Nonhomogeneous Covariance Estimation on the Sphere
Peter Guttorp, Paul Sampson and Dean Billheimer
Sampson and Guttorp (1992)
proposed an approach to estimation of heterogeneous spatial covariance
that allows estimates for any pair of locations in a planar monitoring
field, whether monitored or not. The basic idea is that dispersion (a
concept closely related to covariance) is a smooth function of
geographic coordinates and is locally anisotropic. By transforming the
geographic map appropriately we get a new representation in which
dispersion is a monotone function of distance, i.e., where the
dispersion is homogeneous and isotropic. Consequently, the standard
kriging assumptions of isotropy and homogeneity are met in this
transformed space. In order to generalize the technique outlined above
to the spherical case, we will utilize three components: an approach to
multidimensional scaling on a sphereachieving a representation of
measurement locations in which distance on the sphere corresponds as
closely as possible to dispersion, a method of estimating spherical
covariance in the isotropic homogeneous case, and a method to map
smoothly once set of spherical coordinates into another.
There are two different avenues for generalizing our modeling of
heterogeneous spatial covariance to the sphere: one possibility is to
generate transformed site coordinates on a transformed space that is not
a sphere, but a closed twodimensional manifold embedded in
threedimensional Euclidean space. The difficulty is then to develop
isotropic covariance structures on such manifolds (Darling, 1991,
provides an approach to this). Alternatively, one may map the sphere
into itself. Different continuous mappings of the sphere onto itself can
be used to characterize anisotropic covariance structures on the sphere,
for which currently no characterizations are known. The advantage with
the latter approach is that Jones (1963) characterized covariance
structures on the sphere, utilizing earlier results from Obhukov on
isotropic random fields on the sphere.
Much effort in assessing the effect of the continuing increase in
concentration of many greenhouse gases in the troposphere has been
directed towards assessing changes in global temperature. While this is
undoubtedly not the most sensitive indicator of greenhouse effects (IPCC,
1990), it has the advantage of availability of fairly long measurement
series from large parts of the globe. There are numerous problems with
temperature data. Corrections have to be made for urban development, for
changes in measurement techniques and instrumentation, for changes in
thermometer location and enclosure, and for a variety of other effects
that induce systematic errors into the record. The resulting corrected
data sets (Jones, 1988) have varying degrees of global coverage, and
vastly varying numbers of stations in the equal area regions for which
temperature averages are computed. Typically, each regional average is a
simple average of the corrected data obtained from stations or ships in
that region.
In principle, it should be possible to use the global covariance
procedure given in the previous subsection to produce optimal least
squares predictions of global average temperature. However, the
computational problems appear formidable. Rather, we propose a modified
approach, based on a preliminary regional fit, followed by a global fit
of regional averages. The first step is to use the estimator of
heterogeneous covariance for the plane, using for each region data from
it and its surrounding regions. Missing data will be interpolated using
a Kalman filter approach (Jones, 1984). Using a model due to Solna and
Switzer (1992) we can determine regional trends and anomalies together
with appropriate standard errors.
The second step is to combine the average residuals from the regional
estimates to estimate a global covariance structure, taking into account
the different uncertainties in each regional estimate. We then can do a
linear least squares prediction of global temperature trend and global
annual anomaly with standard errors that better reflect the actual
uncertainty in these estimates.
Another application of this methodology, of particular interest to the
Boeing corporation, is to estimate wind speed and direction around the
globe in order to be able to compute optimal flight paths. This seems
like a good opportunity for universityindustry collaboration: Boeing
atmospheric scientists could provide scientific direction, information
and data, while the NRCSE could provide statistical expertise,
computing, and personnel.
Reference: Barnali
Das dissertation. 

