Inclination Flattening and the Geocentric Axial Dipole Hypothesis

Earth and Planetary Science Letters 233 (2005) 247–261 www.elsevier.com/locate/epsl Frontiers Inclination flattening and the geocentric axial dipole hypothesis

Lisa Tauxe

Scripps Institution of Oceanography, La Jolla, CA 92093-0220, USA Received 26 April 2004; received in revised form 25 January 2005; accepted 31 January 2005 Available online 23 March 2005 Editor: A.N. Halliday

Abstract

William Gilbert first articulated what has come to be known as the geocentric axial dipole hypothesis. The GAD hypothesis is the principle on which paleogeographic reconstructions rely to constrain paleolatitude. For decades, there have been calls for permanent non-dipole contributions to the time-averaged field. Recently, these have demanded large contributions of the axial octupole, which, if valid, would call into question the general utility of the GAD hypothesis. In the process of geological recording of the geomagnetic field, bEarth filtersQ distort the directions. Many processes, for example, sedimentary inclination flattening and random tilting, can lead to a net shallowing of the observed direction. Therefore, inclinations that are shallower than expected from GAD can be explained by recording biases, northward transport, or non- dipole geomagnetic fields. Using paleomagnetic data from the last 5 million years from well-constrained lava flow data allows the construction of a statistical geomagnetic field model. Such a model can predict not only the average expected direction for a given latitude, but also the shape of the distribution of directions produced by secular variation. The elongation of predicted directions varies as a function of latitude (from significantly elongate in the up/down direction at the equator to circularly symmetric at the poles). Sedimentary inclination flattening also works in a predictable manner producing elongations that are stretched side to side and the degree of flattening depending on the inclination of the applied field and a bflattening factorQ f. The twin tools of the predicted elongation/inclination relationship characteristic of the geomagnetic field for the past 5 million years and the distortion of the directions predicted from sedimentary inclination flattening allows us to find the flattening factor that yields corrected directions with an elongation and average inclination consistent with the statistical field model. The method can be tested using sediments deposited in a known field. Application of the elongation/inclination correction method to two magnetostratigraphic data sets from red beds in Asia and Pakistan brings the inclinations into agreement with those predicted from modern GPS measurements and from global paleomagnetic data.

E-mail address: [email protected].

There appears to be no compelling reason at this time to abandon the geocentric dipole hypothesis, which has provided such an excellent working model for so long. D 2005 Elsevier B.V. All rights reserved.

Keywords: geomagnetic field; axial geocentric dipole hypothesis; sedimentary inclination error; paleosecular variation; Asian inclination anomaly

1. Introduction model, but with paleolatitudes predicted from geo- detic and global paleomagnetic data sets. The idea that the Earth’s magnetic field is well approximated by a geocentric axial dipole (GAD) is a very old one. It is central to much of modern 2. The birth of the geocentric axial dipole paleomagnetism which relies on records of the hypothesis geomagnetic field imprinted in rocks. The GAD hypothesis applied to paleomagnetic data provided Prior to his appointment as physician to Queen the first geophysical proof of continental drift and is Elisabeth I, William Gilbert (1544?–1603) inves- still the best way to reconstruct continents with tigated the magnetic properties of spherical speci- respect to paleolatitude and orientation relative to the mens of lodestone he called bterrellae,Q or blittle north pole. Nonetheless, despite centuries of study, Earths.Q He found that iron spikes aligned the limits of the GAD hypothesis are not precisely themselves on the terrellae in unexpected ways known. How much time is required to average the depending on the positions relative to the mag- field to that of a centered dipole? How much netic poles of the terrellae (see Fig. 1a). At the deviation from GAD can be expected from the equator, the spikes were aligned tangent to the time-averaged geomagnetic field? Has the field sphere. As they approached the poles, bthe more always been essentially dipolar, or was it more they are raised up by their versatory nature,Q and complex earlier in Earth’s history? Are discrepancies at the poles, the spikes pointed directly to the between geological and paleomagnetic predictions center of the sphere. the result of bbadQ recording of the magnetic field, Gilbert was not the first to consider the unrecognized crustal deformation, or strongly non- magnetic properties of the Earth and of rocks dipolar ancient magnetic fields? These are old (see [1]), but he seems to have been the first to questions, but have been the subject of much recent make a systematic study. In a great leap of insight effort. (perhaps aided by prior work of Petrus Peregrinus This paper is not a comprehensive review of the in the 13th century), he realized that the behavior history of geomagnetism. For that the reader is of his terrellae was similar to that of the Earth directed to a marvelous paper by Stern [1]. Nor is itself. He used a simple instrument for measuring this a thorough treatment of secular variation or the dip of the Earth’s magnetic field and showed time-averaged field models. We will begin with just how the dip could be transformed into latitude a brief tour of these subjects. We will consider how using a complicated graphical approach. He sedimentary inclination flattening affects the record- exuberantly proclaimed, bWe may see how far ing of the geomagnetic field, then discuss recent from unproductive magnetic philosophy is, how efforts at finding simple detection and correction agreeable, how helpful, how divine! Sailors when methods, in particular the elongation/inclination tossed about on the waves with continuous cloudy method of Tauxe and Kent [2]. Finally, we will weather, and unable by means of the coelestial consider a few case studies where the elongation/ luminaries to learn anything about the place of inclination method brings paleomagnetic data into the region in which they are, with a very slight agreement not only with a simple statistical field effort and with a small instrument are comforted, L. Tauxe / Earth and Planetary Science Letters 233 (2005) 247–261 249

