Wavelet analysis of relative geomagnetic paleointensity at ODP Site

UNCORRECTED PROOF

Wavelet analysis of relative geomagnetic paleointensity atODP Site 983

Yohan Guyodo a;*, Philippe Gaillot b;c;d, James E.T. Channell a

a Department of Geological Sciences, 241, University of Florida, Williamson Hall, Gainesville, FL 32611, USAb CNRS UMR 7516, Imagerie Tectonique, Universite Louis Pasteur, Strasbourg, France

c CNRS UMR 5563, Petrophysique et Tectonique, Universite Paul Sabatier, Toulouse, Franced Laboratoire de Mesures en Forage, ISTEEM, Universite de Montpellier, Montpellier, France

Received 3 July 2000; received in revised form 12 October 2000; accepted 12 October 2000

Abstract

We performed a spectral analysis of a record of relative geomagnetic paleointensity obtained at ODP Site 983,covering the time interval 0^1.1 Ma. The results confirm the presence of significant power at frequencies correspondingto the earth orbital parameters (eccentricity, obliquity, precession). The construction of the evolutionary spectrumallowed us to establish the non-stationarity of the signal at those frequencies. The subsequent use of wavelet techniquesmade it possible to demarcate the time intervals over which orbital frequencies are present in the record. For thoseintervals, the paleointensity record shows some coherency with the isothermal remanent magnetization, normalizer ofthe natural remanent magnetization, and the ratio anhysteretic remanent magnetization/k. These results suggest that theorbital frequencies embedded in the paleointensity record are the expression of lithologic variations, and probably not acharacteristic of the geodynamo itself. Extracting (filtering) these wavelet components from the Site 983 paleointensityrecord indicates, however, that this secondary overprint is of sufficiently low level that it has a minor effect on theoverall character of the record. ß 2000 Elsevier Science B.V. All rights reserved.

Keywords: paleomagnetism; wavelet analysis; magnetic intensity; spectral analysis

1. Introduction

In the past 10 yr, more than 30 sedimentaryrecords of relative paleointensity of variable reso-lutions and durations have been published (e.g.[1^15]). This fast growing database is believed tore£ect the time evolution of the geomagnetic di-pole ¢eld intensity, hence constraining the models

aimed at describing the processes governing thegeodynamo. Evidence for the reliability of sedi-mentary records of relative paleointensity is pro-vided by the high degree of correlation betweenindividual records from di¡erent regions of theworld [15^18,32]. Additional proof comes fromthe covariance of these records with paleointensityproxies derived from cosmogenic radionuclidessuch as 10Be and 36Cl [19^21]. Since geomagneticintensity may vary on a global scale, the use ofrelative paleointensity records as a correlative toolis of great interest. Relative paleointensity stratig-raphy may o¡er a time resolution greater than

0012-821X / 00 / $ ^ see front matter ß 2000 Elsevier Science B.V. All rights reserved.PII: S 0 0 1 2 - 8 2 1 X ( 0 0 ) 0 0 3 1 3 - 7

* Corresponding author. Tel. : +1-352-392-2231;E-mail: guyodo@u£.edu

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N18O techniques, which could be of crucial impor-tance for correlation and dating of climate proxyrecords in sediments. Integration of these paleo-intensity records [15^17] into composite recordshas provided target curves (NAPIS-75, Sint-200,Sint-800) for the last 800 000 yr. Signi¢cant re¢ne-ment of these curves will be achieved with addi-tional high quality records, and the routine use ofquantitative methods of investigation such asjackknife techniques, coherence function spec-trum, and in some cases wavelet analysis.

The scope of our paper is to present the waveletanalysis of a recently published record of relativepaleointensity form Ocean Drilling Program(ODP) Leg 162 Site 983 [1^3]. Site 983 (60.4³N,23.64³W) is of particular interest because it pro-vided a high latitude, high resolution paleomag-netic record (average sedimentation rate of 12.3cm/kyr), characterized by a high density of agecalibration points. The magnetic properties weremeasured on U-channel samples [22], using the 2-G enterprises pass-through magnetometers lo-cated at Gif-sur-Yvette (France) and at the Uni-versity of Florida [23,24]. The natural remanentmagnetization (NRM), the anhysteretic remanentmagnetization (ARM), and the isothermal rema-nent magnetization (IRM) were progressively al-ternating ¢eld-demagnetized, and measured ateach demagnetization step [1^3]. The magnetic

susceptibility (k), the ARM, and the IRM wereused to normalize the NRM for variations in con-centration of magnetic grains and construct pale-ointensity proxies. Comparison of the three esti-mates for several demagnetization steps pointedto the ratio NRM/IRM as the best paleointensityproxy at this site [1^3].

