Advances in Water Resources 75 (2015) 67–79

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Advances in Water Resources

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Return levels of hydrologic droughts under climate change

http://dx.doi.org/10.1016/j.advwatres.2014.11.0050309-1708/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: Department of Civil Engineering, Indian Institute ofScience, Bangalore 560012, India. Tel.: +91 80 2293 2669; fax: +91 80 2360 0290.

E-mail address: pradeep@civil.iisc.ernet.in (P.P. Mujumdar).

Arpita Mondal a, P.P. Mujumdar a,b,⇑a Department of Civil Engineering, Indian Institute of Science, Bangalore 560012, Indiab Divecha Center for Climate Change, Indian Institute of Science, Bangalore 560012, India

a r t i c l e i n f o

Article history:Received 15 March 2014Received in revised form 6 November 2014Accepted 9 November 2014Available online 15 November 2014

Keywords:DroughtsClimate changeExtremesNon-stationaryTransient return levelsDetection

a b s t r a c t

Developments in the statistical extreme value theory, which allow non-stationary modeling of changes inthe frequency and severity of extremes, are explored to analyze changes in return levels of droughts forthe Colorado River. The transient future return levels (conditional quantiles) derived from regionaldrought projections using appropriate extreme value models, are compared with those from observednaturalized streamflows. The time of detection is computed as the time at which significant differencesexist between the observed and future extreme drought levels, accounting for the uncertainties in theirestimates. Projections from multiple climate model-scenario combinations are considered; no uniformpattern of changes in drought quantiles is observed across all the projections. While some projectionsindicate shifting to another stationary regime, for many projections which are found to be non-stationary,detection of change in tail quantiles of droughts occurs within the 21st century with no unanimity in thetime of detection. Earlier detection is observed in droughts levels of higher probability of exceedance.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Traditional assumptions of stationarity to estimate the riskassociated with hydrologic extremes such as droughts have comeunder scrutiny in the light of global climate change. Historicallyobserved tail quantiles of hydrologic events and their uncertaintiesmay undergo non-stationary changes in future because of globalwarming or human effects [45,53,58]. Increasing temperaturesdue to increased greenhouse gas (GHG) emissions are projectedto intensify droughts [12] in the twenty-first century. However,only low to medium confidence is achieved for projection of hydro-logical extremes, because of the limitations of climate models andlack of sufficient observations at smaller scales [24]. In particular,drought events may vary considerably in time and extent [5] andsuch events are extreme and rare by their very nature. In orderto improve drought resilience and/or for long term planning, acomprehensive assessment of changes in drought characteristicsis thus necessary and challenging at the same time.

The Colorado River is one of the most important rivers of theWestern United States sustaining over 30 million people and pro-viding water to seven states of USA and the country of Mexico[44]. Persistent dry conditions are reported for the South-westUSA and the Colorado River Basin [20], which are projected to

intensify in the future [7]. Sensitivity of the runoff generatingsnowmelt processes as well as increasing water demands to chang-ing climate conditions emphasize the need for assessment of cli-mate change impacts in this river basin [36]. General circulationmodels (GCMs) provide future climate simulations for changingGHG emissions, which can be used for future drought riskassessment [6]. The low resolution of GCMs and their inability tosimulate localized processes necessitate the use of a statisticaldownscaling model which is often used together with physically-based hydrologic models for generation of streamflow projections.Christensen et al. [9] used statistically downscaled precipitationand temperature from a multiple-member ensemble of a GCM torun the Variable Infiltration Capacity (VIC) hydrologic model andpredicted decreases in mean runoff in the Colorado River Basin.However, it is recommended that the uncertainties due to lack ofknowledge of physical climate and natural variability be character-ized by using multiple GCMs [6]. Using mean of the simulationsfrom multiple GCMs and scenarios, the average annual precipita-tion and temperature are projected to increase in this river basin,while the mean runoff is projected to decrease by 8.5% by 2050[52]. However, these figures are only with respect to the historicalsimulation of the hydrologic models corresponding to climatemodel runs, and not with respect to the observed naturalizedflows. Miller et al. [44] also use statistically downscaled precipita-tion and temperature to run the VIC model for getting streamflowprojections in the Gunnison River which is a tributary of theColorado River. They use these projections for comparison with

68 A. Mondal, P.P. Mujumdar / Advances in Water Resources 75 (2015) 67–79

paleoreconstruction of streamflow data in terms of regime shifts.Changes in the hydrological extremes in terms of magnitudes orquantiles of droughts with respect to those from the observedflows, and the time in future when such changes are expected,remain unexplored.

Extreme value theory (EVT) provides the theoretical basis formodeling extreme events. Non-stationary extensions to thetraditional stationary extreme value models enable incorporationof effects of one or more physically-based covariates in the param-eters of the statistical extreme value model. These developments inEVT are increasingly being explored to study effects of globalclimate change [4,26,28,29,31,57,59]. It is reported that changesin climate extremes may be more robustly detectable than changesin mean [22,46]. Non-stationary extreme value distributions can beapplied to study such changes in extremes for increasing the rigoras well as for a more physically meaningful analysis [19,28]. Burkeet al. [6] use the Peak-over-threshold (POT) approach of EVT tostudy changes in future meteorological droughts over the UnitedKingdom. We extend their approach to hydrologic droughts inthe Colorado River to additionally report the time of detectionfor tail quantiles of droughts from the non-stationary droughtprojections with respect to the observations, considering the errorsin their estimates. We seek answer to the question of how long thehistorically observed return levels of droughts will hold good forplanning and adaptation. For example, if a drought of 100-yearreturn period is of interest to the water resources manager, wedetermine whether and when the observed historical 100-yeardrought will cease to be a correct measure of risk based on eachstreamflow projection, taking into account the associated uncer-tainties. A similar analysis using the Block Maximum approach ofEVT on the high-flow hydrologic extremes of floods is conductedby Mondal and Mujumdar [49].

In this study, we first determine appropriate statistical extremevalue models for each projection based on a likelihood ratio test.Detection times are computed thereafter. A standardized droughtindex is defined based on 3-month accumulated streamflow inthe Colorado River. A two-component non-stationary POT statisti-cal model based on Poisson-Generalized Pareto (Poisson-GP) distri-bution is used to model changes in the frequency and intensity ofdroughts. Global mean surface air temperature, averaged over mul-tiple GCMs, has been considered as a covariate instead of a simpletime-dependence [6]. Changes in regional mean temperatures atmany locations across the globe are reported to be consistent withthe changes in global mean temperature [43,62].