Fig. 1. (a) Reproduction Gem Gilbert’s de Magnete [3] description of experiments with iron spikes near a globe made of lodestone. The Orbis Virtutis is the region within which the spikes responded to the globe. (b) Lines of flux for a dipole with moment m as a function of radial distance r and angle away from the pole h or equator k. At any point the angle that the field lines make with the local horizontal (tangent to the heavy circle) is the inclination I. (c) Definition of a virtual geomagnetic pole (VGP) as the piercing point of the centered dipole that would give rise to the direction (dashed line) at a given observation site P. and learn the latitude of the place.Q [3]. This was had a rough idea from sailors’ measurements of what the first statement of what has come to be known the deviation from true north was over much of the as the geocentric axial dipole (GAD) hypothesis. Earth. However, he mistakenly assumed that the Gilbert knew little of the physics of magnetism degree of variation was constant in time at a given (let alone trigonometry). Physics tells us that in the special case away from currents and changing electric fields, a magnetic moment m creates a Table 1 magnetic field H which is the gradient of a scalar Table of acronyms and terms potential field Vm (i.e., H=jVm). Vm is a function Symbol Term and definitions of radial distance r and the angle away from the pole GAD Geocentric axial dipole h by (Table 1): m Magnetic moment H Magnetic field md r mcosh Vm Magnetic potential h, k, /, r Co-latitude, latitude, longitude, radius from V m ¼ 3 ¼ 2 : ð1Þ 4pr 4pr center of the Earth D, I Declination, inclination From the potential equation, it is possible to VGP Virtual geomagnetic pole m b Q calculate the field lines produced by the magnetic Pl Harmonic functions of cosh dipole m as shown in Fig. 1b. If we imagine the outer CP88/TK03 Statistical paleosecular variation model of [24] and [2] circle to be the surface of the Earth, the dip of the field b Q V Principal components of a set of directions. lines relative to horizontal ( inclination, I) varies V1 bAverage directionQ, direction along which progressively with latitude k from horizontal at the the data are concentrated. equator to vertical at the pole. We can replace the V3 Direction orthogonal to the average in which intricate graphical approach of Gilbert with the simple the data are least concentrated. V Direction orthogonal to V , V . trigonometric function known as the bDipole For- 2 1 3 E Ratio of variance along V2, V3. mulaQ: tanI=2 tank. f Flattening factor in the inclination error Gilbert knew that the Earth’s magnetic field was formula; ratio of tangents of not that of a perfect bar magnet. In such a field, inclinations of the ambient field versus that compasses would always point to the same pole. He recorded by the sediments. 250 L. Tauxe / Earth and Planetary Science Letters 233 (2005) 247–261 place and even proposed that variation could be used 3. The non-dipole and time-averaged geomagnetic to help constrain longitude. field Measurements of the geomagnetic field made since the time of Gilbert (many by his bsailors tossed about As more paleomagnetic data became available, on the wavesQ [4]) show that the field changes the GAD hypothesis could be examined with more constantly. Yet to first order, Gilbert’s hypothesis rigor. Wilson [12] qualified the appropriateness of has stood the test of time. Indeed, the assumption that the GAD hypothesis suggesting that the best-fitting the geomagnetic field is on average that of a centered model for the time-averaged geomagnetic field was axial dipole is the foundation of the field of that of an boffsetQ dipole, displaced some 191F38 paleomagnetism whereby the magnetic vectors pre- km northward along the rotation axis (dashed line in served in rocks are interpreted as records of ancient Fig. 2). This offset dipole field has the effect of geomagnetic fields (e.g., [5–8]). When averaged over bflatteningQ inclinations at observations sites relative the last few million years, the inclinations of these to what would be produced by a centered dipole directions agreed quite well with those expected from (predicted from the dipole formula). The flattened a dipole field (see solid lines in Fig. 2a,b). inclinations lead to VGPs that are bfar-sided;Q they The simplicity of the notion of a centered dipole plot on the opposite side of the globe from the site of giving rise to an observed direction at a given location observation. P (see Fig. 1c) led Jan Hospers (e.g., [5]) to convert an While an offset dipole is a conceptually simple observed direction (dotted line) to an equivalent pole way of representing the geomagnetic field, it is position. The pole here is the piercing point of the often more advantageous to use an expansion of Eq. centered dipole that would give rise to the observed (1) known as the spherical harmonic expansion. The direction. Equivalent poles of geomagnetic directions full potential equation is beyond the scope of this became known as bvirtual geomagnetic poleQ or VGP paper, which is intended for a general audience, but [9]. Averages of a number of VGPs sufficient to it is important to understand that it is summation of m baverage outQ secular variation are known as paleo- harmonic functions Pl of position (e.g., co-latitude magnetic poles. These appear to bwanderQ away from h, longitude /, and radius r). The shape and the spin axis with increasing age of the rock unit orientation of each term is determined by the degree 0 sampled (e.g., [10]). Wandering paleomagnetic poles l and the order m. The first harmonic P1 is simply 0 were used as early evidence for continental drift and cosh. The second ( P2) is 1/4 (3 cosh+1). Higher are still the best available constraints on paleolatitude order harmonics are increasingly wiggly functions in paleogeographic reconstructions (e.g., [11]). of cosh.

