Comprehensive approaches to 3D inversion of magnetic ... - CGISS

GEOPHYSICS, VOL. 75, NO. 1 JANUARY-FEBRUARY 2010; P. L1–L11, 13 FIGS., 2 TABLES.
10.1190/1.3294766
Comprehensive approaches to 3D inversion of magnetic data
affected by remanent magnetization
Yaoguo Li 1 , Sarah E. Shearer 2 , Matthew M. Haney 3 , and Neal Dannemiller 4
ABSTRACT
Three-dimensional 3D inversion of magnetic data to recover
a distribution of magnetic susceptibility has been successfully
used for mineral exploration during the last decade.
However, the unknown direction of magnetization has limited
the use of this technique when significant remanence is
present. We have developed a comprehensive methodology
for solving this problem by examining two classes of approaches
and have formulated a suite of methods of practical
utility. The first class focuses on estimating total magnetization
direction and then incorporating the resultant direction
into an inversion algorithm that assumes a known direction.
The second class focuses on direct inversion of the amplitude
of the magnetic anomaly vector. Amplitude data depend
weakly upon magnetization direction and are amenable to direct
inversion for the magnitude of magnetization vector in
3D subsurface. Two sets of high-resolution aeromagnetic
data acquired for diamond exploration in the CanadianArctic
are used to illustrate the methods’usefulness.
INTRODUCTION
Quantitative interpretation of magnetic data through inversion for
general distributions of magnetic susceptibility has played an increasingly
important role in mineral exploration in recent years.
Such applications range from district-scale to deposit-scale problems.
Most of the currently available algorithms require knowledge
of magnetization direction, an essential piece of information for carrying
out forward modeling e.g., Li and Oldenburg, 1996; Pilkington,
1997. In most cases, one can assume there is no remanent magnetization,
and the self-demagnetization effect can be neglected.
Consequently, the direction of magnetization is assumed to be the
same as the current inducing field direction. This is a valid assumption
in most cases, as evidenced by many successful applications.
However, there are well-documented cases in which such an assumption
is inadequate because of the presence of remanent magnetization.
The total magnetization direction can be significantly different
from that of the inducing field. Without prior knowledge of the
direction of resultant total magnetization, current inversion algorithms
become ineffective. For instance, simulations by Shearer
2005 indicate that the algorithm by Li and Oldenburg 1996 yields
erroneous results when the error in specified magnetization direction
exceeds 15°. This difficulty has limited the application of these algorithms.
To address this issue, researchers have taken several different
routes. For example, Paine et al. 2001 transform magnetic data into
quantities that resemble the total-field anomaly but have less dependence
on magnetization direction prior to applying 3D inversions.
However, this approach suffers from the inconsistency between the
transformed data, which are not actual magnetic anomalies, and their
inversion using an algorithm designed for magnetic anomalies produced
by induced magnetization. Lelièvre and Oldenburg 2009 expand
the inversion to recover three components of a total magnetization
vector.
We focus on the magnetization direction as extra parameters in the
inversion and have developed two approaches. The first is to estimate
the direction of total magnetization and supply it to the inversion
algorithm, assuming the magnetization direction does not vary
greatly within the target region.Alternatively, we accept the fact that
a single direction cannot be estimated for a particular data set and
therefore opt to directly invert a quantity that is calculated from mag-
Manuscript received by the Editor 8 October 2008; revised manuscript received 29April 2009; published online 25 January 2010.
1 Colorado School of Mines, Center for Gravity, Electrical, and Magnetic Studies CGEM, Department of Geophysics, Golden, Colorado, U.S.A. E-mail:
ygli@mines.edu.
2 Formerly Colorado School of Mines, Center for Gravity, Electrical, and Magnetic Studies CGEM, Department of Geophysics, Golden, Colorado, U.S.A.
Presently Ultra Petroleum Corp., Denver, Colorado, U.S.A. E-mail: sshearer@ultrapetroleum.com.
3 Formerly Colorado School of Mines, Center for Gravity, Electrical, and Magnetic Studies CGEM, Department of Geophysics, Golden, Colorado, U.S.A.
Presently United States Geological SurveyAlaska Volcano Observatory,Anchorage,Alaska. E-mail: mhaney@usgs.gov.
4 Formerly Colorado School of Mines, Center for Gravity, Electrical, and Magnetic Studies CGEM, Department of Geophysics, Golden, Colorado, U.S.A.
Presently Pioneer Natural Resources, Denver, Colorado, U.S.A. E-mail: neal.dannemiller@pxd.com.
© 2010 Society of Exploration Geophysicists.All rights reserved.
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Li et al.
netic data but is weakly dependent upon magnetization direction.
Based on these two methods, we have formulated a comprehensive
approach to interpret any magnetic data affected by significant remanent
magnetization.
We first present three methods to estimate magnetization direction
for use in subsequent inversions and evaluate the performance in interpreting
3D magnetic data with strong remanent magnetization.
Then we present the basics of 3D inversion of amplitude data that are
weakly dependent upon magnetization direction. Both approaches
are illustrated with synthetic and field data sets from diamond exploration.
