Inclination flattening and the geocentric axial dipole

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Earth and Planetary Science Le

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

Editor: A.N. Halliday

Available online 23 March 2005

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 bflatteningfactorQ 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.

0012-821X/$ - s

doi:10.1016/j.ep

E-mail addr

tters 233 (2005) 247–261

ee front matter D 2005 Elsevier B.V. All rights reserved.

sl.2005.01.027

ess: [email protected].

L. Tauxe / Earth and Planetary Science Letters 233 (2005) 247–261248

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

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

paleomagnetism which relies on records of the

geomagnetic field imprinted in rocks. The GAD

hypothesis applied to paleomagnetic data provided

the first geophysical proof of continental drift and is

still the best way to reconstruct continents with

respect to paleolatitude and orientation relative to the

north pole. Nonetheless, despite centuries of study,

the limits of the GAD hypothesis are not precisely

known. How much time is required to average the

field to that of a centered dipole? How much

deviation from GAD can be expected from the

time-averaged geomagnetic field? Has the field

always been essentially dipolar, or was it more

complex earlier in Earth’s history? Are discrepancies

between geological and paleomagnetic predictions

the result of bbadQ recording of the magnetic field,

unrecognized crustal deformation, or strongly non-

dipolar ancient magnetic fields? These are old

questions, but have been the subject of much recent

effort.

This paper is not a comprehensive review of the

history of geomagnetism. For that the reader is

directed to a marvelous paper by Stern [1]. Nor is

this a thorough treatment of secular variation or

time-averaged field models. We will begin with just

a brief tour of these subjects. We will consider how

sedimentary inclination flattening affects the record-

ing of the geomagnetic field, then discuss recent

efforts at finding simple detection and correction

methods, in particular the elongation/inclination

method of Tauxe and Kent [2]. Finally, we will

consider a few case studies where the elongation/

inclination method brings paleomagnetic data into

agreement not only with a simple statistical field

model, but with paleolatitudes predicted from geo-

detic and global paleomagnetic data sets.

2. The birth of the geocentric axial dipole

hypothesis

Prior to his appointment as physician to Queen

Elisabeth I, William Gilbert (1544?–1603) inves-

tigated the magnetic properties of spherical speci-

mens of lodestone he called bterrellae,Q or blittleEarths.Q He found that iron spikes aligned

themselves on the terrellae in unexpected ways

depending on the positions relative to the mag-

netic poles of the terrellae (see Fig. 1a). At the

equator, the spikes were aligned tangent to the

sphere. As they approached the poles, bthe more

they are raised up by their versatory nature,Q and

at the poles, the spikes pointed directly to the

center of the sphere.

Gilbert was not the first to consider the

magnetic properties of the Earth and of rocks

(see [1]), but he seems to have been the first to

make a systematic study. In a great leap of insight

(perhaps aided by prior work of Petrus Peregrinus

in the 13th century), he realized that the behavior

of his terrellae was similar to that of the Earth

itself. He used a simple instrument for measuring

the dip of the Earth’s magnetic field and showed

how the dip could be transformed into latitude

using a complicated graphical approach. He

exuberantly proclaimed, bWe may see how far

from unproductive magnetic philosophy is, how

agreeable, how helpful, how divine! Sailors when

tossed about on the waves with continuous cloudy

weather, and unable by means of the coelestial

luminaries to learn anything about the place of

the region in which they are, with a very slight

effort and with a small instrument are comforted,

Table 1

Table of acronyms and terms

Symbol Term and definitions

GAD Geocentric axial dipole

m Magnetic moment

H Magnetic field

Vm Magnetic potential

h, k, /, r Co-latitude, latitude, longitude, radius from

center of the Earth

D, I Declination, inclination

VGP Virtual geomagnetic pole

Plm bHarmonicQ functions of cosh

CP88/TK03 Statistical paleosecular variation model of

[24] and [2]

V Principal components of a set of directions.

