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Cross- refere nces

Archeomagnetism

Biomagnetism

Paleomagnetism

Rock Magnetism

ENVIRONMENTAL MAGNETISM,

PALEOMAGNETIC APPLICATIONS

Intro ductio n

Environmental magnetism involves magnetic measurements of environ-

mental materials, including sediments, soils, rocks, mineral dust, anthro-

pogenic pollutants, and biological materials. The magnetic properties of

these materials can be highly sensitive to a broad range of environmental

processes, which makes mineral magnetic studies widely useful in the

environmental sciences. More detailed treatments of the breadth of envir-

onmental magnetic research applications, and definitions of magnetic

parameters and their interpretation, can be found in recent review articles

and books (e.g., Verosub and Roberts, 1995; Maher and Thompson 1999;

Evans and Heller 2003; see also Environmental magnetism). Most envir-

onmental magnetic studies aptly focus on environmental interpretations.

Nevertheless, environmental magnetism, whether explicitly stated as

such or not, routinely plays an important role in paleomagnetic studies,

including tectonic, geochronological, and geomagnetic applications. It is

the paleomagnetic applications of environmental magnetism that are in

focus here, especially in relation to studies of sediments and sedimentary

rocks. These paleomagnetic applications are discussed below following

four broad themes:

1. Inter-core correlation and development of relative age models;

2. Development of astronomically calibrated age models;

3. Testing the hypothesis of orbital forcing of the geomagnetic field;

4. Determining the origin of the paleomagnetic signal.

Inter-cor e co rrelation an d developm ent of relati ve

age model s

Sedimentary environments receive inputs of detrital magnetic minerals

that vary with time and that are diluted by varying supply of lithogenic

mineral components and production of biogeni c mineral components,

respectively. Stratigraphic variations in magnetic properties within

sedimentary basins are usually spatially coherent on scales of tens of

meters; in some cases, magnetic property variati ons can be traced over

large portions of an ocean basin. Variations in magnetic properties,

such as the low-field magnetic susceptibility, therefore, provide a fun-

damentally important physical parameter for stratigraphic correlation

among cores from a coherent depositional environment. Variations in

magnetic susceptibility are routinely used for inter-core correla-

tion and provide the basis for development of relative age models in

paleomagnetic studies of geomagnetic field behavior. For example,

classic studies of paleosecular variation (see Paleomagnetic secular

variation ) recorded by postglacial lake sediments depended on inter-

core correlation based on magnetic susceptibility variations (Turner

and Thompson, 1979). The resultant correlation provided independent

confirmation that the same paleomagnetic variations were observed at

the same time in the respective cores. This procedure yielded a coher-

ent relative age model that was then transformed into a numerical age

model by combined use of radiocarbon dating, correlation to historical

secular variation measurements from the Greenwich observatory and

correlation to archaeomagnetic secular variation data (Turner and

Thompson, 1979).

Inter-core correlation, based on magnetic susceptibility variations, is

widely performed in paleomagnetic studies of lacustrine and marine

sediments. For example, thick marine sedimentary sequences are rou-

tinely cored by the Ocean Drilling Program (ODP), and its successor

the integrated ODP (IODP), by taking successive  10-m cores from

a single hole. Modern high-resolution studies require acquisition of

continu ous records, which necessitates coring of multiple holes at the

same site with slight depth offsets among cores from adjacent holes

to ensure complete stratigraphic recovery. A continu ous stratigraphic

record is then “spliced” together by correlating high-resolution mea-

surements of physical properties such as magnetic susceptibility from

multiple cores (Hagelberg et al. , 1992). This routine procedure of

using magnetic susceptibility variations for stratigraphic correlation

provides the basis for high-resolution paleomagnetic studies of sedi-

ment cores as well as for a wide range of other studies. An example

of the type of high-resolution inter-core correlation that is possible

using magnetic susceptibility variations is shown for two ODP holes

from the North Pacific Ocean in Figure E37 . The two holes are sepa-

rated by 00



and were cored at water depths of 2385 m (Hole

883D) and 3826 m (Hole 884D), respectively, on the flanks of Detroit

Seamount. The sediments from the two depositional environments

have different carbonate and volcanic ash concentrations. The mag-

netic susceptibility variations for the two holes, measured at 1-cm

intervals on u-channel samples, therefore have significant differ-

ences (Figure E37a,b ). Regardless, the records can still be readily

correlated and the depths of correlative horizons from one hole can

be transferred into equivalent depths in the other hole (Figure E37c).

The physical correlation between ODP holes 883D and 884D shown

in Figure E37c enabled Roberts et al. (1997) to independently corre-

late relative geomagnetic paleointensity records for the two holes

and to produce a stacked paleointensity record for the North Pacific

Ocean for the last 200 ka. This approach to inter-core correlation

can also help with analysis of finer details of geomagnetic field beha-

vior. For example, the interval between 8.5 and 9.5 m below

seafloor (mbsf ) at ODP Site 884 contains a detailed record of a geo-

magnetic excursion (Roberts et al., 1997). By correlating whole-core

magnetic susceptibility records from three holes cored at Site 884 (Fig-

ure E38a), in the same way as shown in Figure E37, it is possible to

place detailed records of the excursion on the same relative depth scale

(Figure E38b).

