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Multivariate spatial field data are increasingly common and their modeling typically relies on building cross-covariance functions to describe cross-process relationships. An alternative viewpoint is to model the matrix of spectral measures. We develop the notions of coherence, phase, and gain for multidimensional stationary processes. Coherence, as a function of frequency, is a measure of linear relationship between two spatial processes at that frequency. We use the coherence function to illustrate fundamental limitations on a number of previously proposed constructions for multivariate processes, suggesting these options are not viable for data modeling. We also give natural interpretations to cross-covariance parameters of the Matérn class, where the cross-smoothness controls the decay of coherence at infinitely high frequencies, and the cross-range parameter controls the frequency of greatest coherence …