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

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chart was published by Halley in 1700, and was entitled “ A New and

Correct CHART Shewing the Variations of the Compass in the Western

and Southern Oceans as Observed in y

Year 1700 by his Maties Com-

mand by Edmund Halley. ” This map was the first published chart to

include isolines of declinations (called for a while “ Halleyan lines, ” until

Hanssen used the term “isogone” for a contour line of equal declination).

This map became widely used as it allowed navigators to estimate

declination variation when they could not get sight of an astronomical

body (see Clark (2000) for more details).

William Whiston produced the first known charts showing the

direction of magnetic inclination for south-eastern England, in 1719

and 1720. The contours are straight lines and are thus a very smooth

representation of the distribution of inclination (see Howarth (2003)

for more details). In 1768, Wilcke published the first approximately

global inclination map, based on observations made during sea

voyages.

The first chart of magnetic field intensity, covering the Northern

part of South America, was produced by Alexander von Humboldt

(Humboldt and Biot, 1804). On this map the geographic and magnetic

equators are indicated, and five “ isodynamic zones” are defined by the

average number of oscillations per minute of the magnetized needle

used during Humboldt ’s expeditions (Malin and Barraclough, 1991).

To quote Hellman (1909), the first “true ” magnetic field intensity

map was produced by Hansteen (1824) for a region of the Northern

part of Europe. The intensity values shown on this chart are in relative

units (“ Humboldt units ” ), as it was produced before Gauss ’ demonstra-

tion on how to measure the magnetic field in absolute units.

Offici al magnet ic maps

The naval authorities from different countries have published world

magnetic charts for different epochs. The first official chart was of

declination for the epoch 1858, produced by F.J. Evans, under the

superintendence of the hydrographer of the Royal Navy. The first

German chart was published in 1880, with the United States following

in 1882. Other countries, such as France and Spain, have published

copies of the British charts with titles and text translated into the

appropriate language. Hellmann (1909) gives details of the earlier

examples of these charts.

Present-day maps are produced from different geomagnetic field

models for the main field and its secular variation (see Main field mod-

eling ; Geomagnetic secular variation ). Geomagnetic models such as

the World Magnetic Model (WMM) and the Interna tional Geomagnetic

Reference Field (IGRF) only predict the values of that portion of the field

originating in the fluid outer core. In this respect, they are accurate to

within 1



for 5 years into the future, after which they need to be updated.

Maps produced from these models are used widely in civilian navigation

systems.

World Magnetic Model (WMM)

The WMM is a product of the United States National Geospatial-

Intelligence Agency (NGA). The US Nation al Geophysical Data

Center (NGDC) and the British Geological Survey (BGS) produce the

WMM as the standard model for navigation and attitude/heading refer-

encing systems for the US Department of Defense, the UK Ministry of

Defence, the North Atlantic Treaty Organization (NATO), and the

World Hydrographic Office (WHO). The model, associated software,

and documentation are distributed by NGDC on behalf of NGA and

used widely in civilian navigation systems. The model is produced at

5-year intervals, with the current model, WMM-2005, expiring on

December 31, 2009 (McLean et al., 2004).

Internat ional Geom agnetic Refer ence Field (IGRF)

Production of the IGRF is an international collaborative effort relying

on cooperation between magnetic field modelers, institutes, and

agencies responsible for collecting and publishing geomagnetic field

data (see IGRF, International Geomagnetic Reference Field ). The first

model, designated IGRF 1965, was of spherical harmonic degree and

order 8, in both main field and secular variation (Langel, 1987). The

latest versio n of the model— the 10th generation —includes models

of the core field at 5-year intervals from 1900 to 2005, and a secular

variation model for 2005 – 2010. Magnetic charts based on different

generations of IGRF are available on several Web sites, including

http://swdcw ww.kugi.kyoto-u.ac.jp/igrf/; http://www.geomag.bgs.ac.

uk/navi gation.html.

Map s acc uracy

The WMM and IGRF/DGRF geomagnetic models, and the charts pro-

duced from these models characterize only that portion of the Earth ’s

magnetic field that is generated in the Earth ’s fluid outer core (the

main magnetic field). The portions of the geomagne tic field generated

by the Earth ’ s upper mantle and crust, and by the ionosphere and mag-

netosphere, are not represented (see Internal external field separat ion).

Consequently, a magnetic sensor such as a compass or magnetometer

may observe spatial and temporal magnetic anomalies when refer-

enced to these models. In particular, certain local, regional, and tem-

poral magnetic declination anomalies can exceed 10



. Anomalies of

this magnitude are not common but they do exist. Local anomalies

distort the WMM or IGRF predictions, and frequently induce an error

of 3 –4



(ferromagnetic ore deposits; geological features, particularly

of volcanic origin, such as faults and lava beds; topographical features

such as ridges, trenches, seamounts, and mountains; ground that

has been hit by lightning and possibly harboring fulgurites; cultural

features such as power lines, pipes, rails, and buildings; personal items

such as crampons, ice axe, stove, steel watch, hematite ring, or even a

belt buckle).

Mai n field an d secular vari ation maps at the

Ea rth’s surface

To create an accurate main field model, data with good global cover-

age and as low a noise level as possible are needed. The Danish Ørsted

and German CHAMP satellite data sets satisfy these requirements.

Both satellites provide high-quality vector and scalar data at all lati-

tudes and longitudes, but not during all time periods needed for mod-

eling. These satellite data were augmented with groun d observatory

data, which have been available almost continuously over the period

of interest, although with poorer spatial coverage. Used together, satel-

lite and observatory data provide an exceptional quality data set for

modeling the behavior of the main magnetic field in space and time,

and for producing main field magnetic maps for a given epoch. Such

maps at the Earth’s surface field were drawn using a main field model

based on data from the Ørsted magnetic field satellite (Neubert et al.,

2001). For the main field and its secular variation representation some

other models are available, derived from Ørsted, CHAMP and Ørsted-

2/SAC-C measurements, such as IDEMM, CO2, OSVM and OIFM

(see http://www.dsri.dk/Oersted/Field_models/ for more details).