We conclude by discussing the conditions and limitations of
the two approaches and thereby provide explicit guidance on choosing
appropriate strategies for interpreting any magnetic data through
3D inversion.
ESTIMATING MAGNETIZATION DIRECTION
Current 3D magnetic inversion algorithms usually assume a
known magnetization direction and construct the 3D distribution of
magnetic susceptibility or magnitude of magnetization as a function
of 3D position Li and Oldenburg, 1996; Pilkington, 1997. Given
the critical role of magnetization direction, it is reasonable to attempt
to estimate it independently prior to inversion. This is perhaps the
simplest and most straightforward modification to the aforementioned
algorithms. We develop this approach by first examining direction
estimation techniques and then evaluating its utility in 3D inversions.
Many workers recognize the importance of magnetization direction
in interpreting magnetic data. For example, Zietz and Andreasen
1967 examine the relationship between position and intensity
of the maximum and minimum produced by a simple causative
body. Roest and Pilkington 1993 correlate the amplitude of the 2D
total gradient of the magnetic field and the absolute value of the horizontal
gradient of the pseudogravity produced by 2D sources.
Lourenco and Morrison 1973 develop a method based upon the integral
relationships of magnetic moments derived by Helbig 1962.
More recently, Haney and Li 2002 develop a wavelet-based method
for determining magnetization direction in 2D data sets. Dannemiller
and Li 2006 introduce an improvement to Roest and Pilkington’s
1993 method and extend it to 3D cases.
For this paper, we present Helbig’s moment method, wavelet
method, and cross-correlation method. The first two methods directly
explore the relation between the anomaly and magnetization direction
and compute magnetization direction from the data; whereas,
the third method estimates magnetization direction using the
symmetry property of the reduced-to-pole RTP field. In all three
methods, we assume magnetization direction does not vary drastically
for sources within the volume under examination.
Once the direction is estimated, it can be incorporated into a commonly
used inversion algorithm that assumes a known magnetization
direction. The inversion proceeds with the estimated magnetization
direction and attempts to recover an effective susceptibility defined
as the ratio of the magnitude of magnetization over the strength
of the inducing magnetic field H 0 . In the following, we present the
salient features of the estimation methods. We then illustrate the inversion
using such estimates through application to a synthetic example.
Helbig’s moment method
Helbig’s method Lourenco and Morrison, 1973; Phillips, 2005
is based on the integral relations between the moments of a magnetic
anomaly and the magnetic dipole moment developed by Helbig
1962:
xB z x,ydxdy2m x ,
yB z x,ydxdy2m y ,
xB x x,ydxdy2m z ,
where B x , B y , and B z are, respectively, the x-, y-, and z-components of
the magnetic anomaly and where m x , m y , and m z are the three components
of the magnetic moment of the source. Once the magnetic moment
is estimated, it can be used to calculate the inclination and declination
of the magnetization, assuming they are constant within the
source body.Although the integral relationships in equation 1 do not
assume any specific source geometry, we have observed that the
method is best applied to data sets produced by compact source bodies.
Two scenarios arise in practical applications. First, we usually
have only the total-field anomaly data and need to calculate the three
components from the total-field anomaly by the corresponding
wavenumber-domain operators e.g., Pedersen, 1978; Blakely,
1996; Schmidt and Clark, 1998. Difficulties may arise when the
data are acquired in low magnetic latitudes because the conversion
involves a half-reduction to the pole. Therefore, additional efforts
are required near the magnetic equator. Alternatively, vector magnetic
surveys are now becoming available e.g., Dransfield et al.,
2003, and the observed three-component data can be used directly
in the estimation.
Wavelet multiscale edge method
Haney and Li 2002 develop a method for estimating the magnetization
direction in two domensions using multiscale edges of a
magnetic anomaly derived by a continuous wavelet transform. The
multiscale edges correspond to the trajectories of the extrema of the
wavelet transform of the anomaly profile, and their positions in the
x-z-plane are dependent upon the inclination of magnetization.
Tracking the multiscale edges allows one to determine the inclination
of the magnetization in 2D sources.
Given a profile of magnetic data collected at a height z 0 in an ambient
magnetic field with inclination I, a continuous wavelet transform
CWT can be performed using a set of natural wavelets that is
equivalent to the magnetic field produced by a line dipole in a particular
direction Hornby et al., 1999. To estimate magnetization direction,
we use a wavelet whose corresponding dipole orientation
has an inclination of I. Such a wavelet leads to a CWT that is dependent
upon magnetization direction only. Carrying out the calculation
for multiple dilation factors yields the complete wavelet transform.
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Magnetic inversion with remanence
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The magnitude of the multiscale edges depends on the source geometry,
whereas the location depends primarily on the intrinsic
properties of the source, i.e., the magnetization direction I m . For example,
for a line of dipoles, there are four trajectories, given by
sx cot I m 2
3
, x cot I m 3 4
3
, 2
where s is the dilation factor and x is the location along the magnetic
profile. The dipole line is assumed to be directly below x0. Given
a profile of magnetic data, the problem of estimating the direction of
magnetization becomes one of tracking the trajectories of the multiscale
edges and calculating I m by regression according to equation 2.
