Analysis of 11 Myr of geomagnetic intensity variation

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 103, NO. B8, PAGES 17,735-17,748, AUGUST 10, 1998

Analysis of 11 Myr of geomagnetic intensity variation

Catherine G. Constable, Lisa Tauxe, and Robert L. Parker Scripps Institution of Oceanography, University of California at San Diego, La Jolla

Abstract. We have conducted a detailed exploratory analysis of an 11 million year long almost continuous record of relative geomagnetic palcointensity from a sediment core acquired on Deep Sea Drilling Project Leg 73, at Site 522 in the South Atlantic. We assess the quality of the palcointensity record using spectral methods and conclude that the relative intensity record is minimally influenced by climate variations. Isothermal remanence is shown to be the most effective normalizer for these data, although both susceptibility and anhysteretic remanence are also adequate. Statistical analysis shows that the palcointensity variations follow a gamma distribution, and are compatible with predictions from modified palcosecular variation models and global absolute palcointensity data. When subdivided by polarity interval, the variability in palcointensity is proportional to the average, and further, the average is weakly correlated with interval length. Spectral estimates for times from 28.77 until 22.74 Ma, when the reversal rate is about 4 Myr -1, are compatible with a Poisson model in which the spectrum of intensity variations is dominated by the reversal process in the frequency range 1-50 Myr -1. In contrast, between 34.7 and 29.4 Ma, when the reversal rate is about 1.6 Myr -1, the spectra indicate a different secular variation regime. The magnetic field is stronger, and more variable, and a strong peak in the spectrum occurs at about 8 Myr -1. This peak may be a reflection of the same signal as recorded by the small variations known as tiny wiggles seen in marine magnetic anomaly profiles.

1. Introduction

The intensity of the geomagnetic field is intrinsically much more variable than its direction and the full spec- trum of its variations remains poorly characterized. From global models derived from historical measure- ments it is apparent that over the course of the present century the dipole moment of the geomagnetic field has been decreasing at a rate of about 0.05% per year. Since no magnetic intensity measurements survive from ear- lier than 1791 A.D. [Merrill et al., 1996], longer term fluctuations in geomagnetic intensity must be derived from palcomagnetic observations. Although the direc- tion of the geomagnetic field is comparitively straight- forward to determine by palcomagnetic methods, ge- omagnetic palcointensity observations from prehistoric times are more difficult to make, and their quality can be hard to evaluate. The number of reliable palcointen- sity measurements falls off rapidly with increasing age: by the time one reaches as far back as the Cretaceous Quiet Zone (the long normal polarity interval lasting from about 84 to 124 Ma) there are so few data avail- able that it remains uncertain whether the average field strength then was very different from that today. A

Copyright 1998 by the American Geophysical Union.

Paper number 98JB01519. 0148-0227 / 98 /98 JB- 01519509.00

recent review of paleointensity observations is provided by Jacobs [1998].

Paleointensity measurements basically fall into two categories: absolute and relative. Absolute measure- ments are made using some variant of a method in- vented by Thellier and Thellier [1959] that mimics ac- quisition of natural remanent magnetization by impart- ing a thermoremanent magnetization in the laboratory. The method relies on the linear dependence of the in- tensity of thermally activated natural remanence on strength of the magnetic field in which it is acquired, and the possibility of simulating such remanence acqui- sition in the laboratory. It is appropriate for rocks, such as basalts, that acquired their remanence during cooling in an ancient magnetic field, and have not suffered sub- sequent alterations in magnetic mineralogy. It is also suitable for fired archeomagnetic materials. Archeo- magnetic records from globally distributed sites indi- cate that it is not unusual for field intensity to fluctu- ate by a factor of 2 over time intervals of the order of 500 years. Over intervals of a few thousand years it is possible to construct time series of archeomagnetic ob- servations. But to get much longer records is difficult because lava flows tend to be erupted sporadically and are often hard to date with sufficient accuracy to place them in stratigraphic context. That longer records are needed to characterize the spectrum of intensity varia- tions is apparent from data indicating that during geo- magnetic reversals the field typically decreases to 10 or

17,735

17,736 CONSTABLE ET AL.: 11 MYR OF GEOMAGNETIC INTENSITY

20% of its average value during stable polarity epochs [e.g., Roberts, 1995].

The alternative approach is to make estimates of rela- tive field strength in sediments that preserve a record of the magnetic field in depositional or postdepositional re- manent magnetization. These remanences also depend linearly on the magnetizing field, but the exact condi- tions under which the remanence was acquired cannot be simulated in the laboratory. If the magnetic miner- alogy is sufficiently homogeneous throughout a strati- graphic section then one can use magnetic properties measured in the laboratory, such as anhysteretic rema- nent magnetization, isothermal remanent magnetiza- tion or magnetic susceptibility as a means of measuring the concentration of magnetic minerals. Relative inten- sity variations as a function of time can be estimated by normalizing the natural remanence by the laboratory- induced remanence. The degree to which this record accurately reflects relative paleointensity variations will depend, among other things, on the homogeneity of the magnetic properties of the sediment. Of particular con- cern is whether changes in climate exert an influence on the magnetic field estimates because of associated changes in the grain size or composition of the mag- netic component in the sediment. It is often difficult to assess the quality of relative paleointensity records because of the lack of a record from any other site •hat spans the same time interval. This is the case for the record discussed here: to date it is the only one avail- able for this particular time interval. In this paper we develop the use of statistical tools to assess the quality of relative paleointensity records and apply them to an 11 million year long almost continuous record of rela- tive geomagnetic paleointensity during the Oligocene. Our investigations lead us to believe that we have a valid paleointensity record: we go on to characterize the statistical properties of the intensity variations and compare them with predictions from simple models.

2. The 11 Myr Oligocene Data Set

The paleomagnetic and rock magnetic data set ana- lyzed here is derived from a core taken in pelagic sed- iments on Deep Sea Drilling Project (DSDP) Leg 73, Site 522. Magnetostratigraphy was first determined by Tauxe et al. [1983a], and detailed paleomagnetic and rock magnetic records have been presented in earlier pa- pers [Hartl et al., 1993, 1995; Tauxe and Hartl, 1997]. We briefly review the characteristics of the record before proceeding to the new analysis.