Inter-core correlation using environmental magnetic parameters that

are independent of paleomagnetic variations (Figures E37 and E38)

enable powerful direct comparison of details of geomagnetic field

behavior in paleomagnetic analyses of the spectrum of geomagnetic

field behavior, including studies of paleosecular variation, relative

paleointensity of the geomagnetic field, excursions, and polarity tran-

sitions. It should be stressed that this technique of inter-core correla-

tion simply provides a relative stratigraphic age model; development

of numerical age models depends either on direct dating or on correla-

tion of magnetic properties with an appropriate astronomical target

curve, as described below.

Development of astronomically calibrated ag e models

Stratigraphic variations in magnetic properties are climatically con-

trolled in many depositional environments. This can occur, for

256 ENVIRONMENTAL MAGNETISM, PALEOMAGNETIC APPLICATIONS

example, as a result of climatic modulation of the supply of lithogenic

mineral components eroded from a source rock or through dilution of a

roughly constant lithogenic component by a climatically controlled

biogenic mineral component. The result is an environmental magnetic

record that responds coherently to orbitally forced variations in climate

(e.g., Kent, 1982) . In some cases, environmental magnetic parameters

provide such a good representation of an orbitally controlled forcing

parameter (e.g., summer insolation) that the magnetic parameter can

be used to directly date the sediment core (e.g., Larrasoaña et al.,

2003a). Paleomagnetic studies of relative geomagnetic paleointensity

are often conducted on cores where the magnetic properties are clima-

tically controlled. In such cases, age models are routinely developed

Figure E37 Example of inter-core correlation using low-field magnetic susceptibility. (a) Magnetic susceptibility profile for ODP Hole

883D. (b) Magnetic susceptibility profile for ODP Hole 884D. (c) Correlation of magnetic susceptibility profiles (both for u-channel

samples) for ODP holes 883D and 884D, with data from Hole 883D transformed onto the depth scale for Hole 884D (see Roberts et al.,

1997 for details). Such correlations based on environmental magnetic parameters provide an independent basis for detailed correlation

of paleomagnetic features in cores.

ENVIRONMENTAL MAGNETISM, PALEOMAGNETIC APPLICATIONS 257

by correlation of environmental magnetic parameters with astronomi-

cal target curves (e.g., Guyodo et al., 1999; Dinarès-Turrell et al.,

2002). It is preferable to test such age models against, for example,

oxygen isotope records for the same sediment core (e.g., Larrasoaña

et al., 2003a); however, age models are often developed solely by cor-

relating environmental magnetic parameters with astronomical target

curves.

Environmental magnetic parameters can be routinely used not only

for constructing chronologies for individual cores but also for calibrat-

ing geological timescales. While development of the astronomical

polarity timescale (APTS) (Hilgen, 1991; Langereis and Hilgen, 1991)

has been independent of environmental magnetism, extension of the

APTS beyond the Miocene-Pliocene boundary has in some cases relied

on magnetic properties of sediments. For example, magnetic susceptibil-

ity variations have been used to astronomica lly calibrate the Oligocene

and Miocene (Shackleton et al., 1999) via correlation with calculated

records of variations in Earth's orbital geometry (Laskar et al., 1993)

(see Figure E39).

The APTS was developed by identifying the positions of magnetic

reversals in cyclically deposited marine sediments from the Mediterra-

nean Sea, in which organic-rich sapropel deposits provide key tie

points to astronomical target curves (Hilgen, 1991; Langereis and

Hilgen, 1991). Magnetic minerals are highly sensitive to diagenetic

alteration associated with degradation of sedimentary organic matter

during early burial, and variations in the burial of organic carbon has

produced cycles of nonsteady state diagenesis that has left a character-

istic signature in the magnetic properties of these sediments (e.g.,

Larrasoaña et al., 2003b). In many cases, sapropels that were origin-

ally present have been completely removed from the record by

postdepositional oxidation, yet magnetic properties can preserve the

Figure E38 Example of comparison of fine-scale paleomagnetic features based on inter-core correlation using low-field magnetic

susceptibility. (a) Correlation of whole-core magnetic susceptibility profiles for ODP holes 884B, 884C, and 884D in the vicinity of

a geomagnetic excursion (see Roberts et al., 1997). (b) Paleomagnetic inclination records for the geomagnetic excursion for the

characteristic remanent magnetization (ChRM) for discrete samples (holes 884B, C, and D) and for u-channel samples after alternating

field demagnetization at 25 mT (Hole 884D). Susceptibility correlation provides a good independent first-order constraint for tying

together different high-resolution paleomagnetic records of geomagnetic features.

258 ENVIRONMENTAL MAGNETISM, PALEOMAGNETIC APPLICATIONS

characteristic signal associated with the former presence of sapropels

(Larrasoaña et al., 2003b). Environmental magnetic analysis of sedi-

ments that have undergone orbitally controlled variations of nonsteady

state diagenesis can therefore enable development of astronomically

calibrated timescales with age tie points in addition to those that are

identifiable using standard lithological observations.

