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Three-dimensional cross-gradient joint inversion of gravity and normalizedmagnetic source strength data in the presence of remanent magnetization
Junjie Zhou, Xiaohong Meng, Lianghui Guo, Sheng Zhang
PII: S0926-9851(15)00151-2DOI: doi: 10.1016/j.jappgeo.2015.05.001Reference: APPGEO 2763
To appear in: Journal of Applied Geophysics
Received date: 23 December 2014Revised date: 4 May 2015Accepted date: 5 May 2015
Please cite this article as: Zhou, Junjie, Meng, Xiaohong, Guo, Lianghui, Zhang, Sheng,Three-dimensional cross-gradient joint inversion of gravity and normalized magneticsource strength data in the presence of remanent magnetization, Journal of Applied Geo-physics (2015), doi: 10.1016/j.jappgeo.2015.05.001
This is a PDF file of an unedited manuscript that has been accepted for publication.As a service to our customers we are providing this early version of the manuscript.The manuscript will undergo copyediting, typesetting, and review of the resulting proofbefore it is published in its final form. Please note that during the production processerrors may be discovered which could affect the content, and all legal disclaimers thatapply to the journal pertain.
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Three-dimensional cross-gradient joint inversion of gravity and normalized magnetic source
strength data in the presence of remanent magnetization
Junjie Zhoua,b
, Xiaohong Menga,b,*
, Lianghui Guoa,b
, Sheng Zhanga,b
a Key Laboratory of Geo-detection, China University of Geosciences, Ministry of Education,
100083 Beijing, China
b School of Geophysics and Information Technology, China University of Geosciences (Beijing),
100083 Beijing, China
Authors list:
Junjie Zhou: [email protected]
Xiaohong Meng: [email protected] (Corresponding author)
Lianghui Guo: [email protected]
Sheng Zhang: [email protected]
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Abstract
Three-dimensional cross-gradient joint inversion of gravity and magnetic data has the
potential to acquire improved density and magnetization distribution information. This method
usually adopts the commonly held assumption that remanent magnetization can be ignored and all
anomalies present are the result of induced magnetization. Accordingly, this method might fail to
produce accurate results where significant remanent magnetization is present. In such a case, the
simplification brings about unwanted and unknown deviations in the inverted magnetization
model. Furthermore, because of the information transfer mechanism of the joint inversion
framework, the inverted density results may also be influenced by the effect of remanent
magnetization. The normalized magnetic source strength (NSS) is a transformed quantity that is
insensitive to the magnetization direction. Thus, it has been applied in the standard magnetic
inversion scheme to mitigate the remanence effects, especially in the case of varying remanence
directions. In this paper, NSS data were employed along with gravity data for three-dimensional
cross-gradient joint inversion, which can significantly reduce the remanence effects and enhance
the reliability of both density and magnetization models. Meanwhile, depth-weightings and bound
constraints were also incorporated in this joint algorithm to improve the inversion quality.
Synthetic and field examples show that the proposed combination of cross-gradient constraints
and the NSS transform produce better results in terms of the data resolution, compatibility, and
reliability than that of separate inversions and that of joint inversions with the total magnetization
intensity (TMI) data. Thus, this method was found to be very useful and is recommended for
applications in the presence of strong remanent magnetization.
Keywords: Cross-gradient joint inversion; Remanent magnetization; Total magnetic intensity;
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Normalized magnetic source strength; Gravity data
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1. Introduction
The joint inversion technique combines various types of geophysical survey data into a
general processing framework, which can take full advantage of the data complementarities,
reduce the inherent non-uniqueness of the inverse problem, and thus enhance inversion resolution.
For gravity and magnetic data, collocated and regional scale acquisition data are often available in
parallel. Thus, application of a three-dimensional (3D) joint inversion technique to these data has
the potential to produce valuable density and magnetization results with high resolution and
compatibility.