Although the above approach is developed for 2D problems, it can
be applied to 3D data sets with isolated anomalies. In such cases, we
can integrate the data map in each of the two horizontal directions to
synthesize two profiles. Integrating in the easting direction simulates
a profile along a north-south traverse above a 2D source that strikes
east-west. The translation invariant property of magnetic anomalies
means that integrating the anomalous field is equivalent to adding
3D bodies along the strike direction to build up a 2D causative body.
The corresponding magnetization is given by the projection of the
3D magnetization vector in the north-south cross section. Applying
the wavelet estimation to this profile produces the apparent inclination
of the magnetization within the north-south section. Performing
similar operations in the perpendicular direction yields the apparent
inclination in the east-west section. The true inclination and declination
of magnetization in the original 3D source can then be reconstructed
from these two apparent inclinations in the easting and
northing cross sections, respectively.
Crosscorrelation method
The RTP anomaly theoretically has the least asymmetry of all
magnetic anomalies produced by a given causative body. It follows
that the vertical derivative of the RTPanomaly is also least asymmetrical.
It has been shown that the total gradient amplitude of the gradient
vector in three dimensions of the RTPanomaly is the envelope
of the vertical derivative of the anomaly produced under arbitrary inducing-field
and magnetization directions Haney et al., 2003.
The envelope, by definition, is the most symmetric form. Using
these properties, Dannemiller and Li 2006 develop a method to estimate
magnetization by examining the symmetry of various RTP
fields. This method is an extension of the 2D crosscorrelation method
developed by Roest and Pilkington 1993 to three dimensions
and improves upon the latter by using two quantities that have the
same decay rate with distance to the source.As a result, Dannemiller
and Li’s 2006 approach also extends the crosscorrelation method
to anomalies produced by dipping causative bodies.
The method searches for the particular magnetization direction
that yields the maximum symmetry in the resultant RTP field. The
symmetry is measured by the crosscorrelation between the vertical
derivative and total gradient of the RTP anomaly that is calculated
using an assumed magnetization direction. These two quantities
achieve the maximum correlation near the correct magnetization direction.
The key operation of the method is the RTP process. Consequently,
we will encounter difficulties at low magnetic latitudes as before.
Fortunately, several stable RTP methods are available for low latitudes
e.g, Hansen and Pawlowski, 1989; Mendonca and Silva,
1993; Li and Oldenburg, 2001. There is also a trade-off between the
accuracy of estimated inclination and the accuracy of estimated declination.
The accuracy of inclination improves as it approaches
90° or 90°, but the accuracy of the corresponding declination decreases.
However, this does not pose a major problem because the influence
of the declination becomes less important at high magnetic
latitudes.
Synthetic example
To illustrate the direction estimation algorithms and the utility of
the resultant magnetization direction, we apply them to a synthetic
example consisting of a dipping dike buried in a nonmagnetic background.
Figure 1 shows the model and the total-field anomalies with
and without the influence of remanent magnetization. For simplicity,
the effective susceptibility for both cases is set to 0.05 SI. The inducing-field
direction for this model has an inclination of 65°, a declination
of 25°, and a strength of B 0 50,000 nT. The total magnetization
with remanence has an inclination of 45° and a declination of
75°. The negative anomaly toward the northeast of the positive peak
in the data map Figure 1b is primarily related to the altered magnetization
direction.
We first apply Helbig’s method to the data in Figure 1b to estimate
the magnetization direction. To achieve this, we transform the totalfield
anomaly in Figure 1b into three orthogonal components of the
anomalous magnetic field shown in Figure 2.Applying equation 1 to
these components yields an estimate of the magnetic source dipole
moment. The corresponding magnetization direction is I m ,D m
46°,61°.
To apply the wavelet estimation, we first carry out integration in
northing and easting directions, respectively, to simulate two totalfield
profiles in the easting and northing directions Figure 3. The
corresponding apparent inclination values of total magnetization are
51° and 71° in easting and northing sections, respectively. The final
estimate for the magnetization direction in three dimensions is
I m ,D m 49°,68°. Last, we apply the crosscorrelation method.
The vertical and total gradients of computed RTPfield achieve maximum
correlation at I m ,D m 45°,80°, which yields our third estimate
for the magnetization. Figure 4 displays the crosscorrelation
map with the maximum point shown by the plus sign.
Table 1 compares these three estimates with the true values. All
three approaches provide reasonable estimates of the direction. The
largest deviation between true and estimated directions in three dimensions
is 10° Helbig’s method. This is well within the error tolerance
of 15° for the assumed magnetization established by Shearer
2005.
We then use these estimated magnetization directions as input in a
3D inversion algorithm Li and Oldenburg, 2003 to invert the data
shown in Figure 1b. All three inversions produce consistent models
that represent the true dipping slab. For brevity, we only display the
recovered effective susceptibility from the inversion using the magnetization
direction from Helbig’s estimate Figure 5. We can see
that the dip of the anomalous source body is clearly visible, and its
horizontal and vertical extents are well defined. Overall, the recovered
anomalous body is a good representation of the true model.
