Multi-scale Apparent Magnetization Inversion Imaging Method Based on the Theoretical Pseudo-gravity Anomalies

ARTICLE IN PRESS

JID: CHAOS

[m5G;October 18, 2019;11:6]

Chaos, Solitons and Fractals xxx (xxxx) xxx

Contents lists available at ScienceDirect

Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Multi-scale Apparent Magnetization Inversion Imaging Method Based on the Theoretical Pseudo-gravity Anomalies Jun Li a,b,∗, Jinxi Li a,∗, Weilin Li a, Xingxiang Jian a,b, Xin Wu a a b

College of Geophysics, Chengdu University of Technology, Chengdu, 610059, China Key Laboratory of Earth Exploration and Information Technology of Ministry of Education, Chengdu University of Technology, Chengdu, 610059, China

a r t i c l e

i n f o

Article history: Received 9 August 2019 Revised 29 September 2019 Accepted 8 October 2019 Available online xxx Keywords: Inversion imaging Pseudo-gravity anomaly Down continuation Apparent magnetization imaging

a b s t r a c t The complex superposition of magnetic anomalies on the ground caused by multi-source magnetic geologic bodies makes it difficult to invert their geometries and positions. According to the theoretical pseudo-gravity anomalies of the Poisson formula of the gravity and magnetic field, this paper converts the magnetic anomaly dipole source into the centroid anomaly of unipolar source of the gravity, adopts the multi-scale interpolation cutting method to separate the anomaly and utilizes the iterative method to downward continue the multi-scale anomaly to the close-to-source depth of the top interface of each magnetic source so as to acquire the high-resolution magnetic anomaly near the top interface of each source. This magnetization inversion of the anomaly can greatly improve the inversion accuracy and speed. However, the continuation space is always required to meet the first type of boundary conditions of the Laplace equation, which leads to the fact that the close-to-source level may not really reflect the real depth of the anomaly which contains the equivalent effect of depth. Therefore, the method is essentially a fast apparent magnetization imaging method which describes the multi-source and multiscale geological bodies. The increased vertical resolution may help to better define the magnetic interface morphology, and it can also obtain the physical property distribution of the multi-scale magnetic source geological body in three-dimensional space. The method is demonstrated on the synthetic models and on an aeromagnetic data set from the Xisha Trough, South China Sea. © 2019 Published by Elsevier Ltd.

1. Introduction There is usually some correlation between magnetic and gravity anomalies for deep structure and lithology, which reduce the uncertainty during the geological interpretation process. In general, magnetic anomalies are obtained by directional derivative of gravity anomalies based on the principle of potential field of gravity and magnetic anomalies proposed by Poisson in 1826. On this basis, Barnov [1] introduced pseudo-gravity anomalies to seek a unified forward theory of gravity and magnetic anomalies. For 3D inversion of potential field, it’s very difficult to obtain some parameters, for example, relative density and magnetization of rocks and deep structures. Most of the currently available algorithms require the knowledge of magnetization direction. The method of gradient of pseudo-gravity proposed by Roest and Pilkington [15] is used to determine the direction of magnetization.

∗ Corresponding author at: College of Geophysics, Chengdu University of Technology, Chengdu, 610059, China E-mail addresses: [email protected] (J. Li), [email protected] (J. Li).

At present, linear and non-linear inversion methods are mostly applied to source magnetization [2,5–9,13,25,26]. The representative linear methods mainly include: Backus-Gilbert algorithm by Green et al. [6], linear programming by Mottl and Mottlová [12], quadratic linear programming, and Fourier convolution by Bott [2]. With an increasing amount of magnetic data, the algorithm stability becomes weak when either direction solution in the space domain or conversion algorithm in the frequency domain is used. To solve this problem, Yao et al. [25] proposed the local optimization inversion technique in the magnetic interpretation. However, it’s very difficult to solve the hyperscale linear equations because of so many inversion parameters. Zhu et al. [26] introduced the back propagation algorithm of artificial neural networks to invert gravity and magnetic anomalies and built a nonlinear inversion model. Although this algorithm is characterized with power tolerance, it’s hard for fast convergence and exact solution when the effect of overlap appears between positive and negative anomalies of magnetic dipole of multi-source bodies, especially for local magnetic anomalies due to its huge computation. The algorithm adopted in this paper applies Poisson formula to convert the magnetic anomaly dipole source into the centroid

https://doi.org/10.1016/j.chaos.2019.109480 0960-0779/© 2019 Published by Elsevier Ltd.

