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Environmental Modelling & Software 23 (2008) 813e834 www.elsevier.com/locate/envsoft
Automated regression-based statistical downscaling tool Masoud Hessami a,*, Philippe Gachon b,c, Taha B.M.J. Ouarda d, Andre´ St-Hilaire d a Department of Civil Engineering, Shahid Bahonar University of Kerman, Kerman 76169-133, Iran Adaptation and Impacts Research Division, Science and Technology Branch, Environment Canada, Montre´al, Que´bec, Canada c McGill University, Department of Civil Engineering and Applied Mechanics, 817 Sherbrooke Street West, Montre´al, Que´bec H3A 2K6, Canada d INRS-ETE, Chair in Statistical Hydrology, University of Que´bec, 490 de la Couronne, Que´bec G1K 9A9, Canada b
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. SDSM e a 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 Canada 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. Ó 2007 Elsevier Ltd. All rights reserved. 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
* Corresponding author. Tel./fax: þ11 98 341 322 0054. E-mail address:
[email protected] (M. Hessami). 1364-8152/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.envsoft.2007.10.004
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
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methods (Hewitson and Crane, 1996; Wilby et al., 1999), weather pattern approaches (Yarnal et al., 2001), stochastic weather generators (Richardson, 1981; Racsko et al., 1991; Semenov and Barrow, 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 implementation 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 environments (see for example the recent study of Dibike et al., 2007 and Gachon and Dibike, 2007, in coastline areas of northern Canada), and/or for generating large ensembles or transient scenarios. In our study, the statistical downscaling regression-based methods, namely the SDSM model developed by Wilby 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 Section 2, the mathematical formulation of an automated statistical downscaling (ASD) model is presented, followed by Section 3 showing the methodology focussed on model evaluation criteria, the study area, data and predictors selection. The results are then presented in Section 4 in using both the NCEP (National Centre for Environmental Prediction) and two series of GCMs daily predictors. Section 6 presents the main conclusions 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 by Wilby et al., 2002), was developed under the Matlab environment (The Mathworks, 2002). Figs. 1 and 2 show 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 in Wilby et al. (1999): Oi ¼ a0 þ
n X
aj pij ; R0:25 ¼ b0 þ i
j¼1
n X
bj pij þ ei
ð1Þ
j¼1
where Oi is the daily precipitation occurrence, Ri are daily precipitation amounts, pij are predictors, n is number of predictors, a and b are model parameters and ei is modeling error. The modeling of daily temperature is performed in one step: Ti ¼ g0 þ
n X
gj pij þ ei
ð2Þ
j¼1
where Ti is the daily temperature (maximum, minimum or mean) and g is the model parameter. Once the deterministic component is obtained, the residual term ei is modeled under the assumption that it follows a Gaussian distribution: rffiffiffiffiffiffiffiffi VIF ei ¼ ð3Þ z i Se þ b 12 where zi 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: b ¼ Mobs Md VIF ¼
12ðVobs Vd Þ S2e
ð4Þ ð5Þ
where Vobs is the variance of observation during calibration period, Vd is the variance of deterministic part of model output during calibration period, Se is the standard error, Mobs and Md are 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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Fig. 1. Main menu of ASD (automated statistical downscaling) tool.
future climate predictors are standardized using the mean and variance of the current climate predictors). The ridge regression coefficients a for the linear model y ¼ Xa þ e can be calculated from:
of the ridge parameter is often done by iteration and can be somewhat subjective. Hoerl and Kennard (1970) suggest the following guidelines: -
1
a ¼ ðXt X þ kIÞ X t y
ð6Þ
where I is an identity matrix and k is the ridge parameter. When k ¼ 0, a is the least squares estimator. The selection
Plot values of k as a function of a (the so-called ridge trace). Identify the k value for which the system stabilizes. The associated a values provide the general character of an orthogonal system.
Set model configuration
Predictand
NCEP predictors
Select best predictors using stepwise regression / partial correlation
Calibrate model
Generate simulations using NCEP data
Set VIF and bias
Generate scenarios using GCM data
Results evaluation
Fig. 2. ASD architecture.
