both
stations. For Huaxian, the optimal nonstationary model
expressed both log-transformed parameters as linear functions of annual average temperature and annual total precipitation, respectively. And for Xianyang, the log-transformed
location parameter was modelled as linear function of annual
average temperature but the scale parameter was modelled
as a constant. For both stations, the optimal nonstationary
model of using meteorological covariates performed better
than that of the case using time as covariate in terms of
the AIC value and the worm plot.
(2) Different covariates used in nonstationary frequency calculation will lead to different results of the nonstationary
return period and risk analysis. When using time as covariate, there were significant differences between the nonstationary return period and risk under both the EWT and
ENE interpretations and those corresponding to the stationary condition. The nonstationary return period under the
ENE interpretation was a little bit longer than the case of
EWT but they were both much shorter (more frequent event
occurrence) than the stationary case, indicating that the
scarcity of water resources during dry seasons will worsen
over time. However, this result may overstate the risk of a
low-flow event due to inappropriate fitted statistical parameters (moments). There is an evidence of an increase in
low-flow events since the mid-1990s, but the nonstationary
return period and risk were derived under the assumption
that both model parameters monotonously change indefinitely. In contrast, the nonstationary model used the physical variables temperature and precipitation as covariates,
provided more appropriate fitted statistical parameters
(moments) and thus more reasonable nonstationary estimates of return period and risk. On the basis of the Wei
River data, the nonstationary analysis of return period and
risk using meteorological covariates is recommended than
using time for generating information to assist decision
making for the management of water resources during dry
seasons exacerbated by climate change. This conclusion is
likely to apply to the many similar situations around the
world.
(3) Some uncertainties about the nonstationary return period
and risk analysis of extreme hydrological events should be
noted. We used only one of the four Representative
Concentration Pathways adopted by the IPCC for its fifth
Assessment Report, and future research could consider the
others, which have smaller radiative forcing target levels.
Uncertainties exist in the downscaled future scenarios of
meteorological variables used here, and this could be partly
addressed by exploring more downscaling methods and
GCMs. Nonetheless, the purpose of the study was to understand and quantify the differences resulted from using time
T. Du et al. / Journal of Hydrology 527 (2015) 234–250
as the sole covariate and using meteorological variables as
covariates in the nonstationary frequency analysis, rather
than to compare the performance of different GCMs and
their scenarios. Further work can be conducted to investigate the uncertainty of the design low-flow quantile and
how this affects uncertainty of the estimates of nonstationary return period and risk.
Acknowledgements
This research is financially supported by the National Natural
Science Foundation of China (Grants NSFC 51190094 and NSFC
51479139), which are gratefully acknowledged. The contribution
of C.J. Gippel was made whilst visiting the College of Water
Resources and Hydropower Engineering, Wuhan University, supported by the High-End Foreign Expert Recruitment Programme,
administered by the State Administration of Foreign Experts
Affairs, Central People’s Government of the People’s Republic of
China. We are very grateful to the editor and two anonymous
reviewers for their valuable comments and constructive suggestions that helped us to greatly improve the manuscript.
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