Geomagnetic reversal data statistically appraised

Tectonophysics, 31 (1976) 73-91 0 Elsevier Scientific Publishing Company,

73 Amsterdam

-

Printed

in The Netherlands

GEOMAGNETIC REVERSAL DATA STATISTICALLY

R.A.

APPRAISED

REYMENT

Paleontologiska Institutionen, (Submitted

July

28, 1975;

Uppsala Uniuersitet, Uppsala (Sweden)

accepted

for publication

November

13, 1975)

ABSTRACT Reyment, R.A., 31: 73-91.

1976.

Geomagnetic

reversal

data statistically

appraised.

Tectonophysics,

Standard methods of the statistical analysis of time series are applied to available data on geomagnetic reversals. The times between reversals from Kimmeridgian (148 m.y. BP) to Recent display pronounced trend with an increasing rate of occurrence parameter. This is not solely due to the Late Barremian to Late Coniacian quiet period, for significant trend exists in the observations running from Late Maastrichtian (67 m.y. BP) to Recent, although the size of the trend effect decreases steadily. From Middle Eocene (46 m.y. BP) onwards, the rate of occurrence of reversals becomes trend-free, forming some kind of a renewal process (excluding the Poisson process). A comparison of the sequences of times between normal, respectively, reversed polarities for the Early Eocene to Recent shows them to be almost identical. The selected intervals Kimmeridgian to Barremian, Eocene to Oligocene and Oligocene to Miocene are trend-free. The agreement between periods of pronounced non-stationarity in the frequency of reversals and epicontinental transgressions is briefly noted.

INTRODUCTION

The data analyzed in this paper were taken from the published literature; I have attempted to make use of the most recently made determinations of reversals. The pertinent literature is listed in Cox (1969,1973), Pautot and Le Pichon (1973), Larson and Pitman (1972,1975) and Heirtzler et al. (1968). It would lead too far to present a complete account of the battery of statistical methods I have used in the preparation of this paper. All of them are fairly well known and I refer the interested reader to the source literature given in Cox and Lewis (1966), Vere-Jones (1970) and Cox (1962); for an account of the geological applicability of the methods, Reyment (1969,1976) may be consulted. In the following text, I confine myself to sketching briefly the procedure used in each calculation, without developing the statistical background. In order to facilitate reference to the pertinent literature, I have adhered to the notation of Cox and Lewis (1966). As regards the order of analysis, I have first tested a sequence for trend.

74

It is important to identify trend at an early phase of analysis, for it is quite meaningless to compute a mean rate of occurrence of events for data housing significant trend. We say that a series of events showing trend is non-stationary. If trend cannot be shown to exist in the data, the next step is to search for dependence in the sequence of events, using the second-order properties. If the times between events are serially correlated, it may be possible to establish a working hypothesis for the mechanism underlying the processes giving rise to the events. If the events are independently distributed in the statistical sense, the next step requires finding whether the data accord with a renewal process, a special case of which is the Poisson process. In studying a series of events, interest may reside in estimating no more than the mean rate of occurrence of events for a stationary series. To the mind of the statistician, it is more meaningful to search for systematic variations in the pattern of occurrences. This will normally lead to the proposal of an empirical model for the set of observations which will present the main features of the process without detailed knowledge of its mechanisms. Another avenue of approach is by means of special models, based on apriori notions of what is causing the events. Up to the present, this is what seems to have appealed to people studying the properties of geomagnetic reversals. Statistical tests are often made in order to provide support for the theoretical results, but these do not appear to have been applied in a systematic manner (see Cox, 1973, pp. 152-153 for a list of titles). I wish to emphasize that I have not introduced special interpretations into my analysis of the available series of events, which implies that such questions as waxing and waning of magnetic strength, periods of non-polarity, etc. have not been considered, nor could they have been within the existing framework of statistical theory. In other words, the times between events are treated in the same fashion as any other kind of serial data and the switches in polarity are regarded as being point events. NOTATION

AND GENERALITIES

For the purposes of this paper, the reversals in polarity are taken to be distinguishable only by where they are located in time. Capital letters are reserved for denoting random variables and lower case letters for observations. The times to events { Z’ci,} or times between events {X,,,} completely characterize the process. Thus:

where parenthetical indices denote quantities ordered by magnitude. The counting process, N,, denotes the number of events occurring in the interval (0, t]. The subscript o designates a fixed period of observation (e.g. Kimmeridgian-Recent), and n denotes the number of events over the fixed time t,. The II events take place at times:

75

Nt < n if and only if:

and : proW%

< n) = prWTtnj

> t)

n = 1,2,

.. .

