The Cunudiun Jiirrmul i t Sturistics

Lu Revue Cunudienne de Statistique VOl. 26. NIL 3. IYYX. PdgCS 3x3-392

383

On the role of statistics in the palaeomagnetic proof of continental drift Geoffrey WATSON*

Princeton University

Key words and phrases: Unit vectors, Fisher distribution on the sphere, analysis of

AMS 1991 subject classifications: 62H 1 1. dispersion, palaeomagnetism, continental drift, plate tectonics.

ABSTRACT

Magnetized rocks provide a “tape recording” of the more recent history of the earth. I t was this evidence that broke the opposition to continental drift. The nature of the data - samples of unit vectors in three dimensions - demanded new forms of statistical treatment. I hope that my personal account will enable the general statistician to see a rare example of statistics being developed to be a modest help in a major scientific revolution.

RESUME

Les pierres magnetiskes foumissent un “enregistrement” de I’histoire plus recente de la terre. C’est cette Cvidence qui a eu raison de I’opposition 5 la thCorie de la derive des continents. La nature de ces donnCes - des Cchantillons de vecteurs d’unites tri-dimensionnels - exigeait de nouvelles formes de traitement statistique. J’espkre que mon compte-rendu personnel permettra au statisticien gkneral de voir un exemple rare de statistiques developpees afin d’gtre d’une aide modeste lors d’une dkcouverte scientifique majeure.

1. INTRODUCTION

Wegener in the early 1900s saw a “jigsaw fit” if one closed the Atlantic ocean. He also noticed fossil correlations between different continents now far apart. Thus he suggested that once all the continents had been joined together in one supercontinent in the southern hemisphere, which he called Pangaea, but that they had then drifted apart. Geologists were accustomed to think of land masses going up and down, but continental drift was a heresy to all but a few brave souls. The first evidence that broke the opposition to drift came from the study of the magnetization of rocks - palaeomagnetism. I will argue that statistical methods played a very useful role in this.

Gauss had shown by 1839 that the earth’s field was, to a good approximation, that of a geocentric dipole - now inclined at 11.5” to the rotational axis. Thus magnetic north is at the moment about 11.5” away from geographic north, but its position is known to vary gradually with time. In the 1800s, studies on Mt. Vesuvius showed that recent lavas were mostly magnetized parallel to the current field, but at some places in different directions, perhaps due to lightning strikes. Around the turn of the century Pierre Curie observed that, as various magnetizable materials were cooled in a magnetic field, they acquired their magnetization at characteristic temperatures. Thus old lava flows should record the field where and when they cooled down. Some sedimentary rocks also record

*Unli)flunatcly, allcr this articlc was acccptcd. Ccollicy Watson dicd on January 3, IYYX.

384 WATSON Vol. 26, No. 3

the field at the time and place of their creation, but by a different mechanism. Circa 1905, examples of reversals were found, i.e., recent rocks whose direction of magnetization was the reverse of the present-day field. This was mysterious, as it was hard then to believe that the polarity of the earth had switched.

Since the magnetic field undergoes small changes over geologically short times, we can get a picture of these changes by finding rocks of various ages and measuring the magnetization of oriented rock specimens from them - provided, of course, that it can be shown that the rocks faithfully record the field. If so, the earth’s rocks are, in effect, a super-tape-recording of its magnetic history - in fact, of its more general history.

But what generates the earth’s field? The earth’s core is too hot to be magnetized, so it must be due to electric currents - a self-sustaining magnetohydrodynamic dynamo is now considered to be the cause. Roughly, convection (controlled by the temperature gradient and the earth’s rotation) of conductive molten rock leads to currents, which generates the field that we see.

Intense study of palaeomagetism began just after World War 11. For a very readable account, which even a layman can follow, of the evolution of ideas about continental drift and plate tectonics, and the understanding these give us of the earth, see Judson et al. (1976), which has many helpful figures that could not be reproduced here.

