We have
developed a fast total-field anomaly inversion to estimate the magnetization
direction of multiple sources with approximately spherical shapes and known
centres. Our method is an overdetermined inverse problem that can be applied
to interpret multiple sources with different but homogeneous magnetization
directions. It requires neither the prior computation of any
transformation-like reduction to the pole nor the use of regularly spaced
data on a horizontal grid. The method contains flexibility to be implemented
as a linear or non-linear inverse problem, which results, respectively, in a
least-squares or robust estimate of the components of the magnetization
vector of the sources. Applications to synthetic data show the robustness of
our method against interfering anomalies and errors in the location of the
sources' centre. Besides, we show the feasibility of applying the upward
continuation to interpret non-spherical sources. Applications to field data
over the Goiás alkaline province (GAP), Brazil, show the good performance
of our method in estimating geologically meaningful magnetization directions.
The results obtained for a region of the GAP, near to the alkaline complex of
Diorama, suggest the presence of non-outcropping sources marked by strong
remanent magnetization with inclination and declination close to

The magnetic method is one of the oldest geophysical techniques
and plays an important role in mineral and petroleum exploration. This method
underwent great progress after the advent of magnetometers properly developed
for airborne surveys. Nowadays, the combination of modern satellite
positioning systems and improvements in instrumentation and platform
compensation makes the aeromagnetic survey one of the most important data
acquisition techniques due to the ability to cover large areas in a
relatively short period of time

The total field is the most common magnetic data measured in a survey. It is
defined as the Euclidean norm of the magnetic induction produced by all
surrounding magnetic sources. After removing the Euclidean norm of the
magnetic induction predicted by a global model describing the geomagnetic
field and correcting the wide range of undesirable artefacts affecting the
data, the result is a scalar quantity denominated total-field anomaly. By
using the total-field anomalies, the geophysicist can characterize the
magnetic sources in the subsurface and then better define exploration targets

We divide the methods for retrieving the magnetization direction into two
groups. The first one comprises methods that do not impose strong constraints
on the shape of the sources.

The second group of methods to estimate the magnetization direction of the
sources assumes knowledge of the shape of the source. The methods belonging
to this group have led to a few published papers.

In this work, we present a computationally efficient method for inverting the
total-field anomaly produced by multiple sources with approximately spherical
shapes to estimate their magnetization directions. We assume sources with
known centres, which can be provided by Euler deconvolution, for example. The
proposed method is part of the group of methods imposing assumptions about
the shape of the magnetic sources. This assumption about the underlying
sources is able to reduce the non-uniqueness of the problem to a point that
regularization or constraints are not required. Our method can be applied for
estimating the average magnetization direction of multiple sources. It
requires neither that all sources have the same magnetization direction nor
the use of regularly spaced data on a horizontal grid. Besides, our method
also contains flexibility to be implemented in two different numerical
approaches. The first one minimizes an L2-norm, resulting in a linear inverse
problem to obtain a least-squares estimate. The second approach comprises the
iterative minimization of an L1-norm, resulting in a non-linear inverse
problem to obtain a robust estimate. Applications to synthetic data show the
robustness of our method against interfering anomalies and errors in the
location of the sources' centres. Additionally, we show how the upward
continuation can be used to make possible the application of our method to
interpret non-spherical sources. Applications to field data over the
Goiás alkaline province (GAP), Brazil, show the good performance of the
proposed method in estimating geologically meaningful magnetization
directions. The obtained results over a region of the GAP, near to the
alkaline complex of Diorama, suggest the presence of non-outcropping sources
with strong remanent magnetization, corroborating previous works. The
estimated inclinations and declinations are close to

Let

Schematic representation of

For local- or regional-scale magnetic studies, it is very common to consider
that (i) the geomagnetic field

Let us consider that the magnetic sources can be represented
by a set of

Schematic representation of the vector

We assume that the magnetic sources giving rise to the observed data

The least-squares estimate

Both

In a magnetic survey, the measurements are always affected by noise due to
the wide range of experimental errors and inaccuracies that happen in a
geophysical survey. The noise in the observed data

The diagonal of the parameter covariance matrices

The inversion method described above is implemented in version 0.3 of the
open-source Python language library Fatiando a
Terra

Test with the synthetic data (Fig.