Fig. 1 provides an overview of the steps conducted in this study.For observations as well as each of the streamflow projections,monthly flow data are first standardized to obtain a drought indexbased on 3-month accumulated flows. Extreme drought indexvalues are retained using a peak-over-threshold approach. Theseextremes are declustered so that no two extreme values representthe same 3-month drought. Likelihood ratio test is conducted onthe declustered extreme drought index and an appropriate station-ary or non-stationary model is fitted to observations as well aseach of the projections. If the observations are stationary (in thisstudy it is found to be so, as shown later), a constant droughtreturn level would be obtained from the stationary statistical Pois-son-GP model fitted to the observed extreme declustered droughtindex series. Similarly, for stationary projections, a constantdrought return level is obtained. For non-stationary projections,transient ‘effective’ drought return levels [28] would be obtained.These effective return levels are conditional quantiles wherebythe ‘return level’ is allowed to vary from one time period to thenext holding the probability of occurrence constant [28,53]. Forexample, the 100-year transient effective return levels are thequantiles at each time step corresponding to probability ofexceedance p = 1/100 = 0.01. This definition of transient return

levels has been used in earlier studies for risk communication ina non-stationary world [6,15,25,28] and though these quantilesmay not correspond to any one particular return period, they dorepresent meaningful quantities to illustrate the effects of non-stationarity. Projected return levels are compared with theobserved return level through a detection test. Each of these stepsis described in details in the ‘Methodology’ section below. Theremainder of the paper is organized as follows – Section 1describes the observed and model data used, while Section 2provides details on the methodology. The results are discussed inSection 3 and summarized in Section 4.

2. Observed and model data

Natural streamflows (without the effects of regulations) areconsidered in this study as our focus is on changes in droughtquantiles due to global climate change. The projected monthlyflows have been obtained from the streamflow projections datasetprepared by the U.S. Bureau of Reclamation [17,52] for 1950–2099,at 195 sites spanning the Western United States, publicly availableat http://www.usbr.gov/WaterSMART/wcra/flowdata/ (accessedon May 18, 2013). These flow data are obtained from the simula-tions of the physically-based Variable Infiltration Capacity (VIC)macroscale hydrological model [41,64] run with statisticallydownscaled bias-corrected [38] meteorological forcings from the112 projections from the GCMs participating in the World ClimateResearch Program’s Coupled Model Intercomparison Project 3(WCRP/CMIP3). The 112 projections are obtained by simulationsfrom different runs of 16 GCMs with the three emission scenarios– A1B, A2 and B1 [23].

Bias Corrected Spatial Downscaling (BCSD) method is used asthe statistical downscaling model to obtain precipitation and tem-perature at 1/8� � 1/8� latitude-longitude grid across the WesternUnited States covering the major Reclamation basins. Runoff, surfaceand subsurface, simulated by the VIC at 1/8�� 1/8� latitude-longitude grid, are hydraulically routed to each of the 195 locations.Using a statistical downscaling technique in combination with aphysically based hydrologic model is a standard practice for impactassessment studies, and both the BCSD downscaling technique andthe VIC hydrologic model are well-known methods [52].

The BCSD statistical downscaling model in conjunction with VIChas been used to study hydrological changes in the Western UnitedStates also by other studies [8,14,40,44]. There are other studies[7,13] who use the downscaling method of constructed analogues(CA) in conjunction with VIC, though the results of BCSD and CA aremostly reported to be quantitatively similar [39]. We choose Colo-rado River at Lees Ferry (Lat 36� 510 5300, Long 111� 350 1500 W) loca-tion in the state of Arizona, with a drainage area of approximately111,800 sq miles as it has a large drainage area and has longobserved naturalized flow data available. The streamflow projec-tions are obtained without additional efforts on VIC model calibra-tion (for improving the existing level of calibration); however, thestreamflow data that we have used can be used for comparingchanges in flow characteristics with those of the past so as to atleast gain an insight into the nature of such changes [14,44]. More-over, for large basins at monthly and larger scales, the VIC model isable to reproduce streamflows reasonably well [52].

Observed monthly naturalized streamflows in the ColoradoRiver at Lees Ferry for the period 1906–2010, obtained from theUnited States Geological Survey (USGS) observed gage data, areobtained from the website of the Upper Colorado Regional Officeof the United States Bureau of Reclamation (http://www.usbr.gov/lc/region/g4000/NaturalFlow/index.html). Global averagesurface air temperature (tas) anomalies, obtained from the meanof multiple GCMs, are used as covariate. Annual multi-model aver-age detrended tas anomalies relative to the 1980–1999 mean from

Fig. 1. Overview of the steps conducted in the detection analysis for droughts.

A. Mondal, P.P. Mujumdar / Advances in Water Resources 75 (2015) 67–79 69

the GCMs contributing to the IPCC’s Fourth Assessment Report(AR4), are obtained from the IPCC’s data distribution center (avail-able online at http://www.ipcc-data.org/sim/gcm_global/index.html). Each year’s tas anomaly is considered as the covariatefor modeling that year’s drought exceedances.

3. Methodology

3.1. Definition of drought

Monthly observed and projected streamflows are converted to astandardized drought index. The previous 3-months’ accumulatedstreamflow corresponding to each month is converted to thestandardized drought index for that month, by computinganomalies from a baseline 1961–1990 climatology. Thus, themonthly drought index D3 is given by,

D3 ¼ðR3 � Rclim

3 ÞrRclim

3

; R3 ¼X3

i¼1

Ri; ð1Þ

where, Ri is the streamflow for month i, R3 is the accumulatedstreamflow for the previous 3 months including month i, Rclim

3 isthe mean accumulated 3-month streamflow for the period1961–1990, and rclim

R3is the standard deviation of the accumulated

streamflows for the period 1961–1990. This simplistic definitionof drought [2,47,50] represents streamflow deficit with respect tothe reference period and is spatially invariant – both desirable qual-ities for a drought index [27]. A similar definition of meteorologicaldrought, based on accumulated precipitation, was used by Burkeand Brown [5] and Burke et al. [6]. This drought index is structurallysimilar to the Standardized Precipitation Index [42], with theassumption of normal distribution of the 3-month flows for thegiven period. Since accumulated streamflows over 3 monthsare considered, normal distribution may not be an inaccurateassumption here, although normality is not a strict requirementfor using this drought definition [6]. Droughts have significantimpacts only on time scales greater than 1 month [6] which justifiesthe consideration of 3-months period. Moreover, the three monthsperiod also ensures that the probability of zero flows in the period is

negligible. The drought indices are inverted so that positive valuesdenote deficit. Drought indices from all the months are pooled forthe POT analysis following Coles [10].

If the monthly time series of drought indices, denoted by{K1,K2, . . . ,Kd}, is independent and identically distributed (iid) withcommon cdf F, and d is the number of months, then, for a suffi-ciently high threshold u, the excesses Yi = Ki � u, conditional onKi > u, follow a GP distribution [25]. These excesses constitute thePOT (or the partial duration) series. The choice of the threshold ushould be such that it is high enough for the GP-approximationto be valid for the excesses, but not so high as to result in veryfew data points and making the parameter estimation unreliable[25]. The 80th percentile of the empirical distribution of observedmonthly drought indices is chosen as the threshold in this study.To account for temporal dependence in the threshold-exceedanceswithin each 3-month drought, declustering [10] is performed suchthat if two or more months with extreme drought belong to thesame cluster, only the driest month (cluster maxima) is retained.Dry months are also assumed to represent the same 3-monthdrought if there is a gap of less than 3 months between them.

