1. Introduction
Probabilistic assessment of regional climate change for future emission scenarios is challenging as there are multiple sources of uncertainty that have to be taken into account. Uncertainties are related to each step in the cause-and-effect chain, from emissions to climatic effects, and represent, for example, our imperfect understanding of the carbon cycle response, atmospheric feedbacks, and regional dynamical responses that could alter temperature and precipitation changes. In the context of simple climate models, a synthesis of uncertainties along the cause-and-effect chain linking emissions to global-mean temperature changes can be performed (Allen et al. 2009; Meinshausen et al. 2009). Unfortunately, comprehensive uncertainty assessments are usually too computationally intensive to be performed within atmosphere–ocean GCMs (AOGCMs) that provide regional information, although some larger perturbed physics studies exist (e.g., Stainforth et al. 2005).
It is a remarkable property of the climate change signal simulated by AOGCMs that, over the considered range of global-mean temperature increase, various regional surface climate variables are approximately linearly related to global-mean temperature change where scaling coefficients are largely independent of the underlying emission scenario (Santer et al. 1990; Mearns et al. 2001; Mitchell 2003; Giorgi 2008; Solomon et al. 2009). However, despite this striking linearity, nonlinearities could be introduced by feedback processes that are currently not considered in AOGCMs or tipping points that lie outside the simulated temperature range. While the present analyses were focused on temperature and precipitation, Huntingford and Cox (2000) also analyzed relative humidity, zonal and azimuthal wind speed, downward long- and shortwave radiation, rain and snowfall separately, and surface pressure.
The relation enables an efficient way to obtain regional, probabilistic climate projections: the scaling coefficients can be used to derive regional climate projections for emissions scenarios not simulated by AOGCMs, using reduced complexity climate models that only project large-scale aggregated quantities. Furthermore, estimating these scaling coefficients for a large number of different AOGCMs provides a way to capture the spread of sensitivities of regional climate change to changes in global-mean temperature due to different parameterizations and model architectures. In other words, these scaling coefficients can allow a formal treatment of scientific uncertainty in regional climate responses by “parameterizing” structural uncertainties apparent in multi-AOGCM ensemble results.
Apart from this inter-AOGCM variability of scaling coefficients, two other distinct types of variability can be distinguished. First, for the same AOGCM, different ensemble runs of the same scenario might differ in their regional climate, giving rise to what we term “interrun” variability. Second, estimating the scaling coefficients for different scenario runs for the same AOGCM should ideally result in identical scaling coefficients, but generally will not; we call this the “interscenario” variability.
The validity of the scaling approach is supported if these latter two variabilities are small compared to the first inter-AOGCM variability or compared to central estimates of the scaling coefficients. High interscenario variability of the scaling coefficients provides a hint that there might be other processes not closely correlated with global-mean temperature change that influence regional climate change.
Once an uncertainty distribution for the scaling coefficients is determined, it can be coupled to probabilistic projections of global-mean temperature change (or other possible predictors) to finally derive probabilistic regional climate projections (e.g., Dessai et al. 2005).
In addition to global-mean temperature change, both aerosol emissions and greenhouse gas (GHG) forcing have the potential to influence regional precipitation beyond the “slow” effect of induced changes of the surface air temperature (e.g., Ramanathan et al. 2001; Dong et al. 2009; Bala et al. 2010). Thus, these covariates are candidates to explain some of the interscenario variability of the regional hydrological sensitivity (HS) apparent in AOGCMs.
The aerosol effect on precipitation is based on four (micro)physical mechanisms (cf. Ramanathan et al. 2001). Mechanism 1 is the reduction of the amount of incoming solar radiation at the ground by scattering (e.g., sulfate aerosols with a single-scattering albedo of nearly one) or absorption of shortwave radiation [mainly black carbon (BC)]. This direct aerosol effect can result in reduced evaporation that in turn lowers precipitation. The effect is amplified by mechanism 2, the first indirect effect (Twomey 1977), in which an increased number of cloud particles (mainly due to sulfate and organic aerosols) results in brighter clouds reflecting more shortwave radiation back to space. In mechanism 3, the increased number of cloud nuclei also makes them less efficient in releasing precipitation (Albrecht 1989). The increased lifetime of clouds, the second indirect aerosol effect, also contributes to a reduction of shortwave radiation at the surface. In mechanism 4, changes in the atmospheric temperature structure due to absorbing particles lead to an increased atmospheric stability and less convection.
In addition, GHG radiative forcing has the potential to induce changes in precipitation that occur on a short time scale within days or weeks (Allen and Ingram 2002; Wu et al. 2010; Frieler et al. 2011). On global average, this “fast” effect can be understood in terms of the tropospheric energy budget: the perturbed system balances variation in tropospheric radiative cooling with changes in the latent heat released. Carbon dioxide is the most prominent example of a GHG having a higher top of the atmosphere (TOA) than surface forcing. Thus, increasing CO2 concentrations lead to a reduced radiative cooling that is mainly balanced—on a global scale—by a reduction in latent heat release, and therefore, less precipitation (Allen and Ingram 2002; Andrews et al. 2010). Given the considerable energy exchange with the adjacent regions (Lambert and Webb 2008; Andrews et al. 2009), regional precipitation changes cannot simply be described within the framework of the global energy budget without modeling horizontal energy fluxes (cf. Muller and O’Gorman 2011).