The core site is currently in the South Atlantic (26øS, 5øW). At the onset of the Oligocene it was at approxi- mately 33øS [Tauxe et al., 1983b]. Paleomagnetic mea- surements of natural remanent magnetization, anhys- teretic and isothermal remanent magnetization and sus- ceptibility were performed on about 2600 samples taken every 3 or 4 cm down core, a sample about every 4000- 8000 years. Each individual sample averages about 1000 years of geomagnetic intensity variation, the temporal sampling is denser in the earlier part of the record where

the sedimentation rate is about 10 m Myr-X: it gradu- ally decreases to about 5 m Myr -x in the upper part. A timescale for the record was derived by Tauxe and Hartl [1997] based on correlation of the 29 observed field reversals with those in the interval 22.74-34.74

Ma in the Cande and Kent [1995] magnetostratigraphic timescale. The reversal rate is about 4 Myr -1 during the later part of the record and 1.6 Myr -x during the early part. Tauxe and Hartl [1997] used relative pale- ointensity estimates based on averages from three dif- ferent normalization procedures. The record is shown in Figure 1, which also demonstrates the correlation they observed between average field strength and po- larity interval length. Major decreases in paleointensity (DIPs) occur at reversal boundaries and occasionally at intermediate times which may correspond to cryp- tochrons, the lineated tiny wiggles seen in marine mag- netic anomaly records. Some gaps in the record occur in between cores and where the core was too disturbed

to sample or failed to produce meaningful paleomag- netic results. Because of constraints on the sample size used, paleomagnetic directional measurements are too inaccurate to be useful.

3. Quality Assessment of Relative Paleointensity Records 3.1. Time Domain

The relative paleointensity estimates discussed here were derived under the assumption that the natural re- manence Ms(t) is linearly related to the field intensity B(t), that is, at time t we can write

M•v(t) - k(t)B(t) (1)

The factor k(t), designated the magnetic activity re- flects changes in the sedimentary environment with time and is the factor we try to estimate using various dif- ferent magnetic parameters as normalizers. Those dis- cussed here are the anhysteretic remanent magnetiza- tion MA, saturation isothermal remanent magnetiza- tion Mrs, and low field magnetic susceptibility X. Tauxe [1993] summarizes the experimental evidence in favor of sediment magnetization depending linearly on the field in which it is acquired and also discusses the rel- ative merits of the various normalizers. The lack of

theoretical understanding for how Ms is acquired can make it difficult to discriminate among various normal- izers, each of which has its own particular advantages or disadvantages: X is widely regarded as reflecting too great an influence of multidomain and superparamag- netic grains; Mrs is a saturation magnetization that may be orders of magnitude greater than the natural remanence; furthermore, Tauxe [1993] reminds us that even at low concentrations (typical of the Site 522 sedi- ments discussed here) MA acquired in a steady bias field depends nonlinearly on the concentration of magnetic material, and this nonlinearity can generate significant noise in paleointensity records.

Figure 2 shows the natural, anhysteretic, and satu- ration isothermal remanences and initial susceptibility

CONSTABLE ET AL. ß 11 MYR OF GEOMAGNETIC INTENSITY 17,737

(D > (D

8

7

6

5

4-

3-

2-

1-

0 22

12

10-

8-

6-

4-

2-

O-

(o)

24 26 28 30 32 34

22 24 26 28 30 32 34

Time (Ma)

(b)

c

c

Figure 1. (a) Estimated reversal rate as a function of time during the Oligocene. (b) Paleoin- tensity estimates from Tauxe and Hartl (1997) (crosses) and smoothing spline fit to observations to indicate trend in paleointensity (thick line). Field intensity average to unity over entire time interval. Triangles show the lengths of polarity intervals, plotted at the midpoint of the inter- val, with the thin line showing the trend in interval lengths. Note the correlation between long intervals and high field intensity.

as functions of time for our Site 522 record, and Table 1 summarizes the statistical variation in these records.

Each is normalized so that its average over the time in- terval 22.74-34.74 Ma is 1. The very long term trend in the normalizers is related to decreasing clay and in- creasing carbonate content with depth in core. The fact that this trend is less pronounced in the natural remanence is taken to be a reflection of the influence

of geomagnetic field intensity variations on the natural remanence. As Tauxe and Hartl [1997] have discussed, there is a remarkable consistency in the magnetic re- sponse measured by 3//A, Mrs, and X down core, making Site 522 a promising place for the estimation of relative

paleointensity. Their approach was to average relative paleointensity estimates acquired using the three dif- ferent normalizers; here we discriminate further using power and cross-spectral analysis to decide which of the estimates may be better.

3.2. Power Spectral Estimation and Coherences

It is perhaps appropriate to begin this section with a justification for the use of frequency domain methods of analysis for these particular data. Figure 2 shows broad agreement between the various normalizers: the

17,738 CONSTABLE ET AL.: 11 MYR OF GEOMAGNETIC INTENSITY

5.0

2.0

1.0

3.0

20

1.0

I I I

(o)

24 26 28 30 32 34

- b)

24 26 28 30 32 34

5.0

2.0

1.0

24 26 28 30 32 34

3.0 MN __ 20

10

O0

24 26 28 30 32 34

Time (Mo)

Figure 2. Variation in (a) susceptibility, (b) saturation isothermal remanent magnetization, (c) anhysteretic remanent magnetization, and (d) natural remanent magnetization as a function of age at Site 522. Each record is normalized to unit mean. Interpolation is with Akima splines.

Table 1. Correlations Coefficients Between the Various Normalizers and Paleoin-

tensity Estimates

o' MN MA M,.s X BA B,.s B x

MN 0.50 - 0.3736 0.4435 0.4260 0.7523 0.6966 .6865 MA 0.38 - - 0.9630 0.9597 -0.2569 -0.3108 -0.3040 Mr• 0.43 - - - 0.9830 -0.1634 -0.2615 -0.2488

X 0.42 ..... 0.1807 -0.2725 -0.2859 BA 0.50 ..... 0.9742 0.9638 Br, 0.46 ...... 0.9858 B x 0.48 .......

cr is the standard deviation in each parameter. Average value is unity in each case. Other columns give correlation coefficient between the designated parameters.

