  Analyzing massive spatial datasets using Gaussian process model poses
computational challenges. This is a problem prevailing heavily in applications
such as environmental modeling, ecology, forestry and environmental heath. We
present a novel approximate inference methodology that uses profile likelihood
and Krylov subspace methods to estimate the spatial covariance parameters and
makes spatial predictions with uncertainty quantification. The proposed method,
Kryging, applies for both observations on regular grid and irregularly-spaced
observations, and for any Gaussian process with a stationary covariance
function, including the popular $\Matern$ covariance family. We make use of the
block Toeplitz structure with Toeplitz blocks of the covariance matrix and use
fast Fourier transform methods to alleviate the computational and memory
bottlenecks. We perform extensive simulation studies to show the effectiveness
of our model by varying sample sizes, spatial parameter values and sampling
designs. A real data application is also performed on a dataset consisting of
land surface temperature readings taken by the MODIS satellite. Compared to
existing methods, the proposed method performs satisfactorily with much less
computation time and better scalability.