The focus of our work lies in the development of a new scaling approach that—building on and extending previous work (e.g., Giorgi 2008)—integrates the following three aspects:
Estimation of the inter-AOGCM, interscenario, and interrun variability of the scaling coefficients derived from AOGCM simulations;
Consideration of correlations between the temperature and precipitation component of these variations to provide bidimensional uncertainty distributions; and
Consideration of further covariates to broaden the predictor variables beyond global-mean temperature alone.
Our approach is a based on a random effects model (Pinheiro and Bates 2000). This statistical framework is quite established in life sciences. Bayesian versions of this approach have also been applied in climate science (e.g., Tebaldi and Sansó 2009; Furrer et al. 2007).
The methodological steps in this paper, from diagnosis of the AOGCM responses to prediction of exceedance thresholds for regional impacts, are summarized schematically in Fig. 1. In the first part of the following methods section (section 2), the data processing applied to the spatial temperature and precipitation AOGCM output is described (section 2a). The data on additional predictors is presented in section 2b. Subsequently, the random effects statistical model to retrieve uncertainty distributions of the scaling coefficients is introduced. Our basic model version only includes global-mean temperature as a predictor (section 2c). The extended version includes further predictors for the precipitation component (section 2d). Section 2e describes how to sample scaling coefficients from these uncertainty distributions. These scaling coefficients are combined with the joint uncertainty distributions related to global-mean temperature and GHG forcing based on simulations by the reduced complexity climate model, Model for the Assessment of Greenhouse-Gas Induced Climate Change version 6 (MAGICC6) (Meinshausen et al. 2011), as briefly introduced in section 2f. In the results section (section 3), bidimensional distributions of regional temperature and precipitation changes are provided for 31 world regions under the strong mitigation scenario representative concentration pathway with peaking and declining (RCP3-PD) and the highest RCP business-as-usual scenario, RCP8.5 [the RCPs are for use in the Coupled Model Intercomparison Project phase 5 (CMIP5) AOGCM intercomparison exercise in preparation of the Intergovernmental Panel on Climate Change (IPCC) Fifth Assessment Report (Van Vuuren et al. 2011)] (section 3c). Taking Greenland’s surface mass balance (SMB) as an example, section 3d demonstrates the usefulness of bidimensional uncertainty distributions when determining exceedance probabilities for certain impact thresholds. Limitations, sensitivities, and further development of our scaling approach are discussed in section 4. Conclusions are presented in section 5.
Schematic illustration of the analytical steps. After processing the output of the AOGCMs (step 1), the statistical relationship between global temperatures and regional climate changes is estimated (step 2). Sampling the scaling coefficients from the established distributions (step 3) and combining them with global-mean temperature projections (step 4) provides probabilistic projections of mean regional temperature and precipitation changes (step 5). Based on these bidimensional regional uncertainty distributions, it is possible to derive exceedance probabilities for specific regional impacts (step 6) here taking GRL SMB as an example.
Citation: Journal of Climate 25, 9; 10.1175/JCLI-D-11-00199.1
2. Methods
a. AOGCM data preprocessing
Our raw data are obtained from the AOGCM simulations performed as part of CMIP phase 3 (CMIP3) (Meehl et al. 2007b), assessed by the IPCC AR4 (e.g., Meehl et al. 2007a) and stored within the framework of the Program for Climate Model Diagnosis and Intercomparison (PCMDI, http://www-pcmdi.llnl.gov). The available monthly data were averaged to obtain annual data, giving equal weight to each month. We used all available scenario runs listed in Table 1. If there were multiple runs per scenario from a single AOGCM, then we used up to five ensemble members for which temperature and precipitation data were available in our analysis.
Short description of the forcing scenarios considered in this study.
Regional averages were calculated as area-weighted means over all AOGCM grid points falling into the polygons specifying our 31 world regions, which are based on the region definitions by Giorgi and Bi (2005). In addition, we include West and East Antarctica, land and ocean regions, and a region representing the geographical Greenland 
For two AOGCMs—HadCM3 and BCCR BCM2—negative precipitation data appear at individual grid points. The data were set to zero as long as their absolute values were smaller than 10−11 kg m−2 s−1. When absolute values were larger, negative precipitation values were set to missing. Similarly, when precipitation values were larger than 1, we set them to missing, given that precipitation values are generally of the order of 10−7 to 10−5 kg m−2 s−1. As long as the number of missing values for one specific year was smaller than 5% of the total number of grid points falling into the specified areas, regional averages were calculated on the basis of the remaining data. Otherwise, the regional average for that particular year and model run was considered to be missing.
Percentage precipitation and absolute temperature changes were calculated with respect to the corresponding sections of the control runs (as recommended by Mitchell 2003). In most of the cases, this is the preindustrial control run (“picntrl”), while the idealized scenario runs of CCSM3, ECHO-G, CGCM2.3.2, and PCM (full model names listed in Table 2) branch off the present-day control run (“pdcntrl”). We smoothed the raw control run by linear regression before subtracting it from the scenario runs to avoid an artificial inflation of the variability. If the control run was too short, we extended it with the last valid data point in the smoothed time series. For technical reasons we used 10-yr averages of these annual data to reduce the size of the dataset that is otherwise too large for fitting the bidimensional statistical model.
