An alternative 3D inversion method for magnetic anomalies with depth resolution

Abstract

This paper presents a new method to invert magnetic anomaly data in a variety of non-complex contexts when a priori information about the sources is not available. The region containing magnetic sources is discretized into a set of homogeneously magnetized rectangular prisms, polarized along a common direction. The magnetization distribution is calculated by solving an underdetermined linear system, and is accomplished through the simultaneous minimization of the norm of the solution and the misfit between the observed and the calculated field. Our algorithm makes use of a dipolar approximation to compute the magnetic field of the rectangular blocks. We show how this approximation, in conjunction with other correction factors, presents numerous advantages in terms of computing speed and depth resolution, and does not affect significantly the success of the inversion. The algorithm is tested on both synthetic and real magnetic datasets.

Full text

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1021
ANNALS OF GEOPHYSICS, VOL. 49, N. 4/5, August/October 2006
Key words inversion – magnetic field – weighting
function – dipolar approximation – depth resolution
1. Introduction
It is well known how inversion applied to
surface potential field data is a mathematical
ill-posed problem which suffers from ambigui-
ty of solutions (Blakely, 1995). This is due to
the inherent nonuniqueness of the problem and
the inaccuracy and finite number of the poten-
tial field observations.
A common strategy to approach the inverse
problem is to divide the subsoil into homoge-
neous magnetized rectangular blocks. The di-
rection of magnetization is supposed to be
known in advance and equal for each block.
The unknown parameters to be solved are the
module of magnetizations. These parameters
can be solved by means of a linear system with
a number of equations equal to the data avail-
able. Unfortunately the magnetic stations are
generally lower than the blocks, so the linear
system is undetermined and the solution can be
obtained only by simultaneously minimizing an
objective function and the misfit between the
observed magnetic field and the field produced
by the magnetized blocks (Hoerl and Kennard,
1970; Last and Kubik, 1983; Silva and Hoh-
mann, 1983; Guillen and Menichetti, 1984; Li
and Oldenburg, 1996). The main disadvantage
of this strategy is that the definition of the ob-
jective function depends on a priori knowledge
of the source and this makes algorithms non
general, so that a unique approach for inverting
magnetic data cannot be used in every situation.
However the use of the objective function is
necessary in order to reduce the instability of
the solutions.
The algorithm proposed is based on the hy-
pothesis that no information about the source is
available. Thus a general and appropriate objec-
An alternative 3D inversion method
for magnetic anomalies
with depth resolution
Alessandro Pignatelli, Iacopo Nicolosi and Massimo Chiappini
Istituto Nazionale di Geofisica e Vulcanologia, Roma, Italy
Abstract
This paper presents a new method to invert magnetic anomaly data in a variety of non-complex contexts when
a priori information about the sources is not available. The region containing magnetic sources is discretized in-
to a set of homogeneously magnetized rectangular prisms, polarized along a common direction. The magnetiza-
tion distribution is calculated by solving an underdetermined linear system, and is accomplished through the si-
multaneous minimization of the norm of the solution and the misfit between the observed and the calculated
field. Our algorithm makes use of a dipolar approximation to compute the magnetic field of the rectangular
blocks. We show how this approximation, in conjunction with other correction factors, presents numerous ad-
vantages in terms of computing speed and depth resolution, and does not affect significantly the success of the
inversion. The algorithm is tested on both synthetic and real magnetic datasets.
Mailing address: Dr. Alessandro Pignatelli, Istituto Na-
zionale di Geofisica e Vulcanologia, Via di Vigna Murata
605, 00143 Roma, Italy; e-mail: pignatelli@ingv.it

