Introduction
Based on the mathematical theory of the magnetic
induction developed by ,
affirmed that, if U is the
gravitational potential produced by any body
with uniform density ρ and arbitrary shape at
a point (x,y,z), then -∂U∂x
is the magnetic scalar potential produced
at the same point by the same body
if it has a uniform magnetization oriented along x
with intensity ρ.
generalized this idea as a way
of determining the magnetic scalar potential produced
by any uniformly magnetized body in a given direction.
By presuming that this uniform magnetization is due to
induction and that it is proportional to the resulting
magnetic field (intensity) inside the body, he postulated
that the resulting field must also be uniform and
parallel to the magnetization. This uniformity is due to
the fact that the resulting field is defined as the
negative gradient of the magnetic scalar potential.
As a consequence of this uniformity,
the gravitational potential U at points within the
body must be a quadratic
function of the spatial coordinates.
Apparently, was the first one to
postulate that ellipsoids are the only finite bodies
having a gravitational potential which satisfies this
property and hence can be uniformly magnetized
in the presence of a uniform inducing magnetic field.
This property can be extended to other bodies
defined as limiting cases of an ellipsoid
(e.g. spheres, elliptic cylinders). However, all the
remaining non-ellipsoidal bodies cannot be
uniformly magnetized in the presence of a uniform
inducing field.
Another particularity of ellipsoids is that they are
the only bodies which enable an analytical computation
of their self-demagnetization.
The self-demagnetization contributes to a decrease in the
magnitude of the magnetization along the shortest
axes of a body.
This is a function of the body shape and gives rise
to shape anisotropy .
It is well established in the literature that
the self-demagnetization can be neglected if the
body has a susceptibility lower than 0.1 SI
.
On the other hand, neglecting the self-demagnetization in
geological bodies with high susceptibilities (> 0.1 SI)
may strongly mislead the interpretation obtained from
magnetic methods.
This limiting value, however, seems to be determined empirically
and, so far, there has been little discussion about
how it was determined.
demonstrated the importance of the
ellipsoidal model in taking into account the
self-demagnetization and determining reliable
drilling directions on the Tennant Creek field,
Australia.
Later, also showed how the
ellipsoidal model proved to be highly successful in
locating and defining ironstone bodies in the
Tennant Creek field.
provides a good discussion about the
influence of the self-demagnetization in magnetic
interpretation of the Osborne copper–gold deposit,
Australia. This deposit is hosted by ironstone
bodies that have very high susceptibility.
According to , neglecting the effects
of self-demagnetization led to errors of ≈55∘
in the interpreted dip.
Recently, used magnetic modelling
and rock property measurements to show that,
contrary to previous interpretations, the magnetization
of the Candelaria iron oxide copper–gold deposit,
Chile, is not dominated by the induced component.
Rather, the deposit has a relatively weak remanent
magnetization and is strongly affected by
self-demagnetization.
These examples show the importance of the self-demagnetization
and the ellipsoidal model in
producing trustworthy geological models of
high-susceptibility orebodies, which may save significant
cost associated with drilling.
A vast literature about the magnetic
modelling of ellipsoidal bodies was developed in which
are to be found the names of many researchers.
Nevertheless, interest in this subject has not yet died
out, as is evidenced by a list of modern papers in this
field.
Furthermore, the geoscientific community lacks a free easy-to-use
tool to simulate the magnetic field produced by uniformly
magnetized ellipsoids.
Such a tool could prove useful both for teaching
and researching geophysics.
In this work, we present a review of the vast literature
about the magnetic modelling of ellipsoidal bodies and a
theoretical discussion about the determination of the
isotropic susceptibility value above which
the self-demagnetization must be taken into consideration.
We propose an alternative way of determining this value
based on the body shape and the maximum relative error
allowed in the resultant magnetization.
This alternative approach is validated
by the results obtained with numerical simulations.
We also provide a set of routines to
model the magnetic field produced by ellipsoids.
The routines are written in Python language as part of
the Fatiando a Terra , which is an
open-source library for modelling and inversion in geophysics.
We attempt to use the best practices of continuous
integration, documentation, unit-testing and version control
for the purpose of providing a reliable and easy-to-use
code.
Schematic representation of the coordinate systems used to
represent an ellipsoidal body.
(a) Main coordinate system with axes x, y, and z pointing to
north, east, and down, respectively.
The dark grey plane contains the centre (xc,yc,zc; white circle) and two unit vectors, u and w, defining two
semi-axes of the ellipsoidal body. For triaxial and prolate ellipsoids,
u and w define, respectively,
the semi-axes a and b. For oblate ellipsoids, u and
w define the semi-axes b and c, respectively.
The strike direction is defined by the intersection of the
dark grey plane and the horizontal plane (represented in light grey),
which contains the x axis and y axis. The angle ε
between the x axis and the strike direction is called strike.
The angle ζ between the horizontal plane and the dark grey plane
is called dip. The angle η between the strike direction
and the line containing the unit vector u
is called rake. The projection of this line on the
horizontal plane (not shown) is called dip direction
. (b) Local coordinate system
with origin at the ellipsoid centre (xc,yc,zc)
(black dot) and axes defined by unit vectors v1,
v2 and v3. These unit vectors
define the semi-axes a, b and c of triaxial, prolate
and oblate ellipsoids in the same way.
For triaxial and prolate ellipsoids,
the unit vectors u and w shown in (a) coincide
with v1 and v2, respectively. For
oblate ellipsoids, the unit vectors u and w
shown in (a) coincide with v2 and v3,
respectively.
Methodology
Geometrical parameters and coordinate systems
Let (x,y,z) be a point referred to a Cartesian coordinate system with axes
x, y and z pointing to, respectively, north, east and down.
