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We built the model in three steps. First we used the model of Raftery et al.1 for probabilistic forecasting of CO2 emissions. This involves Bayesian hierarchical models for fertility and mortality, and hence population, for GDP, and for carbon intensity for each country. We then built Bayesian time series models of CMIP 5 models forecasts and historical simulations, together with the actual historical temperature anomalies, in order to estimate the bias and measurement error variance of the CMIP 5 models. Finally we took the probabilistic forecast of CO2 emissions from the first step as input, and linked the cumulative emissions to the CMIP 5 model forecasts.

The Coupled Model Intercomparison Project Phase 5 (CMIP 5)8, is a standard experimental protocol for studying the output of coupled ocean-atmosphere general circulation models (GCMs). GCMs take inputs such as future forcings, land use, and make corresponding forecasts of global climate. CMIP 5 models take inputs with the RCPs, converting them into radiative forcings and other climate factors, and make forecasts correspondingly. The data are illustrated in Figure 6.

CMIP 5 models. Both model simulations and the observed data are adjusted such that the mean anomaly between 1861–1880 for each model and the observed data is 0. The black line represents the observed data, while the colored lines are for the CMIP 5 model simulations. In total there are 39 CMIP 5 models, but we include only 10 models in this plot for visual clarity.

In this paper, we use our forecasting emissions trajectories as inputs to the GCMs. Instead of running each GCM for each input trajectory of emissions forecasts for probabilistic forecasts, which would not be feasible, we used the existing forecasts with different scenarios of emissions forecasts and developed statistical models of the relationship between CO2 emissions and model forecasts in CMIP 5 models. We found that the forecasts of each CMIP 5 model match linearly with the cumulative emissions for different RCP scenarios, with different correlation and scale. We took our CO2 emissions forecast as input, and used the linear relationship between global mean temperature and the cumulative emissions to generate probabilistic forecasts of future global mean temperature anomalies.

We generated forecasts based on each CMIP 5 model, and then combined these forecasts as an ensemble. For each model, our forecast is based on two parts. The first part takes CO2 emissions trajectories as input, and uses the linear relationship between cumulative CO2 emissions and global mean temperature predictions to generate the CMIP 5 model temperature forecast. The second part takes the uncertainty and bias of the CMIP 5 model forecast directly, based on the difference between historical simulations and the historical temperature anomalies.

For each CMIP 5 model, the historical simulations are generated similarly under different scenarios. We denote the historical simulation of the global mean temperature anomaly in year t from CMIP 5 model i by xi,t. We take the HadCRUT 4 observed temperature anomalies as the observations of the global surface-mean temperature, and denote them by $y˜t$. Then we denote the difference between the CMIP model backcast and the observed value by $yi,t=xi,t−y˜t$. We denote the true temperature anomalies by $z˜t$. Then $zi,t=xi,t−z˜t$ is the difference between the unobserved true global surface-mean temperature anomaly and the backcasts from the CMIP models. Analysis of the autocorrelation and partial autocorrelation functions of the estimated zi,t time series shows that it is well represented by a first-order autoregressive, or AR(1) model. This leads us to specify the following model:

where

where $Vt2$ is the variance of the measurement error of HadCRUT 4 observations, and is provided in the HadCRUT 4 dataset.

The forecast anomalies are modeled as linearly related to the cumulative emissions. Therefore, we collected the input cumulative CO2 emissions, denoted by cj,t for scenario or RCP j and year t, and the forecast anomalies from each model xi,j,t, where i indexes models, j indexes scenarios and t indexes years. Then we model the historical simulations as:

This relationship is close to being perfectly linear, as can be seen from the simple linear regression results in Table 2.

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