I will present three projects that are related to the modeling of covariance structures on the Euclidean space and on sphere.

Firstly, we propose a method to model isotropic random field on sphere, where a tapered Matérn covariance function is used to capture the local behavior while a nonparametric expansion controls the behavior at large distances. A model selection procedure based on residual sum of squares with penalization is used to reduce over fitting.

Secondly, we address the issue of modeling axially symmetric spatial random fields on sphere with a kernel convolution approach. The observed random field is generated by convolving a latent uncorrelated random field with a class of Matérn type kernel functions. By allowing the parameters in the kernel functions to vary with locations, we are able to generate a flexible class of covariance functions and capture the nonstationary properties. We use precomputation tables to speed up the computation. For regular grid data on sphere, the block circulant property of the covariance matrix enables us to use Fast Fourier Transform (FFT) to get the determinant and the inverse efficiently.

Thirdly, we proposed a semiparametric variogram estimating method through its spectral representation to model the intrinsically stationary random on R². The low frequency part of the spectral density is estimated by solving a regularized inverse problem through quadratic programming. The behavior at high frequencies, however, is modeled via a parametric tail in the form of a power decaying function. The power parameter in the tail is estimated by a log likelihood method.

All proposed methodologies are supplemented with simulation studies and real data analyses.