For this synthetic example, the following observations can be
made. All three estimation methods produce consistent and reliable
estimates of the magnetization direction. Using these estimates, subsequent
inversions produce magnetic models that are excellent representations
of the true model. Thus, the results amply demonstrate

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Li et al.
the validity of the two-step approach. It is also noteworthy, however,
that the estimation methods all rely on the global property of the
magnetic data set. The estimated direction effectively summarizes
the bulk direction of the magnetization in the causative bodies. The
validity of the estimated direction relies on the assumption that actual
direction does not vary significantly. In practice, the applicability
of the approach is therefore limited to anomalies produced by single
a)
sources of compact geometry. For more complex scenarios, and
when the magnetization direction is expected to vary greatly within
the region of interest, an alternative approach is needed.
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Easting (m)
Figure 1. A synthetic example with strong remanent magnetization.
a Dipping body with an effective susceptibility of 0.05 SI. Totalfield
data b with and c without remanent magnetization. The inducing
field has I,D65°,25°, and the total magnetization
direction in the presence of remanence is I m ,D m 45°,75°.
Figure 2. Three orthogonal components of the magnetic anomaly
vector computed from the total-field anomaly in Figure 1. The firstorder
moments of the x- and z-components yield the three components
of the source magnetic dipole and therefore the magnetization
direction.

Magnetic inversion with remanence
L5
INVERSION OF DATA WEAKLY DEPENDENT
ON MAGNETIZATION DIRECTION
Integrated mag (nT)
Integrated mag (nT)
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0
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Easting (m)
3000
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0
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Northing (m)
Figure 3. Two synthesized profiles obtained from the data map
shown in Figure 1b. The east-west profile top is obtained by integrating
the map in the north-south direction; hence, it corresponds to
the anomaly produced by a 2D causative body with a north-south
strike. The north-south profile in the lower panel is obtained by integration
in the east-west direction and corresponds to a 2D body with
a strike in that direction. These profiles are used in wavelet estimation
of the total magnetization direction.
Inclination ( o )
90
75
60
45
0.4 0.42
0.46
0.44
0.48
0.5
0.6
0.52
30
1801501209060
30
0 30 60 90 120 150 180
Declination ( o )
Figure 4. Crosscorrelation map between the total gradient of and vertical
gradient of the RTP field computed from assumed inclination
and declination of the magnetization. The crosscorrelation achieves
maximum near the correct magnetization direction.
0.58
0.54
0.56
We now examine the more complex scenarios where a single estimated
magnetization direction is no longer applicable. Such cases
may arise if a geologic unit has undergone deformation because of
tectonic activities so that the magnetization direction changes significantly
within the source bodies, or when multiple source bodies acquire
remanent magnetization at different times and have significantly
different magnetization directions. The challenge faced in
this case requires a completely new approach that does not rely on
knowledge of magnetization direction. The inversion of data associated
with magnetic anomalies but weakly dependent upon magnetization
direction offers the needed alternative.
The amplitude of the anomalous magnetic field vector and the total
gradient of the magnetic anomaly are independent of the magnetization
direction in 2D problems Nabighian, 1972 because the amplitude
of the magnetic anomaly vector is the envelope of each component
of the vector. The same holds true for the amplitude of the
gradient vector of a 2D field component and the derivative of the
same field component in any direction. Although such a property
does not extend exactly to 3D problems, both quantities are only
weakly dependent on magnetization direction. This is especially so
when the anomaly has been transformed to the vertical component
i.e., half RTP. This property provides the opportunity for direct inversion
of the anomaly amplitude or total gradient to recover the
magnitude of magnetization without knowing its direction.
Shearer and Li 2004 develop such an algorithm by formulating a
generalized inversion using Tikhonov regularization and imposing a
Table 1. Magnetization direction estimated using three
different methods for the dipping-slab data set shown in
Figure 1c. The deviation is defined as the angle between the
true and estimated directions in three dimensions.
Method
Inclination
°
Declination
°
Deviation
°
True value 45 75
Helbig’s method 46 61 9.8
Wavelet method 49 68 6.2
Crosscorrelation 45 80 3.5
a)
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b)
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k
(SI)
0.070
0.063
0.056
0.049
0.042
0.035
0.028
0.021
0.014
0.007
0.000
Figure 5. Effective susceptibility obtained through the inversion of
the synthetic data in Figure 1b with estimated direction of total magnetization
from Helbig’s method shown in Table 1. The inversion is
performed using the algorithm of Li and Oldenburg 1996. a Cross
section at 500 m north. b Plan section at a depth of 125 m. The true
position of the source body is shown by the black outline. The resultant
model is a good representation of the true model. The peak value
of effective susceptibility is consistent with the true value of 0.05 SI.

L6
positivity constraint on the amplitude of magnetization. Shearer
2005 carries out a detailed investigation of the approach and demonstrates
that the amplitude data are far less dependent on magnetization
direction than the total gradient data. Furthermore, the amplitude
data preserve the low-wavenumber content in the data and
therefore retain the signal from deeper causative bodies that is
present in the total-field magnetic anomaly. Consequently, inversion
of amplitude data offers a better alternative than does inversion of total
gradient data. We describe the amplitude inversion below, but
readers are referred to Shearer 2005 for more details. The inversion
of total gradient data is exactly parallel.
