A statistical approach to multi-site multivariate downscaling of daily extreme temperature series

Abstract

Downscaling methods for describing the linkage between global-scale climate variables and local climatic conditions have been frequently used in climate-related impact assessment studies. Previous works, however, have been mainly dealing with downscaling of climatic processes for a single site, but very few studies are concerned with the downscaling of these processes for multi-sites because of the complexity in accurately describing both observed at-site temporal persistence and spatial dependence between different locations. In the present study, a multi-site multivariate statistical downscaling (SD) approach was developed for simulating daily maximum (Tmax) and minimum (Tmin) temperature series at many sites concurrently. The proposed approach consists of a combination of a linear regression component to describe the linkage between global climate predictors and local temperature extremes, and a stochastic component based on a spatial moving average process to reproduce the observed spatial dependence between temperature extremes at different sites. The feasibility of the suggested SD method was assessed using observed daily extreme temperature data available at 10 weather stations located in the southwest region of Quebec and the southeast region of Ontario in Canada, as well as climate predictors from the NCEP/NCAR (National Centers for Environmental Prediction/National Centre for Atmospheric Research) reanalysis dataset for the 1961–1990 period. It was found that the proposed SD approach was able to accurately describe various Tmax and Tmin characteristics, including their spatial and temporal variation as well as their interannual anomalies. In addition, comparison of the results from the proposed multi-site multivariate SD method and one simulation series from the Canadian Regional Climate Model (CRCM) at a 45-km resolution (a dynamic downscaling procedure) has indicated that the suggested SD approach was able to more accurately describe the observed spatial and temporal characteristics of extreme temperature series at the regional scale than the CRCM-based dynamic downscaling method. Copyright © 2011 Royal Meteorological Society

1. Introduction

The main role of the Global Climate Models (GCMs) is to estimate current climate and to project future climate change. Downscaling techniques are required to derive site-scale climate information from the coarser resolution GCM outputs. In general, there are two different types of downscaling approaches. The first type is dynamical downscaling (DD) methods that are based on high-resolution Regional Climate Models (RCMs) or Limited Area Models (Xu, 1999; Laprise, 2008). In addition to their extensive and costly computing requirements, these DD models still have some restricted limitations because their resolutions (in the order of 50 km) are still considered too coarse and, thus, not suitable for local site-scale studies. Further DD drawbacks are associated with the dependence of these models on the later and lower boundary conditions prescribed by the GCM (Kao et al., 1998; Mahadevan and Archer, 1998), and the assumptions of stationarity incorporated within their many sub-grid parameterisation schemes. The second type is statistical downscaling (SD) methods (Nguyen, 2002), which are privileged by their ease of implementation and use. Its inexpensive and simple computational requirement permits the use of different GCMs in the development of climate change scenarios and their associated uncertainties. Furthermore, these SD techniques are quite popular for various types of impact assessment studies since they could be adapted to the climatic conditions for a specific site based on some established statistical relationships between large-scale atmospheric variables (predictors) and local climate variables (predictands). Therefore, the efficiency of the SD results depends on the credibility of the GCM to model the predictors. Indeed, reasonable GCM outputs permit strong relationships between the predictors and predictands and, thus, a skilful SD models. The main limitation of the SD techniques is related to the stationarity assumption of the SD model parameters (Wilby, 1997), which means that the statistical relationships developed for the current climate also hold under the different climatic conditions of future climate. Despite this constraint, SD methods have been commonly used in many different climate change impact studies (Wilby et al., 2002; Gachon and Dibike, 2007; Dibike et al., 2008; Nguyen and Nguyen, 2008).

There are three main different methods within the general SD approach; regression methods (Wilby et al., 1999; Hewitson and Crane, 1996; Hessami et al., 2008), weather pattern approaches (Yarnal et al., 2001), and stochastic weather generators (Richardson, 1981; Semenov and Barrow, 1997). Two practical tools based on SD regression-based methods are developed and adapted for end users: Statistical DownScaling Model (SDSM) (Wilby et al., 2002), and Automated Statistical Downscaling (ASD) (Hessami et al., 2008). These tools give roughly similar results, but ASD performs easily and automatically with a Matlab interface, and improves the problem of predictor selection (Hessami et al., 2004). However, these two approaches were limited to reproduce the characteristics of climate processes (e.g. daily precipitation and temperatures) at a given local site.

In developing SD techniques, an important issue is to respect the temporal variability at a given local site in a given region as well as the spatial dependence observed over the entire region. The latter condition is critical because of the significant effect of the spatial dependence on the results of impact assessment studies (Srikanthan and McMahon, 2001; Qian et al., 2002; Khalili et al., 2007, 2009). For instance, it has been shown that ignoring the spatial dependence of the weather time series at different sites may lead to a strong underestimation of extreme streamflows, especially for summer and autumn seasons (Khalili et al., 2006,, 2011).