For convenience, we denominate this coordinate system as the
main coordinate system (Fig. a).
Let us consider an ellipsoidal body with centre at the point
(xc,yc,zc), orientation defined by the angles
strike ε, dip ζ and rake η
(Fig. a), and semi-axes defined by
positive constants a, b, c (Fig. b).
The orientation angles strike, dip and rake
are commonly used to define the orientation of lines in structural geology
.
The points (x,y,z) located on the surface of this ellipsoidal body
satisfy the following equation:
(r-rc)TA(r-rc)=1,
where r=[xyz]⊤,
rc=[xcyczc]⊤,
A is a positive definite matrix given by
A=Va-2000b-2000c-2V⊤,
and V is an orthogonal matrix whose first, second and third columns
are defined by unit vectors v1, v2 and v3
(Fig. b), respectively.
The matrix V can be defined in terms of three rotation matrices:
R1(θ)=1000cosθsinθ0-sinθcosθ,
R2(θ)=cosθ0-sinθ010sinθ0cosθ
and
R3(θ)=cosθsinθ0-sinθcosθ0001.
For triaxial ellipsoids (i.e. a>b>c) and prolate ellipsoids
(i.e. a>b=c), we define the orthogonal matrix V
as follows:
V=R1π2R2εR1π2-ζR3η.
For oblate ellipsoids (i.e. a<b=c), we define V as follows:
V=R3-π2R1πR3εR2π2-ζR1η.
The orthogonal matrices V used here for triaxial, prolate and oblate
ellipsoids (Eqs. and ) are
different from those used by and .
The magnetic modelling of an ellipsoidal body is commonly performed
in a particular Cartesian coordinate system that is aligned
with the body semi-axes
and has the origin coincident with the body centre
(Fig. b).
For convenience, we denominate this particular coordinate
system as the local coordinate system.
The relationship between the Cartesian coordinates
(x̃,ỹ,z̃) of a point in
a local coordinate system and the Cartesian
coordinates (x,y,z) of the same point in the main
system is given by
r̃=V⊤r-rc,
where
r̃=[x̃ỹz̃]⊤,
r and rc
are defined in Eq. () and the matrix V
(Eqs. 6 and 7) is defined according to the ellipsoid type.
In what follows, quantities referred to the local coordinate system
(Fig. b) are indicated with
the symbol “∼”.
Theoretical background
Consider a magnetized ellipsoid immersed in
a uniform inducing magnetic field H0 (in Am-1)
given by
H0=‖H0‖cosIcosDcosIsinDsinI,
where ‖⋅‖ denotes the Euclidean norm (or 2-norm) and
D and I are respectively the declination and inclination of the
local geomagnetic field in the main coordinate system
(Fig. a).
This field represents the main component of the
Earth's magnetic field, which is usually assumed to be generated
by the Earth's liquid core.
In the absence of conduction currents,
the total magnetic field H(r)
at the position r (Eq. )
of a point referred to the main
coordinate system is defined as follows :
H(r)=H0-∇V(r),
where the second term is the negative gradient of
the magnetic scalar potential V(r) given by
V(r)=-14π∭ϑM(r′)⊤∇1‖r-r′‖dx′dy′dz′.
In this equation, r′=[x′y′z′]⊤
is the position vector of a point located within the volume ϑ,
the integral is conducted over the variables
x′, y′ and, z′ and
M(r′) is the magnetization vector
(in Am-1).
Equation () is valid anywhere,
independently if the position vector r represents
a point located inside or outside the magnetized body
.
Based on 's postulate, let us
assume that the body has a uniform magnetization given by
M=KH†,
where H† is the resultant uniform magnetic field
at any point within the body and K is a
constant and symmetrical second-order tensor representing the
magnetic susceptibility of the body.
This is a good approximation for bodies at room temperature,
subjected to an inducing field H0 with
strength ≤10-3μ0-1 Am-1 ,
where μ0 represents the magnetic constant (in Hm-1).
In this case, the susceptibility tensor K is commonly
represented, in the main coordinate system (Fig. a),
as follows:
K=Uk1000k2000k3U⊤,
where k1>k2>k3 are the
principal susceptibilities and U is
an orthogonal matrix whose columns ui,
i=1,2,3, are unit vectors called
principal directions.
Similarly to the matrix V (Eqs. ,
and ), we define
the matrix U as a function
of given orientation angles ε, ζ
and η depending on the ellipsoid type. For triaxial
and prolate ellipsoids, we define U by
using Eq. (), whereas for oblate ellipsoids
we use Eq. ().
Notice that the orientation angles ε, ζ
and η defining the orthogonal matrix U
may be different from the angles ε, ζ
and η defining the ellipsoid orientation
(Fig. ).
If the principal susceptibilities are different from
each other, we say that the body has an
anisotropy of magnetic susceptibility (AMS).
The AMS is generally associated with the preferred orientation
of the grains of magnetic minerals forming the rock
.
For the particular case in which the principal directions
coincide with the ellipsoid axes, the matrix U is
equal to the matrix V (Eq. 2).
Another important particular case is that in which the
susceptibility is isotropic and, consequently, the principal
susceptibilities k1, k2 and k3 (Eq. )
are equal to a constant χ. In this case, the susceptibility
tensor K (Eq. ) assumes the particular form
K=χI,
where I represents the identity matrix.
By using the magnetization M defined by Eq. (),
the total magnetic field H(r) (Eq. )
can be rewritten as follows:
H(r)=H0+N(r)KH†,
where N(r) is a symmetrical matrix whose
ij-element nij(r) is given by
nij(r)=14π∂2f(r)∂ri∂rj,i=1,2,3,j=1,2,3,
where r1=x, r2=y, r3=z are the elements of
the position vector r (Eq. ),
and
f(r)=∭ϑ1‖r-r′‖dx′dy′dz′.