a) b) -90

-90 90

Fig. 2. Inclinations versus latitude from deep sea sediment cores redrawn from Opdyke and Henry [8]. Solid lines are the trends expected from a geocentric axial dipole, dashed lines are from an boffset dipoleQ of 191 km (see text). (a) Data from normal polarity intervals, (b) data from reverse polarity intervals. L. Tauxe / Earth and Planetary Science Letters 233 (2005) 247–261 251

In a spherical harmonic expansion, the harmonic north pole part pointing to the Greenwich meridian, m m 0 functions are weighted by coefficients gl and hl the contribution would be determined by h1 coef- 1 known as bGauss coefficients.Q These are determined ficient, and if it were at 90 8E, it would be the h1 by fitting the observed geomagnetic field vector data coefficient. Therefore, the total dipole contribution for a particular observation interval. We show three would be the vector sumq offfiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the axial and two 0 2 0 2 1 2 examples in Fig. 3 of the inclinations of the vector equatorial dipole terms or ðg1Þ þðh1Þ þðh1Þ . The fields with their surface harmonics as insets. These are total quadrupole contribution (l=2) combines five the axial (m=0) dipole (l=1), quadrupole (l=2) and coefficients and the total octupole (l=3) contribution octupole (l=3) terms whose contributions are deter- combines seven coefficients. 0 0 0 mined by g1, g2 and g3 respectively. In general, terms for which the difference between If the axial dipole field produced by the harmonic the subscript (l) and the superscript (m) is odd (e.g., 0 0 function in Fig. 3a were turned on its side with the the axial dipole g1 and octupole g3) produce magnetic

a) b)

