Gubbins D., Herrero-Bervera E. Encyclopedia of Geomagnetism and Paleomagnetism

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iron). The Grüneisen parameter is given by (see Grüneisen’s para-

meter for iron and Earth’s core):

g ¼ V



; (Eq. 8)

where, a, K

, K

, C

, and C

are the thermal expansion coefficient,

isentropic and isothermal bulk modulus, and specific heat at constant

pressure and volume. Equation (8) is used to evaluate g at ambient

pressures.

Determining Gru

neisen parameter

The Grüneisen parameter at high pressure is obtained from the mea-

sured bulk sound velocity, C, behind the shock front and is given as

(see Grüneisen’s parameter for iron and Earth’s core):

g ¼

rs

1 þ s  R

2

ð1  sÞ



; (Eq. 9)

when

 ¼ u

=U ¼ðV

 V Þ=V

(Eq. 10)

Figure S39 Shock -wave data for FeO (Jeanloz and Ahrens, 1980),

Fe (McQueen et al., 1970), Fe

and Fe

(McQueen and

Marsh, quoted in Birch (1966); not corrected for porosity)

(heavy curves) compared with the seismologically determined

compression curve of the core. The Fe and FeO data, corrected to

core temperatures, are shown as thin dashed curves. Hugoniots of

mixtures of oxides and Fe corresponding to FeO are given as

thin curves;

(Fe þ Fe

) and (Fe

–Fe

) are

plotted but overlap; open squares are from Al’tshuler and

Sharipdzhanov (1971). Hugoniots for only the high-pressure

phase are shown, and the wu

stite data are corrected for initial

porosity and nonstoichiometry (after Jeanloz and Ahrens (1980)).

Figure S38 Hugoniot shock pressure versus reduced volume

(V/V

) data for g-iron. Upon shock loading, the onset of phase

change which a-iron (bcc) transforms to e-iron (hcp) with a 5%

decrease in volume occurs at 13 GPa. Upon unloading from

23 GPa dynamic transformation from hcp to bcc-iron occurs at

10 GPa. The data are from Bancroft et al. (1956) and Barker and

Hollenbach (1974). Dashed curve (Andrews, 1973) is theoretical

curve.

Figure S37 Measurements of the electrical resistivity of Fe

0.94

under shock loading are compared with previous data for iron and

two iron-silicon alloys, iron, iron alloys, and compounds versus

shock pressure (after Knittle et al. (1986)). Upon extrapolating

these data to the pressure range of core (135–450 GPa), it can be

concluded that these iron-bearing minerals have resistivities of

6

–10

5

O m at core pressures and temperatures.

916 SHOCK WAVE EXPERIMENTS

and



¼ðr=r

Þðc=UÞ: (Eq. 11)

The derivation of (9) assumes that the Hugoniot shock and particle

relation are described by:

U ¼ c

þ su

; (Eq. 12)

where

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

(Eq. 13)

and

s ¼ K

þ 1



=4: (Eq. 14)

By measuring the free-surface velocity profiles of shock states being

overtaken by rarefactions originating at the rear of the iron flyer plate,

the sound velocity c (and hence g) of different states along the princi-

pal Hugoniot and 1373 K preheated Hugoniot of g-iron may be

obtained as a function of density or specific volume (Figure S40).

From a series of free-surface profiles, such as shown in Figure S41,

data on the value of c, and hence g (Ahrens et al., 2002a) can be

obtained, for example, for preheated iron, using apparatus depicted

in Figure S42.

Detection of shock-induced melting

As described above, the bulk sound velocities can be measured for

shock states achieved in molten materials,

c ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

: (Eq. 15)

In the solid regime, the rarefaction wave propagates with the longitu-

dinal elastic velocity which is given by:

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

: (Eq. 16)

Thus by measuring the sound velocity as a function of shock pressure,

the shock pressure upon which the sound velocity decreases sharply,

indicates where the Hugoniot curve enters the regime of trans-

formation from the solid, into the partially molten and then completely

liquid regime. Knowledge of the Grüneisen parameter at high pres-

sure and temperature (9–11) and thermodynamic calculations allow

construction of the high pressure-temperature phase diagram (Brown

and McQueen, 1986; Ahrens et al., 2002b; Nguyen and Holmes,

2004) (see Melting temperature of iron in the core, theory and Melting

temperature of iron in the core, experiment).