Notice that the scalar function f(r)
(Eq. ) is proportional
to the gravitational potential that would be produced by the
ellipsoidal body with volume ϑ if it had a uniform density
equal to the inverse of the gravitational constant.
It can be shown that the elements nij(r) are
finite whether r is a point within or without
the volume ϑ .
The matrix N(r) (Eq. )
is called the depolarization tensor .
The following part of this paper moves on to describe
the magnetic field H(r)
(Eq. ) at points located
both within and without the volume ϑ of the ellipsoidal
body. However, the mathematical developments are conveniently
performed in the local coordinate system (Fig. b)
related to the respective ellipsoidal body.
Coordinate transformation
To continue our description of the magnetic modelling of
ellipsoidal bodies, it is convenient to perform two
important coordinate transformations.
The first one transforms the scalar function
f(r) (Eq. ) from the
main coordinate system (Fig. a)
into a new scalar function
f̃(r̃) referred to the
local coordinate system (Fig. b).
The function f̃(r̃) was first
presented by to describe the gravitational potential
produced by homogeneous ellipsoids.
Later, several authors also deduced and used this function
for describing the magnetic and gravitational fields produced
by triaxial, prolate and oblate ellipsoids
.
It is convenient to use f̃†(r̃)
and f̃‡(r̃) to define
the function f̃(r̃) evaluated,
respectively, at points
r̃ inside and outside the volume ϑ of the
ellipsoidal body.
The scalar function f̃†(r̃)
is given by
f̃†(r̃)=πabc∫0∞1-x̃2a2+u-ỹ2b2+u-z̃2c2+u1R(u)dur̃∈ϑ,
where
R(u)=a2+ub2+uc2+u.
This function represents the gravitational potential
that would be produced by the ellipsoidal body at
points located within its volume ϑ if it
had a uniform density equal to the inverse of the
gravitational constant.
Notice that, in this case, the gravitational potential
is a quadratic function of the spatial coordinates
x̃, ỹ and z̃, which
supported 's ()
postulate about uniformly magnetized ellipsoids.
In a similar way, the function f̃‡(r̃)
is given by
f̃‡(r̃)=πabc∫λ∞1-x̃2a2+u-ỹ2b2+u-z̃2c2+u1R(u)du,r̃∉ϑ,
where R(u) is defined by Eq. () and the
parameter λ is defined according to the
ellipsoid type as a function of the spatial coordinates
x̃, ỹ and z̃ (see Appendix B).
For readers interested in additional information about the
parameter λ, we recommend p. 234,
p. 184 and .
The second important coordinate transformation is defined
with respect to Eq. ().
By properly using the orthogonality of matrix V
(Eq. 2),
the magnetic field H(r)
(Eq. ) can be transformed
from the main coordinate system (Fig. a)
to the local coordinate system (Fig. b) as follows:
V⊤H(r)︸H̃(r̃)=V⊤H0︸H̃0+V⊤N(r)V︸Ñ(r̃)V⊤KV︸K̃V⊤H†︸H̃†,
where the superscript “∼” denotes quantities
referred to the respective local coordinate system.
In Eq. (), the transformed
depolarization tensor Ñ(r̃)
is calculated as a function of the original depolarization
tensor N(r) (Eq. ).
In this case, the elements of Ñ(r̃)
are calculated as a function of the second derivatives of the
function f(r) (Eq. ),
which is defined in the main coordinate system (Fig. a).
It can be shown (Appendix A), however, that the
elements ñij(r̃) of
Ñ(r̃) can also be calculated
as follows:
ñij(r̃)=14π∂2f̃(r̃)∂r̃i∂r̃j,i=1,2,3,j=1,2,3,
where r̃1=x̃, r̃2=ỹ
and r̃3=z̃ are the elements of the
transformed vector r̃ (Eq. )
and f̃(r̃) is given by Eq. ()
or (), depending if r̃ represents a
point located within or without the volume ϑ of the
ellipsoidal body.
Transformed depolarization tensors Ñ(r̃)
Depolarization tensor ц
Let ц be the transformed
depolarization tensor calculated for the case in which
r̃ (Eq. )
represents a point located inside
the ellipsoidal body. In this case,
the elements of ц
are calculated according to Eq. (),
with f̃(r̃) given by
f̃†(r̃) (Eq. ).
As we have already pointed out, the
f̃†(r̃) (Eq. ) is a
quadratic function of the spatial coordinates x̃,
ỹ and z̃. Consequently, the
elements ñij†, i=1,2,3,
j=1,2,3, of ц
do not depend on the elements of the transformed position
vector r̃ (Eq. ).
Also, the off-diagonal elements are zero and
the diagonal elements are given by
ñii†=abc2∫0∞1ei2+uR(u)du,i=1,2,3,
where R(u) is defined by Eq. () and
e1=a, e2=b and e3=c. These elements
are commonly known as demagnetizing factors
and are defined according to the ellipsoid type.
Here, we calculate the demagnetizing factors in the SI
system. Consequently, they satisfy the condition
ñ11†+ñ22†+ñ33†=1, independently of the
ellipsoid type.
It is worth stressing that, according to Eq. (),
the demagnetizing factors ñii† are
constants defined by the ellipsoid semi-axes a, b and c.
Note that, according to Eqs. ()
and (),
N(r)=VцV⊤,
where ц is a diagonal
matrix and V (Eq. 2) is an orthogonal matrix.
This equation shows that, for the particular case
in which r and consequently r̃
represent a point inside the
volume ϑ of the ellipsoid, the elements
ñii†
of ц represent the eigenvalues while the
columns of V represent the eigenvectors of the
original depolarization tensor N(r).
