MasoudHessami ,TahaB.M.J.Ouarda ,Andre´St-Hilaire *,PhilippeGachon Automatedregression-basedstatisticaldownscalingtool

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Automated regression-based statistical downscaling tool

Masoud Hessami

^a,

* , Philippe Gachon

^b,c

, Taha B.M.J. Ouarda

^d

, Andre´ St-Hilaire

^d

aDepartment of Civil Engineering, Shahid Bahonar University of Kerman, Kerman 76169-133, Iran

bAdaptation and Impacts Research Division, Science and Technology Branch, Environment Canada, Montre´al, Que´bec, Canada

cMcGill University, Department of Civil Engineering and Applied Mechanics, 817 Sherbrooke Street West, Montre´al, Que´bec H3A 2K6, Canada

dINRS-ETE, Chair in Statistical Hydrology, University of Que´bec, 490 de la Couronne, Que´bec G1K 9A9, Canada Received 27 August 2005; received in revised form 3 October 2007; accepted 4 October 2007

Abstract

Many impact studies require climate change information at a finer resolution than that provided by Global Climate Models (GCMs). In the last 10 years, downscaling techniques, both dynamical (i.e. Regional Climate Model) and statistical methods, have been developed to obtain fine resolution climate change scenarios. In this study, an automated statistical downscaling (ASD) regression-based approach inspired by the SDSM method (statistical downscaling model) developed by Wilby, R.L., Dawson, C.W., Barrow, E.M. [2002. SDSMea decision support tool for the assessment of regional climate change impacts, Environmental Modelling and Software 17, 147e159] is presented and assessed to reconstruct the observed climate in eastern Canada based extremes as well as mean state. In the ASD model, automatic predictor selection methods are based on backward stepwise regression and partial correlation coefficients. The ASD model also gives the possibility to use ridge regression to alleviate the effect of the non-orthogonality of predictor vectors. Outputs from the first generation Canadian Coupled Global Climate Model (CGCM1) and the third version of the coupled global Hadley Centre Climate Model (HadCM3) are used to test this approach over the current period (i.e. 1961e1990), and compare results with observed temperature and precipitation from 10 meteorological stations of Environment Can- ada located in eastern Canada. All ASD and SDSM models, as these two models are evaluated and inter-compared, are calibrated using NCEP (National Center for Environmental Prediction) reanalysis data before the use of GCMs atmospheric fields as input variables.

The results underline certain limitations to downscale the precipitation regime and its strength to downscale the temperature regime. When modeling precipitation, the most commonly combination of predictor variables were relative and specific humidity at 500 hPa, surface airflow strength, 850 hPa zonal velocity and 500 hPa geopotential height. For modeling temperature, mean sea level pressure, surface vorticity and 850 hPa geopotential height were the most dominant variables. To evaluate the performance of the statistical downscaling approach, several climatic and statistical indices were developed. Results indicate that the agreement of simulations with observations depends on the GCMs atmospheric variables used as ‘‘predictors’’ in the regression-based approach, and the performance of the statistical downscaling model varies for different stations and seasons. The comparison of SDSM and ASD models indicated that neither could perform well for all seasons and months.

However, using different statistical downscaling models and multi-sources GCMs data can provide a better range of uncertainty for climatic and statistical indices.

Keywords:Climate change; Statistical downscaling; GCM; Multiple regression; Eastern Canada

1. Introduction

Climate change scenarios developed from Global Climate Models (GCMs) are the initial source of information for

estimating plausible future climate. However, the spatial resolution of GCMs is too coarse to resolve regional scale effect and to be used directly in local impact studies. Downscaling techniques offer an alternative to improve regional or local estimates of variables from GCM outputs.

Downscaling methods, as reviewed in Wilby and Wigley (1997) and more recently in Wilby et al. (2004) and Mearns et al. (2003), were divided into four general categories: regression

* Corresponding author. Tel./fax:þ11 98 341 322 0054.

E-mail address:[email protected](M. Hessami).

doi:10.1016/j.envsoft.2007.10.004

Environmental Modelling & Software 23 (2008) 813e834

www.elsevier.com/locate/envsoft

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methods (Hewitson and Crane, 1996; Wilby et al., 1999), weather pattern approaches (Yarnal et al., 2001), stochastic weather gen- erators (Richardson, 1981; Racsko et al., 1991; Semenov and Bar- row, 1997; Bates et al., 1998) and limited-area Regional Climate Models (RCMs,Mearns et al., 1995). Among these approaches, regression methods are regularly used because of their ease of im- plementation and their low computation requirements. Statistical downscaling is based on the fundamental assumption that regional climate is conditioned by the local physiographic characteristics as well as the large scale atmospheric state. Based on this assumption, large scale climate fields are related to local variables through a statistical model in which GCM simulations are used as input for the large scale atmospheric variables (or ‘‘predictors’’) to downscale the local climate variables (or ‘‘predictands’’) with the use of observed meteorological data. The major weaknesses of statistical downscaling methods are that the fundamental assumption on which they are based is not verifiable, i.e. the statistical relationships developed for the present day climate also hold under different forcing conditions of plausible future climate (Wilby et al., 2004), and they cannot explicitly describe the physical processes that affect climate. In spite of these limitations, these methods may be helpful for impact studies in heterogeneous envi- ronments (see for example the recent study ofDibike et al., 2007 andGachon and Dibike, 2007, in coastline areas of northern Can- ada), and/or for generating large ensembles or transient scenarios.