V1 bAverage directionQ, direction along which

the data are concentrated.

V3 Direction orthogonal to the average in which

the data are least concentrated.

V2 Direction orthogonal to V1, V3.

E Ratio of variance along V2, V3.

f Flattening factor in the inclination error

formula; ratio of tangents of

inclinations of the ambient field versus that

recorded by the sediments.

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.

L. Tauxe / Earth and Planetary Science Letters 233 (2005) 247–261 249

and learn the latitude of the place.Q [3]. This was

the first statement of what has come to be known

as the geocentric axial dipole (GAD) hypothesis.

Gilbert knew little of the physics of magnetism

(let alone trigonometry). Physics tells us that in the

special case away from currents and changing

electric fields, a magnetic moment m creates a

magnetic field H which is the gradient of a scalar

potential field Vm (i.e., H=�jVm). Vm is a function

of radial distance r and the angle away from the pole

h by (Table 1):

Vm ¼ md r

4pr3¼ mcosh

4pr2: ð1Þ

From the potential equation, it is possible to

calculate the field lines produced by the magnetic

dipole m as shown in Fig. 1b. If we imagine the outer

circle to be the surface of the Earth, the dip of the field

lines relative to horizontal (binclination,Q I) varies

progressively with latitude k from horizontal at the

equator to vertical at the pole. We can replace the

intricate graphical approach of Gilbert with the simple

trigonometric function known as the bDipole For-

mulaQ: tanI=2 tank.Gilbert knew that the Earth’s magnetic field was

not that of a perfect bar magnet. In such a field,

compasses would always point to the same pole. He

had a rough idea from sailors’ measurements of what

the deviation from true north was over much of the

Earth. However, he mistakenly assumed that the

degree of variation was constant in time at a given


place and even proposed that variation could be used

to help constrain longitude.

Measurements of the geomagnetic field made since

the time of Gilbert (many by his bsailors tossed about

on the wavesQ [4]) show that the field changes

constantly. Yet to first order, Gilbert’s hypothesis

has stood the test of time. Indeed, the assumption that

the geomagnetic field is on average that of a centered

axial dipole is the foundation of the field of

paleomagnetism whereby the magnetic vectors pre-

served in rocks are interpreted as records of ancient

geomagnetic fields (e.g., [5–8]). When averaged over

the last few million years, the inclinations of these

directions agreed quite well with those expected from

a dipole field (see solid lines in Fig. 2a,b).

The simplicity of the notion of a centered dipole

giving rise to an observed direction at a given location

P (see Fig. 1c) led Jan Hospers (e.g., [5]) to convert an

observed direction (dotted line) to an equivalent pole

position. The pole here is the piercing point of the

centered dipole that would give rise to the observed

direction. Equivalent poles of geomagnetic directions

became known as bvirtual geomagnetic poleQ or VGP[9]. Averages of a number of VGPs sufficient to

baverage outQ secular variation are known as paleo-

magnetic poles. These appear to bwanderQ away from

the spin axis with increasing age of the rock unit

sampled (e.g., [10]). Wandering paleomagnetic poles

were used as early evidence for continental drift and

are still the best available constraints on paleolatitude

in paleogeographic reconstructions (e.g., [11]).

90

90

-90

-90

a)

Fig. 2. Inclinations versus latitude from deep sea sediment cores redrawn fr

geocentric axial dipole, dashed lines are from an boffset dipoleQ of 191 k

reverse polarity intervals.

3. The non-dipole and time-averaged geomagnetic

field

As more paleomagnetic data became available,

the GAD hypothesis could be examined with more

rigor. Wilson [12] qualified the appropriateness of

the GAD hypothesis suggesting that the best-fitting

model for the time-averaged geomagnetic field was

that of an boffsetQ dipole, displaced some 191F38

km northward along the rotation axis (dashed line in

Fig. 2). This offset dipole field has the effect of

bflatteningQ inclinations at observations sites relative

to what would be produced by a centered dipole

(predicted from the dipole formula). The flattened

inclinations lead to VGPs that are bfar-sided;Q they

plot on the opposite side of the globe from the site of

observation.

