A hierarchical Bayesian approach for the analysis of climate change impact on runoff extremes

HYDROLOGICAL PROCESSESHydrol. Process. (2013)Published online in Wiley Online Library(wileyonlinelibrary.com) DOI: 10.1002/hyp.10113

A hierarchical Bayesian approach for the analysis of climatechange impact on runoff extremes

M. R. Najafi† and H. Moradkhani*Department of Civil and Environmental Engineering, Portland State University, 1930 SW 4th Ave., suite 200, Portland, OR, 97201, USA

*CEnsuiE-m†CuBC

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Abstract:

In this study the impact of climate change on runoff extremes is investigated over the Pacific Northwest (PNW). This paper aims toaddress the question of how the runoff extremes change in the future compared to the historical time period, investigate the differentbehaviors of the regional climate models (RCMs) regarding the runoff extremes and assess the seasonal variations of runoff extremes.Hydrologic modeling is performed by the variable infiltration capacity (VIC) model at a 1/8° resolution and the model is driven byclimate scenarios provided by the North American Regional Climate Change Assessment Program (NARCCAP) including nineregional climate model (RCM) simulations. Analysis is performed for both the historical (1971–2000) and future (2041–2070) timeperiods. Downscaling of the climate variables including precipitation, maximum and minimum temperature and wind speed is doneusing the quantile-mapping (QM) approach. A spatial hierarchical Bayesian model is then developed to analyse the annual maximumrunoff in different seasons for both historical and future time periods. The estimated spatial changes in extreme runoffs over the futureperiod vary depending on the RCMdriving the hydrologic model. The hierarchical Bayesian model characterizes the spatial variationsin the marginal distributions of the General Extreme Value (GEV) parameters and the corresponding 100-year return level runoffs.Results show an increase in the 100-year return level runoffs for most regions in particular over the high elevation areas during winter.The Canadian portions of the study region reflect higher increases during spring. However, reduction of extreme events in severalregions is projected during summer. Copyright © 2013 John Wiley & Sons, Ltd.

KEY WORDS spatial hierarchical Bayesian model; extremes; climate change; downscaling; NARCCAP; Pacific Northwest

Received 13 June 2013; Accepted 8 November 2013

INTRODUCTION

The uprising temperature of the earth due to climate changehas shown to alter the variations in hydro-climate variablesregarding their intensities, frequencies and durations(Hidalgo et al., 2009; Karl and Melillo, 2009; Halmstadet al., 2012). Extreme events, i.e. floods and droughts, aresusceptible to any disturbances in climate cycles (Fieldet al., 2012;Madadgar andMoradkhani, 2013a, b). As such,it is important to provide the policymakers with sufficientknowledge about the probable impacts of climate change onhydrologic extremes and most importantly floods whichcause or threaten damage to properties and lives(Moradkhani et al., 2010). Frei et al. (2006) and Villariniet al. (2012) studied the variability in both observed andclimate model simulated extreme event occurrence. Thepotential for change in the occurrence of extreme events inthe future in conjunctionwith projected climate variability is

orrespondence to: Hamid Moradkhani, Department of Civil andvironmental Engineering, Portland State University, 1930 SW 4th Ave.,te 200, Portland, OR 97201, USAail: [email protected] at Pacific Climate Impacts Consortium, University of Victoria,, Canada

pyright © 2013 John Wiley & Sons, Ltd.

also prevalent in recent studies (Mote and Salathe, 2010;Tryhorn and DeGaetano, 2011).Study of the climate change impact on hydrologic

variables such as runoff is commonly conducted based onhydrologic simulation because runoff amount is notgenerally known at the spatial scale of interest (in this studycomputational grid cells). Therefore, hydrologic model isinitially calibrated based on the observed time series, thenthe model is forced with future climate scenarios (Risleyet al., 2011). The future and historical simulations arecompared to assess the relative impact of climate change(Wood et al., 2004; Jung et al., 2011; Najafi et al., 2011a).Global climate models provide historical as well as futureassessments of the climatologic variables across the globe.However, due to their coarse resolution, attempts have beenmade to nest regional climate models within GCMs andprovide projections at finer scale resolutions. The NorthAmerican Regional Climate Change Assessment Program(NARCCAP) provides data from multiple GCM-RCMcoupled simulations over the majority of the Americancontinent (Mearns et al., 2009).Extreme value theory is commonly utilized in hydrology

and water resources to evaluate the characteristics ofextreme events and has seen more applications over the

M. R. NAJAFI AND H. MORADKHANI

past decade, e.g. (Acero et al., 2010; Kharin et al., 2010;Davison et al., 2012). It is a statistical method to model thetails of the probability distribution of a random variable.Fitting of extreme distributions at individual locations is

restricted by the limited records of hydro-climatic extremesin space and time (Fuentes et al., 2013). Therefore, methodshave been developed to combine extreme data fromdifferent locations to provide a more robust statisticalmodel. Indexfloodmethod (Dalrymple and Survey, 1960) isone approach to improve the accuracy of the estimates and topredict flood at ungagged sites. A homogeneous region isfirst defined and the extreme data at each gage are divided bythe index flood followed by fitting a distribution to thecombined data from all gages. Regional frequency analysis(RFA), however, does not consider the spatial componentsof the point data and cannot incorporate additional variables(i.e. covariates) into the analysis. Besides it is not possible toexplicitly estimate the uncertainties using this approach.Spatial hierarchical Bayesian method is an alternative