Basic algorithm for amplitude inversion
The algorithm starts by calculating the amplitude of the anomalous
magnetic field from the observed total-field anomaly. This is accomplished
by first transforming the total-field anomaly into the
three orthogonal components in the x-, y-, and z-directions. The amplitude
data are given by
B a B a Bx 2 B y 2 B z 2 ,
where B a are the amplitude and B x ,B y ,B z are the transformed magnetic
anomaly vectors. A common approach to obtain the three orthogonal
components is to use the wavenumber-domain expressions
e.g., Pedersen, 1978 when the data are located on a plane. Alternatively,
equivalent-source techniques Dampney, 1969 can be used
to carry out the transformation when ground data are acquired in areas
with high topographic relief. In such cases, the wavenumber-domain
approach, which assumes that all observations lie on a planar
surface, is inappropriate. The amplitude data are then treated as the
input data and inverted to recover the distribution of magnetization
as a function of 3D position in the subsurface. One advantage of the
approach is that it is not limited to a single anomaly nor does it require
that adjacent anomalies have the same magnetization direction.
Therefore, the approach is generally applicable to a wide range
of problems where the source distribution is more complicated.
The basic inversion algorithm follows that of Li and Oldenburg
1996, 2003 in which the Tikhonov formalism is used to trade off
between the data misfit and the structural complexity of the recovered
model. The data misfit is defined as
N
d
i1
B obs pre
ai B ai
i
3
2
, 4
where B obs ai and B pre ai are observed and predicted amplitude data, respectively,
and i are the standard deviation of the amplitude data.
Although we commonly assume a Gaussian distribution for errors in
magnetic field component data, the corresponding errors in the computed
amplitude data no longer follow such a distribution.
The model objective function is chosen as
m s w 2 z 0 2 dv
V
wz 0
x
V
y
V
x
wz 0
y
2
dv
2
dv
Li et al.
z V
wz 0
z
2
dv,
where is the effective susceptibility, defined as the ratio of magnitude
of magnetization over the strength of the inducing field H 0 , 0 is
a reference model, and wz is a depth-weighting function. The inverse
solution is given by the minimization of the total objective
function, consisting of a weighted sum of d and m , subject to the
constraint that the effective susceptibility must be nonnegative:
5
minimize d m
subject to 0, 6
where is the regularization parameter. The positivity is implemented
by using a primal logarithmic barrier method Wright, 1997;
Li and Oldenburg, 2003. The solution is obtained iteratively because
nonlinearity is introduced by the positivity constraint and the
nonlinear relationship between amplitude data and effective susceptibility.
We discuss the basics of this aspect next, but readers are referred
to Shearer 2005 for more details.
We adopt a commonly used model representation that discretizes
the model region in three dimensions into a set of contiguous rectangular
prisms by an orthogonal mesh, and we assume a constant effective
susceptibility value within each prism. Under this assumption,
each component of the magnetic anomaly vector is given by a matrix-vector
product:
d x G x ,
d y G y ,
d z G z ,
where d x B x1 ,¯,B xN T is an algebraic vector holding the
x-components of the anomalous magnetic field; d y and d z are similarly
defined; 1 ,¯, M T is the vector of unknown effective
susceptibility to be recovered; and G x , G y , and G z are the sensitivity
matrices relating the respective components of anomalous magnetic
field to effective susceptibilities. The elements of the sensitivity matrices
quantify the field produced at the ith observation location by a
unit effective susceptibility in the jth prism.
Assuming the susceptibility model is n at the nth iteration, substituting
equation 7 into equation 3 and differentiating B ai with respect
to j yields
n
B ai
B ai
n j ·b ij,
B ai
where B n ai is the predicted anomalous magnetic vector at the ith observation
location by the model n at the nth iteration and where b ij
is the magnetic vector produced at the same location by a unit susceptibility
in the jth prisms. Thus, the sensitivity has a simple and elegant
form given by the inner product of the unit vector of the predicted
magnetic field at the nth iteration and the magnetic field produced
by a unit effective susceptibility in a prism. For computational
purposes, this means we only need to compute and store the three
sensitivity matrices corresponding to the three components of the
field in equation 7 and calculate the sensitivity for the amplitude data
by a sequence of matrix-vector multiplications when solving the
minimization in equation 6.
When calculating the three individual sensitivity matrices, we
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Magnetic inversion with remanence
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must formally specify a magnetization direction. Given that we are
utilizing the weak dependence of amplitude data on magnetization
direction, it is sufficient to use the direction of the current-inducing
field. Thus, the sensitivity is computed as if the magnetization vector
were given by the product of the effective susceptibility and the inducing
field. It is important to note, however, that the interpretation
of the inversion result is independent of the direction of the inducing
field. The strength of the inducing field, on the other hand, defines
the magnitude of the total magnetization if desired.