Figure

Validation test and robustness against interfering anomalies.

By assigning the correct positions of the centres of the simulated bodies, we
invert the noise-corrupted total-field anomaly (Fig.

Figure

Robustness against non-spherical sources.

We repeated the numerical test presented in the previous section (Sect. 3.1),
but using the contaminated total-field anomaly shown in
Fig.

Test with the synthetic data (Fig.

Robustness against non-spherical sources. Noise-corrupted
total-field anomaly produced by each one of the three rectangular prisms
shown in Fig.

In the previous subsections, we applied our method to estimate the
magnetization direction of a rectangular prism whose total-field anomaly is
indicated by B in Fig.

Robustness against non-spherical sources. The blue and red dots
represent, respectively, the results obtained with the least-squares

Figure

We applied our method to interpret these 33 data sets and the results are
shown in Fig.

The greater the distance between the sources and the data, the greater the attenuation of the non-dipolar features, and thus the smaller the difference between the least-squares and robust estimates. In this case, a good practice when applying our method is to perform an upward continuation of the total-field anomaly to be inverted.

In all previous tests with synthetic data, we presume the correct location of
the centre of the sources. However, in real world scenarios, the positions of
the sources cannot be obtained directly and have to be estimated. This
estimation can be done, for example, by using the Euler deconvolution
technique

We simulated a uniformly magnetized sphere (not shown) with centre at

We applied our method to these synthetic data for estimating the
magnetization direction of the simulated spherical body. This application was
done by presuming different locations of the centre of the source along three
orthogonal straight lines which are parallel to the

We can clearly see that the wrong choice of the

Robustness against errors in the centre location. The blue and red
dots represent, respectively, the magnetization direction of a simulated
spherical body obtained with the least-squares

Strongly interfering anomalies. Synthetic prisms (in red) with side
lengths equal to 80, 20, and 70

In this section, we present the performance of our method in recovering the magnetization direction of synthetic sources simulating complex geological scenarios.

Figure

By assigning the correct positions of the centres of the simulated bodies, we
invert the noise-corrupted total-field anomaly
(Fig.

Strongly interfering anomalies. Noise-corrupted total-field anomaly
produced by the synthetic bodies shown in Fig.

Igneous intrusion. 2-D schematic representation of a synthetic
geologic setting composed of a non-magnetic sedimentary package (in grey), an
igneous intrusion (in red) and a basement (in white). The sedimentary package
and basement are semi-infinite along the

Figure

Igneous intrusion. Noise-corrupted total-field anomaly produced by
the synthetic bodies shown schematically in Fig.

Test with synthetic data (Fig.

We applied our method to the total-field anomaly shown in
Fig.

Igneous intrusion. 3-D view of the intrusion (red prisms) and the estimate of the intrusion position by using Euler deconvolution (black point) with a structural index equal to 3. Notice that the Euler solution falls outside the intrusion.

Test with synthetic data (Fig.

In Goiás state, central region of Brazil, there are occurrences of
Cretaceous alkaline rocks along a lineament NW–SE that have been studied
since the 60s. In a broad regional-scale study,

Igneous intrusion. The upper panel shows the true
reduced-to-the-pole anomaly produced by the synthetic bodies shown in
Fig.

The GAP is formed by mafic to ultramafic alkaline rocks presenting a wide
variety of petrographic types

We applied our method to interpret the data located in the area delimited by
the red rectangle shown in Fig.

Application to field data on the Goiás alkaline province (GAP),
Brazil. Simplified geological map of the study area, which is shown as a red
dot on the inset map of Brazil. The inset also shows the Goiás (dark grey
area) and Minas Gerais (light grey area) states. The total-field anomaly over
the area delimited by the red rectangle is shown in
Fig.

Application to field data on the Goiás alkaline province (GAP),
Brazil. Total-field anomaly observed over the area delimited by the red
rectangle in Fig.

Application to field data on the Goiás alkaline province (GAP),
Brazil. Observed total-field anomaly (Fig.