One check that the rock in a given formation has been stably magnetized is due to Graham (1949) and called the conglomerate test. Imagine that a horizontal layer of rock on the top of a hill is all magnetized in a parallel way and subsequently broken up. The broken pieces will roll down the hill, and the resting positions will be unrelated to the direction of magnetization of each boulder if no later event changes their magnetization, i.e., they are stably magnetized. Graham was only able to plot the direction on a projection (usually equal area) of the sphere and see if it looked uniformly random. Clearly a significance test for uniformity is needed here.

Early postwar studies of recent rocks showed that the earth’s field, reversed or not, when averaged over tens of thousands of years, was well approximated by a geocentric dipole lined up with the rotational axis. If the sampling covers this time span, the mean direction of the sample and the formula for a dipole field give us an estimate of the rotational and thus the geographic pole. Even if the Earth’s field had reversed (the simplest explanation of reversals), magnetization of all rocks of the same age on the same continent should be consistent. If these measurements gave different positions for the magnetic pole at different times, one could sketch a curve through the pole positions in increasingly old rocks. Confidence circles were drawn about the pole positions so the curve need only go through these circles. This is called an apparent polar wander path (APWP). Either the pole had moved with respect to the fixed continent over time (this seems unlikely if time averaging is giving a rotational pole) or the continent moved with respect to a fixed pole. The accurate time scale needed came from radioactive dating. This requires the fitting of a linear relation to variables all subject to error - for the most modern account see Kent et al. (1990). But rough scales were used first.

Before this could be sorted out in the 1940s and 1950s, one needed many checks that the rocks faithfully recorded the field, and magnetized rocks of different geological ages from many places on many continents. Better instruments were needed to measure the magnetization, which was often quite weak, especially in sedimentary rocks.

Geophysicists at Cambridge University, Keith Runcorn in particular, were in contact with Sir Ronald Fisher. Runcorn realized that a good statistical treatment was vital, as the directions (the strength is not used) of magnetization of specimens from the same site were quite variable. Other groups simply plotted their data on a spherical projection

1998 STATISTICS AND CONTINENTAL DRIFT 385

and kept sampling until the result was clear, but this requires more measurements than are actually necessary.

Fisher [formally in his 1953 paper, but he used the methods of this paper to do the calculations for Hospers (195 I ) , then a graduate student] suggested the density on the sphere,

for unit-vector observations x. This density is rotationally symmetric about p. When K = 0, the distribution is uniform; as K grows, it peaks more and more about p, which is the modal or mean direction x. Thus K is a concentration measure, the reciprocal of a dispersion measure. Given a sample of n directions XI, x2,. . . , x,,, the maximum- likelihood (ML) estimator of p is clearly the unit vector parallel to R = C xi, R = llRll, and that of K (if large) is n / ( n - R). After giving some basic sampling theory, the rest of Fisher’s paper is used to explain (once more) how to operate with fiduciul probability. In particular, a fiducial confidence cone for p was given - this is the confidence circle about the estimate of the pole position mentioned above. But even more important were his numerical definitions of a “mean” or “preferred” direction and of the “scatter” or “dispersion” about this direction. As we saw above, a formal test of uniformity (of K = 0) and further significance tests (e.g., that two samples had the same mean direction, or the same dispersion) were also needed. I provided these tests and other things.

Edward Irving, a student at Cambridge at the time, feels that Fisher’s contribution, which replaced graphical answers with numerical solutions, put the Cambridge group in front of all competitors. The main competition was Blackett’s group at Imperial College. Blackett, a Noble Prize winner for work on cosmic rays, did not believe in statistics. Moreover, he had no feel for variability, e.g., the difference between taking very many samples at one site and taking a few at many sites.