An initial spectral investigation of the ODP Site983 paleointensity record was performed for theinterval 0^725 ka [2]. The power spectrum re-vealed the existence of signi¢cant power at theearth orbital eccentricity (0.01 kyr31, 100 kyr)and obliquity (0.0244 kyr31, 41 kyr) frequencies.A similar analysis was performed on the bulkmagnetic parameters, and showed that the cyclesat 100 kyr were present in all the records. Thepaleointensity signal at 100 kyr was therefore at-tributed to a lithologic overprint. In contrast, the41 kyr cycles were not observed in the bulk mag-netic parameters, and it was concluded that theobliquity cycles observed in the paleointensity rec-ord may be geomagnetic in origin, and that thegeodynamo may be in£uenced by the orbital ob-liquity. Another spectral analysis on the interval700^1100 ka yielded similar results [3]. The signalto noise ratio in the bulk magnetic parameters ishowever relatively low for these frequencies sothat small intervals with signi¢cant power at or-bital frequencies may have been missed by the

Fig. 1. Relative geomagnetic paleointensity record at ODP Site 983.

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initial Fourier analysis. Given the high resolutionand superior age control in this record, it providesa unique opportunity to try spectral methods ofgreater resolution. The present study concerns theentire paleomagnetic record, covering the last1100 kyr (Fig. 1). Over this interval, we appliedwavelet analysis techniques in search of potential

non-stationarities, and to investigate the possibil-ity of previously undetected coherency betweenthe paleointensity signal and the bulk magneticparameters.

2. Spectral analysis

2.1. Global power spectrum

Fig. 2 represents the global power spectrumcalculated for the paleointensity and the IRM(normalizer) records, along with the associatedcoherence function (squared coherence andphase). The spectrum was obtained using theBlackman^Tukey method and a Bartlett window[25]. The power spectrum calculated for the 0^1100 ka paleointensity record reveals two maximaassociated with frequencies close to those of theearth orbital parameters (Fig. 2). A ¢rst signi¢-cant peak is found at 0.01 kyr31 (100 kyr), and asecond broader peak is found between 0.021kyr31 (48 kyr) and 0.029 kyr31 (35 kyr). In con-trast to the ¢rst analysis performed over the in-terval 0^725 ka [2], the peak at 0.0244 kyr31 (41kyr) is a double peak (¢rst maximum at 0.0240kyr31 and second at 0.0273 kyr31). This could bethe result of some inaccuracies in dating, or theexpression of the various dynamics of the system(e.g. di¡erent levels of climate overprint added toa stochastic geomagnetic signal, or real evolutionof the time scales inherent to the geodynamo). As

Fig. 2. Power spectra of (a) the relative paleointensity recordand (b) he IRM record at ODP Site 983. The sloping linesrepresent the best ¢t estimates for the background (red noise)spectra. The dashed lines correspond to the con¢dence limitsat the 95% con¢dence level. (c) Coherence function (squarecoherence and phase) between the paleointensity record andthe IRM. The dashed line corresponds to signals below the95% con¢dence level.

Fig. 3. Power spectra of the benthic (solid line) and planktic(dashed line) N18O records at Site 983.

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a check, we applied the same spectral analysis tothe benthic and planktic N18O records which wereused to construct the age model (Fig. 3). Theyyielded very clear power spectra, with a sharppeak at 41 kyr showing that the double peak ob-served in the paleointensity power spectrum is notdue to dating inaccuracies. In addition, some co-herency can be observed between the IRM andthe paleointensity for periods close to the eccen-tricity and the obliquity. As already mentioned,no such coherency was observed in the previousspectral analyses of the data [2,3], which might bedue to the fact that we performed our analysisover a much longer interval, therefore slightly in-creasing the signal to noise ratio. However, theIRM spectrum is still very close to the one ofred noise (Fig. 2), suggesting some caution inthe interpretation of this apparent coherence. Abetter understanding of the origin of these signalsmay arise from the reconstruction of the timeevolution of the power spectrum.