While an offset dipole is a conceptually simple

way of representing the geomagnetic field, it is

often more advantageous to use an expansion of Eq.

(1) known as the spherical harmonic expansion. The

full potential equation is beyond the scope of this

paper, which is intended for a general audience, but

it is important to understand that it is summation of

harmonic functions Plm of position (e.g., co-latitude

h, longitude /, and radius r). The shape and

orientation of each term is determined by the degree

l and the order m. The first harmonic P10 is simply

cosh. The second (P20) is 1/4 (3 cosh+1). Higher

order harmonics are increasingly wiggly functions

of cosh.

b)

om Opdyke and Henry [8]. Solid lines are the trends expected from a

m (see text). (a) Data from normal polarity intervals, (b) data from


In a spherical harmonic expansion, the harmonic

functions are weighted by coefficients glm and hl

m

known as bGauss coefficients.Q These are determined

by fitting the observed geomagnetic field vector data

for a particular observation interval. We show three

examples in Fig. 3 of the inclinations of the vector

fields with their surface harmonics as insets. These are

the axial (m=0) dipole (l=1), quadrupole (l=2) and

octupole (l=3) terms whose contributions are deter-

mined by g10, g2

0 and g30 respectively.

If the axial dipole field produced by the harmonic

function in Fig. 3a were turned on its side with the

a) b)

c)

Fig. 3. Examples of surface harmonics (insets) and maps of the associated pa

north pole part pointing to the Greenwich meridian,

the contribution would be determined by h10 coef-

ficient, and if it were at 90 8E, it would be the h11

coefficient. Therefore, the total dipole contribution

would be the vector sum of the axial and two

equatorial dipole terms orffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðg01Þ

2 þ ðh01Þ2 þ ðh11Þ

2q

. The

total quadrupole contribution (l=2) combines five

coefficients and the total octupole (l=3) contribution

combines seven coefficients.

In general, terms for which the difference between

the subscript (l) and the superscript (m) is odd (e.g.,

the axial dipole g10 and octupole g3

0) produce magnetic

-80

-40

0

40

80

tterns for global inclinations. (a) Dipole, (b) quadrupole, (c) octupole.


fields that are asymmetric about the equator, while

those for which the difference is even (e.g., the axial

quadrupole g20) have symmetric fields. In Fig. 3a, we

show the inclinations produced by a purely dipole

field of the same sign as the present-day field. As we

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,

inclinations produced by a purely quadrupolar field

(Fig. 3b) are down at the poles and up at the equator.

The map of inclinations produced by a purely axial

octupolar field (Fig. 3c) are again asymmetric about

the equator with vertical directions of opposite signs

at the poles separated by bands with the opposite sign

at midlatitudes.

The geomagnetic field produced by an offset

dipole is identical to that produced by a combination

of an axial dipole and an axial quadrupole. The offset

of 191 km favored by Wilson [12] is the same as

having a ratio of axial quadrupole to dipole terms

G2=g20/g1

0 of about 6%.

The two decades from 1970 to 1990 saw many

attempts to quantify the contributions of various terms

in the spherical harmonic expansion to the time-

averaged geomagnetic field using a variety of

paleomagnetic data (see, e.g., [13–18]). Most of these

concluded that variations in longitude were indistin-

guishable from noise (the time-averaged field was

controlled primarily by the axial terms (m=0) and

departures from GAD were small with the largest term

being the axial quadrupole with a contribution of at

most 6% (see [19] for an excellent review).