approach which has been recently introduced in theliterature. It considers a spatial model on the parameters ofa univariate extreme value distribution (such as GEV) ateach point (or grid cell). This approach has gained muchattention in climate extreme analyses as it effectivelycombines spatially distributed data. The studies includethe analysis of wind (Fawcett and Walshaw, 2006; Li andShi, 2012), precipitation (Cooley et al., 2007; Schliep et al.,2010; Ghosh and Mallick, 2011; Renard, 2011; Atyeo andWalshaw, 2012) and temperature (Lemos et al., 2009).Cooley and Sain (2010) studied the changes in precipitationextremes using regional climate model data for historicaland future periods. Sang and Gelfand (2009) and Schliepet al. (2010) utilized a spatial autoregressive model of theannual maximum rainfall. Cooley and Sain (2012) andDavison et al. (2012) reviewed and compared severalmodeling approaches for spatial extremes with the applica-tions to rainfall events. Fewer attempts, however, have beenmade to enhance the hydrologic extreme analysis. Lima andLall (2010) employed a hierarchical Bayesian model forstreamflow analysis using the drainage area as the onlyexplanatory variable with no spatial model. Najafi andMoradkhani (2013) analysed the historical extreme runoffdata recorded at gage sites based on a spatial hierarchicalBayesian approach incorporating the elevation and drainagearea as additional covariates. They considered geostatisticalmodeling for spatially characterizing the extreme parame-ters. Nevertheless, analysing grid-based hydrologic vari-ables (e.g. when obtained from distributed hydrologicmodeling) is a challenging task by using the geostatisticalmodels (i.e. variograms) especially for large regions(Schliep et al., 2010).Hydropower is the major energy source in the Pacific

Northwest (PNW) region of USA; hence the possibleimpacts of climate change on its hydro-climate

Copyright © 2013 John Wiley & Sons, Ltd.

characteristics are of concern (Snover et al., 2003).Winter (October–March) precipitation dominates thehydrologic characteristics of PNW. Precipitation in themountains is higher especially on the western slopes of theOlympics and Coast mountain range and the Cascades(Figure 1). The region is climatically divided by the CascadeMountains. In low level elevations such as the low-lyingvalleys west of the Cascades precipitation falls as rain inwinter and little snow pack is accumulated. In intermediateelevations precipitation falls as rain during fall and earlywinter and transits to snowfall during winter followed byspring/summer snowmelt. East of Cascades is snowdominated with spring and summer snowmelt resulting inhigh runoff rates. Consequently the temperature fluctuationsdue to climate change have significant impacts on theintensity, frequency and seasonality of the streamflow(Lettenmaier et al., 1992; Payne et al., 2004). ColumbiaRiver is a predominant hydrologic and water resourcefeature in the PNW. It originates from the Columbia Lake inCanada and discharges in the North Pacific Ocean at itsmouth in Astoria, Oregon.This study addresses the climate change impact on runoff

extremes over the PNW. Individual analyses are performedfor each season using the complete regional climate modelsimulations over the US available at the time when thisresearch was performed. In the section ‘Methodology’ ofthis paper the major steps of the hydrologic extreme analysisare illustrated including the distributed hydrologic modelingand spatial hierarchical Bayesian analysis of extremes. Thesection ‘Application to the runoff extremes over the PacificNorthwest’ describes the application of the method over thePNW study region starting with a description of the grid-based observed data, climate scenarios provided by theNorthAmerican Regional Climate Change Assessment Program(NARCCAP) simulations followed by the downscalingprocedure, hydrologic modeling and parameterization,extreme analysis based on hierarchical Bayesian and thediscussion of results. Summary and conclusions are providedin the ‘Conclusion’.

METHODOLOGY

Hydrologic modeling

The variable infiltration capacity (VIC) hydrologic modelis used in this study for hydrologic simulation usingobservational and RCM gridded data sets. VIC (Lianget al., 1994; Liang et al., 1996) is a semi-distributed landsurface model which has been successfully applied forclimate change studies over the PNW region (Lettenmaieret al., 1992; Matheussen et al., 2000; Payne et al., 2004;Hidalgo et al., 2009) as well as hydrologic forecasts (Yuanet al., 2013). The forcing data including precipitation,maximum and minimum temperature and wind speed are

Hydrol. Process. (2013)

Figure 1. The Pacific Northwest region

SPATIAL HIERARCHICAL BAYESIAN MODELING OF RUNOFF EXTREMES OVER PNW

required to perform the hydrologic modeling along with thewatershed characteristics such as land cover, soil andelevation. Simulations are independently implemented foreach grid cell at a daily or sub-daily time step withoutconsidering the horizontal flow. For routing purposes aseparate model is needed to transport the surface runoff tothe grid outlet and to the flow channel (Lohmann et al.,1998). The VIC model can be run in water balance andenergy balancemodes of operations. Thewater balance doesnot consider the surface energy balance and assumes that thesoil surface temperature and the surrounding air temperatureare equal. VIC model includes a snow algorithm whichconsiders a two-layer formulation of surface and packlayers. It models the surface energy balance at the snow/airinterface along with the ground heat flux from the snowpackinto the ground.The VIC model was initially developed as a soil–

vegetation–atmosphere transfer scheme to be incorporatedin GCMs. The current VIC model considers sub-gridvariability in land cover, soil-moisture storage, precipitationand elevation. Each grid cell is partitioned into snow bands(elevation bands) containing a number of land cover tiles.Each land cover type consists of a vegetation layer, andseveral soil layers (commonly three soil layers are used(Cherkauer et al., 2003)). The land cover tiles are describedbased on leaf area index (LAI), albedo, canopy resistance,


roughness length etc. Spatial variability of infiltration andrunoff generation is simulated using the variable infiltrationcurve within each vegetation class. Penman–Monteithequation governs the potential and actual evapotranspirationwhich is dependent on net radiation and vapor pressuredeficit. The dynamic behavior of soil response to the rainfallevent is signified by the upper soil layers. Gravity controlsthe flow from upper layers to lower ones. Empirically basedArno curve is used to model baseflow.

