Static and Dynamic Constraints on the Estimation of Space-Time Covariance and Wavenumber-Frequency Spectral Fields

Abstract

A straightforward method for estimating the spectral characteristics of spatially distributed fields is to collect a suitable quantity of station-to-station auto- and cross-covariance data, interpolate these onto a regular spatial grid, and perform a Fourier transformation. If the resulting statistics are to be used for the design of operational data processing algorithms, however, it is essential that they satisfy certain criteria of internal consistency, usually expressible via constraints of positive-semidefiniteness of appropriate spectral matrices. In addition, it appears desirable, when static or dynamic relationships among the variables of interest are known, to force the statistical estimates to conform to these. The paper describes some recent research into the statistical analysis of: 1) observed upper-air geopotential and winds, with constraints imposed by the balance equation and by positive-semidefiniteness of the wavenumber spectral matrix; and 2) a Monte-Carlo simulated second-order wave field, with constraints imposed by the dynamical law and by positive-semidefiniteness of the wavenumber-frequency spectral matrix.