SPECTRUM: SPECTRAL ANALYSIS OF UNEVENLY

SPACED PALEOCLIMATIC TIME SERIES

MICHAEL SCHULZ1 and KARL STATTEGGER2

1Universita t Kiel, Sonderforschungsbereich 313, Heinrich-Hecht-Platz 10, D-24118 Kiel, Germany, and2Universita t Kiel, Geologisch-Pala ontologisches Institut, Olshausenstr. 40, D-24118 Kiel, Germany

(e-mail: mschulz@sfb313.uni-kiel.de)

(Received 19 February 1997; revised 21 May 1997)

AbstractA menu-driven PC program (SPECTRUM) is presented that allows the analysis of unevenlyspaced time series in the frequency domain. Hence, paleoclimatic data sets, which are usually irregularlyspaced in time, can be processed directly. The program is based on the LombScargle Fourier trans-form for unevenly spaced data in combination with the Welch-Overlapped-Segment-Averaging pro-cedure. SPECTRUM can perform: (1) harmonic analysis (detection of periodic signal components), (2)spectral analysis of single time series, and (3) cross-spectral analysis (cross-amplitude-, coherency-, andphase-spectrum). Cross-spectral analysis does not require a common time axis of the two processedtime series. (4) Analytical results are supplemented by statistical parameters that allow the evaluation ofthe results. During the analysis, the user is guided by a variety of messages. (5) Results are displayedgraphically and can be saved as plain ASCII les. (6) Additional tools for visualizing time series dataand sampling intervals, integrating spectra and measuring phase angles facilitate the analysis. Com-pared to the widely used BlackmanTukey approach for spectral analysis of paleoclimatic data, the ad-vantage of SPECTRUM is the avoidance of any interpolation of the time series. Generated time seriesare used to demonstrate that interpolation leads to an underestimation of high-frequency components,independent of the interpolation technique. # 1998 Elsevier Science Ltd. All rights reserved

Key Words: Spectral analysis, Harmonic analysis, Cross-spectral analysis, Irregular sampling intervals,Interpolation, LombScargle Fourier transform.

INTRODUCTION

Spectral analysis is an important tool for decipher-

ing information from paleoclimatic time series in

the frequency domain. It is used to detect the pre-

sence of harmonic signal components in a time

series or to obtain phase relations between harmo-

nic signal components being present in two dierent

time series (cross-spectral analysis).

A widely used method for spectral analysis is the

BlackmanTukey method (BT; e.g. Jenkins and

Watts, 1968). See Figure 1. It is based on the stan-

dard Fourier transform of a truncated and tapered

(to suppress spectral leakage) autocovariance func-

tion. The major drawback of this approach is the

requirement of evenly spaced time series

tn1 tn const 8n. In general, paleoclimatic timeseries are unevenly spaced in time, thus requiring

some kind of interpolation before BT spectral

analysis can be performed. As will be outlined

below, interpolation leads to an underestimation of

high frequency components in a spectrum (`redden-

ing' of a spectrum) independent of the employed in-

terpolation scheme.

Since cross-spectral analysis using the Blackman

Tukey method requires identical sampling times for

both time series, that is t1n t2n8n, the compu-

tational eort (interpolation) is considerable if sev-

eral time series with dierent average sampling

intervals have to be analyzed. Furthermore, the in-

terpolation of unevenly spaced time series may sig-

nicantly bias statistical results because the

interpolated data points are no longer independent.

A menu-driven PC program (SPECTRUM) has

been developed in order to avoid these problems.

SPECTRUM is based on the LombScargle Fourier

transform (LSFT; Lomb, 1976; Scargle, 1982, 1989)

for unevenly spaced time series in combination with

a Welch-Overlapped-Segment-Averaging procedure

(WOSA; Welch, 1967; cf. Percival and Walden, 1993,

p. 289) for consistent spectral estimates (Fig. 1).

Hence, unevenly spaced time series can be directly

analyzed by SPECTRUM without preceding interp-

olation. The main features of SPECTRUM include:

(1) autospectral analysis; (2) harmonic analysis

(detection of periodic signal components); (3) cross-

spectral analysis (cross-amplitude-, coherency-, and

phase-spectrum; cross-spectral analysis does not

require a common time axis of the two processed

time series); (4) analytical results are supplemented

by statistical parameters that allow the evaluation of

the results; (5) results are displayed graphically and

can be saved as plain ASCII les; and (6) additional

tools for visualizing time series data and sampling

Computers & Geosciences Vol. 23, No. 9, pp. 929945, 1997# 1998 Elsevier Science Ltd. All rights reserved

Printed in Great Britain0098-3004/97 $17.00+0.00PII: S0098-3004(97)00087-3

929

intervals, integrating spectra and measuring phase

angles facilitate the analysis.