As a powerful tool for the integrated processing of gravity and magnetic data, joint inversion
methods have been widely studied by many researchers for decades; see, for example, the studies
by Menichetti and Guillen (1983), Serpa and Cook (1984), and Zeyen and Pous (1993). In recent
years, a variety of innovative joint methods for gravity and magnetic data have been proposed,
such as layered model inversion (Gallardo-Delgado et al., 2003; Gallardo et al., 2005; Pilkington,
2006), Monte-Carlo inversion (Bosch et al., 2006), cross-gradient inversion based on structural
coupling (Fregoso and Gallardo, 2009; Fregoso et al., 2015; Gallardo, 2004), Gramian constraint
inversion (Zhdanov et al., 2012), geostatistical inversion (Shamsipour et al., 2012), and inversion
based on fuzzy clustering constraints (Carter-McAuslan et al., 2015; Lelièvre et al., 2012; Sun and
Li, 2013). Cross-gradient joint inversion is a practical method that has been applied to electrical
and seismic data (Gallardo and Meju, 2004), and also to other kinds of data combinations
(Gallardo et al., 2012; Hu et al., 2009; Linde et al., 2006). Fregoso and Gallardo (2009) were the
first to extend the cross-gradient joint inversion algorithm to gravity and magnetic data, and their
results showed improvements both in terms of the lateral and depth resolution with high structural
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resemblance. Compared to empirical direct links, statistical correlations, or clustering coupling
measures described in the previous literature, the cross-gradient technique holds the least number
of assumptions and only requires the participant physical models are structurally similar (Lelièvre
et al., 2012; Moorkamp et al., 2011). Although it might not be the most effective, the
cross-gradient measurement has been widely applied to explorations for its feasibility and
reliability, especially for regions where explicit physical property relationships are unclear
(León-Sánchez and Gallardo-Delgado, 2015; Peng et al., 2013; Solon et al., 2014; Wang et al.,
2015; Zhou et al., 2015).
The conventional magnetic inversion, including any magnetic terms involved in joint
inversion, commonly simplified the nonlinear relationship between magnetization and
observational magnetic data to be linear by ignoring the existence of remanent components, and
the total magnetization is assumed parallel to the geomagnetic field vector (Fregoso and Gallardo,
2009, 2015). However, when strong or complicated (i.e., the directions vary within an area)
remanent magnetization is present, the magnetic inversion results will be distorted because of the
simplification measures described above (Li et al., 2010; Shearer, 2005). Considering information
propagation (also error propagation) in the joint framework, the deviation affected by remanence
occurs not only in the inverted magnetic results, but also in the density results. Therefore, it is
necessary to take some considerable measures to reduce the effect of remanent magnetization
when joint inversion is carried out in an area where strong or complicated remanent magnetization
exists.
Several methods can be used to deal with the magnetic inverse problem in the presence of
remanence. The first method is to estimate the magnetization direction before inversion
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(Dannemiller and Li, 2006; Gerovska et al., 2009; Roest and Pilkington, 1993; Shi et al., 2014).
These techniques always require a homogenous distribution of the magnetization directions within
the study area. The second method is to directly invert the magnetization vector distribution.
Lelièvre and Oldenburg (2009) inverted magnetic data for the three components of a subsurface
magnetization vector in a Cartesian or spherical framework. The third method is to convert the
magnetic anomaly into some certain quantity that is insensitive to the magnetization direction, and
then to proceed with the inversion process. Li et al. (2010) and Shearer (2005) inverted the
magnetic anomaly amplitude rather than the conventional total magnetization intensity (TMI) to
reduce the effect of remanent magnetization. Pilkington and Beiki (2013) compared the remanence
sensitivities of the normalized magnetic source strength data (NSS) and other transformations of
magnetic data, and they found that the NSS is minimally affected by the direction of remanent
magnetization. They then adopted the NSS data in a 3D inversion and effectively reduced the
effect of remanent magnetization. Li and Li (2014) carried out an amplitude inversion to generate
a 3D subsurface distribution of the magnitude of the total magnetization vector. Guo et al. (2014)
presented correlation imaging methods for both the NSS data and the total amplitude magnetic
anomaly, and their result showed good correspondence between the source location and anomaly
peak of the NSS data. To summarize, it is suitable to incorporate certain converted quantities into
a joint algorithm to counteract the influence of remanence because of the simple implementation
and the applicability to complicated remanence distributions.
Based on previous work, we present a cross-gradient joint inversion algorithm for gravity and
NSS data that employs several proper constraints techniques to reduce the effect of remanent
magnetization and improve the resolution of the inverted results. We chose to use NSS data
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because such data are less sensitive to the magnetization direction when compared to other
transforms of the magnetic data; hence, these data have a stronger capacity to reduce remanence
effects. Additionally, the NSS has a linear relationship with the total magnetization amplitude,
which makes it much simpler to implement the joint algorithm. In this paper, the NSS definition
and its converting calculation are reviewed first. Then, we propose an improved 3D cross-gradient
joint inversion algorithm for gravity and magnetic data with depth-weightings and physical
property bound constraints. This algorithm can be applied either to the gravity and NSS
combination, or to the gravity and TMI combination. Finally, we test the algorithm on several
synthetic and field data examples to illustrate the comprehensive effectiveness when the
cross-gradient technique and NSS data conversion are employed together.