Very extensive data for iron, centered at normal temperatures (Fig-

ure S43), demonstrates that shock-induced melting begins at

220 GPa (Brown and McQueen, 1986; Nguyen and Holmes, 2004).

The sound velocity of the Hugoniot of other metals is also available

for Al, Cu, Ta, Pb, Mo, W, V, and stainless steel (Dai et al., 2002;

Luo and Ahrens, 2004). Shock temperatures upon melting, for all these

Figure S41 Free-surface velocity profiles from VISAR experiments

in geometry of Figures S33 and S42 on iron preheated to 1373 K.

Shot numbers are shown near each wave profile. Vertical lines

indicate initial unloading arrivals. Peak particle velocities are

within 4% of impedance match solutions. Origin of time axis

is arbitrary. Inset: detailed plot of first release wave arrival for shot

1008 (after Chen (1997)).

Figure S42 Experimental arrangement for conducting VISAR

sound speed measurements in preheated 1373 K iron (after Chen

and Ahrens (1998)).

Figure S40 Thermodynamic Gru

neisen parameter g of liquid iron

versus density. The 1 bar, 1811 K datum is from Anderson and

Ahrens (1994). “This study” with “Chen (1997).”

SHOCK WAVE EXPERIMENTS 917

metals and other materials of geophysical interest were calculated.

These calculations are substantially verified by the, at present, limited

shock temperature and high pressure melting data which are measured

for iron (Williams et al., 1987; Yoo et al., 1993), V (Dai et al., 2001),

FeS and FeS

(Anderson and Ahrens, 1996), and stainless steel

(Ahrens et al., 1990). Moreover, recent studies of melting in the Fe-O

and Fe-S system in diamond anvil apparatus indicate that eutectic

melting temperatures of plausible iron alloys are lowered from those

of iron by <500 K at core pressures (Boehler, 1992, 2000). The above

referenced iron and iron alloys data have indicated the IC-OC melting

temperature of approximately 6000  500 K. These values have pro-

foundly influenced models of the core as discussed in the next section.

Influence of shock wave data on models of the core

Thermal models of the Earth’s core of the 1970s (e.g., Stacey (1977))

were, in part, based on extrapolation of 10 GPa melting data for the

FeS eutectic (Usselman, 1975a, b). Stacey and others modeled

the outer core as having temperature of 3200 K at CMB rising to

4200 K at the IC-OC boundary. The shock temperatures first calculated

by Brown and McQueen (1986) on the basis of their pioneering study of

the shock temperature upon melting along the principal Hugoniot of iron

at 220 GPa (5000– 5700 K) extrapolated to 5800  500 K for melting of

iron at the IC-OC boundary. Later shock temperature measurements

(Figure S44) supported these considerably higher temperatures. The

shock temperatures for iron when applied to the core indicates higher

(by 1000 K) melting points than static high pressure diamond anvil

melting data to 200 GPa (Boehler, 1993). Present models of

the Earth’s core assume IC-OC temperature of 5000–6000 K (e.g.,

Poirier (1994), Anderson and Ahrens (1994), Boehler (2000), Anderson

(2002)) are summarized by the temperature-pressure relation labeled

Earth (representing the outer and inner core) in Figure S44, were influ-

enced by the shock-induced melting data for iron. These rather higher

values lead to rather larger predicted heat flows from the core of

6–10 Tw, compared to the previous assumed values of CMB heat flow

of 4–5 Tw of Matassov (1977). High temperature core values lead

to implausibly short model existence times of the solid Earth core

(Labrosse et al., 2001). This then leads to possible, but still very poorly

constrained models, of radioactive element contents of the core. Such

models of radiogenic heat production lead to higher sustained core

temperatures over the history of the Earth (Buffett, 2003).

As can be seen in Figure S45, as the model values of outer core tem-

peratures are increased from 5000 to 7000 K, the calculated density

deficit (with respect to molten iron at that temperature largely on the

basis of shock wave data for FeS, FeO, and FeSi) goes down from

7 to 3.5% and the wt% of impurity of light element decreases drasti-

cally (see Core, adiabatic gradient and Core composition).

Acknowledgments

I am grateful to William Nellis for a number of suggestions that

improved this manuscript. Research supported by NSF EAR-

0207934. Contribution No. 9039, Division of Geological and Plane-

tary Sciences, California Institute of Technology, Pasadena, CA.

Thomas J. Ahrens

Figure S44 Shock temperature data for iron relative to the

temperatures in the outer and inner core. States between 200

and 250 GPa are assigned to solid e-iron driven to temperatures

above melting line (Lindemann MC, melting curve). States above

270 GPa are believed to represent totally melted material. DAC

MC is diamond anvil melting curve (after Luo and Ahrens (2004)).