Similar procedure is adopted to obtain the standardizeddrought index for each streamflow projection spanning the timeperiod 1951–2099. However, the 1961–1990 mean and standarddeviation of monthly accumulated flows from the projectionsmay not match with those from the observed flows because of biasin the downscaled projections [63]. Thus, to make sure that all theprojections have same 1961–1990 mean and standard deviation,the 1961–1990 simulated mean is subtracted from the projectedmonthly accumulated flows which are then divided by the simu-lated 1961–1990 standard deviation. Thereafter, the observed1961–1990 standard deviation is multiplied to the resultant seriesand the observed 1961–1990 mean is added to it. A similar biascorrection procedure for the mean only was adopted by Ghoshand Mujumdar [18]. Declustered extreme drought indices arefurther obtained from the corrected monthly accumulated flowseries. The extreme value distributions fitted to the observed andprojected drought data described above and the detection method-ology are explained in the next section. The R package ‘extRemes’[19] version 1.64 for the open source statistical programming

70 A. Mondal, P.P. Mujumdar / Advances in Water Resources 75 (2015) 67–79

language R (http://cran.r-project.org/web/packages/extRemes/index. html) is used for fitting the extreme value models and com-putation of return levels and their errors.

3.2. Statistical extreme value models to describe droughts

The POT approach regards the threshold exceedances as‘extreme events’, thus avoiding the restrictions of considering ablock for getting the maxima such as the annual maxima [10].The Poisson-GP distribution is a two-component statisticalextreme value model consisting of a one-dimensional PoissonProcess for the occurrence of a threshold exceedance and a Gener-alized Pareto (GP) distribution for the excesses over the threshold[28]. The GP distribution to describe excesses or POT series ofdeclustered drought indices y, has a cdf given by [10]:

H½y;rðuÞ; n� ¼ 1� 1þ ny

rðuÞ

� ��1=n

; y > 0; 1þ ny

rðuÞ > 0 ð2Þ

which is a two-parameter distribution having the shape parametern, and the scale parameter r(u) > 0 which is constant for a chosenthreshold u. Positive, zero and negative values of the shape param-eter constitute the unbounded heavy-tailed, unbounded light-tailedand bounded, finite tail forms of the GP distribution, respectively.Here, u = 80th percentile of the observed monthly drought indexseries. We use the method of maximum likelihood to estimate theparameters of the GP distribution since it allows variation of theparameters with covariates [28]. If h represents the derivative ofH with respect to y, and b denotes the vector of the GP parameters,b = (r,n), the likelihood function is given by [10]:

LðbÞ ¼YQi¼1

hðyi; bÞ; ð3Þ

where yi denotes the declustered extreme drought index, and Q isthe number of exceedances. Numerical maximizing of the logarithmof L(b) is carried out to obtain the estimates of r and n. Anotherapproach for the threshold-exceedances, namely the Point Processapproach which considers the threshold exceedances as a non-homogeneous Poisson Process, can also be used to model the declu-stered monthly extreme drought indices; we use the more commonPoisson-GP statistical extreme value model [25]. The return levelym, that is exceeded on average once every m observations, condi-tional on ym > u, is obtained by inverting Eq. (2):

ym ¼uþ r

n ½ðmkðuÞÞn � 1� for n – 0

uþ r logðmkðuÞÞ for n ¼ 0

(ð4Þ

where kðlÞ ¼ PrfY > lg is the rate of exceedance estimatedfrom the Poisson distribution for the frequency of threshold-exceedances. For the monthly drought indices, there are 12 valuesper year, so the N-year return level would correspond tom = N � 12. By the delta method [51], the variance of the returnlevel ym is given by:

VarðymÞ ¼ r2ym� ryT

mVrym: ð5Þ

The complete variance–covariance matrix of the Poisson-GPstatistical model is given by:

V ¼kðuÞð1� kðuÞÞ=n 0 0

0 v1;1 v1;2

0 v2;1 v2;2

264

375 ð6Þ

where vi,j denotes the (i, j)th term of variance–covariance matrix ofestimated r(u) and n, and

ryTm ¼

@ym

@kðuÞ ;@ym

@rðuÞ ;@ym

@n

� �¼ rmnðkðuÞn�1Þ; n�1fðmkðuÞÞn � 1g;h

�rn�2fðmkðuÞÞn � 1g þ rn�1ðmkðuÞÞnlogðmkðuÞÞi

ð7Þ

evaluated at the estimated values of r(u) and n. The confidenceintervals can be computed for the N-year return levels using thisvariance assuming that the quantiles approximately follow normaldistribution. There are other approaches to compute the confidenceintervals of return levels such as the profile likelihood approach[10]; however, we use the delta method for its ability to be easilyextended to the non-stationary case [28].

Non-stationarity is introduced in the Poisson-GP statisticalextreme value model by representing the parameters k(u) r(u)and n as functions of the covariate tas. Trends are incorporated inthe parameters of the Poisson distribution through a GeneralizedLinear Model (GLM) framework. If i denotes the month, the non-stationary functional forms of the parameters of the Poisson-GPmodel used are given by:

log kðu; iÞ ¼ k0 þ k1tasðiÞlogrðu; iÞ ¼ r0 þ r1tasðiÞnðiÞ ¼ n

ð8Þ

where the logarithms are taken to ensure positive values of theparameters k and r. Here, temporal dependence is not consideredfor the shape parameter n, since precise estimation of this parame-ter is difficult and it would be unrealistic to model it as a smoothfunction of time [10]. The trends in the k and r parameters canbe interpreted in terms of the corresponding transient or ‘‘effective’’[28] return levels. Trends in the threshold u are also not consideredfor computational simplicity in calculating the errors of the tran-sient effective return levels. The transient m-observation effectivereturn level for the non-stationary case is given by:

ymðiÞ ¼uþ rðu;iÞ

n ½ðmkðu; iÞÞn � 1� for n – 0

uþ rðu; iÞ logðmkðu; iÞÞ for n ¼ 0

(ð9Þ

The transient variance of the m-observation effective return levelcan be computed as a function of the month i by the delta method:

VarðymðiÞÞ � ryTmVrym ð10Þ

where, the complete variance–covariance matrix is given by

V = V1 . . .. . . V2

� �where V1 and V2 denote the variance–covariance

matrix of (k0,k1) and (r0,r1,n) respectively, and ‘. . .’ denotes a vec-tor of zeros of appropriate length, and

ryTm¼

@ym

@k0;@ym

@k1;@ym

@r0;@ym

@r1;@ym

@n

� �¼ rmnkn;rmnkntasðiÞ;r

nfðmkÞn�1g;

�rnfðmkÞn�1gtasðiÞ;�rn�2fðmkÞn�1gþrn�1ðmkÞnlogðmkÞ

�ð11Þ

evaluated at the estimated values of (k0,k1,r0,r1,n) where r and kare given in Eq. (8). Thus, for those projections where significantnon-stationary association exists between the extreme droughtsand tas, the non-stationary effective return levels will also showtrends according to variations in tas.

