COHERENCE FOR MULTIVARIATE RANDOM FIELDS

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. These interpretations provide warnings for particular parameter combinations that imply potentially non-physical relationships between variables. Estimation follows from smoothed multivariate periodogram matrices. We illustrate the estimation and interpretation of these functions on two datasets, forecast and reanalysis sea level pressure and geopotential heights over the equatorial region.