Until recently, few attempts have been made to achieve multi-site climate downscaling. In particular, stochastic weather generators are frequently used to downscale large-scale climate variables at multiple locations (Wilks, 1999; Yarnal et al., 2001). Other approaches have been proposed to downscale climate processes from observed and modelled atmospheric fields using, for instance, the extended Nonhomogeneous Hidden Markov Model (NHMM) (Charles et al., 2004), and the Generalized Linear Models (GLMs) (Chandler and Wheater, 2002; Yang et al., 2006) to simulate multi-site sequences of daily rainfall. This GLM model has been used in many downscaling studies in the UK (Leith, 2005; Frost et al., 2006). More recently, Cannon (2008) uses the Expanded Bernoulli–gamma Density Network (EBDN) with an Artificial Neural Network (ANN) to predict daily precipitation series at multiple sites. Predicted covariances between sites are forced to match observed covariances through the addition of a constraint to the ANN cost function. Multi-site downscaling intercomparison study has been performed by Harpham and Wilby (2005). They compared three models; a Radial Basis Function (RBF) Artificial Neural Network (ANN), Multi Layer Perceptron (MLP) ANN, and a Conditional Resampling Method (using SDSM tool), for downscaling heavy daily precipitation occurrences and amounts at multiple sites. Regarding the reproduction of the interstation correlations, SDSM outperformed the ANNs, which tended to widely overestimate the interstation correlations for the precipitation amounts due to their fully deterministic forcing. However, with SDSM, the interstation correlations were too high in the summer season due to the difficulty of resolving isolated convective storms. Buerger and Chen (2005) compared three regression-based downscaling approaches: randomisation, inflation, and expanded downscaling. For precipitation data, none of these models was able to accurately reproduce the spatial correlations. In fact, the inflation model did not preserve the spatial correlations (they were too strong), and the spatial correlations simulated by the randomisation and expanded downscaling models were too weak.

In view of the above-mentioned issues, the present study presents a multi-site multivariate SD approach for simulating daily maximum and minimum temperatures for different locations, concurrently. The suggested approach differs from existing regression-based downscaling methods (e.g. Buerger and Chen, 2005), by the use of a spatial moving average process in the modelling of the stochastic component of the proposed multiple regression model. More specifically, the observed temporal dependence of daily temperature extremes at each site can be reproduced by the deterministic component of the regression model, and its random component can be described by a spatial moving average model to allow the reproduction of the observed interstation correlations as well as the observed spatial autocorrelations between the downscaled temperature time series at different locations. Furthermore, a comparison has been carried out between the proposed SD model and a DD model based on one series of simulations from the CRCM to see the performance of each model in terms of accurate description of observed temporal and spatial characteristics.

This paper is organized as follows. Section 2 describes the proposed methodology. Section 3 presents the results of the application of the suggested approach for simulating daily temperature extremes for a network of 10 weather stations located in the southwest region of Quebec and southeast region of Ontario in Canada. Finally, Section 4 provides the conclusions of the present study.

2. Multi-site multivariate SD approach

The proposed multi-site multivariate SD model is based on the combination of a deterministic linear regression component and a stochastic component based on a spatial moving average process. The deterministic component describes the linkage between global climate predictors and local temperature extremes, while the spatial moving average process is used to generate spatially autocorrelated random numbers to allow the reproduction of the observed interstation correlations and the observed spatial autocorrelations in the downscaled temperature time series.

The spatial autocorrelation is the degree of spatial dependence between observations of a single variable according to their arrangement in space. It can be computed by spatial statistics such as Moran's I (Moran, 1950; Odland, 1988; Griffith, 2003) expressed as follows:

• (1)

In which, x_i denotes the observed value at location i, x̄ is the average of the x_i over the n locations, and w_ij is the spatial weight between two locations i and j. The spatial autocorrelation concept and the spatial moving average process have been successfully used by Khalili et al. (2007, 2009) to regionalize the Richardson-type Weather Generator (WGEN) (Richardson, 1981). However, in the present paper, the spatial moving average process was used to generate spatially autocorrelated random numbers to reproduce the observed interstation correlations as well as the observed spatial autocorrelations between the downscaled temperature data at different weather stations.

The regression model for Tmax is given in the following, and a similar equation is used for Tmin as well.

• (2)

in which Tmax_{i, m, s} is the maximum temperature on day i in month m and at station s from a network of n stations; α is the jth regression parameter for month m and at station s; p is the value of the jth predictor, from a total of q monthly significant predictors, on day i in month m and at station s; ε is the residual value on day i in month m and at station s. The residuals for Tmax and Tmin are modelled using a multivariate framework of a spatial moving average process as follows:

• (3)

in which, u and u are random numbers drawn from a normal distribution with zero mean and unit standard deviation to simulate Tmax and Tmin, respectively, on day i in month m and at station s; β is the moving average coefficient; and W_m is the weight matrix in month m. In this study, the weight matrix W_m is associated with the monthly correlations of temperature between each pair of stations, and it can be computed by optimisation such that the observed monthly interstation correlations could be accurately described. This matrix can be expressed as follows:

• (4)

In which, W and W are the monthly weight matrices for Tmax and Tmin associated with the monthly correlations between the daily Tmax and Tmin time series respectively at each pair of stations. W or W can be expressed as follows:

• (5)

in which w_{s, r} is the monthly correlation between the daily Tmax or Tmin time series at two stations s and r. The weight between a given station and itself is zero by convention.

W is the weight matrix whose components w_{s, r} are the monthly correlations between the daily Tmax at a given station s and daily Tmin at another station r. W is the weight matrix whose components w_{s, r} are the monthly correlations between the daily Tmin at a given station s and daily Tmax at another station r. The diagonals of W and W are not null because they represent the correlation between daily Tmax and Tmin at a given station.

The extreme eigenvalues of the weight matrix W_m establish the range of the coefficient β, that is , where w_max is the maximum positive eigenvalue, and w_min is the largest negative eigenvalue in absolute value. Notice that the value of the spatial autocorrelation of the generated residuals depends on the selected weight matrix and the value of the moving average coefficient β. In fact, for a given weight matrix, using different values of β from its range, one can obtain residuals with different degrees of spatial autocorrelation and consequently the downscaled temperature time series will exhibit different degrees of spatial autocorrelation. In this study, an optimisation search has been done to find the optimal value of β, which can allow the reproduction of the optimal results by the multi-site SD model in terms of the interstation correlations and the spatial autocorrelations.