The suitability of the non-stationary GEV distribution can betested by the likelihood ratio test [10]. If Mo and M1 representthe stationary sub-model with constant r and n, and the non-sta-tionary statistical extreme value model with parameters r(t) andn respectively, and the corresponding maximized log-lilkelihoodsare given by lo(Mo) and l1(M1), then the null hypothesis of Mo canbe rejected in favor of statistical model M1 at the a level of signif-icance if 2{l1(M1) � lo(Mo)} > ca where ca is the (1 � a)th quantile ofthe Chi-square distribution. The degrees of freedom of theChi-square distribution equal the number of additional parameters

A. Mondal, P.P. Mujumdar / Advances in Water Resources 75 (2015) 67–79 71

in M1. For selecting the best out of many candidate statistical mod-els, the Akaike information criterion (AIC) or the Bayesianinformation criterion [25] can be used. The significance of eachof the parameters k0 and k1 are tested within the GLM framework.In this study, the Poisson model for the rate of threshold exceed-ance is assumed to be stationary and non-stationary if the corre-sponding GP model selected is stationary and non-stationary,respectively.

3.3. Time of detection

The time of detection for the N-year return level of drought isdefined as the point in time (month and year) where there is evi-dence to reject the null hypothesis that the N-year return levelfrom the observed and projected extremes, yo

m and yfm respectively,

are equal, in favor of the alternate hypothesis that the futureN-year return level yf

m is greater than the observed N-year returnlevel yo

m. A similar definition was used by Fowler and Wilby [16]for UK extreme precipitation, though they did not derive the returnlevels from non-stationary extreme value models. As the futurereturn levels larger in magnitude than the observed are more crit-ical for long-term planning than those smaller than the observed,we conduct a one-tailed Student’s t-test (also called Welch t-testwhen the variances are unequal). If the observed droughts arestationary, yo

m and its associated variance r2ymo

is constant. On theother hand, if the projected droughts are non-stationary, yf

m andits variance r2

ymfvary with time and in that case, the time of detec-

tion is one such that the test statistic at any future time step f,

Df ¼yo

m � yfmffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r2ymoþ r2

ymf

q P Zcritical ð12Þ

where Zcritical is the standard normal variate corresponding to the(1 � a)th quantile, and a is the chosen level of significance. Thus,the time of detection is the first time step at which Df P Zcritical, thatis, the transient N-year effective return level at a future time step f issignificantly larger than the observed N-year return level at the astatistical confidence level of (1 � a). Since Zcritical depends on thechosen level of significance, varying degree of confidence in thedetection would give different detection times. We choose a conser-vative 99% confidence level for this study. If the projected droughtshave only increasing trend, detection at any time step f would meanthat the changes in the N-year effective return levels in thesubsequent times would be even more significant. The analysis isterminated in 2099 since projected drought data are available till

Fig. 2. (a) Location of the Colorado River at Lees Ferry (Source of basin map: http://wwdistribution functions (CDFs) of monthly flows from observed (black) and projected (oth

that time. The year of detection is computed for drought magni-tudes of 50-year (probability of exceedance = 0.02) and 100-yearreturn periods (probability of exceedance = 0.01) as they areimportant for long-term planning. Additionally, we also test thesensitivity of the detection results to the uncertainties in returnlevels by considering an alternate scenario where the variance inprojected effective return levels are assumed to be constant andequal to that from the observations, similar to Fowler and Wilby[16]. The next section discusses the results in details.

4. Results and discussion

The location of the Colorado River at Lees Ferry is shown inFig. 2(a). The monthly streamflows from the 112 projections com-pare reasonably well with the observed data for the period of over-lap 1951–2010, as is evident from their cumulative distributionfunctions (CDFs) shown in Fig. 2(b). This indicates that thesestreamflow projections are suitable to be used for regional impactassessment at this location. The multi-model ensemble average, acommon way of combining information from multiple GCM runs,is not considered in the analysis since each of these 112 projectionsof declustered drought index series represents extreme valueswhich will no longer be extreme if an average is taken over all ofthem. Such an average over a large (112) number would tend tobe non-skewed and applying extreme value theory, which typicallydescribes long tails, on such an average would be theoreticallyinappropriate. Moreover, we conduct a detailed analysis on eachof these projections since our aim is to provide a comprehensivescenario of possible changes in return levels of droughts fromwhich the water resources managers can make a choice.

As accumulated flows over the past 3 months is considered ateach month for obtaining the standardized drought index series,the observed data is from 1907 to 2010, while the projected datais from 1951 to 2099. The annual global mean surface air temper-ature anomalies, averaged over multiple model simulations, usedas a covariate in the statistical model for droughts is plotted inFig. 3. The non-linear variations in tas allows the effective returnlevels to have similar variations if the chosen statistical extremevalue model is non-stationary with significant scaling betweenthe extreme drought POT series and tas.

Bias-correction as described in Section 1 is conducted on themonthly accumulated flow series for each projection spanningthe time period 1951–2099, to ensure that their 1961–1990 meanand standard deviation is same as those from observations for

w.usbr.gov/lc/region/g4000/contracts/watersource.html). (b) Empirical cumulativeer colors) data for the period of overlap 1951–2010.

Fig. 3. Multi-model average annual global mean near surface air temperatureanomalies used as a covariate for declustered extreme drought indices.

Fig. 4. Empirical CDF of observed (black) and the 112 projected (other colors)declustered extreme drought indices for the period of overlap 1951–2010.

72 A. Mondal, P.P. Mujumdar / Advances in Water Resources 75 (2015) 67–79

1961–1990. Standardized drought index is thereafter computed foreach month for the observed and projected streamflows. DroughtPOT series for the observed and projected data are then obtainedby considering threshold exceedances and declustering is done toensure that any particular 3-month drought is not representedmore than once in the extreme drought index series. The empiricalCDFs of the declustered extreme drought index series from theobserved and projected flows as plotted in Fig. 4 shows closematch for the period of overlap 1951–2010, confirming that thedeclustered extreme drought index series from these 112 projec-tions can be suitably used for analyzing changes in future extremedrought levels.

The non-parametric Mann–Kendall trend test suggeststhat the null hypothesis of no trend cannot be rejected(p-value = 0.91 > 0.05) at 95% confidence for the observeddeclustered extreme drought index series for 1907–2010, indicat-ing stationarity in the observed drought data. This is furtherexplored by the likelihood ratio test. It is found that the nullhypothesis of a stationary GP distribution (Statistical extremevalue model M0) cannot be rejected at 95% confidence

(p-value = 0.172 > 0.05) against the alternate hypothesis of a non-stationary GP distribution (statistical extreme value model M1)with tas as covariate for the observed declustered extreme droughtindex series. The observed declustered extreme drought index(POT) series is plotted in Fig. 5(a), along with the 0.05th and the0.95th quantiles estimated from the chosen stationary GP-model.The exceedance probability k(u) is constant here since theobserved droughts are considered to be stationary, and is equalto the number of exceedances divided by the number of monthsin the observations. The quantiles from the statistical model arefound to capture the behavior of the extreme series well. A quan-tile–quantile (Q–Q) plot, shown in Fig. 5(b), also shows good matchbetween empirical and statistical model-predicted quantiles. Thus,based on the historical observations, it is natural to obtain a con-stant return level assuming stationarity of extreme droughts. Thedrought index magnitudes corresponding to different returnperiods for the observed data are also shown in Fig. 5(c). Theobserved parameters of the Poisson-GP model and their standarderrors (SE) are given by

r ¼ 0:097ð0:012Þ; n ¼ �0:502ð0:078Þ; k ¼ 0:066ð0:007Þ ð13Þ

The 50-year and 100-year return levels and their associated errorsare computed from the stationary Poisson-GP model fitted to theobserved drought indices, using Eqs. (4) and (5) respectively. Thevariance of k(u) is also constant for the observations and is takeninto consideration in computing the variance of return levels ofdroughts using Eq. (6).