Revisiting the synthetic example
We now return to the synthetic example shown in Figure 1 and illustrate
the approach of amplitude-data inversion. To calculate the
amplitude data, we must first convert the total-field anomaly into
three orthogonal components in x-, y-, and z-directions as shown in
Figure 2. The desired amplitude data shown in Figure 6a are obtained
by equation 3. For comparison, Figure 6b displays the amplitude
data for the same model when the magnetization is aligned with
the inducing field. For consistency, we have added the same amount
of noise to total-field anomalies in both cases prior to conversion to
amplitude data. The high degree of similarity between the two maps
demonstrates the direction insensitivity of the amplitude data. Using
the same mesh as in the previous inversion, we invert the amplitude
data in Figure 6a.
a)
1000
The result is shown in Figure 7, which can be compared with the
model recovered using an estimated magnetization direction Figure
5. The location and spatial extent of the anomalous body is recovered
well, but the dip is less visible.Also, the peak value of the recovered
effective susceptibility is much higher. Overall, however, the
inversion result is much improved, compared to the inability to invert
the original data without the magnetization direction when using
existing algorithms.
The lack of recovered dip is to be expected in this case. The reasons
are twofold. First, much of the dip information is encoded in the
phase of magnetic anomaly, and that information is greatly diminished
in the amplitude data. Second, the particular model does not
have a pronounced elongation in dip direction, and the remaining
phase information in amplitude data is not enough to constrain the
dip. In general, the ability to recover the dip depends on the orientation
of the magnetization relative to the geometry of the causative
body and on its aspect ratio.Although not reproduced here for brevity,
numerical tests have shown that the dip can be recovered when
the depth extent of a causative body is much longer than its width.
The peak value of the recovered effective susceptibility in the amplitude
inversion is much higher than that in the inversion based on
estimated magnetization direction. This difference has been observed
consistently in all synthetic and field data examples that we
have studied. This difference appears to be related to the following
two factors.
First, because of the partial lack of the phase information in amplitude
data, the recovered effective susceptibility tends to spread out
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Figure 6. Comparison of amplitude data computed from the totalfield
anomalies shown in Figure 1. a The amplitude data computed
from the total-field anomaly in Figure 1b when the magnetization is
affected by remanence. b Data computed from the total-field
anomaly in Figure 1c when the magnetization is purely induced.
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Depth (m)
b)
Northing (m)
200
400
1000
800
600
400
200
0
0 200 400 600 800 1000
0 200 400 600 800 1000
Easting (m)
k
(SI)
0.150
0.135
0.120
0.105
0.090
0.075
0.060
0.045
0.030
0.015
0.000
Figure 7. Inversion of the amplitude data in Figure 6a using an amplitude-inversion
algorithm. The model is shown as effective susceptibility,
which is defined by the magnitude of the magnetization
vector divided by the strength of the inducing magnetic field. a
Cross section at 500 m north. b Plan section at 125 m depth. The
outline of the true source body is shown by the solid black line.

L8
Li et al.
more. There is essentially more susceptibility distributed at larger
depth. Correspondingly, a higher peak value is required to reproduce
the data. For example, the depths of the center of mass of the susceptibility
are 200 and 280 m, respectively, in the models from inversion
using estimated direction Figure 5 and from amplitude inversion
Figure 7. The cube of the ratio of the two depths is 0.35. Given
that the amplitude data decay approximately with inverse distance
cubed, this ratio is consistent with the peak susceptibility ratio of
0.45.
Second, every small anomalous feature in the amplitude data can
be reproduced easily, so the data tend to be overfit when we use standard
techniques for estimating the optimal data fit determined by the
regularization parameter. This also leads to a greater peak susceptibility
value in recovered models. However, the geometry of the recovered
source body is similar for the two inversions despite the difference
in peak susceptibility values. Both models can be used to
carry out the final interpretation.
The ability to invert amplitude data has effectively circumvented
the need for magnetization direction as a crucial piece of information.
As a result, we have bypassed the limitation of the first method
that requires a constant magnetization direction within a source
body. This opens the door to applying 3D inversion to interpret a
wide range of data sets, especially those from areas with complex geology
and multiple source regions with variable magnetization directions.
APPLICATION TO FIELD DATA SETS
We now invert two sets of high-resolution aeromagnetic data and
illustrate the application of our methods in their respective scenarios.
The data were acquired by TeckCominco and Diamonds North
over kimberlites on Victoria Island in Northwest Territory, Canada.
The geology in the area is characterized by an Archean granitic
basement overlain by a Proterozoic sedimentary sequence with minor
volcanics, capped by flat-lying Cambrian-to-Devonian carbonate
rocks. The host rocks are largely nonmagnetic, and the kimberlite
bodies stand out in the magnetic surveys as distinct, sharp anomalies
indicative of shallow bodies, compared with the more rounded
anomalies caused by deep features in the basement or within the sedimentary
rocks. Positive and negative anomalies are associated with
kimberlite intrusions, whereas the negative anomalies are produced
by hypabyssal dikes.