(2)

Equations 1 and 2 specify the fundamental relationship between the representation of a series of events by the counting process and its representation by intervals. In essence, the analysis of the interval process {Xi} is the analysis of a time series of positive random variables. In the present study, I have chosen to observe the series for a fixed. number of events, whereby the first’quiet period in the Jurassic has been omitted. Here, the total time of observation t(,) is the observed value of the random variable Ttn ). ANALYSIS

FOR TREND

Graphical analysis

Plots of the cumulative number of events for successively shortened stretches of the series of observations are given in Figs. 1-3. A preliminary inspection of these figures indicates that there is a suggestion of cyclicity in the sequence although, as will be shown later, this is not pronounced enough to show up in the statistical analyses. We note that the slope of the line joining any two points in these cumulative plots indicates the average number of events per unit time for that period. The reason for studying successively shortened sections of the series of events, begun at marked systematic changes in the plot, is that a magnifying effect is achieved which may bring out small but informative details. In Fig. 1, the plot‘from Kimmeridgian to Recent is shown. There is a pronounced jump between Late Coniacian and Late Barremian (the long Cretaceous quiet period) and strong trending from Late Coniacian to Middle Eocene. It is noteworthy that the slope of this trend is the same as that of the trend for the observations between Middle Kimmeridgian and Late Barremian and both stretches are of about the same length. Figure 2 shows the cumulative plot for Campano-Maastrichtian to Recent. There is a change in slope in the Late Paleocene and a jump in the Early Oligocene, on both sides of which the slopes of the plotted points show a slight difference. The plot in Fig. 3 illustrates the sequence for Early Eocene to Recent. This ‘magnification’ brings out the fact that the plot is less regular than would appear from Fig. 1. There are several sectors of regularly spaced events which are separated by jumps in the plot which suggests that longer intervals could

76 OT : 15-

: 0. :

30-

i .

.

4!i-

l :

= w’

60-

::

&

?5-

z . :

6

..

L” 2 w ? 5 : 2

: .

:. : 105-

.

.

. .

120-

:

135-

:.. :

l.

.

150-

165-

MIDDLE

EOCENE

’ : LATE . . .

PALEOCENE

‘*_ LATE .

CONIACIAN . LATE

1eo-

BARREMIAN

:

l. :

195-

. *.

.

’ . MIDDLE

2102252 0.0

I 40.00

I

I

60.00

1 160.00

120.00 MILLIONS

OF

KIHMERICiGIAH I 250.00

YEARS

Fig. 1. Plot of the cumulative time to events, to), for the period Middle Kimmeridgian to Recent. In this figure, as well as Figs. 2-4, the distances between adjacent points are stepped. Features of interest are the big step between Late Coniacian and Late Barremian and the suggestion of cyclicity between Middle Eocene and Recent. Here, u = -8.49.

possibly denote periods of repose and the jumps could have resulted from cyclicity in the process. For the purpose of comparing the empirical properties of the times between normal periods of polarity and those between periods of reversed polarity, the series of reversals from the Middle Paleocene to the present was divided into two subsets. The graphs of the two, sets of points are plotted .alongside each other. The close agreement in the shapes of the two plots intimates that both sequences may be expected to have about the same statistical properties (Fig. 4). Statistical

analysis

The next step is to ascertain whether the trend indicated in the graphs is statistically significant; i.e.~whether or not the series of events under consideration is non-stationary.

0

10

._____ . T l

20-

30-

:. .. .. .. .. .. .

LO-

SO-

.. .. .. .. .