CONSTABLE ET AL.' 11 MYR OF GEOMAGNETIC INTENSITY 17,739

correlation coefficient between each pair of normaliz- ers is greater than 0.95 in each case (Table 1). How- ever, the high correlation reflects in large part the long- period behavior which dominates the variance in each record. At shorter periods there are subtle differences between the normalizers; these may reflect the differ- ent responses of M.•, Mrs, and X to climate-induced changes in concentration and grain size, for example. It is important to understand these factors if we are to be able to make inferences about shorter timescale

variations in intensity. Estimating the power spectrum of variations in each of the normalizers can help iden- tify, for example, grain size or concentration variations linked to Milankovitch cycles. The coherence spectrum is a frequency domain analog of the correlation coeffi- cient and indicates how well correlated two records are

as a function of frequency: we can use this to identify whether the various normalizers respond in the same way on all timescales.

Following modern trends in spectral estimation [see, e.g., Percival and Walden, 1993], we use a direct spec- tral estimator based on the periodogram. It is well known that the raw periodogram suffers from both large variance and bias: variance can be reduced by averaging independent estimates of the spectrum, while bias from spectral leakage is reduced by applying a suitable taper (a multiplicative shaping function) to the observations in the time domain. Thomson [1982] has developed mul- titaper spectral estimation based on the family of pro- late spheroidal wave functions, which concentrate spec- tral energy within a prespecified frequency bandwidth. The prolate spheroidal wave functions are orthogonal so that each one provides an approximately uncorre- lated spectral estimate; typically 2p (where p is the time bandwidth product) of these are averaged to re- duce variance in the final spectral estimate. A new mul- titaper method has recently been described by Riedel and $idorenko [1995] (hereinafter RS) which seeks the taper that minimizes local bias at each point. The RS algorithm finds the appropriate balance between local bias and variance by minimizing the mean square error at each frequency. This means that different amounts of averaging can be done at each frequency, avoiding the possibility of averaging that is too severe in one fre- quency interval, where the spectrum is varying rapidly, and not enough in other parts where the spectrum is relatively fiat. The multitapers for the RS technique are very closely approximated by sine functions, mak- ing them straightforward and efficient to calculate. We have used the RS method for all the spectral and co- herence estimates presented here: however, the results are essentially identical to those obtained by Thomson's method for these data. The sine multitaper method provides less protection from broadband bias than the prolate spheroidal wave functions, but this can usually be remedied by prewhitening for very red spectra or those with a few large peaks.

We begin by looking at the power spectral estimates for Ms, MA, M•s, and X and their interrelationships. For this purpose we use the lower part of the record

spanning the time interval 29.40-34.74 Ma. The record here is more continuous than in the upper part, which suffers from many interruptions because of core distur- bances. The time spacing is not uniform so we ex- perimented with various means of interpolation: linear splines, Akima splines, and cubic splines [Lancaster and $alkauskas, 1986]. Although the core was on average sampled at about 4 kyr intervals in the lower part of the Oligocene, there is the possibility of temporal alias- ing from higher frequency geomagnetic or mineralogi- cal variations: this may well contribute to the rather jittery aspect seen in the records and makes the choice of a suitable interpolant a rather subjective process. If we knew the spectrum a priori we could devise an in- terpolant with suitable spectral properties: failing that, we assessed the differences in the resulting spectra for the various interpolants. Linear interpolation tended to introduce features like the sidelobes from a boxcar filter

in the spectral estimates and produced systematic phase relationships in the cross spectra related to the interpo- lation. Cubic splines were generally more satisfactory and predict a smoother spectral falloff but occasionally generated severe overshoots outside the general range of the record in interpolating between data. We found that Akima splines, which choose the slope on the basis of a nonlinear method suggested by Akima [1970; see also Lancaster and •alkauskas, 1986] were most successful in overcoming this problem, while avoiding the spec- tral contamination found with linear interpolation. We therefore used Akima splines as interpolants for these data, and generated equispaced samples at 0.004 Myr intervals. Figure 3 shows the spectral estimates, from which it is apparent that those for MA, Mrs, and X are broadly similar in structure. The very low frequency signal between about 0 and 5 Myr -• probably reflects the changes in carbonate content mentioned above. All the normalizer spectra show a small peak in power at about 45 Myr -• in approximate correspondence with the period expected for climatic variations associated with precessional changes in Earth's orbit. Although insignificant in comparison with the calculated error bars, it is plausible that such a peak should reflect the climate's control over sediment composition. The spec- trum for the natural remanence MN shows substantially more power at all frequencies than those for the normal- izers. This indicates that a large fraction of the signal recorded by Ms at all frequencies is likely to be in- dependent of magnetic mineralogy. The spectrum for MN also shows a substantial peak in power at about 8 Myr -1.

Further insight can be obtained from the coherence among the various normalizers, plotted in Figure 4. Al- though the coherence is in general high, we see that above 80 Myr -• there is a rapid dropoff in coherence: this reflects in part the influence of the interpolator on the different records and indicates that we should be

suspicious of interpreting geomagnetic variations in this frequency range (note from Figure 3 that the actual variance in these high-frequency fluctuations is quite low). Except at very low frequencies the coherence be-

17,740 CONSTABLE ET AL.- 11 MYR OF GEOMAGNETIC INTENSITY

10-1

10-2

10-3

10-4

Site 522 29.40-54.74 Ma Af=.OO4Ma

0 10 20 50 40 50 60 70 80

Frequency (Myr -1)

Figure 3. Upper solid curve, power spectral density (PSD) estimate for MN, lower curves PSD estimates for X (short-dashed), Mrs (long-dashed), and MA (solid). Each spectral estimate is based on 12 tapers, a time bandwidth product of 6.

tween MA and X is much lower (typically 0.6 versus 0.8) than between MA and Mrs or between Mrs and X- Clearly, the various normalizers have slightly different frequency responses, and we need further information to discriminate the optimal candidate to use as a measure of magnetic activation for the natural remanence.

