Research Summary

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 sphere--achieving 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 two-dimensional manifold embedded in three-dimensional 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 university-industry 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.