Figure S43 Sound velocity upon unloading rarefaction upon

shock compression of a-iron. Solid symbols exhibit a single

solid-liquid transition along Hugoniot at 222  3 GPa (according

to Nguyen and Holmes (2004)). Brown and McQueen’s (1986)

data are shown by open symbols. Open squares (triangles)

represent (explosive driven) shock data (after Nguyen and

Holmes (2004)).

Figure S45 Core density deficit in terms of wt% of three

cosmochemically acceptable elements. Four parallel lines

represent the value of T

at IC-OC (330 GPa) corresponding to

temperatures indicated. Intersection of these parallel lines with

curves for Si, S, and O indicate wt% values of element (after

Anderson and Isaak (2002)).

918 SHOCK WAVE EXPERIMENTS

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of state of solids from shock wave studies. In Kinslow, R. (ed.),

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critical review. Physics of the Earth and Planetary Interiors, 85:

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SHOCK WAVE EXPERIMENTS 919

Cross-references

Core Composition

Core Temperature

Core, Adiabatic Gradient

Grüneisen’s Parameter for Iron and Earth’s Core

Inner Core Composition

Melting Temperature of Iron in the Core, Experimental

Melting Temperature of Iron in the Core, Theory

SPINNER MAGNETOMETER

Introduction

The spinner magnetometer is an instrument for measuring remanent

magnetization (RM) of rock specimens in studies of the magnetic

properties of rocks.

Natural remanent magnetization (NRM) is the magnetic quantity,

which was preserved in the rock during its formation in the presence

of the prevailing magnetic field and then during its long-life existence

when it was exposed to many physical and chemical factors. Thus,

various sorts of RM may exist simultaneously and in different ratio

in the same rock. RM is a vector quantity. If the rock is magnetically

isotropic, the direction of the RM is identical with the direction of

the magnetic field at the time of rock formation. Strong anisotropy

usually causes a difference between the directions of the RM and the

magnetizing field. The spinner magnetometer measures the total

NRM, which is the sum of several RM vectors. The remanent mag-

netizations differ by their formation and have different stabilities.

Magnetization strengths are in the order of 10

–6

1

for weakly

magnetized sediments to 10

1

for strongly magnetized ores.

Typical spinner magnetometer applications are:

Paleomagnetism: Changes in Earth’s magnetic field in geological

history can be investigated through the measurement of the rock’s

remanent magnetization and in the investigation of its stability,

age-dating the rocks, solving some tectonic problems in particular

terrains, dating the development of mineralization of ore deposits,

and many other geological problems.

Archeomagnetism: Changes of Earth’s magnetic field in human

history. These investigations are mostly applicable to dating arche-

ological materials.

Mineralogy: Impurities of ferromagnetic grains in para- or diamag-

netic minerals can be investigated.

Magnetic fabric studies: Measurement of the anisotropy of isother-

mal remanent magnetization (IRM) can help in the separation of

ferromagnetic and paramagnetic fractions of a rock.

Magnetometry: In the interpretation of ground or airborne mag-

netic measurements it is useful to know whether the rock’s magne-

tization is due to its induced or remanent component. Investigation

of RM can help to solve this problem.

Measuring principles

The principle of measurement is very simple, based on electromag-

netic induction law formulated by Michael Faraday as early as 1831.

Rock specimen of defined size and shape, fixed in a specimen holder

rotates at a constant angular speed in the vicinity of a detector.

Provided that the spinning specimen has nonzero magnetic moment,

an AC voltage is induced in the detector. The detector is shielded with

a multilayer permalloy shield. At constant speed of rotation and con-

stant specimen size, the amplitude and phase of the induced signal

depend on the magnitude and direction of RM vector of the specimen.

The voltage is amplified, filtered, digitized, and analyzed. In a

specimen rotating about a single axis inside a pair of detector coils,

the output voltage amplitude is proportional to the projection of the

RM vector to the plane perpendicular to the rotation axis; the compo-

nent of the RM vector parallel to the rotation axis induces no signal.