Triaxial ellipsoids
For triaxial ellipsoids (e.g. a>b>c), the demagnetizing
factors obtained by solving Eq. ()
are given by
ñ11†=abca2-c212a2-b2F(κ,ϕ)-E(κ,ϕ),ñ22†=-abca2-c212a2-b2F(κ,ϕ)-E(κ,ϕ)+abca2-c212b2-c2E(κ,ϕ)-c2b2-c2
and
ñ33†=-abca2-c212b2-c2E(κ,ϕ)+b2b2-c2,
where
F(κ,ϕ)=∫0ϕ11-κ2sin2ψ12dψ
and
E(κ,ϕ)=∫0ϕ1-κ2sin2ψ12dψ,
with κ=a2-b2/a2-c212 and
cosϕ=c/a.
The functions F(κ,ϕ) (Eq. ) and
E(κ,ϕ) (Eq. ) are
called Legendre's normal elliptic integrals of the first and
second kind, respectively. presented a
detailed deduction of the demagnetizing factors ñ11†
(Eq. ), ñ22†
(Eq. ) and ñ33†
(Eq. ). presented similar
formulas. It can be shown that these demagnetizing factors
satisfy the conditions
ñ11†+ñ22†+ñ33†=1
and
ñ11†<ñ22†<ñ33†.
Prolate ellipsoids
For prolate ellipsoids (e.g. a>b=c), the demagnetizing
factors obtained by solving Eq. ()
are given by
ñ11†=1m2-1mm2-112lnm+m2-112-1
and
ñ22†=121-ñ11†,
where ñ33†=ñ22†,
with ñ11† defined in
Eq. () and m=a/b.
The detailed deduction of the demagnetizing factors
ñ11† (Eq. )
and ñ22† (Eq. )
can be found, for example, in .
These formulas were later presented by .
It can be shown that these demagnetizing factors
satisfy the conditions
ñ11†+2ñ22†=1
and
ñ11†<ñ22†.
Oblate ellipsoids
For oblate ellipsoids (e.g. a<b=c), the demagnetizing
factors obtained by solving Eq. ()
are given by
ñ11†=11-m21-m1-m212cos-1m
and
ñ22†=121-ñ11†,
where ñ33†=ñ22†,
with ñ11† defined in
Eq. () and m=a/b.
The detailed deduction of these demagnetizing factors
can be found, for example, in .
These formulas can also be found in .
The only difference, however, is that replaced
the term cos-1 by a term tan-1, according to
the trigonometric identity
tan-1x=cos-1(1/x2+1), x>0.
It can be shown that these demagnetizing factors
satisfy the conditions
ñ11†+2ñ22†=1
and
ñ11†>ñ22†.
Depolarization tensor ч(r̃)
The elements ñij‡(r̃),
i=1,2,3, j=1,2,3, of the transformed depolarization tensor
ч(r̃)
are calculated according to Eq. (), with
f̃(r̃) given by f̃‡(r̃)
(Eq. ).
By following , the diagonal elements
ñii‡(r̃)
and the off-diagonal elements
ñij‡(r̃), i=1,2,3,
j=1,2,3, are given by
ñii‡(r̃)=-abc2∂λ∂r̃ihir̃i+gi
and
ñij‡(r̃)=-abc2∂λ∂r̃ihjr̃j,
where
hi=-1ei2+λR(λ),gi=∫λ∞1ei2+uR(u)du,
where e1=a, e2=b, e3=c and
∂λ∂r̃i
is defined in Appendix B (Eq. ).
The functions gi (Eq. ) are defined according
to the ellipsoid type.
Notice that the elements ñii‡(r̃)
(Eq. ) and
ñij‡(r̃)
(Eq. )
are proportional to the second derivatives of the function
f̃‡(r̃) (Eq. ),
which is harmonic. Consequently, the diagonal elements
ñii‡(r̃) satisfy the condition
ñ11‡(r̃)+ñ22‡(r̃)+ñ33‡(r̃)=0 for any
point r̃ outside the ellipsoid, independently
of the ellipsoid type.
Triaxial ellipsoids
For triaxial ellipsoids (e.g. a>b>c), the functions
gi (Eq. ) are defined as follows:
g1=2a2-b2a2-c212F(κ,ϕ)-E(κ,ϕ),g2=2a2-c212a2-b2b2-c2Eκ,ϕ-b2-c2a2-c2Fκ,ϕ-a2-b2a2-c212c2+λa2+λb2+λ12
and
g3=2b2-c2a2-c212Eκ,ϕ+2b2-c2b2+λa2+λc2+λ12,
where F(κ,ϕ) and E(κ,ϕ) are defined by
Eqs. () and (), but with
sinϕ=a2-c2/a2+λ.
A detailed deduction of these formulas was presented by .
Similar formulas can also be found in .
Prolate ellipsoids
For prolate (e.g. a>b=c) ellipsoids, the functions
gi (Eq. ) are given by
g1=2a2-b232lna2-b212+a2+λ12b2+λ12-a2-b2a2+λ12
and
g2=1a2-b232a2-b2a2+λ12b2+λ-lna2-b212+a2+λ12b2+λ12,
where g3=g2.
These formulas can be obtained by properly manipulating those
presented by .
Oblate ellipsoids
For oblate (e.g. a<b=c) ellipsoids, the functions
gi (Eq. ) are given by
g1=2b2-a232b2-a2a2+λ12-tan-1b2-a2a2+λ12
and
g2=1b2-a232tan-1b2-a2a2+λ12-b2-a2a2+λ12b2+λ,
where g3=g2.
Similarly to the case of prolate ellipsoid shown previously,
these formulas can be obtained by properly manipulating those
presented by .