Among the multitude of geomagnetic models based on satellite

data, POMME, derived from Ørsted and CHAMP data, is used here

to draw maps for three field components. This model provides a

representation of the strongest contributions to the geomagnetic field:

the time-varying core field, the ring current field modulated by the

Dst index, a time-averaged magnetospheric field, the penetration of

the horizontal part of the Interplanetary Magnetic Field (IMF), and

the fields induced by Earth’s rotation in the external fields (Maus

et al., 2004). From POMME coefficients, synthetic data for the main

field components are computed for the whole Earth’s surface, on a grid

of 30

 30

. Each set of synthetic data is then interpolated and plotted

as maps, generally on global scale. In Figures M174-M176 maps of

declination, inclination, and total intensity for the epoch 2002.5, and

their secular variations, derived from the POMME model are shown.

The isolines on the declination map (isogones) show the distribution

676 MAIN FIELD MAPS

Figure M174 Contour maps of declination (degrees) at 2002.5 and secular variation of declination (arc-min y

1

), at the Earth’s surface,

computed from POMME model (Maus et al., 2004).

Figure M175 Contour maps of inclination (degrees) at 2002.5 and secular variation of inclination (arc-min y

1

), at the Earth’s surface,

computed from POMME model (Maus et al., 2004).

MAIN FIELD MAPS 677

of the declination (or of the compass needle eastward or westward from

the true north), the isolines on the inclination map (isoclines) show the

distribution of the inclination (angle between the geomagnetic field

vector and its horizontal projection), and the isolines on the total field

intensity map (isodynamics) show the distribution of the total geomag-

netic field (the scalar value of the geomagnetic field vector). The lines

of equal secular variation (isopods) are used for dates outside the epoch

for which the main field charts have been drawn.

Unfortunately, the same potential theory that allows us to generate

these maps also tells us that, formally, little more about the origin of

the field can be said: the field certainly originates within the Earth,

but where in the Earth cannot be distinguished. However, by looking

at the structure of the field—and making assumptions about its

sources—further inferences can be made. The spectrum of the field

has clearly two parts, first a rapidly declining part down to wave-

lengths of approximately 3000 km, with a much more gentle decline

at shorter wavelengths (see Geomagnetic spectrum, spatial). The pre-

sented maps reflect only the first part of the spectrum corresponding to

the core field. The lithospheric contributions could be included in

extended models, expanded to high degrees. The crustal field is almost

constant in time, which can be inferred from all available marine, aero-

magnetic, and high-resolution Ørsted, CHAMP, and future satellite

data, measured at all times. However, these extended models and maps

would differ significantly in format and form from the current main

field models and maps.

Mioara Mandea

Bibliography

Barraclough, D.R., 2000. Four hundred years in geomagnetic field

charting and modelling. In Schröder, W. (ed.), Geomagnetism.

Research Past and Present. Bremen-Roennebeck: IAGA, pp.

93–111.

Clark, T.D.G., 2000. Edmond Halley’s voyages in the Paramore and

the first isogonic chart of the Earth’s magnetic field. In Schröder,

W. (ed.), Geomagnetism. Research Past and Present. Bremen-

Roennebeck: IAGA, pp. 61–71.

Hansteen, C., 1824. Magnetiske Intensitets-Iagttagelser, Anstillede

paa forskjellige Reiser i den nordlige Deel af Europa. Magazin

for Naturvidenskaberne, 4: 268–316.

Hellmann, G., 1909. Magnetische Kartographie in historisch-kritischer

Darstellung. Abhandlungen des Königlichen Preussischen meteor-

ologischen Instituts 3: 61.

Howarth, R.J., 2003. Fitting geomagnetic fields before the invention of

least squares: II. William Whiston’s isoclinic maps of Southern

England (1719 and 1721). Annals of Science, 60:63–84.

Humboldt, A., von and Biot, J.-B., 1804. Sur les variations du magné-

tisme terrestre à différentes latitudes. Lu par M.Biot à la clsse des

sciences mathématiques et physiques de l’Institut National, le 26

frimaire an 13, pp. 24.

Langel, R.A., 1987. Main field. In Jacobs, J.A. (ed.), Geomagnetism.

London: Academic Press, pp. 249–512.

Mandea, M., 2000. French magnetic observation and the theory at the

time of DE MAGNETE. In Schröder, W. (ed.), Geomagnetism.

Research Past and Present. Bremen-Roennebeck: IAGA, pp.

73–80.

Maus, S., Lühr, H., Balasis, G., Rother, M., and Mandea, M., 2004.

Introducing POMME, the POtsdam magnetic model of the Earth

in CHAMP. In Earth Observation with CHAMP, Results from

Three Years in Orbit. Berlin-Heidelberg: Springer, pp. 293–298.

McLean, S., Macmillan, S., Maus, S., Lesur, V., Thomson, A., and

Dater, D., 2004. The US/UK World Magnetic Model for 2005–

2010, NOAA Technical Report NESDIS-NGDC-1.

Figure M176 Contour maps of total intensity (nT) at 2002.5 and secular variation of total intensity (nT y

1

), at the Earth’s surface,

computed from POMME model (Maus et al., 2004).

678 MAIN FIELD MAPS

Mitchell, A.C., 1937. Chapters in the history of terrestrial magnetism.

II. The discovery of the magnetic declination. Terrestrial Magnet-

ism and Atmospheric Electricity, 42: 241–280.

Nautonnier, G., 1602–1604. Mecometrie de leymant, c’est a dire La

maniere de mesvrer les longitudes par le moyen de l’eymant.

Venes: ches l’autheur, 343 pp.

Neubert, T., Mandea, M., Hulot, G., von Frese, R., Primdahl, F.,

Jorgenson, J.L., Friis-Christensen, E., Stauning, P., Olsen, N., and

Risbo, T., 2001. High-precision geomagnetic field data from the

Ørsted satellite. EOS, 82:81–87.