In the last decade, an entirely new class of models

has become available. These are based on construct-

ing spherical harmonic models for the time-averaged

field from large, globally distributed data sets (e.g.,

[20–23]). The most recent of these was by Hata-

-80-60-40-20020406080

Fig. 4. Map of predicted average inclination from the time-averaged

field models of Hatakeyama and Kono [23].

keyama and Kono [23]. We show the inclination

variations predicted from their normal polarity model

in Fig. 4. The largest departure from a GAD field is

the quadrupole term which is approximately 4% of

the dipole.

4. Statistical paleosecular variation models

From studies of the time-averaged field over the

last 5 million years, it seems that, at least for that

period of time, the field has been dominantly that of

a geocentric axial dipole. At any particular instant in

time, however, there will be significant deviations

owing to the non-axial dipole contributions. This,

combined with distortions in the recording process

(some of which are discussed in the following

section) and decreasing preservation of rocks with

increasing age, makes evaluating the GAD hypoth-

esis increasingly difficult as we go back in time.

Therefore, it would be extremely useful if we had a

way of predicting for a given latitude the distribu-

tions of vectors from the geomagnetic field like the

one that operated over the last 5 million years. Such

statistical predictions could then be compared with

paleomagnetic observations. To find an appropriate

statistical paleosecular variation model, we begin

with the work of Constable and Parker [24] (here-

after CP88).

The CP88 statistical paleosecular variation model

assumes that the time-varying geomagnetic field acts

as bGiant Gaussian ProcessQ (GGP) whereby the

Gauss coefficients glm, hl

m (except for the axial dipolar

term, g10 and in some models also the axial quadrupole

term g20) have zero mean and standard deviations that

are a function of degree l and a fitted parameter a.Once the average dipole moment g1

0, its standard

deviation r10: and a are fixed, realizations of the field

model can be created by drawing the Gauss coef-

ficients from their respective Gaussian distributions.

Geomagnetic vectors can then be calculated for any

given location using the usual transformation from the

geomagnetic potential equation to geomagnetic ele-

ments (see [24] for details).

The principal drawback of the CP88 model is that

it fails to fit the observed scatter in the paleomagnetic

data with latitude. Most of the subsequent variations

(e.g., [23,25–27]) attempted to address this problem


by introducing more fitted parameters, losing the

elegant simplicity of the CP88 model.

The most recent model of the statistical paleosec-

ular variation genre is the TK03.GAD model of Tauxe

and Kent [2]. Like CP88, TK03.GAD has only three

parameters: g10 (set to fit a recent estimate for the long-

term average intensity of the axial dipole [28]), a as

defined in CP88, but fit to the more recent compila-

tion of directional data of [29] and a new parameter bwhich is the ratio of the asymmetric (l+m odd) to the

symmetric (l+m even) Gauss coefficients for a given

l. The term b allows a much improved fit to the

paleomagnetic observations while the model retains

the simplicity of the CP88 model. We show the

variation in r with degree for the two families

(asymmetric and symmetric) in Fig. 5a.

In Fig. 5b, we show the vector end points

calculated from 1000 realizations of the model at 30

8N. The distribution of these vectors predicts what

would be observed at that latitude if we had a large

number of observations of the geomagnetic field or its

paleomagnetic proxies.

In Fig. 6a, we show vector endpoints from 1000

realizations of model TKO3.GAD evaluated at the

equator and seen from the side. The average

direction is north and horizontal, so when we turn

the projection about a vertical axis and look at the

data along the expected direction, we have the

average direction at the center of the diagram as in

Fig. 6b. The projections so far have been of the

Symmetric terms

g1

0

g2

0

g3

0

a)

Asymmetric terms

Fig. 5. (a) Variation of the standard deviation rl as a function of harmon

field model TK03.GAD. All terms have zero mean except the axial di

TK03.GAD at 308N.

vector endpoints, but because the strength of the

ancient geomagnetic field is measured much less

frequently than the direction, paleomagnetic data are

often assumed to be unit length [30]. Unit vectors

can be plotted on equal area projections which

usually are centered on the vertical direction. In

Fig. 6c, we plot the directions from Fig. 6a but have

centered the diagram along the direction expected

from a GAD field (star) at the equator (or

horizontal). What is strikingly apparent in this

diagram is the marked elongation in the distribution

of directions at the equator. This elongation dis-

appears as the observation site approaches the pole.