60? =

70-

2 “0

80-

: m

go-

: z w loo> I= 4

.. .. .. .. .. .. .. . :. .. .. ..

.

EARLY :

: =

OLIGOCENE

:

llO-

: 120-

. .

. :

130-

.

. .

.

lLO-

.

.

.

.

.

LATE .

150-

PALEOCENE .

.

.

.

170

.

-.

160’

,

I

1

00

lL.00

28.00

L2 00

MILLIONS

OF

CAMPANOMAA~TRICHTIAN 56 00

l.,

l

70 00

YEARS

Fig. 2. Plot of the cumulative time to events, t(i), for the period Campano-Maastrichtian to Recent. Note the change in slope between the time before Late Paleocene and that after Early Eocene. Here, u = -4.60.

It is usually outgoing from we denote the the following

good policy to investigate Poissonian considerations Poissonian rate parameter relationship

a sequence of point observations (cf. Cox and Dalrymple, 1967). If as x(t), we have, in the case of trend,

h(t) = e”+@

(3)

where cy is a nuisance parameter. second derivative, to wit: L’(P) = -;nt,

+ L’ti;

An estimate

of /3 can be found by using its

p=0 (4)

78

0 . 10 -

20 -

30 -

l. l. : . ‘. . t

r

l . .. l. . ‘.

l

I: z

do-

%

50-

a” i+i f y > C 5

60-

l.. . l.. l..

-Z. :

m00-

2 2

go-

:. :. .

100-

l.

l. .

.

.. l. :

l

110 120 -

l..

130 -

EARLY

.

160 150

:

l. .

EOCENE

3

,

I

0.0

10.00

20.00

1

,

30.00 MILLIONS

10.00

l.

. ,

50.00

OF YEARS

Fig. 3. Plot of the cumulative time to events for the period Early Eocene to Recent. The parallel stretches of 4-6 events suggest the possibility of pulsations in the series. Here, u just attains significance with a value of -2.02.

and the information function:

w =nC j$-

t,2exP(--Pt,) (pexp(-_@))2

1

;

P#O

(5)

For testing that fl = 0, the ratio: C tj/n -i ld=

t,~i/iFn

to (6)

compares the centzoid of the observed times with the midpoint of the period of observation. The results of the tests for trend applied to the geomagnetic

79 TABLE

I

Tests for trend, Observations

reckoned

from the present

on

Kimmeridgian Late Maastrichtian Middle Paleocene Early Eocene Middle Eocene Middle Paleocene (normal polarities) Middle Paleocene (reversed polarities) Early Eocene (normal polarities) Early Eocene (reversed polarities) Barremian to Kimmeridgian Eocene to Oligocene Oligocene to Miocene

backwards

in time

n

cl

214 168 156 144 132 78

-8.49 -4.60 -3.81 -2.02 4.21 -2.63

*** *** *** * **

highly significant highly significant highly significant significant trend trend-free significant trend

78

-2.59

**

significant

72

-0.78

trend-free

72

-1.32

trend-free

39 108 72

+I.94 -1.50 *.51

trend-free trend-free trend-free

Remarks trend trend trend

trend

reversals, using formula 6, are presented in Table I. Table II contains estimates of 0, using eqs. 4 and 5. A positive value of u for eq. 6 means that the centroid of the events X:ti/n lies above the midpoint of the interval (0, to] which indicates that j3 > 0. This is the situation for a rate of occurrence of events which increases with time. A negative value of u points towards fl< 0 and a rate of occurrence of events that decreases with time. The results of Table I indicate that, on the average, the rate of occurrence of reversals falls off with time, towards the past reckoned, and 0 < 0 for all parts of the sequence with significant trend. This condition persists until Middle Eocene when the times between intervals become trend-free. This is in agreement with Naidu’s (1975) observation concerning a change in the statistical properties of the times between events at 48 m.y. BP. Although the value of u does drop rather sharply from Early Eocene to Middle Eocene, Fig. 3 shows that the change is reflected in a shift in slope which takes place gradually. The TABLE Estimates

II of /3 for trend

Observations

on

Kimmeridgian Late Maastrichtian Middle Paleocene Early Eocene

n

P

w4h)

times initial rate of reversals

214 168 156 144

+X0136 -0.0166 -0.0165 -0.0111

0.134 0.292 0.347 0.559

7.46 3.43 2.88 1.79

80

REVERSED

65;

POLARITIES

0

00 @* “0 0

70-

l0*

.