Ultré-Guérard P., and Mandea, M., 2000. Declination and longitude in

France in the early 17th century. In Schröder, W. (ed.), Geomagnet-

ism. Research Past and Present. Bremen-Roennebeck: IAGA,

pp. 81–92.

Wright, E., 1610. Certaine errors in Navigation, detected and cor-

rected, Printed by Feelix Kingstrõ. London, 354 pp.

Cross-references

Geomagnetic Secular Variation

Geomagnetic Spectrum, Spatial

Geomagnetism, History of

IGRF, International Geomagnetic Reference Field

Internal External Field Separation

Main Field Modeling

MAIN FIELD MODELING

The history of modeling the geomagnetic field stretches back four

centuries, to William Gilbert’s De Magnete, the first book on geo-

magnetism. His assertion, that the Earth is a great magnet, can be

regarded as the first model of the geomagnetic field. The development

of geomagnetic field models is explained by the need to draw magnetic

maps. The practical reasons were mainly connected with navigation:

the use of the magnetic compass and portable sundials that included a

compass needle, the problem of determining longitude at sea, and also

the construction of road maps on land. These practical applications

in turn provided the motivation to seek a more fundamental understand-

ing of the geomagnetic field.

Before modeling the geomagnetic field, some fundamental proper-

ties of the field had to be discovered, like the existence of magnetic

variation or declination, the spatial variation of declination, the exis-

tence of magnetic inclination or dip, and the secular variation. The

history and definitions of these fundamental discoveries are given else-

where. Here, some different ways of modeling the geomagnetic field

since William Gilbert’s first attempt in 1600 are given. A milestone

in main field modeling was achieved in 1839, when Gauss published

in his famous description of the geomagnetic field using spherical

harmonics.

Empirical models: dipoles and magnetized spheres

The first geomagnetic field model was a physical one, the terrella built

out of lodestone and used by Gilbert in his experiments. He investi-

gated the behavior of very small magnetized iron needles on the

surface of such a sphere, and compared the results with what was

known about the magnetic field from direct measurements made with

compass needles and dip needles. Gilbert believed that the positions of

the magnetic poles of the Earth were the same as those of the rotation

poles. He also believed that declination, which with his assumption

would have been zero everywhere, was caused by the effects of the

continents, whose magnetic rocks attracted the needles toward them.

In his experiments Gilbert modeled this effect by using an irregularly

shaped sphere, and he concluded that “the variation in any one place

is constant.”

At about the same time that Gilbert wrote his famous book, a French

cartographer, Nautonnier de Castelfranc came to a similar conclusion

concerning the terrestrial source of the geomagne tic field (Mandea,

2000). However, Nautonnier believed that the positions of the magnetic

and geographic poles were not coincident, and this property explained

the phenomenon of declination. Nautonnier produced the first known

chart showing magnetic poles, magnetic meridians, and the magnetic

equator.

The first attempt to model inclination was published by Edward

Wright in 1610, who gave an empirical expression, obtained from a

geometrical construction, apparently devised to give 0



at the equator,



at the pole, and an acceptable value of dip for the position of

London (note that his tables of inclination were not very accurate for

locations other than London). Wright’s model did not take into

account the secular variation. This was first addressed some years

later, when Henry Bond attempted to take into account the secular var-

iation by using a tilted dipole model, with the magnetic poles moving

around the geographical poles (Barraclough, 2000).

Halley’s theory of the Earth’s magnetic field has been lauded as a

great achievement. In his first paper, Halley used observational data

to examine previous theories (Barraclough, 2000). Finding the theories

lacking in certain respects,Halleyproposed a new model,which explained

the observations qualitatively. Having presented his data, and discussed

the shortcomings of previous theories, Halley turned to his own hypo-

thesis. He suggested that the pattern of declination could be explained

by four magnetic poles, roughly situated at: N1 ð75



N; 129



WÞ;

N2 ð83



N; 6



WÞ;S1ð70



S; 120



EÞ;S2ð74



S; 95



WÞ. He had no means

to quantitatively describe the effects that these poles would have, but he

demonstrated that they did qualitatively explain the pattern of variation

seen in his data. Nine years later he further developed these ideas

by proposing a geomagnetic field model which could incorporate the

secular variation (Halley, 1692). He suggested that the Earth consisted

of an outer shell with poles corresponding to N1 and S1, and an inner

shell with poles corresponding to N2 and S2. The inner shell rotated

with respect to the outer one, and consequently these poles would shift

with time and result in a secular change in declination. At least in part

to provide an observational test of this model, he undertook an exten-

sive survey (accomplished in his three voyages in the Atlantic Ocean

between 1698 and 1701) to remedy the lack of data in ocean areas.

Other priorities evidently prevented him from using these new data to

refine his model, but he used them to produce magnetic charts, far more

useful and important than his four-pole model.

In 1757, Euler introduced a new model, in which the two magnetic

poles were not antipodal (what would today be called an eccentric

dipole model). Using a method of successive approximations he found

the positions of these poles for 1757. During the 19th century, Halley’s

four-pole model was not forgotten. Hansteen collected as many mea-

surements of declination and inclination as he could find and used

them to derive a model with four poles, including their changes in time

(Barraclough, 2000). Although the concept of modeling the geo-

magnetic field using two magnetic axes and four magnetic poles is

today only of historical interest, the data collected and published by

Hansteen have proved to be very valuable in studying how the field

has evolved with time (see Geomagnetism, history of ).

In 1832, Carl Friedrich Gauss and Wilhelm Weber began investigat-

ing the theory of terrestrial magnetism after Alexander von Humboldt

attempted to obtain Gauss’s assistance in making magnetic observation

on a grid of points around the Earth. Gauss was excited by this pro-

spect and by 1840 he had written three important papers on the sub-

ject, all of which dealt with the current theories on terrestrial

magnetism, including Poisson’s equation, the potential theory equation

(in 1812 Poisson discovered that Laplace’s equation is valid only

outside of a solid), and the absolute measure for magnetic force.

