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Emergent constraints use relationships between future and current climate states to constrain projections of climate response. Here we introduce a statistical, hierarchical emergent constraint (HEC) framework in order to link future and current climates with observations. Under Gaussian assumptions, the mean and variance of the future state are shown analytically to be a function of the signal-to-noise ratio between current climate uncertainty and observation error and the correlation between future and current climate states. We apply the HEC to the climate change, snow-albedo feedback, which is related to the seasonal cycle in the Northern Hemisphere. We obtain a snow-albedo feedback prediction interval of (-1.25,-0.58)%/K. The critical dependence on signal-to-noise ratio and correlation shows that neglecting these terms can lead to bias and underestimated uncertainty in constrained projections. The flexibility of using HEC under general assumptions throughout the Earth system is discussed.

Plain Language Summary Reducing the uncertainty in climate projections has been one of the signature challenges in Earth science because simulated future climate states cannot be directly falsified. We propose a hierarchical statistical framework that formally relates projections of future climate to present-day climate and observations. We show that the future-climate estimate is driven by the correlation between future and present climate variability and the signal-to-noise ratio obtained from observations and present climate. This framework is applied to a future northern hemispheric climate projection that is influenced by the snow-albedo feedback, which is an amplification of temperature due to reduced snow extent as a consequence of anthropogenic CO2 emissions. We show that the climate change snow-albedo temperature sensitivity ranges from (-1.25,-0.58)%/K. The flexibility of this approach can be applied more broadly to constrain climate projections across the Earth system.