Journal of Hydrologyfolk.uio.no/chongyux/papers_SCI/jhydrol_43.pdflogical time series is the method of time-varying moment, which assumes that although the distribution function type

Journal of Hydrology 527 (2015) 234–250

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Journal of Hydrology

journal homepage: www.elsevier .com/locate / jhydrol

Return period and risk analysis of nonstationary low-flow seriesunder climate change

http://dx.doi.org/10.1016/j.jhydrol.2015.04.0410022-1694/� 2015 Elsevier B.V. All rights reserved.

⇑ Corresponding author. Tel.: +86 13871078660; fax: +86 27 68773568.E-mail addresses: [email protected] (T. Du), [email protected] (L. Xiong),

[email protected] (C.-Y. Xu), [email protected] (C.J. Gippel), [email protected] (S. Guo), [email protected] (P. Liu).

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, Chinab Department of Geosciences, University of Oslo, P.O. Box 1022 Blindern, N-0315 Oslo, Norwayc Australian Rivers Institute, Griffith University, Nathan, Queensland 4111, Australiad Hubei Provincial Collaborative Innovation Center for Water Resources Security, Wuhan University, Wuhan 430072, China

a r t i c l e i n f o s u m m a r y

Article history:Received 2 December 2014Received in revised form 3 April 2015Accepted 20 April 2015Available online 7 May 2015This manuscript was handled by AndrasBardossy, Editor-in-Chief, with theassistance of Peter F. Rasmussen, AssociateEditor

Keywords:Return periodRiskNonstationarityLow-flowGeneral Circulation Models (GCMs)

Return period and risk of extreme hydrological events are critical considerations in water resources man-agement. The stationarity assumption of extreme events for conducting hydrological frequency analysisto estimate return period and risk is now problematic due to climate change. Two different interpreta-tions 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 anal-ysis by introducing the time-varying moment method into frequency analysis, under the assumption thatthe statistical parameters are functions only of time. This paper aimed at improving the characterizationof nonstationary return period and risk under the ENE interpretation by employing meteorologicalcovariates in the nonstationary frequency analysis. The advantage of the method is that the downscaledmeteorological variables from the General Circulation Models (GCMs) can be used to calculate the non-stationary statistical parameters and exceedance probabilities for future years and thus the correspond-ing return period and risk. The traditional approach using time as the only covariate under both the EWTand ENE interpretations was also applied for comparison. Both approaches were applied to annual min-imum monthly streamflow series of two stations in the Wei River, China, and gave estimates of nonsta-tionary return period and risk that were significantly different from the stationary case. Thenonstationary return period and risk under the ENE interpretation using meteorological covariates werefound 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 towater resources management during dry seasons exacerbated by climate change.

� 2015 Elsevier B.V. All rights reserved.

1. Introduction

Statistical inference is used in hydrological frequency analysisunder the assumption that hydrological events such as flood anddrought are randomly distributed through time. A preconditionfor traditional frequency analysis of hydrological variables is theassumption of stationarity, which means that the controlling envi-ronmental factors such as climate and land cover act to generate ormodify the hydrological variable of interest in the same way in thepast, present and future (Gilroy and McCuen, 2012; Katz, 2013;Benyahya et al., 2014; Charles and Patrick, 2014). However, underthe conditions of climate change, land use change and river

regulation, acting individually or together, the assumption of sta-tionarity is suspected, and existing methodology may no longerbe valid (Katz et al., 2002; Xiong and Guo, 2004; Milly et al.,2005, 2008). To overcome this problem, various approaches havebeen developed for conducting nonstationary hydrological fre-quency analysis (Khaliq et al., 2006). The concept of event returnperiod (or recurrence interval) and the associated risk of occur-rence, with potential consequences of loss of life, social disruption,economic loss and ecological disturbance, are critical considera-tions in the management of water resources, especially with regardto design and operation of hydraulic structures in rivers. Undernonstationary conditions, estimates of return period and risk areambiguous unless the temporally changing environment is explic-itly considered in the analysis.

The most common way of handling nonstationarity in hydro-logical time series is the method of time-varying moment, whichassumes that although the distribution function type of the

T. Du et al. / Journal of Hydrology 527 (2015) 234–250 235

hydrological variable of interest remains the same, the statisticalparameters 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-varyingmoment, it is not difficult to derive a value of a hydrological vari-able for a return period, with a specific design quantile (Olsen et al.,1998; Villarini et al., 2009), i.e. Tt ¼ 1=pt ¼ 1=ð1� FZðzp0 ; htÞÞ,where Tt and pt are the annual return period and exceedance prob-ability, respectively, of the given design quantile zp0

with fittedannual statistical parameters ht, and FZ is the cumulative distribu-tion function of the hydrological variable of interest. However, formany planning and design applications, a measure of return periodthat varies from one year to the next is impractical. To deal withthis issue, various studies have been carried out for return periodestimation 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 thesevarious studies two different interpretations of return period havealready 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 excee-dances (ENE) of the event in T years is 1 (Parey et al., 2007, 2010;Cooley, 2013).

Wigley (1988, 2009) used the EWT interpretation to considerhow nonstationarity can be included in the concepts of return per-iod and risk of extreme events. A normal distribution with a linearincreasing trend in the mean was assumed, and the changes in thereturn period and risk were derived by the technique of stochasticsimulation. Building on the work of Wigley (1988), Olsen et al.(1998) presented a more rigorous mathematical examination ofthe effect of nonstationarity on the concepts of return period andrisk, also using the EWT interpretation. Parey et al. (2007, 2010)introduced the ENE interpretation into the nonstationary frame-work to derive return levels of air temperature in France. A detailedreview and comparison of the two interpretations of return periodcan be found in Cooley (2013). Recently, Salas and Obeysekera(2013, 2014) extended the geometric distribution to allow forchanging exceedance probabilities over time, considering the casesof increasing, decreasing and shifting extreme events. Althoughthese studies vary considerably in their scope and the methodolo-gies employed, they have in common the assumption that the sta-tistical parameters are functions only of time. However, this carriesthe unreasonable implication that the identified pattern of nonsta-tionarity in a hydrological time series will continue indefinitely.Also, while runoff can follow inter-year cyclical patterns, the lackof a direct physical link between time and runoff means that timealone is not quite sufficient as an explanatory variable.

The time-varying moment method can be extended to performcovariate analysis by replacing time with any physical factors thatare known to be causative of the hydrological variable of interest.Using meteorological variables as covariates could be more effec-tive and have clearer physical meaning for modelling return periodand risk under nonstationary conditions than simply using time ascovariate. This physical covariate analysis approach has previouslybeen explored for nonstationary frequency analysis of extremeevents (Coles, 2001; Villarini et al., 2010; López and Francés,2013), but to our knowledge it has not been incorporated witheither the EWT or ENE interpretation of return period in estimatingreturn period and risk of extreme events under nonstationaryconditions.