The multi-site multivariate SD procedure starts with computing the observed residual values Tmax_Residuals and Tmin_Residuals at each station over the calibration period. For a given station s, these residuals are derived from the difference between the observed daily data at this station and the estimated ones given by the regression equation as follows. The following equation is given for Tmax, but similar equation is applied for Tmin as well.

• (6)

in which Tmax is the residual of Tmax, on day i in month m and at station s; and Tmax is the observed daily Tmax value on day i in month m and at station s.

The objective behind the calculation of Tmax and Tmin is to reproduce their statistical properties in the stochastic components, ε and ε, of the multi-site multivariate SD model (Equation (2)). In this study, the residuals of Tmax and Tmin at each station were assumed to follow a normal distribution.

The multivariate spatial moving average process (Equation (3)) is used for computing the values of ε and ε for each day i within each month m at each station s. However, these computed values of ε and ε at a given station s should have the statistical properties of Tmax and Tmin distributions, respectively, at this station. Each set of ε and ε values are thus assumed to follow a normal distribution with the mean and standard deviation equal to those of Tmax and Tmin respectively. The stochastic components ε and ε (Equation (3)) are then added to the deterministic components (Equation (2)) to compute the simulated Tmax and Tmin values at each station. A verification procedure has been implemented to avoid Tmin to be randomly superior to Tmax. Finally, when all the multi-site multivariate SD model parameters are estimated using data from the calibration period, the performance of the resulting calibrated model is tested using data from the validation period.

3. Numerical application

3.1. Data description

In this numerical application, the accuracy and feasibility of the proposed multi-site multivariate SD approach is assessed using the 30-year daily extreme temperature records available from a network of 10 weather stations located in the southwest region of Quebec and southeast region of Ontario (Table I and Figure 1). The station data are partitioned into the calibration period 1961–1975, and the validation period 1976–1990.

Figure 1. Study area with the location of weather stations from Environment Canada, and NCEP/NCAR and CRCM grid points. The NCEP/NCAR grid points correspond to the interpolated values onto the CGCM3 global Gaussian grid (around 3.75° longitude × 3.75° latitude, see DAI CGCM3 predictors, 2008), and the CRCM grid points are on the polar stereographic grid using a horizontal resolution of 45 km. This figure is available in colour online at wileyonlinelibrary.com/journal/joc

Table I. Coordinates (latitude, longitude, and altitude) of the weather stations (i.e. daily Tmin and Tmax from Environment Canada National Archive) used in this study. The location of these stations in Quebec and Ontario provinces (Canada) is shown in Figure 1

Station name	Province	Latitude (°N)	Longitude (°W)	Altitude (m)
L'Assomption	Quebec	45.81	− 73.43	21
Oka	Quebec	45.5	− 74.07	91.4
Stanstead	Quebec	45.02	− 72.1	320
Madawaska	Ontario	45.5	− 77.98	316.4
Chenaux	Ontario	45.58	− 76.68	84.1
Cornwall	Ontario	45.02	− 74.75	64
Ottawa CDA	Ontario	45.38	− 75.72	79.2
St Alban	Quebec	46.72	− 72.08	76.2
Kemptville	Ontario	45	− 75.63	99.4
Morrisburg	Ontario	44.92	− 75.19	81.7

The atmospheric predictor variables used in this study originate from the NCEP/NCAR reanalysis data (Kalnay et al., 1996; Kistler et al., 2001). These predictors have been linearly interpolated to match the Gaussian grids of the third version of the Canadian Centre for Climate Modelling and Analysis Coupled Global Climate Model (CGCM3) (DAI CGCM3 Predictors, 2008). The value in each grid cell corresponds to the value over the centre of the cell defined over an area of 3.75° longitude and approximately 3.75° latitude. As shown in Figure 1, there is an NCEP/NCAR grid point in the middle of the target region and two other nearby. A set containing 25 daily predictors covering the period from 1961 to 2003 (Table II) is available at each of the 3 grid points.

Table II. The significant predictors selected for each month for the computation of Tmax predictand at St Alban station

25 predictor variables	Monthly selection
	January	February	March	April	May	June	July	August	September	October	November	December
Mean sea level pressure	√	√	√	√	√	√	√	√	√	√	√	√
500 hPa airflow strength
500 hPa zonal velocity
500 hPa meridional velocity								√
500 hPa vorticity				√						√	√
500 hPa wind direction
500 hPa divergence	√										√	√
850 hPa airflow strength	√		√		√	√	√	√	√		√
850 hPa zonal velocity		√										√
850 hPa meridional velocity										√
850 hPa vorticity
850 hPa wind direction				√
850 hPa divergence	√									√		√
500 hPa geopotential height
850 hPa geopotential height	√	√	√	√	√	√	√	√	√	√	√	√
Surface airflow strength			√		√	√	√		√
Surface zonal velocity									√
Surface meridional velocity
Surface vorticity		√	√
Surface wind direction
Surface divergence
Specific humidity at 500 hPa		√				√	√	√
Specific humidity at 850 hPa				√	√
Near-surface specific humidity
Mean temperature at 2m

In addition, data from the limited-area nested CRCM (Music and Caya, 2007; Caya and Laprise, 1999) were also evaluated for their accuracy as compared to the observations. The CRCM data used in this study come from the CRCM 4.1.1 version using the Canadian Land Surface Scheme (CLASS) (Verseghy, 1991). The computational domain of this model covers North America (AMNO) with 45-km horizontal resolution (on a polar stereographic grid), and 29 vertical levels. This model is driven by 6-hourly NCEP/NCAR reanalysis with the sea surface temperature and sea ice coming from the Atmospheric Model Intercomparison Project II (Fiorino, 1997 for the AMIP II version) database. This CRCM allows a suitable comparison with the proposed SD approach, because the two downscaling models use the NCEP/NCAR data as inputs.