2.2. Evolutionary power spectrum

One way of assessing the time evolution of atime series power spectrum is to compute the evo-lutionary power spectrum of that series, whichcan be represented as a color diagram displayingages on the horizontal axis, frequencies on thevertical axis, and indexed colors for the powerspectrum. We constructed the evolutionary spec-trum of the ODP Site 983 paleointensity recordby calculating the power spectrum of the timeseries within a sliding window of 300 kyr widthand a translation step of 50 kyr, using the samespectral method as for the global spectrum (Fig.4). The results show that the spectral content ofthe paleointensity record is time dependent. Thelast V800 kyr is the only time interval when thesignal corresponding to the eccentricity (100 kyr,0.01 kyr31) is present. The bandwidth associatedwith the obliquity (41 kyr, 0.0244 kyr31) is ob-served only between V400 ka and V900 ka,and displays a frequency shift around V600 ka.This observation accounts for the two peaks ob-

Fig. 4. Evolutionary power spectrum of the paleointensity record at Site 983. The spectrum was obtained using a sliding windowof 300 kyr width and steps of 50 kyr. The contour lines correspond to the best ¢t estimate of the background spectrum.

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UNCORRECTED PROOFserved in the global power spectrum, which may

be interpreted as a result of the time dependencyof the power spectrum. However, it does not tellus about the origin of the signal, nor if these twopeaks are the expression of the obliquity only, ora combination of several signals including the ob-liquity. The signal in this spectral band drops be-low the background spectrum (as de¢ned by thered noise spectrum associated with this record)around V600 ka, over an interval of V100 kyr.This indicates that the spectral content of the pa-leointensity record varies over time scales shorterthan the window used to calculate the power spec-tra (i.e. 300 kyr). As a consequence, although theevolutionary spectrum represents an improvementrelative to the global approach, it is inappropriatefor resolving the small time scales present in thisrecord. A better resolution of the periodicityaround 41 kyr could be obtained by reducingthe size of the sliding window. However, it wouldgenerate aliasing of the lower frequencies and re-duce the quality of the output at that scale. Inorder to solve this problem, we must use a meth-od providing spectral information that is scaleindependent. The wavelet transform is a goodcandidate for such a task [26^28].

3. Wavelet analysis

3.1. Description of the method

Wavelet analysis provides an automatic local-ization of speci¢c behaviors such as cyclic pat-terns or discontinuities, both in time and fre-quency (e.g. [26,27]). In contrast to the Fouriertransform which is computed using a single timewindow of constant width, the wavelet transformis a multi-scale method that uses narrow windowsat high frequencies and wide windows at low fre-quencies [28]. Therefore, wavelet analysis con-trasts strongly with the classical Fourier trans-form and windowed spectral analysis, which donot detect temporal discontinuities, are not ableto distinguish between continuous low-amplitudeand non-stationary high-amplitude signals, anddo not provide information on the temporal per-sistence of periodicities. The continuous wavelet

transform of a time series f(t) is de¢ned as:

W8 �a; b� � 1��ap

ZT

f �t�8 13ba

� �dt �1�

where 8 is a base wavelet characterized by alength much shorter than the time series f(t),and a, b, and T correspond to the dilatationscales, translation steps, and time length of thewavelet transform, respectively. The wavelet 8(t)must be a function with compact support andzero mean. The second property (admissibilitycondition) ensures that 8(t) has a wiggle (i.e. iswave like), and the ¢rst ensures that it is not asustaining wave. In the present paper, we use theMorlet wavelet [26,29] because its shape is similarto a periodic sinusoidal function, suitable for in-vestigating periodicities (Fig. 5). The Morletwavelet 8(t) = Z31=4eig 0te3t2=2 (in our case g0 = 5)is a complex valued function enabling extractionof information on both the amplitude and phaseof the process being analyzed. The normalizingconstant 1/ka is chosen so that 8a(t) has thesame energy for all scales. The scale parameter(or dilatation factor), a, determines the character-istic frequencies at which the wavelet transform iscomputed. In our case, the dilatation scales werechosen so that the equivalent Fourier periods aregiven by 2kN kt0, where t0 is the sampling rate ofthe time series (Fourier period = 1.224 a) (see[27]). The parameter kNk is chosen so that it is

Fig. 5. The Morlet wavelet used to compute the wavelettransform. The solid line corresponds to the real part of thewavelet function, and the dashed line to its imaginary part.