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2. Methodology
2.1 Normalized magnetic source strength data
The NSS is a quantity that is derived from the eigenvalues of the magnetic gradient tensor
(MGT) components matrix in Cartesian coordinates, and can be expressed by a simple formula
whereby the total magnetization amplitude is normalized by the fourth power of distance between
the observation site and the dipole source location (Beiki et al., 2012; Pilkington and Beiki, 2013;
Wilson, 1985):
2
2 1 3 4
3 m dc md
r , (1)
where d represents the NSS data, 1 , 2 , and 3 are the descending sorted eigenvalues of
the MGT matrix, mc = 10-7
H/m in SI units, r is the observation-source distance, and dm is
the magnitude of dipole magnetization. Obviously, the NSS has the same units as the gradient
tensor (nT/m), and depends on the amplitude while being independent the direction of the
magnetization. Thus, deviation caused by the inconformity between the magnetization direction
and geomagnetic field direction can be reduced theoretically by this operation.
To obtain the NSS, the eigenvalues in Eq. (1) are computed from the 3 × 3 MGT matrix at
each data site. Although the MGT components could be measured directly by magnetic tensor
gradiometers, they are more likely to be transformed in practical surveys from the TMI data by
frequency-domain filters (Schmidt and Clark, 1998). This operation requires that geomagnetic
field direction information is provided, and can be safely conducted in most areas except that
geomagnetic field has shallow inclination. In such a case, the calculation in frequency domain is
unstable, which is similar with the case of reduction to the pole (RTP) transform (Blakely, 1995).
To guarantee the accuracy, It is suggested making proper extensions for the regular-interpolated
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observational grid to reduce the undesired edge effect. Additionally, it is recommended to correct
the TMI to a true potential field before this transform is areas where strong anomalies significantly
perturb the geomagnetic field (Beiki et al., 2012). The NSS forward modeling formula for a given
magnetization model is also provided in Eq. (1), which is adopted to compute the NSS sensitivity
matrix with respect to the magnetization amplitude. The direction information about the
magnetization is weakened by this conversion, so the inversion procedure will be minimally be
affected by remanence. In the following, we use the NSS in the joint inversion to further
investigate its availability in the presence of remanent magnetization.
2.2 Objective function of 3D cross-gradient joint inversion with additional constraints
Under the joint inversion algorithm, the subsurface model is discretized as a regular mesh of
prisms, each of which is assigned fixed density contrast and also total magnetization (both for the
amplitude and the direction). The difference of the directions between geomagnetic field and
magnetization indicates the existence of remanent magnetization. The magnetization directions in
different cells may differ, which shows that more complicated remanent magnetizations exist. The
gravity, transformed or non-transformed magnetic survey data, and the model parameters are
incorporated into a general framework with a corresponding assembled form. The density contrast
and magnetization amplitude vectors 1m and 2m (both arranged in column format) are
regarded as unknowns. These two vectors are combined as 1 2[ , ]T T Tm m m in the joint
framework. The unknown magnetization directions of the model are not considered. Similarly,
observational gravity data 1d and the transformed NSS (or the observational TMI) data 2d are
combined into a data vector 1 2[ , ]T T Td d d . Gravity and the NSS (or the TMI) forward operators
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1g and 2g are replaced by a joint forward operator g .
Generally, the model parameter m of each cell has a realistic physical meaning within a
certain numerical range and should be constrained in the inversion. Various techniques such as the
logarithmic barrier approach (Li and Oldenburg, 2003), gradient projection approach (Lelièvre et
al., 2009; Lelièvre, 2009) and the transform function approach (Kim and Kim, 1999; Lelièvre and
Oldenburg, 2006; Li and Oldenburg, 1996; Moorkamp et al., 2011; Pilkington, 2009) have been
adopted in different inversion schemes to implement this constraint. Here we prefer to adopt the
last method in joint algorithm to convert physical property parameter to a generalized parameter
( )p p m . Then, the inversion procedure can be solved with respect to vector p in the full
numerical space, and the final-obtained model vector m is restricted in the given limits. There
are many choices for the transform function, e.g., the logarithmic transform for positive
constraints or the square function for non-negative constraints. We use a more generic transform to
introduce the bound information, which can be written as an element-wise form (Commer, 2011):
( )1
cp
cp
a bem p
e
, (2)
where a and b are the specified lower and upper limits for ( , )m a b , respectively, and c
is a variable controlling the steepness of the transformation. These parameters can be easily
extended to vector form a , b , and c for cases where bound information is provided for each
cell in detail. This transform is nonlinear but shows approximate linear relationships between m
and p at the central section of the predefined range according to previous research. Hence, the