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1022
Alessandro Pignatelli, Iacopo Nicolosi and Massimo Chiappini
tive function has to be chosen. In this work we
show that the use of the norm minimization of
solution as objective function is a good choice.
However the inversion procedure using the
minimization of norm is influenced by the rapid
decrease of the signal amplitude as a function of
depth, so the solutions tend to be located only in
the upper layers of the earth space model (Fedi
and Rapolla, 1999). So an inversion based on a
simultaneous minimization of the misfit of the
data and the Euclidean norm of the solution
causes the lack of depth resolution. To overcome
this problem it is possible to use a weighting
function (Li and Oldenburg,1996; Portnianguine
and Zhdanov, 2002). Alternatively, Fedi and
Rapolla (1999) have proposed to use a multi-ob-
servation level dataset, arguing that this kind of
data provide depth resolution. However, this
statement is still a subject of some debate (Old-
enburg and Li, 2003). An additional limitation of
inversion methods is related to the long comput-
ing time involved in the calculation of the mag-
netic field of a set of rectangular prisms, which
can be implemented using, for instance, the for-
mulation deduced by Sharma (1986).
This paper describes a new magnetic inver-
sion method in which the magnetic field of a
rectangular prisms is approximated by the field
of a magnetic dipole. This approximation does
not affect significantly the success of the inver-
sion method. Furthermore, this approach makes
it possible, on the one hand, the choice of a
weighting function valid for every magnetic
source geometry and, on the other, to improve
the calculations in terms of computing time. Af-
ter a description of our inversion technique, we
show its efficiency on both synthetic and real
magnetic data sets.
2. Description of the magnetic inversion
method
Let us assume we have an anomaly data set
y of N observations
y=(y1, y2, ..., yN).(2.1)
From the electromagnetic theory, we can obtain
the value of the magnetic field along a generic
direction f produced by a magnetization distri-
bution J(r) as follows:
(2.2)
where rl is the position vector of the observa-
tion point, t is a unit vector in the direction of
magnetization and V is the volume containing
all magnetic sources.
Equation (2.2) can be simplified if a uni-
form magnetization direction, t, for all the dis-
tribution is considered. In fact, in most real cas-
es, a constant direction parallel to the magnetic
ambient field is chosen. This approximation is
reasonable when the most important compo-
nent of the total magnetization vector is the in-
duced one or when the remanent magnetization
is parallel to the earth’s magnetic field. Despite
this simplification, calculation of this integral
can only be achieved analytically if the geome-
try of volume V, and the distribution of magne-
tization J(r) are simple enough.
Theoretically, the purpose of magnetic in-
version is to calculate a magnetization distribu-
tion from the observed magnetic anomaly data.
However, this is not possible because an analyt-
ic solution of the integral eq. (2.2) in the param-
eter J(r) does not exist. For that reason, the
common approach consists of dividing the re-
gion containing magnetic sources into elemen-
tary rectangular prisms cells in which magneti-
zation is assumed to be uniform. Thus, eq. (2.2)
can be rewritten in the following way:
(2.3)
where Ji is the module of the magnetization of
the ith prism, M is the number of blocks, rj is
the position vector of the jth observation point
and xi,jis the discrete version of the Green’s
function associated with eq. (2.2), which de-
pends not only on the block geometry, but also
on the relative position between the ith cell and
the jth observation point.
The discretization process produces a linear
system in which the unknown parameters are
the magnetizations Ji of each cell. Generally,
systems of this kind are underdetermined and
overconstrained (Jackson, 1972), so that opti-
( ) , ,....,1 2
T rx J
ij
jN
ji
i
M
1
∆
==
=/
2t
( )
r
( )
r
−
T
J
r
V
r
V
22
∆
=
l
l
2f
d
#