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possible to cover the Fourier domain of interest(here the one containing periods of 23 kyr, 41 kyr,and 100 kyr) with an appropriate resolution [27].A reading of the wavelet power spectrum can beobtained by constructing a color diagram with theages on the horizontal axis and the scales (or theequivalent Fourier periods) on the vertical axis ;the modulus of the wavelet transform being rep-resented by colored patches [27]. This type of dia-gram is therefore comparable to an evolutionarypower spectrum, and can be revealing about thestructure of a particular process. The main di¡er-ence between the wavelet and the Fourier decom-positions is in the support of the respective basisfunctions. The wavelet transform coe¤cients arein£uenced by local events, while the Fourier co-e¤cients are in£uenced by the function in its en-tire domain. This makes the wavelet spectrum abetter measure of the variance attributed to local-ized events (see [28]). Fig. 6 represents the waveletpower spectra of two synthetic time series x1(t)and x2(t). The ¢rst series x1(t) is a modulatedfunction containing periodic signals at 41 kyr

and 100 kyr. The second function x2(t) was con-structed from a sinusoidal function of period 41kyr in quadrature with x1(t). The wavelet powerspectra in Fig. 6 show the modulations character-izing x1(t), as well as the time intervals over whichthe 41 kyr and the 100 kyr signals are actuallypresent.

In order to facilitate the interpretation of thistype of diagram, one can de¢ne a level abovewhich a maximum in the wavelet spectrum or inthe cross-wavelet power is statistically signi¢cant.It has been shown that each point in the waveletpower spectrum is statistically distributed as a M2

distribution with two degrees of freedom aboutthe background spectrum [27]. The con¢dence lev-el at each scale is therefore the product of thebackground spectrum and the desired signi¢cancelevel (for instance 95% con¢dence) from the M2

distribution. Here, the background spectrum isdetermined by calculating the time-average ofthe wavelet spectrum [27]. Similarly, the con¢-dence levels of the cross-wavelet spectrum can

Fig. 6. Examples of wavelet spectra obtained for synthetic time series, with the ages on the horizontal axis and the equivalentFourier periods on the vertical axis; the modulus of the wavelet transform being represented by indexed colors. This type of dia-gram is therefore comparable in its representation to an evolutionary power spectrum. The ¢rst series contains (a) two periodicsignals (41 kyr, 100 kyr) over di¡erent intervals, which are identi¢able as white patches on the wavelet spectrum (b). The secondseries (c) has a single frequency, which yielded a clear wavelet spectrum (d). The contour lines represent the con¢dence level at95%. The thick black lines on the left and right edges of the spectra indicate the regions of the diagram where the edge e¡ectsbecome important.

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UNCORRECTED PROOFbe derived from the square root of the product of

two M2 distributions [27,30].In addition to the wavelet power spectrum, it is

possible to construct the cross-wavelet spectrumWxy

8 = Wx8 Wy�

8 of two time series x(t) andy(t) from their respective wavelet transformsWx

8 and Wy8 (Wy�

8 is the complex conjugateof Wy

8 ). Subsequently, cross-wavelet power(modulus) and phase can be extracted. Localmaxima in power provide information about the

scales at which coherent events have a signi¢cantcontribution. In Fig. 7 we present the cross-wave-let power and phase of the synthetic series x1(t)and x2(t) from Fig. 6. The diagrams show that thecovariance between the two series is limited to the¢rst half of the records (i.e. where the 41 kyrperiod exists in both series). In addition, thecross-wavelet phase gives the phase relationshipbetween the two series (here equal to Z/2), which

Fig. 7. Cross-wavelet spectrum (a) and phase (b) for the same two time series as in Fig. 6. The two series are coherent only overthe ¢rst half of the record, over which they display a phase lag of Z/2. (c) Cross-wavelet spectrum and phase for the 41 kyrFourier period only.

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UNCORRECTED PROOF

is stable only over the interval containing the 41kyr signal.

If one wants to focus on a speci¢c scale (or theequivalent Fourier period), the cross-wavelet spec-

trum and phase can be represented for this par-ticular scale only (Fig. 7c). This corresponds to areading of the information provided by Fig. 7a,balong a horizontal line placed at 41 kyr.

Fig. 8. (a) Wavelet spectrum of the paleointensity record at Site 983, showing the time intervals where signi¢cant power exists atfrequencies corresponding to the earth orbital parameters. (b) Wavelet spectrum of IRM. (c) Wavelet spectrum of the ratioARM/k (anhysteretic remanence divided by susceptibility).