Despite the scale mismatch between the local observed or SD point-scale information and the grid-cell values from the CRCM (smooth and uniform values over a grid-cell area), the idea behind this comparison is to analyse roughly the added values from the SD approach versus the use of direct RCM grid-cell values. Using an up-scaling approach to pass from the point-scale to the grid-cell area information, by spatial interpolation methods, will smooth the variability of the local climatic variables and introduce some errors or uncertainties related to the subjectively selected interpolation scheme. Hence, this will not allow evaluating the added values from the multi-site SD approach at the local scale and clearly assessing the performance of this SD approach in preserving the spatial and temporal climate information at the local scale. On the other hand, the interpolation of the CRCM grid-values to the station locations can entail different errors and dramatically affects the CRCM outputs. As shown in Figure 1, the CRCM grid points used for the present intercomparison study belong to the nearest grid boxes located in the vicinity of the 10 weather stations (10 green stars in Figure 1).

3.2. Predictors selection

In Equation (2), the selection of the significant atmospheric predictors is the critical factor that could affect the accuracy of the predictand estimation. Different combinations of atmospheric predictors may lead to different results (Huth, 1999; Wilby et al., 2004). In this study, the backward stepwise regression (McCuen, 2003; Hessami et al., 2008) was used to select the significant predictors for both Tmax and Tmin for each month and for each of the 10 selected stations. Backward stepwise regression starts with all candidate variables, tests every variable for statistical significance, and deletes those that are statistically insignificant. Sequences of partial F-test, with degrees of freedom 1 and n − p − 1, are used to decide the addition or elimination of these variables

• (7)

in which n is the number of observations; p is the number of predictors in Equation (2); R_p and R_p−1 are correlation coefficients between the criterion variable and a prediction equation using p and p − 1 variables, respectively. For a given predictor, when F is greater than a critical value defined for a given level of significance, this predictor is retained. The procedure stops adding or dropping when no predictor can be retained or removed.

Furthermore, in this study, the selection of the best combination of predictor variables was carried out on a monthly basis to take into account the monthly variations in the predictor-predictand relationships rather than on an annual basis, for which the combination of predictors is constant through the year, as suggested by some previous studies (Wilby et al., 2002; Hessami et al., 2008). Table II shows an example of the selection of the five most significant predictors identified by the backward stepwise regression for the computation of Tmax predictand at St Alban station. The use of only five most significant predictors derives from need to describe adequately the local climate while achieving the parsimony of the downscaling model as well as to prevent the possible excessive colinearity between predictors when too many variables are selected (Gachon et al., 2005; Dibike et al., 2008; Hessami et al., 2008). Notice that the two predictors, near-surface specific humidity, and mean temperature at 2 m, have been omitted in this study to avoid the issue that the observed temperature data were assimilated within these two NCEP/NCAR predictors. As shown in Table II, the mean sea level pressure and the 850 hPa geopotential height are always identified as significant predictors. However, the optimal combination of the five most significant predictors for each month was found different from one month to another, except for June and July when selected predictors are similar. This difference could indicate the need for considering the different combination of predictors for each month rather than a constant combination for the whole year as mentioned previously.

3.3. Results

To assess the accuracy of the proposed multi-site multivariate SD approach, both numerical and graphical comparisons between observed and simulated results were considered. More specifically, for the numerical comparison, the coefficient of determination (R²), the Mean Absolute Error (MAE), and the Root Mean Square Error (RMSE) were used

• (8)

• (9)

• (10)

in which t is the length of the time series; X_{Obs, i, s} and X_{Sim, i, s} are, respectively, the observed and simulated values for day i and station s. X̄_Obs is the mean of the observed data.

Table III shows the average values of R², MAE, and RMSE based on 100 simulations of daily Tmax and Tmin time series for all stations and for both calibration and validation periods. In general, the high values of R² (larger than 0.9976) and the low values of MAE (less than 0.54 °C) and RMSE (less than 0.66 °C) indicate the very good accuracy of the proposed multi-site multivariate SD procedure.

Table III. Coefficients of determination (R²), MAE, and RMSE between the monthly means of the observed and multi-site simulated daily Tmax and Tmin for both the calibration and validation periods at all stations

Stations	Tmax_Calibration			Tmax_Validation			Tmin_Calibration			Tmin_Validation
	R^2	MAE ( °C)	RMSE ( °C)	R^2	MAE ( °C)	RMSE ( °C)	R^2	MAE ( °C)	RMSE ( °C)	R^2	MAE ( °C)	RMSE ( °C)
L'Assomption	1	0	0.0001	0.9993	0.3343	0.4064	1	0.0564	0.0669	0.9988	0.4448	0.5944
Oka	1	0.0145	0.028	0.9994	0.2642	0.3045	1	0.0483	0.0553	0.9991	0.3384	0.4055
Stanstead	1	0.0023	0.008	0.9995	0.4049	0.4667	1	0.072	0.081	0.9997	0.4079	0.4453
Madawaska	1	0.016	0.0303	0.9992	0.3063	0.3625	1	0.0683	0.0942	0.9988	0.4211	0.5391
Chenaux	1	0.0035	0.0081	0.9992	0.4362	0.5197	1	0.0781	0.0999	0.9976	0.4374	0.6596
Cornwall	1	0.0214	0.0452	0.9989	0.4643	0.5779	1	0.0479	0.0558	0.9991	0.5417	0.6671
Ottawa CDA	1	0	0	0.9996	0.3021	0.399	1	0.0516	0.0675	0.9991	0.4265	0.5618
St Alban	1	0.0018	0.0053	0.9996	0.3139	0.3573	1	0.0564	0.069	0.9977	0.4537	0.5995
Kemptville	1	0.0014	0.004	0.9993	0.3565	0.4078	1	0.0444	0.0547	0.9991	0.2186	0.3402
Morrisburg	1	0	0	0.9985	0.4455	0.5113	1	0.056	0.0652	0.9993	0.1815	0.3194

In terms of accuracy in reproducing the observed temporal dependence of daily Tmax and Tmin series at each station, Table IV shows, as typical example, the comparison between the observed and computed serial autocorrelations of lags 1–3 for Tmax and Tmin at Oka station. The proposed multi-site multivariate SD method was able to capture very well the temporal dependence of the daily Tmax and Tmin time series at this location as indicated by the comparable results between the observed and computed serial autocorrelation values for both calibration and validation periods. Similar results were obtained for the other stations.

