Detection of Change in Flood Return Levels under Global Warmingcivil.iisc.ernet.in/~pradeep/Arpita_JHE.pdf · Levels under Global Warming Arpita Mondal1 and P. P. Mujumdar2 Abstract:

Detection of Change in Flood ReturnLevels under Global Warming

Arpita Mondal1 and P. P. Mujumdar2

Abstract: Using recent advancements in the statistical extreme value theory, this study proposes a methodology for detection of change inflood return levels under climate change. Nonstationary scaling of regional projected peak flows with global warming is first tested by alikelihood ratio test. For nonstationary possible future realizations, the authors then investigate how long the stationary historical designmagnitudes or return levels of floods will remain valid, taking into account the uncertainties in the estimation of observed and projectedreturn levels. Although some flood projections are found to be nonstationary, many are stationary in nature. No coherent change in floodreturn level across the projections is detected in the case study of floods in the Columbia River using available streamflow projections. Mostprojections yield flood quantiles that are not likely to be critical in the coming century. However, for some simulations detection is achieved,with earlier detection in design magnitudes of lower return periods. A possible worst-case scenario considering the maximum of all theprojections shows detection of change in floods of higher return periods in the 21st century. DOI: 10.1061/(ASCE)HE.1943-5584.0001326. © 2016 American Society of Civil Engineers.

Author keywords: Floods; Climate change; Non stationary; Extreme value theory; Detection.

Introduction

Changing patterns of hydrological extremes at regional scales,linked with global warming, are of increasing concern to researchersand policy makers because such extreme events can directly impactsocieties. Tail quantiles of floods such as the N-year return level(for example, the much-used hundred-year flood) and the associateduncertainties, estimated from the historical observations under as-sumptions of stationarity and used in current hydrological designs,may change in the future because of nonstationarity induced byrapid climate change or human intervention (Milly et al. 2008;Rootzén and Katz 2013). Although the notion of nonstationarityapplied to observed time series is questionable (Montanari andKoutsoyiannis 2014), human-induced land-use and/or land-coverchanges, control structures, and anthropogenic global warmingare cited as the most likely causes of nonstationarity in hydrologicalprocesses (Sivapalan and Samuel 2009). Global warming due to in-creased greenhouse gas (GHG) emissions is expected to intensifythe hydrological cycle because of the greater water-holding capacityof a warmer climate (Milly et al. 2002; Meehl et al. 2007a).

There is an increasing risk of floods globally (Hirabayashiet al. 2013; Kundzewicz et al. 2010) under climate changeprojections for the 21st century. Several recent studies report theevidence of human-induced climate change in various elementsof the hydrosphere (Hegerl et al. 2007; Barnett et al. 2008;Min et al. 2011; Willett et al. 2007); though at river-basin scalessuch evidence may not be unequivocally detected and attributed

(Mondal and Mujumdar 2012). A recent study (Pall et al. 2011)also attempts to attribute a stand-alone hydrological extreme event,namely the autumn floods in the United Kingdom in 2000, tohuman-induced global warming. While intensification of extremetemperature or precipitation events caused by human-inducedclimate change is reported for the present as well as the future(Coumou and Rahmstorf 2012), confidence in the projections offloods is only low to medium, owing to limitations of the modelsand lack of observational data at local or regional scales (IPCC2012). Hydrological extremes of floods are, moreover, rare bythe very nature of their occurrence, making inferences about thema challenging problem. The problem of detection of climate changesignals in regional extremes of floods and droughts is thus impor-tant and at the same time difficult to address.

Recent theoretical developments in the statistical extremevalue theory (EVT) allow modeling of effects of physically basedcovariates on the weather and climate extremes, in addition to thetraditional stationary statistical modeling of extremes. Nonstation-ary models are being increasingly explored to analyze changes inhydroclimatic processes influenced by rapid climate change (Katzet al. 2002; Katz 2010; Kharin and Zwiers 2005; Brown et al. 2008;Towler et al. 2010; Sillmann et al. 2011; AghaKouchak et al. 2013;Westra et al. 2013; Kharin et al. 2013). Changes in climate ex-tremes are often reported to be theoretically more robustly detect-able than changes in mean (Hegerl et al. 2004; Min et al. 2009), andthe unique advantages of nonstationary extreme value distributionscan be applied to study such transient extremes both to enhance therigor and to make the analysis more physically meaningful (Katzet al. 2002; Gilleland and Katz 2011; AghaKouchak et al. 2013).Yet, except for extreme precipitation in the United Kingdom(Fowler and Wilby 2010; Fowler et al. 2010; Maraun 2013), devel-opments within the EVT have not been used to study changes infuture transient return levels and their associations with physicalcovariates, or to estimate the time of detection for return levels.Whereas Fowler and Wilby (2010) employ a linear pattern scalingbetween extreme precipitation and global average temperature toobtain future return levels, Fowler et al. (2010) use the nonstation-ary generalized extreme value (GEV) distribution within the EVT

1Research Associate, Dept. of Civil Engineering, Indian Institute ofScience, Bangalore 560012, India (corresponding author). E-mail:[email protected]

2Professor, Dept. of Civil Engineering, Indian Institute of Science,Bangalore 560012, India; Divecha Center for Climate Change, IndianInstitute of Science, Bangalore 560012, India.

Note. This manuscript was submitted on February 23, 2015; approvedon October 2, 2015; published online on March 23, 2016. Discussion per-iod open until August 23, 2016; separate discussions must be submitted forindividual papers. This paper is part of the Journal of Hydrologic Engi-neering, © ASCE, ISSN 1084-0699.

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framework to compare changes in future extreme precipitation re-turn levels with the model-simulated parallel control runs. Fowleret al. (2010) focused only on model simulations because one oftheir aims is to find out important climate model parameters thataffect extreme precipitation.

For hydrological extremes such as floods, it is important to an-alyze the projected global-warming-induced nonstationary changesin return levels, with respect to the observations, so as to test thevalidity of the return levels derived from the historical past, whichare used for hydrological designs with the assumption that the pastcan be a guide to the future. For example, if a hydraulic structure isdesigned for a 50-year return period flood magnitude estimatedfrom the historical observations, and if the future 50-year floodmagnitudes are altered because of climate change, it would be nec-essary to assess when the differences between observed and futurereturn levels become significant such that the observed quantilesmight not be a correct measure of risk.

Some recent studies have indeed proposed new risk measuresfor hydrologic extremes under nonstationary conditions (Rootzénand Katz 2013; Serinaldi 2014) or extended the stationary conceptsof return period to the nonstationary setting (Salas and Obeysekera2014; Cooley 2013). These studies extrapolate the observed non-stationarities to the future design life, thus requiring that nonsta-tionary signals be already discernible in observed hydrologicextremes.

In the context of climate change, there can be nonstationaritiesin future hydrologic extremes even if the observed past is mostlystationary. Climate model-driven physically based hydrologicmodel simulated streamflow projections can be used as possiblerealizations of the future in such cases. Statistically downscaledmeteorological projections from climate models are increasinglybeing used to drive hydrological models to generate correspondingstreamflows by numerous regional impact assessment studieswhich assess changing flood risks (Hamlet and Lettenmaier 1999;Mantua et al. 2010; Coats et al. 2013; Das et al. 2011, 2013; Hamletet al. 2013).