Like many other places around the world, hydrological pro-cesses in China are under the influence of climate change, so theassumption of stationarity of river flow series is not valid for manyrivers (Zhang et al., 2011). The Wei River, the largest tributary of

the Yellow River, is the major source of water supply for the eco-nomic hub of Western China – the Guanzhong Plain. The WeiRiver basin is one of the most important industrial and agriculturalproduction zones in China. However, in recent decades the WeiRiver basin has suffered a significant decrease in streamflow(Song et al., 2007; Zuo et al., 2014), which threatens industrialand agricultural production and socioeconomic development. Theincreasing scarcity of water resources under conditions of climatechange is a serious concern not only in the Wei River basin, butalso across the entire nation. There is an urgent need to providewater resource managers and policy makers with reliable informa-tion on the return period and risk characteristics of the low-flowcomponent of the flow regime of the Wei River under the prevail-ing nonstationary conditions.

For those reasons, the main goal of this paper is to define andapply an integrated approach for understanding and quantifyingthe differences in calculating the nonstationary return period andrisk of extreme low-flow resulted from using time as the solecovariate and using meteorological variables as covariates in thenonstationary frequency analysis of the low-flow series. Note thatthe economic development and human activities might have par-tially contributed in the nonstationarity of extreme hydrologicalevents as identified in other studies (Xiong et al., 2014; Jianget al., 2015b). Thus, in addition to climatic factors, anthropogenicfactors, such as irrigation area, population, and industrial con-sumption, should be considered as covariates in investigating thecauses of nonstationarity of low-flow events. However, theseanthropogenic factors have not, for the moment, been incorporatedin this paper in investigating the nonstationary exceedance proba-bilities of low-flow events for future years for the reason that thevalues of these anthropogenic factors in the future years are verydifficult to be determined, or the estimated values have too biguncertainties even compared to the GCMs estimates of climaticfactors. So, in the present study we would like to limit the explana-tory factors of future low-flow nonstationarity to just climaticfactors.

When using meteorological covariates, GCMs outputs from theCoupled Model Intercomparison Project Phase 5 (CMIP5) providethe statistical parameters of the nonstationary low-flow distribu-tion by substituting downscaled future meteorological variablesinto the derived optimal nonstationary model to extend the excee-dance probabilities into the future. The EWT interpretation ofreturn period requires infinite (or as long as possible) future excee-dance probabilities (Cooley, 2013) but the GCMs outputs are tem-porally finite (most of the GCMs just provide continuouslarge-scale daily predictors to the year of 2100) (Riahi et al.,2011; van Vuuren et al., 2011), so the meteorological covariatescannot be incorporated in estimating return period and risk ofextreme events under nonstationary conditions with the EWTinterpretation. However, the ENE interpretation of return periodrequires just finite future exceedance probabilities, which can befully provided by the GCMs outputs, and thus the meteorologicalcovariates are adopted in the nonstationary return period and riskanalysis with the ENE method. Besides, the traditional approachusing time as the only covariate under both the EWT and ENEinterpretations is also applied in this study for the purpose of com-parison with the approach using meteorological covariates underthe ENE interpretation. The practical application of this approachis illustrated using a case study of the Wei River basin.

This paper is organized as follows. The next section describesthe Wei River basin and the available data sets used in the study.Then the methods for determining the return period (under theEWT and ENE interpretations) and risk under stationary and non-stationary conditions are described, along with a brief outline ofthe methods of nonstationary frequency analysis of low-flow series

236 T. Du et al. / Journal of Hydrology 527 (2015) 234–250

using time-varying moment, and statistical downscaling of meteo-rological variables. The results and discussion of the Wei River casestudy follow, and finally the conclusions about the performance ofthe proposed approach in the case study and its potential for widerapplication are drawn.

2. Study area and data

2.1. General characteristics of the study area

The Wei River, the largest tributary of the Yellow River, origi-nates from the Niaoshu Mountain at an elevation of 3485 m abovemean sea level in the Weiyuan county of Gansu province. The WeiRiver has a length of 818 km and a drainage area of 134800 km2,covering the coordinates of 33�400–37�260N, 103�570–110�270E inthe southeastern part of the loess plateau (Fig. 1). The Wei Riverhas 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’ ofthe Guanzhong Plain of the southern part of the loess plateaubecause of its key role in the economic development of westernChina (Song et al., 2007; Zuo et al., 2014).

The Wei River basin is characterized by semi-arid andsub-humid continental monsoon climate. Average annual precipi-tation 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 precip-itation ranging from 800 to 1000 mm, whereas the northern regionhas a semi-arid climate with annual precipitation ranging from 400to 700 mm. The annual average temperature over the entire basinranges from 6 to 14 �C. The range in the annual potential evapo-transpiration is 660–1600 mm, and the basin average annualactual evapotranspiration from the land surface is about 500 mm.

Fig. 1. Location, topography, hydro-meteorological stations and river systems of the Wethe South China Sea.

The average annual natural discharge of the Wei River is about10 � 109 m3, contributing approximately 17% of the discharge ofthe Yellow River. The annual discharge of the Wei River atLinjiacun station during the decade 1991–2000 was 53.9% less thanthat during the preceding decade, while at Huaxian station nearthe basin outlet it was 50.3% less (Song et al., 2007).

2.2. Data

Four kinds of data were used in this study: observed hydrolog-ical data, observed meteorological data, NOAA National Centres forEnvironmental Prediction (NCEP) reanalysis data, and GCMs out-puts from the CMIP5.

Observed mean daily streamflow data from both Huaxian andXianyang gauging stations (Fig. 1) over the period 1954–2009, pro-vided by the Hydrology Bureau of the Yellow River ConservancyCommission, were the source of the low-flow series (defined hereas the annual minimum monthly streamflow) (Fig. 2a and b).Huaxian hydrological station, located at 109�460E, 34�350N, is about70 km upstream of the junction of the Wei and Yellow rivers(Fig. 1). The catchment area upstream of this station is106500 km2, or about 80% of the total basin area. Xianyang hydro-logical station, located at 108�420E, 34�190N, is about 120 kmupstream of Huaxian station with a drainage area of 46480 km2

(Fig. 1).Temperature and precipitation are two meteorological variables

that are closely related to streamflow and were chosen as covari-ates for nonstationary frequency analysis of low-flow. Observeddaily average temperature and daily total precipitation series from22 stations (Fig. 1) for the period 1954–2009 were obtained fromthe National Climate Center of the China MeteorologicalAdministration (source: http://cdc.cma.gov.cn). Considering thecondition of snowpack at the headwater part of the basin may have

i River basin. The small inset box inside the main China map contains the islands of

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 theobserved 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 low-flow series and winter total precipitation PrepW.

