Three-dimensional cross-gradient joint inversion of gravity and ...

Journal of Applied Geophysics 119 (2015) 51–60

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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 Zhou, Xiaohong Meng ⁎, Lianghui Guo, Sheng Zhang a b

Key Laboratory of Geo-detection (China University of Geosciences, Beijing), Ministry of Education, Beijing 100083, China School of Geophysics and Information Technology, China University of Geosciences, Beijing 100083, China

a r t i c l e

i n f o

Article history: Received 23 December 2014 Received in revised form 4 May 2015 Accepted 5 May 2015 Available online 7 May 2015 Keywords: Cross-gradient joint inversion Remanent magnetization Total magnetic intensity Normalized magnetic source strength Gravity data

a b s t r a c t 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. © 2015 Elsevier B.V. All rights reserved.

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 nonuniqueness 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 ⁎ Corresponding author. E-mail addresses: [email protected] (J. Zhou), [email protected] (X. Meng), [email protected] (L. Guo), [email protected] (S. Zhang).

http://dx.doi.org/10.1016/j.jappgeo.2015.05.001 0926-9851/© 2015 Elsevier B.V. All rights reserved.

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 crossgradient joint inversion algorithm to gravity and magnetic data, and their results showed improvements both in terms of the lateral and

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depth resolution with high structural 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 that 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; Fregoso et al., 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 (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 constraint techniques to reduce the effect of remanent magnetization and improve the resolution of the inverted results. We chose to use NSS data 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 depthweightings 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. 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) component 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): d¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3cm md −λ22 −λ1 λ3 ¼ ; r4

ð1Þ

where d represents the NSS data, λ1, λ2, and λ3 are the descending sorted eigenvalues of the MGT matrix, cm = 10− 7 H/m in SI units, r is the observation-source distance, and md 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 in 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 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

J. Zhou et al. / Journal of Applied Geophysics 119 (2015) 51–60

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.

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

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magnetization amplitude vectors m1 and m2 (both arranged in column format) are regarded as unknowns. These two vectors are combined as m = [mT1, mT2]T in the joint framework. The unknown magnetization directions of the model are not considered. Similarly, observational gravity data d1 and the transformed NSS (or the observational TMI) data d2 are combined into a data vector d = [dT1, dT2]T. Gravity and the NSS (or the TMI) forward operators g1 and g2 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 et al., 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): mðpÞ ¼

a þ becp ; 1 þ ecp

ð2Þ

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,

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.

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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

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.

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integration described above, the objective function of gravity and the NSS data cross-gradient joint inversion in 3D space can be expressed as follows: 2

minimize : φ ¼ kg~ðpÞ−dkC−1 þ kDpk2C−1 þ kWðp−p0 Þk2C−1 p d L 2 3 τx ; subject to : 4 τy 5 ¼ 0 τz

generalized nonlinear least-squares approach (Fregoso and Gallardo, 2009; Tarantola and Valette, 1982), the objective function is solved in an iterative form as follows: −1 T pkþ1 ¼ pk N−1 k nk þ Nk Bk tk ;

ð6Þ

ð3Þ where k denotes the iteration number, Nk and nk are calculated by

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. Cd, CL, and Cp are diagonal covariance matrices for data misfit, smoothness, and smallness terms, respectively. p0 = [pT10, pT20]T 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 τ and its three components τx, τy, and τz at an arbitrary point were first defined by Gallardo and Meju (2004) as ! τ ðx; y; zÞ ¼ ∇m1 ðx; y; zÞ  ∇m2 ðx; y; zÞ;

55

−1 T −1 Nk ¼ PTk GT C−1 d GPk þ L CL L þ Cp ;

ð7Þ

ð4Þ

and ∂m1 ðx; y; zÞ ∂m2 ðx; y; zÞ ∂m1 ðx; y; zÞ ∂m2 ðx; y; zÞ − ∂y ∂z ∂z ∂y ∂m1 ðx; y; zÞ ∂m2 ðx; y; zÞ ∂m1 ðx; y; zÞ ∂m2 ðx; y; zÞ : − τ y ðx; y; zÞ ¼ ∂x ∂z ∂z ∂x ∂m1 ðx; y; zÞ ∂m2 ðx; y; zÞ ∂m1 ðx; y; zÞ ∂m2 ðx; y; zÞ τz ðx; y; zÞ ¼ − ∂x ∂y ∂y ∂x τx ðx; y; zÞ ¼

ð5Þ

To measure the whole model, corresponding vectors τx, τy, and τz 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 non-zero quantities. For the whole model volume, the built-up array in the equality constraint 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 Eq. (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

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.

