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. Paleoclimate reconstructions are increasingly central to climate assessments, placing recent and future variability in a broader historical context. Several estimation methods produce plumes of climate trajectories that practitioners often want to compare to other reconstruction ensembles, or to deterministic trajectories produced by other means, such as global climate models. Of particular interest are “offline” data assimilation (DA) methods, which have recently been adapted to paleoclimatology. Offline DA lacks an explicit model connecting time instants, so its ensemble members are not true system trajectories. This obscures quantitative comparisons, particularly when considering the ensemble mean in isolation. We propose several resampling methods to introduce a priori constraints on temporal behavior, as well as a general notion, called plume distance, to carry out quantitative comparisons between collections of climate trajectories ("plumes"). The plume distance provides a norm in the same physical units as the variable of interest (e.g. ^oC for temperature), and lends itself to assessments of statistical significance. We apply these tools to four paleoclimate comparisons: (1) global mean surface temperature (GMST) in the online and offline versions of the Last Millennium Reanalysis (v2.1); (2) GMST  from these two ensembles to simulations of the Paleoclimate Model Intercomparison Project past1000 ensemble; (3) LMRv2.1 to the PAGES 2k (2019) ensemble of GMST and (4) northern hemisphere mean surface temperature from LMR v2.1  to the Büntgen et al. (2021) ensemble. Results generally show more compatibility between these ensembles than is visually apparent. The proposed methodology is implemented in an open-source Python package, and we discuss possible applications of the plume distance framework beyond paleoclimatology.