inversion can converge quickly with appropriate bound information, especially when a relative
loose range is used. Based on all these vectors and the operator integration described above, the
objective function of gravity and the NSS data cross-gradient joint inversion in 3D space can be
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expressed as follows:
1 1 1
22 2
0: = ( )
:
d L p
x
y
z
minimize g
subject to
C C Cp d Dp W p p
τ
τ 0
τ
, (3)
where g is the transformed forward modeling operator with respect to p , and D is the
combined first- or second-order derivative matrix, which provides the smoothing measure of the
model. dC , LC , and pC are diagonal covariance matrices for data misfit, smoothness, and
smallness terms, respectively. 0 10 20[ , ]T T Tp p p is the combined reference model vector. W is
the depth-weighting matrix used to correct the weight of each cell at different depths. The
objective function is under an equality constraint, which requires that all three components of the
cross gradients are equal to 0 . The cross-gradient vector and its three components x , y ,
and z at an arbitrary point were first defined by Gallardo and Meju (2004) as
1 2(x, y,z) (x, y,z) (x, y,z)m m , (4)
and
1 2 1 2
1 2 1 2
1 2 1 2
(x, y, z) (x, y, z) (x, y, z) (x, y, z)(x, y, z)
(x, y, z) (x, y, z) (x, y, z) (x, y, z)(x, y, z)
(x, y, z) (x, y, z) (x, y, z) (x, y, z)(x, y, z)
x
y
z
m m m m
y z z y
m m m m
x z z x
m m m m
x y y x
. (5)
To measure the whole model, corresponding vectors xτ , yτ , and zτ are arranged in column
form as model vectors and are combined in the equality constraint of the objective function. The
cross-gradient components express the structural similarity of the two models on their orthometric
vertical projection surfaces. When the model gradients have the same or reverse direction, i.e.,
their gradients are parallel, these three components tend to approach zero; otherwise, they take on
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non-zero quantities. For the whole model volume, the built-up array in the equality constraints
term is used to measure the total structural similarity. When the structures are consistent, this array
should approach a null space of 0. For discrete models, the cross gradient is usually computed in
the forward difference scheme. More general finite difference schemes for cross gradients based
on gradient operator meshes are provided by Lelièvre and Farquharson (2013).
In contrast to the previous work of Fregoso and Gallardo (2009), Eq. (3) incorporates a
depth-weighting term that is commonly used in potential field inversion. This term significantly
reduces the tendency for the inverted anomaly to be concentrated near the surface, and it also
attempts to correct structural perturbance during the iterative inversion process. The bound
constraint was also included by employing a transform function, which can reduce the
non-uniqueness remarkably. These measures have been proven necessary and effective for the
cross-gradient joint inversion of gravity and magnetic data both in theoretical studies and practical
applications. Another noteworthy issue is that the contribution of the smallness term is emphasized
to enhance the use of the reference model rather than the smoothness. The model smoothness
requirement is relatively low, and this term only assists as an auxiliary measure and the
depth-weighting matrix is ignored.
2.3 Iterative formulas of inversion
Equation (3) permits one to seek the combined physical property model parameters that
simultaneously satisfy the minimization of data fitting, smoothness, and smallness with structural
consistence. Based on the generalized nonlinear least-squares approach (Fregoso and Gallardo,
2009; Tarantola and Valette, 1982), the objective function is solved in an iterative form as follows:
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1 1
1
T
k k k k k k k
p p N n N B t , (6)
where k denotes the iteration number, kN and
kn are calculated by
1 1 1T T T
k k d k L p
N P G C GP L C L C , (7)
and
1 1 1
0( ) ( )T T T
k k d k L k p kg n P G C p d L C Lp C p p . (8)
Here, G is the combined forward modeling sensitivity matrix with respect to m and kB is
the cross-gradient partial derivative matrix with respect to kp , which can be written as
TTT T
yx zk
ττ τB P
m m m, (9)
where kP is the transform-function derivative matrix of kp and the ith diagonal element is
computed by
2
( )
1
i
i
cp
iicp
b a ceP
e
. (10)
In Eq. (6), vector kt is solved by the damped least squares technique:
11 1max( )T
T k k kk k k k k k k k
B N BB N B I t B N n τ , (11)
where > 0 is a coupling factor and I is the identity matrix. plays an important role in
determining the structural coupling level of the inversion results. Model similarity is enhanced
with increasing values, and it is reduced in turn with decreasing values. This indicates
that the joint inversion is equivalent to a separate one when approaches 0. So a large is
preferred to help obtain a high structural similarity level. One issue is that the matrix on the
left-hand-side brackets of Eq. (11) becomes ill-posed when is too large, which makes the
equation difficult to solve accurately. Thus, numerical experiments should be conducted before
inversion to estimate an optimal value that can achieve both structural resemblance and
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solving accuracy goals. Some tests on gravity and magnetic inversion have shown that a range of
103–10
6 is suitable for most inversion cases.