Spatial hierarchical Bayesian model

Runoff extreme analysis is performed based on thegeneralized extreme value (GEV) distribution:

F y μ;σ; κjð Þ ¼ exp � 1þ κy� μσ

� �h i�1=κ�;

�(1)

where μ,σ and κ are the location, scale and shapeparameters, respectively, and 1 + κ(y �μ)/σ > 0.Depending on different values of κ the GEV distributionwill be Gumbel, Frechet or Weibull distributions. Forκ> 0 the upper tail of the distribution decreases slowly witha power function and never reaches zero, for κ< 0 the GEVdistribution has a bounded upper tail and for κ =0 it has anexponentially decreasing tail (Coles, 2001). GEV distribu-tion models the maximum of sufficiently long series of data.



In hydrologic applications it is commonly used when dataare considered in annual, seasonal or monthly blocks wheretheir block maxima are taken as iid random variables.Using Equation (2) the τ-year return level corresponding

to an extreme event which is expected to be exceeded onceevery τ years (i.e. expected value of an event with anexceedance probability of p = 1/τ) can be obtained by:

zτ ¼ μ� σκ1� � log 1� 1=τð Þð Þ�κ½ �; (2)

In order to deal with the limited records of hydro-climaticextremes in space and time spatial hierarchical Bayesianmodel is developed to collect information from differentlocations to increase the reliability of the estimates. Inaddition this model aims to describe the spatial variations ofmarginal distributions. It consists of a latent process inwhich a spatial model characterizes the parameters of themarginal distribution. The spatial model is commonly basedon variograms in applications where hydro-climate data arepoint-referenced (i.e. geostatistical) and based on theconditional, intrinsic or simultaneous autoregressive models(CAR, IAR, SAR) for data recorded at areal units of grids(i.e. lattice) (Cressie, 1992; Banerjee et al., 2004).The spatial hierarchical model consists of the so-called

data, process and prior stages which are linked throughconditional distributions. In this study the hierarchicalmodeling process is performed separately for each RCM(runoff outputs from VIC driven by RCM) and season aswell as the historical and future time periods.The data stage of the hierarchical model defines the

likelihood function. Let’s denote Yi,t as the maximumannual runoff (mm) at the ith cell in time t. Yi,t follows theGEV distribution with parameters (μi,σi,κi) that makes itconditionally independent of Yj,t (i≠ j). This implies that themaximum series in each grid cell would follow their ownGEV parameters. Therefore the likelihood function is theproduct of Equation (1) over each grid cell:

p1 yijμi;σi; κið Þ ¼ ∏n

i¼1∏T

t¼1

1σi

1þ κiyit � μi

σi

� �� 1=κi�1

� exp

�� 1þ κi

yit � μi

σi

� �� 1=κi�(3)

where T is the total number of years for each of the historicaland future periods and n the total number of cells. In order toimprove the maximum likelihood estimate of extremedistributions for a short record data Martins and Stedinger(2000) recommended a Bayesian prior which restricts thevalues of κ to a reasonable range. Similarly Cooley and Sain(2010) and Schliep et al. (2010) considered this prior in thedata stage of their hierarchical model for precipitationextreme analysis. This prior function is a beta distribution


with a mean of 0.1, supported in the [�0.5,0.5] interval:

π κið Þ ¼ Γ 15ð ÞΓ 9ð ÞΓ 6ð Þ 0:5þ κið Þ8 0:5� κið Þ5 (4)

in which Γ is the gamma function. The product of this priorand the likelihood function defined inEquation (3) constitutesthe data stage of the hierarchical model (i.e. p1 ×π(κi)).The process stage of the model tries to combine extreme

data from different cells. The cell-wise extreme runoff dataobtained from VIC suggests using the conditionallyautoregressive models.Here we assume that each GEV parameter follows a

normal distribution:

p2 μijλμ;Ui;μ; τ2μ� �

¼ NðX’i;qλμ;q’ þ Ui;μ;

1τμ2

�

p2 ψijλψ;Ui;ψ; τ2ψ� �

¼ NðX’i;qλψ;q’ þ Ui;ψ;

1τψ2

�

p2 κijλκ;Ui;κ; τ2κ� � ¼ N X’

i;qλκ;q’ þ Ui;κ;1

τκ2 ��

(5)

with ψi = log (σi). i denotes grid location, q the number ofcovariates and q′ the number of their associated factors(q′= q + 1). Xi,q and λθ;q’ with θ ~μ,ψ, κ represent thecovariates (i.e. explanatory variables associated with eachcell including the geographic coordinates, physical andclimatological characteristics etc.) and their correspond-ing factors, respectively; τ2θ is the precision parameter. Ui,θ

are spatial random effect parameters which account forthe dependencies between GEV parameters.Consider a random vector of Y= (Y1,Y2,…,YN)

