BiasCorrection
Method Description
Global Climate Models (GCMs) have been the primary source of information for constructing climate scenarios, and they provide the basis for climate change impacts assessments of climate change at all scales, from local to global. However, impact studies rarely use GCM outputs directly because climate models exhibit systematic error (biases) due to the limited spatial resolution, simplified physics and thermodynamic processes, numerical schemes or incomplete knowledge of climate system processes . Errors in GCM simulations relative to historical observations are large (RamirezVillegas et al. 2013). Hence, it is important to biascorrect the raw climate model outputs in order to produce climate projections that are better fit for agricultural modeling.
Here we describe three different calibration approaches to produce reliable daily climate for future periods, employed in the new CCAFS Climate Bias Correction Section, as follows:
In addition, briefly describe some observational datasets relevant to agricultural modeling and employed as the historical observations for the calibration methods mentioned here.
1. Bias Correction
The Bias Correction (BC) approach corrects the projected raw daily GCM output using the differences in the mean and variability between GCM and observations in a reference period (Figure 1).
Figure 1. Schematic of the bias correction methodology. BC uses raw model output for the future period, and corrects it using the differences (Δ) between historical reference data from the model and observations. (O_{REF} = observations in the historical reference period; T_{REF} = GCM output from the historical reference period; T_{RAW} = raw GCM output for the historical or future period; T_{BC} = biascorrected GCM output.)
If we assumed the variability as equal both for GCMs and observations, the daily data is simply shifted by the mean bias in the reference period (Hawkins et al., 2013), thus:
Eq. 1 
However, it is possible to apply a more general form of this biascorrection method that corrects not only the mean values but also the temporal variability of the model output in accordance with the observations (Hawkins et al., 2013; Ho et al., 2012):
Eq. 2 
where σ_{T,REF} and σ_{o,REF} represent the standard deviation in the reference period of the daily GCM output and observations, respectively. Note that this biascorrection procedure for the GCM output could be applied to correct both the historical and future periods.
2. Change Factor
In the Change Factor (CF) approach the raw GCM outputs current values are subtracted from the future simulated values, resulting in “climate anomalies” which are then added to the present day observational dataset (Tabor & Williams, 2010).
Figure 2. Schematic of the change factor methodology. CF uses present day observations, corrected using the differences (Δ) between present and future model data. (O_{REF} = observations in the historical reference period; T_{REF} = GCM output from the historical reference period; T_{RAW} = raw GCM output for the historical or future period; T_{BC} = change factorcorrected GCM output.)
When the daily variability is assumed of the same magnitude in the future and reference periods, the method is called “delta method”, and the corrected daily data is:
Eq. 3 
But, the more general form considering changes in variance (Ho et al., 2012), is:
Eq. 4 
where σ_{T,RAW} and σ_{T,REF} represent the standard deviation in the future period of the daily GCM output and observations, respectively.
3. Quantile Mapping
The abovedescribed methods work well for more nonstochastic variables (i.e. temperature). A more sophisticated approach for biascorrecting more stochastic variables (e.g. precipitation and solar radiation) is needed. This is because for example, GCM outputs are known to have a "drizzle problem", that is, too many lowmagnitude rain events as compared to observations (Gutowski et al., 2003). Also, GCMs do not capture realistic interannual variability associated with events such as El Niño and La Niña.
In order to appropriately biascorrect GCM output for monthly totals and wetday frequency, while ensuring realistic daily and interannual variability, we implemented the Quantile Mapping (QM) approach with the qmap library written for R statistical software (Gudmundsson, 2014; Gudmundsson et al., 2012). The quantile mapping technique removes the systematic bias in the GCM simulations and has the benefit of accounting for GCM biases in all statistical moments, though, like all statistical downscaling approaches, it is assumed that biases relative to historical observations will be constant in the projection period (Thrasher et al., 2012).
Observational Datasets
The methods described below must be applied to the historical observations to produce calibrated projections of future climate. Thus, we selected six widely used datasets that could be used to "calibrate" daily outputs of GCMs from the IPCC CMIP5. All datasets are biascorrected versions of existing reanalysis datasets. A reanalysis involves reprocessing observational data spanning an extended historical period using a consistent analysis system, to produce a dataset that can be used for meteorological and climatological studies. In the Table 1 are described some characteristics of these datasets.
Dataset 
Based on 
Period 
Resolution 
Main Reference 
AgCFSR 
The ModernEra Retrospective Analysis for Research and Applications (MERRA). 
19802010 
0.25° × 0.25° 

AgMerra 
The Climate Forecast System Reanalysis (CFSR) 
19802010 
0.25° × 0.25° 

GRASP 
ERA40 
19612010 
1.125° × 1.125° 

Princeton 
Reanalysis1 
19482008 
0.25° × 0.25° 

WFD 
ERA40 
19582001 
0.5° × 0.5° 

WFDEI 
ERAInterim 
19792009 
0.5° × 0.5° 