-40

c) -80

Fig. 3. Examples of surface harmonics (insets) and maps of the associated patterns for global inclinations. (a) Dipole, (b) quadrupole, (c) octupole. 252 L. Tauxe / Earth and Planetary Science Letters 233 (2005) 247–261 fields that are asymmetric about the equator, while keyama and Kono [23]. We show the inclination those for which the difference is even (e.g., the axial variations predicted from their normal polarity model 0 quadrupole g2) have symmetric fields. In Fig. 3a, we in Fig. 4. The largest departure from a GAD field is show the inclinations produced by a purely dipole the quadrupole term which is approximately 4% of field of the same sign as the present-day field. As we the dipole. have seen before (see Fig. 2a), the inclinations are all positive (down) in the northern hemisphere and negative (up) in the southern hemisphere. In contrast, 4. Statistical paleosecular variation models inclinations produced by a purely quadrupolar field (Fig. 3b) are down at the poles and up at the equator. From studies of the time-averaged field over the The map of inclinations produced by a purely axial last 5 million years, it seems that, at least for that octupolar field (Fig. 3c) are again asymmetric about period of time, the field has been dominantly that of the equator with vertical directions of opposite signs a geocentric axial dipole. At any particular instant in at the poles separated by bands with the opposite sign time, however, there will be significant deviations at midlatitudes. owing to the non-axial dipole contributions. This, The geomagnetic field produced by an offset combined with distortions in the recording process dipole is identical to that produced by a combination (some of which are discussed in the following of an axial dipole and an axial quadrupole. The offset section) and decreasing preservation of rocks with of 191 km favored by Wilson [12] is the same as increasing age, makes evaluating the GAD hypoth- having a ratio of axial quadrupole to dipole terms esis increasingly difficult as we go back in time. 0 0 G2=g2/g1 of about 6%. Therefore, it would be extremely useful if we had a The two decades from 1970 to 1990 saw many way of predicting for a given latitude the distribu- attempts to quantify the contributions of various terms tions of vectors from the geomagnetic field like the in the spherical harmonic expansion to the time- one that operated over the last 5 million years. Such averaged geomagnetic field using a variety of statistical predictions could then be compared with paleomagnetic data (see, e.g., [13–18]). Most of these paleomagnetic observations. To find an appropriate concluded that variations in longitude were indistin- statistical paleosecular variation model, we begin guishable from noise (the time-averaged field was with the work of Constable and Parker [24] (here- controlled primarily by the axial terms (m=0) and after CP88). departures from GAD were small with the largest term The CP88 statistical paleosecular variation model being the axial quadrupole with a contribution of at assumes that the time-varying geomagnetic field acts most 6% (see [19] for an excellent review). as bGiant Gaussian ProcessQ (GGP) whereby the m m In the last decade, an entirely new class of models Gauss coefficients gl , hl (except for the axial dipolar 0 has become available. These are based on construct- term, g1 and in some models also the axial quadrupole 0 ing spherical harmonic models for the time-averaged term g2) have zero mean and standard deviations that field from large, globally distributed data sets (e.g., are a function of degree l and a fitted parameter a. 0 [20–23]). The most recent of these was by Hata- Once the average dipole moment g¯ 1, its standard 0 deviation r1: and a are fixed, realizations of the field 80 model can be created by drawing the Gauss coef- 60 ficients from their respective Gaussian distributions. 40 Geomagnetic vectors can then be calculated for any 20 0 given location using the usual transformation from the -20 geomagnetic potential equation to geomagnetic ele- -40 ments (see [24] for details). -60 -80 The principal drawback of the CP88 model is that it fails to fit the observed scatter in the paleomagnetic Fig. 4. Map of predicted average inclination from the time-averaged data with latitude. Most of the subsequent variations field models of Hatakeyama and Kono [23]. (e.g., [23,25–27]) attempted to address this problem L. Tauxe / Earth and Planetary Science Letters 233 (2005) 247–261 253 by introducing more fitted parameters, losing the vector endpoints, but because the strength of the elegant simplicity of the CP88 model. ancient geomagnetic field is measured much less The most recent model of the statistical paleosec- frequently than the direction, paleomagnetic data are ular variation genre is the TK03.GAD model of Tauxe often assumed to be unit length [30]. Unit vectors and Kent [2]. Like CP88, TK03.GAD has only three can be plotted on equal area projections which 0 parameters: g¯ 1 (set to fit a recent estimate for the long- usually are centered on the vertical direction. In term average intensity of the axial dipole [28]), a as Fig. 6c, we plot the directions from Fig. 6a but have defined in CP88, but fit to the more recent compila- centered the diagram along the direction expected tion of directional data of [29] and a new parameter b from a GAD field (star) at the equator (or which is the ratio of the asymmetric (l+m odd) to the horizontal). What is strikingly apparent in this symmetric (l+m even) Gauss coefficients for a given diagram is the marked elongation in the distribution l. The term b allows a much improved fit to the of directions at the equator. This elongation dis- paleomagnetic observations while the model retains appears as the observation site approaches the pole. the simplicity of the CP88 model. We show the Although this tendency was predicted for circularly variation in r with degree for the two families symmetric VGPs [31] and has been noted in the (asymmetric and symmetric) in Fig. 5a. paleomagnetic data for decades (e.g., [32–34]), it is In Fig. 5b, we show the vector end points generally ignored in the paleomagnetic literature. calculated from 1000 realizations of the model at 30 The average direction of the data shown in Fig. 8N. The distribution of these vectors predicts what 6cisheretermedV1. Two other orthogonal would be observed at that latitude if we had a large directions are V3, the direction of least scatter and number of observations of the geomagnetic field or its V2, the direction of elongation as shown in Fig. 6c. paleomagnetic proxies. The elongation parameter E is defined [35] as the In Fig. 6a, we show vector endpoints from 1000 ratio of the variance (calculated as the eigenvalues of realizations of model TKO3.GAD evaluated at the the covariance matrix, see [35]) in the V2 direction equator and seen from the side. The average over that in the V3 direction. We plot elongation and direction is north and horizontal, so when we turn inclination versus latitude predicted by the TKO3.- the projection about a vertical axis and look at the GAD model in Fig. 6d and elongation versus data along the expected direction, we have the inclination in Fig. 6e. This treatment allows us to average direction at the center of the diagram as in have a unique pair of values for elongation E and Fig. 6b. The projections so far have been of the inclination I that is compatible with the statistical

