  In this paper, we develop a method for estimating and clustering
two-dimensional spectral density functions (2D-SDFs) for spatial data from
multiple subregions. We use a common set of adaptive basis functions to explain
the similarities among the 2D-SDFs in a low-dimensional space and estimate the
basis coefficients by maximizing the Whittle likelihood with two penalties. We
apply these penalties to impose the smoothness of the estimated 2D-SDFs and the
spatial dependence of the spatially-correlated subregions. The proposed
technique provides a score matrix, that is comprised of the estimated
coefficients associated with the common set of basis functions representing the
2D-SDFs. {Instead of clustering the estimated SDFs directly, we propose to
employ the score matrix for clustering purposes, taking advantage of its
low-dimensional property.} In a simulation study, we demonstrate that our
proposed method outperforms other competing estimation procedures used for
clustering. Finally, to validate the described clustering method, we apply the
procedure to soil moisture data from the Mississippi basin to produce
homogeneous spatial clusters. We produce animations to dynamically show the
estimation procedure, including the estimated 2D-SDFs and the score matrix,
which provide an intuitive illustration of the proposed method.