Table IV. Observed and multi-site simulated daily Tmax and Tmin serial autocorrelation of lags 1–3 over both the calibration and validation periods at station Oka

Serial Autocorrelations	Station Oka
	Tmax		Tmin
	Calibration	Validation	Calibration	Validation
Observed_lag1	0.96	0.96	0.91	0.9
Simulated_lag1	0.94	0.94	0.85	0.85
Observed_lag2	0.93	0.93	0.85	0.83
Simulated_lag2	0.92	0.92	0.81	0.81
Observed_lag3	0.92	0.92	0.82	0.81
Simulated_lag3	0.91	0.91	0.8	0.8

Regarding the spatial dependence, Figures 2 and 3 show, respectively, the comparisons between the observed and simulated daily Tmax and Tmin interstation correlations in each month for the calibration and validation periods. Each graph represents the correlations of the 45 pairs of stations in each month. In general, a very good agreement between the observed and simulated interstation correlations was found as indicated by the high R² values for the calibration (0.8707 for Tmax and 0.8546 for Tmin) and validation (0.7736 for Tmax and 0.7765 for Tmin). The monthly intervariable correlations between Tmin and Tmax at each station and at each pair of stations are presented in Figure 4. A very good agreement was found between the observed and simulated intervariable correlation results as indicated by the high R² values for both calibration (R² = 0.8629) and validation (R² = 0.8367) steps.

Figure 2. Observed versus simulated daily Tmax interstation correlations in each month for (a) calibration period: 1961–1975, and (b) for validation period: 1976–1990. This figure is available in colour online at wileyonlinelibrary.com/journal/joc

Figure 3. Same as Figure 2 but for the Tmin values. This figure is available in colour online at wileyonlinelibrary.com/journal/joc

Figure 4. Monthly intervariable correlations between Tmin and Tmax at each station and at each pair of stations for (a) the calibration period: 1961–1975, and (b) the validation period: 1976–1990. This figure is available in colour online at wileyonlinelibrary.com/journal/joc

Moreover, a comparison has been made between the observed and simulated daily spatial autocorrelations of Tmax and Tmin computed by Moran's I (Equation (1)). For purposes of illustration, Figure 5 presents the results obtained for Tmin with 365 points are plotted in each graph. The observed daily spatial autocorrelations are averaged over the 15 years of each of the calibration and validation period, and the simulated ones are averaged over the 100 simulations. It can be seen that a very good agreement was obtained between the observed and simulated spatial autocorrelations as indicated by the high R² values (0.8521 for the calibration and 0.8037 for the validation). Comparable results were found for Tmax (not shown) as well with similar high R² values (0.877 for the calibration and 0.8525 for the validation).

Figure 5. Observed versus simulated daily Tmin spatial autocorrelations for (a) the calibration period: 1961–1975, and (b) the validation period: 1976–1990. This figure is available in colour online at wileyonlinelibrary.com/journal/joc

The performance of the proposed multi-site multivariate SD procedure was also assessed based on common temperature indices for each month such as the means and standard deviations (STD) of Tmax and Tmin, the 90th percentile of daily Tmax (Tmax90p), and the 10th percentile of daily Tmin (Tmin10p). Figures 6–11 display the boxplots of these indices computed from the observed data, from the proposed multi-site multivariate SD model, and from the CRCM for both calibration and validation periods. In general, the multi-site multivariate SD model outperformed the CRCM in terms of the accuracy in reproducing these observed statistical properties. More specifically, for the mean values of Tmax and Tmin (Figures 6 and 8), the suggested model gave a very good agreement with the observations for both calibration and validation periods. For the case of the standard deviations of Tmax and Tmin (Figures 7 and 9), a slight overestimation of the median values of the Tmax STD was noted for summer (Jun, Jul, Aug) and September, while for the median values of the Tmin STD an underestimation was observed for winter (Dec, Jan, Feb) and March, and an overestimation for April, May, summer and autumn (Sep, Oct, Nov). Larger discrepancies of the CRCM with respect to the observations were found for the means and STDs of both Tmax and Tmin for almost every month. The bias appears either as an overestimation or an underestimation of the median value and its variability. Furthermore, regarding the results of the Tmax90p and Tmin10p indices (Figures 10 and 11), a slight overestimation of Tmax90p was observed with the multi-site multivariate SD model for summer, an overestimation of Tmin10p for winter and March, and an underestimation of Tmin10p for May, April, and summer for both calibration and validation periods. However, the multi-site multivariate SD model shows better skill as compared to the CRCM. In other words, the CRCM tends to underestimate or overestimate the median and variability of cold and warm extremes of Tmin and Tmax.