a) 0 g b) 1 Asymmetric terms Symmetric terms

0 g N 3 0 g E 2

Fig. 5. (a) Variation of the standard deviation rl as a function of harmonic degree l for asymmetric and symmetric terms for the statistical field model TK03.GAD. All terms have zero mean except the axial dipole term. (b) 1000 vector endpoints from realizations of model TK03.GAD at 308N. 254 L. Tauxe / Earth and Planetary Science Letters 233 (2005) 247–261

c) a) b) N V E 3 E

V 2 D D

d) e)

Fig. 6. (a) Vector end points for 1000 realizations of model TKO3.GAD evaluated at the equator. The color of each point (r, g, b) relates to the component length along N, E ,D. (b) Same data as in panel (a), but projected along the principle direction V1, in this case, parallel to N. (c) 100 data point from (b) projected as unit vectors in an equal area projection. Dashed lines are the eigenvectors V2 and V3, the lengths of which are proportional to the eigenvectors s2, s3 respectively. (d) Variation of elongation (E=s2/s3) (solid line) and inclination (dashed line) as a function of latitude predicted from the TK03.GAD model. (e) Variation of elongation versus inclination from panel (d). field model; applications of this prediction will be in the magnetic recording process and remagnetiza- discussed later in the paper. tion. These processes collectively can be thought of as an bEarth filterQ which takes a given geomagnetic field direction and modifies it in some way. Here we 5. Earth filters consider one example of an Earth filter (see Fig. 7): sedimentary inclination flattening. There are many distortions of the magnetic record In Fig. 7a, we show a set of directions drawn from in Earth materials owing to the accumulating effects the TK03.GAD statistical field model with an average of plate movements, rock deformation, biases inherent inclination of ~458.InFig. 7b, we show that elongate

a) b) c) B

Fig. 7. An example of what geologic processes do to directional distributions. (a) Equal area projection of a set of directions drawn from cloud shown in Fig. 5b. (b) Sedimentation process that produces flattened directions according to sedimentary inclination error formula. (c) Directions produced from flattening filter with f=0.4. L. Tauxe / Earth and Planetary Science Letters 233 (2005) 247–261 255

a) b)

Fig. 8. (a) Directions of remanent magnetizations of specimens from the Soan River in the Potwar Plateau (dots; unpublished data). Triangle is the magnetic field at the time of deposition. (b) Observed inclinations Io from laboratory redeposition of samples under various inclinations of the applied field (If). Data from Tauxe and Kent [38].