GCM predictors
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Coefficients should have reasonable absolute values. Coefficients with improper signs at k ¼ 0 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 judgment. 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 q 1 variables or removing a predictor from the equation containing q variables is: R2q R2q1 ðn q 1Þ F¼ ð7Þ 1 R2q where n is the number of observations, Rq and Rq 1 are correlation coefficients between the criterion variable and a prediction equation having q and q 1 variables, respectively. If F is greater than the critical F 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 and n q 1. A Bonferroni correction (Bonferroni, 1936) is used for the level of significance using the following formula: a1q a¼1 1 2
ð8Þ
where a is the level of significance and q is the number of predictor in the equation. The partial F-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 and j while controlling for third variable k is (e.g. Afifi and Clark, 1996): Rij Rik Rjk Rij;k ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 R2ik 1 R2jk
ð9Þ
where Rij is the correlation coefficient between variables i and j. For partial correlation method, the p-value is used for eliminating any one of the predictors. The p-value is computed by transforming the correlation R to create a t-statistic having n 2 degrees of freedom, where n is the number of observations: R t ¼ qffiffiffiffiffiffiffiffi 1R2 n2
ð10Þ
The probability of the t-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 in Tables 1 and 2 for 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 y against the independent variables x2, x3, ., xm: y ¼ f1 ðx2 ; x3 ; .; xm Þ
ð11Þ
and then we compute the residuals from regressing x1 against the independent variables x2, x3, ., xm: x1 ¼ f2 ðx2 ; x3 ; .; xm Þ
ð12Þ
The correlation between y and x1 controlling x2, x3, ., xm is obtained by computing the correlation between the residuals of the two linear models f1 and f2. 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 2 3 4 5 6 7 8 9
Mean sea level pressure Surface airflow strength Surface zonal velocity Surface meridional velocity Surface vorticity Surface wind direction Surface divergence 500 hPa airflow strength 500 hPa zonal velocity
14 15 16 17 18 19 20 21 22
10
500 hPa meridional velocity
23
11
500 hPa vorticity
24
12
500 hPa geopotential height
25
13
500 hPa wind direction
26
500 hPa divergence 850 hPa airflow strength 850 hPa zonal velocity 850 hPa meridional velocity 850 hPa vorticity 850 hPa geopotential height 850 hPa wind direction 850 hPa divergence Near surface relative humidity Specific humidity at 500 hPa Specific humidity at 850 hPa Near surface specific humidity Mean temperature at 2 m
M. Hessami et al. / Environmental Modelling & Software 23 (2008) 813e834 Table 2 NCEP predictor variables on HadCM3 grid No.
Predictors
No.
Predictors
1 2 3 4 5 6 7 8 9
Mean sea level pressure Surface airflow strength Surface zonal velocity Surface meridional velocity Surface vorticity Surface wind direction Surface divergence 500 hPa airflow strength 500 hPa zonal velocity
14 15 16 17 18 19 20 21 22
10
500 hPa meridional velocity
23
11
500 hPa vorticity
24
12 13
500 hPa geopotential height 500 hPa wind direction
25 26
500 hPa divergence 850 hPa airflow strength 850 hPa zonal velocity 850 hPa meridional velocity 850 hPa vorticity 850 hPa geopotential height 850 hPa wind direction 850 hPa divergence Relative humidity at 500 hPa Relative humidity at 850 hPa Near surface relative humidity Surface specific humidity Mean temperature at 2 m
paper (see their location over eastern Canada in Fig. 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 4 show 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 in Gachon et al. (2005). They are modified to correspond to the characteristics of the Que´bec climate from the STARDEX (Statistical and Regional dynamical Downscaling of Extremes for European regions) SDEIS climate indices software (Haylock, 2004).
817
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. 3 shows the area over eastern Canada where the studied stations are located. We have focused on the following stations located around the Labrador 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 3.7 of 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) and Pope 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 417 278 km at the Equator, reducing to 295 278 km at 45 of latitude. The two GCMs have participated in the CMIP1 (Coupled Model Intercomparison 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 Chapter 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
Fig. 3. Meteorological stations located around the Labrador Sea and the Gulf of St. Lawrence.