In the light of changing climate conditions, although the histor-ically observed data is stationary, the stationarity assumption infuture droughts may not be valid. This is tested for each of thefuture projections using the likelihood ratio test. Out of the 112future drought projections, 29 show no convergence or conver-gence to boundary points of the parameter space in the maximumlikelihood method, making estimation of variances of parameterseither impossible or unreliable. These 29 projections are hencedropped from further analysis. For 43 out of the remaining 83 pro-jections, the null hypothesis of a stationary GP model cannot berejected (p-value > 0.05) against an alternative of a non-stationaryGP model with tas as covariate. For most of these 43 projections,the 50-year and 100-year return levels are lower than those fromthe observed extreme drought data, implying that these projec-tions are less critical requiring no modifications in the designdrought quantiles. However, 15 of these 43 ‘stationary’ projectionsreport 50-year and 100-year return levels greater in magnitudethan those from observations. The details of these 15 projectionsare given in Table 1. The observed GP parameters, 50-year and100-year return levels and their standard errors of estimate arealso given. All these 15 projections have a negative shape parame-ter, similar to the observed drought data. It can be observed thatfor many of these projections, the errors in the estimate of thereturn levels are higher than those in the estimate of observedreturn levels. Using the projected variance of return levels in theWelch t-test in Eq. (12), it is found that the projected 50-yearand 100-year return levels from none of these 15 projections aresignificantly different from those from the observed data. This isbecause large projected variances reduce the test statistic. Whena constant variance in the return level is assumed across the pro-jections for each return period, and that variance is assumed tobe the observed variance of the corresponding return level, sevenprojections report 50-year and 100-year droughts that are signifi-cantly different from those from the observed record. These sevenprojections are shown in bold in Table 1. These projections can beinterpreted to be shifting to a new stationary regime which is sig-nificantly different from that of the observed record.

As an example, the diagnostics of the stationary GP modelfitted to the declustered drought indices from the GFDL-CM2.0

Fig. 5. (a) Observed declustered extreme drought index (POT) series for 1907–2010 (blue ‘o’ markers) along with the 0.05th and 0.95th quantiles (dotted horizontal lines)from the chosen stationary GP distribution (b) Q-Q plot between empirical and GP-model derived quantiles for the observed declustered extreme drought index series (c) Aplot of return level of drought vs return period from the fitted GP-model for the observed declustered extreme drought index series. (For interpretation of the references tocolor in this figure legend, the reader is referred to the web version of this article.)

Table 1Details of the observed and the fifteen stationary extreme drought projections, their corresponding 50-year and 100-year return levels and their standard errors of estimate (SE).The parameters r and n of the GP model and their associated standard error of estimate is also provided along with the p-value from the likelihood ratio test. The projectionsshown in bold have 50-year and 100-year return levels significantly different from those from the observed record at 99% confidence, assuming that the variance in return levels isconstant across the projections and is equal to the variance of the observed drought level of the corresponding return period.

Name Likelihood ratio test p-value r (SE) n (SE) 50-year return level 100-year return level

Observed 0.172 0.097 (0.012) �0.502 (0.078) 0.909 (0.006) 0.918 (0.006)a1b.giss_model_e_r.2 0.940 0.129 (0.013) �0.628 (0.079) 0.931 (0.104) 0.939 (0.110)a1b.giss_model_e_r.4 0.480 0.105 (0.008) �0.558 (0.058) 0.910 (0.078) 0.918 (0.083)a1b.mpi_echam5.3 0.247 0.151 (0.009) �0.649 (0.051) 0.959 (0.068) 0.967 (0.071)a2.mpi_echam5.2 0.225 0.122 (0.007) �0.595 (0.047) 0.929 (0.060) 0.937 (0.064)a2.mpi_echam5.3 0.401 0.136 (0.003) �0.599 (0.024) 0.950 (0.028) 0.959 (0.029)a2.mri_cgcm2_3_2a.3 0.074 0.122 (0.000) �0.651 (0.009) 0.919 (0.002) 0.925 (0.002)a2.ncar_pcm1.3 0.837 0.121 (0.010) �0.591 (0.065) 0.928 (0.087) 0.936 (0.092)b1.gfdl_cm2_0.1 0.100 0.132 (0.000) �0.761 (0.016) 0.910 (0.003) 0.915 (0.003)b1.gfdl_cm2_1.1 0.437 0.157 (0.000) �0.867 (0.003) 0.920 (0.001) 0.924 (0.001)b1.miub_echo_g.2 0.080 0.128 (0.000) �0.677 (0.010) 0.920 (0.002) 0.926 (0.002)b1.mpi_echam5.2 0.804 0.121 (0.011) �0.648 (0.079) 0.916 (0.099) 0.922 (0.104)b1.mpi_echam5.3 0.342 0.135 (0.014) �0.554 (0.074) 0.958 (0.110) 0.969 (0.117)b1.mri_cgcm2_3_2a.2 0.581 0.126 (0.008) �0.699 (0.063) 0.911 (0.072) 0.918 (0.075)b1.ncar_ccsm3_0.3 0.914 0.128 (0.008) �0.590 (0.048) 0.940 (0.064) 0.948 (0.068)b1.ncar_pcm1.3 0.177 0.119 (0.017) �0.582 (0.113) 0.925 (0.145) 0.934 (0.155)

A. Mondal, P.P. Mujumdar / Advances in Water Resources 75 (2015) 67–79 73

B1-Scenario Run#1 is shown in Fig. 6. The projected extremedrought index series (blue ‘o’ markers) and the 0.05th and the0.95th quantiles from the fitted stationary Poisson-GP model(black dots) are shown in Fig. 6(a); the quantile–quantile plotshows a good match between empirical and predicted quantiles(Fig. 6(b)). Drought indices corresponding to different return peri-ods are also shown (Fig. 6(c)).

For 40 out of the 83 projections considered, the null hypothesisof a stationary GP model can be rejected at 95% confidence

(p-value < 0.05) against the alternate hypothesis of a non-stationaryGP-distribution with tas as covariate. Out of these, only 23 projec-tions are critical since the effective return levels are projected toincrease for them, with detection of change in drought level beingachieved within the twenty-first century.

As an example, the fit diagnostics for one of these projections –NCAR-CCSM3.0 B1 Scenario Run#4, are provided in Fig. 7. Similarto Fig. 6, the projected drought index series (blue ‘o’ markers)and the 0.05th and the 0.95th quantiles from the fitted

Fig. 6. (a) Declustered extreme drought index (POT) series for 1951–2099 (blue ‘o’ markers) along with the 0.05th and 0.95th quantiles (dotted horizontal lines) from thechosen stationary GP distribution for GFDL-CM2.0 B1-Scenario Run#1, (b) Q–Q plot between empirical and GP-model derived quantiles for the same projection (c) A plot ofreturn level of drought vs return period from the fitted GP-model for the same projection. (For interpretation of the references to color in this figure legend, the reader isreferred to the web version of this article.)