Irving helped to write the account of Fisher’s palaeomagnetic interests in Joan Box’s (1978) book about her father. Irving (1988) gives a definitive account of the role of palaeomagnetism in the confirmation of continental drift. It is amusing to note that though Fisher and the great geophysicist Jeffreys greatly respected one another, they disagreed strongly over Bayesian inference. Since Jeffreys had proved mathematically, given the mechanisms suggested at that time, that drift could not occur, it gave Fisher great pleasure to help to prove that the continents had in fact drifted. At Runcorn’s suggestion, Fisher used a visit to, and his prestige in, India to have rocks collected there, whose analysis showed that the Indian subcontinent had drifted up from an old position in Pangaea near Madagascar. The collision of the Indian mass with that of Asia is the source of the Himalayas.

2. MY EARLY INVOLVEMENT

In 1953 Fisher visited Australia and later sent me a reprint of his paper. Shortly afterwards Irving and I both joined the Australian National University. Thus I caught from Irving the scientific excitement of this field and saw what further statistical methods were needed. I did not base my methods on fiducial inference.

Firstly a rest for uniformity was needed. On various grounds, it seemed best to reject uniformity if R is “too large”. The distribution problem had been solved exactly and approximately by the physicist Lord Rayleigh for a problem in sound, in two dimensions (1880, 1905) and then in three dimensions (1919). Here R2 = X2+Y2+Z2. The coordinates X, Y, and Z are seen to be uncorrelated and asymptotically n o d . Using spherical polar coordinates, Z = 1 cos 8; and the area element on the sphere is sin 8d6dg. Thus cos 8

386 WATSON Vol. 26, No. 3

is uniform on [ - 1, I ] , so E cos 0 = 0 and E cos’ 0 = 1/3. Symmetries show that asymptotically

3R’lN is x: (chi-square with three d.f.). (2)

This simple consequence of the central limit theorem leads to a significance test, which 1 named after Rayleigh. All I had to do was compute the exact points for N up to 20.

In well-magnetized samples where (1) is often a good approximation, K is rather large, so if we use a spherical polar coordinate system with its z-axis parallel to p, then 0 = 0 at p, and likely 0’s are small. This suggests the approximation of (1) having probability element

1 2n

exp[-K( 1 - cos e ) ] K sin 0d0 . -d+ (3)

Thus 9 is uniform on (0,2n), and approximately, U = K( 1 - cos 0) has density exp(-u), or

(4)

Hence, if we denote the component of R along p by Z (Fisher used X, which would be confusing here),

2 K ( 1 - cos e) is xi.

2 ~ ( 1 - cos 0;) = 2 ~ ( n - Z) has a xi, distribution. ( 5 )

This implies that the ML estimator of K when p is known is n / ( n - Z). So n - Z is a measure of the dispersion of the sample about the true mean direction p. Similarly n - R could be regarded as a measure of the dispersion of the data about the estimated p, rather as C(X, - a)* measures the dispersion of a scalar sample about its estimated mean, while C(X; - p)2 is the dispersion about the true mean p.

Consider the following two identities, the first for real-number data and familiar from elementary statistics, the second from the above ideas for unit-vector data:

both with the analysis of dispersion interpretation [(dispersion about p) = (dispersion about estimated p) + (dispersion of estimated mean about p)], and the distributional statements,

For a normal sample we know that (8) is true exactly. I showed that (9) is approximately true for data from the Fisher distribution with large K - it works very well for K’S over, say 10. Recall what we always say about the effect on x2 of fitting location parameters: d.f. = n - (number of parameters fitted). It works here too, as p needs two numbers to specify it.

The geometric meaning of (6) is seen in a triangle in n dimensions with vertices the origin 0, the point P = (X I , . . . ,x,) and the projection of the vector OP onto the ray through the point (1, . . . , 1). Similarly the geometrical understanding of (7) comes from

1998 STATISTICS AND CONTINENTAL DRIFT 387

a triangle, in thrce dimensions, with sides R , the perpendicular from the end of R down onto a line parallel to p, and the projection of R onto p. From (7) and (9), we have

2K(R - z ) / 2 = F2.2(fl-I) 2 ~ ( n - R ) / 2 ( n - 1)

for testing the null hypothesis that p is the true mean or model direction of the Fisher distribution of the data. It’s like t2 obtained in the same way from (6) and (8).