The phase of the signal depends on the direction of projection of the

RM vector into the plane perpendicular to the rotation axis. To mea-

sure the third RM component, the specimen must be at least once repo-

sitioned by turning it by 90



and re-measured. In practice, the standard

measurement of the RM vector consists of successive measurements in

three, four, or six positions to reduce the measurement errors. The spe-

cimen position change is performed manually or using special auto-

matic manipulator, which turns the specimen between successive

spinning steps around the imaginary body diagonal of a cube which

has threefold symmetry. Thus, the automatic manipulator eliminates

the need of manual specimen positioning during the measuring process

and shortens the measuring time. The other solution is to use a special

specimen holder simultaneously spinning the specimen about two or

three axes. However, this solution usually results in decreasing the

sensitivity of the instrument due to the occurrence of additional

mechanical vibrations caused by the two axes spinning mechanism.

The induced signal is amplitude sensitive proportional to speed of

rotation and phase sensitive due to the band pass filtration of the signal

by the instrument hardware. For this reason a very precise stabil ization

of rotation speed is necessary. A reference signal synchronous to the

specimen rotation is needed for phase calibration, speed control, filtra-

tion, induced signal scanning, and digital processing. The reference

signal is generated by an optocoupler built right on specimen holder

shaft or in some cases into the motor unit. The magnetometer calibra-

tion is a special measurement procedure of the standard, which yields

the gain and phase for calculating the remanence vector components.

Since the specimen is fixed in the specimen holder, the measurement

procedure and specimen position design should eliminate the residual

RM value of the holder. If this is not possible due to principal reasons,

correction for the holder in the case of a weak specimen must be done

by subtracting the empty holder values from the data measured.

In general, the properties of the instrument depend upon the geo-

metric arrangement of the detector relative to the spinning specimen,

the type of detector and its size, homogeneity and geometric precision,

noise of detector, noise of electronic unit, speed of rotation, size and

homogeneity (magnetic and physical) of the specimen, the type of spe-

cimen holder, the influence of external disturbing fields, the resistance

to mechanical shock, and temperature.

Sensitivity of the spinner magnetometer

The signal detector is the heart of the instrument. The design of

the detector together with the preamplifier of the induced signal

significantly determines the sensitivity of the instrument. Since some

rocks have very low amplitude of the RM vector, of the order of

–5

–10

–6

1

, the sensitivity of the instrument should be at least

in the order 10

–6

1

Spinner magnetometer measurements are disturbed by noise, which

can be in three categories: First the thermal noise of the detector (pick-

up coils), second the noise of the electric circuits (preamplifier of the

induced signal and the control unit), and third the noise whose source

may be by various causes (impurities on the specimen holder, vibration

of the pick-up coils, strange external magnetic fields, etc.). The sum of

all of these is called the operating noise, and the sum of the first two

categories is called the basic noise. Other sources of errors occur dur-

ing removing and inserting of the specimen in cases of manual posi-

tion change, or during specimen position change by the automatic

manipulator. The sensitivity, in case we are able to remove all addi-

tional disturbing effects (except for basic noise), is called the limiting

sensitivity or minimum detectable field. For practical use, it is signifi-

cant to work with the standard deviation of repeated measurements of

a weak specimen (at the lower limit of measuring range) using a stan-

dard measuring routine for a particular instrument. This standard

920 SPINNER MAGNETOMETER

deviation characterizes the sensitivity of instrument. The value of the

noise is usually lower than the sensitivity.

The sensitivity of instrument depends on the time of integration, the

prolongation of measurement time (four times results in increasing the

sensitivity two times). The limitation factor is the operating noise. Eva-

luation of the noise and sensitivity of the instrument is an essential step

for instrument selection suitable for a particular application. For mea-

surement of strong rocks specimens, ores for example, the sensitivity

of the instrument is not critical but in the case of weakly magnetized

sediments, it is the most important parameter to determine the suitabil-

ity of the intended method of measurement.

Deviations in specimen size and shape will cause errors in position,

and/or possibly increase of vibration during measurement and thus

decrease the precision of measurement.

The detectors used in classical spinner magnetometers are a pair of

Helmholtz coils, or a fluxgate sensor. Recently, the SQUID detector

working at the temperature of liquid nitrogen, the so-called high

temperature HT-SQUID has been used. HT-SQUID based spinner

magnetometer is a compromise between the classical spinner and the

low-temperature liquid helium cryogenic magnetometer.

The sensitivity of spinner magnetometer with a fluxgate sensor

is about 1.0  10

–4

1

, with a spinning rate of 6 Hz, an integration

time of 24 s (Leaflet Molspin).

The sensitivity of the HT-SQUID spinner magnetometer expressed in

terms of intrinsic noise is 3.0 10

–7

1

(Leaflet F.I.T. Messtechnik).

