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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

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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: zjjs_195@163.com

Xiaohong Meng: mxh@cugb.edu.cn (Corresponding author)

Lianghui Guo: guo_lianghui@163.com

Sheng Zhang: zhangsheng261@163.com

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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.