particles (such as detrital hematite) can produce a) directions that are biased too shallow. Laboratory redeposition experiments led King [36] to the bInclination Error FormulaQ: tanIo=f tanIf where Io, If are the observed and applied field inclinations respectively and f is the so-called bflatteningQ factor. Such a process would result in a distorted directional distribution shown in Fig. 7c. We note that post- depositional compaction can also produce an inclination flattening having the same form (e.g., [37]). That sedimentary inclination flattening occurs in b) c) nature was demonstrated by data from storm deposits retrieved from the Soan River in northern Pakistan 60 f = 4942 [38] which have detrital hematite as the primary 1.0 .3 carrier of magnetic remanence (see Fig. 8a). The .9 geomagnetic reference field when the sediments were deposited had a declination of D=1.58 and an .8 .4 inclination I=508 (triangle in Fig. 8a) while the 59 .7 .5 specimens obtained from the river have a mean .6 direction D¯ =2.0, I¯=26.6, with a cone of confidence a95=1.88 (circles in Fig. 9a). Laboratory redeposition experiments (see Fig. 8b) using these sediments Fig. 9. (a) Average inclination (short dashed) and elongation (long showed that inclination flattening has the form dashed and solid line) of data from Fig. 7c, after transformation to IV (see text). Dashed (solid) portion of the elongation curve has east– predicted by King [36] of tanIo=f tanIf with a value west (north–south) elongation. (b) Heavy hatched line is elongation of f of 0.55 (dashed line). versus inclination from panel (a). Elongation direction (V2) is shown The possibility of inclination flattening biasing by hatch marks. Dashed line is from TK03.GAD [2]. Crossing point results from sedimentary rocks led Jackson et al. [39] (circled) is the elongation/inclination pair consistent with to seek a method for detecting and correcting TK03.GAD model. Light curves are results from bootstrapped samples of the data in panel (a). (c) Histogram of inclinations from inclination flattening. Their method has been success- 1000 bootstrapped crossing points. Bar indicates bounds including fully applied in several studies (e.g., [40–42]). The 95% of the compatible inclinations. Vertical line is the inclination principle reason it has not been universally applied in expected from the ambient magnetic field at the time of deposition. 256 L. Tauxe / Earth and Planetary Science Letters 233 (2005) 247–261 studies where inclination flattening is suspected is that sedimentary inclination flattening and has the func- the method is extremely labor intensive requiring a tional form of tanIo=f tanIf. (3) There are sufficient battery of meticulous laboratory experiments. measurements of the ancient geomagnetic field that, An alternative approach to the inclination flat- once corrected, they will conform to the statistical tening problem was suggested by Tauxe and Kent [2]. model (Monte Carlo simulations suggest that N100 They proposed that one could bcorrectQ sedimentary data points are necessary). If these assumptions are inclination flattening by using the twin tools of the satisfied, the actual elongation–inclination method is inclination error formula and a statistical field model. very fast (a matter of minutes) and can be done with Assuming that inclination flattening from whatever the program find_EI in the pmagl.8 software source (original depositional or compaction related) distribution at: magician.ucsd.edu/software. follows the bunflattening formulaQ tanIV=(1/f) tanIo, one can choose progressively smaller values for f,in order to invert the directions by the unflattening 6. Application of the elongation/inclination method formula and calculate bnewQ mean inclination/elongation pairs as a function of f. We illustrate how such a Tests of the GAD hypothesis for ancient times are method works in practice by inverting the synthetic of two types. The bpredicted direction methodQ (e.g., flattened data from Fig. 7cinFig. 9a. As f decreases, [43,44]) compares paleomagnetic observations to the inclination increases (short dashed line in Fig. 9a) and directions from a given place and time assuming a elongation decreases along the east–west axis (solid reference pole. The bstatistical distribution methodQ line in Fig. 9a). Elongation comes to a minimum value (e.g., [45,46]) assumes that paleomagnetic directions (at f=0.6 in this case) at which the distribution is more obtained from many locations and ages can provide a or less circularly symmetric. Then as f decreases statistical sampling of the ancient geomagnetic field further, the distribution becomes more elongate north– that can then be compared with one generated using a south (long dashed line in Fig. 9a). random sampling of a GAD field. In Fig. 9b, we plot elongation versus inclination An updated compilation of paleopoles for the from Fig. 9a as the heavy hatched line. The direction Atlantic-bordering continents for 0–200 Ma shows of the hatches indicates the direction of V2 or the very good internal agreement with the GAD model, direction of elongation. We have replotted the with only a small 3% quadrupolar contribution [11] elongation–inclination curve from Fig. 6 as a dashed required. Nevertheless, comparison of predicted line, so the inclination at which the heavy line crosses directions with those observed in certain places in the dashed line gives the elongation–inclination pair certain rock types reveals persistent biases in inclina- consistent with model TKO3.GAD, or 538 [we note tion whereby the directions are shallower (more the much improved concurrence of this bcorrectedQ horizontal) than expected. value and that of the hypothetical applied field of One of the puzzling examples of persistent shallow 498]. The light curves are similar results from bias has been in the Cenozoic and Mesozoic bootstrapped samples of the data shown in Fig. 7a. paleomagnetic directions from Central Asia (see, A histogram of inclinations from 1000 such boot- e.g., [43,44,47–49]). Explanations for the discrepan- strapped crossing points are plotted in Fig. 9c. Ninety- cies include sedimentary inclination flattening (e.g., five percent of these lie between 42 and 608 and the [50,49]) and extreme internal deformation of the mode is 498. So, the E/I method here yields an Eurasian plate (e.g., [47]), but most of these studies 60 estimate for inclination of the synthetic data of 4942. attributed the observed inclination shallowing to This estimate is in complete agreement with the significant non-GAD geomagnetic fields (e.g., original inclination of 498 shown as the dashed [43,44,48,51]). vertical line in Fig. 9c, as expected. In order to use non-dipole fields as an explanation Assumptions of the elongation–inclination method for the large discrepancy between expected and are (1) the geomagnetic field in the past produced observed inclinations in the Asian data sets, Si and directions compatible with the TKO3.GAD model. (2) Van der Voo [48] noted that the reference poles are The sole source of flattening in the data is from largely based on results from the UK and North L. Tauxe / Earth and Planetary Science Letters 233 (2005) 247–261 257