0 .

0 .

75-

0

%

600.0

I

I

I

12.00

21,oo MILLIONS

I

36.00 OF

66.00

. 1

0 .O

60.00

YEARS

Fig. 4. Comparison of the cumulative time to events for normal polarities and reversed polarities for the period Middle Paleocene to Recent. The tail in the lower right corner is the cause of the significant trend values (normal polarities, u = -2.63; reversed polarities, u = -2.59). The close agreement in the details of the two plots indicates that the empirical distributions of the intervah between events for both polarities are close.

regressions to the slopes are significantly different. It is worth noting that fl< 0 for all parts of the plot under consideration. The partitioned suites of times between normal, respectively, reversed polarities are, as was to be expected from Fig. 4, close, with significant trend in the Middle Paleocene and trend-free to the Early Eocene (thus underlining the gradual nature of the Early to Middle Eocene shift in rates of occurrences of events). It might well be suspected that the Lang quiet period in the Early and Middle Cretaceous is responsible for the trend effects. That this is not entirely so is demonstrated by the fact that the stretch from Late Maastrichtian to Recent, for example, shows highly significant trend.

81

Examining the estimates of p in Table II, we find that the Kimmeridgian sequence represents an increase in the reversal rate, in comparison with the present, of 7.46 times. From Late Maastrichtian reckoned, the increase is 3.43 times, and from the Middle Paleocene, 2.88 times. Finally, the increase in the reversal rate over the Early Eocene is 1.79 times. It will be shown in a later section that from Middle Eocene onwards, there has been no substantial shift in the rate of occurrence of polarity reversals. STUDY

OF THE TREND-FREE

PART

OF THE

SEQUENCE

The next step in the analysis is to study the properties portion of the sequence of geomagnetic reversals. A natural starting point for this analysis is to consider coefficients, defined as: Pj =

cov(xi,

xi+j)

var(X)

.

’

j = ... -

1, 0,

of the trend-free the serial correlation

1,. . .

5-

IO 15 -

20 .-T

4' 25 2 E

30 -

: >

3540 L5 50 55 -

Fig. 5. Plot of the normalized correlogram (periodogram of serial correlations) for the Middle Eocene sequence and the normal and reversed polarities from Early Eocene. The dotted lines denote the 95% level of significance for the serial correlation coefficients.

82

which gives an approximate means of establishing the presence of dependence between intervals. Providing the marginal distributions are not too highly skewed, the serial correlation coefficients may be considered as observed values of a standardized normal variate, the distribution of which is N[O, l/(n)lj2] ; this is a reasonable approximation for 12> 100. For the Middle Eocene sequence, there is one significant correlation coefficient out of a total of 65 and this may be taken as an indication that the intervals between events are independently distributed. For the subsets of observations for the periods of normal and reversed polarity, respectively, only one serial correlation coefficient out of 35 is significant. Inasmuch as the 5% level of confidence allows one wrongly significant correlation coefficient in 20, this value could be accepted as being fortuitous. However, both significant correlations occur for about the same lag-interval (18, respectively, 19) which could possibly point towards a genuine dependency in the sequence of events. It is noteworthy that the one significant correlation for the entire set of observations shows up at the same relative lag as for the foregoing sequences. The Kimmeridgian to Barremian stretch has no significant correlation coefficients, nor has the Oligocene to Miocene run. A periodogram (here, a normalized correlogram) of the correlation coefficients may be constructed by multiplying eq. 7 by (N --j)l12. Plots of F,(n- j)‘j2 for the three sequences are given in Fig. 5. Specific tests for a Poisson process The serial correlations suggest that the intervals between events are probably independently distributed. Therefore, tests specifically designed for picking up exponentially distributed times between events are now in order. If the events are exponentially distributed, this is a fairly certain indication of a Poissonian distribution; if they are not, then the times between events are TABLE III Mean and coefficient

or variation

Observations

n

+rc,.’