The forcing agents included in the individual CMIP3 AOGCMs and considered in this study as covariates for explaining regional precipitation. We distinguish between GHG forcing, BC, and SOx. Although our analysis does not include the twentieth-century runs, we also show volcanic forcing (VOLC) with model-specific relative amplitudes provided in brackets based on Forster and Taylor (2006) [cf. Table 2 in Meinshausen et al. (2011) and Table 10.1 in Meehl et al. (2007a)]. For sulfate, we provide the background information in brackets, whether the AOGCMs included the direct (D) and/or first (I1) and second (I2) indirect effects, although our method does not distinguish between these cases. Asterisk means that for ECHAM5/MPI-OM and HadCM3, the first indirect effect was assumed in the MAGICC emulations to be equal to the default joint forcing of the first and second indirect effects, given the substantial negative shortwave forcing analyzed by Forster and Taylor (2006) for these models.
We did not log transform the AOGCM precipitation data. Although this transformation provides a way of accounting for positivity constraints (projected precipitation must not become negative), there are potential drawbacks. In particular, a log transformation assumes an exponential relation between changes in the predictors (as, e.g., global-mean temperature) and regional precipitation changes (Watterson 2008). Instead, we assumed both regional temperature and precipitation anomalies to follow a Gaussian distribution, and quantile–quantile plots were used to verify these assumptions [see appendix A for the applied R code and the online database (www.pik-potsdam.de/~frieler/regional_temp_and_precip/) for the associated plots].
b. Data on covariables, aerosols, and GHG forcing
AOGCM simulations include aerosol effects to a varying degree depending on the model (see Table 10.1 in Meehl et al. 2007a), except for the idealized “1pctto2x” and “1pcttp4x” scenarios (forcing scenarios are listed in Table 1). While effects of sulfate aerosols are represented in all AOGCMs, there are only eight CMIP3 models that took BC effects into account. In addition, indirect effects are not generally simulated. Table 2 gives an overview of the aerosol forcing agents and mechanisms included in the different models. Given the relatively short atmospheric residence time of black carbon and sulfate aerosols, we use emissions time series EBC (in megatons per year, or Mt yr−1) and ESOx (megatons of sulfer per year, or MtS yr−1), respectively, as additional predictors in our statistical model. In the framework of the Special Report on Emissions Scenarios (SRES), regional information about SOx emissions is only provided for four regions [Organisation for Economic Co-Operation and Development (OECD); ASIA; Eastern Europe and Former Soviet Union as countries undergoing economic reform (REF); and Africa, Latin America, and Middle East (ALM)]. We map the finer-scale Giorgi regions to these SRES regions as indicated in Table 3. BC is not prescribed in the SRES scenarios. Thus, we use information provided by the individual groups, as described in the supplementary material of Frieler et al. (2011). Similar to the SRES regions, emissions were averaged over the large-scale areas: ASIA, REF, OECDnorth, OECDeurope, OECDsouth, Latin America and Africa (LAM), and Africa and the Middle East (MAF), defined as an aggregation of the Giorgi regions (as listed in Table 3).
Mapping of the Giorgi regions (Giorgi and Bi 2005) and 
Beyond aerosol forcings, we test tropospheric GHG forcing as an additional predictor for precipitation changes. We introduced the covariate ΔRGHG (W m−2) as a weighted sum of TOA forcings in Frieler et al. (2011) to describe the tropospheric fraction of the GHG forcing, that is, the difference between the TOA and the surface radiative forcing. The TOA forcing information stems from the emulation of the AOGCM simulations (Meinshausen et al. 2011) except for the model “BCM2,” which was not emulated but where we used an idealized forcing time series for the 1pctto2x run (3.7 W m−2 at doubling). The availability of these forcing data restricts our dataset to the considered number of AOGCMs and scenarios.
We used a normalized and an unnormalized set of GHG forcings (ΔRGHG), BC, and SOx emissions (EBC and ESOx, respectively) where normalization was done with respect to the multimodel mean of the 1999 values. Normalization is applied to assist the comparison of the effects, which should not affect the results. Table 4 provides the normalization constants. They allow the scaling coefficients for the normalized predictors (see Table 6) to be transformed into scaling coefficients for the absolute values.
Normalization constants used to calculate the normalized predictors. They represent the multimodel mean values of the predictors for the year 1999 (see section 2b).
c. The basic statistical model
This section describes the basic random effects model, including only global-mean temperature change as predictor for the regional changes in temperature and precipitation. In our context the random effects model can be understood as a linear regression model that allows for intermodel, interscenario, or interrun variations of the scaling coefficients. These are handled as random deviations (random effects) from the central scaling coefficients (fixed effects). In contrast to time slice approaches (e.g., Giorgi 2008; Hulme et al. 2000), the regression approach makes use of the full time series of data (as recommended by Mitchell 2003). While a simple least squares fit to the full time series of one (AO)GCM was used by, for example, Mitchell et al. (1999) and Huntingford and Cox (2000), the random effects approach allows to fit one unique model to all time series of the 20 AOGCMs, different scenarios, and different runs per scenario. The method, which to our knowledge has not been used in pattern scaling approaches before, has the advantage of 1) correcting the standard errors of the scaling coefficients for the clustering of the data into different AOGCMs, scenarios and runs; and 2) explicitly estimating the inter-AOGCM, interscenario, and interrun variability of the scaling coefficients. Simply fitting one regression line to all data based on the assumption that the residuals were independent is inappropriate here because the residuals would strongly depend on the considered AOGCM and the associated global-mean temperature change. Following this simple approach would mean that standard errors of the estimated scaling coefficients would not be estimated correctly. In general, ignoring the correlation structure of the residuals means underestimating the actual errors. The random effects model proposed here provides a better representation of the actual structure of the residuals and accounts for it when estimating the standard errors of the central scaling coefficients. Advantage 2 is important as it directly tests to what degree the relationship between global-mean temperature and regional temperature and precipitation changes depends on the scenario (and the run). High and systematic interscenario variations of the scaling coefficients [e.g., faster global-mean temperature change might be associated with higher (or lower) scaling coefficients for regional temperature or precipitation changes] would make the scaling approach questionable as a tool of interpolating/extrapolating AOGCM simulations to new emission scenarios. In this sense quantifying the interscenario (interrun) variability represents a validity test for the scaling approach.