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Table 3 Precipitation indices used to evaluate the performance of statistical downscaling models Indices
Definition
Unit
Time Scale
PRCP1
Percentage of wet days (threshold 1 mm) Mean precipitation amount at wet days Maximum number of consecutive dry days Maximum 3-days precipitation total 90th percentile of rain day amount
%
Season
mm/day
Season
Days
Season
mm mm
Season Season
SDII CDD R3days PREC90
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 2.5 latitude by 2.5 longitude 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 in Tables 1 and 2. Re-gridding and verification of GCM predictors are a necessary part of all statistical downscaling development (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
Table 4 Temperature indices used to evaluate the performance of statistical downscaling models Indices DTR FSL
GSL
FreTh
Tmax90 Tmin10
Definition
Unit
Time Scale
Mean of diurnal temperature range Frost season length: Tmin < 0 C more than 5 days and Tmin > 0 C more than 5 days Growing season length: Tmean > 5 C more than 5 days and Tmean < 5 C more than 5 days Days with freeze and thaw cycle (Tmax > 0 C, Tmin < 0 C) 90th percentile of daily Tmax 10th percentile of daily Tmin
Season
C
Days
Year
Days
Year
Days
Month
Season Season
C C
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 (see Tables 5e8), i.e. are issued from the same physical processes, and this does not constitute a major conceptual 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 of Dibike 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, see Table 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 documented 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 projection). 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 5 shows 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 (see Tables 6e8), through the thermal advection term. For stability and robustness of the downscaling results (see Gachon 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, see Tables 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
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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 2
Predictors Cartwright Goose bay Kuujjuaq Schefferville Causapascal Daniel Harbour Gaspe´ Mont-Joli Natashquan Sept-Iˆles
3 3 2 2 7 2 3 2 2 5
5 9 8 16 9 11 9 9 12 9
9 10 15 17 16 12 10 16 16 12
11 15 16 18 21 18 14 17 17 16
23 23 23 23 23 23 17 23 23 23
2
R
R (ridge)
Predictors
0.24 0.24 0.22 0.32 0.13 0.20 0.17 0.21 0.26 0.24
0.32 0.31 0.33 0.37 0.20 0.27 0.27 0.27 0.35 0.38
3 3 2 2 9 2 12 2 10 2
5 12 12 9 12 11 17 9 12 5
12 15 16 12 16 15 19 12 16 9
19 17 19 16 17 18 22 16 17 16
22 22 22 19 22 22 23 22 22 22
R2
R2 (ridge)
0.27 0.23 0.22 0.25 0.15 0.20 0.18 0.17 0.31 0.22
0.33 0.33 0.33 0.34 0.21 0.27 0.30 0.31 0.36 0.40
For each predictor, the number refers to the atmospheric variables defined in Tables 1 and 2 (for each respective GCM grid).
0.31 (Natashquan) for NCEP data on HadCM3 grid (Table 5). These relatively low explained variances underline the difficulty 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%; see Tables 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 in Tables 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
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 Section 4.1. 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
Table 6 Results of ASD model calibration (1961e1975) for maximum temperature using NCEP predictors interpolated on the CGCM1 and on the HadCM3 grids Station
CGCM1 grid
HadCM3 grid R
Predictors Cartwright Goose bay Kuujjuaq Schefferville Causapascal Daniel Harbour Gaspe´ Mont-Joli Natashquan Sept-Iˆles
1 1 1 1 1 1 1 1 1 1
4 5 4 5 5 4 5 5 5 5
5 6 5 7 16 5 15 10 15 15
17 15 19 18 18 15 18 18 18 18
19 19 24 19 19 19 19 19 19 19
2
0.91 0.95 0.95 0.96 0.95 0.94 0.93 0.94 0.93 0.93
2
R (ridge)
Predictors
0.94 0.96 0.96 0.97 0.96 0.94 0.95 0.96 0.95 0.96
1 1 1 1 1 1 1 1 1 1
5 5 5 3 3 4 3 3 5 5
17 17 11 5 5 5 5 5 7 12
18 18 18 16 16 15 16 12 17 18
For each predictor, the number refers to the atmospheric variables defined in Tables 1 and 2 (for each respective GCM grid).
19 19 19 19 19 19 19 19 19 19
R2
R2 (ridge)
0.89 0.93 0.94 0.95 0.95 0.94 0.94 0.95 0.94 0.93
0.93 0.96 0.96 0.97 0.96 0.94 0.95 0.96 0.94 0.96
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Table 7 Results of ASD model calibration (1961e1975) for mean temperature using NCEP predictors interpolated on the CGCM1 and on the HadCM3 grids Station
CGCM1 grid
HadCM3 grid 2
Predictors Cartwright Goose bay Kuujjuaq Schefferville Causapascal Daniel Harbour Gaspe´ Mont-Joli Natashquan Sept-Iˆles
1 1 1 1 1 1 1 1 1 1
4 5 5 3 5 5 5 5 5 5
5 15 18 5 12 11 9 12 15 15
18 18 19 16 18 18 18 18 18 18
19 19 24 19 19 19 19 19 19 19
2
R
R (ridge)
Predictors
0.93 0.96 0.96 0.97 0.96 0.94 0.96 0.96 0.95 0.96
0.95 0.97 0.97 0.98 0.97 0.95 0.97 0.97 0.96 0.97
1 1 1 1 1 1 1 1 1 1
5 5 3 3 3 5 3 3 5 5
7 7 5 5 5 11 5 5 15 16
17 17 19 18 16 15 18 16 17 18
19 19 23 19 19 19 19 19 19 19
R2
R2 (ridge)
0.94 0.95 0.96 0.96 0.96 0.95 0.96 0.96 0.95 0.96
0.95 0.97 0.97 0.97 0.97 0.96 0.96 0.97 0.96 0.97
For each predictor, the number refers to the atmospheric variables defined in Tables 1 and 2 (for each respective GCM grid).