The detector output voltage is due to a different measuring principle in

comparison with classical spinner independent of the spinning rate.

The measurements with the instrument using the two axes spin-

ning specimen method and HT-SQUID sensor show a sensitivity of

1.0  10

–5

1

(standard deviation) in measurement time of 13 s

(Leslie et al., 2001).

The sensitivity achieved with the standard Helmholtz coils is

2.4  10

–6

1

for a spinning rate of 88 Hz, and 10 s integration

time and about 1.0  10

–5

1

for a slow spinn ing rate of 17 Hz

(Agico, 2001).

Description of Agico spinner magnetometer

The JR-6/JR-6A dual speed spinner magnetometer is an innovated ver-

sion of the instrument series based on classical spinner magnetometer

design with pick-up unit Helmholtz coils. The instrument is equipped

with up to date microelectronic components. Two microprocessors

control and test the speed of specimen rotation, signal gain, acquisition

of the data, carry out digital filtration, and control the autoposition

manipulator. The magnetometer is fully controlled by an external com-

puter via a serial channel RS232C.

The spinner magnetometer JR-6A, equipped with a specimen autop-

osition manipulator and an automatic specimen holder, enables the

automatic measuring of all components of the RM vector.

The low speed of rotation increases the possibility of measuring fra-

gile specimens, soft specimens placed in a perspex container and/or

specimens with considerable deviations in size and shape. The integra-

tion time may be shortened or prolonged by a factor of 2. The repeat

mode enables the repetition of measurement in the current specimen

position without stopping the motor.

A nominal specimen is either a cube with an edge of 20 mm or a

cylinder 25.4 mm (1 in.) in diameter and 22 mm long. Calibration

relates to the nominal volume of the specimen. One side of a cubic

specimen or the base of a cylindrical specimen is marked with an

arrow, which defines the coordinate system of the specimen in which

the RM vector is measured. The components of RM vector are calcu-

lated using Fourier harmonic analysis. The system of the measurement

positions is identical for cubic and for cylindrical specimens. The

sensitivity is 2.4  10

–6

1

for a spinning rate of 87.7 rps and an

integration time of 10 s, and for low-speed rotation of 16.7 rps, the sensi-

tivity is 1.0  10

–5

1

. The accuracy of measurement of the RM

components is 1%,  2.4 mAm

1

. The measuring range of the instru-

ment in full auto range is from 0 (2.4 mAm

1

)to12500Am

1

in seven

decimal ranges.

JR-6/JR-6A spinner magnetometer

measurement procedure

Standard measurement of the RM vector consists of successive mea-

surements in four positions of the specimen with regard to the holder

using the manual specimen holder for four or six positions. The com-

plete measurement yields four values for z component of the RM

vector and two values of the x and y components from which the aver-

age values are calculated. This process eliminates any residual non-

compensated value of the holder RM and reduces the measuring

errors caused by inaccurate shape of the specimen and by instrument

noise. The residual value of the remanent magnetization of the manual

holder for measurement in four or six positions is fully eliminated even

if the correction for the holder was not done. It is due to position

design of manual positioning with respect to the holder.

If less than four positions are used, the residual components of the

holder are not eliminated automatically. We must rely upon the correc-

tion of the holder by subtracting the empty holder values. No correction

is made for errors due to irregular shapes of the specimens. It is there-

fore recommended to reduce the measuring cycle to two positions only,

for approximate measurements.

Sometimes it may be useful to expand the measuring cycle to six

measuring positions, especially in case of specimens of an irregular

shape. If six measuring positions are used, the residual value of the

RM vector of the holder is fully eliminated. All three components

are always measured four times, so the statistics may be better.

Measurement with an automatic holder is available with model

JR-6A. The specimen is fixed in the holder only once; the automatic

manipulator changes the measuring position without the need for the

operator intervention. The further advantage is that the specimen is

not exposed to external magnetic fields during position change, as

the specimen remains closed inside the triple permalloy shield during

the three-position measurement routine. The automatic specimen

holder consists of a partly spherical outer shell and a rotatable inner

spherical core in which the specimen is secured by a plastic screw.