In our study, the statistical downscaling regression-based methods, namely the SDSM model developed byWilby et al.

(2002)and a new tool mainly developed to improve the procedure in the selection of predictors, are assessed to reconstruct the observed climate based extremes (from temperature and precipitation variables only, over the 1961e1990 period), in eastern Canada. The main purpose of this study is to develop and to test a tool capable of performing statistical downscaling automatically from predictor selection to model calibration, scenario generation and statistical analysis of scenarios.

In Section2, the mathematical formulation of an automated statistical downscaling (ASD) model is presented, followed by Section3showing the methodology focussed on model evaluation criteria, the study area, data and predictors selection. The results are then presented in Section4in using both the NCEP (National Centre for Environmental Prediction) and two series of GCMs daily predictors. Section6presents the main conclu- sions and recommendations concerning the assessment of the two statistical downscaling methods (i.e. namely the formal ASD and the original regression method SDSM) studied here.

2. Automated statistical downscaling

An automated regression-based statistical downscaling (ASD) model, inspired by the existing statistical downscaling model (SDSM developed byWilby et al., 2002), was developed under the Matlab environment (The Mathworks, 2002).Figs. 1 and 2show the main menu and the general scheme of the ASD framework to generate climate scenario information, respectively. The model process can be conditional on the occurrence of an event (i.e. for precipitation) or unconditional (i.e. for temperature). Hence, the modeling of daily precipitation involves

the following two steps: precipitation occurrence and precipitation amounts, as described inWilby et al. (1999):

Oi¼a₀þXⁿ

j¼1

a_jpij; R^0:25_i ¼b₀þXⁿ

j¼1

b_jpijþei ð1Þ

whereOiis the daily precipitation occurrence,Riare daily precipitation amounts,pij are predictors,n is number of predictors,aandb are model parameters andei is modeling error.

The modeling of daily temperature is performed in one step:

Ti¼g₀þXⁿ

j¼1

g_jpijþei ð2Þ

where Ti is the daily temperature (maximum, minimum or mean) andg is the model parameter. Once the deterministic component is obtained, the residual termeiis modeled under the assumption that it follows a Gaussian distribution:

ei¼ ffiffiffiffiffiffiffiffi VIF 12 r

ziSeþb ð3Þ

wherezi is a normally distributed random number, Se is the standard error of estimate, b is the model bias and VIF is the variance inflation factor. For calibrating the model, NCEP (National Center for Environmental Prediction, e.g.

Kalnay et al., 1996) reanalysis data must be used. When using NCEP data for scenario generation, VIF and b are, respectively, set to the 12 and 0. When using GCM data for scenario generation, the VIF and the bias can be set automatically using the following equations:

b¼MobsMd ð4Þ

VIF¼12ðVobsVdÞ

S²_e ð5Þ

whereVobsis the variance of observation during calibration period,Vdis the variance of deterministic part of model output during calibration period, Se is the standard error, Mobsand Mdare the mean of observation and the mean of deterministic part of model output during calibration period, respectively.

2.1. Regression methods

Regression-based downscaling methods often use multiple linear regressions, however, the non-orthogonality of the predictor vectors can make the least squares estimates of the regression coefficients unstable. In addition to multiple linear regressions, the present model gives the possibility to use the ridge regression (Hoerl and Kennard, 1970) to alleviate the effect of the non-orthogonality of the predictor vectors.

In this approach, a small bias is introduced to provide more stable estimators (Hoerl and Kennard, 1970). When collinearity exists, for small perturbations in data, the estimates provided by ridge regression are more robust than ordinary least squares (OLS) estimates. The predictor variables should be first standardized to have zero mean and unit variance (the

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future climate predictors are standardized using the mean and variance of the current climate predictors).

The ridge regression coefficients a for the linear model y¼Xaþecan be calculated from:

a¼ ðX^tXþkIÞ¹X^ty ð6Þ

where I is an identity matrix and k is the ridge parameter.

When k¼0, a is the least squares estimator. The selection

of the ridge parameter is often done by iteration and can be somewhat subjective. Hoerl and Kennard (1970) suggest the following guidelines:

- Plot values of k as a function of a (the so-called ridge trace). Identify thekvalue for which the system stabilizes.

The associatedavalues provide the general character of an orthogonal system.

Fig. 1. Main menu of ASD (automated statistical downscaling) tool.

NCEP predictors

Select best predictors using stepwise regression / partial correlation

Generate simulations using NCEP data

GCM predictors Set

VIF and bias Set model configuration Predictand

Calibrate model

Generate scenarios using GCM data

Results evaluation

Fig. 2. ASD architecture.

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- Coefficients should have reasonable absolute values.

- Coefficients with improper signs at k¼0 will have changed to have proper signs (e.g. positive for positive correlation between predictor and predictand).

- The residual sum of squares will not have been inflated to an unreasonable value.

2.2. Predictor selection methods

In SDSM (Wilby et al., 2002), selection of predictors is an iterative process, partly based on the user’s subjective judg- ment. In the present model, we have implemented two methods based on backward stepwise regression (McCuen, 2003) and partial correlation coefficients to select the predictors. Backward stepwise regression starts with all the terms in the model and removes the least significant terms until all the remaining terms are statistically significant. The partial F-test which can be used for either adding a predictor to the equation containing q1 variables or removing a predictor from the equation containingqvariables is:

F¼

R²_qR²_q1

ðnq1Þ

1R²_q ð7Þ

wheren is the number of observations,RqandRq1are correlation coefficients between the criterion variable and a prediction equation having q and q1 variables, respectively.