Again the ML estimator of K , when p is unknown, is n / ( n -R) but if we follow Fisher and set

n - 1 n - R

k = -

then (7) and (8) show that

( 1 1 )

making Fisher’s estimator k seem more sensible and giving a chi-square test statistic for the hypothesis that K has some specific value - or giving a confidence region for K . It may also be used to check whether two samples come from Fisher distributions with the same K. For if they do, (1 1) implies that

k 2 ( n - 1) - 2 ( n - 1) - = - K W n - R ) x;(fl-l) ’

This was misapplied in the 1960s to what was called the “fold test” - see below. Finally notice that (10) may be written, with z = R cos 6, as

kR( 1 - cos 0) = F ~ , Z ( ~ - I ) , (13)

which yields a confidence cone for p. The axis of the cone is @, and the semiangle is 6*, the solution of

kR(1 - cos &) = F;,2(n-I),

the 95% point for this F. Since F ~ , Z ( ~ - I ) is close to x $ / 2 or the standard exponential, we may use this further approximation in most practice. These arguments come from Watson ( 1956a).

One may also do exact frequentist calculations, as in Watson (1956a) and Watson and Williams (1956). Then (13) comes as the first tern in a series, as it does in Fisher’s fiducial argument. Thus the cone can be interpreted in two ways. When K is large, the first term is all that is needed. The second paper mentioned above annoyed Fisher because we said explicitly that we were giving a Neyman-Pearson account. However, he gave my two 1956 papers to Jeffreys, who had then published in the Geophysical Supplement of the Royal Astronomical Society.

My analysis-of-variance - analysis-of-dispersion, as I prefer to call it - approach suggests quick answers to many problems - e.g., given samples of nl and n2 from Fisher distributions with the same K , to test whether they have the same p ’ s . Looking at the triangle with sides RI and R2 and R = R I + R 2 , it is natural to set up the identity, using obvious notation,

n - R = [(nl - R I ) + (n2 - R2)1+ ( R - R I - R2) (14)

388 WATSON Vol. 26, No. 3

TABLE I : Analysis 01' dispcrsion (variancc).

No. 01' Dispersion Mcan Expcchtion 01' Sourcc d.r. (sum 01' squaws) ,%plan: mcan squarc

Bciwccn sitcs

Within siics

2(B - I ) CR;- -R 2(B - I )

Total 2(N - I ) N - R

where R = llRl + Rzll, n = nl + n2. Then 2~ times (14) should be distributionally (approximately)

2 X;(n-l, = [ X & - l ) +X;(n?-I) l + x 2

which leads to 2 ( n - 2 ) ( R - R 1 -Rz) / [ (n l -Rl)+(nz-R2)] being approximately distributed as F ~ . 2 ( ~ - 2 ) . This is our analogue of the two-sample t-test.

It is common practice to take samples (W; from site i) from a number B of sites in the same formation. The within- and between-sites analysis in Watson and Irvin (1957) assumes that a direction x at a site is F(m,w), where the B-site means m are assumed to be a sample from F(pp) . Then it will not surprise statisticians that x is approximately F ( ~ K ) , where

The rest of the argument also parallels the linear case. If there are B sites and W; directions at site i with vector resultant R;, the analysis-of-dispersion table is as shown in Table 1. Here N = C W ; , R is the length of the resultant C R1, and

1 B - 1

W = -(N-E W : / N )

is a weighted average of the W;. Estimates GI and are found by equating mean squares to their expectations. The direction of the resultant Ri is roughly Fisher about p with a concentration which is the reciprocal of (CON)-' + (BB)-'. This sort of analysis is important, e.g., for lavas, where the within-site dispersion is a measure of the accuracy of the magnetic recording and the between-site scatter is a measure of the secular variation and any locally variable subsequent tilting of the formation.

of o and

3. VINDICATION OF WEGENER

With these methods, good and bad data could be separated and polar-wander curves sketched. The APW curves derived from European and North American data are very different. However, if one rotates Europe so that its APW curve matches (in a rough way, with an eye on the confidence circles) that of North America, one finds that the Atlantic is closed just as Wegener suggested long ago. The same sort of matching of the APW curves for other pairs of continents shows that his Pangaea did exist and further allows one to plot the track of the continents from their early to their latest positions.

