Some Considerations on the Continuous Space-time Spectral Analysis of Atmospheric Disturbances

Abstract

In this paper are treated some problems in estimating the continuous space-time spectra with separation into the westward and the eastward moving components using the Fourier transforms of space-time covariance functions, which are applicable to the analysis of the space-time series (aperiodic for both space and time) sampled from a limited area such as the tropical Pacific.
However, at the beginning of this paper, several types of space-time spectra are also discussed in order to make clear the relation of the present study to the previous ones which have been mostly concerned with the discontinuous space-continuous time spectra of the space-time series (periodic for space but aperiodic for time) sampled globally but not locally along latitude circles.
In Chapter I the s p ace-time spectra discontinuous for both space and time with separation into the westward and the eastward moving wave components are first given as the Fourier transforms of the covariance functions defined by the averages over space and time. Then the spacetime spectra discontinuous for space but continuos for time and those continuous for both space and time are derived respectively as the limiting cases of the spectra discontinuous for both space and time. The relation of the continuous to disontinuous spectra is also discussed.
In the statistical approach to space-time series, the space-time spectra are defined as the Fourier tranforms of the space-time covariance functions defined by the averages over the ensembles. It is shown in Chapter II that the general type of space-time spectra appear as the sum of nine defferent types of space-time spectra. If the space-time spectra are absolutely continuous for both space and time, they have space-time spectral density functions.
Statistically speaking, the continuous space-time spectra discussed in Chapter I simply mean the sample spectra of the continuous parameter space-time series which we assume to exist for continuous space and time. However in most cases the space-time series are sampled at equidistant space and time intervals, and therefore we are faced with the problem of wavenumber-frequency aliasing which will be discussed briefly.
The spectra of the space-time series thus sampled are then required. to converge in some probability modes to the true spectral density functions as defined by the ensemble averags. In order for such sample spectra to be consistent estimates to the true spectral density functions, two different space-time spectral windows which smooth the sample spectra are presented. The large sample properties of such smoothed space-time spectra. ara examined and used to derive approximations to their sampling distributions based on x²-distribution.
One of the mo s t important practical problems in the spectral analysis of the continuous parameter space-time series when sampled at equidistant space and time intervals will arise the fact that the very low wavenumberfrequencies appear with much greater sensitivity than others. In Chapter III, two types of high wavenumber-frequency pass filters generated by three different space-time smoothing functions are presented to eliminate the very low wavenumber-frequencies but leave the synoptic scale wave disturbances. In some cases the very high wavenumber-frequencies appear as noise rather than as signal. To eliminate such very high wavenmberfrequencies as well as the very low wavenumber-frequencies but leave the synoptic-scale wave disturbances, a type of intermediate wavenumberfrequency pass filter is also presented. The relatihnship between the filtered and the original space-time spectra is also discussed briefly.