If Fis greater than the criticalF value, the predictor should be included in the equation. The critical F value is defined for a given level of significance and degrees of freedom 1 andnq1. A Bonferroni correction (Bonferroni, 1936) is used for the level of significance using the following formula:

a¼1 1a

2 ¹_q

ð8Þ

whereais the level of significance andqis the number of predictor in the equation. The partialF-test must be computed for every predictor at each step of stepwise regression.

Partial correlation is the correlation between two variables after removing the linear effect of the third or more other variables. The partial correlation between variable i andj while controlling for third variablekis (e.g.Afifi and Clark, 1996):

Rij;k¼ RijRikRjk

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1R²_ik

1R²_jk

q ð9Þ

whereRijis the correlation coefficient between variablesiandj.

For partial correlation method, thep-value is used for eliminat- ing any one of the predictors. Thep-value is computed by trans- forming the correlation R to create a t-statistic havingn2 degrees of freedom, wherenis the number of observations:

t¼ R ffiffiffiffiffiffiffiffi

1R² n2

q ð10Þ

The probability of thet-statistic indicates whether the observed correlation occurred by chance if the true correlation is zero.

In the SDSM model, a recursive algorithm is implemented to compute partial correlation using Eq.(9). This recursive algorithm has a limitation, i.e. when the partial correlation between two variables is computed, the maximum number of controlling variables is 12. However, the number of NCEP predictors used for partial correlation analysis is usually more than 20 (as suggested inTables 1 and 2for NCEP predictors interpolated on the two grids of GCMs, which are used in the following for GCMs based evaluation predictors). In ASD, to control this limitation and fast computation, the following algorithm is used for partial correlation analysis. We compute first the residuals of regressing the response variable yagainst the independent variablesx2,x3,.,xm:

y¼f1ðx2;x3;.;xmÞ ð11Þ

and then we compute the residuals from regressingx1against the independent variablesx2,x3,.,xm:

x1¼f2ðx2;x3;.;xmÞ ð12Þ The correlation betweenyandx1controllingx2,x3,.,xmis obtained by computing the correlation between the residuals of the two linear modelsf1andf2.

3. Methodology

3.1. Model evaluation diagnostic criteria

A core of 11 extremes and climate variability indices has been developed for Que´bec regions in considering their usefulness in the context of Nordic climate (e.g.Gachon et al., 2005), and to analyze frequency, intensity and duration related to extremes of precipitation and temperature. These indices have been used to document the recent observed climate variability related to the precipitation and the temperature regime evolution (i.e. over the 1941e2000 period), and to assess the performance of the statistical downscaling results to reconstruct the current observed period 1961e1990. These provide information on mean and extreme climate for the meteorological stations used in this

Table 1

NCEP predictor variables on CGCM1 grid

No. Predictors No. Predictors

1 Mean sea level pressure 14 500 hPa divergence 2 Surface airflow strength 15 850 hPa airflow strength 3 Surface zonal velocity 16 850 hPa zonal velocity 4 Surface meridional velocity 17 850 hPa meridional velocity

5 Surface vorticity 18 850 hPa vorticity

6 Surface wind direction 19 850 hPa geopotential height 7 Surface divergence 20 850 hPa wind direction 8 500 hPa airflow strength 21 850 hPa divergence 9 500 hPa zonal velocity 22 Near surface

relative humidity 10 500 hPa meridional velocity 23 Specific humidity

at 500 hPa 11 500 hPa vorticity 24 Specific humidity

at 850 hPa 12 500 hPa geopotential height 25 Near surface

specific humidity 13 500 hPa wind direction 26 Mean temperature

at 2 m

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paper (see their location over eastern Canada inFig. 3), as well as help to evaluate the capacity of the statistical models to downscale both the intensity (related to absolute or relative thresholds), duration and frequency in the precipitation and temperature series rather than monthly totals or mean values.

Tables 3 and 4show the climatic indices for precipitation and temperature variables, respectively, which are used as diagnostic criteria to evaluate the performance of statistical downscaling models. Based on daily total precipitation, we use five precipitation indices including percentage of wet days (PRCP1, in that case occurrence was limited to events with amount greater than or equal to 1 mm to avoid the problem in trace measurement and low daily values), mean precipitation amount per wet days (SDII), maximum number of consecutive dry days (CDD), maximum 3-days precipitation total (R3days) and the 90th percentile of rain day amount (PREC90). Based on daily minimum and maximum temperature, we use six temperature indices including the mean of diurnal temperature range (DTR), the frost season length (FSL), the growing season length (GSL), the percentage of days with freeze and thaw cycle (Fr/Th), the 90th percentile of daily maximum temperature (Tmax90) and the 10th percentile of daily minimum temperature (Tmin10).

These indices are presented in more details inGachon et al. (2005). They are modified to correspond to the characteristics of the Que´bec climate from the STARDEX (Statistical andRegional dynamicalDownscaling ofExtremes for European regions) SDEIS climate indices software (Haylock, 2004).

In addition to these indices, we have computed the mean and the standard deviation of observed and simulated monthly values during calibration (1961e 1975) and validation (1976e1990) periods, for total precipitation, maximum, minimum and mean temperature.

