Eigenstructures of Spatial Design Matrices

In estimating the variogram of a spatial stochastic process, we use a spatial design matrix. This matrix is the key to Matheron's variogram estimator. We show how the structure of the matrix for any dimension is based on the one-dimensional spatial design matrix, and we compute explicit eigenvalues and eigenvectors for all dimensions. This design matrix involves Kronecker products of second order finite difference matrices, with cosine eigenvectors and eigenvalues. Using the eigenvalues of the spatial design matrix, the statistics of Matheron's variogram estimator are determined. Finally, a small simulation study is performed.