T in whichYi pertains to the ith cell. One can generate the jointdistribution p(y) using ‘n’ full conditional distributions ofp(yi|yj) where j≠ i. This Markov random field approach callsthe jth cell a neighbor of cell ‘i’ if p(yi|yj) depends on Yj.The standard conditional autoregressive model (CAR) is

a Gaussian MRFwhich defines the conditional distributionsfor each of the random variables (RV) given the RVs inneighboring cells (Besag, 1974; Rue and Held, 2005).Gaussian MRFs can be described by the precision matrix(Q) which is the inverse of the covariance matrix. The mostpopular CAR employment is the pairwise differenceformulation known as the intrinsic autoregressive (IAR)model (Besag and Kooperberg, 1995). In this study thespatial effect parameter is modeled using the multivariateintrinsic autoregressive spatial model which is based on theproximity matrix of W with components of wij (Besag,1974; Banerjee et al., 2004; Schliep et al., 2010). Theproximity matrix is used to weight the similarities betweenvalues at different cells. It is calculated by the neighborhoodstructure of the data in away that if two cells have a commonborder line or notch, the corresponding values in theproximity matrix become one otherwise they are set to zero.Row standardization can be performed to increase the



influence of cells with fewer neighbors (If p(θi|θj) where (j≠ i)depends on θj then j is called a neighbor of cell i). Theprecision is defined byQ=T⊗Q1 where T is a 3× 3 positivedefinite matrix reflecting the dependence informationbetween spatial effect parameters corresponding toμ, σ and κ.

Q1 ¼ Dw �W (6)

Dw is a diagonal matrix with elements of dwii :

dwii ¼ ∑n

j¼1wij (7)

In the third stage of the hierarchical Bayesian model thepriors are designated for the latent parameters which aredefined in the process stage including λθ;q’ and T.

APPLICATION TO THE RUNOFF EXTREMES OVERTHE PACIFIC NORTHWEST

The process of hydrologic extreme analysis over PNW issummarized in Figure 2. Each step is explained in thefollowing sections.

Data

The observational griddedmeteorological data covering theperiod of January 1949 through July 2000 was considered inthis study (Maurer et al., 2002). It includes daily totalprecipitation (mm/day), maximum andminimum temperature(C°) and the average wind speed (m/s) at an average height of2mabove the surface. Therewere 6392 cellswith 1/8th degreeresolutions covering the PNW region including the CRB.North American Regional Climate Change Assessment

Program (NARCCAP) provides a suite of regionalclimate model simulations on a 50 km by 50 km spatialresolution and daily/sub-daily time scale based on theSRES-A2 emission scenario (Mearns et al., 2009). TheA2 scenario predicts large population increases, highcarbon dioxide emissions and weak environmentalconcerns. The RCMs include CRCM, ECP2, MM5I,RCM3, WRFG and HRM3. These are nested insidegeneral circulation models of GFDL, CGCM3, HADCM3and CCSM. Data is available for the historical and futureperiods of 1971–2000 and 2041–2070, respectively. Forhydrologic simulations we were interested in the 3-hourlyprecipitation (kg/m2s), daily maximum and minimumtemperature (K), zonal and meridional wind speeds (m/s)from NARCCAP. At the time when this research wasconducted, data from eight RCM_gcms were availablecovering the historical and future time periods and oneRCM_gcm for the historical period. These includeCRCM_cgcm3, CRCM_ccsm, ECP2_gfdl (for the his-


torical period), HRM3_gfdl, HRM3_hadcm3,RCM3_gfdl, RCM3_cgcm3, WRFG_cgcm3 andWRFG_ccsm.

Downscaling the NARCCAP climate variables

The process of downscaling has been established toaddress the inadequacies of large-scale resolution models.There are two main classes of downscaling procedures:statistical and dynamical. Several studies over the lastseveral years have provided detailed comparisons of bothdownscaling types (Giorgi, 1990; McGregor, 1997;Murphy, 1999; Di Luca et al., 2011; Najafi et al.,2011a). RCMs provided by NARCCAP have 50 kmresolutions, though hydrologic models commonly requiremeteorological variables at a finer resolution. Furtherdownscaling is therefore required to develop the highresolution data (Najafi et al., 2011b; Samadi et al., 2013).In this study the quantile-mapping (QM) approach (Woodet al., 2004) was used for downscaling and bias correctingthe individual RCM variables. However, a more sophis-ticated and accurate approach in bias correction (orpostprocessing) by means of copula functions can be usedfor this as well Madadgar et al., (2012)In the QMmethod the observed and simulated data sets are

each characterized in terms of their full distributions of dailyvalues, a so-called non-parametric approach since it does notrely on adjusting simply the mean, standard deviation, or otherstandard statistical parameters. QM corrects the model outputsdistribution to exclude systematic biases. The correction ismade for each cell at the resolution of the observed data. Itconsiders each percentile of the variable’s distribution andcompares the 1/8° observed grid cell with the closest 50kmresolutionRCMcell (QuintanaSeguí et al., 2010). For both theobserved and simulated data sets, the cumulative distributionfunctions (CDFs) are computed for each month separately.After computing the CDFs, the scaling factor determinedbased on the respective quantile values during the observedperiod are applied for the future period. This is based on theassumption that the correction function remains constant intime. In this study the downscaling of NARCAAP productwas performed for eachmodel following the above procedure.