The system of specimen orientation occurs by specimen rotation

around the imaginary body diagonal of a cube, which has threefold

symmetry. For this reason it is impossible to find a combination

of positions which would eliminate the residual value of the rema-

nent magnetization of the holder as with the traditional four- or six-

orientation magnetometer measurement. It is necessary to rely entirely

upon the correction done by the instrument. This fact may negatively

influence the result, especially in case of very soft and weak specimens

when small impurities, moving obligatory on spherical core, can gen-

erate unrealistic data. Specimens, which differ in size or shape from

nominal values, could cause strong vibrations or imbalance while spin-

ning. Such specimens cannot be measured using automatic holder. But

in this case it is usually possible to use standard manual holder.

Jiří Pokorný

Bibliography

Agico s.r.o., 2001. Instruction Manual for Spinner Magnetometer JR-6/

JR-6A. Brno, Czech Republic.

F.I.T. Messtechnik GmbH. 2001. Leaflet of the HSM2 SQUID-Based

Spinner Magnetometer. Hildesheim, Germany.

Leslie, K.E., Binks, R.A., Lewis, C.J., Scott, M.D., Tilbrook, D.L., and

Du, J., 2001. Three component spinner magnetometer featuring

rapid measurement times. IEEE Transactions on Applied Super-

conductivity, 11(1): 252–255.

SPINNER MAGNETOMETER 921

Natsuhara Gigen Ltd, 2003. Leaflet of the Aspin Spinner Magnetometer.

Osaka, Japan.

Molspin Ltd. Leaflet of the Minispin Fluxgate Magnetometer. Newcastle

on Tyne, England.

Cross-references

Magnetization, Natural Remanent (NRM)

Magnetization, Remanent

STATISTICAL METHODS FOR PALEOVECTOR

ANALYSIS

Our concern is with the statistical description of paleomagnetic vectors

and the estimation of their mean and variance. These vectors may come

from a number of different rock units or archeological samples, repre-

senting a range of acquisition times, and be useful for studies of the

mean paleomagnetic field and paleosecular variation; alternatively,

the vectors may come from individual measurements taken from a

given rock unit or archeological sample, representing the same moment

of acquisition, and be useful for studying the acquisition process itself.

Directional data of a particular polarity are usually analyzed with a

Fisher distribution (1953), and data of mixed polarities are usually

analyzed with a Bingham distribution (1964). Occasionally, other direc-

tional distributions are used. For example, Bingham (1983) considered

the projection of a three-dimensional (3D), scalar-variance Gaussian

distribution onto the unit sphere, something he called the “angular-

Gaussian” distribution. More recently, Khokhlov et al. (2001) consid-

ered a generalization of the angular-Gaussian distribution, one with a

covariance matrix, which they used to analyze directional data from a

number of sites. With respect to intensity data, they have traditionally

been treated separately from paleodirections, analyzed with normal,

log-normal, or gamma distributions. Here, for data of either a particular

polarity or of mixed polarities, we summarize these works, and that

of Love and Constable (2003), who developed a full-vector, scalar-

variance, Gaussian-statistical framework for treating directional and

intensity data simultaneously and self-consistently.

The distributions

In our statistical treatment, each paleomagnetic vector x is considered

to be an independent realization occurring in probability according to a

statistical distribution. In Cartesian coordinates (X, Y, Z ) the probabil-

ity P(x) that x lies within the infinitesimal differential volume

x ¼ dX dY dZ (Eq. 1)

PðxÞ¼

pðxÞd

x; (Eq. 2)

where p(x) is the density function,

pðxÞ¼

PðxÞ: (Eq. 3)

With respect to the many different distributions of probability theory,

the Gaussian occupies the most prominent position. This is due to

the central limit theorem, which, roughly speaking, asserts that the dis-

tribution of the sum of independent, identically distributed random

variables is approximately Gaussian. This theoretical underpinning is

appealing, and, therefore, for the analysis of paleomagnetic vectors,

we consider probability-density functions in a Cartesian three-space

of orthogonal magnetic-field components consisting of nonzero mean

Gaussian distributions. We model vectors recording data of a particular

polarity with a unimodal, Gaussian probability-density function

defined in terms of a mean paleomagnetic vector x

and an associated

scalar variance s

xjx

; s



ð2pÞ

3=2

exp 

ðx  x

Þðx  x



(Eq. 4)

We model vectors of mixed polarities with a bimodal, bi-Gaussian

probability-density function, which, in Cartesian coordinates, is

xjx

; s



ðxjx

; s

Þþp

ðxjx

; s



: (Eq. 5)

The relevant paleomagnetic coordinates are spherical, being the famil-

iar quantities of magnetic intensity, inclination, and declination (F, I, D).