This study investigates whether and when the future return lev-els are likely to be significantly different from the observed returnlevels, taking into account the associated uncertainties. Strictlyspeaking, in the transient case, these are effective return levels (Katzet al. 2002) keeping fixed the probability of exceedance at eachyear, because the one-to-one relation between return period andprobability of exceedance no longer holds as the probability of ex-ceedance changes from year to year. The future return levels areestimated from climate model projections using the theoreticalframework of extreme value theory. A likelihood ratio test is firstconducted for each future realization to assess whether significantnonstationarity exists in the peak streamflow projections. For theprojections having significant nonstationary scaling associationwith rising global average temperature, transient return levels arecomputed from the fitted nonstationary GEV distribution and thetime of detection is computed as the time when significant differ-ences exist between observed and future return levels.

The block-maxima (Coles 2001) approach of the EVT is usedconsidering the yearly spring season maximum one-day stream-flow. The Columbia River at the Dalles is chosen for illustrationsbecause of availability of long historical naturalized flows as wellas daily flow projections under climate change. Multimodel meanglobal average surface air temperature has been considered as acovariate in the nonstationary GEV since it is the most robust in-dicator of human-induced climate change with the strongest attri-bution of causes (Hegerl et al. 2007). A similar detection analysison the low-flow hydrologic extreme of droughts is conducted by

Mondal and Mujumdar (2015), although a different definitionand characterization of extremes is employed in that study.

Observed and Model Data

Climate model simulation based mean runoff in the Columbia Riverin the 21st century is projected to increase by 1.2–3.7% withrespect to the model-simulated historical flows (Reclamation2011). The river flows through a mountainous terrain, which makesit difficult to study the streamflow response to climate change(Burger et al. 2011); increasing temperatures and shifts in stream-flow timing alter flood risks in this river basin (Lee et al. 2009). TheUSGS station Columbia River at the Dalles (45°36′27″ N, 121°10′20″ W) in the state of Oregon, shown in Fig. 1(a), is chosen for theanalysis because it has a large drainage area [∼613; 827 km2

(237,000 mi2)] and long observed naturalized flow data are avail-able for this location. Naturalized flows remove the effects of localhuman interferences; this is important as the focus here is to detectchanges in return levels due to global climate change. Moreover,direct changes in actual human interventions are much harder topredict, especially in long time scales.

Observed daily naturalized streamflow in the Columbia River atthe Dalles from 1916 to 2004 is obtained from a recent version ofthe dataset prepared by Bonneville Power Administration (BPA)(BPA 1993) updated by Naik and Jay (2005). The time period1916–2004 is chosen for comparison with variable infiltrationcapacity (VIC) hydrologic model-simulated historical flows (de-scribed later) to test the ability of the model to reproduce hydro-logical processes for this region. Snowmelt plays a very importantpart in causing extreme streamflows in this river. Earlier snowmelt,which is a direct consequence of increasing temperatures due toclimate change, causes shifts in the streamflow peaks from summerto spring and winter (Stewart et al. 2005; Hidalgo et al. 2009).Hence, the daily maximum streamflow in the spring season(March–May) every year is considered as the block maximum.Statistical extreme value theory requires that the block size, overwhich maximum is considered, should be as large as possible.Strictly speaking, annual maxima seems to be a better choicefor the block maxima approach than a seasonal maxima as theblock size is 365 days in the former and only 91 in the latter(March–May). However, Fig. 1(b) shows that the magnitudes ofthe annual and spring season maxima are comparable. For 52%of the total data, the annual maximum is the same as the springseason maximum. The closeness between the annual and springseason maxima, especially toward the second half of the twentiethcentury, thus justifies the choice of spring season as an appropriateblock for obtaining the maxima.

In the block maxima approach (Coles 2001), daily spring seasonstreamflow values, denoted by the time series fX1;X2; : : : ;Xng,where n = number of days in the spring season of each year,are assumed to be independent and identically distributed (iid) witha common cumulative distribution function (cdf) F1 (Katz 2013). Ifthere exist normalizing constants an > 0 and bn, and the blockmaximum is denoted by Mn ¼ maxfX1;X2; : : : ;Xng such that

Pr

��Mn − bn

an

�≤ x

�→ GðxÞ as n → ∞ ð1Þ

the cdf G is GEV, following the extremal types theorem (Coles2001). In this study, the normalizing constants are absorbed inthe estimated parameters of the GEV distribution. The iidassumption of the parent data, though questionable, is in voguein hydrologic literature, and it can also be shown that the asymp-totic conditions for the EVT to hold can also be proved to exist even

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when the data are dependent, as long as certain mixing conditionshold (Leadbetter et al. 1983). Moreover, the performance of theasymptotic extreme value models in terms of their ability to re-present the observed extremes is tested by the diagnostic checks,as subsequently explained in the ”Results” section.

The VIC simulated historical and future transient projectionsare obtained from the Climate Impacts Group, University ofWashington who serve the aforementioned data at 297 streamflowlocations free of charge to the public, as part of the Columbia BasinClimate Change Scenarios Project (CBCCSP) (Hamlet et al. 2013).The macroscale VIC model is run at a spatial resolution of 1=16°Clatitude and longitude. The VIC simulated historical streamflowsare obtained by running the model with gridded meteorological da-tasets that were constructed from observed station records, using arefinement over the methods described by Hamlet and Lettenmaier

(2005). Fig. 1(c) shows the performance of the historical VIC sim-ulation with respect to the observations in terms of yearly maxi-mum spring flow. The VIC is seen to simulate the springmaxima reasonably well. Hamlet et al. (2013) report that theVIC is found to perform well in the snow-dominated WesternUnited States rivers because of its sophisticated energy-balancesnow model.

Future daily streamflows for the general circulation model(GCM) runs under two emission scenarios—A1B and B1 (IPCC2007)—are obtained from the VIC simulations run with statisti-cally downscaled maximum and minimum temperatures and pre-cipitation. Three statistical downscaling techniques are used byHamlet et al. (2013)—the composite delta (CD) method, the biascorrection and spatial downscaling (BCSD), method and the hybriddelta (HD) method. Although the HD method was devised by the

Fig. 1. (Color) Details of the flood data: (a) location of the Dalles (source of basin map: Washington State Department of Ecology, http://www.ecy.wa.gov/programs/wr/cwp/cwpfactmap.html); (b) observed naturalized annual and spring season (March–April–May) daily maximum flows (cfs);(c) spring season daily maximum flows from observations and VIC simulations run with historical meteorology

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authors to simulate extremes well, neither the CD nor the HDmethod provides transient simulations. Because one of their pri-mary aims was to compute the time of detection under nonstatio-narity, the authors used the BCSD transient simulations for wateryears 1950–2097. The BCSD-downscaled VIC-model simulatedflows are referred to as projections throughout this paper eventhough it may span over the historical period. This is to avoidthe words model simulated, as models are also used to refer tostatistical extreme value models. Although BCSD performs a quan-tile-mapping based bias-correction, it is reported to preserve thetransient time series behavior of GCM simulations and large-scalevariability (Hamlet et al. 2010). The CD and HD methods give sta-tionary statistics associated with a future window (e.g., 2030–2059), and therefore, with the current projected data, only theBCSD method can be used for a detection analysis. The BCSDdownscaled data have been used in earlier works that study changesin floods (Das et al. 2011, 2013) as part of hydrologic impactassessment of climate change.