deterministic part of the model), as suggested in the weak amount of explained variance in Table 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 in Table 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 in Table 10, few differences exist in the RMSE values between the validation and the calibration periods (as shown in Table 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
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 in Tables 9 and 10 (over these two periods, respectively), and through graphical analysis with box plots graphs shown in Figs. 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 Section 4.2. As shown in Table 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
Table 8 Results of ASD model calibration (1961e1975) for minimum temperature using NCEP predictors interpolated on the CGCM1 and on the HadCM3 grids Station
CGCM1 grid
HadCM3 grid 2
Predictors Cartwright Goose bay Kuujjuaq Schefferville Causapscal Daniel Harbour Gaspe´ Mont-Joli Natashquan Sept-Iˆles
1 1 1 1 1 1 1 1 1 1
5 5 5 3 3 5 3 5 2 5
16 15 11 5 5 7 5 9 5 16
18 18 18 15 9 19 18 18 18 18
19 19 19 19 19 21 19 19 19 19
2
R
R (ridge)
Predictors
0.91 0.94 0.94 0.94 0.91 0.91 0.92 0.94 0.93 0.95
0.93 0.95 0.96 0.95 0.93 0.92 0.94 0.95 0.94 0.96
1 1 1 1 1 1 1 1 1 1
5 5 3 5 2 5 3 3 2 5
7 15 5 18 5 17 5 5 5 16
15 18 19 19 19 19 18 16 17 19
For each predictor, the number refers to the atmospheric variables defined in Tables 1 and 2 (for each respective GCM grid).
19 19 23 23 23 23 19 19 19 23
R2
R2 (ridge)
0.90 0.93 0.94 0.94 0.92 0.91 0.92 0.94 0.93 0.95
0.93 0.95 0.96 0.95 0.93 0.92 0.94 0.95 0.94 0.96
M. Hessami et al. / Environmental Modelling & Software 23 (2008) 813e834 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
Mean prec. (mm/day) STD prec. (mm/day) PRCP1 (%) SDII (mm/wet day) CDD (day) R3days (mm) PREC90 (mm/day) Mean Tmax ( C) STD Tmax ( C) Mean Tmin ( C) STD Tmin ( C) Mean Tmean ( C) STD Tmean ( C) DTR ( C) FSLs (day) GSL (day) FreTh (day) Tmax90 ( C) Tmin10 ( C)
HadCM3 grid
SDSM
ASD
SDSM
ASD
0.29 0.53 4.49 1.59 0.98 3.13 1.95 0.01 0.14 0.02 0.43 0.01 0.16 0.48 3.62 2.31 0.90 0.19 0.41
0.16 0.37 5.14 0.44 0.78 4.30 0.72 0.02 0.09 0.01 0.33 0.01 0.13 0.43 3.54 2.60 0.91 0.22 0.38
0.41 0.71 4.00 1.58 1.10 3.32 2.10 0.01 0.15 0.01 0.20 0.01 0.11 0.52 1.23 1.82 0.85 0.25 0.37
0.15 0.40 4.76 0.42 0.87 4.89 0.96 0.02 0.08 0.01 0.20 0.01 0.11 0.55 1.65 2.07 0.83 0.24 0.32
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 in Tables 9 and 10). 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
Mean prec. (mm/day) STD prec. (mm/day) PRCP1 (%) SDII (mm/wet day) CDD (day) R3days (mm) PREC90 (mm) Mean Tmax ( C) STD Tmax ( C) Mean Tmin ( C) STD Tmin ( C) Mean Tmean ( C) STD Tmean ( C) DTR ( C) FSLs (day) GSL (day) FreTh (day) Tmax90 ( C) Tmin10 ( C)
HadCM3 grid
SDSM
ASD
SDSM
ASD
0.41 0.68 5.72 1.16 0.62 4.44 1.26 0.40 0.36 0.55 0.48 0.36 0.23 0.24 6.62 4.78 0.94 0.40 0.89
0.56 0.91 6.68 0.39 0.63 9.58 1.72 0.40 0.32 0.52 0.39 0.35 0.22 0.20 6.82 3.97 0.88 0.45 0.86
0.79 0.96 7.51 1.43 1.61 7.58 2.47 0.48 0.24 0.35 0.30 0.37 0.25 0.58 5.60 2.97 0.80 0.12 0.77
0.57 1.01 6.05 0.45 0.42 11.56 2.07 0.49 0.21 0.33 0.34 0.35 0.22 0.64 4.54 2.07 0.77 0.22 0.78
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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 illustrate 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 in Fig. 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. 13 shows 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
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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 quartiles) is represented by boxes and 1.5 IQR 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).