We do not assume that there is a common scenario effect across all AOGCMs (in the sense that scenarios with a high scaling coefficient—that is, positive rscen or 
It turns out that the interrun variability of the scaling coefficients is very low for some of the considered regions. This causes problems in estimating the correlation between the temperature- and precipitation-related components of the random effects (i.e., rrun and 
The estimated coefficients from the basic model with global-mean temperature being the only predictor for projecting precipitation and temperature changes in 31 world regions. The table provides the fixed effects c (regional temperature change per change in global-mean temperature) and 
d. The extended statistical model
While the above-mentioned model is our starting point, it may be further reduced by 1) excluding predictors or 2) placing further restrictions on the covariance structure of the random effects. Concerning point 1, we used the Bayesian information criterion (BIC; Schwarz 1978) as a model selection criterion. The BIC decreases with improved model versions. It represents a measure of the model fit while adding a penalty term for the number of estimated parameters to avoid overfitting.
Further restrictions of the covariance matrix 
The scaling coefficients from the extended statistical model, including GHG forcing, SOx, and BC emissions as additional predictors. There are separate lines for the fixed effects c, 
In case of the West and East Antarctic Ice Sheets (WAIS and EAIS, respectively), we do not include aerosol effects. We furthermore exclude the GHG forcing component from the WAIS model because of a strong anticorrelation of the random effects. We thus only provide scaling coefficients estimated by the basic model (1) for this region.
As for the basic model, within the extended model framework the scenario-specific random effects are allowed to vary from AOGCM to AOGCM. AOGCM invariant scenario effects could provide a hint to systematic scenario effects that are not explained by the included predictors. Figs. A1 and A2 show the AOGCM-specific scenario effects as estimated by the extended model version. In general, there does not seem to be a substantial common scenario effect in the sense that the random effects for one specific scenario are systematically positive or negative (cf. section 4), although some systematic effects cannot be excluded (e.g., for the SRES B1 scenario in the NEU region). Interscenario variabilities are handled as an unsystematic random component in the following.
e. Sampling the scaling coefficients
To get a sample of the uncertainty distribution associated with the scaling coefficients, we randomly draw from the multivariate normal distribution associated with the estimation of the fixed effects. In addition, we draw from the multivariate normal distributions 
f. Coupling with probabilistic global-mean temperature changes
This section briefly describes our approach to generate (bidimensional) uncertainty distributions of global-mean temperature change ΔTglobal (as well as ΔRGHG). Furthermore, we describe how these distributions are merged with the previously derived scaling coefficients. This allows for obtaining probabilistic projections of regional temperature and precipitation changes for emission scenarios not simulated by the AOGCMs.
We generate uncertainty distributions of ΔTglobal and ΔRGHG using the reduced complexity coupled carbon cycle climate model MAGICC, which has been used within the past IPCC assessment reports and several integrated assessment models (Wigley and Raper 2001). Its reduced complexity makes it highly efficient while being able to closely emulate more complex carbon cycle models [Coupled Climate–Carbon Cycle Model Intercomparison Project (C4MIP)] and global climate models (CMIP3), as shown for the most recent version, MAGICC6 (Meinshausen et al. 2011). To generate our joint distributions of ΔTglobal and ΔRGHG, we use here a setup identical to the “illustrative default” case in Meinshausen et al. (2009): 82 input parameters of MAGICC6 (including climate sensitivity, ocean vertical diffusivity, land–ocean warming ratio, direct SOx forcing, and indirect cloud albedo forcing) were varied using wide uniform a priori distributions or probability distributions that capture the current level of knowledge reflected by the IPCC AR4 in the case of radiative forcing (Forster et al. 2007). The parameter space was then constrained by time series of hemispheric land–ocean temperatures (Brohan et al. 2006) and the observed ocean heat uptake (Domingues et al. 2008). The joint probability density distribution of 82 parameters was then resampled to obtain a posterior climate sensitivity distribution that closely reflects the IPCC AR4 estimate with a best estimate of 3 K and a likely range between 2 and 4.5 K. Carbon cycle feedback strength parameterizations are drawn at random from 10 C4MIP (Friedlingstein et al. 2006) emulations as described in Meinshausen et al. (2011). Finally, the Bayesian approach provides an updated (a posteriori) distribution of annual global-mean temperature changes and ΔRGHG for a given emission scenario.
For deriving regional climate projections, the distributions of global temperature changes and ΔRGHG on the one side, and the regional scaling coefficients on the other side, need to be combined. Here, we employ a simple Monte Carlo technique, sampling independently from the bidimensional or multidimensional distribution of scaling coefficients (as described above) and the (bidimensional) uncertainty distribution of global-mean temperature changes and ΔRGHG. Each vector of scaling coefficients is multiplied by a randomly sampled (paired) time series of global-mean temperature change (and ΔRGHG) to get a bidimensional time series of regional temperature and precipitation changes.