T. Du et al. / Journal of Hydrology 527 (2015) 234–250 237

influence on the nonstationarity of the observed streamflow, apre-check on the form of the precipitation received at the headwa-ter part was carried out. Result indicated that almost all the precip-itation in the three most west meteorological stations at theheadwater part, Yuzhong, Lintao and Minxian, was received inthe form of rainfall, which eliminated the concern about snowpack

influence in this study. Then, the areal average daily series of bothvariables (daily average temperature and daily total precipitation)for the catchments above Huaxian and Xianyang stations weregenerated using the Thiessen polygon method (Szolgayova et al.,2014), and from these the annual average temperature and annualtotal precipitation series (denoted by Temp and Prep, respectively)

238 T. Du et al. / Journal of Hydrology 527 (2015) 234–250

(Xiong et al., 2014) over the period of 1954–2009 were extractedfor each catchment. As expected, there are linear relations betweenthe observed low-flow series and these two annual statistics ofmeteorological 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, seasonalstatistics of precipitation and temperature might be more suitableas covariates than annual statistics for modelling the low-flowevents. Considering most of the observed low-flow events occurredin January and December, winter (December, January andFebruary) average temperature and winter total precipitation(denoted by TempW and PrepW, respectively) were also examined.Correlation plots between the observed low-flow series andTempW and PrepW (Fig. 2g–j) indicated that seasonal statisticshad lower correlation coefficients comparing with annual statisticsfor both stations and thus will not be considered further in thisstudy.

The NCEP reanalysis daily data and GCMs daily data wereemployed to calibrate the statistical downscaling model and derivefuture temperature and precipitation scenarios respectively (Wilbyet al., 2002; Wilby and Dawson, 2007). The NCEP reanalysis datawere used to calibrate the multiple linear regression equationbetween each predictand (daily average temperature and dailytotal precipitation) and NCEP large-scale predictors. Then, futurescenario of the predictand was projected by substituting theGCMs large-scale predictors into the calibrated multiple linearregression equation. The 26 candidate predictors of NCEP reanaly-sis data as described in Wilby and Dawson (2007) for the period of1954–2009 were obtained from the NOAA Earth System ResearchLaboratory (ESRL) (source: http://www.esrl.noaa.gov). TheRepresentative Concentration Pathways (RCPs) are four green-house gas concentration and emissions pathways adopted by theIPCC for its fifth Assessment Report (AR5), each one is namedaccording to radiative forcing target level in watts per squaremetre for year 2100 (van Vuuren et al., 2011). In this study we usedRCP8.5 scenario, which represents the upper bound of the RCPs,but future work could extend our analysis by examining the otherRCPs with smaller radiative forcing target levels. RCP8.5 scenario ischaracterized by increasing greenhouse gas emissions over time,representative of scenarios in the literature that leads to compara-tively high greenhouse gas concentration levels, and does notinclude 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 forthe 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 datasets were interpolated to each meteorological station site. For eachmeteorological station, the grid it locates in and the eight gridsaround it were used for the interpolation of NCEP and GCMs out-puts (large-scale predictors) with the Inverse Distance Weightingmethod (Bartier and Keller, 1996). Then, areal average series ofevery predictor for the catchments above Huaxian and Xianyangstations were calculated using the Thiessen polygon method.

3. Methodology

In this section, firstly, the exceedance probability of a low-flowevent, which is the key element for determining the return periodand risk, is defined. Then, theories about the return period (underboth EWT and ENE interpretations) and risk of low-flow eventsconcerning the future exceedance probability under stationaryand nonstationary conditions are described. Finally, in deriving

the future exceedance probability for the determination of thenonstationary return period and risk of a low-flow event, thetime-varying moment method is employed in the nonstationaryfrequency analysis of the observed low-flow series by using timeor meteorological variables as covariates. When using meteorolog-ical variables as covariates, the downscaled future meteorologicalvariables from the GCMs are used to calculate the statisticalparameters and exceedance probabilities for future years. For thesake of completeness, the methods used in this section are brieflydescribed in the following subsections.

3.1. Exceedance probability of a low-flow event

The low-flow character of the flow regime is denoted by therandom variable Z. Our interest is on the scarcity of waterresources, so we define the design low-flow quantile zp0 which inany given year has a probability p0 that the streamflow is lowerthan this quantile (Fig. 3a). The probability of a flood event thatis higher than a design flood quantile is usually referred to as theexceedance probability, or as the exceeding probability (e.g. Salasand Obeysekera, 2013, 2014). In the case of low-flow (drought)event, the meaning of exceedance or exceeding is that the droughtseverity is exceeded, or the value of the flow statistic is lower thanthe design quantile.

Under stationary conditions, the cumulative distribution func-tion of Z is denoted by FZ(z, h), where h is the constant statisticalparameter set. The focus of this study is to analyse the future evo-lution of return period and hydrological risk of a given designlow-flow decided at a specific year based on the historical observeddata. This specific year is normally defined as the base year or theinitial year and denoted by t = 0. Thus, the given design low-flow att = 0 corresponding to an initial return period T0 can be derived byzp0 ¼ F�1

Z ðp0; hÞ, where p0 = 1/T0, and F�1Z is the inverse function of FZ

(similarly hereinafter). The period of years after the initial year isreferred to as ‘‘future’’. In the present study, we adopt Cooley’smethod (Cooley, 2013) where the initial year t = 0 corresponds tothe last observation year. In the stationary case, which means thecontrolling environmental factors for future years are the sameas the initial year t = 0. h is constant for every year, and the excee-dance probability corresponding to the design quantile zp0 is p0 foreach future year (Fig. 3a), which can be obtained by:

pt ¼ FZðzp0; hÞ ¼ p0; t ¼ 1;2; . . . ;1 ð1Þ

Under nonstationary conditions, the cumulative distributionfunction of Z is denoted by FZ(z, ht), where ht varies in accordancewith time or, more directly, with meteorological variables. In non-stationary case, the design low-flow quantile corresponding to theinitial exceedance probability p0 = 1/T0 can be derived fromzp0¼ F�1

Z ðp0; h0Þ, where h0 is the statistical parameter set of the ini-tial year t = 0. The statistical parameters are time-varying so thefuture exceedance probability corresponding to zp0 is not constantany more. In this case, the temporal variation in the exceedanceprobability corresponding to zp0

can be characterized by the waythe low-flow distribution or, more specifically, the statisticalparameters change through time (Fig. 3b). The exceedance proba-bility for each future year can be obtained by:

pt ¼ FZðzp0; htÞ; t ¼ 1;2; . . . ;1 ð2Þ

Generally, ht in Eq. (2) is derived by taking time as the onlycovariate in nonstationary modelling of the return period and riskanalysis as described in the introduction (Wigley, 1988, 2009;Olsen et al., 1998; Parey et al., 2007, 2010; Cooley, 2013; Salasand Obeysekera, 2013, 2014). In this study, meteorologicalcovariates, which have clear physical meanings, are considered innonstationary modelling. Thus, ht can be derived from the

Fig. 3. Schematic depicting the design low-flow quantile zp0with (a) constant exceedance probability p0, and (b) time-varying exceedance probabilities pt, t = 1, 2, . . .,1.

T. Du et al. / Journal of Hydrology 527 (2015) 234–250 239

downscaled meteorological covariates of GCMs for future years,and then be used to more physically estimate the evolution ofthe exceedance probabilities pt, return period and risk of zp0 inthe future years.