A similar dataset based on VIC simulations at 1=8°C latitude-longitude grid from BCSD-downscaled climate projections isalso prepared and served by the U.S. Bureau of Reclamation(Reclamation 2011) at 195 sites spanning the western UnitedStates. However, the authors chose the Hamlet et al. (2013) datasetbecause of its finer spatial resolution and the additional calibrationefforts taken up in preparing this dataset. In particular, the simu-lations provided are only for the Columbia River Basin, and theDalles is also one of the 11 locations that Hamlet et al. (2013)has used for calibration of the VIC model. The Nash-Sutcliffe ef-ficiency for this site is reported to be 0.91. Also, bias-correctedstreamflows based on a quantile-mapping technique (Hamlet et al.2010) are available for the Dalles because naturalized flows areavailable at this location. the authors used these bias-correctedBCSD-downscaled VIC-simulated daily streamflow projectionsfor their analysis. Yearly spring season maxima are computed fromthis series.

Multimodel mean global average surface air temperature(tas) anomalies are used as covariate in the nonstationary modelfor floods. Annual multimodel average detrended tas anomaliesrelative to the 1980–1999 mean from the models contributing tothe IPCC’s Fourth Assessment Report (AR4), are obtained fromthe IPCC’s data distribution center (Meehl et al. 2007b). Eachyear’s tas anomaly is considered as the covariate for modeling thatyear’s flood magnitude. Observed tas data are used for the 1916–2004 period, and multimodel average projected tas data are used for1950–2097 as a covariate for observed and projected floods,respectively.

For the period of overlap 1951–2004, most of the 14 GCM-scenario combinations available are found to match well withthe observations in terms of the yearly spring maximum flow val-ues, as is evident from the empirical cumulative distribution func-tions (CDF) plotted in Fig. 2. The model-scenario combinationsHadCM-B1 and ECHAM5-A1B are found to simulate the floodspoorly. A two-sample Kolmogorov-Smirnov test shows that thenull hypothesis that the CDF each of these two projections issimilar to the observed CDF can be rejected at a very high confi-dence (p-value ≈ 10−9). Hence, these two combinations aredropped from the subsequent analysis. For 11 out of the remaining12 projections, the same null hypothesis cannot be rejected at 1%significance level (p-value > 0.01). For CNRM-CM3-B1, thoughthe null hypothesis can be rejected at 1% significance level, thep-value (¼ 0.004) is not as low as in the case of HadCM-B1and ECHAM5-A1B and can be an artifact of small sample size(only the period of overlap). Hence, CNRM-CM3-B1 is retained inthe analysis. Additionally, for the CGCM3.1-T47-A1B combination,

maximum likelihood estimation of GEV parameters did not con-verge, because of which this combination was also dropped fromthe analysis. Because all the GCM-scenario combinations areequally likely (IPCC 2007), detection analysis is conducted on eachof the remaining 11 projections individually.

Additionally, two possible multimodel combinations are alsoconsidered: (1) taking median of the 11 projections at each year(cyan in Fig. 2), and (2) taking maximum of the 11 projections ateach year (black in Fig. 2). Clearly, the empirical cdf of themedian of the projected flows does not match with observations.The same can be concluded about the combination of maximumof the 11 projections, except for very high quantiles for whichthere is some match. Thus, the detection analysis for very highquantiles (75-year and 100-year return periods) can be conductedfor this multimodel combination. Considering the highest of all possible flow values, this multimodel projected flow seriesrepresents a worst-case scenario, making the detection resultsconservative.

Methodology

Generalized Extreme Value (GEV) distributions are fitted to theobserved and projected yearly spring maximum floods. The Rpackage extRemes (Gilleland and Katz 2011) version 1.64 forthe open source statistical programming language R has been usedfor the computations. Return levels of projected floods are com-puted from the fitted models for comparison with the return levelsobtained from the models fitted to the observed data. The uncer-tainties in the return level estimates are also computed followingthe methods illustrated below.

Statistical Models to Describe Floods

The GEV distribution, which is the limiting distribution for springseason maximum one-day streamflow as mentioned earlier, hasthree parameters—location μ, scale σ > 0, and shape ξ, and itscumulative distribution function is given by (Coles 2001)

Fig. 2. (Color) Comparison of empirical cumulative distributionfunctions of spring season peak flows from GCMs to those fromobservations (blue) for the period of overlap 1951–2004; two possiblemulti-model combinations are also considered: (1) considering medianof the 11 projections at each year (cyan), and (2) considering maximumof the 11 projections at each year (black)

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Gðx;μ;σ; ξÞ ¼ PðX ≤ xÞ ¼ exp

�−�1þ ξ

�x − μσ

��−ð1=ξÞ�;

where 1þ ξ

�x − μσ

�> 0 ð2Þ

Positive, zero, and negative values of the shape parameter con-stitute the Fréchet (heavy-tailed), Gumbel (unbounded light-tailed),and Weibull (bounded, finite tail) forms of the GEV, respectively.The maximum likelihood estimation (MLE) method is often usedto estimate the parameters of the GEV distribution because it allowsvariation of the parameters with covariates (Katz et al. 2002). If grepresents the derivative of G with respect to x, and β denotes thevector of the GEV parameters, β ¼ ðμ;σ; ξÞ, the likelihood func-tion is given by (Coles 2001)

LðβÞ ¼YQt¼1

gðrt; βÞ ð3Þ

where rt denotes the spring season maximum one-day flow in theyear t, and Q is the number of years. Often, for optimization, min-imization of −LðβÞ is carried out instead of maximizing LðβÞ.Thus, if β̂ denotes the MLE estimate of the true parameter setβ, then

β̂ ¼ ðμ̂; σ̂; ξ̂Þ ¼ argmaxμ;σ;ξLðμ;σ;ξÞ ¼ arg maxμ;σ;ξYTt¼1

gðrt;μ;σ;ξÞ

subject to 1þ ξðrt−μÞ

σ> 0;σ> 0 ð4Þ

The probabilistic quantiles of the GEV distribution or the returnlevels, zð1−pÞ, can be obtained by inversion of Eq. (2)

zð1−pÞ ¼�μ − σ

ξ f1 − ½− logð1 − pÞ�−ξg for ξ ≠ 0

μ − σ log½− logð1 − pÞ� for ξ ¼ 0ð5Þ

where 0 < p < 1, and zð1−pÞ is the return level corresponding to thereturn period (1=p), which is the flood magnitude expected to beexceeded once in every 1=p years on an average (Coles 2001). TheMLE estimate of return levels for the GEV distribution is obtainedby using the MLE estimates of the parameters β̂ ¼ ðμ̂; σ̂; ξ̂Þ inEq. (5) as

dzð1−pÞðμ; σ; ξÞ ¼ zð1−pÞðμ̂; σ̂; ξ̂Þ

¼�μ̂ − σ̂

ξ̂f1 − ½− logð1 − pÞ�−ξ̂g for ξ ≠ 0

μ̂ − σ̂ log½− logð1 − pÞ� for ξ ¼ 0ð6Þ

By the delta method (Oehlert 1992), the variance of this returnlevel is given by

Varð dzð1−pÞÞ ≈ ∇zTð1−pÞV∇zð1−pÞ ð7Þ

where V is the variance-covariance matrix of (μ̂; σ̂; ξ̂) and usingyp ¼ − logð1 − pÞ, and superscript T to denote transpose of matrix

∇zTð1−pÞ ¼�∂zð1−pÞ

∂μ ;∂zð1−pÞ∂σ ;

∂zð1−pÞ∂ξ

�¼½1;−ξ−1ð1 − y−ξp Þ;σξ−2ð1 − y−ξp Þ − σξ−1y−ξp log yp� ð8Þ

evaluated at the MLE estimates (μ̂; σ̂; ξ̂) (Coles 2001). Both∇zð1−pÞ and V are unknown quantities and the authors are only

using a plug-in estimator of right-hand side of Eq. (7). The vari-ance-covariance matrix V is obtained by inversion of the observedinformation matrix evaluated at the maximum likelihood estimatesof the parameters, μ̂, σ̂, and ξ̂. This variance of return level is usedto compute the confidence intervals for dzð1−pÞ assuming it to followthe asymptotic normal distribution.

