In spatial statistics, a wide variety of methods and models have been developed in Euclidean spaces. Additional theory and methods are needed for analyzing processes and phenomena on the sphere, many of which are of utmost importance in the geophysical sciences. Data from global networks of in situ and satellite sensors, for instance, are used to monitor a wide array of important climatological variables such as temperature and precipitation. The goal of this project is to study random processes on the sphere beyond the usual homogeneity assumption using two approaches. The first approach is to consider axially symmetry on the sphere, where the random process exhibits longitudinal symmetry rather than being rotation invariant on the entire sphere. The second approach of this research extends the intrinsic random functions and generalized covariance functions to processes on the sphere. The notion of intrinsic random functions has been developed in order to handle a process with an unknown and possibly non-constant mean function while preserving a linkage to stationarity. The investigators plan to achieve a fundamental understanding of the covariance structures of the non-homogenous processes on the sphere and to develop associated estimation procedures. In addition, these approaches are extended to more sophisticated situations including multivariate random processes and spatio-temporal processes on the sphere. All of these methods and models are applied to global-scale temperature data from surface and satellite sensors.
The models and methods developed here provide new tools for improving our understanding of global-scale phenomena in general and multi-decadal temperature variations in particular. The motivation of this project comes from a desire to understand the statistical characteristics of global-scale temperature variations during the instrumental (1880s onward) and satellite (1979 onward) periods. As a result, two important and widely used data sets are analyzed here:(1) tropospheric temperature data from National Oceanic and Atmospheric Administration satellite-based Microwave Sounding Unit and (2) surface-based instrumental data from the Hadley Centre in the United Kingdom and Climatic Research Unit at the University of East Anglia. The geostatistical analysis of these data sets on the sphere provides profound information on the state of our changing planetary environment. The project also establishes and encourages research and education collaborations between Indiana University and Mississippi State University.