For the whole procedure, observational gravity and the NSS (or the TMI) data are
incorporated along with the reference and initial models into the iterative framework
simultaneously. The intermediate matrices and vectors are solved by Eqs. (7)–(11) at each iteration,
and then, vector p is updated by Eq. (6) until the data misfit and cross-gradient norm satisfy the
given threshold. The inverted density and magnetization results are eventually obtained by Eq. (2).
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3. Synthetic example
To verify the efficiency of the proposed algorithm, we applied it to a simulated area where
remanence cannot be neglected and the magnetization directions vary. The subsurface was divided
into 20 × 20 × 10 regular cells in the x-, y-, and z-directions, respectively, with edge lengths of 50
m. Two prismatic targets were embedded in the homogenous half-space with the same size of 200
× 150 × 150 m3 and a roof depth of 100 m. Their horizontal distance was 100 m. The density
contrasts were assigned as 0 g/cm3 and 1 g/m
3 (equivalent to only having one prism anomaly), and
the magnetization amplitudes were 1 A/m and 0.5 A/m. The magnetization directions were I = 30°,
D = -45° and I = 70°, D = 45° for the two targets, as shown in Fig. 1; furthermore, the
geomagnetic inclination and declination were I = 50°, D = 0°, respectively.
The observational data were acquired on a 36 × 36 regular grid with an elevation of 1 m. As
shown in Fig. 2a and 2b, the synthetic gravity and magnetic responses were modeled with 1%
Gaussian noise added. The RTP transform of the TMI data was conducted by ignoring the
remanence (shown in Fig. 2c). Generally, the RTP data were in accordance with the subsurface
magnetic anomaly when remanence was insignificant. However, in this example, the RTP anomaly
peak values mismatched the central location of the targets because of the existence of remanence
(Guo et al., 2014). The transformed NSS data were also computed (Fig. 2d), and the results were
more coincident with the subsurface anomalies. Therefore, incorporating the NSS data into an
inversion can theoretically reduce the deviation caused by remanent magnetization. Note that the
noise level of the NSS data rose because of the related derivative calculations, so it is preferable to
use low-pass-filtered NSS data in the inversion procedure to get rid of the occurrence of shallow
superfluous local anomalies.
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For simplicity, the density and magnetization bound parameters were fixed as constant for the
whole model. Loose ranges of -0.1–1.1 g/cm3 for the density contrast and -0.1–1.1 A/m for the
magnetization amplitude were chosen which contain the true property values 0, 0.5 and 1 g/cm3
(A/m) with a relative small buffer 0.1. Note that for the lower bound of magnetization, we used
-0.1 rather than 0 to ensure that the open interval includes the true background value 0. Negative
magnetization value within -0.1–0 was approximately regarded as 0, which is equivalent to the
non-magnetism case. Gravity, TMI, and NSS smallness regularization factors of α1 = 3.5 × 10-5
, α2
= 5.0 × 10-6
, and α3 = 1.5 × 10-7
, respectively, were set for building the covariance matrices, and
these values followed those of Fregoso and Gallardo (2009). The smoothness factors served as
auxiliary measures and were set to αs = 105 for all cases. The depth-weighting matrix W were
computed by fitting the kernel decay curves with the approximate functions provided by Li and
Oldenburg (1996). We tested the combination of gravity and the TMI data with and without
cross-gradient constraints, and then, we replaced the magnetic data type by the NSS data for both
the separate and joint inversions. For the coupling factor, an optimal value of 5 × 104 was selected
for some tests to drive the inversion to achieve structural consistency. This value was adopted for
all the synthetic and field data scenarios mentioned below.
The density and magnetization distributions for separate inversions of gravity and the TMI
data are displayed in Fig. 3c and 3d. The density anomaly was recovered well, as was expected,
but the values were lower than the truth data, which is a common flaw in the potential field
generalized inversion. For the magnetization model, the two anomalies were fused into one at the
central area for the low inversion resolution making it difficult to distinguish the independent but
similar anomalies. Additionally, the left anomaly boundary was distorted because of the
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remanence. The separately inverted density and magnetization results were structurally
inconsistent, and the cross-gradient distributions displayed low similarity in the central area (Fig.
4a).
The jointly inverted density and magnetization distributions from gravity and the TMI data
are shown in Fig. 3e and 3f. Compared to the separate inversion case, the resultant magnetization
model appears to have two independent anomalies with blurred boundaries. However, both the
density and magnetization distribution were more intricate than the separate results, which
interferes with the anomaly target identification. The anomaly maximum values were also
inconsistent with the realistic locations. Although high structural resemblance was achieved (Fig.
4b), the inversion reliability was weakened by the existence of remanence. This indicates that the
joint approach should not be used unless the assumptions are consistent with the a priori
geological setting. Furthermore, it is recommended that the effect of remanent magnetization
should be considered in such a joint inversion case.