Figure 6. Box plots of the mean of Tmax for the observed data, the multi-site model, and the CRCM for (a) the calibration period (1961–1975), and (b) the validation period (1976–1990). The boxes correspond to the interquartile range (IQR), the band in the middle of each box to the median value, and the whiskers to the 1.5 × IQR. Outliers are represented by the crosses. Each box plot represents the daily time series obtained at all stations (observation and multi-site model values) or at all used grid points (CRCM values) aggregated over each period of 15 years. This figure is available in colour online at wileyonlinelibrary.com/journal/joc

Figure 7. Same as Figure 6 but for the standard deviation (STD) of Tmax. This figure is available in colour online at wileyonlinelibrary.com/journal/joc

Figure 8. Same as Figure 6 but for the mean of Tmin. This figure is available in colour online at wileyonlinelibrary.com/journal/joc

Figure 9. Same as Figure 6 but for the standard deviation (STD) of Tmin. This figure is available in colour online at wileyonlinelibrary.com/journal/joc

Figure 10. Same as Figure 6 but for the Tmax90p. This figure is available in colour online at wileyonlinelibrary.com/journal/joc

Figure 11. Same as Figure 6 but for the Tmin10p. This figure is available in colour online at wileyonlinelibrary.com/journal/joc

Finally, it was noted that outliers appeared more frequently in the results of the proposed multi-site multivariate SD method. This behaviour could be attributed to the only use of the coarse-scale NCEP/NCAR predictors in this procedure, without taking explicitly into account the regional-scale variables such as surface conditions or diabatic fluxes from the surface needed to capture all range of variability related to the occurrence of temperature extremes. These can come from nonlinear processes and feedbacks linked with frost and thaw conditions of the soil, and/or presence or absence of snow on the ground as indicated by Gachon and Dibike (2007) for northern Canada. As also noted in Hessami et al. (2008), for the simulation of Tmax90p over eastern Quebec in Canada, outliers appear more often for the case of NCEP/NCAR-driven conditions than for the GCM ones.

Tables V–VII show the RMSEs of the standardized seasonal mean and STD of Tmin, and the standardized seasonal Tmin10p, respectively. The RMSE values were computed from the observations and the multi-site multivariate simulations, and from the observations and the CRCM data for both calibration and validation periods for all stations. The seasonal standardising process was carried out by subtracting the climatological mean and dividing by the climatological standard deviation of a given variable (the mean or STD of Tmin, or the Tmin10p) in a given season at a given station, over the 15-year calibration or 15-year validation period. It can be clearly seen that the RMSEs for many stations in the study region are generally smaller for the multi-site multivariate SD than for the CRCM. In particular, over the entire region, the values of the mean RMSE for all seasons were consistently smaller for the multi-site multivariate SD than for the CRCM. Hence, it can be concluded that the proposed multi-site multivariate SD procedure can reproduce more accurately the statistical properties of Tmin than the CRCM. Similar results were found for the RMSEs of the standardized seasonal mean and STD of Tmax, and the standardized seasonal Tmax90p index (not shown). The multi-site multivariate SD model has outperformed the CRCM as well in its accurate description of the statistical properties of Tmin.

Table V. RMSEs of the standardized seasonal Tmin mean computed from the observed data and the multi-site simulations (Obs_Sim), and from the observed data and the CRCM (Obs_CRCM) for both calibration and validation periods

Stations	Standardized Tmin Mean RMSE_Calibration period
	Winter		Spring		Summer		Autumn
	Obs_Sim	Obs_CRCM	Obs_Sim	Obs_CRCM	Obs_Sim	Obs_CRCM	Obs_Sim	Obs_CRCM
L'Assomption	0.50	0.90	0.53	0.73	0.35	0.64	0.52	0.65
Oka	0.46	0.72	0.77	0.77	0.92	1.20	0.54	0.79
Stanstead	0.44	0.77	0.39	0.51	0.51	0.72	0.45	0.62
Madawaska	0.35	0.70	0.66	0.80	0.59	0.86	0.67	0.64
Chenaux	0.33	0.79	0.48	0.72	0.55	0.59	0.41	0.51
Cornwall	0.36	0.74	0.48	0.65	0.48	0.55	0.40	0.56
Ottawa CDA	0.30	0.75	0.51	0.70	0.46	0.58	0.48	0.57
St Alban	0.32	0.86	0.37	0.75	0.49	0.59	0.49	0.66
Kemptville	0.40	0.59	0.57	0.61	0.54	0.88	0.45	0.64
Morrisburg	0.47	0.80	0.60	0.77	0.64	0.72	0.46	0.52
RMSE Mean	0.39	0.76	0.54	0.70	0.55	0.73	0.49	0.62
	Standardized Tmin Mean RMSE_Validation period
L'Assomption	0.53	0.57	0.49	0.79	0.39	0.59	0.66	1.13
Oka	0.60	0.68	0.43	0.97	0.55	0.45	0.37	1.04
Stanstead	0.45	0.64	0.59	0.81	0.88	0.89	0.53	1.07
Madawaska	0.40	0.57	0.41	0.72	0.62	0.65	0.91	0.75
Chenaux	0.44	0.60	0.33	0.90	0.38	0.49	0.62	0.70
Cornwall	0.46	0.70	0.37	1.00	0.49	0.46	0.43	0.96
Ottawa CDA	0.40	0.68	0.37	1.00	0.30	0.46	0.42	0.86
St Alban	0.46	0.75	0.49	0.86	0.51	0.68	0.63	1.18
Kemptville	0.53	0.66	0.49	0.88	0.62	0.60	0.57	0.96
Morrisburg	0.66	0.62	0.56	0.89	0.74	0.81	0.49	0.95
RMSE Mean	0.49	0.65	0.45	0.88	0.55	0.61	0.56	0.96

