A considerable amount of literature has been published on the magnetic modelling of uniformly magnetized ellipsoids since the second half of the nineteenth century. Ellipsoids have flexibility to represent a wide range of geometrical forms, are the only known bodies which can be uniformly magnetized in the presence of a uniform inducing field and are the only finite bodies for which the self-demagnetization can be treated analytically. This property makes ellipsoids particularly useful for modelling compact orebodies having high susceptibility. In this case, neglecting the self-demagnetization may strongly mislead the interpretation of these bodies by using magnetic methods. A number of previous studies consider that the self-demagnetization can be neglected for the case in which the geological body has an isotropic susceptibility lower than or equal to 0.1 SI. This limiting value, however, seems to be determined empirically and there has been no discussion about how this value was determined. In addition, the geoscientific community lacks an easy-to-use tool to simulate the magnetic field produced by uniformly magnetized ellipsoids. Here, we present an integrated review of the magnetic modelling of arbitrarily oriented triaxial, prolate and oblate ellipsoids. Our review includes ellipsoids with both induced and remanent magnetization, as well as with isotropic or anisotropic susceptibility. We also discuss the ambiguity between confocal ellipsoids with the same magnetic moment and propose a way of determining the isotropic susceptibility above which the self-demagnetization must be taken into consideration. Tests with synthetic data validate our approach. Finally, we 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.

Based on the mathematical theory of the magnetic
induction developed by

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

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

Schematic representation of the coordinate systems used to
represent an ellipsoidal body.

Let

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.

Consider a magnetized ellipsoid immersed in
a uniform inducing magnetic field

Based on

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

By using the magnetization

The following part of this paper moves on to describe
the magnetic field

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

It is convenient to use

The second important coordinate transformation is defined
with respect to Eq. (

In Eq. (

Let

Note that, according to Eqs. (

For triaxial ellipsoids (e.g.

For prolate ellipsoids (e.g.

For oblate ellipsoids (e.g.

The elements

For triaxial ellipsoids (e.g.

For prolate (e.g.

For oblate (e.g.

By considering

Let us pre-multiply the uniform internal field

Equation (

For the particular case in which the susceptibility is isotropic,
the susceptibility tensor is defined according to Eq. (

In the case of isotropic susceptibility, the resultant magnetization

Consider a perturbed matrix

By using the concept of vector norm and its corresponding
operator norm

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

Now, consider a confocal ellipsoid with semi-axes

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.

The magnetic field

The code is implemented in the Python language,
by using the NumPy and SciPy libraries

The numerical simulations presented here were generated with
the Jupyter Notebook
(

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

We simulated a triaxial ellipsoid with semi-axes

Figure

We have also simulated 100 different prolate
ellipsoids with semi-axes

Figure

Parameters defining two confocal ellipsoids.

We simulated two confocal ellipsoids by using the parameters
shown in Table

We have computed the total-field anomalies produced by Ellipsoid 1
and Ellipsoid 2 at the same regular grid of

Total-field anomaly (in

In the first case, we used a uniform inducing field

The second inducing field

Total-field anomalies (in

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

Table

Parameters defining a synthetic orebody. This model is
based on that presented by

Total-field anomaly (in

We have calculated the difference between the total-field anomaly

As expected, the differences calculated by using the higher isotropic
susceptibility (Fig.

Difference between the total-field
anomaly calculated with the approximated magnetization

Figure

Finally, Fig.

We present an integrated review of the vast literature about the magnetic modelling of triaxial, prolate and oblate ellipsoids. We also present a numerical simulation confirming the ambiguity between confocal ellipsoids with the same magnetic moment and present 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. Our approach is an alternative to the constant value which seems to be determined empirically and has been used by the geoscientific community. Our alternative approach is validated by the results obtained with numerical simulations. In a future work, it would be interesting to use a similar approach to determine bounds for the maximum relative error in the magnetic field calculated by neglecting the self-demagnetization.

This work also provides a set of routines to model the magnetic
field produced by ellipsoids.
The routines are written in the Python language as part of
the Fatiando a Terra

The current version of our code is freely distributed under the
BSD 3-clause licence and it is available for download at
Zenodo:

Let

By considering the functions

Now, by deriving

Here, we follow the reasoning presented by

Let us consider an ellipsoid with semi-axes

Given

From Eq. (

Let us now consider a prolate ellipsoid with semi-axes

Similarly to the case of a triaxial ellipsoid presented
in the previous section, we are interested in
determining the real number

By properly manipulating Eq. (

In the case of oblate ellipsoids, the procedure for determining
the parameter

The magnetic modelling of triaxial, prolate or oblate ellipsoids
requires not only the parameter lambda defined by Eqs. (

Let us first consider a triaxial ellipsoid. In this case,
the spatial derivatives of

The authors declare that they have no conflict of interest.

Diego Takahashi thanks the Brazilian research funding agency Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) for providing financial support in the form of a scholarship. Vanderlei C. Oliveira Jr. thanks the Brazilian research funding agency Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) for providing financial support in the form of a grant (445752/2014-9). Edited by: Lutz Gross Reviewed by: David Clark and Ralf Schaa