Figures 3g and 3h show the separate inversion results for gravity and NSS data. The
recovered magnetization target was located in accordance with the truth data, and the maximum
value fit the left prism, thus showing effective corrections of the NSS data. The two independent
anomalies again were blurred and appeared to fuse into one, thereby the technique failed to
delineate the two prismatic targets. The NSS data decayed faster than the TMI data with increases
in the observation-source distance, which resulted in a low capacity to reflect sources at distance.
This is illustrated by the broad tails at depth in Fig. 3h. Ultimately, the inversion resolution of the
NSS data needs to be improved with additional information provided by joint inversion.
We also carried out a cross-gradient joint inversion for gravity and NSS data, and the results
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are shown in Fig. 3i and 3j. Apparently, both the density and magnetization distributions showed
noticeable improvements when compared to the results in Fig. 3c–3h. The peak values of the two
models were perfectly located in the center of the prisms, and the anomalies were closer to the true
values. Furthermore, the density anomaly benefited from the magnetic information without
deviation caused by remanence. Additionally, the magnetization results clearly identified two
independent targets in correct positions with sharpened boundaries, and the maximum amplitude
was consistent with the centers of the anomalies. Their cross-gradient distributions are shown in
Fig. 5. It is clear that the cross gradients fall off several orders of magnitude low, thus indicating
high similarity to the results for the joint inversion with the NSS data. Note that a lower
cross-gradient distribution does not mean that it is more reliable. Some unwanted redundant
structure may appear because of errors caused by inconsistencies in the real geological
information and the basic assumptions used for the inversion methodology.
Overall, it was demonstrated that cross-gradient joint inversion of gravity and NSS data could
significantly reduce the influence of remanent magnetization, thereby improving the accuracy and
resolving capacity of the data. It was difficult to obtain reasonable models when only employing
the transformed NSS data or cross-gradient joint inversion technique.
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4. Application to field data
We tested the proposed algorithm on real gravity and magnetic data, which were collected
from a mining area located in a polymetallic metallogenic belt of the Yangtze River, China. The
study area covered 7000 × 3000 m2, and it had flat topography. Continental volcanic formations
are widely exposed in this region. Particularly, there was tremendous volcanic activity including
massive eruptions and intrusions in the late Jurassic Period and early Cretaceous Period. Syenite
and monzonite outcrops occur locally in the area. The volcanic-sedimentary strata can be divided
into several formations based on geological associations. These formations are as follows:
Cretaceous Baitoushan Formation (K1b) of trachyte, vulsinite, and oslporphyry; Gushan
Formation (K1g) of sedimentary tuff, andesite, and quartz diorite porphyry; Jurassic Dawangshan
Formation (J3d) of andesite, breccia andesite, andesitic breccia lava, and diorite; Longwangshan
Formation (J3l) of hornblende andesite, andesitic volcanic breccia, agglomerate, and
trachyandensite. There are also volcaniclastic and lava rocks widely exposed in various formations,
especially in the northwestern part of the study area. The metallic ores such as magnetite,
specularite, and chalcopyrite are mainly present inside of the faults or fracture zones, the intrusive
contact zones, and the depression-uplift structures of the effusive rock basin.
A comprehensive geological profile is shown in Fig. 6, which was inferred through credible
information from field reconnaissance work, rock sampling statistics, drillings, loggings, and
geophysical prospecting databases. This a priori knowledge was later used to verify the
effectiveness of the inversion. The statistics for rock density properties show that most of the
rocks have an intermediate value except for the widely scattered iron ore, which hardly yields an
effective gravity anomaly. However, the slight density differences among the formations and rock
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mass enable one to identify the occurrence and unconformability of the formations. The magnetic
properties show that magnetite has strong and variable magnetism with a modal susceptibility of
33,484 (4π × 10-6
SI) and a modal remanent magnetization of 5389 (10-3
A/m). Syenite and
monzonite have a moderate susceptibility of n × 103 (4π × 10
-6 SI) and remanent magnetization of
n × 102 (10
-3 A/m). The volcanics such as volcaniclastic and lava rocks have slight magnetism
with a modal susceptibility of about 80–800 (4π × 10-6
SI) and a modal remanent magnetization of
about 50–800 (10-3
A/m). The Koenigsberger ratio (Q) statistics indicate that the remanent
magnetization contributes almost equally with the induced component for considerable parts of
volcanic rocks, which strongly suggests that the remanent magnetization should be of concern
because of the widely distributed magnetic ore bodies and volcanics within the area. In summary,
this area is suitable for conducting joint inversion with consideration of the effects of remanent
magnetization.