C=LJ

ni=1

Middle Eocene Early Eocene (normal polarities) Early Eocene (reversed polarities) Barremian to Kimmeridgian Eocene to Oligocene Oligocene to Miocene Kimmeridgian (trend sequence)

P

0.33 m.y. 0.69 m.y.

0.97

0.13 m.y.

0.90

0.94

0.76 0.96 0.75 2.75

m.y. 0.35 m.y. 0.32 m.y. 0.69 m.y.

0.69

____

_

83

distributed in accordance with some kind of a renewal process (Cox, 1962), of which the Poisson process is a special case. Table III contains the means and coefficients of variation for the series of events under consideration. We note that the longer sequences are leptokurtic and right-skewed, which should be borne in mind when interpreting the serial correlations. The series of events from Kimmeridgian to Barremian and from Oligocene to Miocene show only slight kurtosis and skewness. For purposes of comparison, the values for the trending sequence from Kimmeridgian to Recent are included in Table III; note the high value of the coefficient of variation. On the average, an event occurred every 0.33 m.y. from the Middle Eocene onward and, on the average, from the Early Eocene onward, a normal polarity resulted each 0.69 m.y. and a reversed polarity each 0.73 m.y. Inasmuch as we have shown that the sequence of events became stationary in the statistical sense during the’ Early Eocene, these means have some interpretive value; means calculated for a sequence showing trending and nontrending parts are worthless as they constitute a mixture of stationary and non-stationary series of events. (N.B. the fact that 0.33 does not connect directly to 0.69 and 0.73 is no cause for alarm as the series do not cover exactly the same time-span.) All coefficients of variation are less than 1 in Table III (apart from the trending Kimmeridgian to Recent value), which speaks against a Poisson hypothesis. The coefficient for the Middle Eocene sequence is, nevertheless, only slightly less than one, which provides an incentive to be thorough in the selection of tests for a Poisson distribution of the intervals between events. It is interesting to observe that there is an unexpectedly large difference in the coefficients of variation for the intervals between normal, respectively, reversed polarities. The plot of the logarithmic survivor function, which for a Poisson distribution has a straight tail owing to the exponentially distributed times between intervals, also shows a difference in shape between the two polarities (Fig. 6). All of the plots for the sequences investigated have nonlinear tails, but this is less pronounced for the complete sequence from the Middle Eocene, for which the coefficient of variation is just less than one. The empirical survivor function R(x) is defined as: R(x) = 1 -F(x) where F(x) denotes the distribution function, z(x), is defined as:

function

associated

with X. The hazard

z(x) = f(x) jqx)

where f(x) In plots convex log -dz(x)/(dx) ships apply

is the probability density function of X of log R(x), a monotone, non-decreasing survivor function (d/dx log R(x) = -z(x) and a coefficient of variation less than for a monotone, non-increasing hazard.

and R(x) > 0. hazard correspond to a and d2/dx2 log R(x) = one. The converse relationThe shapes of all of the

84 TlME

BETWEEN

0,6-l

0 oa7e

0 99

EVENTS A_.

1 30

_!I?!-

1 90

0.24 0.51 0.76 1.00 1.30 1.50 1.80 2.00 :

2.30 (Y

$

2.50 2.60 3.00 3.30 3.50 3.ao 4.00

.

4.30

4.50

L.80

D

REVERSED

+

N~M&L

l

&LL

POtARlflES WLARiTIES

EVEMTS

*

(E.EOCENE) (E.

EOCENE)

(M. EWENEt

Fig. 6. Plot of the logaritluni’c empirica; survker function for the t=nd-free seetiori from the Middle Eocene, the tiznes betsmzen normd p&rEties and the times between reversed polarities for the Early Eocme.