There are other approaches to compute the confidence intervalsof return levels such as the profile likelihood approach (Coles2001); however, the delta method was used for its ability to beeasily extended to the nonstationary case (Katz et al. 2002). More-over, it is theoretically possible to show asymptotic normality of themaximum likelihood estimates (Coles 2001), making the deltamethod an appropriate choice. Another nonparametric methodfor obtaining uncertainties associated with the return levels com-prises the resampling-based approaches (Kuczera 1999); however,in the nonstationary case, Obeysekara and Salas (2014) show thatthe bootstrap resampling-based approach yields confidence inter-vals of return levels that are very similar to the delta method.Incidentally, Obeysekara and Salas (2014) is the only study in hy-drology that explicitly accounts for uncertainties in quantiles oftime-varying hydrologic extremes.

Nonstationarity is introduced in the form of temporal depend-ence of one or more parameters of the GEV distribution, as a func-tion of a chosen covariate. For example, if the multimodel averageannual global mean surface air temperature is denoted by tas, thenonstationary GEV distribution can have parameters of the form

μðtÞ ¼ μ0 þ μ1tasðtÞ;σðtÞ ¼ σ; ξðtÞ ¼ ξ ð9Þ

The parameter μ1 represents the slope of the linear trend in thecenter of the GEV distribution.

Additionally, proportionate changes in the scale or the shapeparameters could also be considered; however, here the temporaldependence in the location parameter is consider only becauseof its intuitive appeal (Towler et al. 2010), the realistic possibilityof it being a smooth function of the covariate (Coles 2001) and thesimplicity in getting its precise estimates and derivatives. Linearfunctional form is assumed for the GEV parameters for simplicityin computation of standard errors of return levels.

The parameters are obtained by MLE, as in the stationary case.Thus, if cβns denotes the MLE estimate of the true parameter setβns ¼ ðμ0;μ1; σ; ξÞ, then

cβns ¼ ð bμ0; bμ1; σ̂; ξ̂Þ ¼ arg maxμ0;μ1;σ;ξL½μðtÞ;σ; ξ�

¼ arg maxμ;σ;ξYTt¼1

g½rt;μðtÞ; σ; ξ�

subject to 1þ ξ½rt − μðtÞ�

σ> 0;σ > 0 ð10Þ

The trends in the location parameter can be interpreted in termsof the corresponding transient or effective (Katz et al. 2002) returnlevels. The transient (1 − p)th quantile of the nonstationary GEVdistribution can thus be expressed as a function of the covariate byinverting Eq. (2) considering the location parameter μðtÞ for theparticular time step (year) t

zð1−pÞðtÞ ¼�μðtÞ − σ

ξ f1 − ½− logð1 − pÞ�−ξg for ξ ≠ 0

μðtÞ − σ log½− logð1 − pÞ� for ξ ¼ 0ð11Þ

The MLE estimate of these time varying effective return level isgiven by

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dzð1−pÞðtÞ ¼�μ̂ðtÞ − σ̂

ξ̂f1 − ½− logð1 − pÞ�−ξ̂g for ξ ≠ 0

μ̂ðtÞ − σ̂ log½− logð1 − pÞ� for ξ ¼ 0ð12Þ

The variance of the return level will also vary from one time stepto another as a function of the covariate as

Var½ dzð1−pÞðtÞ� ≈ ∇zTð1−pÞV∇zð1−pÞ ð13Þ

where V is the variance-covariance matrix of (μ0, μ1, σ, ξ)and

∇zTð1−pÞ ¼�∂zð1−pÞ

∂μ0

;∂zð1−pÞ∂μ1

;∂zð1−pÞ∂σ ;

∂zð1−pÞ∂ξ

�¼ ½1; tasðtÞ;−ξ−1ð1−y−ξp Þ;σξ−2ð1−y−ξp Þ−σξ−1y−ξp logyp�

ð14Þevaluated at the estimated values of ( bμ0, bμ1, σ̂, ξ̂). In general, if thelocation parameter has a linear variation with the covariate, then theeffective return levels would also have similar variations.

The suitability of the nonstationary GEV distribution can betested by the likelihood ratio test (Coles 2001). IfMo is the station-ary submodel with constant μ, σ, and ξ, M1 is the nonstationarymodel with parameters μðtÞ, σ, and ξ, and loðMoÞ and l1ðM1Þare the maximized log-likelihoods for Mo and M1, respectively,then the null hypothesis of Mo can be rejected in favor of validityof M1 at the α level of significance if 2fl1ðM1Þ − loðMoÞg > cαwhere cα is the (1 − α) quantile of the chi-square distributionwhose degree of freedom equals the number of additional param-eters in M1.

Time of Detection

For the nonstationary projected floods that lead to transient returnlevels, a definition of time of detection of change in return levelswas used in a manner similar to that used by Fowler and Wilby(2010) for United Kingdom extreme precipitation. The time of de-tection for the N-year return level is defined as the point in time(year) when there is evidence to reject the null hypothesis of nochange in return levels at 90% confidence level. That is, the pointin time when the null hypothesis that the N-year return level fromthe observed and projected extremes, bz0 and bzf respectively, areequal, can be rejected in favor of the alternate hypothesis thatthe projected N-year return level bzf is greater than the observedN-year return level bz0.

A one-tailed Student’s t-test is performed here because the fu-ture return levels larger in magnitude than the observed are ofgreater concern from the hydrological design point of view ascompared to those smaller than the observed. If the observedfloods are stationary, the observed N-year return level bz0 andits associated variance σ2

zo is constant. For stationary projections,the projected N-year return level bzf and its associated variance σ2

zfis constant as well. On the other hand, if the projected floods arenonstationary, the future N-year return level bzf and its variance σ2

zfvary with time and in that case, a one-tailed Student’s t-test is usedto estimate the detection year such that the test statistic at anyfuture time step f is

Df ¼ bzf − bzoffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiσ2zf þ σ2

zo

q ≥ Zcritical ð15Þ

where Zcritical is the standard normal variate corresponding tothe (1 − α)th quantile, where α denotes the chosen level of

significance. This test provides a standard normal distributionand is based on signal-to-noise ratio (Fowler and Wilby 2010).Thus, the year of detection is the first year at which Df≥Zcritical,that is, the transient N-year return level for year f is significantlylarger than the observed N-year return level at the α significancelevel. Zcritical depends on the chosen level of significance; so vary-ing degree of confidence in the detection would give differentdetection times. A 90% confidence level is chosen for the testsin this study.

If the extremes have increasing trend, detection at any year fwould mean that the changes in the N-year return levels in the sub-sequent years would be even more significant. Here, σ2

zf and σ2zo are

estimated by the delta method as described earlier. The analysis isterminated in 2097 because projected flood or drought data are notavailable beyond that point. For the stationary projections, Df isconstant and its value is significant only if the projected flood re-turn levels are significantly higher than the observed (one-tailedtest). The year of detection is computed for 10-year and 50-yearreturn levels of floods for each of the 11 projections. For the multi-model combination with maximum of the 11 projections at eachyear which simulate the high tail reasonably well, detection timesare computed for 75-year and 100-year floods. It may be noted thatalthough the observed and projected data are of longer lengths thanthe return periods, these return levels are still important for hydro-logic designers. They may be interested in specific values corre-sponding to particular probabilities of exceedance as obtainedfrom the fitted statistical models, and not necessarily based on sim-ple extrapolation of extreme data. The results are discussed in detailin the following section.