For data preparations, observational gravity and magnetic data were properly processed by
low-pass filtering (Wang et al., 2014), and the results are shown in Fig. 5a and 5b. The ambient
field inclination and declination were about I = 46.7°, D = -4.4°, respectively. The NSS data were
transformed and filtered with the expectation of mitigating the remanence effect (Fig. 5c). The
subsurface was divided into 10 × 20 × 10 cubic cells with edge lengths of 400 m, 350 m, and 170
m in the north, east, and depth directions, respectively. Loose density and magnetization bound
constraints were set as -0.2–0.2 g/cm3 and -1–10 A/m in consideration of both the statistical rock
sampling records and inversion converging behavior. Optimal smallness factors of α1 = 1 × 10-6
,
α2 = 1 × 10-5
, and α3 = 1 × 10-8
were chosen by separate inversions for gravity, TMI, and NSS
cases to build the covariance matrices. The initial and reference model were set to zero, and the
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maximum iteration was 6. After the inversion, we extracted the profile AB (solid black line in Fig.
4) from the results to make comparisons with the different inversion cases.
Figures 6a and 6b show the separate inversion results and Figs. 7c and 7d show the joint
inversion results for the gravity and TMI data, respectively. In relation to the geological formation
information, both density results revealed anomalies raised by syenite–monzonite intrusive bodies
and volcanic breccia–lava bodies at low resolution, while the magnetic profile showed little
accordance with the geological formation information. The separately inverted magnetization
model seemed to be affected by remanence because of its deviation with the rock mass at depth,
especially in the northwestern area. A low anomaly occurred near the Longwangshan Formation
(J3l) in the jointly inverted model, which is less reliable and not compatible with the geology. Then,
separate inversions of gravity and NSS data were conducted, the results of which are shown in Fig.
7e and 7f. The magnetization distribution showed a high magnetism layer extending horizontally
at depth, and its uplifting location in the northwest was in accordance with an inferred
paleovolcanic vent. The NSS data resulted in better model performance in regards to the
horizontal location of the causative body. However, the anomalous volume was much different
from the density results and the former magnetic results. As described in the previous section,
surface NSS data have difficulty reflecting deep sources, so the continuous and significant high
magnetization anomaly was regarded as unreliable. This can also be proven by known geological
knowledge. Figures 7g and 7h show the joint inversion results for gravity and NSS data. This
density model was in better agreement with the geological settings than the former results, and it
illustrates the benefits of including the elaborate complementary magnetic data. More
improvements appeared in the magnetization section, which represents the syenite–monzonite
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intrusive bodies and the volcanic breccia-lava bodies, and the data reproduced the high
magnetization anomalies well with higher resolution. Figure 8 shows the cross-gradient
distributions on the profile produced by the inversions employing the TMI or NSS data. It is clear
that the joint inversion results were more compatible than those from the separate cases, as shown
by the effectiveness of cross-gradient constraints. The main advantage of this joint inversion of
gravity and NSS data is that the quality of both the density and magnetization results is heightened
in terms of the resolution and reliability. It is obvious that the inverted density and magnetization
results were in good agreement with the existing geological information.
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5. Conclusions
Three-dimensional cross-gradient joint inversion of gravity and NSS data was studied for the
purpose of reducing the remanence effect and acquiring better results with improved resolution.
Since the commonly used assumption that all magnetic anomalies are the result of induced
magnetization fails when remanence is an issue, we first transformed TMI data to NSS, then
employed it to conduct the joint inversion to mitigate the unwanted effects of remanence. The
cross-gradient approach was proven effective, and it can be widely applied to obtain more
compatible models with better resolution. To improve the joint algorithm to obtain even better
results, some additional modules such as depth-weightings and bound constraints were
incorporated; these help to further reduce the inherent non-uniqueness. A coupling factor was also
introduced in the iterative formula to achieve high structural similarity. Synthetic and field data
inversion examples demonstrated that this algorithm could effectively reduce the effects of
remanent magnetization and produce inverted density and magnetization results that are closer to
real geological information. Comparatively, the cross-gradient joint inversion with TMI data
definitely brings about deviations due to the effects of remanence, and the separately inverted
magnetization model from the NSS data showed a low ability to recover causative bodies at depth.
It is unlikely to perform better if the NSS data and cross-gradient constraints are not employed
simultaneously in such a case. To summarize, when significant remanence exists, implementation
of the proposed joint method for gravity and NSS data can produce results that are more reliable.
It should also be noted that this algorithm might be improved further by introducing additional
borehole data that can enhance the resolution, especially the depth resolution.
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Acknowledgement
We sincerely thank Peter G. Lelièvre, Luis A. Gallardo and Emilia Fregoso for the valuable
suggestions and helpful discussions to improve this paper. We are grateful for the financial support
of the National Natural Science Foundation of China (No. 41374093 and No. 41474106), Beijing
Higher Education Young Elite Teacher Project (YETP0650), the Major National scientific research
and equipment development project (ZDYZ2012-1-02-04), and the national 863 Project (No.