In this case, the differential volume element is transformed accord-

ing to

d X dY dZ ! F

cos I dF dI dD; (Eq. 6)

and the Gaussian probability-density function is

xjx

; s



¼ p

F; I; DjF

; I

; D

; s



¼ F

cos Iqðxjx

; s

Þ;

(Eq. 7)

where

qðxjx

; s

Þ¼

ð2pÞ

 exp 

ðF cos I cos DF

cos I

cos D



 exp 

ðF cos I sin D  F

cos I

sin D



 exp 

ðF sin I  F

sin I



: (Eq. 8)

The quantity F

is the magnetic intensity, or the Euclidean length, of

the mean vector x

,andI

and D

are the inclination and declination of

the mean vector. In spherical coordinates the bi-Gaussian probability-

density function is

xjx

; s



¼ p

F; I; DjF

; I

; D

; s



cos Iqðxjx

; s

Þþqðxjx

; s



(Eq. 9)

If we define y to be the off-axis angle between a particular unit paleo-

magnetic vector and the mean unit vector,

x ¼

jxj

and

; (Eq. 10)

then

cos y ¼

x 

: (Eq. 11)

For both the Gaussian and bi-Gaussian cases the vectorial variance

is taken to be spherically symmetrical. That is, the three Cartesian

vectorial components are assumed to be independent and to have equal

922 STATISTICAL METHODS FOR PALEOVECTOR ANALYSIS

scalar variance. Such a situation is sometimes described as being one

of “isotropic” variance. More generally, however, a Gaussian dis-

tribution can be defined in terms of a covariance matrix, where the

Cartesian vectorial variance is ellipsoidal, the components having pos-

sibly different variances and, even, correlation. Such a situation that

is sometimes described as being one of “anisotropic” variance. Of

course, because the anisotropic distribution has a larger number of

degrees of freedom, it will always fit a given dataset at least as well

as the isotropic distribution. In either case, however, it is worth

remarking that the Gaussian distributions are idealizations. We do

not expect that they will fit all paleovector datasets, since the data

themselves result from a myriad of physical processes that in all like-

lihood cannot be completely distilled down to simple mathematical

descriptions. Instead, the utility of statistical distributions is as bench-

marks for comparison, and in that sense, the isotropic unimodal

and bimodal Gaussian distributions, being relatively mathematically

simple, are the most practically attractive.

Marginal forms

For paleomagnetic datasets consisting wholly of coincident intensity

and directional measurements, the Gaussian distribution (7) and the

bi-Gaussian distribution (9) are of obvious utility. However, in most

circumstances, paleomagnetic data consist of only parts, or mixtures

of different parts, of the full paleomagnetic vector. Then, what we

need are the appropriately marginalized probability-density functions

corresponding to the underlying Gaussian distributions. So, for exam-

ple, most paleomagnetic data are only directional, they consist of

inclination-declination pairs with no associated absolute paleointensity.

To analyze such data we need the joint probability-density function

for inclination and declination, obtained by integrating (7) and (9) over

all intensities,

I; DjI

; D

; ðs=F



ðF; I; DÞdF: (Eq. 12)

Alternatively, if we are analyzing data from an azimuthally unoriented

borecore, providing (say) intensity-inclination data, then we need the

marginal density function

F; IjF

; I

; s



ðF; I; DÞdD: (Eq. 13)

If the data consist only of inclinations then we integrate (13) over all

intensities, etc. In each case, we integrate over the vectorial compo-

nents that are either not available or are not needed. For reference,

all require d integrations are given in Love and Constable (2003). In

Figure S46 we show the intensity, inclination, declination, and off-axis

angular distributions corresponding to the Gaussian distribution (7) for

a variety of different dispersions and mean inclinations.

Intensity

In recent years the analysis of paleointensity data, be they from within

a particular epoch or spanning a much longer geological period of

time, has become the subject of increasing interest to researchers. It

is useful, therefore, to compare such data to the intensity distribution

corresponding to the 3D Gaussian distributions. For both unimodal

and bimodal cases the intensity density function is the same, obtained

by integrating either (7) or (9) over all angles,

FjF

; s



¼ s

1



1=2



 exp 







sin h



;

(Eq. 14)

which is a special case of the n-dimensional Rayleigh-Rician distribu-

tion. This function is invariant with respect to change in sign of inten-

sity, although, of course, intensity is, by convention, taken to be a

positive quantity. As an aside, we note that distributions of this type

have application to digital communications and the radar identification

of targets surrounded by Gaussian clutter.