America. A non-zero axial octupolar contribution with 2000 of kilometers of displacement over the last 20 the same sign as the dipole makes directions in mid- million years to explain the Subei data or an average northern latitudes shallower than expected from a rate of over 100 mm/yr. Such a rate is larger than GAD field (see Fig. 3c). These directions, when any present-day rate within China. It would be converted to paleomagnetic poles will be bfar-sided.Q particularly surprising for the location of Subei If this reference pole is then used to predict directions which currently has a very small rate of deformation in Asia, the predicted directions will be too steep. The in European coordinates (see GPS data of [54] in effect would be amplified by the fact that the actual Fig. 10). directions observed in Asia in the same octupolar field To bring the Potwar Plateau data into accord with will also be shallower than expected. Typical con- Indian plate motion as inferred from the paleomag- 0 tributions of g¯ 3 called for are between 10 and 20% of netic poles compiled by Besse and Courtillot [11] the average axial dipole. would require some 1600 km of undetected northward In order to explore the problem of the Asian transport over the last 10 million years. In European inclination anomaly, we will use two data sets coordinates, there must be over 2000 km of crustal originally obtained for magnetostratigraphic purposes shortening in 10 million years for an average rate of as these are much larger than those used for over 200 mm/yr, again much larger than the present- determining pole positions. The insets to Fig. 10 day rates of northward motion with respect to stable show the Oligo–Miocene data from Subei of Gilder et Europe determined by GPS measurements (see Fig. al. [52] and the middle Miocene data from the Potwar 10). These are approximately 40 mm/yr. Plateau of Tauxe and Opdyke [53] [N.B.: for the latter, Extreme crustal shortening is therefore difficult to we have only used the 105 sites with a95V5 and Nz3, reconcile with the present-day observations or with in order to reduce the effect of sources of scatter other palinspastic reconstructions. Large octupolar contri- than sedimentary inclination flattening]. The inset butions would have the uncomfortable consequence equal area projections are superposed on the map of of rendering useless the paleomagnetic poles which present-day crustal deformation of China based on currently are the primary means of placing latitude global positioning system (GPS) data (in the Eurasian bands on plate reconstructions. The alternative reference frame) of Wang et al. [54]. The GPS vectors explanation for shallow bias in the Asian sedimentary from around the Subei area show virtually no north- data would be some form of inclination flattening ward motion of that block with respect to Europe either from original depositional processes (e.g., while those from the Indian subcontinent show some [2,49,52]) or subsequent compaction (e.g., [50]). 40 mm/yr northward motion. From the foregoing, we see that there are at least The average of the 222 Subei paleomagnetic three completely different explanations for the shal- directions has a declination of D¯ =356.18 and an low bias observed in the Asian red sediments. Some inclination of I¯=43.78. Assuming that the location of way of discriminating among the competing hypoth- the study (presently located at 39.58N, 94.78E) has eses would be useful. Fortunately, as we have seen been fixed to the European coordinate system and from the discussion of statistical field models and the taking the 20 Myr pole for Europe from [11] (81.48N, section on the elongation/inclination (E/I) method of 149.78E), the expected inclination is 638. Similarly, Tauxe and Kent [2], there is more to paleomagnetic the Potwar data set has an average of D¯ =344, J=33.7. data than just the average direction. The shape of the The European pole position for 10 Ma from [11] is distribution changes with latitude as well as the 85.08N, 155.78E giving a predicted inclination at the inclination, varying from fairly elongate at the equator location of the Potwar data (33.38N, 72.38E) of 538. to circularly symmetric at the poles. Following Tauxe The pole position for India at 10 Ma (85.88N, and Kent [2], we show the histogram of inclinations 231.18E) predicts an inclination of 48.48. from the E/I method applied to the Subei data in Fig. The Subei and Potwar sediments are typical of 11a which yields an estimate for inclination of the 69 Asian sedimentary units in having inclination values Subei data of 6356. This estimate is in complete that are substantially shallower than expected. A agreement with the expected inclination if the Subei northward transport mechanism would require over region had remained fixed in European coordinates 258 L. Tauxe / Earth and Planetary Science Letters 233 (2005) 247–261