74 A. Mondal, P.P. Mujumdar / Advances in Water Resources 75 (2015) 67–79

non-stationary GP-model (black dots) are also shown Fig. 7(a). Forthe non-stationary case, the lack of homogeneity in the distribu-tional assumptions necessitates modification of the data for plot-ting the model diagnostics such as the Q–Q plot [10]. As astandard practice, diagnostic checks are applied to a standardizedversion of the data conditional on estimated parameters of the dis-tribution. Standard exponential distribution is used here to obtainthe standardized data for the Q–Q plot corresponding to thenon-stationary projections. The Q–Q plot for NCAR-CCSM3.0 B1Scenario Run#4, shown in Fig. 7(b), suggests that the non-stationary GP model is appropriate. For this projection, the actualnumber of exceedances per year (blue ‘o’ markers), and that fromthe mean of the non-stationary Poisson distribution are also shownin panel (c) of Fig. 7. The number of exceedances per year esti-mated from the mean of the non-stationary Poisson distribution,shown in black in Fig. 7(c), has increasing trends because of thetrend parameter k1. The p-value for testing the null hypothesis thatk1 = 0 against the alternative that k1 – 0 is 0.02 < 0.05, suggestingthat the non-stationary Poisson model is appropriate.

Effective return levels and their uncertainties are similarlyobtained for each of the 23 critical ‘non-stationary’ projections.The details of these projections are given in Table 2. The p-valueof the likelihood ratio test, the parameter estimates and theirstandard errors are provided. The projected extreme droughtsare also found to have negative shape parameters, like theobserved data. The Poisson rate parameters k0 and k1 and theirstandard errors or p-values are not mentioned in the table forbrevity. Time of detection for each of the 50-year and 100-yeareffective return levels, that is, transient quantiles corresponding

to probabilities of exceedance of 0.02 and 0.01 respectively, isestimated by the one-tailed Welch t-test described in Section3.3, taking into account the transient variance of these quantilesfrom each projection. It is found that detection always occurs inthe month of Jan since the flows are usually least for this month.The year of detection for each of the 23 projections obtained bythe Welch t-test is given in Table 2. Each of these projections isequally likely and no uniform time of detection is reportedacross all the projections. For some projections, detection occursas early as 2020s. As mentioned earlier, we have not computed amulti-model ensemble average year of detection, since a simpleaverage may not always be adequate in combining informationfrom multiple GCM simulations for regional hydrologic impactassessments [18]. Instead, Table 2 provides a comprehensive sce-nario of possible changes in extreme drought levels dependingon which projection is realized. Across all the non-stationaryprojections, detection occurs earlier for magnitudes of extremeswith higher probability of exceedance, indicating that the effectsof climate change on extreme droughts are more critical fordesigns based on more frequent events. Fowler and Wilby [16]also report earlier detection for extreme precipitation with lowerreturn periods, though their conclusion is not based on non-stationary extreme value theory. Additionally, the sensitivity tothe chosen variance of effective return levels is explored, byusing the constant variance of the observed drought returnlevels. The year of detection for each projection, computed bythe Student’s t-test for the constant variance case is also pro-vided in Table 2. Detection mostly occurs later in the constantvariance case.

Fig. 7. (a) Declustered extreme drought index (POT) series for 1951–2099 (blue ‘o’ markers) along with the 0.05th and 0.95th quantiles (dotted horizontal lines) from thechosen non-stationary GP distribution for NCAR-CCSM3.0 B1-Scenario Run#4, (b) Q–Q plot between empirical and GP-model derived quantiles for the same projection (c) Thenumber of exceedances per year for the projected droughts (blue ‘o’ markers) with that from the mean of the fitted non-stationary Poisson distribution (black ‘o’ markers) forthe same projection. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Table 2Details of the non-stationary extreme drought projections and their corresponding year of detection for 50-year and 100-year return levels (a) considering transient variance foreach projection, (b) keeping the variance constant and equal to the observed variance in return levels of extreme droughts. The parameters r0, r1 and n of the non-stationary GPmodel and their associated standard error (SE) of estimate is also provided along with the p-value from the likelihood ratio test.

Name Likelihood ratio test p-value r0 (SE) r1 (SE) n (SE) Year of detection

(a) Different variances (b) Constant variance

50-year RL 100-year RL 50-year RL 100-year RL

a1b.cccma_cgcm3_1.1 1.279E�02 �2.103 (0.112) 0.055 (0.011) �0.756 (0.093) 2097 Not detected Not detected Not detecteda1b.cnrm_cm3.1 2.546E�05 �1.950 (0.080) 0.152 (0.000) �0.962 (0.080) 2045 2053 2052 2059a1b.gfdl_cm2_0.1 6.386E�04 �2.263 (0.126) 0.158 (0.058) �0.690 (0.075) 2056 2062 2057 2059a1b.ipsl_cm4.1 5.960E�03 �2.076 (0.116) 0.074 (0.016) �0.745 (0.092) 2056 2063 2061 2069a1b.miroc3_2_medres.2 5.948E�07 �1.954 (0.000) 0.137 (0.000) �1.012 (0.003) 2062 2071 2068 2077a1b.miub_echo_g.1 3.244E�05 �2.103 (0.174) 0.143 (0.069) �0.682 (0.093) 2018 2019 2016 2021a1b.miub_echo_g.2 9.175E�03 �1.995 (0.103) 0.046 (0.011) �0.749 (0.080) 2023 2037 2037 2052a1b.mpi_echam5.1 3.147E�03 �2.232 (0.110) 0.175 (0.023) �0.825 (0.105) 2075 2079 2075 2080a1b.mri_cgcm2_3_2a.2 2.763E�04 �2.193 (0.088) 0.244 (0.014) �0.859 (0.085) 2052 2053 2053 2058a1b.mri_cgcm2_3_2a.4 1.048E�04 �2.194 (0.089) 0.157 (0.000) �0.882 (0.080) 2081 2089 2087 2095a1b.ncar_pcm1.3 3.178E�02 �2.088 (0.135) 0.131 (0.035) �0.839 (0.119) 2090 2094 2072 2075a2.cccma_cgcm3_1.1 2.764E�04 �2.060 (0.102) 0.094 (0.000) �0.843 (0.088) 2071 2081 2079 2090a2.cccma_cgcm3_1.5 1.043E�04 �2.117 (0.097) 0.095 (0.014) �0.689 (0.071) 2029 2036 2036 2043a2.ipsl_cm4.1 2.714E�04 �2.237 (0.110) 0.126 (0.021) �0.580 (0.070) 2027 2027 2028 2029a2.miub_echo_g.1 1.091E�08 �2.010 (0.065) 0.173 (0.019) �0.935 (0.073) 2047 2053 2052 2059a2.mpi_echam5.1 2.676E�03 �2.144 (0.113) 0.147 (0.011) �0.873 (0.107) 2075 2082 2080 2087a2.ncar_ccsm3_0.4 5.281E�06 �1.906 (0.031) 0.104 (0.005) �0.872 (0.024) 2020 2032 2030 2040b1.cnrm_cm3.1 1.692E�05 �1.940 (0.100) 0.116 (0.005) �0.844 (0.089) 2024 2032 2032 2038b1.csiro_mk3_0.1 7.427E�05 �2.544 (0.120) 0.212 (0.016) �0.695 (0.096) 2096 2097 2094 2096b1.miroc3_2_medres.2 6.974E�03 �2.064 (0.116) 0.073 (0.037) �0.724 (0.074) 2038 2045 2045 2053b1.miub_echo_g.1 1.198E�04 �2.378 (0.135) 0.156 (0.053) �0.538 (0.067) 2038 2038 2040 2040b1.ncar_ccsm3_0.4 3.517E�05 �2.297 (0.100) 0.174 (0.033) �0.726 (0.074) 2063 2068 2065 2068b1.ncar_ccsm3_0.6 2.996E�03 �2.099 (0.033) 0.129 (0.000) �0.856 (0.026) 2065 2072 2071 2079