Hydrologic modeling

The hydrologic simulation was carried out using theVIC model (version 4.1.2.c) driven by the 1/8th degreeresolution observational and downscaled NARCCAPmeteorological data. Modeling was performed in a dailytime step for the historical time period of 1971–2000 andfuture period of 2041–2070 by considering three layers ofsoil. The calibrated parameters including the soil depth,baseflow drainage and infiltration capacity of the soillayers along with the vegetation and snow band data wereobtained from the Land Surface Hydrology Research


Figure 2. Flowchart of the modeling procedure


Group, University of Washington (Maurer et al., 2002).The geographic information of each grid cell, soilparameters and the initial soil moisture conditions arestored in the soil parameter file. The land cover types,number of vegetation tiles and their coverage in each gridcell as well as other vegetation parameters such as rootdepth etc. are contained in the vegetation parameter file.The VIC model is capable of disaggregating each grid cellinto elevation bands in order to lapse the related averagetemperature, pressure and precipitation. The snow bandfile contains information on each elevation band used bythe snow model to account for the topographicalinfluences on snow pack accumulation and ablation. Fiveelevation (snow) bands were considered to bettercharacterize the snow processes at each grid cell.Simulations were made based on the water balance modeindicating that the surface temperature was set to thesurrounding air temperature. Several variables wereestimated for this study region including the evapotrans-piration, canopy interception of liquid water, moisturecontent of each soil layer, snow water equivalent, snowpack depth and fractional area of snow cover. In thisstudy we analysed the resulting runoff (mm) at each cellfor the hydrologic extreme analysis.


RESULTS AND DISCUSSION

The grid-based annual maximum runoffs obtained fromeach RCM data were separately extracted for the historicaland future periods and all seasons including winter (DJF),spring (MAM), summer (JJA) and fall (SON). The spatialhierarchical Bayesian modeling of extremes was thenperformed for each data set. Both the historical(1971–2000) and future (2041–2070) time periods consistof 29 years of data (i.e. T = 29 in Equation 3) while there arein total 6392 cells with 1/8° resolution in the study region (i.e. n = 6392 in Equation. 3). Maximum likelihood estimatesof the GEV parameters for each cell were considered asstarting points for the inference process in the spatialhierarchical Bayesian model. As discussed in the previoussections the locationμ, scale σ and shape κ parameters of thegeneralized extreme value distribution were spatiallycharacterized by a multivariate intrinsic autoregressive(IAR) model. Furthermore the latitude, longitude, 30-yearaverage daily precipitation and temperature at each cell wereconsidered as potential covariates. These variables were alltransformed to have a mean of zero and standard deviationof one. According to Equation (5) values of precisionparameters were set constant τ2θ ¼ 4 100 1600ð Þ which


Table I. Model selection based on the deviance informationcriterion

Models DIC pD

Model1 290 071.9 10 746.96Model2 287 712.3 10 650.56Model3 288 995 10 700.37Model4 288 478.2 10 529.89

Model1: includes latitude and longitude as the covariatesModel2: includes latitude, longitude and 30-year average precipitation asthe covariates (having the lowest DIC value therefore chosen for the restof the analyses)Model3: includes latitude, longitude and 30-year average temperature asthe covariatesModel4: includes latitude, longitude and 30-year average precipitation andtemperature as the covariates


allowed the spatial effect parameters (Ui,θ) to define most ofthe variability in the spatial model.In the third stage of the hierarchical Bayesian model,

priors were defined over the latent parameters of λθ;q’ and T.Uninformative Gaussian distributions were considered forregression coefficients λθ;q’ with zero mean and variance of5. The intercept terms λθ,0 were assigned Gaussiandistributions with means equal to the maximum likelihoodestimates of the corresponding GEV parameters at each cellwith variance 50. VagueWishart prior with three degrees offreedom was considered for the precision matrix T withdiagonal of (0.04, 8, 80)T (Cooley and Sain, 2010).Metropolis–Hastings within Gibbs sampler was used

for parameter estimation (Casella and Robert, 1999;Banerjee et al., 2004; Rue and Held, 2005). To ensureconverging to stationary posterior distributions, simula-tions were performed for 65 000 iterations with a burn inperiod of 50 000 iterations. In order to break thedependencies between draws and improve the mixing ofthe posterior samples in the Markov Chain, thinning wasconsidered where every 15th sample draw was kept. Twoparallel chains were generated each having differentinitial values. MCMC convergence was assessed based ontrace plots (i.e. plot of the iteration number versus thevalue of the sampled parameter) and the scale reductionfactor which should decline to one as the number ofiterations increases indefinitely (Gelman and Rubin,1992). The chains were merged to produce the posteriordistributions of the GEV parameters as well as the latentparameters. The probability distributions of the 100-yearreturn level runoffs were calculated based on theestimated posterior distributions of the GEV parameters.Several models were constructed using different covar-

iates in the process stage including temperature, precipita-tion, latitude and longitude. Model selection was performedusing the deviance information criteria (DIC) (Spiegelhalteret al., 2002). The posterior distribution of the deviancestatistic is obtained using the likelihood function of:

D Yitjθið Þ ¼ �2∑i∑tlogp Yitjμi;ψi; κið Þ (12)

where θi= (μi,ψi,κi)

An arbitrary scaling function can be added toEquation (12)which is not considered here as it has no influence on therelativedifferencesbetweenDICsacross themodels (Banerjeeet al., 2004). The posterior expectation of the devianceD andthe effective number of parameters pD are calculated by:

D ¼ D Yitjθið Þ (13)

pD ¼ D� D Yitjθ� �

(14)


The deviance information criterion (DIC) is thendetermined by:

DIC ¼ Dþ pD (15)

DICs were calculated based on the posterior samples ofeach model. After comparing models with differentcovariates the model with latitude, longitude and 30-yearaverage precipitation presented the lowest DIC, henceselected for extreme runoff analysis driven by differentRCMs (Table I).