Testing the hypothesis of orbital forcing of the

geomagnetic field

Long, continuous records of vector variability of the geomagnetic field

are needed to understand long-term field evolution and to constrain

models of field generation. While it is straightforward to determine

the horizontal and vertical components of the paleomagnetic vector

in sediments, estimating the intensity of the ancient geomagnetic field

has often proved problematical (see Paleointensity, relative, in sediments).

The problem can be overcome in ideal sediments by normalizing the nat-

ural remanent magnetization (NRM) by an appropriate parameter that can

remove the effects of variation in magnetic grain size and magnetic

mineral concentration. Environmental magnetic parameters such as the

low-field magnetic susceptibility, the anhysteretic remanent magnetiza-

tion, and the isothermal remanent magnetization are routinely used to nor-

malize the NRM to determine the relative geomagnetic paleointensity.

It is accepted that the geomagnetic field is generated by dynamo

action within the Earth's electrically conducting fluid outer core (where

buoyancy-driven convection generates a self-sustaining dynamo). Cal-

culations indicate that any energization of the dynamo by external orbital

forcing is insufficient (by at least an order of magnitude) to control the

geomagnetic field (e.g., Rochester et al., 1975). Nevertheless, recent

recovery of long, continuous high-resolution vector paleomagnetic

records has provided the type of geomagnetic time series needed for test-

ing whether there is any relationship between the field and orbital forcing

parameters. Surprisingly, several paleomagnetic records of both the

intensity (e.g., Tauxe and Shackleton, 1994; Channell et al., 1998) and

direction (e.g., Yamazaki and Oda, 2002) of the field have resurrected

the suggestion that orbital energy could drive the geodynamo.

In the case of orbital periodicities embedded within relative paleo-

intensity records, periodograms of these records suggested that orbital

periodicities were not present in the environmental magnetic para-

meters used for relative paleointensity normalization. This was taken

to suggest that statistically significant signals at Earth orbital periods

in the relative paleointensity records indicate orbital energization of

the geomagnetic field (Channell et al., 1998). However, when Guyodo

et al. (2000) performed wavelet analysis on both the relative paleoin-

tensity signals and on the magnetic parameters used for normalization,

they found statistically significant periodicities in both the paleointen-

sity estimate and in the normalization parameter during the same

time intervals. Guyodo et al. (2000) concluded that, overall, there is

a subtle climatic contamination of the magnetic parameters that was

not removed in the normalization process. The claim of orbital modu-

lation of the geomagnetic field by Channell et al. (1998) was therefore

retracted.

In another intriguing case, Yamazaki and Oda (2002) reported a link

between the paleomagnetic inclination and orbital parameters

and explained it by using a model based on long-term regions of non-

dipole field behavior at the Earth's surface. Horng et al. (2003) studied

a core that spanned the same age interval from the same region of

nondipole field behavior and found no such relationship. Roberts

et al. (2003) used the data of Yamazaki and Oda (2002), and, while

they found the same spectral peaks, they demonstrated that these

peaks were not statistically significant. Even when it is considered

that sediments might record a forcing signal in a nonlinear manner,

comparison of the spectral signal of Yamazaki and Oda (2002)

with that of the orbital solution of Laskar et al. (1993) demonstrates

a lack of coherence between these signals (Figure E40a). In contrast,

a demonstrably orbitally controlled environmental magnetic signal

(Larrasoaña et al., 2003a) is convincingly coherent with orbital para-

meters (Figure E40b). Roberts et al. (2003) therefore concluded that

the hypothesis of (partial) orbital energization of the geomagnetic field

remains undemonstrated. Environmental magnetic signals have played

a fundamental role in the debate concerning this interesting paleomag-

netic application.

Figure E39 Example of how an environmental magnetic parameter (low-field magnetic susceptibility) can be used to calibrate

geological time. The example is from the Early Miocene in ODP Hole 929A from Ceara Rise (Shackleton et al. , 1999), with the

astronomical target curve being the eccentricity-tilt-precession (ETP) parameter that was extended from the data of Laskar et al.

(1993). Differences between the curves result from numerous factors (see Shackleton et al., 1999).

ENVIRONMENTAL MAGNETISM, PALEOMAGNETIC APPLICATIONS 259

Determining the origin of the paleomagnetic signal

Rock magnetic and environmental magnetic investigations constitute a

fundamentally important component of most modern paleomagnetic

studies. This is because the fidelity of a measured paleomagnetic signal

is directly related to the magnetic mineral(s) and the magnetic grain size

distribution of the magnetic mineral(s) within a rock. For example, it

is generally expected that an assemblage of detrital single-domain or

pseudosingle-domain magnetic grains in an undisturbed sediment will

faithfully record the geomagnetic field at or near the time of deposition.

On the other hand, similar sediments that are magnetically dominated by

multidomain grains would not be expected to faithfully record field

information dating from the time of deposition. To complicate matters,

paleomagnetically important minerals can authigenically grow within

some sediments. In such cases, constraining the time of magnetic

mineral growth is crucially important for interpretation of paleomag-

netic data. Two examples of authigenic iron sulfide minerals, greigite

(Fe

) and monoclinic pyrrhotite (Fe

), are given here to illustrate

the importance of rock magnetic and environmental magnetic studies

in support of paleomagnetic studies (see Iron sulfides).

