SDSM downscaling technique

There is a systematic complication with the daily ensembles of the ECHO-G model which provides daily predictors based on a 360-day calendar, while the climate simulations in this study are based on a standard calendar system (Gregorian calendar). Therefore the time-dimension slicing of GCM-predictor is executed in the daily-predictor simulations by using a linear interpolation, where the predictor at time , which is the unknown value between time and that give predictors at upper and lower values and , respectively, is calculated as :

This approach is likewise used in the interpolation of the observation, when their time scale does not fit the predictor time scale.

Downscaling using conventional statistical tools

where are the estimated regression coefficients and ̂ are the normalized predictors (Chen et al. 2010b) defined by:

where is the original variable at time , ̅ is the mean of the variable for the simulation period, and is the standard deviation of . In the next step a uniformly distributed random number is generated. A precipitation is taken on the wet day when ever . The amount of precipitation on that day can be conveyed by the cumulative distribution function described as:

where is the z score at day , is the regression coefficient for each month estimated by model optimization, and is an error term in the normal distribution. The daily precipitation amount is then computed by:

where is the normal cumulative distribution function and is the empirical distribution function of the daily precipitation amount . The occurrence and amount of precipitation are calculated by the same GCM-predictor. With the set of random numbers , the occurrence and amount of rainfall is subsequently generated for the wet days, resulting finally in a daily precipitation time-series for one realization. The downscaling of temperature is performed using the same approach; however, the occurrence formulation in Eq. (3.4) is based on the outcome of the rainfall simulation, i.e. whether a day is wet or dry.

There are two types of climate parameters used in SDSM, unconditional and conditional variables (Wilby et al. 2002). Unconditional predictands, like temperature and wind speed, are those that are computed directly over the regression transfer model with the regional-scale predictors. A conditional variable, like precipitation, is a dependable parameter which hinges on an intermediate condition such as the probability of wet or dry days, as pointed out above.

Consequently, the temperatures and the precipitation are modeled here with unconditional and conditional-variable predictions, respectively.

Although the standard SDSM-downscaling is usually applied to NCEP –reanalysis predictors, the method can also be applied and calibrated with other (observed) climate data (e.g. Lavers et al.

2007, Lavers et al. 2008, Koukidis and Berg 2009). Raje and Mujumdar (2011) evoked that the downscaling model should be fitted directly to the 20^th-century control run of a GCM (20c3m) and using the regression-relations obtained in this way for projecting the future climate under various SRES. Since the 20c3m- and the future SRES- database from the CMIP3-archive also differ from the NCEP- reanalysis database, the climate projection based on the SRES-projections of the CMIP3-database requires the direct fitting with 20c3m data in order to optimally utilize all SRES variables. Accordingly, the SDSM- default NCEP- dataset is replaced

∑ ̂

(3.4)

̂ ̅

(3.5)

∑ ̂

(3.6)

(3.7)

by the various ocean-atmospheric predictor variables (discussed in the previous section) of the 20c3m-datasets in the CMIP3-archive.

The various standard procedures implemented in the SDSM-4.2 software to downscale a climate variable at a particular station are shown in Figure 3.5. Following the enumeration of the SDSM manual (Wilby and Dawson 2007), the process of statistical downscaling of a daily weather series can essentially be carried out following the four subsequent steps:

Figure 3.5. Steps of the SDSM- climate scenario generation process (Wilby et al. 2002)

1) Selection and screening of predictor variables: In the first step of SDSM modeling, the atmospheric predictors are analyzed and selected. Before the SDSM downscaling is applied to the CMIP3-dataset, the validation of the GCM and the screening of the numerous predictor variables is accomplished with MAGIC/SCENGEN and the subsequent cross-correlation analysis, as discussed in the previous section.

Actually, this selection and screening process is divided into two sub-steps; firstly, the primary selection of the best GCM-predictors by cross-correlation analysis and, secondly, their exact screening (including the choice of the atmospheric layer) by forward stepwise regression.

The average correlation power, i.e. correlation coefficients, of the numerous climate predictors of the ECHO-G GCM with the three climate variables from all observing stations in the study region are shown in Figure 3.6. The magnitudes of the correlation demonstrate that the local Tmin is mostly related to the wind vector (ua and va), Tmax to air temperature (ta) and precipitation to humidity (hu).

Figure 3.6. Average correlation power of ECHO-G climate predictors with the local climate in the study area.

Based on the ranks of these correlation scores, the top 10 correlative predictors (maximum number of predictors that can be used in SDSM) of the GCM, as listed in Table 3.12 are selected for the second screening sub-step, which is forward stepwise multi-linear regression, to find the best combination of predictors, which can improve the regression model significantly (based on a significance level α of 0.05) (Wetterhall et al. 2006). As shown also in Table 3.12, the effective number of predictors found in this way for temperature and precipitation is reduced to about 8 and 6 predictors, respectively. More specifically, for the temperature prediction the humidity (hu) and air temperature (ta) are the best predictors, whereas heat flux (hf) and humidity (hu) are best for the prediction of precipitation.

Table 3.12. Initial selection of GCM-predictors based on correlation rank and their subsequent screening by SDSM’s statistical-significant screening process.

initial selection

(by cross correlation) screening in SDSM

(by forward stepwise regression)

temperature precipitation temperature precipitation

Uas Hus hus.60000 Hfss

Ua Hfss ta.92500 hus.100000

Va Hfls tas.2 hus.70000

Vas Tas tasmin.2 hus.92500

Rlds Tasmax ua.20000 tas.1

Hus Rlus va.100000 tasmax.1

Hfls Pr va.92500

Psl Uas Vas

Hfss Ua

Rlut Rlut

2) Model calibration: In the second step, the available climate predictors are fitted with multi-regression to formulate the final climate predicting model at a single site. Thus the normally used NCEP/NCAR reanalysis data is replaced by the calibrated 20c3m (1971-2000) GCM – (ECHO-G) predictors from the CMIP3 database. This predictors set is used to generate the occurrence of rainfall, amount of rainfall and the temperatures. The precipitation is computed conditioned on the occurrence of a wet day, with the effective amount determined by an exponential regression model (Kilsby et al. 1998), whereby observed rainfall is regressed on various observed atmospheric predictor variables (altitude, pressure, flow, vorticity). The parameters of the regression model are estimated through the efficient dual simplex algorithm of Narula and Wellington (1977).

3) Synthesis of observed data: In the third step more realistic weather data - more in line with