Analysis of GEV parameters

Figure 3 shows the posterior mean of the GEV locationparameter of runoff extremes for each 1/8th degree gridcell. Legend is truncated for better visualization. On theleft, results show the historical period of 1971–2000 andon the right the future period of 2041–2070. Theestimated location parameter values are higher on thewest side of the Cascade Range and on the Coast Range(geographical locations are shown in Figure 1). Theyshow the decrease from winter to summer and thenincrease during fall. In the Rocky Mountains fromCanada down to Idaho and Montana the values arelowest in winter and as the snowmelt starts they increaseduring spring and reach the highest in summer followedby a decrease in fall. For other parts of the PNWthe models show different results but overall the values ofthe location parameters are higher on the east of CascadeRange during winter and on the east of the PNW duringspring and summer. The results for future time period(figures on the right column) indicate an overall increasefor location parameter for all seasons except for summer.In particular, the parameter values increase on the west ofCascade Range during winter and on the RockyMountains in northern parts during spring. The resultsshow a decrease in parameters during summer especially


(a) (b)

(c) (d)

Winter Spring

Summer Fall

Winter Spring

Summer Fall

Winter Spring

Summer Fall

Winter Spring

Summer Fall

Winter Spring

Summer Fall

Winter Spring

Summer Fall

Winter Spring

Summer Fall

Winter Spring

Summer Fall

(e) (f)

(g) (h)

Winter Spring

Summer Fall

Winter Spring

Summer Fall

Winter Spring

Summer Fall

Winter Spring

Summer Fall

Winter Spring

Summer Fall

Winter Spring

Summer Fall

Winter Spring

Summer Fall

Winter Spring

Summer Fall

Figure 3. Posterior estimate mean of the location parameter for each grid shown for the current period of 1971–2000 on left and future period of2041–2070 on right for (a) CRCM_ccsm, (b) CRCM_cgcm3, (c) HRM3_gfdl, (d) HRM3_hadcm3, (e) RCM3_cgcm3, (f) RCM3_gfdl, (g) WRFG_ccsm

and (h) WRFG_cgcm3


on high elevations including the Cascade Range, CoastRange and Rocky Mountains. The Olympics Mountainson the northwest of the state of Washington shows the


highest values during the winter compared to the otherseasons except for the HRM3_hadcm3 model data.Increasing trend is seen for the future time period



during winter in most of the models except the onementioned previously along with RCM3_cgcm3,RCM3_gfdl and WRFG_ccsm which indicate adecrease in the winter season and an increase in fallfor the latter.The posterior distribution mean of the GEV scale

parameter follows a similar pattern to the locationparameter (not shown). Accordingly the HRM3_hadcm3,RCM3_cgcm3 and RCM3_gfdl show contradictoryresults on the Coast Range for the future time periodwhen compared with other RCM data inputs.Considering the shape parameter (not shown here)

most of the models agree that on the west side ofCascade Range the shape parameters are lowest duringwinter, except for the Olympics Mountains, and highestduring fall. However, results show highest values on theRocky Mountains on north and east of the region duringwinter which reaches the lowest in summer. East of theCascade Range presents highest values during fall andlowest in spring.

Spatial and temporal variations of runoff extremes

As explained previously, the RCM results were down-scaled and bias corrected using the UW observed data.These were used as forcing data to the VIC model and, theannual maxima of the estimated runoff (mm) over each gridcell were modeled using a GEV distribution based on aspatial hierarchical Bayesian model. The GEV parametersincluding the location, scale and shape parameters were then

Figure 4. The biases and their corresponding frequencies of the 100-year retuobservation min


used to calculate the 100-year return level runoff for eachgrid cell. Figure 4 shows the biases and their correspondingfrequencies of the 100-year return level runoff posteriormeans of different RCMs during 1971–2000 (i.e. observa-tion minus simulation). Overall models seem to agree wellfor all seasons except HRM3_hadcm3 which underesti-mates the extremes in winter and overestimates them duringspring. The spatial variation of bias (not shown here)demonstrates that this model differs from others in theRockies during winter and west of Cascades during springand summer. Also CRCM_cgcm3 and RCM3_cgcm3overestimate the 100-year return level runoffs in the Rockiesduring spring; however, marginal differences are still seen.During winter season the RCMs slightly underestimate therunoff values over the west of the Cascades except forWRFG_ccsm which indicates overestimation. On the northof the Rockies the RCMs slightly overestimate the runoffsexcept HRM3_hadcm3 which underestimates them and onthe east they mostly underestimate the values. Over the eastof the Cascades they show underestimation. During springseason the RCMs underestimate the runoff values over thewest of the Cascades except for HRM3_hadcm3 whichindicates overestimation. The situation is reversed on theOlympics Mountains. On the north of the Rockies theRCMs overestimate the runoffs except CRCM_ccsm whichunderestimates them and on the east they mostly underes-timate the values except for the HRM3_hadcm3. Over theeast of the Cascades they show underestimation except forthe HRM3_hadcm3. During summer season the RCM3

rn level runoff posterior means of different RCMs during 1971–2000 (i.e.us simulation)



underestimate the runoff values over the west of theCascades except for CRCM_cgcm3, HRM3_gfdl andHRM3_hadcm3 which indicate overestimation. On thenorth of the Rockies the RCMs underestimate the runoffsexcept HRM3_hadcm3 which overestimates them and onthe east they mostly underestimate the values. Over the eastof the Cascades they show underestimation. During fallseason the RCM3 underestimate the runoff values over thewest of the Cascades except for the south of the state ofOregon where they overestimate the return levels. On thenorth of the Rockies the RCMs underestimate the runoffsexcept HRM3_hadcm3which overestimates them. Over theeast of the Cascades they show underestimation. Bias ateach cell was calculated using the posterior mean of theestimated runoff extremes, then the medians of all biaseswere foundwhich is shown inTable II. Results show that theoverall biases of the RCMs are low. In winter, spring andsummer HRM3_hadcm3 has the largest bias and in fall the