Table VI. Same as Table V but for the standardized seasonal Tmin STD

Stations	Standardized Tmin STD RMSE_Calibration period
	Winter		Spring		Summer		Autumn
	Obs_Sim	Obs_CRCM	Obs_Sim	Obs_CRCM	Obs_Sim	Obs_CRCM	Obs_Sim	Obs_CRCM
L'Assomption	0.77	0.85	0.75	0.80	0.93	0.78	0.43	1.08
Oka	0.93	1.08	0.37	0.60	0.85	1.18	0.54	1.01
Stanstead	0.67	0.97	0.60	0.69	0.55	0.84	0.34	0.68
Madawaska	0.67	1.06	0.48	0.56	0.58	1.11	0.40	0.93
Chenaux	0.86	1.05	0.43	0.65	0.89	1.17	0.49	0.95
Cornwall	0.63	0.95	0.49	0.74	0.38	0.71	0.33	0.94
Ottawa CDA	0.64	0.91	0.57	0.71	0.54	1.00	0.27	0.95
St Alban	0.55	0.90	0.80	0.93	0.78	0.99	0.57	1.06
Kemptville	0.81	1.05	0.52	0.66	0.44	0.98	0.41	1.06
Morrisburg	0.80	1.04	0.63	0.78	0.61	1.01	0.48	1.09
RMSE Mean	0.73	0.98	0.56	0.71	0.66	0.98	0.42	0.97
	Standardized Tmin STD RMSE_Validation period
L'Assomption	0.77	0.77	0.53	0.79	0.68	0.86	0.68	0.98
Oka	0.55	0.63	0.53	0.75	0.48	0.84	0.68	0.91
Stanstead	0.50	0.67	0.49	1.01	0.55	0.85	0.52	0.83
Madawaska	0.55	0.59	0.32	0.93	0.46	0.95	0.80	0.86
Chenaux	0.99	0.99	0.39	0.95	0.62	0.79	0.74	0.89
Cornwall	0.69	0.87	0.37	0.78	0.46	0.91	0.59	0.83
Ottawa CDA	0.46	0.69	0.46	0.87	0.61	0.90	0.64	0.95
St Alban	0.51	0.63	0.38	0.77	0.62	1.01	0.51	0.81
Kemptville	0.59	0.75	0.43	0.86	0.55	0.85	0.77	1.09
Morrisburg	0.67	0.71	0.48	1.03	0.84	0.89	0.72	1.01
RMSE Mean	0.63	0.73	0.44	0.87	0.59	0.89	0.66	0.92

Table VII. Same as Table V but for the standardized seasonal Tmin10p

Stations	Standardized Tmin10p RMSE_Calibration period
	Winter		Spring		Summer		Autumn
	Obs_Sim	Obs_CRCM	Obs_Sim	Obs_CRCM	Obs_Sim	Obs_CRCM	Obs_Sim	Obs_CRCM
L'Assomption	0.63	0.79	0.72	1.01	0.81	0.80	0.55	1.13
Oka	0.69	1.00	0.56	0.79	1.10	1.22	0.74	1.11
Stanstead	0.55	0.91	0.60	0.67	0.58	0.81	0.57	0.95
Madawaska	0.69	0.82	0.68	0.90	0.77	0.89	0.71	1.10
Chenaux	0.60	0.91	0.57	0.89	0.64	0.96	0.47	1.20
Cornwall	0.60	0.95	0.55	0.84	0.55	0.68	0.62	1.05
Ottawa CDA	0.60	0.89	0.55	0.82	0.59	0.86	0.58	1.12
St Alban	0.51	1.02	0.79	0.87	0.69	0.93	0.59	1.11
Kemptville	0.58	0.86	0.53	0.86	0.46	0.96	0.59	1.20
Morrisburg	0.76	1.05	0.77	1.05	0.68	0.87	0.49	1.05
RMSE Mean	0.62	0.92	0.63	0.87	0.69	0.90	0.59	1.10
	Standardized Tmin10p RMSE_Validation period
L'Assomption	0.79	0.73	0.58	0.92	0.84	1.03	0.43	0.80
Oka	0.79	0.70	0.55	0.99	0.76	1.03	0.44	0.90
Stanstead	0.62	0.99	0.51	1.05	1.13	0.91	0.53	0.91
Madawaska	0.52	0.76	0.51	0.81	0.96	1.03	0.86	0.82
Chenaux	0.84	0.81	0.36	0.97	0.86	0.94	0.80	0.76
Cornwall	0.70	0.82	0.38	1.00	0.79	0.90	0.51	0.77
Ottawa CDA	0.63	0.82	0.51	0.97	0.67	0.91	0.45	0.71
St Alban	0.77	1.12	0.46	0.83	0.77	1.18	0.51	0.97
Kemptville	0.79	0.53	0.45	1.10	0.98	0.98	0.65	1.00
Morrisburg	0.89	0.66	0.56	1.10	1.09	1.13	0.74	0.81
RMSE Mean	0.73	0.79	0.49	0.98	0.89	1.00	0.59	0.85

For purposes of illustration, Figures 12, 13, and 14 show the interannual anomalies of the standardized seasonal mean and STD of Tmax, and the standardized seasonal Tmax90p index, respectively, for both calibration and validation periods. The standardising process is the same as that for the spatial variation, but in this case, these figures show, for each of the 15 years presented in the x-axis, the mean of the standardized variables computed over all 10 stations. It can be clearly seen that the multi-site multivariate SD method can accurately reproduce the temporal variability of these Tmax statistical properties over the entire 15-year period of both calibration and validation. No significant differences are noted, except for the summer of the 13th year of the calibration period (Figure 14(a)), which shows an overestimation of the standardized Tmax90p index, and for the autumn of the 8th year of the validation period, which shows an underestimation of the standardized Tmax STD (Figure 13(b)). The CRCM-simulated values were able to describe the general trend of these interannual anomalies, but displayed a higher discrepancy with respect to the observed values than the multi-site multivariate SD model. Similar results were found for the interannual anomalies of the mean and STD of Tmin, and the Tmin10p index as well (not shown).