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An alternative 3D inversion method for magnetic anomalies with depth resolution
mization techniques become necessary in order
to evaluate the blocks magnetization.
We use the Levemberg and Marquardt tech-
nique (Press et al., 1992) minimizing simulta-
neously the norm of the solution and the misfit
between the observed and the computed field.
By means of this iterative technique it is possi-
ble to introduce the positivity constraint by set-
ting to zero the negative magnetization values.
The iterative process is stopped when there are
no significant improvements in data conver-
gence. The effect of the minimization of the
norm is the concentration of the solution in the
upper layers of the cells instead of around the
real depth due to the decay of the kernel ampli-
tude with depth. Our method is designed to
minimize this problem.
Let us multiply and divide each term of the
sum by a weighting function
(2.4)
where zij is the distance between the ith dipole
and the altitude of jth station.
Replacing Ji/w(zij) by Jri and w(zij)xij by Xij,
the linear system to be solved becomes
(2.5)
By solving this new linear system with the iter-
ative method we obtain the set Jri. The initial
guess model is the one with all the magnetiza-
tions null.
Through this procedure, a geometrical dis-
tribution of weighted magnetizations for every
block is achieved. However, it is not convenient
to recover Ji by simply multiplying Jri by w(z).
In fact, due to simultaneous minimization of the
misfit between the observed data and the com-
puted field together with the euclidean norm of
solution assuming only positive values for mag-
netization modules, we obtain a partial fit of the
data during the inversion procedure.
The main task of this work has been to find
an appropriate weighting function and a method
for correcting magnetization values, as ex-
plained in detail in the following paragraphs.
( )
r
, ,..., 1 2
T X J
ij
jN
ji
i
M
1
∆
==
=/
( )
r
( )
( )
, ,...,
1 2
T w z xw z
J
jN
jij ij
ij
i
i
M
1
∆
==
=/
2.1. The weighting function
Equation (2.3) can be used to calculate the
magnetic field produced by a given set of mag-
netized blocks. Usually, the geometrical factors
xi,j are computed using the analytical formula
outlined by Sharma (1986), which calculates
the magnetic anomaly produced by a uniformly
magnetized prismatic block. Unfortunately, this
approach has the great disadvantage of being
CPU-time consuming.
To reduce computational time, we have re-
placed Sharma’s analytical expression with the
field produced by magnetic dipole located at
the geometrical center of each elementary cell.
The dipolar approximation is valid if the dis-
tance between the observation point and the
source is much higher than the source size, a re-
quirement that is ensured if a relatively small
cell size is chosen.
The real advantage of the dipole approxima-
tion, however, is that we can define the weight-
ing function as the dipolar field decay
(2.6)
where zijis the vertical distance between the ith
dipole and the altitude of jth station.
2.2. Correction of the magnetization values
The values of Jri obtained by inverting eq.
(2.5) by means of Levemberg and Marquardt
technique (Press et al., 1992) are the obtained
final parameters which best fit simultaneously
the anomaly field and the euclidean norm of the
solution.
In order to minimize only the misfit of the
data, let’s write the final magnetizations Ji
the following way:
fin
(2.7)
where Ai are corrective factors to be deter-
mined. We assume that these factors depend on-
ly on the depth to the dipole and not on its hor-
izontal position, that is
, ,...,1 2 , ,...,1 2
JA w J
i
iMjN
,
i
f
i ji
===
( ), ,..., 1 2, ,..., 1 2
w zziMjN
ij ij
3
===
1023

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1024
Alessandro Pignatelli, Iacopo Nicolosi and Massimo Chiappini
(2.8)
where ziis the depth of the jth dipole measured
from the horizontal plane z=0.
In order to compute the magnetization of
each block, we just need the set of parameters:
A1(z1), A2(z2), ..., Ah(zh), where h is the number
of layers along the z-axis chosen to perform the
inversion. To compute the right factors, it is
necessary a new data misfit minimization in
which the following equation must be satisfied:
(2.9)
Now, the number of unknown parameters,
Ai(zi), is lower than the number of data and the
problem of undertermination has been over-
come. So we do not need the minimization of
the norm together with the minimization of the
misfit of the data. After this second inversion,
we can obtain the magnetization value of the ith
block by applying
.
(2.10)
3. Application of the method
3.1. A synthetic case
The proposed inversion method has been
tested using a synthetic and a real data sets. For
the synthetic test we have constructed two mag-
netic sources with a complex geometry located
at different depths, with the aim of reproducing
actual geological structures (fig. 1a). The mag-
netization intensity is of 2.5 A/m for the shal-
low source and 5.0 A/m for the deep one.
The magnetic anomaly produced by these
structures has been calculated on a 21×21 km2
regular grid with a cell size of 1 km, at an alti-
tude of 100 m (fig. 1b). In addition, to simulate
a measured anomaly field, a Gaussian noise
with 0 mean and standard deviation of 1 nT was
added to the calculated anomaly. The inversion
was then performed on a 12×12×5 km lattice
, ,...,1 2, ,...,1 2
J J A w
i
iMjN
i
f
iij
===
( ) ( )
A z w z x J
ii
, ,..., .1 2
yjN
i
M
ij ijij
1
==
=/
)
i
, ,...,1 2(
A A ziM
i
==
with a 300×300×250 m elementary block cell,
which has been selected in order to not match
exactly the synthetic sources. On the left hand
side of fig. 2, the result of the inversion previ-
ous to the correction of magnetization values is
shown. The various slices represent horizontal
cross sections of the lattice at different depths.
The position of the synthetic sources is outlined
using thick black lines. The magnetization val-
ue correction was then applied according to eq.
(2.10), and the new calculated source distribu-
tion is shown on the right hand side of fig. 2.
This graphical representation is particularly
suitable to easily compare both results at the
same depths. Without the correction factors, the
solution appears more elongated on the vertical
axis, whereas with our method a more reliable
location of the sources, along the vertical axis,
and a better characterization of the magnetic
properties is achieved.
3.2. A real case
To test our method on a real case, we have
chosen a data set acquired during a ground-based
magnetic survey conducted for environmental in-
vestigations (Marchetti et al., 1998). In this case
study, the parameters of the sources and their ex-
act location are well known in advance, therefore
it is an excellent test for our purposes. A set of 12
empty 55-gallons steel drums were buried at a
depth of 4.5 m (fig. 3a). The magnetic anomaly
field produced by the drums was measured at
0.75 m above the ground and the interpolated
grid with a cell size of 1 m is shown in fig. 3b.
The inversion was then applied using a 22×
×25×10 m lattice with a 1×1×0.5 m elemen-
tary block cell. Without any correction, the so-
lution of the inversion is shown in fig. 4 (left
hand side). The obtained magnetic sources
show an elongated pattern along the vertical ax-
is which is similar to the one observed in the
synthetic case. The application of our algorithm
improves the source characterization, as shown
on right hand side of fig. 4, where the solution
is more compact. The obtained source magneti-
zation is about 3 A/m, which is several orders of
magnitude lower than the magnetization of
steel (Ravat, 1996; Marchetti et al., 1998). This