For this purpose we turn to the coherence between M•v and the various normalizers. Equation (1) tells us that, absent any geomagnetic intensity variations, a good normalizer will be highly correlated with M•v: departures from a coherence of unity reflect the frac- tion of the variance that must be attributed to noise

at that frequency. Therefore a justifiable means of se- lecting the best normalization parameter is to treat the part of M•v generated by intensity variations as noise and choose the parameter that is most coherent with the natural remanence. Figure 5 shows coherence and phase between M•v and each of the normalizers. The long-dashed line on Figure 5(b) is the level below which if2 should fall 95% of the time when there is no linear relationship between the two parameters. Mrs is clearly most coherent with M•v and ranks as our favored can-

didate for normalizer. This is in agreement with the conclusions of Hartl et al. [1993] who chose M•.• on the grounds of simplicity, namely, that it generated the least amount of variation in geomagnetic intensity. We note that most of the candidate normalizers show significant coherence with M•v at many frequencies: the fact that the variations are roughly in phase and the phase is rel- atively stable up to about 60 Myr- • suggests a common physical cause. At higher frequencies there is very little power in the rock magnetic variations: we found both coherence and phase above 60 Myr -• to depend on the interpolator used and are therefore very wary of mak- ing any physical interpretations of the spectra at high frequencies.

Thus on the basis of Figure 5 we choose M•s as the optimum norrealizer for providing relative paleointen- sity esti•nates. Tauxe and Wu [1990] have suggested that a further criterion for the selection of relative

paleointensity records should be that after normaliza- tion the coherence between the selected normalizer and

the estimated relative intensity should lie below the 95% confidence level for coherence of uncorrelated sig-

CONSTABLE ET AL.' 11 MYR OF GEOMAGNETIC INTENSITY 17,741

1.0

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Mrs, X

I I I I I I I

0 40 8o 120

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Figure 4. normalizers.

0.8

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0 40 80 120

Frequency Myr -• Coherences between the various candidate

nals. Their argument is that if norrealizer and intensity record exhibit significant coherence, then the natural remanence may not have been adequately corrected for environmental effects. This test has been advocated as

a means of assessing relative paleointensity data quality [Tauxe, 1993], but we reject it in this case for the follow- ing reasons. Figure 6 shows the coherence between Brs (our relative paleointensity estimate given by MN/Mr•)

and Mrs. Although the coherence is substantially re- duced by the normalization, ?2 does lie above the 95% significance level at a number of frequencies. However, the phase spectrum is extremely unstable and rarely lies near 0 ø as we might expect if the relative intensity vari- ations were controlled by the same factors as control Mrs. Furthermore, the spectrum of variations in Mrs typically has an order of magnitude less variance than that for Brs, so that even if inadequately compensated environmental variations are making a contribution to Brs, the variance in the estimate from this source will be only about 10% of the signal of interest. As an aside, we note that if we had simply chosen the normalizer with least coherence with its corresponding relative paleoin- tensity estimate in this case we would have used MA. As we noted earlier, this choice would have provided the noisiest paleointensity estimates, a strong argument in favor of using the alternative criteria we suggest here.

A remaining question of interest is how sedimenta- tion rate affects the spectral estimates discussed in this

150

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Figure 5. (a) Phase and (b) coherence spectra between MN and Mn (solid), MN and Mrs (dashed), and MN and X (short-dashed). Long-dashed line in Figure 5b gives level below which coherence should fall 95% of time when the two signals have no linear relationship.

17,742 CONSTABLE ET AL.' 11 MYR OF GEOMAGNETIC INTENSITY

c-

150

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-15o

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08

06

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00i • I • 0 20 40 60 80

Frequency Myr- 1

Figure 6. (a) Phase and (b) coherence between pale- ointensity estimate Brs and normalizer Mrs. Note that although some significant coherence is still present, Fig- ure 2 shows there is an order of magnitude less power in Mrs.

paper and whether variations in rate associated with climate change might generate peaks in the spectrum of the kind seen in A/l•v at about 8 Myr -• We be- lieve that we can rule out large-scale spectral variations generated by this process for the following reason. Sed- imentation rate is basically controlled by the amount of carbonate coming into the system. The susceptibil- ity is strongly influenced by carbonate concentration, with low susceptibility corresponding to high carbonate concentration. Thus if the large peak in M•v is caused by accummulation rate fluctuations, we should expect a corresponding spectral peak in X reflecting variations in sedimentation rate within that frequency band, and strong coherence between M•v and X in the same fre- quency band. No such features are observed. We cannot rule out inaccuracies in our spectral estimates because of sedimentation rate fluctuations but suspect that the primary influence of such variations would be to smear out any concentrations of power into nearby frequencies rather than concentrating them in a particular band at low frequency.

4. Statistical Analysis of the Paleointensity Record

We turn now to an analysis of the relative paleoin- tensity variations and geomagnetic field behavior. The record presented in Figure I is the longest single time se- ries of geomagnetic intensity variations currently avail- able and can provide us with a baseline for geomagnetic intensity variations on millions of years timescales.

We begin with an assessment of the statistical distri- bution of the observations and comparisons with various theoretical distributions. Figure 7a shows an estimate of the probability density function for the observations and Figure 7b compares the empirical cumulative distri- bution function with that of a gamma distribution func- tion (b). Also plotted in Figure 7c are the difference (or residuals) of the empirical distribution from Gaussian, lognormal, and gamma distribution functions. We find that the distribution that best fits the observations is a

gamma distribution whose probability density function is given by [see, e.g., Rice, 1995]

f(b)- bø•-te -kt' b > O, (2) -

where

x>O. (3)

The parameter c• is called a shape parameter and ,k a scale parameter for the gamma density. The implica- tion is that varying c• changes the shape of the density, whereas varying X corresponds to changing the units of measurement. The mean value for this distribution is

•u - s/X, and the variance cr 2 - c•/X 2. One consequence of these relationships is that the observed standard de- viation in geomagnetic field strength is approximately proportional to its average value. This feature is illus- trated for our data set in Figure 8a, where we plot •B, the standard deviation in Brs, as a function of average

_

field strength, Brs, within each polarity interval. There _

is a strong correlation between Brs and •B. Also shown in Figure 8b is the weak correlation observed between polarity interval length and average field strength. The correspondence of high field variability with longer po- larity intervals seems at first somewhat counterintuitive: we investigate this further in the next section.