The ages of the kimberlite intrusions range from 250 to 300 million
years J. Lajoie, personal communication, 2004. The distinct
magnetic signature of these kimberlite dikes above a quiet magnetic
background makes high-resolution magnetic surveys an ideal exploration
tool in this area. However, the highly variable orientation of
these anomalies is produced by magnetization that is dominated by
strong remanence with variable direction. As a result, the quantitative
interpretation of the anomalies is difficult to achieve through
current 3D inversion algorithms. For this reason, the data are ideal
for testing our new approaches.
We investigate data from two different areas with variable complexity
in anomaly patterns. The first data set contains a single anomaly
associated with a kimberlite dike; the second contains several
different anomalies with large variations in magnetization direction.
The data set containing a single anomaly is shown in Figure 8a.
The inducing field has an inclination of 86.7° and declination of
26.3°. Given the high magnetic latitude and dominant negative
anomaly, it is clear that the kimberlite has strong remanence, and the
total magnetization is nearly in the opposite direction to the inducing
field. Figure 8b displays the amplitude data computed from these
data. Given the single anomaly in this data set, we can estimate the
direction first and then invert the total-field data or we can directly
invert the amplitude data. We present both for comparison.
The results of estimation are listed in Table 2. The estimated values
for inclination are similar, but the declination varies greatly. This
is expected, given the inclination is close to 90°. When used in an
inversion, the error in declination does not strongly affect the final
result either. Using the direction estimated from multiscale edges,
inversion of total-field data Figure 8a is shown in one cross section
at 300 m north and one plan section at 50 m depth in Figure 9. The
result from inverting the amplitude data is shown in the same format
in Figure 10. The two inversions recover models with similar geometry
but differing peak susceptibility, as noted in the preceding section.
Both effectively image a compact magnetic body. It has a northwest
strike and a strike length of approximately 250 m, and it is located
at 50 m depth to the center. This result is consistent with the
presence of a kimberlite dike.
The second data set is shown in Figure 11a. A number of dipolar
total-field anomalies with differing orientations occur throughout
the survey area. The map has been rotated clockwise by 34°; the inducing
field has an inclination of 86.7° and a nominal declination of
7.7°. Two types of anomalies dominate the data set: several smaller,
more compact, and high-frequency anomalies surrounding areas
a)
Northing (m)
b)
Northing (m)
600
400
200
0
600
400
200
0
0 200 400 600 800
0 200 400 600 800
Easting (m)
4
3
2
1
0
1
2
3
4
5
6
7
8
T
(nT)
7.0
6.5
6.0
5.5
5.0
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
B a
(nT)
Figure 8. a Total-field magnetic anomaly T over a kimberlite
dike. The inducing field is in the direction of I86.7° and D
26.3°. Judging from the negative anomaly in the center, the presence
of strong remanent magnetization is apparent. b The corresponding
amplitude, B a , of the anomalous field vector.

Magnetic inversion with remanence
L9
of broad, lower-frequency anomalies. The orientation of the anomalies
indicates the total magnetization direction varies greatly from
anomaly to anomaly. Thus, it is unlikely that we can invert this data
set with a single magnetization direction. Overlapping anomalies
also mean that separately inverting each anomaly by first estimating
a magnetization direction is not feasible. We resort to the second approach,
i.e., we invert the amplitude of the anomalous magnetic field
and recover the magnitude of magnetization in the form of an effective
susceptibility. The computed amplitude data are shown in Figure
11b.
The effective susceptibility recovered from the inversion of the
amplitude data in Figure 11b is shown in Figure 12 as a volume-rendered
image with an overlain translucent color display of the amplitude
data. There are five main anomalous bodies of high susceptibility
in the recovered model. The two broad, elongated bodies resemble
kimberlite dikes known in this area, whereas the more compact bodies
oriented vertically resemble kimberlite pipes.
DISCUSSION
Table 2. Magnetization direction estimated using three
different methods for the field data set shown in Figure 8a.
Method
Inclination
°
Declination
°
Helbig’s method 84.7 70.0
Wavelet method 89.3 1.8
Crosscorrelation 87.4 26.0
The methodology developed in this paper for inverting magnetic
data in the presence of remanent magnetization consists of two approaches.
The first approach directly addresses the issue of unknown
magnetization direction and estimates it using several existing and
newly developed algorithms. The data are then inverted using existing
magnetic inversion algorithms. The second approach circumvents
the need for reliable knowledge of magnetization direction
and, instead, inverts directly the amplitude of the anomalous field to
recover the magnitude of the magnetization.
Figure 13 summarizes the three routes to the inversion of magnetic
data. For a given data set, the first question to be answered is
whether the data are affected by strong remanent magnetization that
is not aligned with the current inducing field. If the answer is no, then
any standard inversion algorithms for 3D magnetic inversion can be
applied. If the answer is yes, the data set should be inverted by using
one of two approaches depending on the complexity of the magnetic
anomaly. The criterion for choosing which method to use is whether
a single magnetization direction is a valid assumption and can be estimated.
If the answer is yes, then the method based on direction estimation
should be used. In practice, this means that only a single
compact anomaly is present, although rare cases of multiple anomalies
with the same magnetization direction may exist. If the answer is
no, then the amplitude-data inversion method should be used. Such
cases include a single anomaly produced by a complex source body
or multiple anomalies with different orientations.
