Julien Emile-Geay , Greg Hakim , Frédéri Viens, Feng Zhu , Daniel E. Amrhein

Paleoclimate reconstructions are increasingly central to climate assessments, placing recent and future variability in a
broader historical context. Several estimation methods produce ensembles of climate trajectories that practitioners often want to compare to other ensembles, or to deterministic trajectories produced by other methods such as global climate models. Of particular interest are so-called “offline” data assimilation (DA) methods, which have recently been adapted to paleoclimatology, and lack an explicit estimate of reconstruction error covariability in time. As a result, offline DA ensemble members are not true system trajectories. We show that this atemporality obscures quantitative comparisons, particular when considering the ensemble mean in isolation. We propose several parametric resampling methods to introduce a priori constraints on temporal covariance among ensemble members. We also propose a general framework to carry out quantitative comparisons between ensembles using a “plume distance” framework, which provides a norm in the same physical units as the variable of interest. We apply these tools to three paleoclimate questions: (1) Comparing global mean surface temperature in the online and offline versions of the Last Millennium Reanalysis; (2) Comparing global mean surface temperature from these two ensembles to simulations of the Paleoclimate Model Intercomparison Project past1000 ensemble; and (3) Comparing northern hemisphere mean surface temperature from the offline DA ensemble to the Büntgen et al. (2021) ensemble. The proposed methodology is implemented in an open-source Python package, inviting re-use and extensions. We also discuss possible applications of the plume distance framework to a broad array of problems where ensemble comparisons arise.