Adaptive filtering and maximum entropy spectra with application to changes in atmospheric angular momentum

Abstract

The spectral resolution and statistical significance of a harmonic analysis obtained by low-order maximum entropy methods (MEM) can be improved by subjecting the data to an adaptive filter. This adaptive filter consists of projecting the data onto the leading temporal empirical orthogonal functions obtained from singular spectrum analysis (SSA). The combined SSA-MEM method is applied both to a synthetic time series and a time series of atmospheric angular momentum (AAM) data. The procedure is very effective when the background noise is white and less so when the background noise is red. The latter case obtains in the AAM data. Nevertheless, we detect reliable evidence for intraseasonal and interannual oscillations in AAM. The interannual periods include a quasi-biennial one and a low-frequency one, of 5 years, both related to the E1 Nifio/Southern Oscillation. In the intraseasonal band, separate oscillations of about 48.5 and 51 days are ascertained. , b]. MEM has the advantage of being able to find sharp spectral peaks which the fast Fourier transform (FFT) [cf. Bloomfield, 1976] or lag-autocorrelation methods [Blackman and Tukey, 1959] fail to isolate. However, use of MEM with a number of lags large enough to resolve closely lying peaks in the frequency spectrum usually results in spurious spectral features as well. Getting rid of the spurious peaks often reduces the valuable resolution too.The resolution of spectral peaks in a short, noisy time series will obviously be improved if some of the noise in the series can be removed. The trick is to do this without losing a significant portion of the signal. In this paper we show how the resolution of low-order MEM can be significantly improved by adaptive filtering.The adaptive filter is provided by singular spectrum analysis (SSA) [see Broomhead and King, 1986;Fraedrich, 1986;Vautard and Ghil, 1989]. Although the word "spectrum" is often used to denote a Fourier spectrum, the spectral decomposition of a time series in terms of orthonormal bases other than sines and cosines is obviously possible. The particular interest of SSA is that the basis functions in terms of which the data are decomposed are determined from the time series itself. The geophysical significance of these basis functions, called temporal empirical orthogonal functions (T-EOFs), and of their time-dependent coefficients, the temporal principal components (T-PCs), has attracted substantial interest recently [Vautard and Ghil, 1989; Ghil and Mo, 1991a, b; Rasmusson et al., 1990].Although the T-EOFs are interesting in their own right, we propose here to use them in order to refine a more traditional form of Fourier analysis for two reasons: first, scientists are accustomed to visualizing sinusoidal oscillations; second, additional information can be extracted from the T-PCs by applying MEM to them. In particular, use is made of the fact that the power spectral density of a stationary process is equal to the sum of the power spectra of its T-PCs [Vautard and . SSA also provides the statistical dimens...

Interannual and interdecadal variability in United States surface-air temperatures, 1910-87

“…In M-SSA, these oscillations may span all or some of the time seiles considered. Previous studies have shown that SSA is successful at reliably isolating even weak oscillations in short, noisy climatic seiles (e.g., Rasmusson et al, 1990;Penland et al, 1991;Keppenne and Ghil, 1992). A pair of oscillatory ST-EOFs and their ST-PCs are in quadrature with each other (Ghil and Mo, 1991a, b), thus facilitating the visualization of the oscillation's main phases.…”

“…This method provides high-resolution spectral estimates, but it can yield spurious peaks. Penland et al (1991) have shown that SSA prefiltering can eliminate the spurious peaks, while maintaining the high resolution of MEM. To further corroborate the spectral peaks found by MEM, the multi-taper method (Thomson, 1982) and BlackmanTukey method (Kay, 1988) were also applied.…”

Spatio‐temporal variability in a mid‐latitude ocean basin subject to periodic wind forcing

“…Each time series is prefiltered data-adaptively by projection onto the leading empirical orthogonal functions (EOFs) provided by its singular-spectrum analysis (SSA). The prefiltered time series is then analyzed by the maximum-entropy method (MEM; Penland et al, 1991) and the results are further checked by the multi-taper method (MTM; Thomson, 1982;Mann and Lees, 1996) or the Monte Carlo version of SSA (Allen and Smith, 1996).…”

Decadal-scale variability of warm season temperature in Beijing over the past 2650 years

“…Using SSA to remove the influences of noise, non-linear and low frequency trends, the time series was decomposed with a window-length of 80 (M=80), the Burg covariance estimation [13], and Heuristic significant tests [6]. The 80 principle components (PCs) so obtained are plotted as the rank of their relative powers in Figure 2(a).…”

“…The method is particularly efficient when combined with other techniques and has been widely used in paleoclimatic studies [7][8][9][10]. In this study, MTA [11][12][13] and WSA [14] are used to explore the spectral features of the relevant SSA components.…”