Please cite this article as: J. Li, J. Li and W. Li et al., Multi-scale Apparent Magnetization Inversion Imaging Method Based on the Theoretical Pseudo-gravity Anomalies, Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109480

ARTICLE IN PRESS

JID: CHAOS 2

[m5G;October 18, 2019;11:6]

J. Li, J. Li and W. Li et al. / Chaos, Solitons and Fractals xxx (xxxx) xxx

anomaly of unipolar source of the gravity based on the pseudogravity anomaly theory and utilizes the interpolation cutting method to perform the multi-scale separation of three-dimensional space upon the anomalies. At the same time, the multi-scale anomalies are downward continued by means of iterative method to obtain multi-space and high-resolution anomalies below the observation space, and the fast apparent magnetization inversion are performed upon the anomalies of the depth. The validity and feasibility of the method are verified by model tests and measured data. In comparison, this method requires minimal prior information and can be applied to the inversion of complex magnetic anomalies with multi-space and multi-field sources.

(1)

where (u,v) are the wave numbers in the x, y coordinate system; T(x, y) and T˜ (u, v ) are magnetic anomalies respectively in spatial and frequency domains. For a magnetic anomalies source, it is assumed that the original magnetic inclination and declination are I1 and D1, respectively. Its direction cosines on surface is (P1 , Q1 , R1 ). I0, D0 and (P0 , Q0 , R0 ) represent magnetic inclination, declination and direction cosines of rock strata. If the magnetization direction of the same body varies from I1 to I2 , with D2 declination, its direction cosines are (P2 , Q2 , R2 ). Then the conversion formula between two different magnetization directions can be expressed as:

Z˜a (u, v ) =



+v

2

 √ i(P0 u + Q0 v ) + R0 u2 + v2   √ i(P2 u + Q2 v ) + R2 u2 + v2  · T˜ (u, v ), · √ i(P1 u + Q1 v ) + R1 u2 + v2

When the magnetic inclination changes from I1 to I2 , the new direction cosines (P2 , Q2 , R2 ) of the same magnetic source body are given by



P2 = cos I2 cos D2 Q2 = cos I2 sin D2 , R2 = sin I2

(5)

(6)

With the substitution from (6) through (2), the reduction-tothe-pole frequency Z˜⊥ (u, v ) is obtained by

Z˜⊥ (u, v ) =



u2 + v2

 √ i(P0 u + Q0 v ) + R0 u2 + v2

·

1  · T˜ (u, v), √ i(P1 u + Q1 v ) + R1 u2 + v2

(7)

The relationship between magnetic potential U and gravitational potential V satisfies Poisson’s equation,

U =−

1 M · gradV, 4 π Gρ

(8)

where M is total magnetization vector, G is gravitational constant, and ρ is the residual density of magnetic source. Supposing the unit vector on X, Y, and Z axes are i, j and k, respectively, then M is established

M = Mx i + My j + Mz k,

(9)

where Mx , Mx and Mz are the project of magnetization vectors in the direction of X, Y, and Z axes, respectively. The gravitational potential gradient is

gradV =

∂V ∂V ∂V i+ j+ k, ∂x ∂y ∂z

(10)

Therefore, form (8) is

1 U =− Gσ



 ∂V ∂V ∂V M + M + M , ∂x x ∂y y ∂z z

(11)

If I = 90°, the magnetic and gravitational field intensities in the gradient are

 ∂U

˜ ∂ z = Z⊥ (u, v ) , ∂ V = g˜ u, v ( ) ∂z

(12)

The pseudo-gravity anomalies in frequency domain is

(2)

g˜(u, v ) =

where i is an imaginary unit (i.e. i2 = -1). The direction cosines (P1 , Q1 , R1 ) of a magnetic source body are given by