In Allgemeine Theorie des Erdmagnetismus (1839) Gauss showed

that there can only be two poles in the globe and used the Laplace

equation to aid him with his magnetic field calculations (see Gauss,

Carl Friedrich).

MAIN FIELD MODELING 679

Mathematical models: spherical harmonic analysis

The Earth’s magnetic field is continuously changing, in such a way

that it is impossible to predict accurately what the field will be at

any point in the very distant future (for example, declination is accu-

rately described by geomagnetic models to within 1



for 5 years).

By measuring the magnetic field, its spatial variation and changes over

a period of years, can be observed. This structure and variation can

most conveniently be expressed in terms of spherical harmonics (see

Harmonics, spherical). A review of the technique’ s development can

be also found in Langel (1987) or Merrill et al. (1996).

The two Maxwell equations relating to the magnetic field are:

r

H ¼

J þ

]

(Eq. 1)

r

B ¼ 0 (Eq. 2)

where

H is the magnetic field,

B is the magnetic induction,

J is the

current density, and ]

D=]t is the electric displacement current density.

In a region that does not contain sources of magnetic field (from the

Earth’s surface up to about 50 km), it is reasonable to assume that

J ¼ 0 and ]

D=]t ¼ 0, so r

H ¼ 0, meaning that the vector field

is conservative in the region of interest, and the magnetic field

can be expressed as

H ¼

rV , where V is a scalar potential. Because

B ¼ m

H above the Earth’s surface (where m

¼ 4p  10

7

1

), it

follows that r

H ¼ 0 and that V has to satisfy Laplace’s equation:

V ¼ 0 (Eq. 3)

In spherical coordinates ðr; y; fÞ, which are the radial distance, the geo-

centric colatitude, and the longitude, Laplace’s equation takes the form:

]

rVðÞþ

siny

]

siny



sin

]

¼ 0 (Eq. 4)

Equation (4) has, as a solution, a product of three expressions (the first

is to be only a function of r, the second only a function of y, and the

third only a function of f) and can be solved by separation of vari-

ables. A completely general solution is provided by spherical harmo-

nics, and can be written as:

V ðr; y; fÞ¼a

n¼1

m¼0

n;m

cos mf þ B

n;m

sin mf





nþ1



þ C

n;m

cos mf þ D

n;m

sin mf







n;m

ðcos yÞ

(Eq. 5)

where A

n,m

, B

n,m

, C

n,m

, D

n,m

are called spherical harmonic coefficients,

and P

n;m

ðcos yÞ are the associated Legendre polynomials. In geomag-

netism the partially normalized Schmidt functions P

ðcos yÞ are used.

The P

ðcos yÞ are functions of colatitude only. They are quasisinu-

soidal oscillations having n  m þ 1 nodes. Figure M177 illustrates

the variations of some of these functions.

The surface harmonics P

ðcos yÞsin mf and P

ðcos yÞcos mf

divide the surface of the sphere into regions defined by the intersec-

tions of latitudinal zones and longitudinal sectors. The three situations

are: (i) when m ¼ 0, the surface harmonics are described by the

Legendre polynomials and they are referred to as zonal harmonics;

(ii) when n ¼ m, the surface harmonics are referred to as sectorial

harmonics, and (iii) when 0 < m < n, the surface is divided in

2mðn  m þ 1Þ regions and the surface harmonics are referred to as

tesseral harmonics. Some examples are given in Figure M178.

Application of spherical harmonics to the Earth’s magnetic field

involves writing the magnetic scalar potential as the sum of two

contributions:

V ¼ V

int

þ V

ext

(Eq. 6)

where V

int

and V

ext

are the internal and external scalar potential,

respectively (see Internal external field separation). These two poten-

tials can be represented by spherical harmonic expansions:

int

¼ a

max

n¼1



nþ1

m¼0



cosðmfÞþh

sinðmfÞ





cosðyÞ



(Eq. 7)

ext

¼ a

max

n¼1



m¼0



cosðmfÞþs

sinðmfÞ





cosðyÞ



(Eq. 8)

where the reference radius is a ¼ 6 371:2 km. Measurements of the

magnetic field are used to estimate the so-called Gauss spherical

harmonic coefficients (ðg

; h

Þ for internal sources, and ðq

; s

Þ for

external sources), which in principle uniquely describe the geomag-

netic field outside source regions. The summations in Eqs. (7) and

(8) are truncated at maximum degree N

max

and N

max

, respectively.

In practice

B is not measured everywhere on the Earth’s surface.

The coefficients ðg

; h

Þ and ðq

; s

Þ are obtained by a least-squares

fit to the data. Gauss originated this method, and with the small

number of observatory measurements he had at that time, he took

max

¼ 4 and N

max

¼ 0. Even today, there are fewer than 200 mag-

netic observatories, and they are very unevenly distributed. A pro-

blem for main field modeling is that magnetic observatories, as

well as being poorly distributed over the globe, may be situated in

areas with large local crustal magnetic anomalies. In contrast, low-

altitude satellites in near-polar orbits rapidly provide excellent global

data coverage and are much less affected by crustal magnetic fields,

which are considerably attenuated at satellite altitude. A problem

with magnetic satellite data is that the full vector measurements need

very accurate orientation of the magnetometer. Originally it was

hoped (in the POGO and OGO missions) that one could measure

the field strength j

Bj, which is much easier, and a very high density

of measurements would then amount to knowing the vector field.

Unfortunately, it was shown that even if j

Bj is known exactly every-

where, a unique field model cannot be obtained (Khokhlov et al.,

1999). There exist pairs of magnetic fields

and

satisfying

r

H ¼ 0 and r

H ¼ 0, outside the considered sphere and such

that j

j¼j

j everywhere on the sphere, but

6¼

. These two

Figure M177 Legendre polynomials P

ðxÞ up to n ¼ 5.

680 MAIN FIELD MODELING

fields would give exactly the same values of total intensity, but two

different models. The details of nonuniqueness of fields based on

total field observations are given by Backus (1970). Empirically it

has been observed that fitting MAGSAT intensity data produces

models whose

B can differ from the vector field components

measured by the satellite by some thousands of nanoteslas near the

equator.