Fig. 10. Base map is from Wang et al. [54] of present-day crustal deformation of China. Inserts are directional data of Gilder et al. [52] from Subei and from Tauxeand Opdyke 1531 from the Potwar Plateau.

for the last 20 million years as it is today from the a) b) GPS measurements. 69 49 63 41 We repeat the E/I method on the data from the 56 34 Potwar Plateau in Fig. 10b and get an estimate for 49 inclination of 4131.Thisisconsistentwiththat predicted from the paleomagnetic pole from India of Fraction Besse and Courtillot [11]. Furthermore, the corrected data are also compatible with the present-day northward motion of India with respect to Europe of 40 mm/yr observed from the GPS data of Wang et al. Inclination (I') Inclination (I') [54]. Fig. 11. Elongation–inclination results from data shown in Fig. 10. By using the distribution of paleomagnetic direc- Expected inclination from European reference pole is shown as a tions instead of inclinations alone, it is possible to test vertical line. (a) Data from Subei. (b) Data from Potwar. which of the various competing explanations for L. Tauxe / Earth and Planetary Science Letters 233 (2005) 247–261 259 shallow directions is the most consistent. In the case ter of the time-averaged field is poorly constrained in of the data from Subei and the Potwar Plateau, we find the past. More data are necessary to assess the general that inverting the directions assuming a sedimentary validity of the model derived from the last 5 million inclination flattening model yields data distributions years. that are both compatible with current statistical field models as well as with present-day crustal deformation measurements. Acknowledgements The E/I method has been applied successfully in several other recent studies. Krijgsman and Tauxe [55] I gratefully acknowledge stimulating conversations used it on Neogene sediments from Spain and Crete. with Cathy Constable, Catherine Johnson, Dennis The correction brought the average inclination into Kent, Ken Kodama and Wout Krijgsman. Careful excellent agreement with those predicted from their reviews by Ted Irving, Subir Banerjee and Martin tectonic contexts. Also, noting that the reconstruction Frank improved the manuscript as did suggestions by of Atlantic bordering continents in the Triassic the editor, Alex Halliday. I also appreciate the advice predicts paleolatitudes in conflict with those observed of Julie Bowles and Kristin Lawrence. This work was in the Triassic sedimentary sequences preserved in the partially funded by NSF Grant EAR-0003395. Triassic rifts extending from southern US to Green- land, Kent and Tauxe [56] used the E/I method to bring all the Triassic inclinations into a consistent References temporal/spatial context. [1] D.P. Stern, A millennium of geomagnetism, Rev. Geophys. 40 (3) (2002) 1007. 7. Conclusions [2] L. Tauxe, D.V. 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