P1 = cos I1 cos D1 Q1 = cos I1 sin D1 , R1 = sin I1

(4)

P2 = 0 Q2 = 0 , R2 = 1

The concept of pseudo-gravity anomaly was first proposed by Baranov in 1957 and it was obtained by means of vertical integration of magnetic anomalies. This gravity anomaly converted from magnetic anomalies is not a real measured gravity anomaly, and it is called theoretical pseudo-gravity anomaly. Baranov put forward a six-azimuth algorithm under special circumstances. When this method is used, it must be assumed that the normal magnetic field of the earth is in the same direction as that of the magnetization vector of the ore bodies, and a special coordinate system must be established. Then, in 1967, Hasegawa put forward the eight-azimuth algorithm on this basis. Later, Xie et al. [21]improved this algorithm that is not limited by the above conditions and the calculated coordinates can be arbitrarily selected. However, the above algorithms are all deduced and calculated in the spatial domain, which is intricate and complicated. The algorithm for theoretical pseudo-gravity anomaly by means of frequency domain is given below. In general, gravity or magnetic anomalies on surface are detected on the base of the rock density or magnetic differences. The transform of measured magnetic anomalies between spatial and frequency domains can be expressed as:

u2

P0 = cos I0 cos D0 Q0 = cos I0 sin D0 , R0 = sin I0



2.1. Pseudo-gravity anomaly theory

√

The direction cosines (P0 , Q0 , R0 ) of rock strata are given by

If I2 = 90°, i.e. the magnetic inclination is perpendicular to the reduction-to-the-pole direction, then Eq. (6) is

2. Multi-scale apparent magnetization inversion imaging method

T˜ (u, v ) = F [T (x, y )],



(3)

√ Gρ u2 + v2 1 √ √ 2π M i(P0 u + Q0 v ) + R0 u2 + v2 i(P1 u + Q1 v ) + R1 u2 + v2 · T˜ (u, v ),

(13)

The spectral formula of pseudo-gravity anomalies can be obtained by the inverse Fourier transform:

g(x, y ) = F −1 [g˜(u, v )],

(14)

Please cite this article as: J. Li, J. Li and W. Li et al., Multi-scale Apparent Magnetization Inversion Imaging Method Based on the Theoretical Pseudo-gravity Anomalies, Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109480

ARTICLE IN PRESS

JID: CHAOS

[m5G;October 18, 2019;11:6]

J. Li, J. Li and W. Li et al. / Chaos, Solitons and Fractals xxx (xxxx) xxx

2.2. Multi-scale separation of magnetic anomalies

(n represents the number of cutting radius).

A multi-cutting method for dividing regional gravity field from local gravity field was proposed by Cheng et al. [3] based on the theoretical models which are utilized to investigate various anomaly separation methods. Later, on this basis, Wen and Cheng [20] further introduced the interpolation cutting method for identifying regional and local fields of magnetic anomaly. Some scholars applied multiple interpolation cutting method to practical examples with good results [19,23]. Synthesizing the above theories and methods, we introduce the separation method of theoretical pseudo-gravity anomaly through multi-scale interpolation cutting method: G (x, y), R (x, y) and L (x, y) are defined to represent the measured magnetic anomaly data, large-scale magnetic anomaly data and small-scale magnetic anomaly data at coordinates respectively. The relationship among them shall be:

L(x, y ) = G(x, y ) − R(x, y ),

(15)

Eq. (16) was defined by Cheng et al. [3]

R(x, y ) = aG(x, y ) + bA(x, y ),

(16)

where A (x, y) represents the average value of some observations at a distance of r from the calculation point (x, y), and a and b are the weighting coefficients. By using the above method for iterative calculation, the following can be obtained

lim |Rn−1 (x, y ) − Rn (x, y )| → 0,

n→∞

3

(17)

⎧ g0 (x, y, 0) = L1 (x, y ) ⎪ ⎪ ⎪ g1 (x, y, 0) = L2 (x, y ) − L1 (x, y ) ⎪ ⎪ ⎪ ⎪ g2 (x, y, 0) = L3 (x, y ) − L2 (x, y ) ⎪ ⎪ ⎪ .. ⎨ .