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1-OBSERVATION (1961-1975) 3-OBSERVATION (1976-1990) 4-ASD (1976-1990) 2-ASD (1961-1975)
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.
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1-OBSERVATION (1961-1975) 3-OBSERVATION (1976-1990) 4-ASD (1976-1990) 2-ASD (1961-1975)
Fig. 6. Box plots of mean for monthly mean temperature model using 100 simulations based on NCEP predictors interpolated on HadCM3 at Schefferville, over the calibration (1961e1975) and the validation (1976e1990) periods.
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1-OBSERVATION (1961-1975) 3-OBSERVATION (1976-1990) 4-ASD (1976-1990) 2-ASD (1961-1975)
Fig. 7. Box plots of mean for monthly minimum temperature model using 100 simulations based on NCEP predictors interpolated on HadCM3 at Schefferville, over the calibration (1961e1975) and the validation (1976e1990) periods.
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Fig. 8. Box plots of mean and standard deviation for monthly precipitation models during 1961e1990 at Schefferville.
simulation time series only differ by 1e2 days in most cases, and no important bias in the simulated values is apparent. However, in all cases, both ASD and SDSM overestimate the highest values of CDD with also excessive outliers. Also, all downscaling results are quite similar with no obvious better skill with NCEP driven conditions compared to GCMs ones. Fig. 15 shows the maximum 3-days precipitation total
index (R3days) with an underestimation of the median values for spring season by all models using CGCM1 predictors and with an overestimation of extremes in all seasons and whatever driven conditions or downscaling models. Median values of the 90th percentile of observed daily rain amounts (PREC90), shown in Fig. 16, are generally well reproduced with a maximum bias of 4 mm/day (with SDSM using
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Fig. 9. Box plots of mean and standard deviation for monthly maximum temperature models during 1961e1990 at Schefferville.
HadCM3 predictors in summer season). All models overestimated the median spring, summer and autumn values. All observed values were below 24 mm, irrespective of season, and singular simulated values over this threshold are quite common (with outliers ranging between 19 and 40 mm), suggesting a strong dispersion from the stochastic component of the two models. Again no obvious differences are suggested in
all results whatever the driving conditions, i.e. NCEP or GCMs. For the climate indices of temperature, Fig. 17 shows that the median of the diurnal temperature range (DTR) is well reproduced with a 0e3 C bias in the simulated values compared to the observed ones, except poorest results with SDSM over summer, and with SDSM and ASD over winter
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Fig. 10. Box plots of mean and standard deviation for monthly minimum temperature models during 1961e1990 at Schefferville.
driven, respectively, by CGCM1 and HadCM3. Generally in winter, the two models overestimate the DTR, both in median and higher values. Fig. 18 shows the observed and simulated frost season length index, relatively well reproduced by ASD and SDSM models for the median values. Here, SDSM using CGCM1 data provides poorest estimation of FSL indices. However, all simulated indices overestimate the variability with respect to the observed one, as well as the number of
lowest values. Fig. 19 shows the observed median value of the GSL index was around 100 days whereas the simulated values were 90, 110, 108 and 105 days for SDSM (CGCM1), ASD (CGCM1), SDSM (HadCM3) and ASD (HadCM3), respectively. All downscaling series overestimate the variability with the poorest results coming from SDSM driven by CGCM1, with strong shift of the statistical distribution (mainly lower values than observed). In that case, ASD strongly improves
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Fig. 11. Box plots of mean and standard deviation for monthly mean temperature models during 1961e1990 at Schefferville.
the corresponding values from the same series of predictors, by comparison with SDSM. The observed and simulated FreTh index shown in Fig. 20 suggests that all models using HadCM3 data provide better estimation for this index, especially with the ASD model. The stronger bias appears in winter and in July and August in terms of overestimated variance and singular values, where the Frost/Thaw observed cycle is weaker. Also in
that case, strong compatibility from results driven by NCEP for both ASD and SDSM is revealed. The 90th percentile of daily maximum temperature (Tmax90) shown in Fig. 21 suggests that all models overestimated the winter median, with more often the best results simulated by ASD using HadCM3 predictors. For the spring, ASD using CGCM1 estimated also very well the median value. We can also notice that excessive outliers are
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Fig. 12. PRCP1 precipitation index at Schefferville during 1961e1990 using 100 simulations based on NCEP and GCM predictors.