Thus by about 1960 continental drift was believed by most geologists. At that time the state of palaeomagnetism and its physical & statistical techniques and geological applications was summarized by the books of Irving (1964) and Collinson et al. (1967,

STATISTICS AND CONTINENTAL DRIFT

but actually the proceedings of a 1964 conference). The statistical methods are simply those given above.

Of course, finding an explanation of the continental motions was an essential part of this acceptance. This leads to the general subject of plate tectonics, which would take us too far afield. Initially it came from the discovery (Vine and Mathews 1963) of a sequence of bands of alternating magnetic polarity on either side of the Mid-Atlantic Ridge. The first notion was that new sea floor was being created at the ridge and being marked by the many reversals of the earth’s field. The widths of the bands were tied to the rate of spreading and the time between reversals. This was supposed to be the way these two continents were pushed apart; similar ridges were found in the other seas. But the picture quickly becomes much more complicated. Statistics continues to be useful, however - see, e.g., Chang (1993).

4. LATER STATISTICAL DEVELOPMENTS

Though I did a lot of refereeing for geophysical journals, job changes after 1956 meant that I lost personal contact with palaeomagnetists, so that I did not write practical articles. I did only theoretical and expository writing in this area from the 1960s to the 1980s. In this period my students M.A. Stephens and R.J.W. Beran did important work. My new scientific motivation was bird navigation, and so I worked mainly on the circle, not the sphere. In the early nineties a Princeton geophysics student, Michel Debiche, raised many questions about technical problems in some of the many papers written in the interim. He got me back into palaeomagnetism. We wrote two joint papers (Watson and Debiche 1992, Debiche and Watson 1995) and started some more that I eventually wrote with other people because he had no time to spare. Email has since allowed me to work with many people around the world.

Many statistical papers of practical significance were written in the interim by geo- physicists, almost all based on the Fisher assumption. For example when samples are found from drill cores, one does not know how the sample has rotated about the axis of the drill hole. Thus the declination of its magnetization is lost, so one has only in- clination data. The literature for this problem may be traced through Enkin and Watson (1996). The Fisher distribution was first questioned in the U.S. literature. Of course data dispersions that looked non-Fisherian were seen early on, and were usually explained by the way the sample was obtained. Often data points diametrically opposite the bulk of the data were seen too - the most extreme outliers possible! These were also explained away in various ways before polar reversals were recognized as genuine events. I had students run simulations to check the robustness of my tests. It is clear from elementary geometry that one or two outliers can make little difference to a mean direction - it is robust - but the estimator of K is not.

Among the statisticians who got interested then in directional statistics were K.V. Mardia (see, e.g., his book, 1972) and N.I. Fisher (see, e.g., his book, 1987, which is filled with practical advances). Malcolm Clark spent time with geophysicists and wrote, e.g., about the estimation of APW paths (Thompson and Clark 1981). Thompson also collaborated with Michael Prentice on this topic and on plate rotations (e.g., Thompson and Prentice 1987). Jupp and Kent (1987) described general splining methods for points on the sphere and on the rotation group. But the major contributor to scientific questions was Ted Chang (from 1986 onwards; see, e.g., Chang 1986, 1993), who wrote a number of important theoretical and applied papers about estimating the rotational movements of plates. His 1993 paper, written for statisticians, gives authoritative detail about plate reconstructions. We can therefore leave this difficult subject for simpler ones.