g j (x, y, 0) = L j+1 (x, y ) − L j (x, y ) ⎪ ⎪ ⎪ .. ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ ⎩gn−1 (x, y, 0) = Ln (x, y ) − Ln−1 (x, y ) gn (x, y, 0) = Rn (x, y )

,

(19)

where gj (x, y, 0) represents the ground residual magnetic source gravity anomaly at scale j ( j = 0, 1, 2, 3 · · · n). 2.3. Apparent magnetization imaging inversion In order to obtain the high-resolution magnetization inversion results, the gravity anomalies of the above-mentioned separated magnetic sources at various scales are extended downward by iterative method [21,22]. If a certain irregular magnetic source body with the top interface burial depth of h obtained above is regarded as being composed of M × N slender vertical prisms with different sizes, irregularities and different centroid positions, the frequency spectrum of the magnetic source gravity anomaly can be expressed as follows:



4 au g˜(u, v ) = 2π G · sin uv 2 ·



· sin

bv 2



M×N  e−hr (1 − e−dhr ) ρ j · e−i(ux0 j +vy0 j ) , r

(20)

j=1

Thus, a large-scale anomalous field with a cutting radius of r can be obtained

R(x, y ) = lim Rn (x, y ), n→∞

(18)

By subtracting the large-scale field R (x, y) from the observed potential field anomaly G (x, y), the small-scale field L (x, y) can be obtained from Eq. (15) Generally speaking, the larger the cutting radius r is, the larger the geological body scale reflected by the anomaly being cut is. Therefore, multi-scale magnetic source gravity anomalies g(x, y) can be separated by choosing different cutting radius r1 , r2 , r3 …rn

  −i (ux0 j +vy0 j ) in which ρ˜ (u, v ) = M×N and r = u2 + v2 . The j=1 ρ j · e bottom edge lengths of the prism are respectively a and b, the burial depth of the top interface of the prism is h, and the height is dh. If the anomaly information is extended downward to the depth of hj by means of iteration method, the high resolution magnetic source gravity anomaly g(x, y, hj ) at the corresponding depth of hj of the magnetic body of this scale can be obtained. If the continuation depth does not exceed the burial depth of the top interface of the magnetic source, it can be ensured that the theoretical pseudo-gravity anomaly is always harmonic and continuous during the downward continuation. If the above continuation depth is exactly the upper top surface depth of the magnetic source body, i.e.,

Fig. 1. (a) Space distribution map of three prismatic models; (b) plane map (top view).

Please cite this article as: J. Li, J. Li and W. Li et al., Multi-scale Apparent Magnetization Inversion Imaging Method Based on the Theoretical Pseudo-gravity Anomalies, Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109480

JID: CHAOS 4

ARTICLE IN PRESS

[m5G;October 18, 2019;11:6]

J. Li, J. Li and W. Li et al. / Chaos, Solitons and Fractals xxx (xxxx) xxx

Fig. 2. Magnetic anomaly and pseudo-gravity anomaly. (a) Ground magnetic field value; (b) Pseudo-gravity anomaly.

Fig. 3. Ground residual pseudo-gravity anomaly. (a) a small scale (0–200 m) residual pseudo-gravity anomaly; (b) a large scale (20 0–50 0 m) residual pseudo-gravity anomaly.

Fig. 4. Residual pseudo-gravity anomaly at the top interface of magnetic source. (a) Residual pseudo-gravity anomaly at 25 m depth; (b) Residual pseudo-gravity anomaly at 450 m depth.

Please cite this article as: J. Li, J. Li and W. Li et al., Multi-scale Apparent Magnetization Inversion Imaging Method Based on the Theoretical Pseudo-gravity Anomalies, Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109480

JID: CHAOS

ARTICLE IN PRESS

[m5G;October 18, 2019;11:6]

J. Li, J. Li and W. Li et al. / Chaos, Solitons and Fractals xxx (xxxx) xxx

5

Fig. 5. Residual apparent density imaging inversion at the top interface of magnetic source. (a) Residual apparent density imaging inversion at 25 m depth; (b) Residual apparent density imaging inversion at 450 m depth.