2014AA06A613, No. 2013AA063901-4 and No. 2013AA063905-4).
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Figures
Fig. 1. Expansion view of the synthetic model. Two prismatic bodies and their magnetization
direction are demonstrated. Dashed lines outline the location of the geology targets. Arrows and
their length illustrate the total magnetization direction and corresponding orthogonal projection in
the specific plane.
Fig. 2. Observational responses of the synthetic model and the transforms of the TMI data. (a)
Gravity data, (b) TMI data, (c) RTP transform of the TMI data, and (d) NSS transform of the TMI
data. The white line is the top view of the vertical profile AB.
Fig. 3. Cross section AB of the true model and inversion results. (a) True density and (b)
magnetization model. (c) Density and (d) magnetization model for the separate inversion of
gravity and TMI data; (e) density and (f) magnetization for the joint inversion of gravity and TMI
data; (g) density and (h) magnetization model for the separate inversion of gravity and NSS data;
(i) density and (j) magnetization for the joint inversion of gravity and NSS data.
Fig. 4. Cross-gradient amplitude distributions of profile AB for the four inversion cases,
which were shown in Fig. 3; (cd) is for the separate inversion of gravity and TMI data, (ef) is for
the separate inversion of gravity and NSS data, (gh) is for the joint inversion of gravity and TMI
data, and (ij) is for the joint inversion of gravity and NSS data.
Fig. 5. Low-pass-filtered (a) gravity, (b) TMI, and (c) its transformed NSS data from a
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metallic deposit region in China. The black line indicates the profile AB, and white points marked
BH1 and B2 are the locations of borehole collars.
Fig. 6. Geological cross section AB used to evaluate the inversion performance. The blue
lines stand for the locations of the boreholes, and red boxes indicate the known orebodies. 1 –
Quaternary, 2 – trachyte (Baitoushan Fm.), 3 – sedimentary tuff and andesite (Gushan Fm.), 4 –
andesite, Breccia andesite, and andesitic breccia lava (Dawangshan Fm.), 5 – volcanic breccia and
lava, 6 – hornblende andesite, andesitic volcanic breccia, agglomerate, and trachyandensite
(Longwangshan Fm.), 7 – monzonite, 8 – syenite.
Fig. 7. Inversion results of the field data presented in Fig. 5. (a) Density and (b)
magnetization results for the separate inversion of gravity and TMI data; (c) density and (d)
magnetization results for the joint inversion of gravity and TMI data; (e) density and (f)
magnetization results for the separate inversion of gravity and NSS data; (g) density and (h)
magnetization results for the joint inversion of gravity and NSS data.
Fig. 8. Cross-gradient amplitude distributions for the four inversion cases, which were shown
in Fig. 7; (ab) is for the separate inversion of gravity and TMI data, (cd) is for the separate
inversion of gravity and NSS data, (ef) is for the joint inversion of gravity and TMI data, and (gh)
is for the joint inversion of gravity and NSS data.
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Figure 1
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Figure 2
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Figure 3
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Figure 4
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Figure 5
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Figure 6
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Figure 7
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Figure 8
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Table
Table 1. Statistics for the magnetic properties of the rock samples within the study area.
Rocks
Sample
number
Susceptibility (4π × 10-6
SI) Remanent magnetization (10-3
A/m)
Q Density (g/cm3)
Range Modal value Range Modal value
Magnetite 154 25000–200,000 33,484 2000–200,000 5389 <1 >3.5
Hematite - - 2394 - 387 - >3.5
Syenite 179 600–7200 1500 300–1400 600 - -
Monzonite 187 2900–5900 4000 250–2000 700 - -
Andesite - 300–3500 - - 775 - 2.5–3.8
Volcaniclastic rock (K1g) 82 62–107 80 50–75 61 1.55 2.65
Lava rock (K1g) 25 94–781 511 95–286 229 0.91 2.65
Volcaniclastic rock (J3d) 211 micro–973 156 micro–290 117 1.52 2.65
Lava rock (J3d) 145 65–2662 223 72–576 129 1.17 2.65
Volcaniclastic rock (J3l) 48 micro–3455 89 micro–799 79 1.8 2.67
Lava rock (J3l) 42 281–811 319 465–558 475 3.02 2.67
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Highlights
A 3D cross-gradient joint inversion algorithm for gravity and NSS data is proposed.
The NSS data were incorporated to reduce the remanent magnetization effect.
Depth-weightings and bound constraints were also included in the inversion.
The method was validated successfully using synthetic and field data.
The method was found to improve both the resolution and reliability of the inversion results.