Before doing so we ask first whether the gamma model is a good one for other paleointensity obser- vations, and second, whether it agrees with predic- tions from statistical models of paleosecular variation.

aZ. [OOS] (hereinafter TU) omplea database of absolute paleointensity measurements that span the last 300 Ma and studied the statistical proper- ties of observations from the past 5 Myr. They conclude on the basis of a X 2 test on the distribution of virtual dipole moments (VDMs) that a lognormal distribution is a better fit to the observations than a normal distri-

bution. McFadden F/McElhinny [1982] suggested that a truncated normal distribution for the dipole moment (to force positivity on the observed paleointensity data)

CONSTABLE ET AL. ß 11 MYR OF GEOMAGNETIC INTENSITY 17,743

0.12

0.04 . , 522 (c) 0.08

... ....... 0.00 ; ........... 0.04 j • ', ."'•- ' ...... 0.00

-0.04 • / -0.04

1.0

1 2 3 4 5 3 4 5

I t

_

1 2

0.8

0.6

0.4

0.2

0.0

o 1 2

(b)

0 4 5 0 1 2 3 4 5

Figure ?. (a) Estimate of the probability density function for Brs data from 522 (solid line) and for virtual dipole moments from worldwide compilation of absolute paleointensity data (TKU) for the past 5 Myr (dashed line). (b) Empirical cumulative distribution function for 522 (solid), TKU (dashed), and maximum likelihood estimate of gamma distribution for 522 data (short dashed). (c) Residuals of 522 data empirical cumulative distribution function to gamma (solid), Gaussian (dashed), and lognormal (short dashed) distribution functions. (d) Same as Figure 7c for TKU data. Note different scale.

101

_c 10ø

• 10 -1

10-2 5 10 15 20 25 ,30 ,35

B (,u,T)

14

12

10

8

6

4

2

0 0 5 10 15 20 25 30 35

B

Figure 8. (a) Standard deviation in field intensity as a function of average field intensity within each polarity interval. (b) Polarity interval length plotted as a function of average field intensity Brs during the interval. Correlation coefficient is 0.64 in linear domain.

17,744 CONSTABLE ET AL. : 11 MYR OF GEOMAGNETIC INTENSITY

provide the best fit to their earlier more limited dataset. We abstracted data for the age interval 0-5 Ma from Tanaka et al.'s compilation and carried out a similar analysis to that for the 522 data (see Figures 7a, 7b, and 7d). There are 216 globally distributed observa- tions of VDM. We find again that a gamma distribu- tion provides the best fit to the observations and that the dispersion in the data is proportional to the VDM magnitude, although it is not possible to carry out a de- tailed analysis by polarity interval because of the sparse data distribution. The similarity between the absolute and relative paleointensity distributions seen in Figures 7a and 7b is remarkable given the differences in exper- imental procedure, and the small number of absolute measurements available (216 versus 2238 for Brs). We should also keep in mind the limitations of comparing distributions of time-varying field intensity from a sin- gle site with globally distributed VDMs.

The above discussion does not take into account un-

certainty in the relative intensity observations due to measurement noise. McFadden and McElhinny [1982] accommodated this in their modeling by allowing for a 10% uncertainty in paleointensity observations, a figure derived from work by McElhinny and $enanayake [1982] on the reproducibility of paleointensity measurements. As McFadden and McElhinny note, a constant percent- age uncertainty will generate a skewed paleointensity distribution with a long tail at high values. Although measurement error will contribute to the correlation be-

tween field variability and average paleointensity, the variance is too high for this to be the entire cause. The standard deviation in Br8 is 46% of the mean paleoin- tensity, suggesting that the dominant cause of the vari- ability seen in the distribution is changing geomagnetic field intensity.

We turn now to statistical models of paleosecular variation and their predictions about paleointensity vari- ations. Constable and Parker [1988] hereinafter CP88 modelled the secular variation by a statistical process in which secular variation can be simulated by statistical variability in each of the Gauss coefficients representing the geomagnetic field in a spherical harmonic expansion. The CP88 model was isotropic (that is independent of location) except for long-term average contributions to the axial-dipole and axial-quadrupole parts of the field. The spatial power spectrum of nondipole field variations was white at the core-mantle boundary, consistent with what is seen for the present geomagnetic field. Subse- quently, it has been shown that this background model fails to describe two distinct features seen in the secu-

lar variation over the past 5 Myr: these are the gradual increase in dispersion of virtual geomagnetic pole po- sitions with latitude, and the large variance found in paleointensity observations. A number of modifications of the background version of CP88 have been suggested that explain one or other of these features [Kono and Tanaka, 1995; Hulot and Gallet, 1996; Quidelleur and Courtillot 1996; Kono and Hiroi, 1996]. The two fea- tures can be simultaneously accommodated by allowing additional freedom in the model in the form of increased

variance in the axial-dipole and non-axial-quadrupole

field contributions, somewhat modifying the spectrum of the field at the core-mantle boundary (model CJ98) [Constable and Johnson, 1998]. CJ98 is in agreement with the TKU data for 0-5 Ma shown in Figure 7, and the distribution of intensity variations it predicts are once again well described by a gamma distribution. Pre- dictions from CJ98 for the site of the 522 data predict too large a variance in the data: however, modifications to the model parameters produce a more satisfactory agreement. The lava flow data sets from which the CJ98 model is derived are designed to provide random sam- ples in time, rather than a time series of observations like that obtained for 522. Thus one reason for the ex-

cess variance may be the lack of any allowance for tem- poral correlations in the statistical model. Although it may seem plausible that smoothing inherent in the remanence acquisition process will also result in an un- derestimate of the geomagnetic variance, the structure of the intensity spectrum derived in the next section can be used to argue against this. Most of the variance is concentrated at periods that are long compared with the filtering effect expected from the sedimentation pro- cess. An alternative explanation for the excess variance predicted by the model is that the same parameters may not be appropriate to describe the field 30 Myr ago as for the past 5 Myr.