A. Mondal, P.P. Mujumdar / Advances in Water Resources 75 (2015) 67–79 75

As an example, the projected 50-year and 100-year effectivereturn levels (bold lines) and their 90% confidence interval (thinlines) for a stationary projection – GFDL-CM2.0 B1-Scenario

Run#1 (blue), and a non-stationary projection – NCAR-CCSM3.0Run#4 (red), are shown in Fig. 8. The observed (black) return levelsand their confidence intervals are also shown. The year of detection

76 A. Mondal, P.P. Mujumdar / Advances in Water Resources 75 (2015) 67–79

at 99% confidence for the non-stationary projection is shown by avertical line for each of the two cases of exceedance probabilities.The non-stationary projection shows non-linear increases in theeffective return levels, scaled with similar variations in the covar-iate. Fig. 9 shows the same plot as Fig. 8, but for the constant var-iance case.

It can be observed that the non-stationary projections in Figs. 8and 9 do not start at the same return level as the observations inspite of correcting for their biases in mean and standard deviationof accumulated flows. However, if the extreme drought indices arefurther bias-corrected for each return level to match with the sta-tionary observations, it might unrealistically impose stationarityon the projections as well. For example, the statistical model willnot be able to give a constant return level till 2004 and transientreturn levels immediately after that. Moreover, we do a one-taileddetection test here since the focus is on transient effective returnlevels which exceed the observed return level as they are more crit-ical from the hydrologic design point of view. For example, if along-term plan is adopted assuming a particular constant droughtquantile from the observations and the projected quantiles areonly lower, future changes in infrastructure or planning may notbe necessary. Thus, the projected flows where the return level inthe period of overlap is lower than that in the observations arenot critical in this analysis. The choice of the one-tailed test affectsdetection times.

It is also to be noted here that the monthly extreme droughtindices are derived based only on regional conditions which areobtained by statistical downscaling from the large-scale GCMsimulations. The physically-based hydrologic model (VIC) is runwith these downscaled regional conditions; monthly streamflowscorresponding to GCM runs are thus generated. The peak-over-threshold drought series, that is the declustered extremedrought index series, is obtained from only the monthly series ofgenerated streamflows, by standardization. Thus, the definition of

Fig. 8. Observed (black) and projected (other colors) (a) 50-year and (b) 100-year extremlines) estimated by the delta method. GFDL-CM2.0 B1-Scenario Run#1 extreme droughtthat from NCAR-CCM3.0 B1-Scenario Run#4 is non-stationary leading to increasing eftemperature. The time of detection for the non-stationary projection is shown by the veScenario Run#1 4 projected return levels is found to be small compared to the variance

drought does not impose an association between the extremedroughts and global average temperature. If the drought index wasunrealistically sensitive to temperature by definition, all the extremedrought projections would have been non-stationary because of largetrends in temperature, which is not the case here as many projectionsare found to be stationary (Table 1). It is also worth mentioning herethat the climate models are run with changing external forcings thatare likely to result in a non-stationary output. However, for regionalimpact assessment studies, use of statistical downscaling along withphysically-based hydrologic models may provide additional filteringleading to stationary projections.

In the statistical extreme value model, global average tempera-ture is used as a covariate for only defining the non-stationarity,which is first tested by the likelihood ratio test. Non-stationaritycould have been otherwise considered by allowing the locationparameter of the GEV distribution to be a function of time; forexample, Fowler et al. [15]. However, following Burke et al. [6]and Westra et al. [62] we choose global average temperatureinstead of simple time dependence, since changes in global averagetemperature are non-linear. Global average temperature is knownto have the strongest attribution of human-induced causes andrepresents a long-term response of the climate system to anthro-pogenic forcings [21]. The GCMs are also known to simulatesmooth fields such as temperature, at large scales quite reliably.The averaging over multiple model runs cancels the climate noisedue to internal variability and reflects the long-term trendresponse [55]. Fowler and Wilby [16] considered local processes(extreme precipitation) to be scaling with global average tempera-ture. Burke et al. [6] used global average temperature as a covariateto study regional extreme meteorological droughts in the UK.Westra et al. [62] also considered global average temperature asa covariate for annual extremes (precipitation) at local, pointscales. Thus, the choice of global average temperature as covariateis justified in this study.

e effective drought return levels (bold lines) and their 95% confidence interval (thinprojection series is stationary (blue) leading to constant value of return levels whilefective return levels with increasing multi-model average global mean surface airrtical line of corresponding color. The variances of the stationary GFDL-CM2.0 B1-s of the same from observed or NCAR-CCSM3.0 projected droughts.

Fig. 9. Same as Fig. 8, keeping the variance in the projected extreme effective drought return levels constant and equal to the observed variance of the corresponding returnlevels.

A. Mondal, P.P. Mujumdar / Advances in Water Resources 75 (2015) 67–79 77

A similar analysis on detection of change in return levels offloods is conducted in a parallel study [49] where hydrologicextremes, namely, annual maximum daily streamflows in theColumbia River, are modeled using the alternate approach ofblock-maxima within the non-stationary extreme value theory.Although the statistical test for detection of change in return levelsof floods is similar to that used in this paper for droughts, the wayhydrologic extremes are defined, the non-stationary statisticalmodels fitted, and computation of return levels and theiruncertainties are much different in that study. Regional extremestreamflow projections are obtained from climate model simula-tions using a different dataset. For floods in the Columbia Rivertoo, change in return levels are detected within the twenty-firstcentury, though less conspicuously, and detection of change occursearlier for more frequent extremes, similar to the present study.