3.2. Return period using EWT interpretation

Under stationary conditions, if X is denoted as the random vari-able representing the year of the first occurrence of a low-flow thatexceeds (i.e. is lower than) the design quantile, then a low-flow Zexceeding 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� p0Þx�1p0; x ¼ 1;2; . . . ;1 ð3Þ

Noting that Eq. (3) is derived on the assumptions of indepen-dence and stationarity, then the expected value of X, i.e. the returnperiod (expected waiting time interpretation) of the low-flowexceeding the design quantile zp0 under stationary conditions, is:

T ¼ EðXÞ ¼X1x¼1

xf ðxÞ ¼ 1=p0 ð4Þ

Under nonstationary conditions, the exceedance probabilitycorresponding to zp0 is no longer constant (Fig. 3b), and then thegeometric probability law considering time-varying exceedanceprobabilities pt is (Cooley, 2013; Salas and Obeysekera, 2013,2014):

f ðxÞ ¼ PðX ¼ xÞ ¼ ð1� p1Þð1� p2Þ . . . ð1� px�1Þpx

¼ px

Yx�1

t¼1

ð1� ptÞ; x ¼ 1;2; . . . ;1 ð5Þ

The EWT-return period T of the low-flow exceeding the designquantile zp0

under nonstationary conditions is thus:

T ¼ EðXÞ ¼X1x¼1

xf ðxÞ ¼X1x¼1

xpx

Yx�1

t¼1

ð1� ptÞ ð6Þ

3.3. Return period using ENE interpretation

Under stationary conditions, if M is denoted as the random vari-able representing the number of exceedances in T years, thenM ¼

PTt¼1IðZt < zp0

Þ, where I(�) is the indicator function. M followsa binomial distribution (Cooley, 2013):

f ðmÞ ¼ PðM ¼ mÞ ¼T

m

� �pm

0 ð1� p0ÞT�m ð7Þ

It follows that the expected value of M is 1:

EðMÞ ¼XT

t¼1

p0 ¼ Tp0 ¼ 1 ð8Þ

And thus the return period (expected number of exceedances inter-pretation) of the low-flow exceeding the design quantile zp0

understationary conditions is T = 1/p0.

Under nonstationary conditions, the exceedance probability isnot constant and M does not follow a binomial distribution. In thissituation the expected number of exceedances is expressed as(Cooley, 2013):

EðMÞ ¼XT

t¼1

E½IðZt < zp0Þ� ¼

XT

t¼1

PðZt < zp0Þ ¼

XT

t¼1

FZðzp0; htÞ

¼XT

t¼1

pt ð9Þ

The ENE-return period T of the low-flow exceeding the design quan-tile zp0

under nonstationary conditions can thus be derived by set-ting Eq. (9) equal to 1 and solving:

240 T. Du et al. / Journal of Hydrology 527 (2015) 234–250

XT

t¼1

pt ¼ 1 ð10Þ

3.4. Hydrological risk

In practical application of hydrological frequency analysis, themanagement question is often framed as one of risk, whereby,for a design life of n years, the hydrological risk R is the probabilityof a low-flow event exceeding the design value zp0 before or at yearn. This risk can be derived from the perspective of complement,which means that there is no exceedance during the design lifeof n years. Under the assumption of independence and stationarity,the probability of the complement is (1 � p0)n. Then the hydrolog-ical risk under stationary conditions is (Haan, 2002):

R ¼ 1� ð1� p0Þn ð11Þ

In recent years, hydrological risk analysis under nonstationaryconditions has become increasingly popular (Rootzén and Katz,2013; Salas and Obeysekera, 2014; Condon et al., 2015; Serinaldiand Kilsby, 2015) and provides a different view from return periodand design level for designers by combining the basic informationof design life n and exceedance probabilities pt. As with the station-ary case, for a design life of n years, the probability of a low-flowexceeding the design quantile zp0 before or at year n under the cir-cumstances of time-varying exceedance probabilities pt is:

R ¼ 1� ½ð1� p1Þð1� p2Þ . . . ð1� pnÞ� ¼ 1�Yn

t¼1

1� ptð Þ ð12Þ

3.5. Nonstationary frequency analysis of low-flow series

The standard way of calculating the nonstationary returnperiods under the EWT and ENE interpretations and the risk of adesign quantile zp0 corresponding to an initial return period T0 isby Eqs. (6), (10) and (12), respectively. An important part of theprocedure is derivation of time-varying exceedance probabilitiespt (Eq. (2)) for future years. This relies on determination of the rela-tionships of the statistical parameters of the low-flow distributionto the explanatory variables, which is normally referred to asnonstationary frequency analysis. Several studies have explorednonstationary return period and risk analysis of extreme eventsusing only time as covariate in the nonstationary frequency

Table 1Summary of the distributions used to model the low-flow series in this study.

Distribution Probability density function

Gamma f Zðzjl;rÞ ¼ 1ðlr2Þ1=r

2zð1=r

2�1Þe�z=ðlr2 Þ

Cð1=r2Þ

z > 0;l > 0;r > 0Weibull f Zðzjl;rÞ ¼ rzr�1

lr exp � zl

� �rh iz > 0;l > 0;r > 0

Gumbel f Zðzjl;rÞ ¼ 1r exp � z�l

r� �

� exp � z�lr

� � � ��1 < z <1;�1 < l <1;r > 0

Logistic f Zðzjl;rÞ ¼ 1r exp � z�l

r� � � �

f1þ exp � z�lr

� � g�2

�1 < z <1;�1 < l <1;r > 0Lognormal f Zðzjl;rÞ ¼ 1ffiffiffiffiffi

2pp

r1z exp � ½logðzÞ�l�2

2r2

n oz > 0;l > 0;r > 0

Pearson-IIIf Zðzjl;r;jÞ ¼ 1

rjljjCð1=j2Þz�llrjþ 1

j2

� � 1j2�1

exp � z�llrjþ 1

j2

� �h ir > 0;j–0; z�l

lrjþ 1j2 P 0

GEV f Zðzjl;r;jÞ ¼ 1r 1þ j z�l

r� � ð�1=jÞ�1 exp � 1þ j z�l

r� � �1=j

n�1 < l <1;r > 0;�1 < j <1

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 variableshave physical meaning and therefore have more convincibleexplanatory power to be used as covariates. Thus, in the method pre-sented here, we derive the time-varying exceedance probabilities pt

for future years by employing meteorological covariates in the non-stationary frequency analysis of low-flow events and incorporatingthe statistical downscaling of future meteorological variables fromGCMs. As explained before, when meteorological covariates are usedfor describing the nonstationary frequency of low-flow events, onlythe ENE interpretation can be adopted for the nonstationary returnperiod and risk analysis, for the current GCMs outputs are providedonly for a finite time period and unable to meeting the indefinitedata requirement by the EWT interpretation (see Eq. (6)).