Although this tendency was predicted for circularly

symmetric VGPs [31] and has been noted in the

paleomagnetic data for decades (e.g., [32–34]), it is

generally ignored in the paleomagnetic literature.

The average direction of the data shown in Fig.

6c is here termed V1. Two other orthogonal

directions are V3, the direction of least scatter and

V2, the direction of elongation as shown in Fig. 6c.

The elongation parameter E is defined [35] as the

ratio of the variance (calculated as the eigenvalues of

the covariance matrix, see [35]) in the V2 direction

over that in the V3 direction. We plot elongation and

inclination versus latitude predicted by the TKO3.-

GAD model in Fig. 6d and elongation versus

inclination in Fig. 6e. This treatment allows us to

have a unique pair of values for elongation E and

inclination I that is compatible with the statistical

N

E

D

b)

ic degree l for asymmetric and symmetric terms for the statistical

pole term. (b) 1000 vector endpoints from realizations of model

NE

D D

E

a)

e)d)

b) c)

V3

V2

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

discussed later in the paper.

5. Earth filters

There are many distortions of the magnetic record

in Earth materials owing to the accumulating effects

of plate movements, rock deformation, biases inherent

a) b)

Fig. 7. An example of what geologic processes do to directional distributio

shown in Fig. 5b. (b) Sedimentation process that produces flattened directio

produced from flattening filter with f=0.4.

in the magnetic recording process and remagnetiza-

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

consider one example of an Earth filter (see Fig. 7):

sedimentary inclination flattening.

In Fig. 7a, we show a set of directions drawn from

the TK03.GAD statistical field model with an average

inclination of ~458. In Fig. 7b, we show that elongate

Bc)

ns. (a) Equal area projection of a set of directions drawn from cloud

ns according to sedimentary inclination error formula. (c) Directions

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].

f = 1.0

.8

.9

.7

.6.5

.4

.3

494260

c)b)

a)

Fig. 9. (a) Average inclination (short dashed) and elongation (long

dashed and solid line) of data from Fig. 7c, after transformation to I

(see text). Dashed (solid) portion of the elongation curve has east–

west (north–south) elongation. (b) Heavy hatched line is elongation

versus inclination from panel (a). Elongation direction (V2) is shown

by hatch marks. Dashed line is from TK03.GAD [2]. Crossing poin

(circled) is the elongation/inclination pair consistent with

TK03.GAD model. Light curves are results from bootstrapped

samples of the data in panel (a). (c) Histogram of inclinations from

1000 bootstrapped crossing points. Bar indicates bounds including

95% of the compatible inclinations. Vertical line is the inclination

expected from the ambient magnetic field at the time of deposition


particles (such as detrital hematite) can produce

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 inclina-

tion flattening having the same form (e.g., [37]).

That sedimentary inclination flattening occurs in

nature was demonstrated by data from storm deposits

retrieved from the Soan River in northern Pakistan

[38] which have detrital hematite as the primary

carrier of magnetic remanence (see Fig. 8a). The

geomagnetic reference field when the sediments were

deposited had a declination of D=1.58 and an

inclination I=508 (triangle in Fig. 8a) while the 59

specimens obtained from the river have a mean

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

showed that inclination flattening has the form

predicted by King [36] of tanIo=f tanIf with a value

of f of 0.55 (dashed line).

The possibility of inclination flattening biasing

results from sedimentary rocks led Jackson et al. [39]

to seek a method for detecting and correcting

inclination flattening. Their method has been success-

fully applied in several studies (e.g., [40–42]). The

principle reason it has not been universally applied in

V

t

.


studies where inclination flattening is suspected is that

the method is extremely labor intensive requiring a

battery of meticulous laboratory experiments.