3.2. Study area, data and predictors selection

Fig. 3shows the area over eastern Canada where the studied stations are located. We have focused on the following stations located around the Labra- dor Sea and the Gulf of St. Lawrence: Cartwright, Goose bay, Kuujjuaq, Schefferville, Causapscal, Daniel Harbour, Gaspe´, Mont-Joli, Natashquan and Sept-Iˆles. For statistical downscaling, we have used the following data:

the daily meteorological data from Environment Canada stations, i.e. maximum, minimum and mean temperature, and precipitation, corresponding to homogenized and rehabilitated values developed by Vincent and Mekiz (e.g.

Vincent et al., 2002; Vincent and Me´kis, 2004) as predictands and three series of daily normalized predictors, from NCEP reanalysis and from two GCMs independent outputs (i.e. CGCM1 and HadCM3), for the period of 1961e1990.

The availability of two series of GCM predictors constitutes an opportunity to test the two statistical models in using two independent data, and to evaluate the uncertainties of the results associated with two GCM structures and parameterizations.

CGCM1 is the first Generation of the coupled Canadian Global Climate Model (e.g.Flato et al., 2000). The atmospheric component of CGCM1 has 10 vertical levels and a horizontal resolution of approximately 3.7of latitude and longitude (about 400 km). HadCM3 is a coupled atmosphereeocean general circulation model developed at the Hadley Centre and described by Gordon et al. (2000)andPope et al. (2000). The atmospheric component of HadCM3 has 19 levels with a horizontal resolution of 2.5 of latitude by 3.75 of longitude, which is equivalent to a horizontal resolution of about 417278 km at the Equator, reducing to 295278 km at 45 of latitude.

The two GCMs have participated in the CMIP1 (Coupled Model Intercompar- ison Project, Phase 1) with climate simulations beginning around 1860 or 1900 (for HadCM3 and CGCM1, respectively) in using historical estimates of greenhouse gases and sulphate aerosols concentration (see Table 8.1 in Chap- ter 8 and Table 9.1 in Chapter 9,IPCC, 2001). The two runs from CGCM1 and HadCM3 come from the first member of the ensemble runs (i.e. 3/1 member(s), for CGCM1/HadCM3).

The ASD and SDSM models for each station, month and season were run using NCEP predictors to calibrate the models before using the two corresponding CGCM1 and HadCM3 predictors properly, over the 1961e1990 time-window. Hence, the NCEP series of predictors have been re-gridded, i.e. interpolated on the two GCMs grids because the grid-spacing and/or Table 2

NCEP predictor variables on HadCM3 grid

No. Predictors No. Predictors

1 Mean sea level pressure 14 500 hPa divergence 2 Surface airflow strength 15 850 hPa airflow strength 3 Surface zonal velocity 16 850 hPa zonal velocity 4 Surface meridional velocity 17 850 hPa meridional velocity

5 Surface vorticity 18 850 hPa vorticity

6 Surface wind direction 19 850 hPa geopotential height 7 Surface divergence 20 850 hPa wind direction 8 500 hPa airflow strength 21 850 hPa divergence 9 500 hPa zonal velocity 22 Relative humidity

at 500 hPa 10 500 hPa meridional velocity 23 Relative humidity

at 850 hPa

11 500 hPa vorticity 24 Near surface

relative humidity 12 500 hPa geopotential height 25 Surface specific humidity 13 500 hPa wind direction 26 Mean temperature

at 2 m

Fig. 3. Meteorological stations located around the Labrador Sea and the Gulf of St. Lawrence.

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coordinate systems of reanalysis data sets do not correspond to those of the two GCM outputs (for example the NCEP/NCAR reanalysis has a grid-spacing of 2.5latitude by 2.5longitude instead of the grid-spacing of CGCM1 and HadCM3 as previously mentioned). It is a reason why two series of statistical downscaling simulations incorporate the NCEP predictors interpolated on the two different grids, as suggested inTables 1 and 2. Re-gridding and verification of GCM predictors are a necessary part of all statistical downscaling de- velopment (and time-consuming). The interpolation procedure to the GCM grids rather than the NCEP/NCAR reanalysis grid is mainly motivated as we use the GCM predictors for the climate change simulations (i.e. the main issues for downscaling methods used to developed high resolution climate scenarios), and raw GCM information must be preserved for the downscaling process. As regularly used in SDSM, the interpolation has been carried out in this manner since we have more confidence in finer resolution data interpolated to a coarser resolution than we do in interpolating coarse-resolution (i.e. GCM) data to a finer resolution. In doing the reverse interpolation, i.e.

from the coarser GCM grid to the finer NCEP/NCAR grid, an ‘‘artificial’’

higher resolution set of predictor variables is generated with high risk to create an unreliable physical information especially over high gradient in atmospheric values (i.e. part issued from the interpolation process). Also, this interpolation procedure must be realized both for the current and for the future periods. Hence, this process modifies the original GCM predictor without any potential added values and/or physical changes, i.e. not fully representative of the atmospheric circulation changes simulated by the GCM at its original resolution.

Gachon et al. (2005)have analyzed in detail the results from the NCEP driven statistical downscaling models with the two series of predictors interpolated on the HadCM3 and CGCM1 grids, and no significant differences have

been found in general. In other words, the interpolation procedure from the NCEP grid to the GCM grids is more accurate and more physically based, in spite of the fact that direct intercomparison is not fully viable. However, in general, we use most of the time the same predictors and when the predictors are different they are strongly correlated (seeTables 5e8), i.e. are issued from the same physical processes, and this does not constitute a major concep- tual problem for the intercomparison of all downscaling results. As shown in the following, the interpolation procedure has weak influence on the predictor values and on the downscaling results, with respect to the relevance and the reliability of the raw GCM predictors, which plays the key role in the accuracy of the downscaled variables (see for example, the recent results ofDibike et al., 2007, using two independent GCMs predictor series).