Spherical harmonic analysis is an appropriate way of modeling the

geomagnetic field. Considering only the part of the geomagnetic field

whose sources lie beneath the Earth’s surface, Eq. (7) is used. How-

ever, this equation describes both the main field (with sources in the

core) and the crustal field (with sources in the Earth’s crust).

The Gauss coefficients can be interpreted in terms of “sources”

at the Earth’s center. The first term, g

, is associated with the

geocentric dipole oriented along the vertical axis, with dipole

moment g

4pa

(see Geomagnetic dipole field). The next two

terms, g

, h

, correspond to geocentric dipoles oriented along the

two horizontal axes. The magnitude of the geocentric dipole is given

by M ¼ð4pa

Þððg

þðg

þðh

1=2

, and it is tilted at roughly



from the rotation axis. The potential for a dipole falls off as r

2

and the strength of the field components as r

3

. Similarly, for n ¼ 2

the terms represent the geocentric quadrupole (potential falls off

as r

3

, and the strength as r

4

), for n ¼ 3 the terms represent the geo-

centric octupole (potential falls off as r

4

, and the strength as r

5

), and

so forth.

The first three Gauss coefficients ðg

, g

, h

Þ represent the dipole

field, while the remaining terms represent the nondipole field. The

lowest degree terms correspond to the largest wavelength features of

the field, as can be appreciated by considering the zonal harmonics.

The first Gauss coefficient has a cos y dependence, so at distance

r the “wavelength” of the associated magnetic feature is 2pr. Generally

speaking the term g

has a wavelength of 2pr=n.

The internal field at the Earth’s surface contains clearly defined

components from the core at least up to degree N

¼ 13, beyond

which they begin to become dominated by those from the crust.

The maximum associated wavelengths for the crustal components

are of the order of a few thousand kilometers. The relative impor-

tance of the two internal contributions is clearly illustrated by using

the normalized power in spherical harmonics of the internal geomag-

netic field. The spectrum of a model is constructed by plotting the

contribution of each degree n of the spherical harmonic expansion

to the mean square field over the Earth’s surface (see Geomagnetic

spectrum, spatial ). The power at degree n is computed from the

equation (Lowes, 1974):

¼ n þ 1ðÞ

ðg

þðh



: (Eq. 9)

High-quality data sets, such as those provided by the magnetic satellite

surveys MAGSAT, Ørsted, and CHAMP (Langel and Estes, 1985,

Figure M178 Surface harmonics showing three cases, with n ¼ 7 and (a) m ¼ 0(zonal harmonics); (b) n ¼ m (sectorial harmonics), and

(c) for 0 < m < n, when the surface is divided in 2mðn  m þ 1Þ regions (tesseral harmonics), with m ¼ 4 in this example.

MAIN FIELD MODELING 681

Neubert et al., 2001, Reigber et al., 2002) have allowed the computa-

tion of spherical harmonic models up to high degree and order.

Figure M179 shows the power spectra from models obtained from

MAGSAT and Ørsted data, and also the difference between them

(Langlais et al., 2003). The dipole term obviously stands alone, the

main field dominates the signal for degrees less than 13, while the

crustal field dominates for degrees larger than 15.

The Gauss coefficients are also time-dependent. Indeed, the tem-

poral variations of the geomagnetic field have an extremely wide spec-

trum, ranging over more than 20 orders of magnitude (Langel, 1987).

In order to resolve time variations of internal origin, the secular varia-

tion, about 1 year of continuous observations is needed. If the secular

variation is assumed to be constant over short timescales, it can be

included in Eq. (7) by adding a secular-variation potential, V

, trun-

cated to N

max

¼a

max

n¼1



nþ1

m¼0

ðt T

cosðmfÞþ

sinðmfÞ



ðcosðyÞÞ

(Eq. 10)

where

and

are the time derivatives of the internal Gauss coeffi-

cients. T

denotes the reference time (i.e., the epoch of the main field

model), and t is the considered epoch.

Since the geomagnetic field changes in space and time, magnetic

observations must be made continually and models have to be gener-

ated to accurately represent the magnetic field. For example, the Inter-

national Geomagnetic Reference Field (IGRF) models are adopted

every 5 years (see IGRF, International Geomagnetic Reference Field )

by the International Association of Geomagnetism and Aeronomy

(Mandea and Macmillan, 2000). These models are simply truncated

series (up to degree and order 10 or, more recently, 13). An extreme

cutoff like this is, of course, unrealistic, and downward continuation

of the models to the core-mantle boundary is unstable. Other methods,

including stochastic inversion, have been used to produce reasonable

models of the field at the core-mantle boundary (see Time-dependent

models of the main magnetic field).

Mathematical models: other methods

The mathematical properties of the spherical harmonic representation

make it a convenient method for describing the magnetic field. How-

ever, this representation does not have direct physical significance;

some important limitations are linked to the precision in observations,

their distribution and to the use of truncated spherical harmonic expan-

sions. Indeed, the truncation of the spherical harmonic series followed

by straightforward least squares inversion leads to solutions that are

strongly dependent on the chosen truncation level. Changing the trunca-

tion level modifies all the coefficients, due to spatial aliasing of the

higher-order harmonics (Whaler, 1986). Some alternative modeling meth-

ods exist, such as harmonic spline modeling (Shure et al.,1982).This

approach produces models smoothed according to some objective criteria

(e.g., minimizing the complexity of some well-defined parameters of the

field, such as the energy outside the core, the mean-square radial field

component at the core-mantle boundary, the surface gradient of the radial

field, or imposing a lower bound on Ohmic losses in the core)

which are nevertheless, consistent with the data. Short-wavelength

fields originating in both the core and the crust are suppressed.

More details about this method can be found elsewhere (e.g., Par-

ker and Shure, 1982; Shure et al., 1982; Gubbins, 1983).

To represent both the global and the regional pattern of the magnetic

field and to overcome the limitations of spherical harmonic analysis, a

new method to model the Earth’s magnetic field is desirable. Wavelet

techniques may play a role in the solution of this challenging problem.