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An alternative 3D inversion method for magnetic anomalies with depth resolution
apparent mismatch is due to the peculiar shape
of the source, with non-magnetic or empty
space inside the drums. Therefore, the estimat-
ed magnetization value MZwould represent a
bulk magnetization for all the source volume, as
given by
(3.1)
where M(x,y,z) represent the distribution of
magnetization in the volume V occupied by the
steel drums.
V
( , , )d
M x y zM
V
1
V
=
#
1025
Fig. 1a,b. a) Synthetic magnetic source; b) total-field anomaly produced at an altitude of 100 m above the sur-
face. Grid spacing: 1 km.
Fig. 2. Results of the inversion using synthetic data. The column on the left-hand side shows the solution with-
out the magnetization values correction, whereas the right-hand side reports the results calculated after the mag-
netization correction. The slices represent horizontal cross sections of the lattice at various depths. The inversion
was performed on a 12×12×5 km lattice with 300×300×250 m elementary cells. The thick black lines outline the
cross sections of the true source.
1
2
a
b

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Alessandro Pignatelli, Iacopo Nicolosi and Massimo Chiappini
4. Conclusions
We have proposed a new method for the 3D
inversion of magnetic anomalies when a priori
knowledge on the sources and a multi-observa-
tion level dataset are not available. We have
shown that it is possible to estimate the location
and magnetization intensity of magnetic sources
by means of a discretization of the subsoil into
elementary parallelepipeds whose magnetic ef-
fect is approximated by a dipolar field. The
main advantage of this approach, besides a great
improvement in computing speed, is that it al-
lows the choice of an analytic weighting func-
tion through which the source is obtained with
good depth resolution. Compared with other
methods, this weighting function does not de-
pend on the geometry of the elementary cells,
Fig. 3a,b. a) Steel drums buried at a depth of about 4.5 m; b) magnetic anomaly field produced at an altitude
of 0.75 m above the surface. Grid spacing: 1 m.
Fig. 4. Results of the inversion using real data. The column on the left-hand side shows the solution without
the magnetization values correction, whereas the right-hand side reports the results calculated after the magne-
tization correction. The slices represent horizontal cross sections of the lattice at various depths. The inversion
was performed on a 22×25×10 m lattice with 1×1×0.5 m elementary cells.
4
3
a
b

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An alternative 3D inversion method for magnetic anomalies with depth resolution
and therefore is valid for every discretized re-
gion. Finally, an adjustment on the estimation of
the unknown parameters is proposed in order to
further improve our knowledge about the loca-
tion and magnetization intensity of the sources.
We have showed the results on a synthetic mag-
netic field dataset.
Acknowledgements
Isabel Blanco Montenegro and an anonymous
referee provided thorough comments on the man-
uscript. We are also thankful to the Professor
Dhananjay Ravat for all his helpful comments.
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(received January 16, 2006;
accepted June 20, 2006)