Internal magnetic field and magnetization
By considering r̃ as a point within the
volume ϑ of the ellipsoid and using the Maxwell postulate
about the uniformity of the magnetic field H(r)
inside ellipsoidal bodies, we can use Eq. ()
for defining the resultant uniform magnetic field H̃†
inside the ellipsoidal body as follows:
H̃†=I+цK̃-1H̃0,
where I is the identity matrix and
ц is as defined in the
previous section.
Let us pre-multiply the uniform internal field H̃†
(Eq. ) by the transformed
susceptibility tensor K̃
(Eq. ) to obtain
M̃=K̃I+цK̃-1H̃0=I+K̃ц-1K̃H̃0,
where M̃
represents the transformed magnetization, as can be easily
verified by using Eqs. () and ().
The matrix identity used for obtaining the second line of Eq. () is given by p. 151.
Equation () can be easily generalized for the case in
which the ellipsoid has also a uniform remanent magnetization
M̃R. Let us first consider that
the uniform remanent magnetization satisfies the condition
H̃A=K̃-1M̃R,
where H̃A represents a hypothetical
uniform ancient field. Then, if we assume that H̃0,
in Eqs. () and (),
is in fact the sum of the inducing magnetic field H̃0
and the hypothetical ancient field H̃A, we obtain the
following generalized equation:
M=ΛKH0+MR,
where
Λ=VI+K̃ц-1V⊤.
Despite the coordinate system transformation represented by the
matrix V (Eq. 2),
Eq. () is consistent with that given by
Eq. 38. It shows the combined effect of the
anisotropy of magnetic susceptibility (AMS) and the shape anisotropy.
The AMS is represented by the susceptibility tensor K
(Eq. ) and reflects the
preferred orientation of the magnetic minerals forming the
body. The susceptibility tensor appears in Eq. (),
defined in the main coordinate system (Fig. a),
and (), defined in the local coordinate system
(Fig. b).
The shape anisotropy is represented, in Eq. (), by the
depolarization tensor ц and
reflects the self-demagnetization associated to the body shape.
Notice that the resultant magnetization M
(Eq. ) does not necessarily have
the same direction as the inducing field H0
(Eq. ).
The angular difference between the resultant magnetization
and the inducing field depends on the combined effect of the
anisotropy of magnetic susceptibility and the shape anisotropy.
For the particular case in which the susceptibility is isotropic,
the susceptibility tensor is defined according to Eq. ().
In this case, the magnetization M (Eqs. and ) is
referred to the main coordinate system (Fig. a),
and the matrix Λ (Eq. )
can be rewritten as follows:
M=ΛχH0+MR,
and
Λ=VI+χц-1V⊤.
Despite the coordinate transformation represented by
matrix V (Eq. 2),
this equation is in perfect agreement with those presented
by Eqs. 13–15.
The first term, depending on the inducing field H0
(Eq. ),
represents the induced magnetization whereas the term depending on
MR is the remanent magnetization.
Equation () reveals that, as pointed out by
many authors
e.g.,
the induced magnetization opposes the inducing field
if it is parallel to an ellipsoid axis,
independently of the ellipsoid type. Otherwise, the
magnetization is not necessarily parallel to
the inducing field.
If we additionally consider that χ≪1, the matrix
Λ (Eq. )
approaches to the identity and the magnetization
M (Eq. )
can be approximated by
M˘=χH0+MR,
which is the classical equation describing the resultant
magnetization in applied geophysics p. 89.
Notice that, in this particular case, the induced magnetization
is parallel to the inducing field H0 (Eq. ),
whether it is parallel to an ellipsoid axis or not.
Usually, Eq. ()
is considered a good approximation for χ≤0.1 SI.
Although this value has been largely used in the literature,
there have been few empirical and/or theoretical investigations
about it.
Relationship between χ and the relative error in M˘
In the case of isotropic susceptibility, the resultant magnetization
M (Eq. ) may be determined by solving the
following linear system:
Λ-1M=χH0+MR,
where, according to Eq. (),
Λ-1=VI+χцV⊤.
As we have already pointed out, the approximated
magnetization M˘ (Eq. )
represents the particular case in which the matrix
Λ (Eq. ),
and consequently the matrix Λ-1
(Eq. ), are close to the identity.
Consider a perturbed matrix δΛ-1 given by
δΛ-1=Λ-1-I
and, similarly, a perturbed magnetization vector δM
given by
δM=M-M˘.
By using these two equations, we may rewrite that of the approximated
magnetization M˘ (Eq. ) as follows:
Λ-1-δΛ-1M-δM=χH0+MR.
Now, by subtracting the true magnetization
M (Eq. )
from this linear system (Eq. )
and rearranging the terms, we obtain the following linear
system for the perturbed magnetization δM
(Eq. ):
δM=-δΛ-1M.
By using the concept of vector norm and its corresponding
operator norm , we may use Eq. () to write the following inequality:
‖δM‖‖M‖≤‖δΛ-1‖.
where ‖δM‖ and ‖M‖
denote Euclidean norms (or 2-norms) and the term
‖δΛ-1‖ denotes the matrix 2-norm
of δΛ-1.
By using Eqs. ()
and ()
and the orthogonal invariance of the matrix 2-norm
, we define
‖δΛ-1‖ as follows:
‖δΛ-1‖=χñmax†,
where ñmax† is the demagnetization
factor associated with the shortest ellipsoid semi-axis.
For a triaxial ellipsoid,
ñmax†≡ñ33†
(Eq. ), for a prolate ellipsoid,
ñmax†≡ñ22†
(Eq. ), and,
for an oblate ellipsoid,
ñmax†≡ñ11†
(Eq. ).
It is worth stressing that, independently of the ellipsoid
type, ñmax† is
a scalar function of the ellipsoid semi-axes.