Next, it is enlightening to consider the limiting form of the intensity

distribution where s  F

. It is approximately that for a one-

dimensional normal distribution,

FjF

; s



ﬃﬃﬃﬃﬃﬃ

exp 

F  F



; (Eq. 15)

which McFadden and McElhinny (1982) have suggested might be

appropriate for paleointensity studies, after truncation of negative

intensities. Note that (15) is not a log-normal distribution and it is

not a gamma distribution, each of which have been employed on occa-

sion in the analysis of paleointensity data. However, because it is

directly linked to the 3D Gaussian distributions (7) and (9), which

can be used for directional analyses as well, and because it applies

to a complete and proper range of mean intensities and dispersions,

(14) is suitable for paleointensity studies, even (say) during periods

of reversal when one can expect that the mean intensity would be

small, but the vectorial dispersion would be large.

Off-axis angle

For the Gaussian distribution, the marginal density function for off-

axis angle is

yjðs=F



sin y exp 



 1 þ



cos

exp



cos

"#(

 1 þ erf

ﬃﬃﬃ



cos y





cos y

)

;

(Eq. 16)

and the marginal density function for off-axis angle corresponding to

the bi-Gaussian distribution is just

yjðs=F



sin y exp 



sin

 1 þ



cos

: (Eq. 17)

For the limiting case where s=F

 1, the unimodal off-axis angular

probability-density function (16) is approximately

yjðF

=sÞ



/ sin y exp



cos y

; (Eq. 18)

this corresponding to the Fisher distribution so often used by the

paleomagnetic community for unimodal directional. For the bi-

Gaussian off-axis angular probability-density function (16), and in

the same limit, the off-axis angular density function is approximately

yjðF

=sÞ



/ sin y exp



cos

; (Eq. 19)

STATISTICAL METHODS FOR PALEOVECTOR ANALYSIS 923

Figure S46 Examples of the marginal probability-density functions p

for the Gaussian distribution (7). (a) Intensity F, with

vectorial-mean intensity F

, and with vectorial dispersions s of 7.5, 15, 30, and 60 mT, shown, respectively, by solid, long-dashed,

short-dashed, and dotted lines. (b) Inclination I, (c) declination D, and (d) off-axis angle y with vectorial mean direction

ðI

; D

Þ¼ð45



; 0



Þ, and with relative vectorial dispersions s=F

of 0.25, 0.5, 1, and 2, shown, respectively, by solid, long-dashed,

short-dashed, and dotted lines. Examples (e-h) are for different mean inclinations, but only the (f) inclination and (g) declination density

functions are affected; for vectorial-mean values of ðF

; D

; sÞ¼ð30 mT; 0



; 7:5 mTÞ the solid, long-dashed, short-dashed, and dotted

lines are for vectorial-mean inclinations I

of 0



,30



,60



, and 90



, respectively.

924 STATISTICAL METHODS FOR PALEOVECTOR ANALYSIS

this corresponding to the Bingham distribution so often used by the

paleomagnetic community for bimodal directional analysis. As with

our comment about intensity distributions, because they are directly

linked to the 3D Gaussian distributions, and because they apply to

the complete range of possible vectorial dispersions, (16) and (17)

are suitable for most paleodirection studies.

Maximum-likelihood estimation

In fitting paleomagnetic data to a particular Gaussian distribution,

thereby yielding a measure of mean and variance, a convenient method

is that of maximum-likelihood; for a general review see Stuart et al.

(1999). With this formalism, the likelihood function is constructed

from the joint probability-density function for the existing dataset.

In our case we use the Gaussian density functions and/or their

appropriate marginalizations to construct the likelihood, which, in its

most general form, for all normally encountered types of data groups,

is just

L ¼

FID

j¼1

; I

; D



k¼1

; D

ðÞ

l¼1

; I

ðÞ

m¼1

ðÞ

n¼1

ðÞ: (Eq: 20)

Figure S47 Comparison of maximum-likelihood fits of the 3D bi-Gaussian distribution to the (positive, top) Hawaiian data and

(negative, bottom) Re

union data covering the past 5 Ma. Both the probability-density functions and the histograms of the data are

shown for (a) intensity, (b) inclination, (c) declination, and (d) off-axis angle. Note that the Re

union data, particularly the declination

and off-axis angle data, are fitted better than the Hawaiian data, also note the sizable difference in mean inclination between these two

sites.

STATISTICAL METHODS FOR PALEOVECTOR ANALYSIS 925