The nonstationary low-flow series was modelled using thetime-varying moment method, which was built under theGeneralized Additive Models in Location, Scale and Shape(GAMLSS) framework (Rigby and Stasinopoulos, 2005; Xionget al., 2014). Various probability distributions have been suggestedfor 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 dataof 34 streams and found the Gumbel and Pearson-III distributionsfitted the data well and were more representative than theLognormal and Pearson-V distributions. Eratakulan (1970) foundGamma and Weibull were the first two distributions to be selectedin modelling the low-flow series of 37 stations in the MissouriRiver basin. Hewa et al. (2007) introduced the GEV distributioninto the frequency analysis of low-flow data from 97 catchmentsof Victoria, Australia. Liu et al. (2015) tested six distributions inmodelling the annual low flows of the Yichang station, China undernonstationary conditions and found the GEV distribution gave thebest fit. Based on these studies, five two-parameter distributions,i.e. Gamma (GA), Weibull (WEI), Gumbel (GU), Logistic (LO), andLognormal (LOGNO), and two three-parameter distributions, i.e.Pearson-III (P-III) and GEV, that widely used in modellinglow-flow data were considered as candidates in this study(Table 1). Considering that the shape parameter j of P-III andGEV 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 nonstationaritiesin both the location l and scale r parameters were examinedthrough monotonic link functions g(�) (Table 1). The optimal

Moments Link functions

EðZÞ ¼ lSDðZÞ ¼ lr

g1ðlÞ ¼ lnðlÞg2ðrÞ ¼ lnðrÞ

EðZÞ ¼ lC 1rþ 1� �

SDðZÞ ¼ lffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC 2

rþ 1� �

� C 1rþ 1� � 2q g1ðlÞ ¼ lnðlÞ

g2ðrÞ ¼ lnðrÞ

EðZÞ ¼ lþ cr ’ lþ 0:57722rSDðZÞ ¼ pffiffi

6p r ’ 1:28255r

g1ðlÞ ¼ lg2ðrÞ ¼ lnðrÞ

EðZÞ ¼ lSDðZÞ ¼ pffiffi

3p r ’ 1:81380r

g1ðlÞ ¼ lg2ðrÞ ¼ lnðrÞ

EðZÞ ¼ w1=2el

SDðZÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwðw� 1Þ

pel

w ¼ expðr2Þ

g1ðlÞ ¼ lg2ðrÞ ¼ lnðrÞ

EðZÞ ¼ lCv ¼ rCs ¼ 2j

g1ðlÞ ¼ lnðlÞg2ðrÞ ¼ lnðrÞg3ðjÞ ¼ jo

EðZÞ ¼ l� rjþ r

j g1

SDðZÞ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig2 � g2

1

q=j

gm ¼ Cð1�mjÞ

g1ðlÞ ¼ lg2ðrÞ ¼ lnðrÞg3ðjÞ ¼ j

T. Du et al. / Journal of Hydrology 527 (2015) 234–250 241

nonstationary model was selected by penalizing more complexmodels 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 isthe number of independently adjusted parameters within themodel, and theoretically �1 < AIC <1. The model with the small-est AIC value was considered the optimal one.

While the AIC value identifies the optimal model, it is not ameasure of model performance. Goodness-of-fit of the selectedoptimal model was assessed qualitatively by the worm (Buurenand Fredriks, 2001) and the centile curves diagnostic plots, andquantitatively using the statistics of the Filliben correlation coeffi-cient (denoted by Fr) (Filliben, 1975) and the Kolmogorov–Smirnov(KS) test (denoted by DKS) (Massey, 1951). Fr and DKS are calculatedby Eqs. (14) and (16), respectively:

Fr ¼ CorðS;BÞ ¼Ps

i¼1ðSðiÞ � �SÞðBi � �BÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPsi¼1ðSðiÞ � �SÞ2

Psi¼1ðBi � �BÞ2

q ð14Þ

where S(i) are the ordered residuals derived by sorting U�1[FZ(zi, hi)],1 6 i 6 s in ascending order. U�1 is the inverse function of the stan-dard normal distribution and s is the length of the observation per-iod (similarly hereinafter). Bi are the standard normal order statisticmedians calculated from U�1(bi) where bi are derived by:

bi ¼1� bs

ði� 0:3175Þ=ðsþ 0:365Þ0:5ð1=sÞ

i ¼ 1i ¼ 2;3; . . . ; s� 1

i ¼ s

8><>: ð15Þ

Fr have the range of (0,1] and a Fr bigger than the critical value Faindicates that the nonstationary model passes the goodness-of-fittest.

DKS ¼max1�i�sjGi � GðiÞj ð16Þ

where Gi are the empirical cumulative probabilities calculated fromi/(s + 1), and G(i) are the ordered theoretical cumulative probabili-ties 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 valueDa indicates that the nonstationary model passes thegoodness-of-fit test.

To summarize, the main steps in deriving time-varying excee-dance probabilities pt for future years are:

(i) Nonstationary modelling of the observed low-flow series byusing time alone or meteorological variables as covariates.

(ii) Calculating the design low-flow quantile zp0 correspondingto an initial return period T0 from the quantile functionzp0¼ F�1

Z ðp0; h0Þ, where p0 = 1/T0 and h0 is the fitted statisti-cal parameter set of the initial year t = 0.

(iii) Deriving time-varying exceedance probabilities pt corre-sponding to zp0

for future years t = 1, 2, . . .,1 (time ascovariate) or t = 1, 2, . . ., tmax (meteorological covariates)from Eq. (2), where ht is calculated by extending the optimalnonstationary model of step (i) into the future under respec-tive case of covariates.

3.6. Statistical downscaling model (SDSM)

GCMs are a tool for predicting future time series of meteorolog-ical variables, thereby extending the time-varying exceedanceprobabilities of Eqs. (10) and (12). The coarse spatial resolutionof GCMs data restricts its direct application to local impact studies(Wilby et al., 2002; Wilby and Dawson, 2007), but this problem can

be overcome by a technique known as downscaling. The statisticaldownscaling model (SDSM) provided by Wilby et al. (2002) is adecision support tool for assessing local climate change impactsusing a robust statistical downscaling technique that combines aweather generator and multiple linear regression. SDSM has beenwidely used in research related to climate change (Wilby andDawson, 2013) and is fully described in Wilby et al. (2002) andWilby and Dawson (2007).

Theoretically, the downscaling process with SDSM is eitherunconditional (as with temperature) or conditional on an event(as with precipitation amounts). For unconditional process, a directmultiple linear regression equation between the unconditional pre-

dictand yUCi and the chosen normalized large-scale predictors u j

i onday i is constructed as (Wilby et al., 2003; Wetterhall et al., 2006):

yUCi ¼ c0 þ

Xl

j¼1

cjuji þ e ð17Þ

where cj are the estimated regression coefficients deduced by theleast square method, l is the number of chosen predictors and e isa normally distributed stochastic error term.