WATSON Vol. 26, No. 3

Of course, the subject of statistics has now changed from the Fisherian paradigm it had when I began. This is largely due to the ready availability of cheap computing, and especially of easy graphics. The popularity of the bootstrap was due not only to this, but to a growing critical attitude toward the automatic assumption of specific distributions. The major papers on the bootstrapping of directions are by Fisher and Hall (see, e.g., 1989). There is a problem here because there are no exact pivots. Cabrera and Watson (1996) use it to reduce bias in estimates of angles, a surprisingly neglected problem - see also Debiche and Watson (1995). But so far it has not been taken up by most geophysicists.

The palaeomagnetist McFadden wrote many statistical papers dealing with some of the newer problems that had arisen. Perhaps the most significant was McFadden and Jones (1 98 I), showing that McElhinney’s “fold test” (1 964) is invalid. Imagine what will happen if a horizontal stratum that is uniformly and stably magnetized is later folded. If specimens are measured at various sites along the strata, their directions will be more scattered after folding than before. If, then, the data from the folded formation are rotated arithmetically about a horizontal axis so that the normal to the bed moves to the vertical, the scatter should decrease. If one computes k for each percentage of unfolding, its value should increase. If the magnetization took place after the folding, the reverse will be true. If the magnetization took place at some time while the fold was forming k will peak in the middle somewhere. McElhinney (1964) suggested that the concentrations of the two sets of directions as measured by k’s should be compared with my F-test [see (12) above] for two independent samples. This is of course an invalid use of the test. But the idea of unfolding through an angle until the data were as nearly parallel as they could be had become popular with geophysicists and led to many papers. Debiche drew my attention to this, and I had the idea of using simulation to justify it. This meant making it a percentage-of-unfolding estimation, not a testing, problem. Watson and Enkin (1 993) used samples from Fisher distributions to get many curves of the concentration k at successive unfolding angles. Tauxe and Watson (1994) used the bootstrap and a different measure: an eigenvalue of the dispersion matrix of the data.

Advised by D.V. Lindley, Hext (1963) wrote an important paper on the estimation of second-order tensors, the motivating example being the estimation of the magnetic susceptibility W, which is a symmetric 3 x 3 matrix. If a magnetic substance is placed in a field H, the induced magnetism is I = WH. One may choose various directions for H, each time measuring I. Thus, not only is there an estimation problem, but also a design problem. How best to choose the H directions? The statistical model is simply a linear model with additive Gaussian errors. Hext’s paper used propagation of errors to get confidence intervals. The huge literature about susceptibility was summarized in Jelenik (1978). Much more recently, the geophysicists Constable and Tauxe (1990) used simulation methods to get more reliable results for real palaeomagnetic problems. They found that actual confidence regions could be larger than Hext’s methods suggested and that the actual errors might be non-Gaussian. Hence they suggested use of the bootstrap.

The original magnetization of a rock is often overlaid by later magnetizations. Then the final magnetization is the sum of several vectors of varying lengths. Suppose that they can gradually be removed - by (say) heating to higher and higher temperatures. Then the magnetization of a specimen can be viewed as a curve in three dimensions. If the original magnetization were the hardest to remove, we would want the initial direction of the curve. Many methods have been suggested to estimate this space curve, notably by Kirschvinck (1980) and by Kent et al. (1983).

Thus palaeomagnetism has a large number of reliable statistical methods today that

1998 STATISTICS AND CONTINENTAL DRIFT 39 1

it can call upon. However, it has become very technical. I doubt that any statistician coming to it now could find the excitement that I felt 40 years ago.

REFERENCES Box, J. (Fishcr) (197X). The Lije ofu Scienri.sr. Wilcy, Ncw York, 512. Cahrcra, J., and Watson. G.C. ( IYY6). Simulation mcthods for thc rncan and mcdian hias reduction in parmetric

Chang, T. (19x6). Sphcrical rcgrcssion. Ann. Siurisr. 14, YO4-924. Chang, T. (1993). Sphcrical rcbycssion. and thc statistics of tcctonic platc rcconswctions. Inrernuf. Srurisr. Rev.