Results

A nonparametric Mann-Kendall trend test reveals no significanttrend in the observed spring season maximum flows at 90%confidence (p-value ¼ 0.227). Thus, it is expected that a station-ary GEV would be the appropriate fit for the observed series.In the likelihood ratio test, indeed the null hypothesis of a station-ary GEV (Model M0) cannot be rejected (p-value ¼ 0.996 > 0.1,considering 90% confidence) in favor of an alternate hypothesisnonstationary GEV (Model M1) where the location parameteris nonstationary. The alternate nonstationary model M1 considersglobal tas as a covariate. Thus, it is evident that the stationarityassumptions in deriving flood quantiles from the observed pastis appropriate. A quantile-quantile plot (not shown) also showsthat the fitted stationary GEV model is able to reproduce thequantiles of the observed extreme series well. The median,0.05th, and 0.95th quantiles estimated from the fitted stationaryGEV model are shown with respect to the observed extremesin Fig. 3(a).

For each of the 11 projected flood extremes and the multimodelcombination (which considers the maxima of these 11 projectionsevery year), the likelihood ratio test is first conducted and the quan-tiles are estimated from the chosen distributions. Details of the fit-ted statistical models are given in Table 1. The nonstationarymodels are selected for 3 out of 11 projections for which the nullhypothesis of a stationary GEV distribution can be rejected at 90%confidence (p-value < 0.1) in favor of the alternate hypothesis of anonstationary GEV distribution where the extreme flows vary withchanges in multimodel average tas anomalies. For all these threeprojections, the Mann-Kendall test also shows a significant trend at90% confidence. The multimodel combination also has a signifi-cant trend, at 90% confidence, according to the Mann-Kendall test(p-value ¼ 0.00048). A nonstationary GEV distribution is selected

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for this extreme series as well. Though the selected GEV modelmay underestimate the middle quantiles, very high quantiles wouldstill be reasonably reliable for this series since the tail of the seriesshow good match with the observations.

In general, the fitted models to the projections, stationary ornonstationary, are able to capture the quantiles reasonably well.As an example, Figs. 3(b and c) show the performance for twoprojections. For both CNRM-CM3-A1B, which is found to be

Fig. 3. (Color) Performance of the fitted GEV distributions for (a) observed; (b) non-stationary CNRM-CM3-A1B; (c) stationary PCM1-A1Bprojected yearly spring season maximum flows

Table 1. Details of the GEV Models Fitted to the Observed Data As Well As Each of the Projections

Flood series

Stationary GEV model Nonstationary GEV model Likelihoodratio test p-valueμ σ ξ nllh μ0 μ1 σ ξ nllh

Observed 580872.0 140971.2 −0.095 1202.9 580875.2 9.3e-06 140964.9 −0.095 1202.9 0.996CCSM3-A1B 543522.3 158536.4 −0.15 1974.3 543522.4 −0.09 158536.5 −0.15 1974.3 0.996CNRM-CM3-A1B 562217.3 153718.2 −0.096 1976.9 562216.6 50980.01 140114.3 −0.053 1968.4 4.06e-5ECHO-G-A1B 534966.8 116379.1 −0.059 1940.7 534966.7 6624.785 114918.7 −0.057 1940.3 0.38HadCM-A1B 563186.6 129521.3 0.069 1958.2 563445.8 7088.795 129521.1 0.070 1957.9 0.467PCM1-A1B 519666.8 114606.5 0.046 1941.2 519666.8 −0.296 114606.8 0.046 1941.2 0.995CCSM3-B1 543317.0 127353.1 −0.053 1956.3 543316.3 0.488 127353.3 −0.053 1956.3 0.999CGCM3.1-T47-B1 569572.9 127723.4 −0.11 1945.1 569572.7 17748.23 120223.9 −0.095 1942.3 0.02CNRM-CM3-B1 563432.0 150905.8 −0.077 1980.1 563018.5 19675.8 150906.2 −0.089 1977.7 0.028ECHAM5-B1 521206.8 126340.2 −0.197 1935.9 521207.1 −0.527 126340.4 −0.197 1935.9 0.994ECHO-G-B1 529235.0 121738.1 −0.130 1935.9 529234.9 0.1771 121738.0 −0.130 1935.9 0.996PCM1-B1 541534.6 121580.3 −0.006 1952.7 541534.5 0.1824 121580.2 −0.005 1952.7 0.999Multimodel extremeprojection

801722.4 140520.1 −0.279 1939.2 801722.2 23467.98 140520.4 −0.289 1936.5 0.021

Note: Details for the multimodel extreme combination (considering maximum of the 11 projections at each year) are also shown. Bold text denotes theprojections for which the null hypothesis of stationarity can be rejected at >90% confidence against the alternative that there is a nonstationary scalingrelationship between the floods and global warming.

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nonstationary, and PCM1-A1B, which is found to be stationary, themedian, 0.05th and 0.95th quantiles are found to capture the blockmaximum series well.

The 10-year and 50-year effective return levels are estimatedfrom the fitted distributions for each of the projected series. Thesereturn periods are widely used in hydrology. The uncertainty asso-ciated with each of these return levels for the observed and pro-jected floods is estimated by the delta method to compute their95% confidence interval.

The return levels and their confidence intervals are plottedin Fig. 4. For the stationary projections, both the 10-year andthe 50-year return levels are found to be smaller or insignificantlygreater than those from observations, as judged by the one-tailedStudent’s t-test, leading to no detection. For the nonstationaryprojections, it is observed that the projected transient return levelsvary nonlinearly because of nonlinear variations in tas. An in-creasing trend in the projected return levels can also be seenfor all the three nonstationary projections. It can be noted in Fig. 4that the nonstationary projections do not start at the same return

level as the observed data even though streamflow data used inthis study are bias-corrected as mentioned earlier. If the floods 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 until 2004 (the end ofobserved record) and transient return levels right after that. More-over, a one-tailed detection test is conducted in this study becausethe focus is on transient return levels that are greater than theobserved return level as they are more critical from the hydrologicdesign point of view. For example, if a hydraulic structure is de-signed assuming a particular constant flood return level from theobservations and the projected return levels are only lower, futurechanges in design may not be necessary. In Fig. 4, most of theprojections, though not starting at exactly the same level as theobserved return level, are not statistically significantly differentfrom the observed return level. None of the stationary projectionsresults in a return level that is significantly higher than the ob-served return level.

Fig. 4. (Color) Observed (black) and projected (other colors): (a) 10-year; (b) 50-year return levels for floods and their 95% confidence intervalsestimated by the delta method (thin lines of corresponding color); year of detection for non-stationary projections shown by vertical lines

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Time of detection for each return level is estimated by the one-tailed Student’s t-test for the nonstationary projections. Time ofdetection is shown by the vertical lines in Fig. 4 having colorsof corresponding projections. For the 10-year return level,detection occurs at 2061 for CNRM-CM3-A1B and at 2092 forCNRM-CM3-B1 projections. For the 50-year return level, forCNRM-CM3-A1B, detection occurs late at 2094, while forCNRM-CM3-B1, it does not occur until 2097, though the differ-ence between observed and projected return levels is found to beincreasing. Thus, if these projections are realized in future, floodquantiles are likely to increase significantly beyond the detectiontimes. For a 100-year return level, none of the 11 projections showsdetection within the 21st century (figure not shown). Because it isonly beyond the time of detection that the changes in projectedreturn levels are significantly different from the observed return lev-els calling for alternate assessments of risk and modifications inexisting designs, for this river basin, the flood quantiles are thusunlikely to be critical for many projections.

