Statistical inference on spatiotemporal processes is a fundamental problem in many fields including Ecology, Oceanography, and Climatology. Of particular interest to the paleoclimate community is the study of Climate Field Reconstructions (CFRs) with seasonal to annual resolution spanning the last several millennia. CFRs attempt to recover spatiotemporal fields of climate variables, using proxy records of past climate variability, and have emerged as important tools for studying the mechanisms of climate change. Motivated by assessing differences between CFRs, we propose a new method for evaluating the differences in the distributions of two spatiotemporal processes by using the notions of data depth and functional data. Our test is robust, computationally efficient, distribution free and and has a convenient asymptotic distribution. We apply our test to study global and regional proxy influence on a Data Assimilation based CFR by comparing its background and analysis states. We find that there is a steadily increasing divergence between the state’s distributions over time, indicating increasing proxy influence, and that proxy influence can extend far beyond collection sites.