For each station, five predictor variables were selected using a stepwise regression. Prior to the stepwise regression, the predictor variables 22 (near surface relative humidity, seeTable 1), 25 (near surface specific humidity) and 26 (mean temperature at 2 m) have been removed from NCEP data interpolated on CGCM1 grid (in order to avoid the problem in using the equivalent predictor variables from CGCM1 output in which strong biases have been docu- mented in early and end of the winter, due to the simplistic bucket model in this version of the Canadian Model; e.g. IPCC, Chapter 8, 2001). To make comparison more consistent, the variables 24 (near surface relative humidity, see Table 2), 25 (surface specific humidity) and 26 (mean temperature at 2 m) have been also removed from NCEP data interpolated on HadCM3 grid. Also, we have had access to the specific humidity for the CGCM1 output rather than the relative humidity for the HadCM3 output.

Specific and relative humidity are not interchangeable, but they are strongly correlated. As the two are highly correlated to the occurrence of precipitation, as their synchronous variation is dependent to the saturated phase of water vapour in the air, the use of relative or specific humidity gives the same results for the downscaling of precipitation. Hence, no differences are found depending on which is chosen (this may not be the case for future climate pro- jection). In fact, the combination of humidity variables at different levels is more often important for the precipitation process (occurrence and intensity) than the single value of humidity taken individually at only one level.

The downscaling model parameters were obtained by multiple linear regressions with separate regressions for each month using daily observed and predictors data.Table 5shows that the most commonly used predictor variables for precipitation modeling are specific/relative humidity at 500 hPa, surface airflow strength, 500 hPa geopotential height and 850 hPa zonal velocity.

For modeling temperature, mean sea level pressure, surface vorticity and 850 hPa geopotential height seem to be the most important predictor variables, which are physically plausible because those are strongly associated to strong modification of temperature characteristics in the boundary layer (seeTables 6e8), through the thermal advection term.

For stability and robustness of the downscaling results (seeGachon et al., 2005), for each application of ASD and SDSM models, 100 simulations were performed to produce 100 synthetic series of daily precipitation and mean, minimum and maximum temperature. Differences between these 100 realizations do not reflect the full range of internal variability because only the stochastic component differs between each run. The deterministic component (i.e.

controlled by the atmospheric variables) follows the same evolution in each run because only one realization of the predictor variables at daily scale exists in each case (either the NCEP or GCMs data).

Each downscaled series is accumulated into monthly totals and averaged over the 100 realizations, and then compared in the following with the observed series for climate mean and standard deviation variables. For climate indices evaluation, each series is accumulated into monthly, seasonal or yearly totals (according to the considered time scale of each indices, seeTables 3 and 4).

All the 100 realizations are then compared with the observed series to evaluate the range of the stochastic component (i.e. to analyze the spreading and outliers in the results with the aid of box plot graphs). The comparison is performed over the period of ASD and SDSM calibration (1961e1975) and over an independent verification/validation period (1976e1990), using two series of NCEP predictors (i.e. same atmospheric fields but interpolated on the two GCMs grids as mentioned previously). For the downscaling values using GCMs predictors, the complete period 1961e1990 is used to compare all results. The criteria for results comparison are computing the amount of model explained variance (R²) and Root Mean Squared Error (RMSE) for the Table 3

Precipitation indices used to evaluate the performance of statistical downscaling models

Indices Definition Unit Time Scale

PRCP1 Percentage of wet days (threshold1 mm)

% Season

SDII Mean precipitation amount at wet days

mm/day Season

CDD Maximum number

of consecutive dry days

Days Season

R3days Maximum 3-days precipitation total mm Season PREC90 90th percentile

of rain day amount

mm Season

Table 4

Temperature indices used to evaluate the performance of statistical downscaling models

Indices Definition Unit Time Scale

DTR Mean of diurnal temperature range

C Season

FSL Frost season

length:Tmin<0C more than 5 days

andTmin>0C more than 5 days

Days Year

GSL Growing season

length: Tmean>5C more than 5 days

and Tmean<5C more than 5 days

Days Year

FreTh Days with freeze and thaw

cycle (Tmax>0C,Tmin<0C)

Days Month

Tmax90 90th percentile of dailyTmax C Season Tmin10 10th percentile of dailyTmin C Season

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estimated statistics and climatic indices. ASD takes the output of each monthly simulation and corresponding monthly observation, and then computes the mean amount of explained variance. For computing RMSE, the observed and estimated indices are averaged at the respective time scale over the 100 simulations.

4. Results

Before analyzing the results with GCMs predictors, the series downscaled from the NCEP predictors are needed to evaluate the performance of the ASD and the SDSM models in comparison with the observed precipitation and temperature series, for both the calibration and the validation periods (i.e. cross-validation procedure), as shown in Section4.1.

4.1. Calibration (1961e1975) and validation (1976e 1990) using NCEP predictors

Over all stations, Tables 5e8 summarize the results of model calibration with the ASD model (over the period 1961e1975) for precipitation, maximum, minimum and mean temperature using NCEP predictors interpolated on the CGCM1 and on the HadCM3 grids. For the downscaling of precipitation, the amount of explained variance (R²) varies from 0.13 (Causapscal) to 0.32 (Schefferville) for NCEP data on CGCM1 grid and it varies from 0.15 (Causapscal) to

0.31 (Natashquan) for NCEP data on HadCM3 grid (Table 5).