Application to field data on the Goiás alkaline province (GAP),
Brazil. Total-field anomaly observed over the Montes Claros de Goiás
alkaline complex (Fig.

Application to field data on the Goiás alkaline province (GAP),
Brazil. Observed total-field anomaly (Fig.

For verifying the plausibility of the estimated inclinations and
declinations, we used them to reduce the observed total-field anomaly
(Fig.

By using the estimated magnetization directions obtained from the simple
dipolar total-field anomaly shown in Fig.

We present a computationally effective method for estimating the magnetization direction of multiple sources with approximately spherical shapes by inverting the total-field anomaly produced by them. Our method assumes that the sources have uniform magnetization and that the positions of their centres are known. Prior knowledge about the source sizes is not required. Our method can be applied for determining the average magnetization direction within multiple sources with different magnetization directions. Besides, it can be directly applied to interpret irregularly spaced total-field anomaly data measured on uneven surfaces and requires no prior transformation like reduction to the pole, total gradient or total magnitude anomalies. The method also contains flexibility to be implemented in two different numerical approaches. The first one is based on the minimization of the L2-norm of the residuals between the observed and predicted total-field anomalies. This approach results in a linear inverse problem for obtaining a least-squares estimate of the magnetization vector components of the sources. The second approach is based on the minimization of the L1-norm of the residuals between the observed and predicted total-field anomalies, leading to a non-linear inverse problem for obtaining a robust estimate of the magnetization vector components of the sources.

The results obtained with the synthetic data simulating a spherical source with a known centre show the good performance of our method in retrieving the true magnetization direction. Tests with synthetic data produced by simulated sources that violate the premises assumed by our method show the robustness of our method against interfering anomalies and against errors in the location of the centre of the source. The results show that our method is sensitive to errors in the horizontal location of the centre of the source. On the other hand, it is insensitive to errors in the depth of the centre of the source. Additionally, we show how the upward continuation can be used to make possible the application of our method for interpreting non-spherical sources producing total-field anomalies with non-dipolar features. These non-dipolar features can greatly affect the results obtained with the least-squares estimate, especially when the data are near to the source. Applications to field data over the Goiás alkaline province (GAP), Brazil, show that our method can be a powerful tool for interpreting real geological scenarios. Our estimates near to the alkaline complex of Diorama suggest the presence of non-outcropping sources with strong remanent magnetization, corroborating previous works. This estimated magnetization direction leads to very plausible RTP anomalies not only over the region near to the complex of Diorama, but also over the alkaline complex of Montes Claros de Goiás. These results show that the non-outcropping sources near to the alkaline complex of Diorama have almost the same magnetization direction of those ones in the alkaline complex of Montes Claros de Goiás, strongly suggesting that these sources have emplaced at depth within almost the same geological time interval.

Although the upward continuation seems to be useful for overcoming the difficulties in the interpretation of strongly non-dipolar total-field anomalies, there will always be a limit for using this technique. The interpreter should always verify the quality of the estimated magnetization direction by using, for example, a reduction to the pole. One might think that the high sensitivity of our method to uncertainties in the horizontal coordinates of the centres of the sources is a drawback. This is not true because these coordinates are generally well estimated by the Euler deconvolution. The high sensitivity of our method to errors in horizontal locations of the centres of the sources suggests that the horizontal coordinates of the sources' centres could also be estimated by inversion. On the other hand, the insensitivity of our method to errors in the depth of the sources suggests that the sources' depth could not easily be estimated by inversion and would need some a priori information.

The authors would like to thank the Editor and the referees for their constructive review. The authors would also like to thank the government of the state of Goiás, Brazil, for permission to use the real aeromagnetic data set. D. P. Sales is particularly grateful to CPRM for permission to work in this research. V. C. F. Barbosa was supported by a fellowship from CNPq (Conselho Nacional de Desenvolvimento e Tecnológico) and L. Uieda was supported by a scholarship from CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior). Additional support for the authors was provided by FAPERJ (Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro) under contracts E-26/103.175/2011 and E-26/111.152/2014 and CNPq under contract 445752/2014-9. Edited by: R. Gloaguen Reviewed by: J. Ebbing, P. G. Lelièvre, and one anonymous referee