Figure 12. Inter-annual anomalies of standardized seasonal Tmax mean over (a) the calibration period (1961–1975), and (b) the validation period (1976–1990). This figure is available in colour online at wileyonlinelibrary.com/journal/joc

Figure 13. Inter-annual anomalies of standardized seasonal Tmax STD over (a) the calibration period (1961–1975), and (b) the validation period (1976–1990). This figure is available in colour online at wileyonlinelibrary.com/journal/joc

Figure 14. Inter-annual anomalies of standardized seasonal Tmax90p over (a) the calibration period (1961–1975), and (b) the validation period (1976–1990). This figure is available in colour online at wileyonlinelibrary.com/journal/joc

Finally, in addition to the evaluation of the performance of the proposed multi-site multivariate SD model based on a number of common temperature indices that are important for various climate-related impact assessment studies, the present paper includes also an assessment of the summer heat spell index. Many studies have been carried out on this extreme phenomenon especially after the disastrous European summer heat wave of 2003 (WHO, 2003; Beniston and Diaz, 2004; Beniston and Stephenson, 2004; Schär et al., 2004; Gachon et al., 2005; Khaliq et al., 2005, 2006, 2007). In particular, as mentioned by Khaliq et al., 2007, the assessment of the heat spells depends upon the type and length of the available data, time of the year, and the sector impacted by the heat spells. The present analysis focuses on the extreme summer temperatures, particularly affecting public health in the study area. Heat wave duration index (HWDI3days) has therefore been calculated for the summer of both the calibration and validation periods. The HWDI3days index computes the total of days in sequences superior to three days, where Tmax is superior to the calendar day mean, calculated on a five-day window centred on each calendar day during the calibration or the validation period, supplemented by 3 °C (Gachon et al., 2005; Drouin et al., 2005). Figure 15 shows the interannual anomalies of the standardized HWDI3days for both the calibration and validation periods. The mean of the standardized HWDI3days is computed over all 10 stations and presented for each of the respective 15-year period. These results confirm that the proposed SD model is able to reproduce adequately the occurrence and temporal variability of the observed heat waves over both calibration and validation periods, except for the 13th year of the calibration period in which an overestimation of the standardized HWDI3days was found due to the overestimation of Tmax90p index for the summer of this year, as noted previously (Figure 14(a)). The CRCM-simulated values were able to reproduce the interannual anomalies, but with less skill than the SD results, especially for the calibration period.

Figure 15. Inter-annual anomalies of standardized HWDI3days over (a) the calibration period (1961–1975), and (b) the validation period (1976–1990). This figure is available in colour online at wileyonlinelibrary.com/journal/joc

4. Conclusions

The present study has proposed a multi-site multivariate statistical approach to downscaling of daily extreme temperature series at many sites concurrently. The proposed procedure was based on a combination of a regression model to describe the linkage between global climate predictors and local temperature extremes, and a stochastic component based on a spatial moving average process to reproduce the observed interstation correlations and spatial autocorrelations of the daily temperature extreme time series at different sites. The novel feature of this hybrid method lies in the use of the spatial moving average process to model the stochastic part of the regression model. This spatial moving average framework was found more suitable for the simulation of temperature time series for a large number of sites without complex computational requirements as compared to the multivariate autoregressive framework that has been used in most previous investigations. Furthermore, the backward stepwise regression was successfully used to identify, for both Tmax and Tmin, the significant climate predictors from the available NCEP/NCAR reanalysis dataset.

Results of an illustrative application using available climate data in the southwest region of Quebec and the southeast region of Ontario in Canada have indicated the feasibility and accuracy of the proposed multi-site multivariate SD procedure. More specifically, it has been demonstrated that the suggested method was able to accurately reproduce various statistical properties of the Tmax and Tmin time series at a local site as well as the spatial consistency over the entire study region, including their temporal and spatial correlations.

In addition, comparison of the results from the proposed multi-site multivariate SD method, and the outputs from the CRCM (a DD procedure), has indicated that the multi-site multivariate SD approach was able to more accurately describe the observed spatial-temporal characteristics of extreme temperatures at many sites in the study region, including the interannual anomalies.

In summary, the proposed SD approach has been shown to be able to solve the challenging statistical simulation problem related to the modelling of multiple temperature extreme time series in the framework of climate change; that is, its ability to accurately describe both the at-site temporal persistence and the spatial dependence of temperature extreme processes as well as the linkage between large-scale climate predictors and local site temperature characteristics. Finally, due to its simple computation and implementation processes the proposed SD method could be considered as an efficient and practical tool for the downscaling of multi-site temperature extreme series for various climate-related impact assessment studies.

The proposed multi-site multivariate SD method could be extended to include other meteorological variables due to the general flexibility of the spatial moving average model. More specifically, the spatial moving average framework could be extended to include additional variables without increasing the computational requirement for a larger weight matrix. Further research is currently underway to develop a statistical approach to multi-site downscaling of daily precipitation processes at different locations concurrently.

Acknowledgements

The authors would like to acknowledge financial support from the Natural Sciences and Engineering Research Council of Canada (Special Research Opportunity Program) for the project entitled ‘Probabilistic assessment of regional changes in climate variability and extremes’. Also, the authors acknowledge the ‘Fond Québécois de Recherche sur la Nature et les Technologies’ for its funding of this research, and the Data Access Integration (DAI, see http://loki.qc.ec.gc.ca/DAI/) team for providing the data. The DAI data download gateway is made possible through collaboration among the Global Environmental and Climate Change Centre (GEC3), the Adaptation and Impacts Research Division (AIRD) of Environment Canada, and the Drought Research Initiative (DRI). The authors also acknowledge Ouranos consortium for the access of the CRCM time series data which have been generated and supplied by Ouranos' Climate Simulation Team.