This combination of scaling coefficients and global-mean temperature projections assumes them to be independent. We test for independence using scenario-specific estimates for the scaling coefficients from our extended model. To derive associated AOGCM- and scenario-specific values of global-mean temperature change, we average over all annual data points falling in the period from 2091 to 2100 and belonging to one AOGCM–scenario (SRES B1, SRES A1B, SRES A2, and commit). We calculate the correlation between these global-mean temperature values and the vector containing the AOGCM–scenario-specific estimates of 
3. Results
In this section, we first show examples of regional temperature and precipitation data we extracted from AOGCMs (see section 3a). Then, we present the results provided by the statistical models described above. We focus on the uncertainty distribution of scaling coefficients (section 3b) and show the associated distribution of regional climate changes under the RCP8.5 and RCP3-PD emission scenarios (section 3c). The associated impacts on Greenland’s contribution to sea level rise (SLR) from surface mass balance changes are described in section 3d.
a. Diagnosed AOGCM data
We chose the regional temperature and precipitation changes over Greenland 
Greenland 
Citation: Journal of Climate 25, 9; 10.1175/JCLI-D-11-00199.1
As in Fig. 2, but for the EAS region. The EAS region is defined by Giorgi and Bi (2005).
Citation: Journal of Climate 25, 9; 10.1175/JCLI-D-11-00199.1
Over Greenland, we find a linear relationship between global-mean temperature change and regional precipitation changes over the main part of the simulations. But some AOGCMs show a change of the slope at the end of some scenario runs, for example, in year 2100 in the SRES scenarios when concentrations were held constant afterward or in the idealized runs after stabilization (see arrows in Fig. 2d).
In the EAS region, global-mean temperature changes alone cannot explain the decline in precipitation during the twentieth century and the following increase within the SRES simulations of some AOGCMs. Aerosol emissions might provide an explanation for the nonlinear relation. In contrast to Greenland, aerosol emissions have strongly increased during the region’s industrialization in the twentieth century. This explanation is particularly supported by the fact that the decrease in precipitation shown for some AOGCMs over the EAS region during the twentieth century does not appear in the corresponding idealized 1pctto2x and 1pctto4x scenarios that do not include aerosol effects.
Fast damping effects of tropospheric GHG forcings may provide one possible explanation for the change in regional hydrological sensitivity over Greenland during the stabilization periods of the different scenarios runs. Based on CCSM3 simulations, Bala et al. (2010) found that the fast effect of CO2 forcing on precipitation can be more than 50% of the total response. In contrast to our results, they found a slight increase in precipitation over Greenland as fast response to CO2 forcing. Unfortunately, the stabilization periods simulated by the CCSM3 model and stored in the PCMDI archive are very short. Therefore, it is not possible to assess whether this AOGCM would show a slower increase in precipitation during these periods.
b. Resulting scaling coefficients
Figure 4 shows the uncertainty distribution of the global-mean temperature scaling coefficients obtained by fitting the basic statistical model [Eq. (1)] to the regional changes in temperature and precipitation as provided by 20 CMIP3 AOGCMs for the Greenland and EAS areas. We also calculated individual pairs of scaling coefficients by fitting separate linear regression models to the temperature and precipitation data for each individual model run. The scaling coefficients calculated from the 20C3M runs (see red triangles in Fig. 4) show a high variability without a systematic pattern in Greenland. In the EAS region, the precipitation scaling coefficients are systematically lower for the 20C3M run compared to the other scenarios. This is in line with Fig. 3 showing decreasing precipitation for the twentieth century and increases afterward. This systematic effect is probably due to tropospheric aerosol effects that strongly increase within the twentieth century. While the unsystematic effects might be due to 1) a lower signal to noise ratio and 2) the fact that the 20C3M run is especially short in comparison to the SRES runs, there might be other systematic effects not included in our model, for example, solar or volcanic forcing. For that reason, we left out the 20C3M run when determining our regression coefficients.
Scaling coefficients for 
Citation: Journal of Climate 25, 9; 10.1175/JCLI-D-11-00199.1
The extended model shows that the high interscenario variability may be largely explained by varying aerosol loadings. The interscenario variability for precipitation in the East Asian region decreased from 1.9 to 0.4 for the temperature-related scaling coefficient when aerosol emissions and GHG forcing are included in the regression. Starting from the comprehensive version that includes ΔRGHG, EBC, and ESOx as additional predictors for the precipitation component [Eq. (2)], the model was stepwise reduced whenever this decreased the BIC. In the case of 
Regional precipitation changes for the (a) 
Citation: Journal of Climate 25, 9; 10.1175/JCLI-D-11-00199.1
The remaining interscenario variability of the scaling coefficients is generally lower than the inter-AOGCM variability but not completely negligible (in the order of 50% for temperature and in the order of 30% for precipitation when taking GHG forcing and aerosol effects into account). For temperature, the standard deviation of the interscenario variability is in the order of 5%–10% of the estimated scaling coefficients. The interrun variability is estimated to be about one order of magnitude smaller than the inter-AOGCM variability and might be neglected given the other contributions to the overall uncertainty of the scaling coefficients.
In some regions the scaling coefficients of the additional predictors show a considerably larger inter-AOGCM variability than the multimodel mean absolute value. This indicates that the effects are not consistent within the 20 AOGCMs.