It has to be kept in mind that detection, in the context ofanthropogenic climate change, can have different implications.Barnett et al. [1], for example, performed a formal, fingerprint-based detection and attribution analysis and concluded thathuman-induced signals are already discernible (time of detectionin 1980s) in the hydrological changes in Western United States.Their aim is to compare observed patterns of changes with thoseexpected from anthropogenic runs (historical) and analyzewhether the changes could have been caused by natural internalvariability alone, which is estimated from pre-industrial cli-mates. For regional hydrology, the detection of anthropogeniceffects depends on the goal of the analysis as well as on theregion, variable and scale considered. Mondal and Mujumdar[48] conduct a similar fingerprint-based detection and attribu-tion analysis on monsoon precipitation and streamflow for arain-fed river basin in India and conclude that unequivocal attri-bution is not possible within the twentieth century. A late timeof emergence of the climate change signal with respect to theinternal climate variability is also reported by Maraun [37] forEuropean precipitation. Time of emergence is significantlydependent on the background with which a comparison of theobservation is made.

In this study, the aim is to compare future (and not historical)return levels of droughts under climate change with those fromhistorical observations (and not to pre-industrial climate) sincethe observed return levels might be currently in use for hydrologicdesigns. If the future droughts are significantly non-stationary dueto association with global warming and if the future return levelsare higher than the return levels estimated from observations,modifications in hydrologic systems may be necessary. Asomewhat similar question is addressed by Fowler and Wilby[16] for UK precipitation who also report detection late in thetwenty-first century, though they use the less-rigorous scalingfactor approach, and not the extreme value theory.

4.1. General remarks

There are some studies which question the analysis of climatemodel derived output within the non-stationary extreme valueframework and argue that non-stationarity in such model simula-tions are a result of models than time series [32]. It is to be notedhere that the aim of using climate models in this study is to obtaina range of plausible future projections taking into account a broad,large-scale view of evolution of climate under external forcings[33,34]. Non-stationary extreme value theory has earlier beenapplied to climate model outputs [15,60,61,65]. Use of multipleprojections is recommended to address the uncertainties associ-ated with the lack of knowledge about the climate system and nat-ural variability [55]. Thus, each of these projections is not a one-stop solution, and in fact many of the projections are indeed foundto be stationary. The primary aim of this study is to propose amethodology for examining the presence of non-stationarity inthe extreme drought projections and more importantly, fordetection of change in drought levels, given that there is non-sta-tionarity in the future projections. Figs. 8 and 9 do not suggestour final recommendation, but provide examples of such detectionfor stationary and non-stationary projections. In this regard, Tables1 and 2 provide a more comprehensive summary of our findings,presenting results from the entire range of projections.

78 A. Mondal, P.P. Mujumdar / Advances in Water Resources 75 (2015) 67–79

The definition of drought used in this study is in similar lines asthat of Burke et al. [6] and Burke and Brown [5]. Accumulatedstreamflow over three months is considered so that zero flow inany period is unlikely. Residual seasonality and autocorrelationmay inflate redundancy in the monthly drought indices; however,it is to be noted that the extreme (POT) drought indices, on whichthe analysis is performed, are duly declustered to remove anydependence or redundancy. Moreover, pooling drought indicesfrom all months increases our sample size for reliable estimationof parameters in MLE.

The ‘parent’ data from which the POT series is derived isassumed to be iid; such an assumption may not be valid for themonthly drought indices. There are several studies in hydroclima-tology where such an assumption is made for the extreme valuetheory to be valid, not necessarily testing for the independenceof the parent data [25,28,30,59]. It is argued that the asymptoticextreme value distributions do not change even if the parent dataare dependent, as long as certain relatively weak mixing conditionshold good [35]. Additionally, whether the asymptotic GP distribu-tion provides a reasonable description of the tail behavior ofextreme drought indices can be further tested by diagnostic plots.The quantile–quantile plots in Figs. 5(b), 6(b) and 7(b) show thatthe GP-models perform reasonably well. Alternate methods suchas using different thresholds for obtaining iid parent data and get-ting the threshold exceedances such as that of Bernardara et al. [3]can be explored to overcome the limitations of the iid assumption,though such work falls outside the scope of the present study.

Also, in the non-stationary case, the correspondence betweenthe return period T and probability of exceedance p will not bevalid and the effective return levels do not necessarily correspondto any particular return period, though they represent meaningfulquantiles. The standard definitions of the return period (orequivalently return levels) can be extended to the non-stationarycase based on either the waiting-time or the expected number ofoccurrences [11]. Alternate risk measures in the non-stationarysituations are only recently being explored in hydroclimatologicalapplications. For example, the waiting-time based definition orreturn period is adopted for hydrological applications by Salasand Obeysekera [54]. However, Rootzén and Katz [53] point outthat such a definition is sensitive to developments after the designlife which may not be relevant. Serinaldi [56] propose a probabilityof failure based design as an alternative to the return period basedapproach, and the ‘design life level’ proposed by Rootzén and Katz[53] can be shown to be a special case of this approach. The ‘mini-max design life level’ defined by Rootzén and Katz [53] is an exten-sion of the concept of effective return levels. Rootzén and Katz [53]mention that use of effective return levels is akin to redefining theflood plain from one time period to the other. Our study provides amethod to assess whether such changes in return levels (andequivalently changes in flood plain or drought measures) aresignificant, i.e. a hydrological designer may choose to shift thedesign value only at the time of detection where the projectedtransient return level is significantly different from that observedin the historical past.

5. Conclusion

Hydrological extremes such as droughts are expected to bealtered through land and atmospheric processes that are linkedphysically to a warming climate. Under changing climate condi-tions, traditional assumptions of stationarity to derive designquantiles of droughts for long term planning may not hold good.Within the EVT framework, we investigate changes in longreturn-period droughts in the Colorado River using an ensembleof regional drought projections based on hydrologic simulationsdriven by multiple climate model-scenario combinations. Future

50-year and 100-year drought return levels may shift to a newstationary regime, or undergo transient changes due to non-stationarity induced by climate change, depending on the climateprojection realized in the future. We additionally report time ofdetection for 50-year and 100-year effective drought return levelsbased on the non-stationary projections. No uniform pattern ortime of change is observed across all the projections, though earlierdetection is observed unanimously for extreme droughts of higherprobability of exceedance.

It can be noted that for the hydrologic projections used, the VICmodel calibration may be an important issue that needs attention[52]. This limitation is expected to be addressed [52] in the upcom-ing version of the streamflow projections by the United StatesBureau of Reclamation with the hydrologic simulations based onthe Coupled Model Intercomparison Project 5 (CMIP5) climatemodel runs. Also, other statistical distribution could be used formodeling the extreme drought indices which may result in some-what different detection results. We choose the GP distributionbecause of the theoretical arguments for its suitability to modelthreshold exceedances [10]. A possible extension of this workcould be along the lines of addressing the various uncertaintiesin the detection results.

Our findings imply that the assumption that the historical pastcan provide a valuable clue to the future will not remain correctbeyond the detection times, depending on which future climate-model hydrologic projection is realized. For such non-stationaryprojections, modifications in future hydrologic designs arenecessary for long term planning.

Acknowledgments

The authors thank the U.S. Bureau of Reclamation for makingpublic the projected statistically-downscaled and VIC-simulateddaily and monthly streamflow projections for the Western UnitedStates for the CMIP3 GCMs. They also thank Rick Katz and Dan Coo-ley for helpful discussions through email, the editor, the epony-mous reviewer Francesco Serinaldi and one anonymous reviewerfor providing valuable comments on the paper.

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