For conditional process, a conditional probability of predictandoccurrence xi on day i is directly expressed as a multiple linear

regression equation of u ji as:

xi ¼ g0 þXl

j¼1

gjuji ð18Þ

where gj are the estimated regression coefficients. If xi 6 ri, whereri is a uniformly distributed random number (0 6 ri 6 1), condi-tional predictand yC

i occurs with the amount of:

yCi ¼ F�1½UðZiÞ� 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

expression of Zi ¼ k0 þPl

j¼1kjuji þ e, where kj are the estimated

regression coefficients.Specifically, the downscaling of daily average temperature and

daily total precipitation was carried out according to the uncondi-tional and conditional processes, respectively. A correlation analysis(Wilby and Dawson, 2007; Hessami et al., 2008) between each pre-dictand and alternative NCEP large-scale predictors indicated thatmean sea level pressure, 500 hPa geopotential height, 500 hPa east-ward wind, 850 hPa air temperature, and near-surface air tempera-ture had higher correlations with daily average temperature thanother predictors and thus were selected for the downscaling of dailyaverage temperature. For daily total precipitation, one additionalpredictor, 850 hPa specific humidity, was included. The SDSM mod-els for both daily average temperature and daily total precipitationwere optimized with the respective selected NCEP predictors. Then,daily average temperature and daily total precipitation for the per-iod of 1954–2009 were simulated by the weather generator in theSDSM driven by the NCEP reanalysis predictors. The simulationresults were assessed by the Nash–Sutcliffe efficiency (NSE)between the simulated and observed Temp and Prep. NSE is calcu-lated as (Nash and Sutcliffe, 1970):

NSE ¼ 1�Ps

i¼1ðYobsi � Ysim

i Þ2

Psi¼1ðY

obsi � YmeanÞ

2 ð20Þ

where Yobsi are the observed meteorological variables, Ysim

i are the

simulated values corresponding to Yobsi , Ymean is the mean value of

Yobsi . 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. thecalibrated multiple linear regression equation, is assumed to betransferrable to future projection period (Wilby et al., 2002;Wilby and Dawson, 2007, 2013; Wetterhall et al., 2006; Mullanet al., 2012). Then with the input of the GCMs large-scale predic-tors, future scenarios of daily average temperature and daily totalprecipitation can be derived. Daily average temperature and dailytotal precipitation for the period of 2010–2099 generated by thescenario generator in the SDSM driven by the GCMs predictorswere then used to calculate the statistics of Temp and Prep forthe 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 andXianyang stations declined through time, with irregular scatter(Fig. 2a and b). Significant decreasing trends were detected by

Fig. 4. Summary of different distributions with different nonstationary models fitted tostations. Location l and scale r parameters modelled as functions of time in (a) and (b), abox is the optimal nonstationary model for respective case of covariate analysis. (For intthe web version of this article.)

the Mann–Kendall test (Mann, 1945; Kendall, 1975; Li et al.,2014) with the statistics ZMK = �2.71 and ZMK = �3.63 for Huaxianand Xianyang, respectively, compared to the critical value ofZ1�a/2 = 1.96 at a = 0.05. More detailed analysis of the nonstation-arity was undertaken as part of modelling the low-flow seriesunder the GAMLSS framework.

When the low-flow series were modelled using time as covari-ate, for Huaxian, the AIC values suggested that the Weibull (WEI)distribution (with logarithmic link functions for both the locationl and scale r parameters) was the optimal distribution, with bothlog-transformed parameters modelled as linear functions of time(Fig. 4a). While for Xianyang, the Gamma (GA) distribution (withlogarithmic link functions for both the location l and scale rparameters) was the optimal distribution, also with bothlog-transformed parameters modelled as linear functions of time(Fig. 4b). Indeed, the WEI and GA distributions performed verycomparable for both stations (Fig. 4a and b). However, we wouldstrictly follow the selection criterion of AIC for determining theoptimal nonstationary model for each station.

the observed low-flow series from Huaxian (left panel) and Xianyang (right panel)nd meteorological variables Temp and Prep in (c) and (d). The expressions in the red

erpretation of the references to colour in this figure legend, the reader is referred to

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 pointsshould 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 (theblue 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 greyregion 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 greyregion 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) 234–250 243

For Huaxian, sporadic worm points were without 95% confi-dence intervals (Fig. 5a), while for Xianyang, all the worm pointswere within the 95% confidence intervals (Fig. 5b), indicatingacceptable consistency between the selected model and theobserved low-flow data. The vast majority of the points werewithin the 5% and 95% centile curves for both stations(Fig. 5c and d) indicating that the model captured the variabilityof the data. The statistics of the Fr and the DKS (Table 2) also indi-cated that the selected nonstationary model was an adequate fitto the low-flow series for respective station.

In fitting the low-flow series to nonstationary models with theannual average temperature Temp and annual total precipitationPrep as covariates, the most complex model expressed both linkfunction-transformed statistical parameters l and r as linear func-tions of Temp and Prep. This process also included all possible sim-pler sub-models. The AIC values suggested that the WEIdistribution (with logarithmic link functions for both the locationl and scale r parameters) was the optimal distribution for bothHuaxian and Xianyang stations (Fig. 4c and d). For Huaxian, thelog-transformed l and r were modelled as linear functions ofTemp and Prep, respectively. And for Xianyang, thelog-transformed l was modelled as linear function of Temp, butr 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 ofAIC for determining the optimal nonstationary model for each sta-tion. Besides, the two seasonal meteorological covariates (winteraverage temperature and winter total precipitation) were alsoexamined for modelling the observed low-flow series. However,the AIC values of the optimal nonstationary models with the sea-sonal covariates were 870.9 and 831.7 for Huaxian and Xianyang,respectively, which were larger than the cases of using the annualcovariates with the AIC values of 863.8 and 817.1 for Huaxian andXianyang, respectively (Table 2). This further proved the rationalityof the selection of the annual covariates rather than winter covari-ates in modelling the nonstationarity of the observed low-flow ser-ies in this study.

All the worm points were within the 95% confidence intervalsfor both stations, indicating perfect consistency between theselected model and the observed low-flow data (Fig. 6a and b).The vast majority of the points were within the 5% and 95% centilecurves (Fig. 6c and d) indicating that the model captured the vari-ability of the data. The statistics of the Fr and the DKS (Table 2) alsoindicated that the selected nonstationary model was an adequatefit to the low-flow series for respective station. In addition, for eachstation, the optimal nonstationary model using meteorological

Table 2Summary 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 ascovariates. The critical values of the Filliben correlation coefficient is Fa = 0.978 and the KS test is Da ¼ 1:36=

ffiffiffiffiffiffi56p

� 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 Estimated parameters(standard errors)

AIC values Filliben correlationcoefficient Fr

KS statistic DKS

Huaxian station Nonstationary Weibulla (t as covariate) la ¼ 7:5266ð0:1652Þlb ¼ �0:0229ð0:0066Þra ¼ 0:7664ð0:2600Þrb ¼ �0:0145ð0:0082Þ

872.4 0.982 0.086

Nonstationary Weibullb (Temp and Prep as covariates) la ¼ 14:3504ð1:3879Þlb ¼ �0:7795ð0:1455Þra ¼ �0:6740ð0:2096Þrb ¼ 0:0020ð0:0003Þ

863.8 0.983 0.074

Xianyang station Nonstationary Gammac (t as covariate) la ¼ 7:1722ð0:1524Þlb ¼ �0:0244ð0:0055Þra ¼ �0:7104ð0:1978Þrb ¼ 0:0112ð0:0061Þ

828.5 0.986 0.099

Nonstationary Weibulld (Temp as covariate) la ¼ 15:7336ð1:4568Þlb ¼ �0:9804ð0:1553Þra ¼ 0:5528ð0:1054Þ

817.1 0.993 0.077

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 theobservation 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 wormplots (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% and75% 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 thedark 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 theweb version of this article.)