Collinson, D.W., Crccr, K.M., and Runcom. S.D.K. ( 1967). Merhiid.~ in Pulueomugnetism, Dcvclop. Solid

Constahle, C.G., and Tauxc. L. (IYYO). Thc bwLstrap l'or magnctic susfcptihility tcnsiws. 1. Geophys. Res.

Dchichc, M.G., and WaLson, G.S. (1995). Conlidcncc limits a d hias cometion for cstimating angles between

Enkin. R.. and Watson, G.S. (1%). Statistical analysis ol' palwomabmdc inclination data, Geophys. J. Int.

Fishcr. N.I., and Hall, P.G. (1989). Btxttsvap conlidcncc rcgions for dircctimal data J. Amer. Srurisr. Soc., 84.

Fishcr, R.A. (1953). Dispcrsion on a sphcrc. Proc. Roy. Soc. kind<~n Ser. A. 217, 295-305. Graham, J.W. (1949). Thc stability and signilicancc of mabmtism in scdimcnlary nxks. J. Geophys. Rex, 54,

Hall, P.. WaLson, G.S., and Cahrcm J. ( 19x7). Kcmcl dcnsity cstimation with sphcrid data. Biomerriku (4).

Hcxt. G.R. (1963). Thc cstimation of sccond ordcr tcnsom with rclakd ksb and dcsibms. Biomerriku, 50,

Hospcrs, J. (1951). Rcmancnt magnctism ol' rocks and thc history of thc gcomagnctic licld. Nurure, 168.

Hospcrs, J., and Van Andcl (IY6X). Palacomagnctic data l'mm E U K ~ and NcHth Amcrica and their bearing on

Irving, E. ( 1964). Puluerimugneti.sm. und its upplicuririn rri Geiil~igicul und Ceriphysicul Problems. Wiley, New

Irving. E. (1988). Thc palacomagnctic conlirmation ol'contincncal drift. E m , 44.04. %W7, W, IoOti-1009. Judson, S., Dcl'l'cycs. K., and Hargavcs, R. ( 1976). Phvsicul Geology. Prcnticc-Hall, Englcwixnl Clirrs, NJ,

560. Kcnt, J.A., Bridcn, J.C.. and Mardia, K.V. (IYS3). Lincar and planar swctuICs in o d e d muldvariate data as

applicd to progrcssivc dcmagnctixation ol' placomagwtic nmancncc. Geophys. J. Roy. Astron. Soc., 75. 59342 1.

Kcnt, J.. Watson. G.S.. and Onstott. T. (IYYO). Fitting straight lincs and plancs with an applicatitm to radiometric dating. Eurih Plunet Sci. ,!.err., 97, 1-17.

Kirschvinch. Mardia, K.V. ( 1972). McElhinncy, M.W. (1964). Thc statistical signilicmcc of the fold tcst in ptllxomabwtism. Ceophys., J. Roy.

Asrron. Soc., 8, 33%340. McFaddcn, P.L., and Joncs. D.L. (1981). Thc fold (cst in pdwomabmdsrn. Geoph.ys. J. Roy. Astron. Soc., 67,

53-58, Prcnticc, M.J. (19x7). Fitting smooth paths to n)tational dilk J. R q Sfurisr. Soc. Ser. C, 36, 325-331. Raylcigh, Lord (IXXO). On thc resultant ol' a largc numhcrs ol' toms ol' thc same frcqucncy of arbitrary phase.

Raylcigh, Lord (IYO5). Thc prohlcm of random walk. Nurure, 72, 318. Raylcigh. Lord (1919). On the prohlcm ol' random vibrations and ol' random flights in onc. two and three

Tauxc. L., and WaLson, G.S. (1994). Thc hlds (cst - an cigcnimalysis. Eurrh Pluner. Sci. Leu., 122. 331-

Tauxc, L.. Klyslra, N.. and Constilhlc. C.G. ( I Y Y I ) . Btw)LsWdp stahtics for pulitcomagncdc data. 1. Geophys.