Fig. 14. CDD precipitation index at Schefferville during 1961e1990 using 100 simulations based on NCEP and GCM predictors.
more common with NCEP driven conditions than with GCMs ones, especially in spring, summer and fall. Fig. 22 shows that the 10th percentiles of daily minimum temperature were generally well reproduced by SDSM and ASD models, except in winter, summer and fall when SDSM is driven by CGCM1. For
example, for the summer, SDSM using CGCM1 predictors showed a strong bias, with a median of 3.2 C compared to observed value of 0.7 C. Also, excessive outliers and values around 0 C are present in the simulated values of ASD in the fall season.
Fig. 13. SDII precipitation index at Schefferville during 1961e1990 using 100 simulations based on NCEP and GCM predictors.
Fig. 15. R3days precipitation index at Schefferville during 1961e1990 using 100 simulations based on NCEP and GCM predictors.
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Fig. 18. FSLs temperature index at Schefferville during 1961e1990 using 100 simulations based on NCEP and GCM predictors. Fig. 16. PREC90 precipitation index at Schefferville during 1961e1990 using 100 simulations based on NCEP and GCM predictors.
In Table 11, the comparison of RMSE indicates that ASD provides better results in general (i.e. strong majority of the results) than SDSM except for mean precipitation amount and its standard deviation, intensity per wet days, consecutive dry days, 90th percentile of rain day amount, standard deviation of both minimum and maximum temperature and diurnal temperature range, with CGCM1 predictors. For other indices and
all results from HadCM3 predictors, ASD gives clearly the best results (i.e. lowest RMSE) as suggested also in the box plots graphs (Figs. 12e22). For temperature, ASD gives also the best results and strongly improves the biases in extremes compared to SDSM when this model is driven by CGCM1. This reveals the usefulness of Eqs. (4) and (5) to set automatically the VIF and the bias when generating scenarios from GCM data by the model whose coefficients are obtained
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1-OBSERVATION 2-SDSM(NCEP on CGCM1 grid) 3-ASD(NCEP on CGCM1 grid) 4-SDSM(NCEP on HADCM3 grid)
Fig. 17. DTR temperature index at Schefferville during 1961e1990 using 100 simulations based on NCEP and GCM predictors.
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Fig. 19. GSLs temperature index at Schefferville during 1961e1990 using 100 simulations based on NCEP and GCM predictors.
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Fig. 20. FreTh temperature index at Schefferville during 1961e1990 using 100 simulations based on NCEP and GCM predictors.
Fig. 21. Tmax90 temperature index at Schefferville during 1961e1990 using 100 simulations based on NCEP and GCM predictors.
Fig. 22. Tmin10 temperature index at Schefferville during 1961e1990 using 100 simulations based on NCEP and GCM predictors.
M. Hessami et al. / Environmental Modelling & Software 23 (2008) 813e834 Table 11 RMSE of the estimated statistics and climatic indices during 1961e1990 period at Schefferville based on GCMs predictors CGCM1
Mean prec. (mm/day) STD prec. (mm/day) PRCP1 (%) SDII (mm/wet day) CDD (day) R3days (mm) PREC90 (mm) Mean Tmax ( C) STD Tmax ( C) Mean Tmin ( C) STD Tmin ( C) Mean Tmean ( C) STD Tmean ( C) DTR ( C) FSLs (day) GSL (day) FreTh (day) Tmax90 ( C) Tmin10 ( C)
HadCM3
SDSM
ASD
SDSM
ASD
0.33 0.52 6.73 1.19 0.29 5.24 1.29 3.40 0.74 3.22 0.56 3.36 0.64 0.57 14.35 11.70 3.40 2.48 3.13
0.41 0.70 5.05 1.54 0.58 3.28 1.59 0.52 1.06 0.53 1.18 0.51 1.09 1.60 2.76 7.19 1.40 0.99 0.92
0.79 1.11 5.16 2.03 1.27 8.75 3.12 1.59 0.90 0.92 0.80 0.98 0.72 1.29 2.95 4.40 2.24 1.50 1.21
0.54 0.60 3.69 1.43 0.68 2.25 1.40 0.74 0.49 0.67 0.30 0.69 0.37 0.41 0.07 3.14 1.06 0.40 1.16
