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.

气候场重建（CFR）尝试使用诸如树木年轮，冰芯和珊瑚等气候代理来估算过去的气候变量时空场。数据同化（DA）方法是获得CFR的最新且有希望的新方法，该CFR可以最佳地将气候代理与气候模型输出融合在一起。尽管基于DA的CFR的应用日益广泛，但对于被吸收的代理在多大程度上改变了气候模型数据的统计特性，人们知之甚少。为了解决这个问题，我们提出了一种基于功能数据深度的健壮且计算效率高的方法，以评估两个时空过程的分布差异。我们通过比较背景和分析状态，将我们的测试应用于研究基于DA的CFR中的全球和区域代理影响，将其视为时空场的两个样本。我们发现由于代理的同化，分析状态与基于气候模型的背景状态发生了显着变化。此外，即使代理服务器收集站点以外的区域，分析状态和背景状态之间的差异也会随着代理服务器数量的增加而增加。我们的方法允许我们表征代理的附加值，指示分析状态在何时何地与背景状态不同。