INQUA Sub-Commission on Data-Handling Methods

Newsletter 16: July 1997

SPECTRUM: A PC-program for spectral analysis of unevenly spaced time series

2) Universität Kiel
Geologisch-Paläontologisches Institut
Olshausenstr. 40
D-24118 Kiel, Germany

Spectral analysis is an important tool for deciphering information from palaeoclimatic time series in the frequency domain. It is used to detect the presence of harmonic signal components in a time series or to obtain phase relations between harmonic signal components being present in two different time series (cross-spectral analysis). In practice a problem arises because in most cases data points are unevenly spaced along the time axis, whereas the majority of computer programs for spectral analysis require evenly spaced observations in time. Therefore, one usually interpolates a time series to some evenly spaced interval. However, this interpolation is equivalent to a low-pass filtering (Horowitz, 1974; IEEE Transactions on Acoustics, Speech and Signal Processing 22, 22-27) and may lead to a severe underestimation of high frequency components in a spectrum (`reddening' of a spectrum) independent of the employed interpolation scheme.

In order to avoid these problems a menu-driven PC program (SPECTRUM) has been developed. SPECTRUM is based on the Lomb-Scargle Fourier-Transformation for unevenly spaced time series, which is equivalent to the fit of sine- and cosine functions to a time series by means of a least-squares criterium. Hence, unevenly spaced time series can be directly analysed without preceding interpolation. Input data is read from unformatted ASCII files. SPECTRUM can perform:

Harmonic Analysis

Purpose:

Detection of periodic signal components in a noise background

Method:

Tests after Fisher and Siegel

Features:

Subtraction of strong signal components (e.g., quasi-periodic peaks) that may mask weak harmonic components

Autospectral Analysis

Purpose:

Variance distribution as function of frequency of a single time series

Method:

Welch-Overlapped-Segment-Averaging Method

Features:

- various windows (Welch, Hanning, Triangular, Blackman-Harris)

- linear or logarithmic ordinate scales

- linear detrending of the data

- confidence intervals for various levels of significance

- option for integrating parts of a spectrum

Cross-spectral Analysis
SPECTRUMs cross-spectral analysis module does not require a common time axis of the two processed time series.

Cross-Amplitude Spectrum

Purpose:

Determination of frequencies where two time series have common variance

Coherency-Function

Purpose:

Measure of the extent to which the second time series may be predicted from the first time series by a linear least-squares relationship

Features:

- confidence intervals are based on the coherency distribution function

- test for significant coherency

- bias correction for inherent coherency overestimation

Phase-Function

Purpose:

Measure of the time delay (lead / lag) between two time series as function of frequency, assuming a linear system

Features:

- option to invert a time series (equivalent to a 180° phase shift)

- confidence intervals

- marker to measure phase angle on screen

- option to align processed time series

Analytical results are supplemented by statistical parameters that allow the evaluation of the results. During the analysis the user is guided by a variety of messages. Results are displayed graphically and can be saved as plain ASCII files. In addition, SPECTRUM can produce script files for the GNUPLOT (Version 3.6) graphics program. This allows to plot results on a variety of output devices including postscript and Windows metafiles. Additional tools for visualizing time series data and sampling intervals further facilitate the analysis.

The hardware requirements for SPECTRUM are: 1) MS-DOS compatible PC with at least a 80386 CPU and numeric coprocessor, 2) VGA (or above) graphics adapter, 3) > 580 kBytes of available conventional memory/about 1 Mbytes EMS memory, and 4) MS-DOS 5.0 or higher. SPECTRUM is freeware and can be obtained via anonymous ftp from infosrv.rz.uni-kiel.de (directory: /pub/sfb313/mschulz). The compressed program can be found in the file SPEC20B.ZIP. (Make sure to set the file transfer mode to BINARY before copying the file. In order to unpack the ZIP- archives one needs either UNZIP.EXE or PKUNZIP.EXE. The former is also available from the above ftp-server.) Installation of the program proceeds by copying the archive to an appropriate directory and unzipping it using the '-d' option. Running SPC.BAT starts the program. In case that SPECTRUM refuses to work there is usually not enough conventional memory available; hints for troubleshooting are provided in the README.1ST file. A detailed documentation in postscript format can be found in the archive SPEC20D.ZIP. This manual is a preprint of the following manuscript: "Schulz, M. and Stattegger, K.: Spectral analysis of unevenly spaced paleoclimatic time series. Computers and Geosciences (in press)". This reference should be cited in publications using SPECTRUM for time series analyses.