5. Geomagnetic Intensity Spectra

The almost continuous nature of these sedimentary paleointensity records makes them good candidates for studying the power spectrum of geomagnetic intensity variations at very long periods. We divide the record into parts from 22.74 to 28.77 Ma and from 29.40 to 34.74 Ma. In addition to being separated from each other by a large temporal gap (where disturbed core material gave a poor quality magnetic record), it can be seen in Figure I that the paleointensity record in the lower section is different in character from that in the

upper part: the average field strength is greater, and there are larger paleointensity variations. These appear to be associated with the long reverse polarity inter- val C12R. In the earlier part the reversal rate is about 1.6 Myr -•, while from 22.74 to 28.77 Ma it is 4 Myr -•. As we shall see, the differing reversal rates apparently correspond to different characteristic spectra associated with intensity variation.

Figure 9 shows geomagnetic power spectral estimates for 22.74-28.77 Ma. The record has been further subdi-

vided into five separate sections, because of data gaps we considered too long to interpolate. The lengths of these sections range from 0.45 to 1.44 Ma. Within each subsection we again used Akima spline interpolation to uniform sampling every 0.006 Ma. The spectra are dominated by low-frequency variations. Inspection of the time series shows that the major DIPs in this part of the record tend to occur at reversal boundaries: we

questioned whether the intensity spectrum here is dom- inated by the reversal process. As a test we invoke a simple model for the geomagnetic intensity spectrum associated with reversals. At very long periods we can

CONSTABLE ET AL. ß 11 MYR OF GEOMAGNETIC INTENSITY 17,745

10 -2

• 10 -3

o

o

._c

10 -4

10 -5

1 t t t t t

t i t i tl tl

0 10 20 30 40 50 60 70 80

Frequency Myr -1

Figure 9. Power spectral density (PSD) estimates for the six paleointensity subsections in time interval 22.74-28.77 Ma. Heavy dashed line is the spectrum predicted from the Poisson reversal model described in the text, with rate 4 Myr -1.

think of magnetic field reversals as being modeled by a rectangular waveform with amplitude A (the intensity of the field at the site) and average frequency of rever- sals • (see Figure 10). Field reversals last for a time cr and are supposed to occur randomly as in a Poisson process, a model that we know to be reasonably satis- factory for time periods of a few million years [see e.g., McFadden, 1984; McFadden and Merrill, 1984]. We suppose that during reversal the intensity drops to zero and remains there until the reversal is completed. The geomagnetic intensity power spectrum associated with such a process is derived in the appendix and is

4Ae-X•

S(f)- )•2 + 4•.2f2

( •sin2•'fa) x I - e-X•[cos2•rfa + 2•rf ] . (4) The predictions of this model with appropriate val-

ues (A = 4 Myr -1, A = 1.0, a = 30 kyr) for our data set are shown by the dashed line in Figure 9. It falls right in the middle of our estimates in terms of the amount of power expected. The positions of the large troughs in the theoretical power spectrum are controlled by the parameter a, and we surmise that the absence of troughs in the estimated spectra arises from fluctuation

in the actual length of reversals and spectral leakage in the estimation procedure. Tauxe and Hartl [1997] in a spectral analysis of individual core section records noted a tendency for pulsation in the 30-50 kyr period range. Although it is tempting to conclude this may be related to the typical time associated with a reversal in each core, their records did not actually include any reversals. The short lengths of the records they ana- lyzed make the peaks they observed barely significant: our more complete analysis of longer records indicates that such apparent pulsations are of an ephemeral na- ture. We conclude that the spectra are entirely consis- tent with the notion that between 0 and 50 Myr -1 the

Figure 10. Sample of rectangular waveform used in the simple Poisson reversal model.

17,746 CONSTABLE ET AL.' 11 MYR OF GEOMAGNETIC INTENSITY

geomagnetic intensity variations are controlled by rever- sals of the geomagnetic field. Barton [1982] suggested that there is a drop in power in the geomagnetic secular variation signal at frequencies lower than 100 Myr -•, before the power due to magnetic reversals kicks in at low frequency. We see no evidence for this here.

Following the same recipe for the time interval 29.40- 34.74 Ma, we derive the spectra in Figure 11. Here we find quite different behavior from that predicted by the simple intensity model (which for this part of the record was computed with parameters • - 1.6 Myr -• resulting in about a factor of 2 less power than in the upper part). A bump in the power spectrum centered at about 8 Myr -• reflects repeated DIPs seen in palcoin- tensity that Hartl et al. [1993] associated with the cryp- tochrons seen in the marine magnetic anomaly record. This bump in the spectrum is dominantly controlled by the field behavior in C12R which has many cryp- tochrons: however, when we exclude C12R from our analysis, we still find (as for the whole of the 29.40- 34.74 Ma section) that although the bump is absent there is substantially more power overall in the geo- magnetic intensity variations.

What causes the cryptochrons found in some parts of the marine magnetic anomaly records remains un- resolved \Cande and Kent, 1992]: they may be ei- ther large-scale intensity fluctuations in the geomag- netic field or very short polarity intervals that cannot be fully resolved by the marine magnetic anomaly record. If they are short polarity intervals then they do not fit within the simple Poisson process intensity model that we have used here. Predictions for the spectrum when the reversal rate is 10 Myr- • still do not produce enough power in the spectrum or the bump seen at 8 Myr-•. An alternative is to consider the cryptochrons as aborted reversals, brief excursions away from the dipolar field state perhaps reaching the opposite polarity, before re- bounding to the initial polarity state. The directional data considered by Hartl et al. [1993] might offer some support for this view, but if these are aborted reversals we must explain why there are so many consecutive un- successful ones and why the secular variation during this time has so •nuch more power associated with it. We seem to be forced to conclude that from 29.40 to

34.74 Ma a different mode of secular variation is in op- eration at Site 522 than in the later part of the record.

10-1

10 -2

10-3

0_ 4 1

10 -5

Sire 522 Poleoinfensif7 29.40-54.74 Me, bf=.004

\ /

\ / \ / \

/ \

!