Greigite grows within anoxic sedimentary environments that sup-

port active bacterial sulfate reduction. Dissolved sulfide forms within

pore waters (or within an anoxic water column), which then reacts

with detrital iron-bearing minerals, causing their progressive dissolu-

tion, resulting in the authigenic formation of pyrite (FeS

). Pyrite is

the end product of sulfidization reactions, with greigite forming as a

precursor to pyrite. Greigite can survive in sediments for long periods

of geological time and can retain a stable paleomagnetic signal. In

early studies involving greigite, it was interpreted to have formed dur-

ing early diagenesis within tens of centimeters of the sediment-water

interface (e.g., Tric et al., 1991; Roberts and Turner, 1993). Subse-

quently, however, sedimentary greigite has been found to frequently

carry a late diagenetic paleomagnetic signal that makes it a highly pro-

blematical mineral in many paleomagnetic applications, including mag-

netostratigraphy (e.g., Florindo and Sagnotti, 1995; Horng et al.,1998;

Sagnotti et al., 2005) and tectonics (e.g., Rowan and Roberts, 2005).

Paleomagnetic field tests, in particular the fold test, become extremely

important for constraining the timing of magnetizations carried by grei-

gite (e.g., Rowan and Roberts, 2005). The reversals test can be mislead-

ing and false polarity stratigraphies, resulting from patchy or variably

timed remagnetizations (e.g., Florindo and Sagnotti, 1995; Horng

et al., 1998; Rowan and Roberts, 2005; Sagnotti et al., 2005), can easily

go unidentified if careful rock magnetic and environmental magnetic

investigations are not conducted in support of paleomagnetic studies.

Similar to the case with greigite, it is often assumed in paleomag-

netic studies that pyrrhotite forms during early diagenesis and that it

can carry a stable syndepositional paleomagnetic signal. Geochemical

literature, however, indicates that the formation of pyrrhotite is kineti-

cally inhibited at ambient temperatures typical of early burial and that

pyrrhotite cannot form during early diagenesis (e.g., Schoonen and

Barnes, 1991; Lennie et al., 1995). This presents an apparent conun-

drum because Horng et al. (1998) reported that pyrrhotite carried an

identical paleomagnetic signature to detrital magnetite in sedimentary

rocks from Taiwan. Horng and Roberts (2006) carried out a source

to sink study in southwestern Taiwan and demonstrated that the pyr-

rhotite in the studied sediments is detrital in origin and that it is

sourced from metamorphic rocks in the Central Range of Taiwan.

While it is now recognized that pyrrhotite can carry a detrital magne-

tization (Horng and Roberts, 2006), and while it can also carry late

diagenetic remagnetizations (e.g., Jackson et al., 1993; Xu et al.,

1998; Weaver et al., 2002), pyrrhotite clearly cannot carry an early

diagenetic magnetization. Knowing which type of magnetization is

being carried by monoclinic pyrrhotite grains within a sediment is of

crucial importance in arriving at a correct paleomagnetic interpretation.

These two examples involving different magnetic iron sulfide minerals

demonstrate that the assumptions made in many paleomagnetic studies

are incorrect. They also make clear the need for careful rock magnetic

and environmental magnetic investigations in support of determining

the origin, nature, and age of the magnetization in paleomagnetic

studies.

Summary

While most environmental magnetic studies appropriately focus

on environmental interpretations, environmental magnetism routinely

plays an important role in a wide range of paleomagnetic applications.

These applications include correlation of paleomagnetic features in

suites of sediment cores, development of astronomically calibrated

age models, testing the hypothesis of orbital forcing of the geomag-

netic field, and determining the origin of the paleomagnetic signal.

Figure E40 Example of a test of the hypothesis of orbital forcing of

the geomagnetic field. (a) Coherency for the ETP parameter from

the data of Laskar et al. (1993) and the inclination record of

Yamazaki and Oda (2002). There is no statistically significant

coherency at the 100-ka period (or at other Milankovitch

periodicities). (b) Coherency for the ETP parameter from the data

of Laskar et al. (1993) and the Saharan dust record of Larrasoan

et al. (2003a) for the last 3 million years. Orbital forcing of the

dust record is clearly demonstrated, with statistically significant

coherency with the orbital solutions at the 99% confidence level

(see Roberts et al., 2003 for details). This comparison suggests that

the hypothesis of orbital forcing of the geomagnetic field is

undemonstrated.

260 ENVIRONMENTAL MAGNETISM, PALEOMAGNETIC APPLICATIONS

The importance of environmental magnetism to this range of

paleomagnetic applications therefore makes it fundamentally valuable

to paleomagnetism.

