  Multivariate spatial field data are increasingly common and whose 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,
can be seen to be a measure of linear relationship between two spatial
processes at that frequency band. 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 real data.
We also give natural interpretations to cross-covariance parameters of the
Matern class, where the smoothness indexes dependence at low frequencies while
the range parameter can imply dependence at low or high frequencies. 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. Examining these functions lends insight that would otherwise be
difficult to detect and model using standard cross-covariance formulations.