!

I

i

i

i

\

\

\

\

\

\

\

/

/

/

i

I

I

I

10-6 . • t t i • • t • 0 10 20 50 40 50 60 70 80

Frequency Myr -1

Figure 11. Power spectral density (PSD) estimates of palcointensity variations for 29.40-34.74 Ma (solid line). Poisson model predictions, with rate 1.6 Myr -•, are given by the dashed line.

CONSTABLE ET AL.' 11 MYR OF GEOMAGNETIC INTENSITY 17,747

6. Conclusions

We have revisited the relative geomagnetic paleoin- tensity record acquired at DSDP Site 522 that spans the time interval 22.74-34.74 Ma. Using coherence and spectral analysis, we have evaluated the various can- didates for normalizing the natural remanence. We choose the one that has the highest coherence with the natural remanence as likely to produce the best record (in this case saturation isothermal remanence). The low spectral power in the variations in the normalizers compared with that in the natural remanence suggests that there is a high signal-to-noise ratio in our rela- tive paleointensity estimates. We reconsider Tauxe and Wu's [1990] suggestion that coherence between normal- izer and relative intensity estimates exceeding the 95% confidence level for uncorrelated signals should lead to rejection of relative paleointensity records and recom- mend that the phase spectrum should be used as an additional diagnostic, along with the relative power lev- els in natural remanence and normalizer.

The relative paleointensity observations are shown to be consistent with being drawn from a gamma dis- tribution, as are absolute measurements of VDMs for the time period 0-5 Ma, and simulations from statis- tical models for paleosecular variation. The average value of field intensity within each polarity interval is strongly correlated with its variability and weakly cor- related with interval length. For the time period 28.77- 22.74 Ma the spectrum of intensity variations between frequencies 0 and 50 Myr-1 is dominated by the reversal process which takes place on average 4 times every mil- lion years. In the earlier part of the record, which has a lower reversal rate, there is substantially more power in the secular variation, and a peak in spectral power at about 8 Myr -•. This peak is probably a reflection of the same signal recorded as cryptochrons or tiny wig- gles in marine magnetic anomaly profiles. These cryp- tochrons cannot be conveniently classified as short po- larity intervals of the same kind as seen elsewhere: if they are aborted reversals we need an explanation for the large number of consecutive unsuccessful reversals and the increased power in secular variation.

Appendix' An Intensity Spectrum

The magnetic field is modeled as of constant intensity, except during the times of a transition from one polar- ity state to the other, when it is known that the field strength drops by a large factor, in our model to 0. The reversal model is a Poisson process, one in which at any given instant the probability per unit time of a transi- tion to the opposite polarity state is constant. The field intensity X(t) is constant (taken to be unity for simplic- ity) outside the transition intervals. When a transition occurs, X drops to zero and stays there for a period of time a, then the field returns to full strength with reversed polarity. The chance of a further transition is the same during the transition period, so that occasion- ally transitions may overlap and produce a longer than normal zero-strength interval.

To find the spectrum of this process, we first calculate the autocovariance function. We start at a point in the record where the field is at full strength: X(0) -- 1. The chance of finding such a point at random is defined to be P+; because the other state of the field is 0:

E(X(t)) = P+ (A1)

where E is the expectation operation. The autocovari- ance of the process (without removing the mean value of the record) is

= (A2)

and initially we consider only t > 0. As noted, the value of X is either 0 or 1, and by definition X(0) - 1. Given that the initial point is nontransitional, the expectation of X(t) can be expressed as lx probability (t is outside one of the intervals of zero intensity). We can write this explicitly thus: Prob (no transitions up to time t) q- Prob (one transition in the interval (0, t- a) and none in (t-a,t)) q- Prob (two transitions in the interval (0, t-a) and none in (t-a, t)) + .... To find A(t) in (A2) this series must be multiplied by P+, the probability that the first sample is unity. So in an obvious notation

A(t) P+[P(0, t)+ P(1, t-a)P(0, a)

+P(2, t-a)P(0, a) + ... ]. (A3)

If s > 0 the theory of Poisson processes [Rice, 1995] shows

•nsn

P(n, s) = n! e-X* (A4) where A is the rate parameter of the process (average number of reversals per unit time). To calculate A ex- plicitly, we must consider separately two cases: t < a and t _> a. When t _> a, (A4) applies, and so

A(t) = P+[e-Xt + • 'V•(t- a)'•e-X(t-•)e-X*] n! n--1

P+e -x*. (AS)

If t < a the probabilities P(n, t-a) are all zero for n > 0 because otherwise X(0) = 0 contrary to assumption. So from (A3), when t < a

A(t) = P+e -xt . (A6)

It is easily seen that as t becomes large, A(t) must tend toward P•; thus from (A5)

z(x(t)) = e+ =

The conventional autocovariance R(t) is formed from the series after the mean has been removed: then

R(t) - A(t) - E(X) 2 - A(t) - P_•. (A8)

Combining (A5), (A6) and (A7), and noting that R is always an even function of t, we find the autocovariance of X'

17,748 CONSTABLE ET AL. ß 11 MYR OF GEOMAGNETIC INTENSITY

R(t) - e -x(Itl+•) - e -2x• ---- e•

The one-sided PSD of the process is then

S(f) - 2 f_• d•e-.27rift]•(t) 4Ae -x•

__

-- A2_i_4•r2 f2

x l1 -e-*•[cos 2•rf• + * sin •[•1• 2wf •] ' (Ale)

Obvious refinements to the model would include a

nonzero intensity during the transition state. This change has no effect on the form of the spectrum (Ale), because we have removed the mean. Another modifica-

tion might be to allow a more complicated shape for the transition to low intensity to replace the sudden step. We can accommodate this behavior simply by convolv- ing the step with a shape factor g(t); the new spectrum would then be

S -

where •0 is the Fourier transform of g.

Acknowledgments. This analysis was funded by NSF grant EAR-9706019. We thank reviewers Jeff Love and Den- nis Kent and associate editor Andy Jackson for constructive comments and suggestions that improved the manuscript.

References

Akima, H., A new method of interpolation and smooth curve fitting based on local procedures, J. Assoc. Corn- put. Mach., 17, 589-602, 1970.