85 TABLE

IV

Tests for Poisson Observations

processes

on

Middle Eocene Early Eocene (normal polarities) Early Eocene (reversed polarities) Eocene to Oligocene Barremian to Kimmeridgian Oligocene to Miocene Kimmeridgian

n

0:

0,

132 72

0.54 0.23

1.88 2.64

*** ***

1.88 2.64

*** ***

5.48 12.25

*** ***

72

0.53

2.26

***

2.26

***

8.09

***

108 39 72 214

0.47 0.17 0.05 3.23

1.81 1.43 1.85 0.77

*** * ***

1.81 1.43 1.85 3.23

*** * *** ***

***

WZ

0:

8.09 *** 2.53 * 5.64 *** 17.79 ***

of these tests are listed in Table IV. They indicate quite clearly that the Poisson hypothesis is rejected. Thus, although the intervals between events appear to be independently distributed, they are not exponentially distributed. Table IV also includes values for the Kimmeridgian to Recent set for comparison. Further useful information on the times between events can be obtained from an analysis of the periodogram related to the spectral density function (cf. Bartlett, 1963). Although this might be thought superfluous in the present study, owing to the indication of independent times between events yielded by the serial correlation coefficients, there is a need to back up this information in view of the fact that it was not accepted without reservation. The periodogram, or power spectral density, may be written as: I(w) = WX)f+ where f+(o), f+(w) =+[l

(0)

( (8)

the positive spectral density function, is: + 2,Fi

Pjhj

codj~)]

(?r>

w>O)

(9)

and the Xi’s are suitably chosen weights. A comparative plot of eq. 9 for the series of normal and of reversed polarities is given in Fig. 7. There is fair agreement in the shapes of the curves and both of them oscillate around a value of l/n for a renewal process. Plots of the normalized cumulative periodogram for the Middle Eocene sequence of events (Fig. 8) and the normal and reversed polarities superimposed (Fig. 9) should approximate a straight line if they derive from a renewal process. This is roughly so and there is no trend, as shown by u of eq. 6, the values for which are listed in Table V. Table V also contains values for D”,, D;, 0: and Wi for these sets of observations, none of which is near significance. The test for independence between intervals amounts here to seeing whether I(w,), (p = 1, 2, . . . . ; n,), for successive values of p, accords with a Poisson process; this makes use of the fact that the spectral transformation reduces the test

86

I.30

37 10

l

NORMAL

0 REVERSED

POLARITIES

i//

POLARITIES

Fig. 7. Juxtaposed plots for the normal and reversed polarities (Early Eocene to Recent) of the positive spectral density function f+(w,), plotted against p. where p = 1, 2, .. . . ino.

for a renewal process to a test-for a Poisson process. Consequently, the analysis of the periodogzam provides fur&herapport for a renewal hypothesis. The stretches Eocene to Oligocene and Oligocene to Miocene are trend-free with non-parametric sWtis far from significance (Table VI). The Kimmeridgian to Barren&n sketch shows significant trend for the values of the periodogram tid a significant figure for I?$. The values for the entire sequence of events are significant. The ratio I( CJ, )/f+ ( W, ) for the Middle @cene sequence gave x& = 11 which, through falling short of the 5% level of significance, is further evidence

87 CUMULATIVE 0.20

0.0

PERIOOOGRAM

I Iwp)

0.60

0.40

1

O,W

o-

6. 9-

12

15

. :

16

21 26

. *. . . .

. l

. .

*. . -. .

36

‘. 19

‘. .

62

‘. b5

. .I . .

46

*.

51

.

: . *. .

5L

55I-

‘. .

6C,-

MIDDLE

63-

I

66

* .

EOCENE

I

I

I

Fig. 8. Plot of the cumulative periodogram Z(C+) against p for the Middle Eocene series of events. There is no significant trend and the tail of the plot is reasonably straight.

TABLE

V

Tests for trend and independence

of the periodogram for the Eocene to Recent sequence

Observations on

u

0;:

0;

0:

W?I

Middle Eocene Early Eocene (normal polarities) Early Eocene (reversed polarities)

0.52 0.59

0.45 0.35

0.66 0.53

0.66 0.53

0.32 0.32

0.72

0.47

0.74

0.74

0.63

--

.-

88

0.0

CUMULATIVE O.&O I

0.20

PERIO~OGRAM 0.60

Itwp) 0.80

1.00

7 0

2-a .