from NCEP data. It is important to mention that SDSM results are obtained by a skilled user and the best VIF and the bias are obtained by trial and error. However, comparable results are obtained by ASD with ease very quickly. This is an important advantage of ASD. Therefore, as suggested in Table 11 by comparison with Tables 9 and 10 with NCEP predictors, no systematic decrease of performance in using GCMs predictors instead of NCEP ones, especially for precipitation with similar performance for mean amount and majority of precipitation indices, whereas the downscaled temperatures and their extremes are better simulated by NCEP driven conditions. For standard deviation at monthly scale, the results are mostly similar (i.e. inter-annual variability of maximum and minimum temperature). Finally, in strong majority of the results from GCMs variables, the best downscaled values are obtained when the statistical models were driven by HadCM3 predictors, as in the recent study of Dibike et al. (2007) in northern Canada. 4.3. Model calibration using ridge regression All of the results previously shown were obtained using multiple linear regressions because that component of ASD is more akin to the one provided by SDSM, which is the basis of comparison. The implementation of the ridge regression is also briefly investigated here. We have used ridge regression with the five predictors selected by the stepwise regression (Tables 5e8) and we obtained values of explained variance equal to those obtained with multiple linear regressions. However, when ridge regression is implemented using all predictors, model explained variances increase. Tables 5e8 include also the explained variance when ASD model calibration is based on ridge regression for each predictand. For each station, all predictors have been used except the
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predictor variables 22, 25 and 26 from NCEP data interpolated on CGCM1 grid and the variables 24, 25 and 26 from NCEP data interpolated on HadCM3 grid, as suggested in Section 3. When using ridge regression, it can be seen from Table 5 that the precipitation model explained variances have been greatly improved. For example when using NCEP data interpolated on the HadCM3 grid, the precipitation model explained variance was 0.34 for Schefferville with the use of the ridge regression. These values were 0.25 when multiple linear regression is used with five predictors. However, it can be seen from Tables 6e8, slight improvement for temperature models based on ridge regression. Hence, the ridge regression mainly improved the explained variance for the precipitation process in which low correlation between predictors and predictand is common, with strong probability to have collinearity between predictors, as too many variables and parameters in the model are highly correlated. Also, it allows the user to implement a model without discriminating predictors a priori. The user can thus decide to select all predictors or some predictors based on experience and their physical characteristics without considering the non-orthogonality, which is inherently taken into account in the ridge regression algorithm.
5. Discussion The choice of predictor variables is one of the most influential steps in the development of statistical downscaling scheme (both with automatic and manual selection) because the decision largely determines the character of the downscaling results. It is essential to remember that predictors relevant to the local predictand should be adequately reproduced by the host climate model at the spatial scales used to condition the downscaled response. Prior knowledge of climate model limitations is essential when screening potential predictors, i.e. predictors have to be chosen on the balance of their relevance to the target predictand and their accurate representation by climate models (Wilby and Wigley, 2000; Wilby et al., 2004). In our case, we have undertaken GCM verification for few important predictors of interest, as low air temperature and specific and relative humidity near the surface. Strong biases have been found and in order to prevent the propagation of errors into the downscaling process (see Gachon et al., 2005; Dibike et al., 2007; Gachon and Dibike, 2007), we have excluded these variables in the list of potential predictors. In using or not an automatic process for the selection of predictors, the scrupulous analysis of the candidate predictors is needed as those must be strongly correlated with the target variable, makes physical sense and captures multiyear variability (Wilby et al., 2004). Further works are needed in that matter, in order to systematically evaluate the accuracy of candidate predictors and incorporate other independent data series with other GCMs, especially those recently released by IPCC (International Panel on Climate Change) and PCMDI (Program for Climate Model Diagnosis and Intercomparison) and incorporated in the Fourth Assessment Report of IPCC (2007).