244 T. Du et al. / Journal of Hydrology 527 (2015) 234–250

Fig. 8. Projected meteorological variables from different GCMs for the future period2010–2099. The ensemble average is the arithmetic average value of all theindividual GCMs: (a) annual average temperature Temp for Huaxian station (�C); (b)annual total precipitation Prep for Huaxian station (mm); (c) annual averagetemperature Temp for Xianyang station (�C).

T. Du et al. / Journal of Hydrology 527 (2015) 234–250 245

covariates had a smaller AIC value than the optimal model usingtime alone as covariate (Fig. 4 and Table 2), confirming the neces-sity and effectiveness of employing physical covariate analysis inthe nonstationary frequency analysis of low-flow series.

4.2. Statistical downscaling of temperature and precipitation

For the optimal nonstationary model with Temp and Prep ascovariates, the future time series of these two variables allowedcalculation of the time-varying parameters lt and rt for futureyears, enabling estimation of the time-varying exceedance proba-bilities pt into the future. Since for Xianyang station only Tempwas selected in the optimal nonstationary model (Table 2), thedownscaling of Prep for this station will thus not be needed. TheNash–Sutcliffe efficiency (NSE) between the simulated andobserved Temp and Prep (Fig. 7) suggested an adequate result forTemp, but the result for Prep was not as good. This was expected,as it is acknowledged that the downscaling of precipitation is moreproblematic than temperature. Other authors have reported simi-lar findings (Wilby and Dawson, 2007; Chen et al., 2012; Yanget al., 2012). We considered the simulation result acceptable forthis purpose.

The projected annual time series of Temp and Prep for the futureperiod 2010–2099 for the seven GCMs (Fig. 8) showed strongincreasing trends for Temp of both stations (around 0.0596 �C peryear for the ensemble average for Huaxian, and 0.0573 �C per yearfor Xianyang), while the ensemble average value of Prep forHuaxian was stable at around 470 mm.

4.3. Nonstationary return period and risk using time as covariate

Having determined the optimal nonstationary model using timealone as covariate with estimated parameters l0 and r0 (Table 2),the nonstationary return periods T of the design low-flow quantilezp0

corresponding to the specified initial return period T0 werecomputed under both the EWT and ENE interpretations of returnperiod using Eqs. (6) and (10), respectively. For Huaxian station,the nonstationary return period T of zp0

under the ENE interpreta-tion was a little bit longer than the case of EWT but they were bothmuch shorter than the specified T0 (Fig. 9a). For example, whenT0 = 50 years, the values of T under nonstationarity were only18.6 and 22 years for the EWT and ENE interpretations, respec-tively. 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. 7. Comparisons between simulated and observed meteorological variables for the observation period 1954–2009: (a) annual average temperature Temp for Huaxianstation (�C); (b) annual total precipitation Prep for Huaxian station (mm); (c) annual average temperature Temp for Xianyang station (�C). All variables were simulated by theweather generator in the SDSM driven by the NCEP reanalysis predictors.

246 T. Du et al. / Journal of Hydrology 527 (2015) 234–250

t = 0. Using 50 years as the basis for water resources planning deci-sions for this exceedance event would be imprudent because non-stationarity of the low-flow series suggests that the value of Tequals to 19 or 22 years might be more appropriate. ForXianyang station which has relative smaller drainage area, the caseof nonstationary T was very similar to that of Huaxian(Fig. 9a and b), confirming the necessity of considering nonstation-arity in the return period analysis of the Wei River basin.

Given the initial return period T0 and design life n, the hydrolog-ical risk R of the design low-flow quantile zp0

for stationary andnonstationary conditions was calculated using Eqs. (11) and (12),respectively. Considering the risk results of the two stations werevery similar (Fig. 9c and d), we took the Huaxian result for illustra-tion. Under both stationary and nonstationary cases risk increasedwith n, but for any T0, the R for nonstationary conditions washigher than that for stationary conditions. For example, whenT0 = 50 and n = 40 years, the risks for the stationary and nonsta-tionary conditions were 55.4% and 96.8%, respectively (Fig. 9c).The implication is that, for a design low-flow quantile zp0

corre-sponding to T0 = 50, if stationarity was incorrectly assumed, theprobability of low-flow event lower than zp0

occurring before orat year n = 40 would be 55.4%, whereas in reality, because of non-stationarity, the probability would be 96.8%.

4.4. Nonstationary return period and risk using meteorologicalcovariates

Having determined the optimal nonstationary model usingmeteorological covariates with estimated parameters l0 and r0

Fig. 9. Nonstationary return period T and hydrological risk R of the Wei River design lowENE interpretations, where the future exceedance probabilities pt of Eqs. (6), (10) an(Fig. 4a and b). (a) and (b) are relations of the nonstationary return period T and the initiadesign life n for zp0

with different initial return periods T0 (the dashed lines are the risconditions).

(Table 2) and downscaled future Temp and Prep (Fig. 8), the nonsta-tionary return period T of the design low-flow quantile zp0

corre-sponding to the specified initial return period T0 was computedunder the ENE interpretation of return period using Eq. (10). ForHuaxian station, the comparison of T and T0 under the ENE inter-pretation with meteorological covariates (Fig. 10a) was quite dif-ferent to those under the EWT and ENE interpretations with timecovariate (Fig. 9a). In the case of the ENE interpretation with mete-orological covariates, for T0 < 20, the nonstationary T was slightlylonger than the stationary T0, while for T0 P 20, the nonstationaryT was shorter than the stationary T0 (Fig. 10a), but to a lesserdegree than the case of using time covariate (Fig. 9a). For example,when T0 = 10 years, the value of T under the correct assumption ofnonstationarity was 12 years, while when T0 = 50 years, the valueof T under the correct assumption of nonstationarity was 33 years.This characteristic of the nonstationary T can be more clearlyreflected from Xianyang station (Fig. 10b) where the values of Tcorresponding to T0 = 10 and T0 = 50 were 20 and 39 years, respec-tively. In this case, the implications for water resources planningdecisions based on an incorrect assumption of stationarity woulddepend on the magnitude of T0.

Given the initial return period T0 and design life n, the hydrolog-ical risk R of the design low-flow quantile zp0 for stationary and non-stationary 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 nonstationaryreturn periods (Fig. 10a and b). For Huaxian station, for n < 20, theR for nonstationary conditions was slightly lower than the R for sta-tionary conditions, while for n P 20, the R for nonstationary

-flow quantile zp0(corresponding to the initial return period T0) under the EWT and

d (12) are derived from the optimal nonstationary model with time as covariatel return period T0; (c) and (d) are nonstationary hydrological risks R as a function ofk under stationary conditions and the solid lines are the risk under nonstationary

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 theoptimal 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 stationaryconditions 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 Tand R are not as smooth as the case of Fig. 9.

