  Quantifying the risk of global warming exceeding critical targets such as 2.0
K requires reliable projections of uncertainty as well as best estimates of
Global Mean Surface Temperature (GMST). However, uncertainty bands on GMST
projections are often calculated heuristically and have several potential
shortcomings. In particular, the uncertainty bands shown in IPCC plume
projections of GMST are based on the distribution of GMST anomalies from
climate model runs and so are strongly determined by model characteristics with
little influence from observations of the real-world. Physically motivated
time-series approaches are proposed based on fitting energy balance models
(EBMs) to climate model outputs and observations in order to constrain future
projections. It is shown that EBMs fitted to one forcing scenario will not
produce reliable projections when different forcing scenarios are applied. The
errors in the EBM projections can be interpreted as arising due to a
discrepancy in the effective forcing felt by the model. A simple time-series
approach to correcting the projections is proposed based on learning the
evolution of the forcing discrepancy so that it can be projected into the
future. These approaches give reliable projections of GMST when tested in a
perfect model setting, and when applied to observations lead to well
constrained projections with lower mean warming and narrower projection bands
than previous estimates. Despite the reduced uncertainty, the lower warming
leads to a greatly reduced probability of exceeding the 2.0 K warming target.
