Downscaling using the MLR- model 3.5
6) Climate projection: The last step of the MLR downscaling procedure is the climate projection over the 21 st –century, using climate predictors from the multi-domain
GCM-ensemble of step 2, and following the three SRES scenarios A1B, A2 and B1, in the 20c3m calibrated MLR-model (step two). The results of this analysis are presented in Section 3.6.4.
In this study, the experiments using the MLR downscaling approach are separated into two parts, i.e., single-domain and multi-domain downscaling, as listed in Table 3.14. The single domain downscaling model employs ECHO-G domain to predict daily climate-series. The MLR-daily-daily is used to downscale MLR-daily-daily GCM to MLR-daily-daily climate time-series. The MLR-MLR-daily-daily-monthly uses the same domain but sums up the daily values for monthly aggregates. Additionally, the MLR-monthly is used to downscale MLR-monthly GCM-predictor data into MLR-monthly climate series. These three experiments are used to demonstrate the performance of the single-domain MLR model, in comparison with the SDSM and LARS-WG models that are likewise applied with ECHO-G.
Another climate database that is employed in the MLR experiment is a High-Resolution GCM (HiRes GCM) which provides finer gird information for downscaling. The HiRes GCM is applied in the investigations of multi-domain downscaling by comparing with it with other mixes of GCM-domains in the multi-domain downscaling experiments that employ different combinations of CMIP3 GCMs (ECHO-G, BCCR, ECHAM5, GISS and PCM) and Hi-Res GCM, as listed in the last column of Table 3.14.
3.5.2 Multi-domain GCM ensembles
The studies of climate change impacts on the hydrology at the regional scale such as a watershed basin increases the demand for high-resolution future predictors from global climate models to operate the hydrological model (Fowler et al. 2007, Vliet et al. 2011). Unfortunately, the available GCMs usually provide only a coarse resolution of future climate projections. Moreover, the uncertainty in the projected climate using a single GCM or even an embedded RCM is not yet
Table 3.14. Single- and multi-domain- downscaling experiments with the multiple linear regression (MLR) model to downscale local climate in the study area.
variable experiments name input resolution output
resolution GCM model 1.single domain
downscaling
MLR-daily-daily daily 2.5°x2.5° daily ECHO-G MLR-daily-monthly daily 2.5°x2.5° monthly ECHO-G MLR-monthly monthly 2.5°x2.5° monthly ECHO-G
2.multi-domain downscaling
HiRes MLR monthly 0.5°x0.5° monthly HiRes GCM multi-domain MLR monthly 2.5°x2.5° monthly
CMIP3 (ECHO-G, BCCR, ECHAM5, GISS,PCM) multi-domain
+HiRes MLR
monthly 2.5°x2.5°+
0.5°x0.5° monthly
CMIP3 (ECHO-G, BCCR, ECHAM5, GISS,PCM) + HiRes GCM
resolved (Vliet et al. 2011) and using only an individually-running, daily GCM is not warranted to fulfill the requirements of a reliable hydrological simulation (Salathé 2005). Zhu et al. (2008) and Fordham et al. (2012) indicate that climate predictions using statistical downscaling of multi-model ensemble using a combination of the predictors from various climate multi-models can partly overcome the weakness of an individual climate model and strengthens the prediction results.
Therefore, the use of multi-model technique, which combines different variables of GCMs (Tebaldi and Knutti 2007), is proposed here to enhance the projection of the future climate state.
Use of multi-domain and high-resolution GCMs 3.5.2.1
As introduced in Section 3.3.3, multi-model ensembles are generated under the CMIP3 project (Meehl et al. 2007) which share a common time-scale and supply predictors of 20 individual GCMs under the mutual SRES greenhouse gas emission scenarios (Nakicenovic and Swart 2000) of the IPCC AR4 report (Solomon et al. 2007). Another experiment with a multi-ensemble model is the perturbed physics ensemble (PPE) (Murphy et al. 2004, Stainforth et al. 2005), where the climate predictors are produced from multi distributions of the model parameters and which are considerably larger than the number of CMIP3- multi-model ensembles (Murphy et al.
2004). However, because of the huge number of adjustable climate parameter, there are still considerable uncertainties in the climate predictions (Solomon et al. 2007). Therefore, in this study only the multi-domain GCM-output from the CMIP3 database is selected.
Downscaling of GCM- predictors should carefully consider the model domain, size and the resolution (Giorgi and Mearns 1991, 1999). The resolution of the GCM is also another factor to consider when performing regional climate studies (Giorgi and Mearns 1991, 1999, Mearns et al.
2003). Accordingly, high-resolution, 0.5o x 0.5o grid climate models have been developed (e.g.
Saarikko and Carter 1996, New et al. 1999, New et al. 2000, Mearns et al. 2003). The advantage of using such high-resolution models in the present study is also indicated by Table 3.15, which shows the correlation coefficients between the predictors from both the coarse multi-domain GCMs and the high-resolution GCM and the local climate variables in the study are. One can thus notice that for most climate variables the correlation coefficients of the high-resolution GCM are higher than those of the simple coarse multi-domain GCMs.