Table II. Overall bias (obs-sim) of the regional c

CRCM_ccsm

CRCM_cgcm3

HRM3_gfdl

HRM3_hadcm3

Winter 0.123 0.31 0.109 2.561Spring 0.692 0.518 0.782 �2.697Summer 0.085 �0.041 �0.138 �0.535Fall �0.088 0.021 0.082 �0.002

Figure 5. Change in the posterior mean of 100-year return level runoff (mm)current period of 1971–2000. (a) CRCM_ccsm, (b) CRCM_cgcm3, (c) H

WRFG_ccsm and (h


RCM3_cgcm3 has the largest one. Considering relativeseasonal changes, highest values are seen during spring forall models, while lowest ones occur primarily in fall andoccasionally in summer (for CRCM_ccsm, RCM3_cgcm3and WRFG_cgcm3).Figure 5 shows the posterior mean of the change in

100-year return level runoff (future-historic) for allmodels. Although the results vary between models fordifferent regions especially the ones that are close to thecoast, they show overall increase during winter, springand fall and decrease during summer. On winter themodels predict increase over the Cascades, the Rockies inparticular on south of Canada and north of US. Modelsshow contradictory results for the Olympics Mountainsand Coast Range, for example CRCM_ccsm,CRCM_cgcm3 and WRFG_cgcm3 present increasingrunoff over Olympics Mountains while others presentreversed scenario. The models agree on the reduction of

limate models compared with the observation

ECP2_gfdl

RCM3_gfdl

RCM3_cgcm3

WRFG_ccsm

WRFG_cgcm3

0.326 0.096 0.655 �0.041 0.3760.617 1.115 0.822 1.093 0.9410.142 0.128 0.071 0.206 0.0510.079 0.097 0.149 0.035 0.058

estimated for each grid cell for the future period of 2041–2070 versus theRM3_gfdl, (d) HRM3_hadcm3, (e)RCM3_cgcm3, (f) RCM3_gfdl, (g)) WRFG_cgcm3


Figure 6. Posterior standard deviation of the estimated 100-year return level runoff (mm) for each grid shown for the current period of 1971–2000 on leftand future period of 2041–2070 on right for (a) CRCM_ccsm, (b) CRCM_cgcm3, (c) HRM3_gfdl, (d) HRM3_hadcm3, (e) RCM3_cgcm3, (f)

RCM3_gfdl, (g) WRFG_ccsm and (h) WRFG_cgcm3


Copyright © 2013 John Wiley & Sons, Ltd. Hydrol. Process. (2013)

Figure 6. Continued


Copyright © 2013 John Wiley & Sons, Ltd. Hydrol. Process. (2013)

(g)

(h)

Winter Spring

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Figure 6. Continued


runoff extremes over the Cascades and the Rockies insummer. They also project an increase over the west ofthe Cascades and the Rockies during fall. The projectionsshow increases in runoff extremes over the Rockies andwest of Cascades especially in Canada during spring. Infact the highest predicted increase is seen for this regionfor this period. The increase in runoff extremes especiallyon high altitudes can be explained by the increase inextreme rainfall resulting from increased humidity due totemperature rise, change of snowmelt time from mid-summer to early summer and spring and increasedpossibilities of rain on snow processes due to increasedchance of rain instead of snow.The posterior standard deviations of the 100-year return

level runoffs are shown in Figure 6. Overall the values arehigher over the west of Cascades during winter and overthe Rockies on the north during summer. The results are


lowest on the center and south of the region. In the futuretime period the standard deviations of the posteriordistributions increase. In particular the results indicateincreases over the west of Cascades and the Rockiesduring spring and fall.Figure 7 (left) presents the posterior means of the

precipitation coefficients for all models, seasons and timeperiods. Comparison between the results for the historicalversus future time periods indicates overall increaseduring the winter, spring and fall and decrease duringsummer for GEV location and scale parameters. Never-theless, HRM3_gfdl, HRM3_hadcm3, RCM3_cgcm3 andRCM3_gfdl show different trends depending on theseason and parameter. This could be one of the reasonswhy these models presented contrary results regarding theestimates of the 100-year return level runoffs. Consider-ing the shape parameter, the results are more diverse than


(a)

(b)

(c)