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Here, G is the combined forward modeling sensitivity matrix with respect to m and Bk is the cross-gradient partial derivative matrix with respect to pk, which can be written as  T B ¼ ∂τx ∂m

∂τTy

∂τTz ∂m

∂m

T Pk ;

ð9Þ

where Pk is the transform-function derivative matrix of pk and the ith diagonal element is computed by P ii ¼

ðb−aÞcecpi ð1 þ ecpi Þ2

:

ð10Þ

In Eq. (6), vector tk is solved by the damped least squares technique:   3 max Bk N−1 BTk k 4Bk N−1 BT þ I5tk ¼ Bk N−1 k k k nk −τk ; β 2

ð11Þ

where β N 0 is a coupling factor and I is the identity matrix. β plays an important role in determining the structural coupling level of the inversion results. Model similarity is enhanced with increasing β values, and it is reduced in turn with decreasing β values. This indicates that the joint inversion is equivalent to a separate one when β approaches 0. So a large β 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 β is too large, which makes the equation difficult to solve accurately. Thus, numerical experiments should be conducted before inversion to estimate an optimal β value that can achieve both structural resemblance and solving accuracy goals. Some tests on gravity and magnetic inversion have shown that a range of 103–106 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). 3. Synthetic example

Fig. 5. Low-pass-filtered (a) gravity, (b) TMI, and (c) its transformed NSS data from a 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.

and −1 T −1 ~ nk ¼ PTk GT C−1 d ½g ðpk Þ−d þ L CL Lpk þ Cp ðpk −p0 Þ:

ð8Þ

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/m3 (equivalent to only having

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, and 8 — syenite.

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Table 1 Statistics for the magnetic properties of the rock samples within the study area. Rocks

Magnetite Hematite Syenite Monzonite Andesite Volcaniclastic rock (K1g) Lava rock (K1g) Volcaniclastic rock (J3d) Lava rock (J3d) Volcaniclastic rock (J3l) Lava rock (J3l)

Sample number

154 – 179 187 – 82 25 211 145 48 42

Susceptibility (4π × 10−6 SI)

Remanent magnetization (10−3 A/m)

Range

Modal value

Range

Modal value

25,000–200,000 – 600–7200 2900–5900 300–3500 62–107 94–781 Micro–973 65–2662 Micro–3455 281–811

33,484 2394 1500 4000 – 80 511 156 223 89 319

2000–200,000 – 300–1400 250–2000 – 50–75 95–286 Micro–290 72–576 Micro–799 465–558

5389 387 600 700 775 61 229 117 129 79 475

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 b, 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

Q

Density (g/cm3)

b1 – – – – 1.55 0.91 1.52 1.17 1.8 3.02

N3.5 N3.5 – – 2.5–3.8 2.65 2.65 2.65 2.65 2.67 2.67

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-passfiltered NSS data in the inversion procedure to get rid of the occurrence of shallow superfluous local anomalies. 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

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.

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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, which 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 was 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 d. 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 remanence. The separately inverted density and

magnetization results were structurally inconsistent, and the crossgradient 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 f. 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. Fig. 3g and h shows 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 are shown in Fig. 3i and j. Apparently, both the density and magnetization distributions showed noticeable improvements when compared to the results in Fig. 3c–h. 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. 4. Application to field 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.

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

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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 (Table 1) 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 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 × 10 3 (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 b. 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 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. Fig. 6a and b shows the separate inversion results and Fig. 7c and d shows 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 f. 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

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performance in regard 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. Fig. 7g and h shows 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 intrusive bodies and the volcanic breccia–lava bodies, and the data reproduced the high magnetization anomalies well with higher resolution. Fig. 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. 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 nonuniqueness. 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. Acknowledgment 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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