However, using only a high-resolution GCM for the prediction of climate probably provides relative poor performance, when compared with the use of a regression downscaling (transfer model) technique (Kidson and Thompson 1998). Hence, to improve the prediction confidence,
the use of a multi-domain GCM is suggested (Mearns et al. 2003). Therefore, in this study both multi-domain and high-resolution GCMs are employed to improve the climate projections.
Based on the idea that every predictor in a multi-model ensemble probably provides a potential predictor of future climate (Semenov and Stratonovitch 2010), the predictor-sets from the CMIP3- multi-domain- and from the High-Resolution GCM are, consequently, treated equally by screening all the predictors under the same criteria, i.e. by not weighting them. Accordingly, a total set of 1340 multi-domain predictors belonging to five coarse-grid GCMs, with 256 GCM-variables each, and another set of five predictors from the high resolution, HiRes GCM (see detail in Section 3.3.1) are employed in the predictor-screening process.
Table 3.15. Average correlation between monthly CMIP3 multi-domain GCM-predictors, as well as of the high-resolution GCM and the observed climate variables during years 1971-1999.
GCM type predictor average correlation
Tmax Tmin HMD SLR PCP %Wet
CMIP3
multi-domain optimal
predictor 0.47 0.89 0.54 0.65 0.56 0.66
HiRes GCM
Cld 0.10 0.59 0.63 0.57 0.64 0.77
Dtr 0.29 0.85 0.55 0.08 0.45 0.55
Pre 0.08 0.37 0.71 0.46 0.76 0.76
Tmp 0.71 0.91 0.33 0.26 0.21 0.28
Vap 0.48 0.89 0.67 0.07 0.53 0.62
Note: The optimal parameters are highlighted in bold italics
Comparative evaluation of multi-domain downscaling 3.5.2.2
To investigate the performance of the multi-domain MLR-regression downscaling model, comparative experiments of this technique applied to climate predictor from the various sets of GCMs as discussed have been carried out. The experiments are separated into four cases, as shown in Table 3.16, belonging to two main groups, i.e. single-domain (Nos.1 and 2) and multi-domain (Nos.3 and 4) MLR- models. In each of these two groups the coarse 2.5° x 2.5° GCM-set, as indicated is supplemented by the 0.5° x 0.5° High-Resolution GCM (Nos. 2 and 4). The single-domain MLR- model corresponds to the MLR-monthly models of Table 3.14. The results of these four experiments applied to the study region and the corresponding prediction skills of these four MLR-downscaling options will be presented later in the validation section.
Table 3.16. List of downscaling experiments with various combinations of single- and multi-domain GCMs with the MLR- downscaling model to evaluate the performance of climate prediction as a function of the characteristics of the parent GCM(s).
experiments domain resolution GCM-model
number of MLR predictors 20th
century future scenario sub model n 1.single-domain
MLR (MLR-monthly) single
domain 2.5°x2.5° ECHO-G 21 - 21
2.single-domain single
domain 0.5°x0.5° Hi-Res GCM 5 5 5
multi-domain MLR 3.multi-domain
multi-domain MLR multi
domain 2.5°x2.5° CMIP3 multi-domain GCM
(ECHO-G,BCCR, 256 - 256
ECHAM5,GISS,PCM) 4.multi-domain+HiRes multi
domain 2.5°x2.5°+
0.5°x0.5° CMIP3 (ECHO-G,BCCR, 256+5 5 256+5
multi-domain MLR ECHAM5,GISS,PCM) + Hi-Res
3.5.3 MLR- core module
Multi-linear regression (MLR) is the core of the long-term climate downscaling methodology proposed in this study. This MLR-technique is similar to the model used in the data reconstruction (see Section 2.3) in that MLR formulates a linear relationship between an observed climate predictand and a set of atmospheric GCM- predictors , which can be written as:
where is the dependent climate variable vector, is the independent GCM predictor vector, are the unknown regression coefficients and is the error term or noise. These coefficients of the regression model are determined by using a least squares technique and solving the set of (normal) equations by means of a QR-decomposition (Wilkinson and Rogers 1973, Chambers and Hastie 1992).
The optimal number of predictors ( ) is chosen based on the analysis of the goodness of fit of the MLR- model to the observed data. More specifically, the Akaike information criterion (AIC) as described in Eq. (2.3) is used. By changing the size of predictor-set in the model, following the rank of predictors (as mentioned in screening process section 3.3.4), the optimal number of predictors ( ) in the MLR- model is that which provides the lowest AIC value.
To further optimize the MLR- model, four techniques of predictor selection in this so-called method of stepwise regression, i.e. exhaustive search, forward and backward stepwise and sequential replacement (Miller 2002), are comparatively examined in the model calibration.
Employing years 1971-1985 time series of observed climate variable and GCM-predictors, the MLR-models are calibrated by using four predictor-sets obtained with these variants of stepwise regression. The best technique is chosen from the lowest values of the Bayesian information criterion (BIC) of the calibrated MLR models, which is defined as (Schwarz 1978):
where is the maximized value of the likelihood function for the calibrated model, is the number of predictors including the intercept and is the number of observation data points.
