  Estimation of the covariance structure of spatial processes is of fundamental
importance in spatial statistics. In the literature, several non-parametric and
semi-parametric methods have been developed to estimate the covariance
structure based on the spectral representation of covariance functions.
However,they either ignore the high frequency properties of the spectral
density, which are essential to determine the performance of interpolation
procedures such as Kriging, or lack of theoretical justification. We propose a
new semi-parametric method to estimate spectral densities of isotropic spatial
processes with irregular observations. The spectral density function at low
frequencies is estimated using smoothing spline, while a parametric model is
used for the spectral density at high frequencies, and the parameters are
estimated by a method-of-moment approach based on empirical variograms at small
lags. We derive the asymptotic bounds for bias and variance of the proposed
estimator. The simulation study shows that our method outperforms the existing
non-parametric estimator by several performance criteria.
