Empirical Orthogonal Representation of Time Series in the Frequency Domain. Part I: Theoretical Considerations

Difficulties in using conventional cross-spectrum analysis to explore atmospheric wave disturbances have indicated the need for some extension of the usual technique. It is suggested here that the eigenvectors of the cross-spectrum matrix be used for interpreting such data. The method is analogous to the use of empirical orthogonal functions applied to band-pass filtered time series. However, the eigenvectors of the cross-spectrum matrix contain additional information concerning phase which is not available from the eigenvectors of the covariance matrix. It is possible to generate a new set of time series which are mutually uncorrelated within a pre-selected frequency interval and which have the same combined variance in the frequency interval as the original set of time series. These new series are obtained by applying the eigenvectors of the cross-spectrum matrix to a set of complex time series involving the original time series and their time derivatives. The application and physical interpretation of the technique are discussed. Examples of the technique applied to atmospheric data are given in a second part of this study.