Additionally, the time of detection for the multimodel extremeprojection series for very high quantiles is computed, namely75-year and 100-year return levels. Fig. 5 shows the transient pro-jected return levels, the observed return levels and their 95% con-fidence intervals. Nonlinear increases are observed in the transientreturn levels. Time of detection is shown by the vertical lines. If thisworst-case scenario is realized in the future, detection occurs asearly as 2027 for the 75-year return level. For the 100-year returnlevel, detection occurs at 2065 in the worst-case scenario.

For all the nonstationary cases, it is observed that detection oc-curs earlier for magnitudes of extremes with lower return periods.Fowler and Wilby (2010) also report earlier detection for extreme

precipitation with lower return periods, though their conclusionis not based on nonstationary extreme value theory. The time ofdetection is sensitive to the choice of statistical significance.Detection times may also be sensitive to the duration over whichextreme flows are considered. The authors have used maximumdaily streamflows as they constitute one of the most commonlyused measures of floods and is appropriate for the block-maximumapproach of the EVT. The focus of this study is on extremes and theuse of nonstationary EVT, the essence of which lies in discardingmost of the data and retaining only those which theoretically tendsto follow the long-tailed statistical extreme value models (Katz et al.2002). Accumulated flows over greater durations of 7 or 15 daystend toward being more nonskewed; the larger the duration overwhich the sum or average is taken, the more theoretically inappro-priate it would be to use the long-tailed extreme value statisticalmodels as has been used in this study. Though a similar analysiswith maximum three-day or five-day flows may report differentyears of detection, the overall nature of return levels and their var-iances are unlikely to be altered.

Discussion

Nonstationarities of other different forms or in other parametersof the extreme value distributions can be considered in thisstudy. Global average temperature anomalies were considered asa covariate because the aim was to analyze the effects of globalwarming. The projected streamflows are derived based on regionalconditions that are obtained by statistical downscaling from thelarge-scale GCM simulations. The physically based hydrologic

Fig. 5. (Color) Observed and multi-model extreme projection (combination of the 11 projections taking the maximum at each year): (a) 75-year;(b) 100-year return levels for floods and their 95% confidence intervals estimated by the delta method; the year of detection is shown by the verticalline

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model (VIC) is run with these downscaled regional conditionsto generate streamflows corresponding to GCM runs. The block-maxima flood series, that is, the spring season maximum one-day flow series, is obtained from daily generated streamflows.Thus, there is no direct imposition of effects of global average tem-perature in the generation of hydrologic extremes.

In the statistical extreme value model, global average tempera-ture is used as a covariate only for defining the nonstationarity,which could have been also considered by allowing the locationparameter of the GEV distribution to be a function of time (Fowleret al. 2010). However, following Burke et al. (2010) and Westraet al. (2013) the authors chose global average temperature insteadof simple time dependence because since global average temper-ature variations are nonlinear. Global average temperature is knownto be the most robust indicator of long-term response of the climatesystem to anthropogenic effects (Hegerl et al. 2007). Increases inglobal-specific humidity, an indicator of the total amount of precip-itable water responsible for causing extreme precipitation events, aswell as changes in regional mean temperatures at many locationsacross the globe are reported to be consistent with the changes inglobal mean temperature (Mears et al. 2007; Westra et al. 2013).Smooth fields such as temperature are represented reliably in theGCMs at large scales. Averaging over multiple model runs cancelsthe noise from internal climate variability and reflects the long-termtrend response (Santer et al. 2007). Fowler and Wilby (2010) alsoconsidered local extreme precipitation to be scaling with globalaverage temperature. Burke et al. (2010) used global average tem-perature as a covariate for analyzing regional extreme meteorologi-cal droughts in the United Kingdom. For local, point-scale annualextremes (precipitation) Westra et al. (2013) also considered globalaverage temperature as a covariate.

Whether the hydrologic extremes of floods are indeed signifi-cantly associated with global average temperature is tested bythe statistical likelihood ratio test and the return levels are estimatedonly from appropriate statistical models. This does not force anyparticular year’s flood magnitude to be related to that year’s globalaverage temperature, but tests whether the long-term trends inglobal average temperature can be significantly associated withthe long-term trends in floods.

Further, though the hydrological projections used in this studyare derived with well accepted practices with sufficient efforts onmodel calibration, the statistical downscaling model or the hydro-logic model may not be completely error-free and might induceadditional uncertainties into the analysis. The BCSD method,for example, involves disaggregation to smaller spatial and tempo-ral scales, which may lead to unreliable estimates of the variabilityof hydrologic extremes (Hamlet et al. 2010). Statistical downscal-ing, as opposed to physical downscaling, may have biases, particu-larly for extremes; however, hydroclimatologic variables, which arehighly nonsmooth, may not be robust across different model for-mulations and choices even in physical downscaling (e.g., regionalclimate models). Also, observed concentrations at this point in timeand their increases beyond the emission scenarios considered mayhave significant effects on the detection results. The primary goal ofthis study is to introduce the method of detection of change in re-turn levels of floods under global warming, and for the purpose ofillustration, this case study is chosen with only these two emissionscenarios for their projected daily streamflow data availability.

A similar analysis on detection of change in return levels ofdroughts is conducted in a parallel study (Mondal and Mujumdar2015) where standardized extreme drought indices in the ColoradoRiver, based on three-month accumulated flows, are modeledusing the alternate approach of peak-over-threshold within thenonstationary extreme value theory. Although the statistical test

for detection of change in return levels of droughts is similar tothat used in this paper for floods, the way hydrologic extremesare defined, the nonstationary statistical models fitted, and compu-tation of return levels and their uncertainties, in these two studies,are much different.

Alternate views exist on the notion of nonstationarity and itsinterpretation for stochastic models and real-world phenomenon(Koutsoyiannis 2006, 2013), stressing that nonstationarity mustnot be confused with change, and that an actual observed timeseries should not be termed stationary or nonstationary. Onemay argue that the future cannot be predicted in deterministic terms(Montanari and Koutsoyiannis 2014), thereby questioning the useof deterministic climate model simulations. In this study, the pur-pose of using such simulations is to obtain physics-based plausiblefuture scenarios considering large scale evolution of the climatesystem, and hence, each of the projections used here do not denotea prediction of how the system should behave in future.

Also, increasing model complexity in the form of nonstationarymodels (with larger number of parameters, for example) with aview to reduce bias may lead to larger uncertainty (Montanariand Koutsoyiannis 2014; Serinaldi and Kilsby 2015), and it is rec-ommended that stationary models be retained as benchmark forevery other competing model of higher complexity (Serinaldiand Kilsby 2015). Indeed, this study also finds that several projec-tions are found to be stationary and lead to flood quantiles that arenot any more critical than those observed.

It is also worth noting, going a step backward, that some studiesquestion the very basic procedure of approximating a natural ob-served phenomenon such as floods by a random variable (Klemes1986, 2000), stressing that different floods may be caused by differ-ent generating physical mechanisms. Although such considerationsare duly necessary, in absence of adequate quality and quantity ofdata, hydrologists are often compelled to extrapolate tails to pre-scribe rare flood quantiles for infrastructure design and return peri-ods and return levels have served well in the past as design tools.How long the nonstationarities should be continued in the futureremains a modeling choice that has to be guided by adequateknowledge of the physical system. A thorough exploration alongthese lines falls outside the scope of the present study and providesa direction worthy of pursuing in the future.