Recently, a new representation of the magnetic field on the sphere has

been sought, the chosen approach making a direct relation between

spherical harmonics and wavelets. In Holschneider et al. (2003) the

main result is a theoretical description of the wavelets on the sphere

in order to use them in field modeling. The first comparisons between

the spherical harmonic and wavelet bases show that wavelets are able

to reproduce the spherical harmonics well.

Any kind of geomagnetic field modeling can be described as seek-

ing a solution to optimize the cost function “error þ smoothness”.In

general, the smoothness can be associated with some (physically plau-

sible) norms. It is in general implemented through some a priori

damping of the basis functions in the inversion. As it stands, it does

not depend on the family of basis functions in which the actual numer-

ical computations are carried out. Therefore, in the limit of infinitely

many functions and without any numerical errors, it is possible to

get the same models, whether spherical harmonics, wavelets, or any

other complete family of functions are used. However, given finite

resources it does matter, since for a fixed number of basis functions,

wavelets have better approximation properties than spherical harmo-

nics. Moreover, in modeling strongly heterogeneous fields, such as

those due to external currents, wavelets may play an additional role,

since they allow a more geometric description of the fields.

Mioara Mandea

Bibliography

Backus, G.E., 1970. Nonuniqueness of the external geomagnetic field

determined by surface intensity measurements. Journal of Geophy-

sical Research Space Physics, 75: 6339–6341.

Barraclough, D.R., 2000. Four hundred years in geomagnetic field

charting and modelling. In Schröder, W. (ed.) Geomagnetism.

Research Past and Present. Bremen-Roennebeck: IAGA, pp. 93–111.

Gauss, C.F., 1839. Allgemeine Theorie des Erdmagnetismus, Resultate

aus den Beobachtungen des magnetischen Vereins im Jahre 1838.

Printed by Wilhelm Weber, Leipzig.

Figure M179 Geomagnetic field spectra up to degree n ¼ 20 at

the Earth’s surface, from two internal models, computed from

MAGSAT (stars) and Ørsted (filled circle) data. The difference

between these two spectra is also shown (filled diamond).

682 MAIN FIELD MODELING

Gubbins, D., 1983. Geomagnetic field analysis I—stochastic inversion.

Geophysical Journal of the Royal Astronomical Society, 73:641–652.

Halley, E., 1692. An account of the causes of the change of the varia-

tion of the magnetical needle; with an hypothesis of the structure of

the internal parts of the Earth. Philosophical Transactions of the

Royal Society of London, 195: 208–221.

Holschneider, M., Chambodut, A., and Mandea M., 2003. From global

to regional analysis of the magnetic field on the sphere using wavelet

frames. Physics of the Earth and Planetary Interiors, 135: 107– 124.

Khokhlov, A., Hulot G., and Le Mouël, J.-L., 1999. On the Backus

effect–II. Geophysical Journal International, 137: 816–820.

Langel, R.A., 1987. Main field. In Jacobs, J.A. (ed.) Geomagnetism.

London: Academic Press, pp. 249–512.

Langel, R.A., and Estes, R.H., 1985. The near-Earth magnetic field at

1980 determined from MAGSAT data. Journal of Geophysical

Research, 90: 2495–2509.

Langlais, B., Mandea, M. and Ultré-Guérard, P., 2003. High-resolution

magnetic field modeling: application to MAGSAT and Ørsted data.

Physics of the Earth and Planetary Interiors, 135:77–91.

Lowes, F.J., 1974. Spatial power spectrum of the main geomagnetic

field and extrapolation to the core. Geophysical Journal of the

Royal Astronomical Society, 36: 717–730.

Mandea, M., 2000. French magnetic observation and the theory at the

time of DE MAGNETE. In Schröder, W. (ed.), Geomagnetism:

Research Past and Present. Bremen-Roennebeck, IAGA, pp. 73–80.

Mandea, M., and Macmillan, S., 2000. International geomagnetic reference

field—the eighth generation. Earth Planets Space, 52:1119–11 24.

Merrill, R.T., McElhinny, M.W., and McFadden P.L., 1996. The

Magnetic Field of the Earth. London: Academic Press, p. 531.

Neubert, T., Mandea, M., Hulot, G., von Frese, R., Primdahl, F.,

Jorgenson, J.L., Friis-Christensen, E., Stauning, P., Olsen, N. and

Risbo, T., 2001. High-precision geomagnetic field data from the

Ørsted satellite. EOS, 82:81–87.

Parker, T.L., and Shure, L., 1982. Efficient modelling of the Earth’s

magnetic field with harmonic splines.

Geophysical Research Letters,

9: 812–815.

Reigber, Ch., Luehr, H., and Schwintzer, P., 2002. CHAMP mission

status. Advances in Space Research, 30: 129–134.

Shure, L., Parker, R.L., and Backus, G.E., 1982. Harmonic splines for

geomagnetic modelling. Physics of the Earth and Planetary Inter-

iors, 28: 215–229.

Whaler, K.A., 1986. Geomagnetic evidence for fluid upwelling at the

core-mantle boundary. Geophysical Journal of the Royal Astro-

nomical Society, 86, 563–586.

Cross-references

Gauss, Carl Friedrich (1777–1855)

Geomagnetic Dipole Field

Geomagnetic Spectrum, Spatial

Geomagnetism, History of

Harmonics, Spherical

IGRF, International Geomagnetic Reference Field

Internal External Field Separation

Time-dependent Models of the Geomagnetic Field

MAIN FIELD, ELLIPTICITY CORRECTION

The first spherical harmonic analyses of the geomagnetic main field

ignored the ellipticity of the Earth. This was not a serious omission,

since the radial departures of the Earth’s surface from a sphere never

amount to more than 0.17%. However, as the quality of data improved,

it was thought that they might justify inclusion of a correction for

ellipticity.