These relatively low explained variances underline the diffi- culty to downscale the precipitation regime compared to the temperature. However, in the case of daily rainfall, any positive explained variance is valuable, as this corresponds to a correlation between 0.36 and 0.56, which for daily climatic time series is quite respectable, in particular considering the stochastic character of daily rainfall. In the case of temperature, the high values of explained variance (>89%; seeTables 6e8) indicate the greater skill to downscale the temperature regime than for precipitation.

Using the same set of five predictors selected by the ASD model illustrated inTables 5e8, an example of the results of model calibration with SDSM is given for Schefferville station (not shown in a table). For this station, when using NCEP data on CGCM1 grid, the amounts of explained variance (R²) are 0.18, 0.77, 0.82 and 0.65 for precipitation, maximum temperature, mean temperature and minimum temperature, respectively. These values are slightly different when using NCEP data on HadCM3 grid (0.12, 0.76, 0.76 and 0.64). Hence, as suggested in Tables 5e8, ASD provides higher values for the amounts of explained variance than SDSM. The difference between the amounts of explained variance is that SDSM computes a mean value of the amounts of explained variance over 12 months, but ASD takes the output of 12 monthly models and corresponding observations, and then computes the

Table 5

Results of ASD model calibration (1961e1975) for precipitation using NCEP predictors interpolated on the CGCM1 and on the HadCM3 grids

Station CGCM1 grid HadCM3 grid

Predictors R² R²(ridge) Predictors R² R²(ridge)

Cartwright 3 5 9 11 23 0.24 0.32 3 5 12 19 22 0.27 0.33

Goose bay 3 9 10 15 23 0.24 0.31 3 12 15 17 22 0.23 0.33

Kuujjuaq 2 8 15 16 23 0.22 0.33 2 12 16 19 22 0.22 0.33

Schefferville 2 16 17 18 23 0.32 0.37 2 9 12 16 19 0.25 0.34

Causapascal 7 9 16 21 23 0.13 0.20 9 12 16 17 22 0.15 0.21

Daniel Harbour 2 11 12 18 23 0.20 0.27 2 11 15 18 22 0.20 0.27

Gaspe´ 3 9 10 14 17 0.17 0.27 12 17 19 22 23 0.18 0.30

Mont-Joli 2 9 16 17 23 0.21 0.27 2 9 12 16 22 0.17 0.31

Natashquan 2 12 16 17 23 0.26 0.35 10 12 16 17 22 0.31 0.36

Sept-Iˆles 5 9 12 16 23 0.24 0.38 2 5 9 16 22 0.22 0.40

For each predictor, the number refers to the atmospheric variables defined inTables 1 and 2(for each respective GCM grid).

Table 6

Results of ASD model calibration (1961e1975) for maximum temperature using NCEP predictors interpolated on the CGCM1 and on the HadCM3 grids

Cartwright 1 4 5 17 19 0.91 0.94 1 5 17 18 19 0.89 0.93

Goose bay 1 5 6 15 19 0.95 0.96 1 5 17 18 19 0.93 0.96

Kuujjuaq 1 4 5 19 24 0.95 0.96 1 5 11 18 19 0.94 0.96

Schefferville 1 5 7 18 19 0.96 0.97 1 3 5 16 19 0.95 0.97

Causapascal 1 5 16 18 19 0.95 0.96 1 3 5 16 19 0.95 0.96

Daniel Harbour 1 4 5 15 19 0.94 0.94 1 4 5 15 19 0.94 0.94

Gaspe´ 1 5 15 18 19 0.93 0.95 1 3 5 16 19 0.94 0.95

Mont-Joli 1 5 10 18 19 0.94 0.96 1 3 5 12 19 0.95 0.96

Natashquan 1 5 15 18 19 0.93 0.95 1 5 7 17 19 0.94 0.94

Sept-Iˆles 1 5 15 18 19 0.93 0.96 1 5 12 18 19 0.93 0.96

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mean amount of explained variance. When using a similar methodology, the amount of explained variances is the same for the two models.

Overall the NCEP data on CGCM1 and HadCM3 grids provide similar results for model calibration, as well as for model validation, with only few differences for precipitation and no differences for temperature. In order to analyze in more detail the effect of interpolation of NCEP predictors over the two GCMs grids on the downscaling results and the various performance of the ASD model over the calibration and the validation periods, one representative station (i.e. Schefferville) is used to evaluate RMSE criteria inTables 9 and 10(over these two periods, respectively), and through graphical analysis with box plots graphs shown inFigs. 4e7. In these last figures, only results using NCEP predictors interpolated on to the HadCM3 grid are shown, as Figs. 8e22 using both the two series of NCEP and of GCMs predictors over the all 1961e1990 period are shown and discussed in Section4.2.

As shown inTable 9, the RMSE of the estimated statistics and climatic indices for the calibration period is quite similar between the two series of results with NCEP predictors interpolated over the two GCMs grids, with negligible differences in most cases (mainly below 0.5 for all indices and maximum of 2 days for FSL). The slight differences appear for precipitation amount and occurrence (i.e. wet days) as the stochastic part is much higher in that case (i.e. compared to the

deterministic part of the model), as suggested in the weak amount of explained variance inTable 5, and the random process between simulated series is much higher as the reliable predictors are less strongly correlated with the precipitation as compared to the equivalent ones for temperatures. As also shown inTable 9, the RMSE for temperature indices is quite weak in all cases, suggesting a strong capacity to downscale the temperature variables, with similar performance between SDSM and ASD. As shown inTable 10, few differences exist in the RMSE values between the validation and the calibration periods (as shown inTable 9), in spite of slight increase in RMSE for precipitation indices, but this is not systematic for other indices.

