In many environmental applications (e.g., sections 2.3.1 and 2.4.3) there is
need for a covariance estimation technique that does not assume isotropy
or homogeneity. Sampson and Guttorp (1992) introduced an estimation
technique that
utilizes the fact that environmental monitoring data are taken
over time, and so provide sequences from which one can compute
empirical spatial covariances (consistent with the objective analysis
approach of atmospheric science). Given empirical correlation estimates
among the monitored sites, we use multi-dimensional scaling to transform
the geographic coordinates-the G-plane ()-into a space where distance
corresponds to spatial dispersion, defined as
for a variance-standardized field
at arbitrary geographic locations
and
. We call this the D-space
(usually also
in analyses to date).
Hence, in the D-space the correlation structure is isotropic and
homogeneous, so standard variogram estimation techniques apply. We
use a pair of thin-plate splines to map the coordinates of the G-plane into
those of the D-space. Given any two points in the G-plane, we obtain an
estimate of their correlation by mapping them into the D-space,
computing their D-space distance, and reading the correlation off the
estimated variogram. Simulation studies indicate that this method yields
reasonable results both for homogeneous and heterogeneous situations
(Meiring, 1995), and it has been successful in dealing with both spatial
(Sampson and Guttorp, 1992; Guttorp et al., 1992) and spatio-temporal
(Guttorp et al., 1994; Meiring, 1995) atmospheric pollution processes.
The problem of estimating spatially varying cross-covariances, on the
other hand, has not been adequately addressed. In particular, it is
important to take into account the asymmetry in space of
cross-covariances (e.g., certain chemical transformations are more likely
to occur directionally in time, so at a fixed time the covariance between
SO at location
and SO
at location
is not the same as that
between SO
at location
and SO
at location
).
Recent statistical models in the literature for analysis of spatial-temporal fields have involved interpolation of both a space-time mean field or trend and the residuals or deviations from the mean field (Le and Petkau, 1988; Guttorp et al., 1992; Loader and Switzer, 1992; Oehlert, 1993). The simplest models assume that the spatial and temporal correlation structures of these residuals are separable, i.e., the spatial correlation structure is the same at all times. Such an assumption is unwarranted for hourly ozone data on small to medium spatial meso-scale (Eder et al., 1993; Guttorp et al., 1994). Rather, there is a periodic diurnal structure, with strong spatial covariance during the high ozone periods in the afternoon, and very weak covariance during low ozone times at night. To model the joint spatial-temporal correlation structure, we have considered the spatial interpolation of site-specific temporal correlation models (Sampson et al., 1994), and more recently a ``multivariate'' approach for joint analysis of each of the 24 hours of the day. The asymmetry of cross-covariance discussed above becomes even more pronounced for space-time covariance.