Barton, C.E., Spectral analysis of palaeomagnetic time se- ries and the geomagnetic spectrum, Philos. Trans. R. Soc. London, Ser. A, 306, 203-209, 1982.

Cande, S.C., and D.V. Kent, Ultrahigh resolution marine magnetic anomaly profiles: A record of continuous pale- ointensity variations, J. Ceophys. Res., 97, 15,075-15,083, 1992.

Cande, S.C., and D.V. Kent, Revised calibration of the ge- omagnetic polarity timescale for the late Cretaceous and Cenozoic, J. Ceophys. Res., 100, 6093-6095, 1995.

Constable, C., and C.L. Johnson, Paleosecular variation, pa- leointensity, and the geodynamo, EOS Trans., 79, (17), Spring Meet. Suppl., S7, 1998.

Constable, C. G., and R. L. Parker, Statistics of the geo- magnetic secular variation for the past 5 My., J. Ceophys. Res., 95, 11569-11581, 1988.

Hartl, P.D., L. Tauxe and C. Constable, Early Oligocene ge- omagnetic field behavior from DSDP Site 522, J. Ceophys. Res., 98, 19,649-19,665, 1993.

Hartl, P.D., L. Tauxe and T. Herbert, Earliest Oligocene in- crease in South Atlantic productivity as interpreted from rock magnetics at Deep Sea Drilling Project Site 522, Pa- leoceanography, 10, 311-325, 1995.

Hulot, G., and Y. Gallet, On the interpretation of virtual ge- omagnetic pole (VGP) scatter curves, Phys. Earth Planet. Inter., 95, 37-53, 1996.

Jacobs, J.A., Variations in the intensity of the Earth's mag- netic field, Ceophys. Surv., 19, 139-187, 1998.

Kono, M., and K. Hiroi, Paleosecular variation of field inten- sities and dipole moments, Earth Planet. Sci. Lett., 139, 251-262, 1996.

Kono, M., and H. Tanaka, Mapping the gauss coefficients to the pole and the models of paleosecular variation, J. Geomagn. Ceoelectr., J7, 115-130, 1995.

Lancaster, P., and K. •alkauskas, Curve and Surface Fit- ting, Academic, San Diego, Calif.,1986.

McElhinny. M.W. and W.E. Senanayake, Variations in the geomagnetic dipole, 1; The past 50,000 years, J. Geo- magn. Geoelectr., 3d, 39-51, 1982.

McFadden, P.L., Statistical tools for the analysis of geomag- netic reversal sequences, J. Geophys. Res., 89, 3363-3372, 1984.

McFadden, P.L., and M.W. McElhinny, Variations in the geomagnetic dipole 2: statistical analysis of VDMs for the past 5 million years, J. Geomagn. Geoelectr., 3d, 163-189, 1982.

McFadden, P.L. and R.T. Merrill, Lower mantle convec- tion and geomagnetism, J. Geophys. Res., 89, 3354-3362, 1984.

McFadden, P.L., and R.T. Merrill, Inhibition and geomag- netic field reversals, J. Geophys. Res., 98, 6189-6199, 1993.

Merrill, R.T., M.W. McElhinny, and P.L. McFadden, The Magnetic Field of the Earth: Paleomagnetism, the Core and the Deep Mantle, Academic, San Diego, Calif.,1996.

Percival, D.B., and A.T. Walden, Spectral Analysis for Phys- ical Applications: Multitaper and Conventional Univari- ate Techniques, Cambridge Univ. Press, New York,1993.

Quidelleur, X., and V. Courtillot, On low degree spherical harmonic models of paleosecular variation, Phys. Earth Planet. Inter., 95, 55-77, 1996.

Rice, J. A., Mathematical Statistics and Data Analysis, 2nd Ed., Duxbury, Boston, Mass.,1995.

Riedel, K., and A. Sidorenko, Minimum bias multiple taper spectral estimation, IEEE Trans. on Signal Process., d3, 188-195, 1995.

Roberts, A.P., Polarity transitions and excursions of the geo- magnetic field, Rev. Geophys., U.S. Nat. Rep. Int. Union Geod. Geophys. 1991-199d, 33, 153-160, 1995.

Tanaka, H., M. Kono, and H. Uchimura, Some global fea- tures of paleointensity in geological time, Geophys. J. Int., 120, 97-102, 1995.

Tauxe, L., Sedimentary records of relative paleointensity of the geomagnetic field: Theory and practice, Rev. Geo- phys., 31, 319-354, 1993.

Tauxe, L. and P. Hartl, 11 million years of Oligocene geo- magnetic field behaviour, Geophys. J. Int., 128, 217-229, 1997.

Tauxe, L., and G. Wu, Normalized remanence in sediments of the western equatorial Pacific: relative paleointensity of the geomagnetic field?, J. Geophys. Res., 95, 12,337- 12,350, 1990.

Tauxe, L., P.J. Tucker, N.P. Petersen, and J.L. LaBrecque, Magnetostratigraphy of Leg 73 sediments, Paleogeog., Pa- leoclimat. Paleoecol., d2, 56-90, 1983a.

Tauxe, L., J. Besse, and J. LaBrecque, Paleolatitudes from DSDP 73 sediments cores: implications for the apparent polar wander path for Africa during the late Mesozoic and Cenozoic, Geophys. J. R. astr. Soc., 73, 315-324, 1983b.

Thellier, E. and O. Thellier, Sur l'intensit• du champ mag- n•tique terrestre dans la pass• historique et g•ologique, Ann. Geophys, 15, 285-378, 1959.

Thomson, D.J., Spectrum estimation and harmonic analy- sis, Proc. IEEE, 70, 1055-1096, 1982.

C. G. Constable and R.L. Parker,Institute for Geophysics and Planetary Physics, Scripps Institution of Oceanography, La Jolla, CA 92093-0225 (e-mail: [email protected]).

L. Tauxe, Geosciences Research Division, Scripps Institu- tion of Oceanography, La Jolla, CA 92093-0220.

(Received September 11, 1997; revised March 17, 1998; accepted April 30, 1998.)