An alternative approach to the inclination flat-

tening problem was suggested by Tauxe and Kent [2].

They proposed that one could bcorrectQ sedimentary

inclination flattening by using the twin tools of the

inclination error formula and a statistical field model.

Assuming that inclination flattening from whatever

source (original depositional or compaction related)

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

formula and calculate bnewQ mean inclination/elonga-

tion pairs as a function of f. We illustrate how such a

method works in practice by inverting the synthetic

flattened data from Fig. 7c in Fig. 9a. As f decreases,

inclination increases (short dashed line in Fig. 9a) and

elongation decreases along the east–west axis (solid

line in Fig. 9a). Elongation comes to a minimum value

(at f=0.6 in this case) at which the distribution is more

or less circularly symmetric. Then as f decreases

further, the distribution becomes more elongate north–

south (long dashed line in Fig. 9a).

In Fig. 9b, we plot elongation versus inclination

from Fig. 9a as the heavy hatched line. The direction

of the hatches indicates the direction of V2 or the

direction of elongation. We have replotted the

elongation–inclination curve from Fig. 6 as a dashed

line, so the inclination at which the heavy line crosses

the dashed line gives the elongation–inclination pair

consistent with model TKO3.GAD, or 538 [we note

the much improved concurrence of this bcorrectedQvalue and that of the hypothetical applied field of

498]. The light curves are similar results from

bootstrapped samples of the data shown in Fig. 7a.

A histogram of inclinations from 1000 such boot-

strapped crossing points are plotted in Fig. 9c. Ninety-

five percent of these lie between 42 and 608 and the

mode is 498. So, the E/I method here yields an

estimate for inclination of the synthetic data of 494260.

This estimate is in complete agreement with the

original inclination of 498 shown as the dashed

vertical line in Fig. 9c, as expected.

Assumptions of the elongation–inclination method

are (1) the geomagnetic field in the past produced

directions compatible with the TKO3.GAD model. (2)

The sole source of flattening in the data is from

sedimentary inclination flattening and has the func-

tional form of tanIo=f tanIf. (3) There are sufficient

measurements of the ancient geomagnetic field that,

once corrected, they will conform to the statistical

model (Monte Carlo simulations suggest that N100

data points are necessary). If these assumptions are

satisfied, the actual elongation–inclination method is

very fast (a matter of minutes) and can be done with

the program find_EI in the pmagl.8 software

distribution at: magician.ucsd.edu/software.

6. Application of the elongation/inclination method

Tests of the GAD hypothesis for ancient times are

of two types. The bpredicted direction methodQ (e.g.,[43,44]) compares paleomagnetic observations to the

directions from a given place and time assuming a

reference pole. The bstatistical distribution methodQ(e.g., [45,46]) assumes that paleomagnetic directions

obtained from many locations and ages can provide a

statistical sampling of the ancient geomagnetic field

that can then be compared with one generated using a

random sampling of a GAD field.

An updated compilation of paleopoles for the

Atlantic-bordering continents for 0–200 Ma shows

very good internal agreement with the GAD model,

with only a small 3% quadrupolar contribution [11]

required. Nevertheless, comparison of predicted

directions with those observed in certain places in

certain rock types reveals persistent biases in inclina-

tion whereby the directions are shallower (more

horizontal) than expected.

One of the puzzling examples of persistent shallow

bias has been in the Cenozoic and Mesozoic

paleomagnetic directions from Central Asia (see,

e.g., [43,44,47–49]). Explanations for the discrepan-

cies include sedimentary inclination flattening (e.g.,

[50,49]) and extreme internal deformation of the

Eurasian plate (e.g., [47]), but most of these studies

attributed the observed inclination shallowing to

significant non-GAD geomagnetic fields (e.g.,

[43,44,48,51]).