Thompson, R.. and Clark. R.M. (19x1). Fitting polar wandcr paths. Phys. f i r t h Pluner. Interiors, 27. 1-7.

csti mation. To appcar in J. Srurisr. P l u m Inf

61(2), 290-316.

Earth Gcophys. 3. Elscvicr. Amstcrdilm. MW.

YS(B6). XlX34395.

unit vcctors with applications to palacomagnctism. J. Geophys. Res. 100. 24,405-24.429

126,495-504.

996-1 002.

I3 1-1 67

74, 751-762.

353-373.

1 1 1 1 .

thc origin ol' hc North Atlantic &can. Tecronriphy.sics, 6, 4754%).

York, 399.

Philos. Mug. (5). 10. 73-78.

dimcnsions. Phikos. Mug.. 6(37), 321-347.

341.

R ~ s . 96( 87). I 1.723-1 1.740.

392 WATSON Vol. 26, No. 3

Thompson, R.. and Prcnticc. M.J. (1987). An alkrnalivc mcthod Tor cstimating linitc plate rotations. Phw.

Van Dcr Vcw, R. ( I HI). Phancroxoic palacomagnctic polcs l'mm E u N ) ~ and North Amcrica and comparisons

Vinc, F.J., and Ma~hcws, D.H. (1963). Magnctic anomalics ovcr txcan ridgcs. Nature, 1Y9, W749. Watson. G.S. ( 1956a). Analysis ol' dispersion on a sphcrc. Monthly Notices Roy. Astron. Soc. Geriphvs. Suppl.

WaLson, G.S. ( 1956h). A tcst for mndomncss ol' directions. Month/v Notices R0.v. Aslron. Soc. Geriphv.~. Suppl.

Watson, G.S., (I W)). Mcm signilicancc tcsL. on thc sphcrc. Birimetrika, 47 (Part.. I and 2). 874 I. Wacson. G.S., (lY65). Equattwial disvihutions on a sphcrc. Birimetriku, 52 (Parts I and 2). IY3-201. Watson. G.S. (IY6H). Oricntaiion statistics in thc w t h scicnccs. Bull. Geol. Inst. Univ. Uppsulu N.S. (9). 2,

Watson, G.S. (lYY4). Distribution of anglcs hctwcn unit vcctcns and the multiple comparison pn&lcm l'or unit

Watson, G.S.. and R.J. &ran (lY67). Tcsting a sequcncc of unit v c c t m for serial cimlation. J. Geophy. Res.

Watson. G.S.. and Dcbichc. M. (1992). Testing whcthcr scvcral Fisher disvibutions have Ihe samc ccntcr.

Watson, G.S., and Enkin. R. (IWy3). Thc fold k s t in palacomabwtism as a paramclcr estimation pmblem.

Watson. G.S., and Irving, E. (1957). Statistical mchods in rock magnctism. Monthlv Notices Roy. Astron. SIJC.

Watson, G.S., ilnd Lutx, T. ( I 9x8). EITCCLS of long-tcrm variation on hc licquency spectrum of thc gcomagnctic

Wcgcncr, A. (1925). The Origin if Continents und Oceuns (English wansl. by J.G.A. Skcrl). Methucn. London.

Eurth Plunet. 1nierior.v. 4X. 7943.

with contincntal mconstructions. Rev. Geriphys.. 2H. 167-206,.

(4). 7, 15.3-159.

(4), 7, 160-161.

7.349.

vcctors.

(22). 72. 5655-5659.

Geophys. J. Int., I (W, 225-232.

Geophys. Rex Lett. 20. 2 135-2 137.

Geriphys. Suppl. (6). 7. 2HY-30().

rcvcrsal rccod. k i t . Nuture, 334. 240-242.

212.

Received 16 October 19% Revised 15 Ma.y I997 Accepted 4 June 1997

Depunmenr of Muthematics Princeton Universir?,

Princeton, New Jersey U. S. A. 08544- loo0