We now discuss two distinct applications for the scaling coefficients given in Tables 5 and 6. In the first application, it becomes possible to derive regional probabilistic climate response estimates for any particular fixed warming level. Using our “temperature only” model version provided in Table 5, any particular warming level can be transformed into a regional temperature and precipitation change and its uncertainty. For instance, we may multiply the provided “fixed effects” scaling coefficients for temperature (second column of Table 6) by 2 to yield the mean temperature change under a 2°C global warming level, recently adopted as a climate target by the Copenhagen Accord. As an example, the North Asia–Siberia region (NAS) is likely to be 3.18°C (±0.56°C) warmer at a 2°C global-mean warming level. However, we need the triplet of global-mean warming level, GHG forcing, and aerosol emissions to account for the nonlinearities in the precipitation patterns. For the 2°C warming level, the associated fixed effect by temperature on precipitation has to be multiplied by 2 and merged with the scaled GHG forcing, BC, and sulfate emission values.
In the second application, we use the scaling coefficients in conjunction with uncertainty distributions of global-mean temperatures and GHG forcings to derive transient probabilistic regional projections, as described in section 2f and demonstrated in the following sections.
c. Probabilistic regional projections
The combination of probabilistic global projections with the scaling coefficient now allows us to derive regional climate changes, as shown in Fig. 6 for Greenland and other world regions in Fig. 7. All projections are based on the extended model versions, that is, the scaling coefficients given in Table 6.
Regional temperature and precipitation changes for 
Citation: Journal of Climate 25, 9; 10.1175/JCLI-D-11-00199.1
Joint distributions of mean regional temperature and precipitation changes in 2100 with respect to preindustrial levels under the low (RCP3-PD, blue) and high (RCP8.5, orange) scenarios. Boxes indicate the inner 80% of the marginal distributions. The labeling of the axes is the same in all subpanels and given in the legend. The region definitions are based on Giorgi and Bi (2005) and amended by those for the WAIS and EAIS separated by the Transantarctic Mountains.
Citation: Journal of Climate 25, 9; 10.1175/JCLI-D-11-00199.1
Several implications for future regional climate change seem noteworthy. First, if the new RCP scenarios are truly spanning the full range of literature scenarios—which was one of their design criteria—then the lower (RCP3-PD) and higher (RCP8.5) scenario results shown in Fig. 7 do span the range from strong mitigation to high-emission no-climate policy cases. With the possible exception of South America (CSA and SSA), Southeast Asia (SEA), and south Australia (SAU), a temperature increase in excess of 8°C by 2100 cannot be excluded in any of the regions under the high RCP8.5 scenario. Second, even the low RCP3-PD emission scenario that implies strong and immediate mitigation action results in best-guess land warming at or beyond 2°C, although median global warming remains below 2°C.
Our analysis re-enforces earlier diagnoses (Meehl et al. 2007a) with regards to the regional distribution of precipitation changes under climate change. While precipitation is increasing for Greenland and other polar regions, models suggest strongly decreasing precipitation for the Mediterranean region (MED) and possibly Central America (CAM). However, the sign of the precipitation change is not clear for the Amazon because of the divergent result from the underlying set of AOGCMs. Confirming earlier studies, our results show that the occurrence of an Amazon dieback as both consequence and cause of a strong reduction in precipitation (Cox et al. 2004; Jones et al. 2009) are rather unique to the HadCM3 and HadGEM1. For the very dry Sahel zone (SAH), the high uncertainty in relative precipitation changes reflects the very small current absolute precipitation levels that easily result in high relative deviations from preindustrial levels. Note that these regions are not strictly defined climatic regions and therefore subregions with more pronounced precipitation changes might exist.
Overall, the projected median changes in precipitation (and the associated uncertainty ranges) based on the extended model are very similar to the basic model results for most of the regions. Differences between the basic and the extended models are most pronounced for the EAS and MED regions and 
In all our projections, we do not include any residual variation. That is, the distributions describe the uncertainties of the projected mean changes without taking into account internal variability.
d. Example of Greenland Ice Sheet contribution to sea level rise
While accurate simulation of ice sheet surface mass balance usually requires higher spatial resolution than provided by the AR4 AOGCM runs, Gregory and Huybrechts (2006) developed a method for calculating mass balance changes in terms of ice sheet averaged temperature and precipitation changes from AOGCM simulations by combining them with high-resolution climate model runs and an ice sheet mass balance model. Their model uses the integrated sum of expected positive degree days, based on annual averages of temperature, and different degree-day factors (DDFs) for snow and ice to calculate surface melt and the resulting runoff. The model was used to calculate the change of surface mass balance for a range of regional climate changes ΔT and ΔP/P0, where changes were described with respect to “present day” values (T0 and P0) averaged over the 1960–90 period. Increasing precipitation produces a negative sea level contribution because of higher accumulation rates. Increasing temperatures produce a positive contribution because of accelerated melting rates.