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UNCORRECTED PROOF3.2. Wavelet power spectrum of ODP Site 983

paleointensity record

We computed the wavelet transform of the rel-ative paleointensity record at ODP Site 983. Thewavelet power spectrum, modulus of the wavelettransform, is shown on Fig. 8a. Essentially, thefeatures initially observed in the evolutionarypower spectrum are depicted with more detail.The signal for the 100 kyr period (possibly theeccentricity signal) is restricted to the last V750kyr and shows some modulations, with threemaxima centered at V150 ka, V400 ka, andV650 ka. The signal at 41 kyr (obliquity) is ob-served over relatively short periods of time. Threemain intervals can be distinguished: 400^550 ka,650^825 ka, and 950^1100 ka. Furthermore, theapparent shift in frequencies observed in the evo-lutionary power spectrum does not appear on thewavelet spectrum. In contrast, it reveals the exis-tence of a complex structure over the interval400^550 ka, characterized by scales with equiva-lent Fourier periods ranging from 10 to 70 kyr.Therefore, only a small fraction of the intensityseems to be carried by the 41 kyr period signalover this interval, which accounts also for theshift observed in the evolutionary power spec-trum. A few patches corresponding to the 23kyr signal (precession) are also present in thewavelet power spectrum, essentially aroundV250 ka, V500 ka, and V1050 ka. These inter-vals were barely detected by the evolutionaryspectrum. In addition, several small-scale featuresof short duration are scattered though the waveletspectrum. These features correspond to intervalsin the paleointensity record characterized by largeamplitude variations such as for instance the in-tervals 30^60 ka, or 100^160 ka (Figs. 1 and 8).These intervals of major change in relative paleo-intensity do not correspond to parts of the pale-ointensity records containing frequencies similarto the orbital obliquity.

In order to understand the origin (climaticoverprint or orbital forcing in the geodynamo)of the periodic signals detected by the wavelettransform, we performed the same analysis onthe IRM (Fig. 8b), which was used to normalizethe NRM to yield the paleointensity proxy. As for

the spectral method involving the calculation ofcoherence functions, a match between the waveletspectrum of the paleointensity record and the oneof the IRM indicates an incomplete normalizationof the NRM. We also calculated the wavelettransform of the ratio ARM/k (Fig. 8c), whichcan be used as a proxy for relative changes ingrain size of magnetite, the principal remanencecarrier. Fig. 8b,c shows signi¢cant power in fre-quencies corresponding to the orbital parametersover time intervals similar to those of the paleo-intensity record (Fig. 8a). The existence of someintervals over which the IRM and the paleointen-sity records depict similar wavelet power spectrasuggests that part of the lithologic component ofthe NRM was not completely removed by thenormalization with IRM. The similarity betweenthe paleointensity record and ARM/k (Fig. 8c)suggests that the NRM does not depend on themagnetic ¢eld and the magnetic concentrationalone, but also on the magnetic grain size. Thiswould explain why the IRM, in£uenced by theconcentration of a particular grain size fraction,and not as sensitive to grain size as ARM/k,would not be able to fully normalize the NRMat orbital frequencies. A quantitative assessmentof this covariance can be obtained by calculatingthe cross-wavelet spectrum between the paleoin-tensity record and the magnetic parameters, fol-lowing the same philosophy as when computingcoherence function spectra from the global spec-tral analysis.

3.3. Cross-wavelet spectra

We calculated the cross-wavelet spectrum of therelative paleointensity record versus the IRM andthe ratio ARM/k, and subsequently extractedcross-wavelet powers and phases. Instead of plot-ting these results in the form of contoured dia-grams we have selected only the relevant scalesfor clarity (i.e. 23 kyr, 41 kyr, and 100 kyr). Theresults show signi¢cant covariance between therelative paleointensity record and both the IRMand the ratio ARM/k in speci¢c intervals of thetime series (Fig. 9). For each frequency, this co-variance is observed over the same time intervalsas those showing signi¢cant power in the paleo-

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intensity. When a high correlation is observed, thepaleointensity signal tends to have inverse phasewith respect to IRM, while it is in phase withARM/k. This may be the consequence of a sec-ondary dependency of the NRM on the magneticgrain size. In order to test the plausibility of thishypothesis, we constructed a simple syntheticmodel in which the NRM was simulated by atime series :