In Eq. (), the ratio
‖δM‖‖M‖-1 represents the
relative error in the approximated magnetization
M˘ (Eq. ) with respect
to the true magnetization M (Eqs.
and ).
Given a target relative error ϵ and an ellipsoid
with given semi-axes, we may use the inequality
represented by Eq. ()
to define
χmax=ϵñmax†,
which represents the maximum isotropic susceptibility
that the ellipsoidal body can assume in order to guarantee
a relative error lower than or equal to ϵ.
For isotropic susceptibilities greater than χmax,
there is no guarantee that the relative error in the
approximated magnetization M˘ (Eq. )
with respect to the true magnetization M
(Eqs. and )
is lower than or equal to ϵ. The geoscientific community
has been using χmax=0.1 SI as a limit value for
neglecting the self-demagnetization and, consequently,
use magnetization M˘ (Eq. )
as a good approximation of the true magnetization M
(Eqs. and ).
Equation (), on the other hand, defines χmax
as a function of the ellipsoid semi-axes, according to a
user-specified relative error ϵ.
Ambiguity between confocal ellipsoids with the same magnetic moment
There is a fundamental non-uniqueness of ellipsoidal bodies, analogous to the
equivalence of concentric spheres with the same magnetic moment.
To show this ambiguity, let us first
consider a reference ellipsoid which is immersed in a uniform inducing
field and has semi-axes a, b and c, isotropic susceptibility χ
and no remanence.
The magnetization of this ellipsoid, defined in the local coordinate system,
can be obtained by using Eqs. (), ()
and ()
as follows:
M̃=χI+χц-1H̃0.
In this case, the magnetic moment P̃, defined in the
local coordinate system, is given by
P̃=ϑM̃,
where ϑ=43πabc is the ellipsoid volume.
From Eqs. () and (),
we can easily show that, if the inducing field H̃0
is parallel to a semi-axis ei, where i=1,2,3, e1=a, e2=b,
e3=c, only the ith component Pi of the magnetic moment P̃
is non-null, and is given by
Pi=ϑχH01+χñii†,
where H0 is the intensity of the inducing field H̃0
and the demagnetizing factor ñii† is defined
by Eq. (), ()
or (), according to the ellipsoid type.
Now, consider a confocal ellipsoid with semi-axes a′=a2+u,
b′=b2+u and c′=c2+u, where u is a
positive real number. From Eq. (), we can define the
isotropic susceptibility χ′ that is necessary to this confocal
ellipsoid produce the same magnetic moment P̃ (Eq. )
as follows:
χ′=Piϑ′H0-ñii†Pi,
where ϑ′=43πa′b′c′ and
ñii† is the new demagnetizing factor computed for the
confocal ellipsoid by using Eq. (), ()
or (), according to the ellipsoid type.
It can be shown that this confocal ellipsoid produces the same magnetic field
as the reference ellipsoid.
This ambiguity between confocal ellipsoids with the same magnetic moment
has already been pointed out by Clark (2014). It occurs for the
particular case in which the uniform inducing field is parallel
to an ellipsoid axis and there is no remanence.
Otherwise, the magnetic field produced by the confocal ellipsoids
will be different due to the shape anisotropy.
External magnetic field and total-field anomaly
The magnetic field ΔH(r)
produced by an ellipsoid at external points is calculated from
Eqs. () and () as the difference
between the resultant field H(r)
and the inducing field H0:
ΔH(r)=Vч(r̃)V⊤M,
where ч(r̃) is
the transformed depolarization tensor whose elements
ñii‡(r̃) and
ñij‡(r̃) are defined,
respectively, by Eqs. () and ().
ΔH(r)
represents the magnetic field produced by a uniformly magnetized
body located in the crust.
Equation () gives the magnetic field
(in Am-1) produced by an ellipsoid. However,
in geophysics, the most widely used field is the magnetic
induction (in nT). Fortunately, this conversion can
be easily done by multiplying Eq. ()
by km=109μ0, where μ0 represents the
magnetic constant (in Hm-1).
For geophysical applications, it is preferable to
calculate the total-field anomaly produced by the
magnetic sources. This scalar quantity is given by
ΔT(r)=‖B0+ΔB(r)‖-‖B0‖,
where B0=kmH0
and ΔB(r)=kmΔH(r),
with H0 and ΔH(r) defined,
respectively, by Eqs. () and ().
In practical situations, however,
‖B0‖>>‖ΔB(r)‖
and, consequently, the following approximation is valid :
ΔT(r)≈B0⊤ΔB(r)‖B0‖.
Numerical simulations
All the code developed for generating the results presented in the following
sections, as well as the code developed for generating additional numerical simulations,
can be found at the folder code of the online repository
https://github.com/pinga-lab/magnetic-ellipsoid.
Demagnetizing factors
We simulated a triaxial ellipsoid with semi-axes
a0=1000 m, b0=700 m and c0=200 m.
Then we used this ellipsoid as a reference to generate
100 different triaxial ellipsoids and calculate
their demagnetizing factors ñ11†,
ñ22† and ñ33†
by using Eqs. (),
() and ().
The semi-axes of these 100 ellipsoids are given by
a=a0+ub0, b=b0+ub0 and
c=c0+ub0, where 0≤u≤10.
Notice that, for u=0, the resulting semi-axes are equal to those
of the reference ellipsoid.
The larger the variable u, the larger the resulting semi-axes
a, b and c, but the smaller the relative difference between
them.
Consequently, the resulting ellipsoids obtained from the semi-axes
a, b and c become more spherical as u increases.
In this case, the demagnetizing factors ñ11†
(Eq. ),
ñ22† (Eq. )
and
ñ33† (Eq. )
tend to 1/3 e.g..