Andrew P. Roberts

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Cross-references

Environmental Magnetism

Iron Sulfides

Paleointensity Relative in Sediments

Paleomagnetic Secular Variation

ENVIRONMENTAL MAGNETISM, PALEOMAGNETIC APPLICATIONS 261

EQUILIBRATION OF MAGNETIC FIELD,

WEAK- AND STRONG-FIELD DYNAMOS

The ne cessity of equilibr ation of a grow ing

magnet ic field

It is enlightening to consider the theory of the Earth's dynamo as two

separate constituents: kinematic and dynamic problems. The kinematic

problem concerns conditions under which an electrically conducting

flow can produce a growing magnetic field. Alternatively, the kine-

matic dynamo may be regarded as an instability problem where a

purely hydrodynamic flow becomes unstable to infinitesimal distur-

bances of magnetic field. In contrast, the dynamic dynamo problem stu-

dies the equilibration of a kinematically growing magnetic field by

considering the magnetic feedback on the flow (for example, Bassom

and Gilbert, 1997).

In the kinematic dynamo problem, let u denote the velocity of non-

magnetic convection in the Earth 's fluid core at a given Rayleigh num-

ber R, while the magnetic field B is described by the dynamo equation

]

] t

B ¼r u  BðÞþlr

B ; (Eq. 1)

together with the condition

r B ¼ 0 : (Eq. 2)

In Eq. (1), l represents the magnetic diffusivity of the fluid. Equations (1)

and (2) admit a solution with the generated magnetic field of the form

B ðr ; t Þ¼ Bð rÞ e

s t

(E q. 3 )

together with an appropriate set of boundary conditions for B (M offa tt ,

1978). When U, the typical speed of the convective flow u,issmall,any

initial magnetic field would decay to zero as t !1ðReal½ s< 0Þ ;

when U is sufficiently large, Real½ s> 0 and an initial magnetic field

would grow without bound, i.e.,

j Bj!1; as t !1: (E q. 4 )

The critical threshold at the onset of dynamo action takes place at

some particular Rayleigh number, R

, when Real ½ s¼ 0 (for example,

Zhang and Busse, 1989). Physically speaking, however, the kinematically

growing magnetic field at R > R

must be brought to an equilibrium,

presumably by nonlinear effects.

Equilibr atio n: sub critical and supercr itical bifurcati on

Mathematically speaking, the equilibration of a kinematically growing

magnetic field is described by the momen tum equation in a rotating

fluid which may be written as

2 O  u ¼

r p þ

ðr BÞ B þ f ; (Eq. 5)

where p is the pressure, m is the magnetic permeability, r is the fluid den-

sity, f represents forces such as buoyancy, and O is the angular velocity of

the Earth. For an infinitesimal magnetic field, the magnetic (Lorentz) force

ð 1= rmÞðr BÞ B is too weak to alter the convective flow u and two

problems, one pertaining to nonmagnetic convection and the other to kine-

matic dynamo, are nearly decoupled. A growing magnetic field ultimately

leads to a sufficiently large magnetic force which suppresses the convec-

tive flow and brings the generated magnetic field to an equilibration.

The equilibration of a kinematically growing magnetic field is

associated with a bifurcation from the corresponding nonmagnetic

convective state. There exist at least two different sequences of the

bifurcation, which are sketched in Figure E41 . In the first possible

sequence, the linear convective instability occurs at R ¼ R

. When the

system is driven harder at larger R, the typical amplitude U of nonmag-

netic convection increases. Dynamo action takes place at R ¼ R

when

convection is vigorous enough. The equilibration of the generated

magnetic field is achieved when R > R

as a supercritical bifurcation.

In this case, larger values of the Rayleigh number R are required to

power the convection-driven dynamo with extra magnetic dissipation.

The second possible sequence of bifurcation is subcri tical (or back-

ward), which is also shown in Figure E41. In comparison to a super-

critical bifurcation, the nature of equilibration is quite different,

which is perhaps more relevant to a rapidly rotating system like the

Earth's core dynamo. When R is slightly smaller than R

, the nonmag-

netic convective system is linearly stable but is unstable to finite-ampli-

tude perturbations of a magnetic field. In consequence, the backward

branch of the bifurcation (the dashed curve for R < R

in Figure E41)

is unstable and cannot be physically realized. By implication, the Lor-

entz force resulting from dynamo action has a destabilizing effect which

relaxes the inhibiting effect of rapid rotation on nonmagnetic convec-

tion. When R is close to R

, the amplitude of the generated magnetic

field suddenly jumps and the equilibrated magnetic field is characterized

by a large amplitude. In this case, the usefulness of the kinematic study

determining the onset of dynamo action is highly limited.

Eq uilibra tion: a-quench ing in turbul ent dynamos

The equilibration of magnetic field in turbulent dynamos can be mod-

eled through a mechanism called a-quenc hing. In this process we

separate the generated magnetic field B into large-scale axisymmetric

( B

) and small-scale nonaxisymmetric (

B ) parts

B ¼ B

B; (Eq. 6)

where B

is generated by an a -effect in connection with interactions

between small-scale turbulent flows and magnetic fields (Moffatt,

1978). The strength of the a-effect would be suppressed when the

back-reaction of the generated magnetic field on the flow is signifi-

cantly strong. This reduced a would prevent the unlimited growth of

the mean magnetic field B

and bring it to an equilibration. This

process is usually described by

Figure E41 Conjectured structure of bifurcation diagram for the

equilibration of convection-driven dynamos. Here R

is the

critical Rayleigh number for the onset of conve ction and R

is the

another critical Rayleigh number at which dynamo action takes

place according to kinematic dynamo theory. Both supercritical

(the solid line) and subcritical dynamo (the dashed line)

bifurcations are shown.