Figure 7. Posterior distribution mean (left) and standard deviation (right) of the precipitation coefficient for all RCMs confronting current and future timeperiods; results shown for (a) mean, (b) scale and (c) shape parameters


the previous two GEV parameters. Precipitation coeffi-cient decreases in winter except for HRM3_hadcm3,RCM3_cgcm3, RCM3_gfdl and WRFG_cgcm3. Inspring it increases except for CRCM_ccsm,CRCM_cgcm3 and WRFG_cgcm3. In summer itdecreases except for HRM3_hadcm3, RCM3_gfdl andWRFG_cgcm3, and in fall it increases except forHRM3_gfdl. Figure 7 (right) presents the posteriorstandard deviations of the precipitation coefficients. Withregard to the location and shape parameters the overallseasonal trends correspond to the ones for the posterior


means, although a model which shows increase in meandoes not necessarily do so for the standard deviation.Furthermore for the shape parameter the overall standarddeviation values decrease in the future during spring.Considering the scale parameter the standard deviationsof the precipitation coefficients are highest forCRCM_cgcm3 during winter in the historical periodand RCM3_gfdl during fall for the future period.The hierarchical model spatially characterizes the

extreme distribution parameters (namely the location,scale and shape parameters). Using this method the



uncertainties related to all parameter estimates as well asthe return levels can be easily obtained. In additionthrough the Bayesian spatial model the information fromneighboring cells are combined together resulting in amore robust model. Comparison of the point-by-pointMLE estimated extreme parameters and the return levelswith the ones from hierarchical model have beenperformed by Cooley and Sain (2010) and Schliep et al.(2010). Their results showed spatially smoother estimatesobtained from the hierarchical model especially regardingthe shape parameter. In addition the inferred shapeparameter values are within the range commonly foundfor precipitation data. These were also detected in thisstudy. The estimated 100-year return level runoffs fromMLE and hierarchical model vary depending on theregion. In addition to this we performed anotheranalysis to evaluate the performance of the hierarchicalmodel in conditions where a grid cell experiences norunoff (i.e. because of the snow season etc.). Using theobserved data a number of 344 cells were detectedwith zero runoff during the winter season. Consideringthese cells the hierarchical model results wereevaluated which presented reasonable values. Usingthe posterior means of the estimated parameters thesimulated 100-year return level runoffs from thesecells have a 95% quantile of (0.0059mm, 0.0255mm)with a maximum value of 0.0282mm which is close tozero. This is due to the fact that the hierarchical modelcombines data from neighboring cells thus resulting inmore robust estimates.

CONCLUSIONS

The impact of climate change on runoff extremes overthe Pacific Northwest was studied in this paper. NineRCMs including CRCM_cgcm3, CRCM_ccsm,ECP2_gfdl (for the historical period), HRM3_gfdl,HRM3_hadcm3, RCM3_gfdl , RCM3_cgcm3,WRFG_cgcm3 and WRFG_ccsm covering the histori-cal and future time periods of 1971–2000 and2041–2070 were considered for this analysis. Hydro-logic modeling was performed after downscaling of theprecipitation, maximum and minimum temperature andwind speed using the quantile-mapping approach.Variable Infiltration Capacity (VIC) which is adistributed hydrologic model was used to provide dailyrunoff estimates (mm) for each cell of 1/8th degreeresolution. Spatial hierarchical Bayesian model wasthen applied on the cell-wise extreme runoff (mm) forboth time periods and for all seasons.The comparison between different RCMs regarding

the changes in extreme runoff showed varying out-comes especially for the HRM3-hadcm3. This high-


lights the importance of multi-modeling techniques byevaluating the performance of model ensembles duringa training period, assigning spatially distributed weightsto each model and providing a multi-model averageresult for future projections. The hierarchical Bayesianmodel identified the spatial variations in the marginaldistributions of the GEV parameters and the corre-sponding 100-year return level runoffs. The posteriordistributions of the latent variables provided informa-tion about the significance of each covariate on theextreme analysis in each season. Overall outcomesshowed increases in the estimated 100-year return levelrunoffs for most seasons particularly over the highelevation areas during winter. The Canadian portions ofthe study region reflected higher increases duringspring. Summer indicated reduction of extreme eventsin most areas.In this study stationarity assumption was assumed for

downscaling by applying similar delta changes for eachquantile obtained in historical period to the futureRCMs. The water budget version of the VIC hydro-logic model was considered which assumes that the soilsurface temperature is similar to the air temperature.Although more computationally intensive, the energybudget mode solves the complete water balance whilesimulating the surface energy fluxes such as sensibleheat, latent heat and ground heat to account for the totalincoming radiation fluxes. The spatial hierarchicalBayesian model was based on the conditional indepen-dence of runoff extremes at each cell.This study was conducted based on the CMIP3 climate

data under A2 emission scenario. In future studies, otherclimate scenarios fromCMIP5 (Taylor et al., 2012) could beanalysed after dynamically downscaling the coarse resolu-tion climate scenarios. The analyses similar to current workmay be used for risk assessment and storm water systemdesigns and eventually adaptation strategies. In next studieswe would focus more on analysing the uncertainties fromdifferent models or scenarios, investigate the possibilitiesformulti-modeling (Najafi et al., 2011c; Najafi et al., 2011a)or the combination of multimodeling with data assimilation(Moradkhani et al., 2012; DeChant and Moradkhani, 2012;Parrish et al., 2012) in particular regarding extreme andquantify the robustness of our extreme analysis.

ACKNOWLEDGEMENT

Partial financial support for this project was provided bythe National Science Foundation, Water Sustainabilityand Climate (WSC) program (grant no. EAR-1038925).The authors thank the NARCCAP for providing the dataused in this article. NARCCAP is funded by the NationalScience Foundation, the US Department of Energy, the



National Oceanic and Atmospheric Administration andthe US Environmental Protection Agency Office ofResearch and Development.

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A hierarchical Bayesian approach for the analysis of climate change impact on runoff extremes