(a) Comparison between the demagnetizing factors
ñ11† (in red),
ñ22† (in green) and
ñ33† (in blue)
produced by 100 triaxial ellipsoids with semi-axes
a=a0+ub0, b=b0+ub0 and
c=c0+ub0, where 0≤u≤10 and
b0=700 m.
The demagnetizing factors were calculated
by using Eqs. (),
() and ().
(b) Comparison between the demagnetizing factors
ñ11† (in red) and
ñ22† (in green)
produced by 100 prolate ellipsoids with semi-axes
a=mb0 and b=b0, where 1.02≤m≤10 and
b0=1000 m.
The demagnetizing factors were calculated
by using Eqs. () and ().
(c) Comparison between the demagnetizing factors
ñ11† (in red) and
ñ22† (in green)
produced by 100 oblate ellipsoids with semi-axes
a=mb0 and b=b0, where 0.02≤m≤0.98 and
b0=1000 m.
The demagnetizing factors were calculated
by using Eqs. () and ().
The horizontal black line represents the value 1/3.
Figure a shows the calculated
demagnetizing factors ñ11† (in red),
ñ22† (in green) and
ñ33† (in blue) for the
100 triaxial ellipsoids.
The result shows that the relative difference
between the demagnetizing factors is large for
small values of u and decreases as u increases.
In this case, all demagnetizing factors tend to 1/3, according
to what we know from theory.
Also, Fig. a confirms that the demagnetizing
factors satisfy the condition
ñ11†<ñ22†<ñ33†
independently of the value of u.
We have also simulated 100 different prolate
ellipsoids with semi-axes a=mb0 and b=b0,
where 1.02≤m≤10 and b0=1000 m,
and calculate their demagnetizing factors
ñ11† and ñ22†
by using Eqs. () and (), respectively.
Similarly, we simulated 100 different oblate
ellipsoids with semi-axes a=mb0 and b=b0,
where 0.02≤m≤0.98 and b0=1000 m,
and calculate their demagnetizing factors
ñ11† and ñ22†
by using Eqs. () and (), respectively.
Figure b and c show the results obtained for
the 100 prolate and the 100 oblate ellipsoids, respectively.
As expected from theory, the demagnetizing factors
ñ11† (red line in Fig. b)
and
ñ22† (green line in Fig. b)
calculated for the prolate ellipsoids are close to 1/3 for m close to 1.
Also, these demagnetizing factors satisfy the condition
ñ11†<ñ22† for all values
of m.
The result obtained for the oblate ellipsoids (Fig. c)
are also in perfect agreement with theory.
The demagnetizing factors
ñ11† (in red) and
ñ22† (in green), which were calculated by using
Eqs. () and (),
respectively,
are close to 1/3 for m close to 0 and satisfy the condition
ñ11†>ñ22† for all values
of m.
Confocal ellipsoids
Parameters defining two confocal ellipsoids.
Parameter
Ellipsoid 1
Ellipsoid 2
Unit
Semi-axis a
900
≈1676.31
m
Semi-axis b
500
1500
m
Semi-axis c
100
≈1417.74
m
Coordinate of the centre xc
0
0
m
Coordinate of the centre yc
0
0
m
Coordinate of the centre zc
1500
1500
m
Orientation angle ε*
45
45
∘
Orientation angle ζ*
10
10
∘
Orientation angle η*
-30
-30
∘
Isotropic susceptibility χ
1.2
≈0.014
SI
* Defined in Fig. a.
We simulated two confocal ellipsoids by using the parameters
shown in Table .
The semi-axes of Ellipsoid 2 were defined as a2+u,
b2+u and c2+u, where a, b, and c
are the semi-axes of Ellipsoid 1 and u=2×106 m.
We have computed the total-field anomalies produced by Ellipsoid 1
and Ellipsoid 2 at the same regular grid of 200×200 points
located on a horizontal plane at z=0 m by using two different
inducing fields H0.
Total-field anomaly (in nT) produced by the
synthetic bodies Ellipsoid 1 and Ellipsoid 2, both
defined in Table .
The synthetic data produced by these confocal ellipsoids
were calculated on a regular grid of 200×200 points
at the constant vertical coordinate z=0 m.
These data were calculated with a uniform inducing field
parallel to the semi-axis a of the confocal ellipsoids.
In the first case, we used a uniform inducing field H0 which is
parallel to the semi-axis a of the confocal ellipsoids and
has inclination ≈-4.98∘, declination ≈15.38∘
and intensity H0≈18.7 Am-1.
The total-field anomaly produced by Ellipsoid 1 by using this
inducing field is shown in Fig. .
In this case, Ellipsoid 2 produces the same total-field
anomaly as Ellipsoid 1.
The isotropic susceptibility of Ellipsoid 2 was calculated with
Eq. () and consequently its magnetic moment is equal
to that of Ellipsoid 1.
Notice that the volume of Ellipsoid 2 is approximately 79 times greater
than that of Ellipsoid 1, whereas the isotropic susceptibility
of Ellipsoid 1 is approximately 85 times greater than that
of Ellipsoid 2.
This result illustrates the ambiguity between the field produced
by confocal ellipsoids with the same magnetic moment.
The second inducing field H0 used is oblique to
the semi-axes of the confocal ellipsoids, has the same
intensity as the other one, but a different direction.
In this case, the inclination and declination of H0
are, respectively, -30 and 60∘.
Figure a and b show the total-field anomalies
produced, respectively, by Ellipsoid 1 and Ellipsoid 2
by using this new inducing field H0.
Notice that by using this oblique inducing field,
the total-field anomalies produced by the confocal ellipsoids
are different from each other due to the shape anisotropy.
The differences are shown in Fig. c.