Using the joint probabilities we produced for the RCP scenario runs, we can now calculate exceedance probabilities for various isolines of surface mass balance change by integrating over the samples that fall above a certain threshold (see Fig. 6). Results for 0, 0.62, 1, 1.5, and 2 mm SLR yr−1 are given in Table 7. In the reference period 1960–90, the surface mass balance is estimated to be −0.62 mm yr−1, a gross gain of the GRL Ice Sheet (that was historically balanced by an associated loss by dynamical processes), so that a 0.62 mm yr−1 change in contribution to SLR computed here would lead to an absolute SMB of zero. A zero SMB is the point at which no gross mass gain of the GRL could occur, even under zero dynamical ice sheet losses. These calculations do not account for SLR because of changes in dynamical losses of the Greenland Ice Sheet. These lead to a net loss of ice mass, even at a SMB of zero, and could possibly lead to substantially higher changes in SLR as observations suggest (e.g., Van den Broeke et al. 2009).
Percent exceedance (mm yr−1) probabilities for different contributions to SLR from changes in GRL’s SMB. The calculations are based on the parametric scaling approach coupled to the impact function provided by Gregory and Huybrechts (2006).
4. Limitations, sensitivities, and outlook
This section highlights sensitivities, limitations, and possible future extensions of the presented scaling approach. In particular, difficulties in interpreting our results are highlighted.
a. Limitations of the scaling approach
Coupling the uncertainty distributions of global-mean temperature to the regional scaling coefficients can lead to an extrapolation of the available AOGCM simulations beyond the calibration range of the statistical model. In particular, the RCP8.5 emission scenario leads to a sample of global-mean temperature trajectories that partly lies outside of the range simulated by the CMIP3 AOGCMs. This hampers the estimation of exceedance probabilities, especially for this high temperature range because new processes (e.g., vegetation changes, loss of sea ice) might come into play at these levels. The forthcoming CMIP5 ensemble will allow for testing the validity of this extrapolation. Similarly, the results for the new RCP3-PD scenario will provide new insights into the validity of the scaling for the low scenarios, in which temperatures might peak in the course of the twenty-first century.
Estimation of the change in regional precipitation that is induced by tropospheric forcing is hampered by the high correlation of global-mean temperature change and GHG forcing. Therefore, some of the temperature-induced changes might be attributed to the forcing component and vice versa. While our estimates are based on the relatively short stabilization periods available from the CMIP3 data, the situation will improve with analysis of the RCP3-PD simulations planned for the CMIP5 project. This scenario will comprise a period of declining GHG forcings, thereby breaking the close correlation between temperature and forcing (see Fig. 8).
Relationship between global-mean temperature change and our approximation of the tropospheric GHG forcing; see main text for details. Both the global-mean temperature change and the tropospheric forcing are given by the median of their probabilistic projections provided by MAGGIC6 for the RCP3-PD (blue) and RCP8.5 (orange) scenario.
Citation: Journal of Climate 25, 9; 10.1175/JCLI-D-11-00199.1
In addition, there will be simulations in which CO2 is instantaneously quadrupled (Taylor et al. 2011). These idealized CMIP5 runs will provide an ideal setting for distinguishing the fast forcing-induced effect on precipitation from the “long term” response to global-mean temperature change.
Our approximation of the tropospheric GHG forcing is based on the assumption that the tropospheric forcing is proportional to the TOA forcing. While this is supported by Andrews et al. (2010) for CO2, it needs further validation. Our approximation might be improved by future studies providing more information about the tropospheric fraction of the GHG forcing, particularly for non-CO2 GHGs.
The CMIP3 AOGCM simulations were not designed to fully sample modeling uncertainty by, for example, systematically varying model parameters; therefore, they might lead to overconfident projections. Our quantification of uncertainties due to different model structures and parameterizations might also be overconfident because different AOGCMs are considered as independent although their development might not have been completely independent. Structural similarities and identical components of code might reduce their effective number to less than 20 (Pennell and Reichler 2011). In addition, our study does not allow any statement about the validity of the AOGCM projections. It provides a kind of summary of their projections but does not weight the AOGCMs according to implemented feedback processes, their representation of aerosol effects, or the agreement with observations. For example, aerosol forcings were only partially implemented in most CMIP3 AOGCMs. Dynamic vegetation and sea ice models were not coupled and stratospheric processes were not well resolved in almost all cases. This might lead to systematic biases in regional projections [e.g., Betts et al. (2004) for vegetation feedbacks and Ineson and Scaife (2008) for regional responses to stratospheric processes].
Our scaling coefficients are coupled to global-mean temperature and GHG forcing projections by the simple climate model MAGICC6 based on “illustrative default assumptions” about climate sensitivity and its uncertainties (Meinshausen et al. 2009). For example, choosing different climate sensitivity distributions to ours (that closely reflects the IPCC AR4 estimate for climate sensitivity) would lead to different results throughout all regions.
Handling the interscenario variability as an unsystematic uncertainty component might be a limitation for some regions. In case of the temperature scaling coefficients, the “commit” scenario seems to be most problematic (Fig. A1): The temperature scaling coefficients for this scenario shows relatively strong deviations that tend to be systematically (i.e., persistent over most of the AOGCMs) lower or higher than zero in some regions. This may be due to the stabilization character of the commitment run with changing land–ocean warming gradients. The effect could also be caused by aerosols: In many regions, the aerosol loading reaches a high level at the end of the twentieth century. This high level is held constant for the commitment run, providing relatively strong cooling under a relatively low GHG-induced warming.