S1�t� � �0:8 S2�t� � 0:2 S3�t��S4�t�

where S2(t) represents a model of magnetic con-centration (depicted by the IRM) and S3(t) amodel of magnetic grain size (depicted by the ra-tio ARM/k), and S4(t) the geomagnetic ¢eld. Thetwo other series, S2(t) and S3(t) were constructedfrom red noise spectra with the addition of twosinusoidal signals of periods 100 kyr and 41 kyr.The synthetic paleointensity record was then cal-

culated by normalizing S1(t) with S2(t) (Fig. 10).We also calculated the coherence functions be-tween the paleointensity model S5(t) and the pa-rameters S2(t) and S3(t). Fig. 10 indicates that aslight grain size dependency of the NRM wouldinduce signi¢cant covariance of the relative pale-ointensity with IRM, as well as with the ratioARM/k. In addition, the paleointensity wouldbe in phase with the ratio ARM/k and in opposi-tion of phase with the IRM. The coherence func-tions were calculated over the entire interval, sincethe periodic signals built into the series weresteady, and provided results showing strong cor-relations. There is no doubt that increasing thecomplexity of the model, by involving modulationof the periodic signals, di¡erent noise spectra, andabrupt changes in lithology and sedimentationrates would provide more realistic results suchas those presented in Fig. 9. In any case, thisvery simple model provides a good, and rather

Fig. 9. Cross-wavelet spectra (black lines = power and gray lines = phase) of (a) the paleointensity record versus the IRM and (b)the paleointensity record versus the ratio ARM/k. We selected the scales corresponding to the earth orbital parameters.

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simple explanation of the cross-wavelet spectraobtained for the Site 983 relative paleointensityrecord. One positive outcome of this model isthat an incomplete normalization of the NRMover speci¢c time intervals would not diminish

the overall character of the paleointensity record.Indeed, the series S4(t) (¢eld model) and S5(t)(paleointensity model) display very similar fea-tures, and are correlated with a high correlationcoe¤cient of 0.75 (Fig. 10). This conclusioncomes in support of other studies [14,16^18]showing that paleointensity records with di¡erentlithologies and sedimentation histories, includingthe present dataset, can be correlated over highdistances with a large degree of con¢dence.

4. Conclusion

The present study con¢rms prior analysesshowing the existence of periodic signals em-bedded into the paleointensity record at ODPSite 983 [1^3]. These signals correspond to theearth orbital eccentricity (100 kyr), obliquity (41kyr), and precession (23 kyr). Calculation of theevolutionary power spectrum for the paleointen-sity record over the interval 0^1.1 Ma establishedthe non-stationarity of these periodic signals,which display amplitude variations and are re-stricted to speci¢c time intervals. The durationof these amplitude modulations appears to besmaller than the sliding window used to constructthe evolutionary spectrum (300 kyr), calling forthe use of more suitable spectral methods.

The use of wavelet analysis allowed us to de¢newith greater precision the intervals over which theorbital frequencies are present in the record. Thesignal for the 100 kyr period (eccentricity) is re-stricted to the interval 0^750 ka and shows somemodulations, with three maxima centered atV150 ka, V400 ka, and V650 ka. The signalat 41 kyr (obliquity) is observed over relativelyshort periods of time. Three main intervals canbe distinguished: 400^550 ka, 650^825 ka, and950^1100 ka. The apparent absence of the 100kyr period prior to V750 ka, in the so-called 41kyr world, may re£ect the diminished in£uence ofthe eccentricity in climate proxy records. Othersigni¢cant scales of variability were also recog-nized in the paleointensity record, resulting fromthe high complexity of the ¢eld which seems to beessentially non-stationary (see [31]). For the orbi-tal frequencies, the paleointensity record shows

Fig. 10. Synthetic model aimed at explaining the observedcoherency between the paleointensity record and the mag-netic properties. (a) Models for S1(t) (NRM), S2(t) (IRM),S4(t) (geomagnetic ¢eld), and S5(t) = S1(t)/S2(t) (relative pale-ointensity). Notice that the paleointensity variations are verysimilar to the model for geomagnetic variations. (b) Coher-ence transforms of model paleointensity versus model IRM(in black) and versus model ARM/k.