Note that the difference between AIC and BIC is that the BIC is used to select a proper model among the various candidate models, while the AIC value is used, as mentioned above, to find the optimal size of predictors in the selected model.
Among the four models obtained with the different predictor-selection techniques, the BICs show that the exhausted-search variant provides the best MLR- model. Therefore, this method, which is the simplest method but take the longest modeling time, is applied here for the optimization of the regression-equations (transfer-model) in this downscaling model. The algorithmic flow diagram of the MLR model is illustrated in Figure 3.8.
In model validation, the downscaling data- time period is separated by a ratio 50%:50% into a calibration- (1971-1985) and verification- (1986-1999) period. The modeling skill is evaluated by the root mean square error and Nash-Sutcliffe coefficient for the predicted climate variables, i.e.
min. and max. temperature, rainfall, probability of wet day, humidity and solar radiation. It should be noted that the MLR-model to downscale the precipitation is conditionally optimized by setting the particular minimum value of rainfall to zero. However, the minimum predicted rainfall is limited to 0.5 mm/day, which means that if the former is less than this limit, the
(3.9)
(3.10)
corresponding day is set to dry state. The MLR- downscaling model and the optimizing technique are written in the R-programming language (R Development Core Team 2011) and the whole script has more than 2500 lines of code.
3.5.4 Predictor optimization
Since the prediction skills of a particular GCM/MLR-downscaling combination depend on the individual parent GCM-model used, an optimal selection of the predictors in the multi-ensemble GCM is necessary (Kar et al. 2012). The methodology used purpose is described in the following sub-sections.
Predictor-selection in the MLR- model 3.5.4.1
The primary aim of predictor-screening is to provide the downscaling/forecasting model the best set of appropriate predictors (Wilby et al. 2002). Here the most challenging process is to consider that set of predictors, which is able to best represent the climate character in the downscaled study area (Winkler et al. 1997, Charles et al. 1999b). In recent studies of climate prediction, optimization techniques such as the use of filtered data (Yun et al. 2005) and the ranking score (Chakraborty and Krishnamurti 2009) are usually employed to enhance the performance of multi-ensemble downscaling. Accordingly, the predictor-selection proposed here uses also two main filters, i.e. domain size and optimal seasonal scheme, and is based on screening and ranking of climate predictors. The modules of selection and optimization are sketched in Figure 3.9, and all these processes are integrated into the MLR- core model.
In the selection process, firstly, a cross-correlation analysis is done which defines the rank of the predictors by their correlation power. These ranked predictors are then arranged from highest to lowest rank. Then, increasing subsets of the ranked GCM-predictors are employed in the MLR- model. While the number of predictors is increased, the selection of reliable predictors and the optimization of multi-model ensemble can be possibly improved by giving more the regression model more choices. However, when there are more predictors, the prediction model gets more complicated and errors will increase. Therefore, the verification power of the various model- variants is measured, as discussed, by the Akaike's information criterion (AIC) which, in simple words, defines a compromise between good model performance and low number of predictors.
Practically, to find the best scheme for downscaling, the maximum size of a predictor-set starts with five predictors and is increased at intervals of five up a maximum of 30 predictors. This upper limit of 30 predictors was found beforehand from the optimal AIC value for the fits of the MLR- model to the climate variables at each climate site in the study region.
Moreover, to find the best seasonal scheme, the seasonal regression is formulated according to four schemes, i.e., single season (s0), two-season (s2), three-season (s3) and four-season (s4), similar to what has been described in the seasonal variation section 2.4.1. Another process in choosing the best scheme in seasonal regression is further explained in the next section.
By using the set of predictors as mentioned above, the predictors of the MLR-model are fitted to the observed predictands in the calibration period and this fitted model is then verified in the verification period. The accuracy of each predictor-set with specific number of predictors and the seasonal scheme are evaluated based on R2 - and AIC- values. However, when using AIC alone in the predictor-evaluation for large and complex GCM-domains, the selection procedure takes a long processing time and may lead to overfitting (Bozdogan 2000, Claeskens and Hjort 2008). Consequently, to optimize a large predictor-set, the model-accuracy of the MLR- model is estimated from the residual error in validation by choosing an initial number of predictors (predictor set with k = 5, 10, …, 30) in the downscaling process. The Nash-Sutcliffe coefficient
Figure 3.8. Schematic diagram of the optimization of the single-domain MLR- downscaling- model for the selection of the best predictors for use in the final downscaling model.
(NS) is then employed to measure the model-accuracy in this process, by finding the best This predictor set is later selected for finding the exact number of predictors, but now using the best R2 - and AIC- values.
For example, based on the best NS for the k = 5, 10, … ,30 predictor sets, .the sizes of the domain for predicting the maximum temperature and the precipitation at station 48459 are optimal for k=5 and k=10 predictors, respectively. These numbers are then used to limit the exact number of predictors further using the AIC-criteria. The two panels of Figure 3.10
local climate obsi...obsn