In order to use non-dipole fields as an explanation

for the large discrepancy between expected and

observed inclinations in the Asian data sets, Si and

Van der Voo [48] noted that the reference poles are

largely based on results from the UK and North

http://magician.ucsd.edu/software/


America. A non-zero axial octupolar contribution with

the same sign as the dipole makes directions in mid-

northern latitudes shallower than expected from a

GAD field (see Fig. 3c). These directions, when

converted to paleomagnetic poles will be bfar-sided.QIf this reference pole is then used to predict directions

in Asia, the predicted directions will be too steep. The

effect would be amplified by the fact that the actual

directions observed in Asia in the same octupolar field

will also be shallower than expected. Typical con-

tributions of g30 called for are between 10 and 20% of

the average axial dipole.

In order to explore the problem of the Asian

inclination anomaly, we will use two data sets

originally obtained for magnetostratigraphic purposes

as these are much larger than those used for

determining pole positions. The insets to Fig. 10

show the Oligo–Miocene data from Subei of Gilder et

al. [52] and the middle Miocene data from the Potwar

Plateau of Tauxe and Opdyke [53] [N.B.: for the latter,

we have only used the 105 sites with a95V5 and Nz3,

in order to reduce the effect of sources of scatter other

than sedimentary inclination flattening]. The inset

equal area projections are superposed on the map of

present-day crustal deformation of China based on

global positioning system (GPS) data (in the Eurasian

reference frame) of Wang et al. [54]. The GPS vectors

from around the Subei area show virtually no north-

ward motion of that block with respect to Europe

while those from the Indian subcontinent show some

40 mm/yr northward motion.

The average of the 222 Subei paleomagnetic

directions has a declination of D=356.18 and an

inclination of I=43.78. Assuming that the location of

the study (presently located at 39.58N, 94.78E) has

been fixed to the European coordinate system and

taking the 20 Myr pole for Europe from [11] (81.48N,149.78E), the expected inclination is 638. Similarly,

the Potwar data set has an average of D=344, J=33.7.

The European pole position for 10 Ma from [11] is

85.08N, 155.78E giving a predicted inclination at the

location of the Potwar data (33.38N, 72.38E) of 538.The pole position for India at 10 Ma (85.88N,231.18E) predicts an inclination of 48.48.

The Subei and Potwar sediments are typical of

Asian sedimentary units in having inclination values

that are substantially shallower than expected. A

northward transport mechanism would require over

2000 of kilometers of displacement over the last 20

million years to explain the Subei data or an average

rate of over 100 mm/yr. Such a rate is larger than

any present-day rate within China. It would be

particularly surprising for the location of Subei

which currently has a very small rate of deformation

in European coordinates (see GPS data of [54] in

Fig. 10).

To bring the Potwar Plateau data into accord with

Indian plate motion as inferred from the paleomag-

netic poles compiled by Besse and Courtillot [11]

would require some 1600 km of undetected northward

transport over the last 10 million years. In European

coordinates, there must be over 2000 km of crustal

shortening in 10 million years for an average rate of

over 200 mm/yr, again much larger than the present-

day rates of northward motion with respect to stable

Europe determined by GPS measurements (see Fig.

10). These are approximately 40 mm/yr.

Extreme crustal shortening is therefore difficult to

reconcile with the present-day observations or with

palinspastic reconstructions. Large octupolar contri-

butions would have the uncomfortable consequence

of rendering useless the paleomagnetic poles which

currently are the primary means of placing latitude

bands on plate reconstructions. The alternative

explanation for shallow bias in the Asian sedimentary

data would be some form of inclination flattening

either from original depositional processes (e.g.,

[2,49,52]) or subsequent compaction (e.g., [50]).