0

L-

.

0

. 6-

0 .O .

0

6-

. .

0 0 .D .

lo-

0

12-

.

0

.O .O .

lb cz ci ‘6D z -

0 .

0

CI 0

16-

.

0

20-

0.

22-

0 . 0

ZINORMAL 26-

. 0 . 0

REVERSED

POLARITIES POLARITIES

l

. 0

o

. 0 . 0. 0

26-

. 0. .

30-

0 .

0 .

32-

. 3L-

0 .

36

Fig. 9. The cumulative periodogram for the normal and reversed polarities, superimposed. Early Eocene.

TABLE VI Tests for trend and independence of the periwkgram

for selected stretches of events

Observations on

U

0;

D,

D:

Bf

Kimmeridgian Barremian to Kimmeridgian Eocene to Oligocene Oligocene to Miocene

2.03 * 2.12 * 0.47 1.33

0.25 0.03 0.48 0.25

1.46 * 0.94 0.69 0.93

1.46 * 0.94 0.69 0.93

3.16 *** 1.73 *** 0.54 1.10

_-.

89

in support of a renewal hypothesis (Cox and Lewis, 1966, p. 168). Naidu’s (1971,1975) claim for dependence between intervals might seem to be a consequence of his including part of the trending portion of the series of events in his calculations. However, for the entire sequence from the Kimmeridgian onwards, there is only one significant serial correlation (at lag 2) out of 106 (remembering that this set of observations is greatly skewed and highly kurtosic). The evidence for serial correlation yielded by the present analysis is slight for the stationary part of the sequence (cf. Crain and Crain, 1970 and Crain et al., 1969). SUMMARY

OF THE STATISTICAL

ANALYSIS

Consideration of the entire sequence of 214 events showed that the stretch from Kimmeridgian to Paleocene contains significant trend, with a gradual increase in the rate of occurrence of reversals towards the present. The section from Kimmeridgian to Barremian is trend-free, although it shows some deviations from a renewal process. In the Early Eocene, trend disappears from the reversal pattern and the series becomes stationary. The pattern of the times between events from Early Eocene to the present approximates some kind of a renewal process, but not a Poisson process. CONCLUDING

REMARKS

It is interesting to speculate on the possible reason for the change from nonstationarity to stationarity in the pattern of geomagnetic reversals in the Early Eocene. The period of non-stationarity coincides rather well with the opening of the South Atlantic (Reyment and Tait, 1972; Reyment et al., 1975) and the vast epicontinental transgressions associated therewith, these latter apparently being related to profound changes in the morphology of the ocean floor. The last trans-Saharan epicontinental transgression took place in Early Paleocene time after which the South Atlantic has been the site of a gradual, unidirectional regression (cf. LePichon, 1968, pp. 107,116,117). Various theoretical models have been devised for explaining geomagnetic reversals, among them those of Cox (1973, p. 147), Ramberg (1972) and Naidu (1975). If I have understood the proposed mechanisms of Ramberg’s and Cox’ models correctly, it is conceivable that they would give rise to a series of events approximating the properties of a renewal process. (See Cox’ (1962, p. 71) remarks on the superposition of renewal processes.) A statistical analysis can, of course, never be more precise than the data to which it is applied. It should therefore be appreciated that the results accounted for in this study apply specifically to the geomagnetic data as they stand at the moment and, should new determinations come to light, and such seems to be fully possible (cf. Rea and Blakely, 1975), then a new set of analyses could well yield a different picture.

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ACKNOWLEDGMENTS

This work was carried out as part of the project Mid-Cretaceous Events of the International Geological Correlation Programme. It was supported by Grant 2320-61 of the Swedish Natural Science Research Council and Computing Grant 104320 of Uppsala University. A computer programme supplied by IBM, and modified for the present purposes (based on Lewis, 1964) was used for doing the calculations and making the plots.

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