262 EQUILIBRATION OF MAGNETIC FIELD, WEAK- AND STRONG-FIELD DYNAMOS

]

] t

¼r

a B

ð 1 þ q jB



 l

rrB

; (Eq. 7)

where l

is a turbulent magnetic diffusivity, q > 0 and n often takes

the value 2 (Jepps, 1975; Roberts and Soward, 1992). The a -quenching

formulation has been widely used in dynamo simulations and one of

its major advantages is that it can simulate the equilibration of the ge-

nerated magnetic field and the essential dynamo processes without

reference to the difficult dynamics of strong nonlinear interaction

between the flow and the Lorentz force described by the momentum

equation (5). The corresponding bifurcation in the a-quenc hing dynamo

is typically supercritical because the dynamic effect of rotation is not

taken into account.

Weak- and strong -field dyn amos

In rapidly rotating systems like the Earth's fluid core, magnetohydro-

dynamic dynamos may be divided into two different categories:

weak-field dynamos and strong-field dynamos (for example, Gubbins

and Roberts, 1987; Zhang and Schubert, 2000). They are usually

defined on the basis of the relative importance of the Lorentz and Cor-

iolis forces in a convection-dri ven dynamo. In the case of weak-field

dynamos, the dynamic role of the Lorentz force plays an insignificant

role and the viscous force must be sufficiently strong to offset the part

of the Coriolis force that cannot be balanced by the pressure gradient.

In the case of strong-field dynamos, the dynamic role of viscosity is

taken over by the Lorentz force.

A widely used criterion for the distinction between weak-field and

strong-field dynamos is associated with the Elsasser number L defined as

L 

2 jOj mrl

;

where B

is an average or a representative value of the amplitude of

the magnetic field generated by a dynamo. Equations (1) and (5) yield

an estimate for the relative importance of the Lorentz and Coriolis

forces

jð mrÞ

 1

ðr BÞB j

j 2 O  u j

¼ Oð LÞ; (Eq. 8)

where inertial and viscous terms are neglected. In estimating the value

of the Elsasser number L, we often take B

as a spatially or temporally

averaged magnetic field. A strong-field dynamo is one in which

L  Oð1 Þ while a weak-field dynamo is one in which L < Oð 1Þ .

However, it should be pointed out that this definition is not entirely

satisfactory because a large Elsasser number does not necessarily

imply a strong interaction between the convective flow and the gener-

ated magnetic field.

An alternative criterion for the distinction between strong- and

weak-field dynamos is based on the dynamic effect of the generated

magnetic field. A strong-field dynamo is defined as one in which visc-

osity is dynamically unimportant so that the scale of convection is

large as a direct consequence of the dynamic effect of the generated

magnetic field. In other words, in a strong-field dynamo the structure

and scale of the convective flow are highly sensitive to variations in

the generated magnetic field. Similarly, we may define a weak-field

dynamo as one in which the dynamic effect of the generated magnetic

field can be treated as a perturbation. In other words, in a weak-field

dynamo the structure and scale of the convective flow are controlled

by the effects of viscosity and rotation and are insensitive to variations

in the generated magnetic field. When one turns off dynamo action,

convective motions remain largely unchanged for a weak-field dynamo

but change almost completely for a strong-field dynamo.

Keke Zhang

Bibliography

Bassom, A.P., and Gilbert, A.D., 1997. Nonlinear equilibration of a

dynamo in a smooth helical flow. Journal of Fluid Mechanics,

343: 375–406.

Gubbins, D., and Roberts, P.H. 1987. Magnetohydrodynamics of the

Earth's core. In Jacobs, J.A. (ed.), Geomagnetism, Vol. 2.

London: Academic Press, pp. 1–183.

Jepps, S.A., 1975. Numerical models of hydrodynamic dynamos.

Journal of Fluid Mechanics, 67: 625–646.

Moffatt, H.K., 1978. Magnetic Field Generation in Electrically Con-

ducting Fluids. Cambridge: Cambridge University Press.

Roberts, P.H., and Soward, A.M., 1992. Dynamo theory. Annual

Review of Fluid Mechanics, 24 : 459–512.

Zhang, K., and Busse, F., 1989. Convection driven magnetohydrody-

namic dynamos in rotating spherical shells. Geophysical and Astro-

physical Fluid Dynamics, 49:97–116.

Zhang, K., and Schubert, G., 2000. Magnetohydrodynamics in rapidly

rotating spherical systems. Ann. Rev. Fluid Mech, 32: 409–443.