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signi¢cant and stable covariance with the IRMand ARM/k over speci¢c intervals of the record.This did not appear in previous spectral investi-gations of the paleointensity record [2,3] per-formed using standard spectral methods. Ourstudy suggests that it is probably due to the lim-itations imposed by the global Fourier analysis,which is not always able to detect low-amplitude,locally distributed signals. Our wavelet analysissuggests the presence of a secondary lithologicoverprint in the paleointensity record due to anincomplete normalization of the NRM. The re-sults seem to be explained by simple models in-volving a slight magnetic grain size dependency ofthe NRM. The model shows also that the litho-logic overprint does not drastically modify thepaleointensity record, which remains very similarto the actual ¢eld model (Fig. 10).

Quanti¢cation of the amount of climatic over-print in the paleointensity record is not an easytask, nor is the normalization of this secondarycomponent. In an attempt to estimate this over-print, we removed the orbital signals from thepaleointensity record. As a ¢rst step, we extractedthe wavelet components at (23 þ 5) kyr, (41 þ 5)kyr, and (100 þ 5) kyr. Then, we weighted thewavelet components with the cross-wavelet pro-¢les of paleointensity versus ARM/k obtainedfor those frequencies, in order to restrict the ¢lter-ing to the time intervals where a covariance be-tween the two records exists. Evidently, the signalextracted contains information on both the over-print and the geomagnetic ¢eld. Indeed, the truegeomagnetic signal contains variability at thesetime scales, as would any stochastic signal, whichmeans that its power spectrum overlaps the one ofARM/k. The output signals were then substractedfrom the initial paleointensity variations. Fig. 11ashows the power spectrum of the resulting ¢lteredsignal, which does not seem to have any dominantFourier period, compared with the global power

spectrum of the original paleointensity record.The ¢ltered paleointensity record does not seemto di¡er considerably from the initial record (Fig.11b), with an overall di¡erence of about V7%.The main changes reside in a reduction of some ofthe large amplitude variations present in the ini-tial paleointensity record (Fig. 11b). The wavelettransform of the ¢ltered paleointensity record(Fig. 11c) shows patches that are more uniformlydistributed than for the initial paleointensity rec-ord (Fig. 8a). In particular, the regions of thespectrum corresponding to higher frequencies ap-pear more clearly. This indicates the lesser domi-nance of speci¢c oscillations, and that the ¢lteredsignal can be compared to a stochastic process.However, the extracted wavelet components arelikely to contain some true geomagnetic signalover some intervals, or may have ignored someof the lithologic overprint (due to the restrictedbandwidth of the ¢ltering). An actual correctionof the paleointensity signal would require quanti-¢cation of the dual dependency of the NRM onthe IRM and ARM/k, which is probably not con-stant throughout the time interval investigated.The paleointensity record at Site 983 has beenshown to be correlatable at high resolution toother paleointensity records from the Labradorsea [18] and as far a¢eld as the South Atlantic[32], suggesting that the lithologically controlledsecondary overprint does not strongly a¡ect thecharacteristic features of the paleointensity rec-ord, and probably does not impair its usefulnessfor stratigraphic correlation. However, for aquantitative assessment of the level of lithologicoverprint in the record, multiple records, fromvarious lithologies and di¡erent oceans need tobe compared.

Fig. 11. Filtering of the paleointensity record for the Fourier periods 23 kyr, 41 kyr, and 100 kyr. (a) The power spectrum of theoriginal paleointensity record (in black) compared to that for the ¢ltered paleointensity record (in gray) which is close to itsbackground spectrum. (b) Comparison of the ¢ltered (in gray) and the original (in black) paleointensity records. The ¢ltered rec-ord was obtained by substraction of the wavelet components at 23 kyr, 41 kyr, and 100 kyr from the initial paleointensity record.In order to facilitate the comparison, the records were slightly smoothed using a singular spectrum analysis with three principalcomponents. (c) Wavelet spectrum of the ¢ltered paleointensity record, to be compared to the wavelet spectrum in Fig. 8a.6

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UNCORRECTED PROOFAcknowledgements

The authors thank Alain Mazaud and CarlRichter for their review of the manuscript. Dis-cussions with Christopher Torrence resulted inimprovement of our wavelet program. This re-search was partly supported by the US NationalScience Foundation (OCE 97-11424, EAR 98-04711).[RV]

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Documents

Wavelet analysis of relative geomagnetic paleointensity at ODP Site