Concluding Remarks

Climate change can bring about significant nonstationarity inregional hydrological extremes reflected through variations in re-turn levels of floods and droughts used in hydrological designs.Traditionally, return levels are estimated from historical observa-tions, based on assumptions of stationarity. Though some studiesassess changes in return levels considering different future time sli-ces and fitting different distributions to those slices, such studies donot truly account for persistent nonstationarity. Theoretical devel-opments in the extreme value theory allow modeling of extremesconsidering the effects of physically based covariates.

Using these developments, changes in flood return levels wereanalyzed and compared with the same from historical observationsfor the Columbia River Basin. Statistically downscaled VIC-simulated streamflow projections are converted to extreme floodsand the analysis is carried out on each projection individually be-cause all projections represent possible future trajectories. Futureflood return levels are computed from appropriate fitted distribu-tions, as judged by a statistical test. For the nonstationary projec-tions, the authors further attempt to address the question: how longcan the observed return levels adequately represent future risks for

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hydrological designs pertaining to floods? It is found that manyfuture projections are stationary and the projected flood return lev-els are not significantly greater than those obtained from observeddata. For some simulations, detection of change in 10-year and50-year return levels occurs within the twentieth century, with ear-lier detection in floods of lower return periods. Additionally, themultiple projections are combined to produce a worst-case scenarioconsidering the maximum of all the projections in a given year. Inthe worst-case, for high quantiles, detection can occur as early as2027 for the 75-year return level.

Thus, this study provides an approach to test the future utility ofthe paradigm of stationarity in hydrological designs for extremes offloods for projections under climate change, taking into account theassociated uncertainties, and highlights the necessity for robust as-sessment and communication of hydrologic risk.

Acknowledgments

The authors thank the CBCCSP team, University of Washington,for making available the projected statistically downscaled andVIC-simulated daily streamflow projections for the ColumbiaRiver Basin for the CMIP3 models. They also thank Rick Katz, EricGilleland, and Dan Cooley for helpful clarifications through e-mail,and the editor, associate editor, and three anonymous reviewerswhose comments helped improve the manuscript significantly.

References

AghaKouchak, A., Easterling, D., and Hsu, K. (2013). Extremes in achanging climate: Detection, analysis and uncertainty, Springer,Netherlands.

Barnett, T. P., et al. (2008). “Human-induced changes in the hydrology ofthe western United States.” Science, 319(5866), 1080–1083.

BPA (Bonneville Power Administration). (1993). “1990 level modifiedstreamflow (1928–1989).” Portland, OR.

Brown, S., Caesar, J., and Ferro, C. (2008). “Global changes in extremedaily temperature since 1950.” J. Geophys. Res., 113(D5), 1984–2012.

Burger, G., Schulla, J., and Werner, A. (2011). “Estimates of future flow,including extremes, of the Columbia River headwaters.” Water Resour.Res., 47(10).

Burke, E. J., Perry, R. H., and Brown, S. J. (2010). “An extremevalue analysis of UK drought and projections of change in the future.”J. Hydrol., 388(1), 131–143.

Coats, R., et al. (2013). “Projected 21st century trends in hydroclimatologyof the Tahoe basin.” Clim. Change, 116(1), 51–69.

Coles, S. (2001). An introduction to statistical modeling of extreme values,Springer, London.

Cooley, D. (2013). “Return periods and return levels under climate change.”Extremes in a changing climate: Detection, analysis, and uncertainty,Springer, New York, 97–114.

Coumou, D., and Rahmstorf, S. (2012). “A decade of weather extremes.”Nat. Clim. Change, 2(7), 491–496.

Das, T., et al. (2013). “Increases in flood magnitudes in California underwarming climates.” J. Hydrol., 501, 101–110.

Das, T., Dettinger, M., Cayan, D., and Hidalgo, H. (2011). “Potentialincrease in floods in California’s Sierra Nevada under future climateprojections.” Clim. Change, 109(S1), 71–94.

Fowler, H., and Wilby, R. (2010). “Detecting changes in seasonal precipi-tation extremes using regional climate model projections: Implicationsfor managing fluvial flood risk.” Water Resour. Res., 46(3), W03525.

Fowler, H. J., Cooley, D., Sain, S. R., and Thurston, M. (2010). “Detectingchange in UK extreme precipitation using results from the climatepre-diction.net BBC climate change experiment.” Extremes, 13(2),241–267.

Gilleland, E., and Katz, R. W. (2011). “New software to analyze howextremes change over time.” Eos Trans. Am. Geophys. Union, 92(2),13–14.

Hamlet, A. F., et al. (2010). “Final project report for the Columbia basinclimate change scenarios project.” ⟨http://warm.atmos.washington.edu/2860/report⟩ (Dec. 2013).

Hamlet, A. F., et al. (2013). “An overview of the Columbia basin climatechange scenarios project: Approach, methods, and summary of keyresults.” Atmos. Ocean, 51(4), 392–415.

Hamlet, A. F., and Lettenmaier, D. P. (1999). “Effects of climate change onhydrology and water resources in the Columbia River basin.” J. Am.Water Resour. Assoc., 35(6), 1597–1623.

Hamlet, A. F., and Lettenmaier, D. P. (2005). “Production of temporallyconsistent gridded precipitation and temperature fields for thecontinental United States.” J. Hydrometeorol., 6(3), 330–336.

Hegerl, G. C., et al. (2007). “Understanding and attributing climatechange.” Climate Change 2007: The Physical Science Basis. Contribu-tion of Working Group I to the Fourth Assessment Rep. of the Intergov-ernmental Panel on Climate Change, S. Solomon, et al. eds.,Cambridge University Press, Cambridge, U.K.

Hegerl, G. C., Zwiers, F. W., Stott, P. A., and Kharin, V. V. (2004).“Detectability of anthropogenic changes in annual temperature and pre-cipitation extremes.” J. Clim., 17(19), 3683–3700.

Hidalgo, H., et al. (2009). “Detection and attribution of streamflow timingchanges to climate change in the western United States.” J. Clim.,22(13), 3838–3855.

Hirabayashi, Y., et al. (2013). “Global flood risk under climate change.”Nat. Clim. Change, 3(9), 816–821.

IPCC. (2012). “Summary for policymakers.” Managing the Risks ofExtreme Events and Disasters to Advance Climate Change Adaptation.A Special Rep. of Working Groups I and II of the IntergovernmentalPanel on Climate Change, C. B. Field, et al. eds., Cambridge UniversityPress, Cambridge, U.K., 1–19.

IPCC (Intergovernmental Panel on Climate Change). (2007). “Climatechange 2007: The physical science basis.” Contribution of WorkingGroup I to the 4th Assessment Rep. of the Intergovernmental Panelon Climate Change, Cambridge University Press, Cambridge, U.K.

Katz, R. W. (2013). “Statistical methods for nonstationary extremes.”Extremes in a changing climate: Detection, analysis and uncertainty,A. AghaKouchak, D. Easterling, and K. Hsu, eds., Springer, Netherlands,

15–37 .Katz, R. W. (2010). “Statistics of extremes in climate change.” Clim.

Change, 100(1), 71–76.Katz, R. W., Parlange, M. B., and Naveau, P. (2002). “Statistics of extremes

in hydrology.” Adv. Water Resour., 25(8), 1287–1304.Kharin, V., Zwiers, F., Zhang, X., and Wehner, M. (2013). “Changes in

temperature and precipitation extremes in the CMIP5 ensemble.” Clim.Change, 119(2), 345–357.

