On the consistency of non-parametric estimation of the spatial spectrum

The problem of estimating the spectra of a stochastic tempo-spatial field is of considerable interest, owing to their numerous applications in many fields. Indeed a wide selection of estimators are described in the literature, ranging from the classical to the advanced and sophisticated. We investigate the statistics of the straight-forward, classical, non-parametric spatial spectrum estimator. We show that if the field is temporally-ergodic, then the spatial spectrum estimator can be made consistent by employing appropriate temporal averaging. This result is particularly interesting in light of the well-known fact that the analogous, classical temporal spectrum estimator, is inherently inconsistent