The Earth’s surface is closely approximated by an oblate spheroid

with eccentricity 0.082. This affects both the position of an observation

and the observation itself, since the horizontal component, for example,

is measured tangential to the spheroid rather than perpendicular to the

line joining the site to the Earth’s center. Early analyses that took account

of this ellipticity (Schmidt, 1898; Adams, 1900; Jones and Melotte,

1953) first determined coefficients for a spherical Earth, then adjusted

these for ellipticity. A better method (Cain et al., 1963) was based

directly on the original observations.

Adjustment from geodetic to geocentric coordinates

A closer approximation to the Earth’s surface than a sphere is an

oblate spheroid. That adopted by the International Astronomical Union

and subsequently used with the International Geomagnetic Reference

Field (see IGRF, International Geomagnetic Reference Field) has flat-

tening f ¼ 1/298, where f ¼ 1–b/a. Here a denotes the equatorial

radius (6378.388 km) and b the polar radius (6356.912 km). This cor-

responds to a mean radius a

¼ (a

1/3

¼ 6371.2 km and ellipticity

e ¼(1–b

)

1/2

¼0.082. Departures from this spheroid are well-known

and determined, but are too small to be of concern in the context of

geomagnetism.

Observations made at or near the surface of the Earth are referred to

geodetic coordinates, with north and east tangential to the surface and

vertical perpendicular to it. (Satellite data are usually provided in geo-

centric coordinates.) Geodetic latitude, f

, is the angle between the

equatorial plane and the vertical. Height, h, is measured vertically from

the surface (see Figure M180). In geocentric coordinates, positio n is

defined by r, f, l, where r denotes the radial distance from the center

of the Earth, f denotes latitude—the angle between the equatorial

plane and a line drawn from the center of the Earth, and l denotes

longitude measured east from the Greenwich meridian. In geocentric

coordinates, north and east are perpendicular to the radial direction

and vertical is along it. The direction of east and l are the same in both

coordinate systems.

Two operations are needed to convert the observations from geo-

detic to geocentric coordinates. (1) The site must be converted from

h, f

, l,tor, f, l. This may be done using the relations

¼ h

þ 2hA þða

cos

þ b

sin

Þ=A

;

cos f ¼ cos f

cos d þ sin f

sin d

Figure M180 Geodetic (f

, h) and geocentric (f, r) coordinates of

point P.

MAIN FIELD, ELLIPTICITY CORRECTION 683

and

sin f ¼ sin f

cos d  cos f

sin d;

where d ¼ f

f,

cos d ¼ðh þ AÞ=r;

sin d ¼½ða

 b

Þcos f

sin f

=Ar

and

A ¼ða

cos

þ b

sin

1=2

(2) The magnetic elements X, Y, Z, the geodetic north, east and vertically

downward geomagnetic components, respectively, must be rotated to

, Y, Z

, the geocentric equivalents. The appropriate relations are

¼ X cos d  Z sin d

and

¼ Z cos d þ X sin d:

The treatment of nonorthogonal elements may be deduced from that

described in main field modeling (q.v.)

Once the data are in geocentric coordinates, a standard spherical

harmonic analysis may be performed. Magnetic values synthesized

from spherical harmonic coefficients may, if required, be converted

back into geodetic coordinates.

Discussion

Since the advent of computers, it is just as easy to take account of the

Earth’s ellipticity as to ignore it, so the adjustment is now universally

incorporated as outlined above. Whether or not it is justified is debata-

ble. Precomputer analysts (Adams, 1900; Jones and Melotte, 1953)

concluded that the additional labor was not justified. Before satellite

data were available it was thought that there might be an advantage

when upward extrapolation was involved (Kahle et al., 1964; Cain

et al., 1965) or when external as well as internal coefficients were

being determined (Winch, 1967). For modeling the main magnetic

field of internal origin, the correction for ellipticity is harmless but

probably unnecessary (Malin and Pocock, 1969).

A related point, which is probably only of academic interest, con-

cerns the separation of the field into internal and external parts (see

Internal external field separation). The boundary between internal

and external should be the surface of the Earth, but, using the above

method, the boundary is the reference sphere, with radius a

, which

passes some 14 km above the poles and is about 7 km below the surface

at the equator. Any sources within the region between the ellipsoid and

the reference sphere would be wrongly classified. For example, crustal

sources near the equator would be deemed to be external. It would not

help to increase the radius of the reference sphere to a, since any

magnetic fields due to currents in the lower polar atmosphere would then

be considered to be of internal origin.

Stuart R.C. Malin

Bibliography

Adams, J.C., 1900. Terrestrial Magnetism. Scientific Papers, Vol. 2.

Cambridge: Cambridge University Press, pp. 243–640.

Cain, J.C., Daniels, W.E., Hendricks, S.J., and Jensen, D.C., 1965. An

evaluation of the main geomagnetic field, 1940–1962. Journal of

Geophysical Research, 70: 3647–3674.

Jones, H.S., and Melotte, P.J., 1953. The harmonic analysis of the

Earth’s magnetic field for epoch 1942. Monthly Notices of the Royal

Astronomical Society, Geophysical Supplement, 6: 409–430.

Kahle, A.B., Kern, J.W., and Vestine, E.H., 1964. Spherical harmonic

analyses for the spheroidal Earth. Journal of Geomagnetism and

Geoelectricity, 16: 229–237.

Malin, S.R.C., and Pocock, S.B., 1969. Geomagnetic spherical harmo-

nic analysis. Pure and Applied Geophysics, 75:117–132.

Schmidt, A., 1898. Der magnetische Zustand der Erde zur Epoch 1885.0.

Archiv der deutschen Seewarte, Hamburg, 21(2): 76 pp.

Winch, D.E., 1967. An application of oblate spheroidal harmonic func-

tions to the determination of geomagnetic potential. Journal of

Geomagnetism and Geoelectricity, 19:49–61.