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Therefore, the best performing predictors or combination of predictors for current climate (NCEP or GCMs) does not necessarily mean those predictors are the most suitable for future changed climate conditions. For example, Murphy (1999) found that a moisture predictor was not selected when it is expected that changes in moisture will influence precipitation, and Charles et al. (1999) found that relative humidity but not specific humidity produced downscaled estimates consistent with RCM projections. In our study, the downscaling procedure of precipitation and temperature is not only based on circulation variables (i.e. represented by geopotential, vorticity or the wind component) but also includes other variables such as temperature (through geopotential heights at various levels and specific/relative humidity near the mid-troposphere) and moisture variables (specific/relative humidity). This is in agreement with other studies (Wilby et al., 1998; Huth, 1999, 2002, 2004; Gachon et al., 2005) which have shown that the use of combined predictors (circulation and temperature) is superior to that of any single predictor when downscaling temperature and/or precipitation. This approach also avoids the assumption that changes in surface climate elements due to the enhanced greenhouse effect can be derived from changes in circulation alone, as this for example can lead to unrealistically low temperature change estimates (as suggested in Huth, 2004). Also, as suggested by Crane and Hewitson (1998) and Wilby et al. (1998) including humidity not only improves the transfer function predictions for the present climate but also reduces the stationarity issue noted above, as the dynamics and the humidity fields become separate predictor variables in the downscaling function. As shown in the recent study of Gachon and Dibike (2007), the SDSM model is able to simulate reliable and plausible changes in mean values as well as probabilities of extreme temperatures, in some specific locations in northern Canada, from mainly the same sets of available predictors (from CGCM2 and HadCM3) and using two emission scenarios (i.e. SRES A2 and B2). Over the period 2070e2099, the results demonstrate that the SDSM model is able to capture the major part of the temperature change signal, with a plausible climatic regime for higher warming in winter than in summer and in A2 than in B2 runs. The combination of relevant atmospheric predictors in the downscaling process is able to take into account most key factors of the temperature change signal, with strong convergence in the magnitude and the timing of the changes in all results. However, some potential underestimation of the warming from the downscaled signal remains, as the regression method is inherently conservative in the presence of non-stationarity in the climate system. Also, while empirical downscaling gives the first-order response to the regional or local climate change, the regression-based method is unable to incorporate local-scale feedbacks, in particular the radiative effect based on cloud cover feedbacks. This leads to an underestimation of the change in the local temperature regime, in particular in summer. In other seasons, stronger signals from changes in synoptic scale forcing are better taken into account in the statistical downscaling scheme (from wind, geopotential and vorticity)
for all areas analyzed in northern and northeastern Canada. However, the potential effect of regional sea-ice processes and change in land cover (i.e. in particular the snow cover duration) is not well captured in the downscaling process. As seaice formation and retreat are coarsely simulated at the regional scale by most GCMs, such feedbacks are not adequately taken into account in the remaining predictors. Nevertheless, as suggested in previous studies on a regression-based approach, the use of relevant combination of predictors and not only a single one or issued from the same physical processes is crucial, if possible representing different combinations of circulation, temperature and humidity predictors. 6. Conclusion In this paper, an automated statistical downscaling methodology was presented using a backward stepwise regression for predictors’ selection. Like other regression approaches, the results indicate the strength of statistical downscaling for modeling temperature and less success for precipitation. When modeling precipitation, the most commonly used 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 statistical downscaling approach, several climatic and statistical indices have been analyzed. The results indicate that the agreement of simulations with observations depends on the statistical model and the GCM data, and the performance of the statistical downscaling model varies for different stations and between seasons. From this study, the result comparison of SDSM and ASD models has shown that neither model consistently outperforms the other for the precipitation regime. For temperature, ASD outperforms SDSM with HadCM3 predictors, and in the majority of the time with CGCM1 predictors. Overall results suggest also that HadCM3 driven conditions give more often the best results, as revealed in other recent study over Canada (Gachon et al., 2005; Dibike et al., 2007). The quality of SDSM results depends mostly on the skill of the user; however, similar results can be obtained using ASD with ease very quickly. ASD reduces the problem of predictor selection and it is computationally more efficient than SDSM and it is capable of performing all steps of statistical downscaling automatically. At the same time, the user can interfere in the process and is not limited to implementing this tool merely as a black box. However, using different statistical downscaling models can provide a better estimation of uncertainty for simulated climatic and statistical indices. The use of multi-sources GCM input together has the potential to provide better results. This also allows to increase our confidence in projections for future climate changes scenarios when applying the statistical models to a wide range of climate models to evaluate the uncertainties associated with different GCM structures (as suggested in the guidelines of Wilby et al., 2004). Future studies will benefit from this feature by using
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other method, as a non-linear approach like artificial neural network which can use redundant information from multisources GCM data to reduce the observation noises. Also, with this new regression-based method, complementary analyses and evaluation should benefit from the downscaling results developed over other areas where the climate regime is different, especially to test its usefulness with the ridge regression for the downscaling of precipitation in using other type of candidate predictors as those used in our study. In that respect, the ASD model has been recently applied over monsoon areas in Africa with encouraging results in reconstructing the occurrence of observed precipitation and in using more extended series of predictors. Nevertheless, further works are needed to evaluate in depth the fundamental assumption of statistical downscaling, i.e. the stability of the relationships between predictors and predictand in altered climate, and there are some ways to test their plausibility and consistency, as suggested in Frias et al. (2006), Gachon and Dibike (2007) and Huth (2004).
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