As show in Figs. 4e7, results for monthly mean values of precipitation, and of maximum, mean and minimum temperatures indicate that ASD replicate observed inter-monthly and inter-annual variability faithfully, except for precipitation in January, February and March where strong changes in the variability (IQR and extreme values) are not well captured by the model. For other months and in particular for temperatures, the performance of the ASD model is almost as good over the verification/validation period as it is over the calibration period, indicating that the empirical model has not been overfit to the data (i.e. due mainly to the following factors: number and type of selected predictors, statistical model, stability of the relationships between predictand/predictors according to

Table 7

Results of ASD model calibration (1961e1975) for mean temperature using NCEP predictors interpolated on the CGCM1 and on the HadCM3 grids

Cartwright 1 4 5 18 19 0.93 0.95 1 5 7 17 19 0.94 0.95

Goose bay 1 5 15 18 19 0.96 0.97 1 5 7 17 19 0.95 0.97

Kuujjuaq 1 5 18 19 24 0.96 0.97 1 3 5 19 23 0.96 0.97

Schefferville 1 3 5 16 19 0.97 0.98 1 3 5 18 19 0.96 0.97

Causapascal 1 5 12 18 19 0.96 0.97 1 3 5 16 19 0.96 0.97

Daniel Harbour 1 5 11 18 19 0.94 0.95 1 5 11 15 19 0.95 0.96

Gaspe´ 1 5 9 18 19 0.96 0.97 1 3 5 18 19 0.96 0.96

Mont-Joli 1 5 12 18 19 0.96 0.97 1 3 5 16 19 0.96 0.97

Natashquan 1 5 15 18 19 0.95 0.96 1 5 15 17 19 0.95 0.96

Sept-Iˆles 1 5 15 18 19 0.96 0.97 1 5 16 18 19 0.96 0.97

Table 8

Results of ASD model calibration (1961e1975) for minimum temperature using NCEP predictors interpolated on the CGCM1 and on the HadCM3 grids

Cartwright 1 5 16 18 19 0.91 0.93 1 5 7 15 19 0.90 0.93

Goose bay 1 5 15 18 19 0.94 0.95 1 5 15 18 19 0.93 0.95

Kuujjuaq 1 5 11 18 19 0.94 0.96 1 3 5 19 23 0.94 0.96

Schefferville 1 3 5 15 19 0.94 0.95 1 5 18 19 23 0.94 0.95

Causapscal 1 3 5 9 19 0.91 0.93 1 2 5 19 23 0.92 0.93

Daniel Harbour 1 5 7 19 21 0.91 0.92 1 5 17 19 23 0.91 0.92

Gaspe´ 1 3 5 18 19 0.92 0.94 1 3 5 18 19 0.92 0.94

Mont-Joli 1 5 9 18 19 0.94 0.95 1 3 5 16 19 0.94 0.95

Natashquan 1 2 5 18 19 0.93 0.94 1 2 5 17 19 0.93 0.94

Sept-Iˆles 1 5 16 18 19 0.95 0.96 1 5 16 19 23 0.95 0.96

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different climate regimes and collinearity between predictors).

However, the spreading of high extreme values in the ASD results for precipitation with respect to observed data suggests a well known problem of poorest representation of extreme events and observed variability from regression-based statistical downscaling method in particular for precipitation (e.g.

Wilby et al., 2004), as for SDSM (see Gachon et al., 2005).

For temperatures, including extreme values, ASD performs relatively well (box plots not shown) with similar results compared to the SDSM model (see the RMSE inTables 9 and 10).

4.2. Analysis of observed and estimated statistics and climatic indices using CGCM1 and HadCM3 predictors:

example at Schefferville station

One representative station is also used in this section to il- lustrate the results from ASD and SDSM in using GCMs predictors, and in comparing the RMSE values as well as box plots graphs of basic variables and climate indices with respect to observed data over the complete 1961e1990 period. In the box plots graphs, the results in using NCEP predictors are also included as a reference for this baseline period (i.e. calibration/validation in a reanalysis mode), and compared with results driven by GCMs (i.e. evaluation in a climate mode).

Figs. 8e11 compare the monthly observed and estimated mean and standard deviation of precipitation and temperatures, and Figs. 12e22 compare the seasonal or annual observed and estimated values of climate indices. Table 11 provides the RMSE criteria from SDSM and ASD results based on GCMs predictors (both CGCM1 and HadCM3) for each basic variable and climate index.

For basic variables (monthly mean and standard deviation of precipitation and temperature shown in Figs. 8e11), the performance of models can vary on a month-to-month basis.

However, for temperature, the results with SDSM and ASD are more often better with HadCM3 predictors than those using CGCM1 (as shown in Figs. 9e11). As noted earlier, the NCEP driven downscaling results for temperature are not dependent on the interpolation process over the two GCMs grids, neither on the downscaling models. For precipitation, the performance is equivalent between the two downscaling models as in the recent study of Gachon et al. (2005). In that case, all results are similar in terms of median and IQR estimated values in using NCEP or GCMs predictors, with over-spreading in extremes of daily precipitation as shown in Fig. 8.