Kharin, V. V., and Zwiers, F. W. (2005). “Estimating extremes in transientclimate change simulations.” J. Clim., 18(8), 1156–1173.

Klemes, V. (1986). “Dilettantism in hydrology: Transition or destiny?.”Water Resour. Res., 22(9S), 177S–188S.

Klemes, V. (2000). “Tall tales about tails of hydrological distributions. I.”J. Hydrol. Eng., 10.1061/(ASCE)1084-0699(2000)5:3(227), 227–231.

Koutsoyiannis, D. (2006). “Nonstationarity versus scaling in hydrology.”J. Hydrol., 324(1), 239–254.

Koutsoyiannis, D. (2013). “Hydrology and change.” Hydrol. Sci. J., 58(6),1177–1197.

Kuczera, G. (1999). “Comprehensive at-site flood frequency analysis usingMonte Carlo Bayesian inference.”Water Resour. Res., 35(5), 1551–1557.

Kundzewicz, Z. W., Hirabayashi, Y., and Kanae, S. (2010). “River floods inthe changing climate—Observations and projections.” Water Resour.Manage., 24(11), 2633–2646.

Leadbetter, M. R., Lindgren, G., and Rootzen, H. (1983). Extremes and re-lated properties of random sequences and processes, Springer, New York.

Lee, S.-Y., Hamlet, A. F., Fitzgerald, C. J., and Burges, S. J. (2009). “Opti-mized flood control in the Columbia River basin for a global warmingscenario.” J Water Resour Plann. Manage., 10.1061/(ASCE)0733-9496(2009)135:6(440), 440–450.

© ASCE 04016021-11 J. Hydrol. Eng.

J. Hydrol. Eng., 2016, 21(8): 04016021

Dow

nloa

ded

from

asc

elib

rary

.org

by

Indi

an I

nstit

ute

of S

cien

ce B

anga

lore

on

04/1

6/19

. Cop

yrig

ht A

SCE

. For

per

sona

l use

onl

y; a

ll ri

ghts

res

erve

d.

Mantua, N., Tohver, I., and Hamlet, A. (2010). “Climate change impactson streamflow extremes and summertime stream temperature and theirpossible consequences for freshwater salmon habitat in WashingtonState.” Clim. Change, 102(1–2), 187–223.

Maraun, D. (2013). “When will trends in European mean and heavy dailyprecipitation emerge?” Environ. Res. Lett., 8(1), 014004.

Mears, C. A., Santer, B. D., Wentz, F. J., Taylor, K. E., and Wehner, M. F.(2007). “Relationship between temperature and precipitable waterchanges over tropical oceans.” Geophys. Res. Lett., 34(24), L24709.

Meehl, G. A., et al. (2007a). “Understanding and attributing climate 814change.” Climate Change 2007: The Physical Science Basis. Contribu-tion of Working Group I to the 4th Assessment Rep. of the Intergovern-mental Panel on Climate Change, S. Solomon, et al. eds., CambridgeUniversity Press, Cambridge, U.K.

Meehl, G. A., et al. (2007b). “The WCRP CMIP3 multi-model dataset: Anew era in climate change research.” Bull. Am. Meteorol. Soc., 88(9),1383–1394.

Milly, P. C., et al. (2008). “Stationarity is dead: Whither water manage-ment?.” Science, 319(5863), 573–574.

Milly, P. C. D., Wetherald, R. T., Dunne, K. A., and Delworth, T. L. (2002).“Increasing risk of great floods in a changing climate.” Nature,415(6871), 514–517.

Min, S.-K., Zhang, X., Zwiers, F. W., Friederichs, P., and Hense, A. (2009).“Signal detectability in extreme precipitation changes assessed fromtwentieth century climate simulations.” Clim. Dyn., 32(1), 95–111.

Min, S.-K., Zhang, X., Zwiers, F. W., and Hegerl, G. C. (2011). “Humancontribution to more-intense precipitation extremes.” Nature, 470(7334),378–381.

Mondal, A., and Mujumdar, P. P. (2012). “On the basin-scale detection andattribution of human-induced climate change in monsoon precipitationand streamflow.” Water Resour. Res., 48(10), W10520.

Mondal, A., and Mujumdar, P. P. (2015). “Return levels of hydrologicdroughts under climate change.” Adv. Water Resour., 75, 67–79.

Montanari, A., and Koutsoyiannis, D. (2014). “Modeling and mitigatingnatural hazards: Stationarity is immortal!” Water Resour. Res.,50(12), 9748–9756.

Naik, P. K., and Jay, D. A. (2005). “Estimation of Columbia River virginflow: 1879 to 1928.” Hydrol. Processes, 19(9), 1807–1824.

Obeysekera, J., and Salas, J. D. (2014). “Quantifying the uncertainty ofdesign floods under nonstationary conditions.” J. Hydrol. Eng.,10.1061/(ASCE)HE.1943-5584.0000931, 1438–1446.

Oehlert, G. W. (1992). “A note on the delta method.” Am. Statistician,46(1), 27–29.

Pall, P., et al. (2011). “Anthropogenic greenhouse gas contribution toflood risk in England and Wales in autumn 2000.” Nature, 470(7334),382–385.

Reclamation. (2011). “West-wide climate risk assessments: Bias-correctedand spatially downscaled surface water projections.” U.S. Dept. of theInterior, Bureau of Reclamation Technical Service Center, Denver.

Rootzén, H., and Katz, R. W. (2013). “Design life level: Quantifying risk ina changing climate.” Water Resour. Res., 49(9), 5964–5972.

Salas, J. D., and Obeysekera, J. (2014). “Revisiting the concepts of returnperiod and risk for nonstationary hydrologic extreme events.” J. Hydrol.Eng., 10.1061/(ASCE)HE.1943-5584.0000820, 554–568.

Santer, B. D., et al. (2007). “Identification of human-induced changesin atmospheric moisture content.” Proc. Natl. Acad. Sci., 104(39),15248–15253.

Serinaldi, F. (2014). “Dismissing return periods!.” Stochastic Environ. Res.Risk Assess., 29, 1179–1189.

Serinaldi, F., and Kilsby, C. G. (2015). “Stationarity is undead: Uncertaintydominates the distribution of extremes.” Adv. Water Resour., 77,17–36.

Sillmann, J., Croci-Maspoli, M., Kallache, M., and Katz, R. W. (2011).“Extreme cold winter temperatures in Europe under the influence ofNorth Atlantic atmospheric blocking.” J. Clim., 24(22), 5899–5913.

Sivapalan, M., and Samuel, J. M. (2009). “Transcending limitations ofstationarity and the return period: Process-based approach to flood es-timation and risk assessment.” Hydrol. Processes, 23(11), 1671–1675.

Stewart, I. T., Cayan, D. R., and Dettinger, M. D. (2005). “Changes towardearlier streamflow timing across western North America.” J. Clim.,18(8), 1136–1155.

Towler, E., Rajagopalan, B., Gilleland, E., Summers, R. S., Yates, D., andKatz, R. W. (2010). “Modeling hydrologic and water quality extremesin a changing climate: A statistical approach based on extreme valuetheory.” Water Resour. Res., 46(11), W11504.

Westra, S., Alexander, L. V., and Zwiers, F. W. (2013). “Global increasingtrends in annual maximum daily precipitation.” J. Clim., 26(11),3904–3918.

Willett, K. M., Gillett, N. P., Jones, P. D., and Thorne, P. W. (2007).“Attribution of observed surface humidity changes to human influence.”Nature, 449(7163), 710–712.

© ASCE 04016021-12 J. Hydrol. Eng.

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