Both methods can effectively construct the source distribution for
a compact source body that meets the assumption of a constant magnetization
direction. However, when multiple source bodies are
present with varying magnetization directions, amplitude-data inversion
proves to be much more versatile. The price we pay for that
ability is, of course, the partially missing phase information in the
data.As a result, the dip of the recovered source distribution may not
be clearly imaged if the causative body does not have a pronounced
a)
Depth (m)
b)
Northing (m)
0
50
100
150
200 300 400 500 600
500
400
300
200
200 300 400 500 600
Easting (m)
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
k
(SI)
Figure 9. Effective susceptibility recovered by inverting the totalfield
magnetic anomaly in Figure 8a. The inversion uses the magnetization
direction estimated by the multiscale edge method. a Cross
section at 300 m north. b Plan section at 50 m depth.
a)
Depth (m)
b)
Northing (m)
0
50
100
150
200 300 400 500 600
500
400
300
200
200 300 400 500 600
Easting (m)
k
(SI)
0.020
0.018
0.016
0.014
0.012
0.010
0.008
0.006
0.004
0.002
0.000
Figure 10. Effective susceptibility recovered by inverting the amplitude
anomaly in Figure 8b. a Cross section at 300 m north. b Plan
section at 50 m depth.

L10
Li et al.
a)
Northing (m)
b)
Northing (m)
1600
1400
1200
1000
1600
1400
1200
1000
400
200
0 200 400
400
200
0 200 400
Easting (m)
6
4
2
0
2
4
6
8
10
14 12
16
18
20
22
24
T
(nT)
26
24
22
20
18
16
14
12
10
8
6
4
2
0
B a
(nT)
Figure 11. a The total-field anomaly data over a group of kimberlites
on Victoria Island, Northwest Territories, Canada. The 600
1000-m data have been rotated 34° clockwise and regridded at a
10-m spacing. The rotated data map has an inducing field with an inclination
of 86.7° and a nominal declination of 7.7°. b The corresponding
amplitude of anomalous magnetic vector.
0
50
100
150
200 –500
–250
0
Easting (m) 250
29.4
24.6
19.8
14.9
10.1
5.24
0.407
500
Ba (nT)
Figure 12. Volume-rendered inversion results of the recovered effective
susceptibility with a translucent color display of the amplitude
data shown in Figure 11b. The view is from the southeast. The color
scale indicates the amplitude data in nT; the effective susceptibility
is cut off at 0.025 SI. Five major magnetic sources are recovered. The
two broad, elongated sources resemble kimberlite dikes known in
this area; the more compact, vertically oriented bodies resemble
kimberlite pipes in the area.
Invert
total field anomaly
Magnetic
susceptibility
No
Yes
Estimate
magnetization
direction
Invert
total field anomaly
Total-field anomaly
Remanent
magetization
Single anomaly
Magnitude of
magnetization
elongation along its dip. Computationally, the cost of estimating a
magnetization direction is negligible compared to the subsequent inversion
that uses the estimation. Therefore, this method incurs a total
computational cost similar to 3D inversion of purely induced magnetic
data. The amplitude inversion must generate and use three sensitivity
matrices; hence, the total cost is approximately three times
that of the first method.
Our approach deals with the difficulty caused by unknown magnetization
direction. In exploration problems, one such occurrence
may be the result of the presence of remanent magnetization. Although
our approach enables the inversion of data to obtain a 3D distribution
of magnitude of magnetization, it does not separate induced
and remanent components. Another leading cause of unknown
magnetization direction is the self-demagnetization effect in
highly magnetic environments such as banded iron formations. The
methods developed in this paper may apply to interpretation of data
affected by self-demagnetization effect. This aspect is under investigation.
CONCLUSION
Yes
We have developed a comprehensive set of methods to tackle the
problem of inverting magnetic data in the presence of remanent
magnetization that alters the direction of the total magnetization.
Given these methods, we have a set of tools at our disposal for interpreting
magnetic data in the presence of significant remanent magnetization.
Coupled with existing 3D inversion algorithms, we can
confidently state that most of the magnetic data acquired in exploration
problems can be interpreted quantitatively by constructing 3D
distributions of either magnetic susceptibility of the magnitude of
magnetization. This will further enhance the applicability and effectiveness
of 3D magnetic inversion.
ACKNOWLEDGMENTS
Compute
amplitude data
Invert
amplitude data
Figure 13. The three routes to the inversion of magnetic data. If the
data are not affected by remanent magnetization, any standard inversion
algorithms for 3D magnetic inversion can be applied. Otherwise,
the data should be inverted by using one of the two approaches
developed here.
We thank Jules Lajoie, TeckCominco, and Diamond North for
providing the data used in the study. We also thank Misac Nabighian
and Jeff Phillips for many discussions. Finally, we thank Richard
Lane, Peter Lelièvre, and an anonymous reviewer for detailed comments
that improved the clarity of the presentation. This work was
No

Magnetic inversion with remanence
L11
partly supported by the Gravity and Magnetics Research Consortium,
sponsored by Anadarko, BGP, BP, Chevron, ConocoPhillips,
and Vale. Partial funding support was provided by KORES.
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