AbstractsMathematics

There have been noticeable advancements in developing parametric covariance models for spatial and spatial-temporal data in climate science. However, literature on covariance models for processes on the surface of a sphere is still sparse, due to its mathematical difficulties. In this dissertation, we study random fields and spatial-temporal covariance functions on the surface of a sphere. At first, smooth climate variables need smooth covariance functions. We develop a methodology to construct parametric covariance functions using the great circle distance for spatial processes, geared towards smooth processes on the surface of a sphere. We integrate a non-differential process over a small neighborhood on the surface of a sphere, which result in a smoother process. The resulting model is isotropic and positive definite on the surface of a sphere with the great circle distance, with a natural extension for nonstationarity case. Extensive numerical comparisons of our model, with a Mat?rn covariance model using the great circle distance as well as the chordal distance, are presented. Next, utilizing the one-to-one mapping between the Euclidean distance and the great circle distance, isotropic and positive definite functions in a Euclidean space can be used as covariance functions on the surface of a sphere. However, this approach may result in physically unrealistic distortion on the sphere especially for large distances. We consider several classes of covariance functions on the surface of a sphere, defined with either the great circle distance or the Euclidean distance, and investigate their impact upon prediction. We demonstrate that covariance functions originally defined in the Euclidean distance may not be adequate for some global data. Finally, climate variables often vary in both space and time and it has become popular to model multiple processes jointly. We consider the extension of the bivariate Mat?rn covariance models for spatial-temporal processes on the surface of a sphere. Since data sets have large dimension, a number of challenges arise when performing parameter estimation and prediction. To overcome the computational challenges, we consider the Discrete Fourier Transformation (DFT). We present a method to compute the approximate likelihood efficiently for the case of regularly spaced data of large dimension. Advisors/Committee Members: Jun, Mikyoung (advisor), Subba Rao, Suhasini (committee member), Katzfuss, Matthias (committee member), Saravanan, Ramalingam (committee member).