Cross-references

Core Motions

Core, Magnetic Instabilities

Geodynamo

Magnetoconvection

EULER DECONVOLUTION

Introduction

Magnetic surveys are widely used in mineral and oil exploration to

delineate geological structure and locate targets of interest (see Mag-

netic anomalies for geology and resources). They are also used in

environmental work to locate buried metal objects such as unexploded

ordnance, drums, pipework, and the like. In all of these applications,

we need a quick means of turning magnetic field measurements into

estimates of magnetic source body location and depth. Euler

deconvolution is one such method. Others include Werner decon-

volution (Werner, 1955; Hartman et al., 1971) and the Naudy (1971)

method.

The method may be applied to survey profile data or gridded data. It

operates on a data subset extracted using a moving window. In each

window, Euler's equation (see below) is solved for source body coor-

dinates. The source body geometry is specified by a structural index

(SI) ranging between 0 (contact of great depth extent) and 3 (dipole

or sphere). Possible source bodies and their structural indices are tabu-

lated below. Intermediate bodies have noninteger SIs and the Euler for-

mulation is only approximate. The method does not require any

assumptions about direction of exciting field, direction of magnetiza-

tion, or the relative magnitudes of induced and remanent magneti-

zation of the sources. But it appears to work best on data after

reduction to pole (q.v.).

History

Hood (1963) proposed that Euler's equation might be used with verti-

cal magnetic gradient survey measurements to determine source depth.

Ruddock et al. (1966) were awarded a US patent describing a graphi-

cal method that exploited Euler's equation to determine source depth

from magnetic gradient measurements. Thompson (1982) described a

fully developed profile method (called EULDPH). It was widely used

within the Gulf and Chevron Oil Companies and also by Durrheim

(1983) and Corner and Wilsher (1989) to delineate the magnetic mar-

kers in the Witwatersrand Basin in the search for gold. Thompson also

developed the theory for the case of gridded data. Reid et al. (1990)

EULER DECONVOLUTION 263

implemented Thompson's method for gridded data and extended the

theory to the case of zero structural index. They also coined the term

“Euler deconvolution” by analogy with the established Werner decon-

volution technique.

Theory

Strictly speaking, Euler deconvolution is only valid for homogeneous

functions. A function f (v) of a set of variables v ¼ (v

, v

,...) is homo-

geneous of degree n,if

f ðtvÞ¼t

f ðvÞ (Eq. 1)

where t is a real number, and n is an integer. This is really a statement

of the scaling properties of f (v ). If f has a differential at v, then

f ðvÞ¼nf ðvÞ: (Eq. 2)

This is Euler's equation, and Euler deconvolution solves it in appropri-

ate cases.

If a field F can be expressed in the form

F ¼ A=r

(Eq. 3)

F will be homogeneous of degree –n. For convenience we define the

SI as N (¼n, i.e., the negative degree of homogeneity).

Thompson (1982) showed that Euler's equation could usefully be

written in the form

ðx  x

ÞqT=qx þðy  y

ÞqT=qy þðz  z

ÞqT=qz ¼ NðB  TÞ;

(Eq. 4)

where (x

, y

, z

) is the position of a magnetic source whose total field

T is detected at (x, y, z). The total field has a regional value of B. N is

the structural index.

Strictly speaking, Euler deconvolution is only valid for homoge-

neous fields. These are only obtained from particular idealized bodies

(see Table E2) with integer structural index. The fields of most real

bodies do not meet this strict criterion and the method is therefore an

approximation in practice. Nevertheless, it has been found useful

enough to be applied in a wide variety of cases.

Implementation

The process is implemented as follows.

1. Interpolate line data to a constant spacing, or multiline or point data

to a uniform grid. Use an interpolation interval which represents the

data well but does not overinterpolate. The interval should be no

more than the depth of the shallowest magnetic sources thought

to exist in the area.

2. Calculate along line (x) and vertical (z) gradients for profile data

and x, y, and z gradients for gridded data, using finite difference

or Fourier methods. If measured gradient are available, interpolate

them to the same pattern as the field data.

3. Choose a suitable subset size or “window.” Grid windows are

normally square. The window length should be greater than half

the depth to the deepest magnetic sources sought. Since the method

assumes that any window contains effects from only one source

body, good resolution requires windows to be as small as possible.

Typical grid windows might be 7  7to10 10 grid points. If

necessary, for deep sources, it may be necessary to revisit 1

above, apply an anti-alias low-pass filter and desample the profile

or grid.

4. Move the sample window through the profile or grid, using

small overlapping steps. At each step, use all the data points in

the current window to solve Eq. (4) above for x

, y

, z

and T

for whatever values of structural index N seem relevant to the

situation.

The method generates a solution for each window position, regardless

of the presence (or lack) of any nearby magnetic source. In addition,

the method effectively assumes that the effects of only one source

are present in any window. Interference effects give rise to festoons

of misleading solutions trailing away from valid solutions that cluster

along source body edges. Various strategies have been devised to elim-

inate unhelpful solutions. They include:

1. Eliminate solutions with high estimation uncertainties (often nor-

malized by depth).

2. Eliminate solutions from poorly conditioned matrices.

3. Identify and select for clusters of solutions.

4. Only solve in windows showing positive field curvature (Fairhead

et al., 1994).

