Evaluating proxy influence in assimilated paleoclimate reconstructions -- Testing the exchangeability of two ensembles of spatial processes

Climate field reconstructions (CFR) attempt to estimate spatiotemporal fields of climate variables in the past using climate proxies such as tree rings, ice cores, and corals. Data Assimilation (DA) methods are a recent and promising new means of deriving CFRs that optimally fuse climate proxies with climate model output. Despite the growing application of DA-based CFRs, little is understood about how much the assimilated proxies change the statistical properties of the climate model data. To address this question, we propose a robust and computationally efficient method, based on functional data depth, to evaluate differences in the distributions of two spatiotemporal processes. We apply our test to study global and regional proxy influence in DA-based CFRs by comparing the background and analysis states, which are treated as two samples of spatiotemporal fields. We find that the analysis states are significantly altered from the climate-model-based background states due to the assimilation of proxies. Moreover, the difference between the analysis and background states increases with the number of proxies, even in regions far beyond proxy collection sites. Our approach allows us to characterize the added value of proxies, indicating where and when the analysis states are distinct from the background states.