We introduce a method for spatiotemporal data fusion and demonstrate its performance on three constructed data sets: one entirely simulated, one with temporal speech signals and simulated spatial images, and another with recorded music time series and astronomical images defining the spatial patterns. Each case study is constructed to present specific challenges to test the method and demonstrate its capabilities. Our algorithm, BICAR (Bidirectional Independent Component Averaged Representation), is based on independent component analysis (ICA) and extracts pairs of temporal and spatial sources from two data matrices with arbitrarily different spatiotemporal resolution. We pair the temporal and spatial sources using a physical transfer function that connects the dynamics of the two. BICAR produces a hierarchy of sources ranked according to reproducibility; we show that sources which are more reproducible are more similar to true (known) sources. BICAR is robust to added noise, even in a “worst case” scenario where all physical sources are equally noisy. BICAR is also relatively robust to misspecification of the transfer function. BICAR holds promise as a useful data-driven assimilation method in neuroscience, earth science, astronomy, and other signal processing domains.