With respect to precipitation, the scaling coefficients related to global temperature in the ocean region are systematically smaller than zero for the 1pctto4x scenario, while they are systematically larger than zero for the commit scenario (Fig. A2). The reason for that is unclear, but the effect might be due to inappropriate weightings when deriving the tropospheric forcing for each GHG (Frieler et al. 2011).
b. Application of the bidimensional approach to subregions or seasons
Our projections for annual averages are based on monthly data; therefore, they may be easily applied for long-term projections of seasonal changes. For example, it might be interesting to distinguish between winter and summer temperature over Greenland, where wintertime warming is stronger than summertime warming (Meehl et al. 2007a). In this setting, the inter-AOGCM (interscenario and interrun) variations of the seasonal scaling coefficients might also be correlated across seasons. This can be handled by fitting a bidimensional random effects model where one component describes summertime temperature changes while the other one describes the winter temperature changes. In an analogous way, it is also possible to obtain a bidimensional uncertainty distribution of temperature or precipitation scaling coefficients of neighboring regions. For instance, one component of the model could describe annual temperature changes in the western subregion of Antarctica and the second describes temperature changes of the eastern part. Even an extension to more than two dimensions is possible. A 12-dimensional model could describe a complete annual cycle of scaling coefficients, but it may require some restrictions concerning the considered correlations structures to allow convergence of the fitting routine.
5. Conclusions
A new scaling approach to calculate probabilistic projections of regional temperature and precipitation changes is introduced. Our method allows the quantification and the separation of uncertainty components related to the inter-AOGCM, interscenario and interrun variability of the scaling coefficients. Furthermore, estimates of the correlation between the uncertainty distributions of the temperature and precipitation scaling coefficients are obtained.
Our approach captures the possible correlations between the uncertainty distributions associated with the temperature and precipitation components as extracted from the CMIP3 archive of AOGCM experiments. It is thus suitable for coupling with impact functions that depend on the bivariate input. In particular, it is possible to calculate the exceedance probabilities for different isolines in the two-dimensional temperature–precipitation plane. In the case of Greenland’s surface mass balance where the effect of increasing temperatures can be partly balanced by increasing precipitation, the strong correlation between both components of the uncertainty distribution leads to reduced probabilities of exceeding certain SLR isolines.
Samples of scaling coefficients and regional temperature and precipitation changes for 31 world regions, including the whole land and ocean area, are available in the online archive (http://www.pik-potsdam.de/primap/regional_temp_and_precip). The methods presented here provide an efficient way to make probabilistic projections of future regional temperature and precipitation change; therefore, they may be useful for impact studies being prepared as part of the Working Group II contribution to the IPCC AR5.
Acknowledgments
KF and MM were supported by the UFOPLAN project (FKZ 370841103) by the German Federal Environment Agency. We thank Reto Knutti and Julie Arblaster for initial NCL code examples to diagnose the AOGCM data, which were the early basis for our R-based diagnostics routines. We thank Tom Wigley for his collaboration on MAGICC and SCENGEN. We acknowledge the modeling groups, the Program for Climate Model Diagnosis and Intercomparison (PCMDI) and the WCRP’s Working Group on Coupled Modelling (WGCM), for their roles in making available the WCRP CMIP3 multimodel dataset. Support of this dataset is provided by the Office of Science, U.S. Department of Energy.
APPENDIX A
Model Code
Here we provide a schematic representation of the R code used to fit the basic model [cf. Eq. (1)] (the mathematical notation is given in the main text).
Scenario-specific random effects associated with the regional temperature change per global-mean temperature change [rscen given in Eq. (2)] as estimated from the extended model versions. One cross represents the estimate associated with one specific AOGCM. (a) 1pctto4x, (b) 1pctto2, (c) commit, (d) SRES B1, (e) SRES A1B, and (f) SRES A2.
Citation: Journal of Climate 25, 9; 10.1175/JCLI-D-11-00199.1
As in Fig. A1, but associated with the regional precipitation change per global-mean temperature change.
Citation: Journal of Climate 25, 9; 10.1175/JCLI-D-11-00199.1
APPENDIX B
BC and SOx Emissions
Figures B1 and B2 show the SOx and BC emissions for the future RCP3-PD and RCP8.5 scenarios we used for the future projections. Data are given for the large-scale emission regions we considered within the model fitting.
The SOx emissions (Mt yr−1) for ALM, the region of the OECD, and REF. Orange: RCP8.5; blue: RCP3-PD.
Citation: Journal of Climate 25, 9; 10.1175/JCLI-D-11-00199.1
As in Fig. B1, but for the BC emissions (Mt yr−1) for the OECD subregions, REF, ASIA, LAM, and MAF as defined by Table 3.
Citation: Journal of Climate 25, 9; 10.1175/JCLI-D-11-00199.1
APPENDIX C
Individual Contributions to Precipitation Changes
Figure C1 shows the individual contributions of GHG forcing, BC and SOx emissions, and global-mean temperature change on precipitation changes over the 
Individual contributions to regional precipitation change over (a) GRL and (b) EAS. Blue lines represent the results for the RCP3-PD scenario. Orange lines represent results for the RCP8.5 scenario. Short-dashed lines represent the temperature-induced changes as estimated from the extended model (see Table 3). Dotted line represents the GHG-induced change as estimated by the same statistical model. Long-dashed line represents the contribution of sulfate aerosols. Dotted–dashed lines represent BC-induced changes. Thick solid line represents the overall response given by the sum of all components. Thin solid line represents the temperature-induced change as estimated from the basic statistical model. All lines represent the medians of the associated probabilistic projections of the mean changes with respect to preindustrial values.
Citation: Journal of Climate 25, 9; 10.1175/JCLI-D-11-00199.1
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