Cross-references

Harmonics, Spherical

IGRF, International Geomagnetic Reference Field

Internal External Field Separation

Main Field Modeling

MANTLE, ELECTRICAL CONDUCTIVITY,

MINERALOGY

The electrical conductivity profile in the mantle can be obtained by

magnetotelluric (q.v.) and geomagnetic deep sounding (q.v.) methods

(Figure M181). Although the profiles obtained can have significant

differences from each other, the following features can be seen in the

most conductivity models. Conductivity in the upper mantle is rela-

tively low, i.e., 10

4

–10

2

1

. It increases with increasing depth

to the top of the lower mantle. At the top of the lower mantle, conduc-

tivity is 10

1

. Conductivity does not increase significantly in the

lower mantle. Some studies have shown that the electrical conductivity

of the uppermost mantle to 100 km depth is about 10

2

10

1

known as the mantle high conductive layer (HCL). Local variations of

electrical conductivity are large at shallow depths, and become smaller

with increasing depth.

Figure M181 The electrical conductivity of the upper part of the

mantle. Long dashed curve: north Pacific region, dotted curve:

northeastern China, short dashed curve: Canadian shield,

one-dotted dashed curve: southwestern United States, two-dotted

dashed curve: Hawaii. Thick solid curve: laboratory electrical

conductivity model (Xu et al., 2000).

684 MANTLE, ELECTRICAL CONDUCTIVITY, MINERALOGY

Geochemical and petrological studies suggest that the major consti-

tuent of the uppermost mantle is olivine, with an approximate compo-

sition of (Mg

0.9

0.1

)

SiO

. With an increasing pressure, olivine

transforms to its high-pressure polymorph, wadsleyite at about

410 km depth (olivine-wadsleyite transition). Wadsleyite transforms

to a further high-pressure polymorph, ringwoodite, at about 520 km

depth (wadsleyite-ringwoodite transition). Ringwoodite dissociates to

(Mg,Fe)SiO

perovskite and (Mg,Fe)O ferropericlase at 660 km depth

(postspinel transition). These minerals form a network in each region

and therefore, the electrical conductivity of these minerals would be

primarily responsible for that of the mantle itself. Additionally, some

minor phases such as silicate melt that have high conductivity may

also have important roles.

Electrical conduction mechanisms

In metals, free electrons transfer electric charges. However, the mantle

minerals are essentially ionic solids in which most electrons are bound

to ions. Therefore, carriers other than free electrons play an important

role in transferring electric charges. For example, if an electron is

released from an ion, an electric hole is created in the ion, and the hole

itself can migrate to carry an electric charge.

For the migration of carriers other than a free electron, defects play

important roles in crystalline ionic solids. Here, defects in ferromagne-

sian silicates are briefly described using the Kroger-Vink notation in

which the base character refers to the defect species, that is, ion, elec-

tron, or hole. An electron or an electron hole is expressed by element

symbols, “e” and “h,” respectively. The vacancy of an ion is expressed

by “V”. The subscript refers to the site, that is, Mg, Si, O, or an inter-

stitial site that is expressed as “I”. The dots or slashes as superscripts

refer to the number of positive or negative excess charges compared

to the site in a normal lattice. “x” as a superscript refers to no excess

charge. In this notation, a released electron or a hole is expressed as

“e

” or “h

”, respectively. The normal Mg

2þ

,Si

4þ

2þ

ions in their

own sites are expressed as Mg

,Si

, and O

respectively.

Fe ions usually substitute the Mg site, which is expressed as Fe

The Fe ion can have 3þ charges. The Fe

3þ

ion in the Mg site has

an extra plus charge and is expressed as Fe

. The vacancy has a

charge with an opposite sign to that of the ion that should occupy

the site. The vacancies for Mg, Si, and O sites are expressed as V

0000

, and V

, respectively. A small cation can exist interstitially in

the crystalline lattice. A typical interstitial ion is H

, which is

expressed as H

The major mantle minerals contain significant amounts of Fe and,

therefore, electron hopping from Fe

2þ

to Fe

3þ

ions could be the domi-

nant electron conduction mechanism (Figure M182). An electron has

to jump across an energy barrier, DE

and, therefore the electrical

conductivity, s, is usually expressed as

s ¼ s

expðDE

=kTÞ (Eq. 1)

where k is the Boltzmann constant and T is absolute temperature. The

is called the activation energy. The preexponential term s

could

be a function of temperature; however, it is often regarded as a con-

stant with a limited temperature range. For electron hopping, the

energy barrier should be low.

Electron hopping can be r egarded as the migration of Fe

(Fig-

ure M182). The surrounding oxygen ions and cations, respectively,

are attracted and repulsed by the excess positive charge,

which causes a distortion of the lattice in the neighborhood (Figure

M183). The migration of Fe

is associated with this lattice distor-

tion. The unit of Fe

and its induced lattice distortion is called a

small polaron. Migration o f a small polaron is a thermally activated

process:

s ¼ s

expðDE

=kTÞ (Eq. 2)

where DE

is the activation energy for the small polaron. A small

polaron is less mobile than an electron itself, resulting in DE



. DE

ranges from 0.5 to 2 eV for mantle minerals.

The charges are carried by Fe

and, therefore, the electrical

conductivity should be a function of the populations of Fe

and

. Electrical conductivity thus increases with increasing total Fe

content. With increasing oxygen partial pressure, the population of

increases; therefore, the electrical conductivity of the mantle

minerals increases with increasing oxygen partial pressure even with

constant Fe content.

The bodily motions of ions transfer electric charges, which is called

ionic conduction. The best known material for ionic conduction is zir-

conia (ZrO

), in which the oxygen ion is highly mobile and carries

minus charges whereas the zirconium ion is very immobile and retains

the crystalline lattice. In the case of ferromagnesian silicate or oxide,

the migration of Mg

via V

can be an effective conduction

mechanism and is regarded as the migration of V

(Figure M184).

Figure M182 Electrical conduction by electron hopping between

and Fe

. It can be regarded as migration of the hole.

Figure M183 Small polaron model. The Mg

(and Si

) and

ions are repulsed and attracted by the excess plus charge of

, which induces distortion of the lattice.

MANTLE, ELECTRICAL CONDUCTIVITY, MINERALOGY 685