From the foregoing, we see that there are at least

three completely different explanations for the shal-

low bias observed in the Asian red sediments. Some

way of discriminating among the competing hypoth-

eses would be useful. Fortunately, as we have seen

from the discussion of statistical field models and the

section on the elongation/inclination (E/I) method of

Tauxe and Kent [2], there is more to paleomagnetic

data than just the average direction. The shape of the

distribution changes with latitude as well as the

inclination, varying from fairly elongate at the equator

to circularly symmetric at the poles. Following Tauxe

and Kent [2], we show the histogram of inclinations

from the E/I method applied to the Subei data in Fig.

11a which yields an estimate for inclination of the

Subei data of 635669. This estimate is in complete

agreement with the expected inclination if the Subei

region had remained fixed in European coordinates

6356

6941

34

49

a)

Inclination (I')Inclination (I')

Fra

ctio

n

b)

Fig. 11. Elongation–inclination results from data shown in Fig. 10.

Expected inclination from European reference pole is shown as a

vertical line. (a) Data from Subei. (b) Data from Potwar.

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

GPS measurements.

We repeat the E/I method on the data from the

Potwar Plateau in Fig. 10b and get an estimate for

inclination of 413149. This is consistent with that

predicted from the paleomagnetic pole from India of

Besse and Courtillot [11]. Furthermore, the corrected

data are also compatible with the present-day north-

ward motion of India with respect to Europe of 40

mm/yr observed from the GPS data of Wang et al.

[54].

By using the distribution of paleomagnetic direc-

tions instead of inclinations alone, it is possible to test

which of the various competing explanations for


shallow directions is the most consistent. In the case

of the data from Subei and the Potwar Plateau, we find

that inverting the directions assuming a sedimentary

inclination flattening model yields data distributions

that are both compatible with current statistical field

models as well as with present-day crustal deforma-

tion measurements.

The E/I method has been applied successfully in

several other recent studies. Krijgsman and Tauxe [55]

used it on Neogene sediments from Spain and Crete.

The correction brought the average inclination into

excellent agreement with those predicted from their

tectonic contexts. Also, noting that the reconstruction

of Atlantic bordering continents in the Triassic

predicts paleolatitudes in conflict with those observed

in the Triassic sedimentary sequences preserved in the

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

temporal/spatial context.

7. Conclusions

The geocentric axial dipole hypothesis, first

articulated over 400 years ago, serves as the founding

principle of paleogeographic reconstructions that rely

on paleomagnetically determined paleolatitudes.

Despite the attractiveness of the GAD hypothesis,

there have been persistent calls for permanent non-

dipole contributions to the time-averaged field. Over

the last decade, these have called for larger and larger

contributions of the axial octupole, which, if valid,

call into question the utility of the GAD hypothesis.

There are inherent biases in geological recording of

the geomagnetic field. Many of these lead to a net

shallowing of the observed direction. The shallow

bias could thus be explained by recording artifacts as

well as permanent non-dipole contributions to the

time-averaged field.

By applying a statistical field model consistent

with the last 5 million years which predicts not only

the average direction but also the shape of the

distribution, one can sometimes differentiate among

the possible explanations for shallow bias. It is shown

here that the shallow bias in Mesozoic and Cenozoic

Asian red beds is plausibly attributable to sedimentary

inclination flattening. However, the statistical charac-

ter of the time-averaged field is poorly constrained in

the past. More data are necessary to assess the general

validity of the model derived from the last 5 million

years.

Acknowledgements

I gratefully acknowledge stimulating conversations

with Cathy Constable, Catherine Johnson, Dennis

Kent, Ken Kodama and Wout Krijgsman. Careful

reviews by Ted Irving, Subir Banerjee and Martin

Frank improved the manuscript as did suggestions by

the editor, Alex Halliday. I also appreciate the advice

of Julie Bowles and Kristin Lawrence. This work was

partially funded by NSF Grant EAR-0003395.

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Inclination flattening and the geocentric axial dipole