We directly exploit the stochasticity of the internal variability, and the linearity of the forced response to make global temperature projections based on historical data and a Green’s function, or Climate Response Function (CRF). To make the problem tractable, we take advantage of the temporal scaling symmetry to define a scaling CRF characterized by the scaling exponent H, which controls the long-range memory of the climate, i.e. how fast the system tends toward a steady-state, and an inner scale τ≈2\documentclass[12pt]{minimal}
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\begin{document}$$\tau \approx 2$$\end{document} years below which the higher-frequency response is smoothed out. An aerosol scaling factor and a non-linear volcanic damping exponent were introduced to account for the large uncertainty in these forcings. We estimate the model and forcing parameters by Bayesian inference which allows us to analytically calculate the transient climate response and the equilibrium climate sensitivity as: 1.7-0.2+0.3\documentclass[12pt]{minimal}
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\begin{document}$$1.7^{+0.3} _{-0.2}$$\end{document} K and 2.4-0.6+1.3\documentclass[12pt]{minimal}
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\begin{document}$$2.4^{+1.3} _{-0.6}$$\end{document} K respectively (likely range). Projections to 2100 according to the RCP 2.6, 4.5 and 8.5 scenarios yield warmings with respect to 1880–1910 of: 1.5-0.2+0.4K\documentclass[12pt]{minimal}
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\begin{document}$$1.5^{+0.4}_{-0.2}K$$\end{document}, 2.3-0.5+0.7\documentclass[12pt]{minimal}
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\begin{document}$$2.3^{+0.7}_{-0.5}$$\end{document} K and 4.2-0.9+1.3\documentclass[12pt]{minimal}
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\begin{document}$$4.2^{+1.3}_{-0.9}$$\end{document} K. These projection estimates are lower than the ones based on a Coupled Model Intercomparison Project phase 5 multi-model ensemble; more importantly, their uncertainties are smaller and only depend on historical temperature and forcing series. The key uncertainty is due to aerosol forcings; we find a modern (2005) forcing value of [-1.0,-0.3]Wm-2\documentclass[12pt]{minimal}
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\begin{document}$$[-1.0, -0.3]\, \,\,\mathrm{Wm} ^{-2}$$\end{document} (90 % confidence interval) with median at -0.7Wm-2\documentclass[12pt]{minimal}
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\begin{document}$$-0.7 \,\,\mathrm{Wm} ^{-2}$$\end{document}. Projecting to 2100, we find that to keep the warming below 1.5 K, future emissions must undergo cuts similar to RCP 2.6 for which the probability to remain under 1.5 K is 48 %. RCP 4.5 and RCP 8.5-like futures overshoot with very high probability.
