MTM is a method for calculating spectral estimates. Several routines for deriving MTM estimates are discussed and developed byJonathanof UNC's Department of Geological Sciences.
A paper on MTM is a good introduction to the method and code:
Lees, J. M. and J. Park (1995). Multiple-taper spectral analysis: A stand-alone C-subroutine: Computers & Geosciences: 21, 199-236.
A simple set of subroutines in ANSI-C are presented for multiple taper spectrum estimation. The multitaper approach provides an optimal spectrum estimate by minimizing spectral leakage while reducing the variance of the estimate by averaging orthogonal eigenspectrum estimates. The orthogonal tapers are Slepian np prolate functions used as tapers on the windowed time series. Since the taper functions are orthogonal, combining them to achieve an average spectrum does not introduce spurious correlations as standard smoothed single-taper estimates do. Furthermore, estimates of the degrees of freedom and F-test values at each frequency provide diagnostics for determining levels of confidence in narrow band (single frequency) periodicities. The program provided is portable and has been tested on both Unix and Macintosh systems.
This paper is available via ftp.
Lees, J. M.(1995): Reshaping spectrum estimates by removing periodic noise: application to seismic spectral ratios, Geophys. Res. Lett., 22(4), 513-516.
An automated method for removing line spectrum elements embedded in colored spectra is presented. Since smooth spectrum estimates are desired, line spectra tend to smear out over an effective smoothing window. This introduces a bias in the spectrum estimation that can seriously degrade determination of signal-to-noise ratios, spectral deconvolution or any other operation where spectrum shape is important in analysis. Multi-taper analysis provides a simple algorithmic approach to this problem and a simple method of determining where spectral peaks are both significant and contain signal power is suggested. While the method is completely general, an illustration of the technique applied to seismic signals is provided. Examples include estimation of signal-to-noise ratio at the high frequency array at, Pinyon Flat, CA. A comparison of noise spectra line segments and signal spectra line spectra reveals similarities associated with instrument noise and shallow resonances that are stimulated by incoming seismic signals. Identification and removal of the resonances provides a better means of estimating background noise spectrum for the purposes of modeling earthquake source spectra and path effects associated with attenuation.
Mann, M. E. & Lees, J. M. (1996): Robust Estimation of Background Noise and Signal Detection in Climatic Time Series, Climate Change, 33, 409-445.
We present a new technique for isolating climate signals in time series with a characteristic ``red'' noise background which arises from temporal persistence. This background is estimated by a ``robust'' procedure that, unlike conventional techniques, is largely unbiased by the presence of signals immersed in the noise. Making use of multiple-taper spectral analysis methods, the technique further provides for a distinction between purely harmonic (periodic) signals, and broader-band (``quasiperiodic'') signals. The effectiveness of our signal detection procedure is demonstrated with synthetic examples that simulate a variety of possible periodic and quasiperiodic signals immersed in red noise. We apply our methodology to historical climate and paleoclimate time series examples. Analysis of a $\approx$ 3 million year sediment core reveals significant periodic components at known astronomical forcing periodicities and a significant quasiperiodic 100 year peak. Analysis of a roughly 1500 year tree-ring reconstruction of Scandinavian summer temperatures suggests significant quasiperiodic signals on a near-century timescale, an interdecadal 16-18 year timescale, within the interannual El Nino/Southern Oscillation (ENSO) band, and on a quasibiennial timescale. Analysis of the 144 year record of Great Salt Lake monthly volume change reveals a significant broad band of significant interdecadal variability, ENSO-timescale peaks, an annual cycle and its harmonics. Focusing in detail on the historical esimated global-average surface temperature record, we find a highly significant secular trend relative to the estimated red noise background, and weakly significant quasiperiodic signals within the ENSO band. Decadal and quasibiennial signals are marginally significant in this series.
Following are screen shots of MTM and images of various program outputs.
The MTM software and documentation is available at our public FTP site.
1996 Jonathan M. Lees
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