T. Du et al. / Journal of Hydrology 527 (2015) 234–250 247

conditions was higher than the R for stationary conditions (Fig. 10c).For example, when T0 = 50 and n = 10 years, the risks for the station-ary and nonstationary conditions were 18.3% and 14.1%, respec-tively. 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 atyear n = 10 would be 18.3%, whereas in reality, because of nonsta-tionarity, the probability of this event was 14.1%. In contrast, whenT0 = 50 and n = 40 years, the nonstationary risk probability was79.0%, which was much higher than the stationary case of 55.4%.This characteristic of the nonstationary R can be more clearlyreflected from Xianyang station (Fig. 10d) where the values of R cor-responding to T0 = 50, n = 10 and T0 = 50, n = 40 were 5.8% and68.4%, respectively. In this case, the implications for risk associatedwith water resources planning based on an incorrect assumption ofstationarity would depend on the chosen design life n.

4.5. Discussion

The nonstationary return periods and risks of low-flow eventsusing either time or meteorological variables as covariates wereclearly different from those where the incorrect assumption of sta-tionarity was applied (Figs. 9 and 10). This result demonstrates theimportance of considering nonstationarity when estimating returnperiod and hydrological risk. There were also large differencesbetween the results for the two kinds of covariate under the non-stationary framework.

When using time alone as covariate, under both the EWT andENE interpretations, both the nonstationary return period and

hydrological risk values suggest that, in the future, the occurrenceof low-flow events will be a more serious problem compared withthat suggested by analysis based on the assumption of stationar-ity (Fig. 9). While when using temperature and precipitation ascovariates in the nonstationary model under the ENE interpreta-tion, the comparison of return period and risk of low-flow eventswith the stationary model depends on the magnitude of the initialreturn period and the length of the design life (Fig. 10). Usingtime as covariate, the fitted moments of the observed low-flowseries, i.e. the mean E(Z) and standard deviation SD(Z), derivedfrom the fitted statistical parameters l and r according to therelations in Table 1 monotonously decrease with time for bothstations (Fig. 11 black lines). However, there has been a notice-able upward movement in the annual minimum monthly stream-flow since the mid-1990s (Fig. 2a and b), which contradicts thepatterns of sustained decrease in E(Z) and SD(Z) over time(Fig. 11 black lines). Therefore, simply using time alone as covari-ate and assuming the statistical parameters monotonouslychange indefinitely is inappropriate. In contrast, the nonstation-ary model using temperature and precipitation as covariates pro-vides better model performance (Figs. 4–6 and Table 2) and morereasonable statistical parameters and fitted moments (Fig. 11 redlines). Overall, the analysis suggests that the nonstationary returnperiod and risk of a low-flow event derived by a model thatincludes meteorological variables would produce more reliableinformation than time covariate to assist decision making in themanagement of water resources during naturally dry periods thatare being progressively exacerbated over time by the effects ofclimate change.

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 meteorologicalcovariates (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.)

248 T. Du et al. / Journal of Hydrology 527 (2015) 234–250

5. Conclusions

Two interpretations of return period, i.e. the expected waitingtime (EWT) and expected number of exceedances (ENE), were con-sidered to explore the nonstationary return period and risk of theannual minimum monthly streamflow in the Wei River, China.Under the ENE interpretation, meteorological variables wereemployed in the nonstationary frequency analysis and downscaledfuture meteorological variables from the CMIP5 GCMs outputswere used for deriving the future statistical parameters of thelow-flow distribution and the exceedance probabilities for futureyears. This approach was compared with the case of using timeas covariate in the frequency analysis under both the EWT andENE interpretations. A number of conclusions that have implica-tions for assessing risk in water resources management in the sit-uation of nonstationary river flow data were drawn as follows.

(1) The annual minimum monthly streamflow series of bothHuaxian and Xianyang stations in the Wei River from 1954to 2009 exhibited a strong nonstationarity. When using timeas covariate, the Weibull and Gamma distributions (bothwith logarithmic link functions for the two model parame-ters) were the best distributions for modelling the observedlow-flow series of Huaxian and Xianyang, respectively.Significant nonstationarities were detected in both modelparameters and the optimal nonstationary model expressedboth log-transformed parameters as linear functions of time.When using meteorological variables as covariates, theWeibull distribution (with logarithmic link functions forthe two model parameters) was the best distribution for bothstations. For Huaxian, the optimal nonstationary modelexpressed both log-transformed parameters as linear func-tions of annual average temperature and annual total precip-itation, respectively. And for Xianyang, the log-transformedlocation parameter was modelled as linear function of annualaverage temperature but the scale parameter was modelledas a constant. For both stations, the optimal nonstationarymodel of using meteorological covariates performed betterthan that of the case using time as covariate in terms ofthe AIC value and the worm plot.

(2) Different covariates used in nonstationary frequency calcu-lation will lead to different results of the nonstationaryreturn period and risk analysis. When using time as covari-ate, there were significant differences between the nonsta-tionary return period and risk under both the EWT andENE interpretations and those corresponding to the station-ary condition. The nonstationary return period under theENE interpretation was a little bit longer than the case ofEWT but they were both much shorter (more frequent eventoccurrence) than the stationary case, indicating that thescarcity of water resources during dry seasons will worsenover time. However, this result may overstate the risk of alow-flow event due to inappropriate fitted statistical param-eters (moments). There is an evidence of an increase inlow-flow events since the mid-1990s, but the nonstationaryreturn period and risk were derived under the assumptionthat both model parameters monotonously change indefi-nitely. In contrast, the nonstationary model used the physi-cal variables temperature and precipitation as covariates,provided more appropriate fitted statistical parameters(moments) and thus more reasonable nonstationary esti-mates of return period and risk. On the basis of the WeiRiver data, the nonstationary analysis of return period andrisk using meteorological covariates is recommended thanusing time for generating information to assist decisionmaking for the management of water resources during dryseasons exacerbated by climate change. This conclusion islikely to apply to the many similar situations around theworld.

(3) Some uncertainties about the nonstationary return periodand risk analysis of extreme hydrological events should benoted. We used only one of the four RepresentativeConcentration Pathways adopted by the IPCC for its fifthAssessment Report, and future research could consider theothers, which have smaller radiative forcing target levels.Uncertainties exist in the downscaled future scenarios ofmeteorological variables used here, and this could be partlyaddressed by exploring more downscaling methods andGCMs. Nonetheless, the purpose of the study was to under-stand and quantify the differences resulted from using time

T. Du et al. / Journal of Hydrology 527 (2015) 234–250 249

as the sole covariate and using meteorological variables ascovariates in the nonstationary frequency analysis, ratherthan to compare the performance of different GCMs andtheir scenarios. Further work can be conducted to investi-gate the uncertainty of the design low-flow quantile andhow this affects uncertainty of the estimates of nonstation-ary return period and risk.

Acknowledgements

This research is financially supported by the National NaturalScience Foundation of China (Grants NSFC 51190094 and NSFC51479139), which are gratefully acknowledged. The contributionof C.J. Gippel was made whilst visiting the College of WaterResources and Hydropower Engineering, Wuhan University, sup-ported by the High-End Foreign Expert Recruitment Programme,administered by the State Administration of Foreign ExpertsAffairs, Central People’s Government of the People’s Republic ofChina. We are very grateful to the editor and two anonymousreviewers for their valuable comments and constructive sugges-tions that helped us to greatly improve the manuscript.

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