For the climate indices of precipitation, the percentages of wet days (PRCP1) calculated by season have median observed values varying between 26 and 49 days (Fig. 12). Most of the simulated time series from GCMs predictors underestimated the median observed percentage during all four seasons, except in winter with ASD and in summer with both SDSM and ASD driven all by HadCM3 predictors. In general, the only simulated values that appear to be less biased were the ASD model using these predictors (model 9 inFig. 12) during the winter, spring and autumn. Box and whiskers plots in Fig. 12 also show that the simulated values generally have a greater variance than the observed values, with excessive outliers in general. In most cases, comparable skill is obtained between NCEP driven conditions and in using GCMs predictors.Fig. 13shows that for simple daily intensity index (SDII) all models driven by GCMs overestimated the median values for all seasons with bias on the order of 2 mm/wet day. Over all seasons, the ASD model tends to outperform SDSM in using NCEP predictors as no systematic higher skill is suggested from GCMs downscaling results. Also, systematic outliers in estimated values of SDII are suggested from all downscaling results. For the maximum number of consecutive dry days (CDD shown in Fig. 14), median values of observation and

Table 9

RMSE of the estimated statistics and climatic indices during calibration period at Schefferville based on NCEP predictors (interpolated on the CGCM1 and HadCM3 grids)

CGCM1 grid HadCM3 grid

SDSM ASD SDSM ASD

Mean prec. (mm/day) 0.29 0.16 0.41 0.15

STD prec. (mm/day) 0.53 0.37 0.71 0.40

PRCP1 (%) 4.49 5.14 4.00 4.76

SDII (mm/wet day) 1.59 0.44 1.58 0.42

CDD (day) 0.98 0.78 1.10 0.87

R3days (mm) 3.13 4.30 3.32 4.89

PREC90 (mm/day) 1.95 0.72 2.10 0.96

MeanTmax(C) 0.01 0.02 0.01 0.02

STDTmax(C) 0.14 0.09 0.15 0.08

MeanTmin(C) 0.02 0.01 0.01 0.01

STDTmin(C) 0.43 0.33 0.20 0.20

MeanTmean(C) 0.01 0.01 0.01 0.01

STDTmean(C) 0.16 0.13 0.11 0.11

DTR (C) 0.48 0.43 0.52 0.55

FSLs (day) 3.62 3.54 1.23 1.65

GSL (day) 2.31 2.60 1.82 2.07

FreTh (day) 0.90 0.91 0.85 0.83

Tmax90 (C) 0.19 0.22 0.25 0.24

Tmin10 (C) 0.41 0.38 0.37 0.32

Table 10

RMSE of the estimated statistics and climatic indices during validation period at Schefferville based on NCEP predictors (interpolated on the CGCM1 and HadCM3 grids)

CGCM1 grid HadCM3 grid

SDSM ASD SDSM ASD

Mean prec. (mm/day) 0.41 0.56 0.79 0.57

STD prec. (mm/day) 0.68 0.91 0.96 1.01

PRCP1 (%) 5.72 6.68 7.51 6.05

SDII (mm/wet day) 1.16 0.39 1.43 0.45

CDD (day) 0.62 0.63 1.61 0.42

R3days (mm) 4.44 9.58 7.58 11.56

PREC90 (mm) 1.26 1.72 2.47 2.07

MeanTmax(C) 0.40 0.40 0.48 0.49

STDTmax(C) 0.36 0.32 0.24 0.21

MeanTmin(C) 0.55 0.52 0.35 0.33

STDTmin(C) 0.48 0.39 0.30 0.34

MeanTmean(C) 0.36 0.35 0.37 0.35

STDTmean(C) 0.23 0.22 0.25 0.22

DTR (C) 0.24 0.20 0.58 0.64

FSLs (day) 6.62 6.82 5.60 4.54

GSL (day) 4.78 3.97 2.97 2.07

FreTh (day) 0.94 0.88 0.80 0.77

Tmax90 (C) 0.40 0.45 0.12 0.22

Tmin10 (C) 0.89 0.86 0.77 0.78

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Fig. 4. Box plots of mean for monthly precipitation model using 100 simulations based on NCEP predictors interpolated on HadCM3 at Schefferville, over the calibration (1961e1975) and the validation (1976e1990) periods. The red lines represent the median values, the Interquartile Range (IQR, i.e. 25th and 75th quar- tiles) is represented by boxes and 1.5IQR by whiskers. The red crosses correspond to outliers. (For interpretation of the references to colour in figure legends, the reader is refered to the web version of this article).

Mean (ºC)Mean (ºC)Mean (ºC)

1 2 3 4

-20 -10

Jan

1 2 3 4

-20 -15 -10

Feb

1 2 3 4

-15 -10 -5

Mar

1 2 3 4

-8 -6 -4 -2 0 2 4

Apr

1 2 3 4

2 4 6 8 10

May

1 2 3 4

8 10 12 14 16

18 Jun

1 2 3 4

15 20

Jul

1 2 3 4

15

20 Aug

1 2 3 4

5 10 15

Sep

1 2 3 4

-2 0 2 4 6

Oct

1 2 3 4

-10 -8 -6 -4 -2

Nov

1 2 3 4

-20 -15 -10

Dec

1-OBSERVATION (1961-1975) 3-OBSERVATION (1976-1990) 2-ASD (1961-1975) 4-ASD (1976-1990)

Fig. 5. Box plots of mean for monthly maximum temperature model using 100 simulations based on NCEP predictors interpolated on HadCM3 at Schefferville, over the calibration (1961e1975) and the validation (1976e1990) periods.