The paper is organized as follows: the next three

sections provide the mathematical background of

the methods implemented in SPECTRUM.

Subsequently, the eect of dierent interpolation

schemes on spectral estimates is discussed, and

nally, two examples will be given. A description of

Figure 1. Computational steps in univariate spectral analysis. Left column shows estimation of spec-trum of SPECMAP oxygen isotope stack (top; Imbrie and others, 1984) by Welch-Overlapped-Segment-Averaging (WOSA) method. Estimated spectrum results from averaging (in this example)three raw spectra. Right column shows steps performed in BlackmanTukey (BT) method. Estimatedautospectrum is Fourier transform of truncated autocovariance function (acvf). Master parameters thatcontrol results are number of segments in WOSA and truncation point of acvf (M) in BT. Unevenly

spaced time series can be directly processed using WOSA method, but not by BT method.

M. Schulz and K. Stattegger930

the installation and usage of SPECTRUM is pro-vided in the Appendix. The paleoclimatic time series

used in this paper reect late Pleistocene climatevariability as documented by marine sedimentaryrecords. The generated time series have properties

(length, sampling interval) similar to these data sets.SPECTRUM can, of course, be applied to datareecting other time-scales, for example time series

documenting Holocene climate variability.

UNIVARIATE SPECTRAL ESTIMATION

Scargle (1982, 1989) developed a discrete Fouriertransformation (DFT) that can be applied to evenlyand unevenly spaced time series. Let xn xtn,n = 1, 2,..., N denotes a discrete, second-order

stationary time series with zero mean. The DFT isthen given by:

Xk Xok F0Xn

Axn cos o ktn 0

iBxn sin oktn 0, 1awith

o k 2pfk > 0, k 1, 2,:::, K , tn 0 tn tok1b

F0o k 1=

2p exp fioktf to kg 1c

Aok X

n

cos2 oktn 01=2

,

Bok X

n

sin2 oktn 01=2

1d

and

tok 12ok

arctan

Xn

sin 2oktnXn

cos 2oktn

264375: 1e

The constant t ensures time invariance of theDFT, that is a constant shift of the sampling times(tn4tn+T0), will not aect the result because sucha shift will produce an identical shift inEquation (1e), that is t4 t + T0 and thereforecancel in the arguments of Equations (1a) and (d)(Scargle, 1982). Furthermore, Scargle (1982) showedthat this particular choice of t makesEquations (1a)(1e) equivalent to the t of sine-and cosine functions to the time series by means ofleast squares. The latter was already investigated by

Lomb (1976) in conjunction with spectral analysis,and therefore, the method is referred to as LombScargle Fourier transform (LSFT). The response of

a Fourier transformation to a time shift should be aphase shift of the Fourier components. The factorexpfioktf tokg in Equation (1c) producessuch a phase shift depending on the time tf. Note

that Equation (1c) diers from the phase factor

given by Scargle (1989; Eq. II.2 therein). It is, how-

ever, identical to the factor used in his algorithm.For univariate spectral analysis, tf is set to zero.

Since tf allows a virtual shift of a time series alongthe time axis, it can be used to align two time series

in cross-spectral analysis (see later).

The least squares approach of the LSFT can be

considered as follows. Let

xfk tn ak sin oktn bk cos o ktn 2

be a discrete model for a signal component of x(tn)with frequency fk. The LSFT minimizes the sum of

squares J(fk) of the dierences between the model

from Equation (2) and the data:

J fk min!XNn1

xtn xfk tn

2, k 1, 2,:::, K : 3

An important aspect of the LSFT involves thechoice of K, that is the number of frequencies used

in Equation (2). Although there is no principal limit

for K, it can be anticipated that a nite-length timeseries will only result in a nite amount of statisti-

cally independent Fourier components and, hence,

frequencies in Equation (2). Using Monte-Carlo ex-periments, Horne and Baliunas (1986) showed that

in the situation of an evenly spaced time series of

length N, the number of independent frequencies in fNyq, fNyq is N fNyq 1=2Dt denotes theNyquist frequency according to the sampling theo-

rem (e.g. Bendat and Piersol, 1986, p. 337)) and isthus identical to a standard Fourier transformation.

The same holds true for unevenly spaced time

series, where the samples are almost uniformly dis-tributed along the time axis