These results confirm numerically what was pointed out by
: confocal ellipsoids with
properly scaled isotropic susceptibilities, no remanence and
the same magnetic moment produce different magnetic fields at the
same external points, unless the inducing field happens to lie along one
of their axes.
Total-field anomalies (in nT) produced by (a) Ellipsoid 1
and (b) Ellipsoid 2, both defined in Table .
The synthetic data produced by these confocal ellipsoids
were calculated on a regular grid of 200×200 points
at the constant vertical coordinate z=0 m.
These data were calculated with a uniform inducing field
which is oblique to the semi-axes of the confocal ellipsoids.
(c) Difference between the total-field anomalies shown in (b) and (a).
Simulation of a geological body
We simulated an ellipsoidal body similar to the Warrego orebody,
which was the resource on which the well-known Warrego mine developed in Tennant Creek, Australia. After nearly a decade as
one of the most important gold and copper mines in Australia, the
Warrego mine was closed in late 1989.
According to , the Warrego orebody is a combination of
two major and several small ironstone lodes, which are discrete bodies
comprised predominantly of magnetite or hematite above the base of
oxidation.
represented the Warrego orebody as a triaxial
ellipsoid having a high isotropic susceptibility.
In this case, the self-demagnetization strongly impacts the
magnetic modelling of this body.
Table shows the parameters defining
a synthetic orebody which is based on that presented by
to represent the Warrego orebody.
Figure shows the total-field anomaly
ΔT(r) (Eq. ) produced
by the synthetic body on a regular grid of 100×100
points at a constant vertical coordinate z=0 m.
The total-field anomaly varies from ≈-71 nT
to ≈482 nT, resulting in a peak-to-peak amplitude
of ≈553 nT, and was calculated by using the
true magnetization M defined in Eqs. () and ().
Parameters defining a synthetic orebody. This model is
based on that presented by to simulate
the Warrego orebody, Tennant Creek field, Australia.
Parameter
Value
Unit
Semi-axis a
490.7
m
Semi-axis b
69.7
m
Semi-axis c
30.0
m
Coordinate of the centre xc
0
m
Coordinate of the centre yc
0
m
Coordinate of the centre zc
500
m
Orientation angle ε1
-34.0
∘
Orientation angle ζ1
66.1
∘
Orientation angle η1
45.0
∘
Isotropic susceptibility χ
1.69
SI
x component of the inducing field B0 2
32610
nT
y component of the inducing field B0 2
0
nT
z component of the inducing field B0 2
39450
nT
1 Defined in Fig. a.
2 Defined in Eq. ().
Total-field anomaly (in nT) produced by the synthetic orebody
defined in Table . The synthetic data
are calculated on a regular grid of 100×100 points
at the constant vertical coordinate z=0 m.
We have calculated the difference between the total-field anomaly
ΔT(r) (Eq. ) calculated with the
true magnetization M (Eqs. and )
and that calculated with the approximated magnetization
M˘ (Eq. ).
The differences were calculated by using the synthetic body
defined in Table , but with three
different isotropic susceptibilities.
Figure a, b
and c show the differences calculated
by using, respectively, isotropic susceptibilities χ=1.69 SI
(Table ), χ1=0.1 SI and
χ2=0.116 SI.
As expected, the differences calculated by using the higher isotropic
susceptibility (Fig. a) are very large.
The peak-to-peak amplitude is ≈40 nT
and represents ≈8% of the peak-to-peak amplitude of
the total-field anomaly shown in Fig. .
Difference between the total-field
anomaly calculated with the approximated magnetization M˘
(Eq. ) and with the
true magnetization M
(Eqs. and ).
The total-field anomalies are in nT and were
calculated with Eq. (),
on a regular grid of 100×100 points,
at the constant vertical coordinate z=0 m.
The differences are produced by the synthetic orebody defined in
Table , but with different
isotropic susceptibilities:
(a) the isotropic susceptibility defined in
Table ,
(b) an isotropic susceptibility χ=0.1 SI and
(c) an isotropic susceptibility χ=0.116 SI.
This last value was calculated
with Eq. (), by using ϵ=8%.
Figure b shows the differences calculated
by using χ1=0.1 SI.
It is commonly accepted that, for bodies having isotropic susceptibilities
lower than or equal to 0.1 SI, the self-demagnetization can be neglected and,
consequently, the magnetization M˘ (Eq. )
is a good approximation of the true magnetization M
(Eqs. and ).
In our test, the use of χ1=0.1 SI leads to a relative error
‖δM‖‖M‖-1≈0.7%
(Eq. ) in the magnetization.
The peak-to-peak amplitude of the differences in the total-field anomaly
(Fig. b) is ≈0.2 nT,
which represents ≈0.6% of the peak-to-peak amplitude of the
total-field anomaly calculated by using the true magnetization M
(Eqs. and ).
Finally, Fig. c shows the differences
calculated by using χ2=0.116 SI.
This value was calculated by using Eq. () with a
target relative error ϵ=8% and the ñmax†
defined by Eq. ().
By using this isotropic susceptibility, it is expected that the
calculated relative error
‖δM‖‖M‖-1
(Eq. ) in the magnetization be
lower than or equal to the target relative error ϵ=8%.
In this test, the use of χ2=0.116 SI leads to a relative error
‖δM‖‖M‖-1≈0.8%
(Eq. ) in the magnetization.
The peak-to-peak amplitude of the differences in the total-field anomaly
(Fig. c) is ≈0.3 nT,
which represents ≈0.7% of the peak-to-peak amplitude of the
total-field anomaly calculated by using the true magnetization M
(Eqs. and ).
In this case, the use of an isotropic susceptibility greater than the usual
limit 0.1 SI does not mislead the magnetic modelling
dramatically. On the contrary, it shows small discrepancies in the
magnetic modelling and validates Eq. ().