5. Eliminate solutions that fall beyond their data window.

Recent developments and ongoing research

Stavrev (1997) has applied differential similarity transforms to Euler

deconvolution. This unconventional approach shows promise of pro-

gress in the area of multiple interfering sources. Barbosa et al.

(1999) developed a means of determining the structural index for an

isolated source body by correlating the observed field with the derived

“background” field. Mushayandebvu et al. (2001, 2003) added rota-

tional invariance to the scaling rule and found it possible to solve for

strike, dip, and magnetization contrast or the magnetization-thickness

product. Hansen and Suciu (2002) have developed a formulation for

multiple sources. But all sources in one solution run must have the

same structural index Nabighian and Hansen (2001) applied a Hilbert

transform as well as rotational invariance and have found it possible to

solve for structural index and dip. They also showed that this form of

Euler deconvolution is equivalent to Werner deconvolution (see

above).

The following problems are unsolved (this list is almost certainly

incomplete):

1. Develop a reliable and robust means of solution selection.

2. Pursue the application of seismic migration approaches to collap-

sing the festoon (spray) solutions onto their primary points.

3. Establish the bounds of applicability of the method to real data by

working on realistic models based on the real Earth.

4. Further explore the potential of differential similarity transforms as

applied to Euler deconvolution. They appear to offer scope for

advance in multiple-source analysis.

Alan B. Reid

Table E2 Possible source bodies and their structural indices

Source body Magnetic structural index

Dipole (sphere, small, compact

body)

Thin pipe (vertical—kimberlite;

horizontal—pipeline); thin bed

fault

Thin sheet edge (dyke, sill) 1

Contact with depth extent  depth 0 (requires modified equation)

264 EULER DECONVOLUTION

Bibliography

Barbosa, V.C.F., Silva, J.B.C., and Medeiros, W.E., 1999. Stability

analysis and improvement of structural index estimation in Euler

deconvolution. Geophysics, 64:48–60.

Corner, B., and Wilsher, W.A., 1989. Structure of the Witwatersrand

Basin derived from interpretation of the aeromagnetic and gravity

data. In Garland, G.D. (ed.), Proceedings of Exploration ‘87: Third

Decennial International Conference on Geophysical and Geo-

chemical Exploration for Minerals and Groundwater. Ontario

Geological Survey, Special Volume 3, pp. 532–546.

Durrheim, R.J.,1983. Regional-residual separation and automatic inter-

pretation of aeromagnetic data. Unpublished MSc thesis, Univer-

sity of Pretoria.

Fairhead, J.D., Bennett, K.J., Gordon, D.R.H., and Huang, D., 1994.

Euler: beyond the “black box” .In64th Annual International Meet-

ing. Los Angeles, CA: Society of Exploration Geophysicists,

Expanded Abstracts, pp. 422–424.

Hansen, R.O., and Suciu, L., 2002. Multiple-source Euler deconvolu-

tion. Geophysics, 67: 525–535.

Hartman, R.R., Teskey, D.J., and Friedberg, J.J., 1971. A system for

rapid digital aeromagnetic interpretation. Geophysics, 36: 891– 918.

Hood, P.J., 1963. Gradient measurements in aeromagnetic surveying.

Geophysics, 30: 891–902.

Mushayandebvu, M.F., van Driel, P., Reid, A.B., and Fairhead, J.D.,

2001. Magnetic source parameters of two-dimensional structures

using extended Euler deconvolution. Geophysics, 66: 814–823.

Mushayandebvu, M.F., Lesur, V., Reid, A.B., and Fairhead, J.D.,

2003. Grid Euler deconvolution with constraints for two-dimen-

sional structures. Geophysics, 69: 489–496.

Nabighian, M.N., and Hansen, R.O., 2001. Unification of Euler and

Werner deconvolution in three-dimensions via the generalized Hil-

bert transform. Geophysics, 66: 1805–1810.

Naudy, H., 1971. Automatic determination of depth on aeromagnetic

profiles. Geophysics, 36: 717–722.

Reid, A.B., Allsop, J.M., Granser, H., Millett, A.J., and Somerton,

I.W., 1990. Magnetic interpretation in three dimensions using Euler

deconvolution. Geophysics, 55:80–91.

Ruddock, K.A., Slack, H.A., and Breiner, S., 1966. Method for deter-

mining depth and falloff rate of subterranean magnetic disturbances

utilising a plurality of magnetometers. US Patent 3,263,161, filed

March 26, 1963, awarded July 26, 1966, assigned to Varian

Associates and Pure Oil Company.

Stavrev, P.Y., 1997. Euler deconvolution using differential similarity

transformations of gravity or magnetic anomalies. Geophysical

Prospecting, 45: 207–246.

Thompson, D.T., 1982. EULDPH—a new technique for making com-

puter-assisted depth estimates from magnetic data. Geophysics, 47:

31–37.

Werner, S., 1955. Interpretation of magnetic anomalies at sheet-like

bodies: Sveriges Geologiska Undersökn ing. Årsbok 43(1949),

No. 6, Series C, No. 508.

Cross-references

Magnetic Anomalies for Geology and Resources

Reduction to Pole

EULER DECONVOLUTION 265