  Spatio-temporal covariances are important for describing the spatio-temporal
variability of underlying random processes in geostatistical data. For
second-order stationary processes, there exist subclasses of covariance
functions that assume a simpler spatio-temporal dependence structure with
separability and full symmetry. However, it is challenging to visualize and
assess separability and full symmetry from spatio-temporal observations. In
this work, we propose a functional data analysis approach that constructs test
functions using the cross-covariances from time series observed at each pair of
spatial locations. These test functions of temporal lags summarize the
properties of separability or symmetry for the given spatial pairs. We use
functional boxplots to visualize the functional median and the variability of
the test functions, where the extent of departure from zero at all temporal
lags indicates the degree of non-separability or asymmetry. We also develop a
rank-based nonparametric testing procedure for assessing the significance of
the non-separability or asymmetry. The performances of the proposed methods are
examined by simulations with various commonly used spatio-temporal covariance
models. To illustrate our methods in practical applications, we apply it to
real datasets, including weather station data and climate model outputs.