Fig. 6. Distribution map of apparent magnetization at the top interface of magnetic source. (a) Apparent magnetization distribution map at 25 m depth; (b) Apparent magnetization distribution map at 450 m depth.

h = h j , the depth parameter in (Eq. 20) shall satisfy h = 0. So there is:

ρ˜ (u, v ) =

g˜(u, v ) 2π G · u4v sin( au ) · sin( b2v ) · 1r (1 − e−dhr ) 2

,

(21)

By performing inverse Fourier Transform on the above formula, the residual density value of the top surface of the magnetic source body can be obtained:



ρ (x, y ) = F

−1

g˜(u, v ) 2π G · u4v sin( au ) · sin( b2v ) · 1r (1 − e−dhr ) 2



,

(22)

In accordance with the Poisson formula (Eq. 4), it can be deduced that the magnetic susceptibility and density have the following linear relationship:

κ (x, y ) = ρ (x, y ) · Fig. 7. Comparison of the apparent magnetization (density) inversion results and the theoretical values of three models. The green lines are the true values of three models; the dash line is the apparent magnetization (density) at 200 m depth; the red dot dash line is the apparent magnetization (density) at 600 m depth.

κ , ρ

(23)

By substituting the calculated real density ρ (x, y) of the magnetic source gravity into the Eq. (23), the real magnetic susceptibility κ (x, y) of the magnetic source body can be obtained. Meanwhile, the total magnetization M of the magnetic source body can

Please cite this article as: J. Li, J. Li and W. Li et al., Multi-scale Apparent Magnetization Inversion Imaging Method Based on the Theoretical Pseudo-gravity Anomalies, Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109480

ARTICLE IN PRESS

JID: CHAOS 6

[m5G;October 18, 2019;11:6]

J. Li, J. Li and W. Li et al. / Chaos, Solitons and Fractals xxx (xxxx) xxx

Fig. 8. Reginal tectonic map of the South China Sea.XST, Xisha Trough.

Fig. 9. Bouguer gravity (a) and aeromagnetic (b) anomaly map of the Xisha Trough.

be calculated according to the Eq. (24).

M = κ (x, y ) ·

T0

μ

,

nitude of its own magnetization, but also the fluctuation of the magnetic interface of the same depth.

(24)

where T0 is total induced magnetization intensity; μ is the permeability of a substance. However, it shall be emphasized that the burial depth of the top interface of the magnet source is usually unknown. Therefore, the actual downward continuation depth may not reach or exceed the actual depth of the top interface of the magnetic source body, which will result in the depth parameter h = 0 (in Eq. (21)). If it is still calculated according to Eq. (22), the depth shall be regarded as the top surface depth of the magnetic source body, which is an apparent depth, and the value of the calculated magnetization is smaller or larger than its real value. In this paper, the inversion magnetization is also called the apparent magnetization. At the same time, when the magnetic source body contains strong residual magnetization, it is difficult to accurately acquire the magnetic field T0 , and usually the normal field of the measurement area is taken. This apparent magnetization may reflect not only the mag-

3. Synthetic examples To describe the algorithm process and imaging effect in detail, three prismatic models are created as shown in Fig. 1. The forward geometric parameters and physical parameters of each model are shown in Table 1. The ground magnetic field value is calculated according to the parameters (Fig. 2a), and the gravity anomaly of the ground magnetic source of the model is calculated through the theoretical pseudo-gravity anomaly (Fig. 2b). The gravity anomalies of the ground magnetic source are separated by the cutting radius of 200 m and 500 m respectively, and the gravity anomalies of the ground residual magnetic source at a small scale (0–200 m) and at a large scale (20 0–50 0 m) in space are obtained (Fig. 3). The small-scale anomaly is extended downward by 25 m through an iterative method to obtain the pseudogravity anomaly data at the top interface of the model 2 and the

Please cite this article as: J. Li, J. Li and W. Li et al., Multi-scale Apparent Magnetization Inversion Imaging Method Based on the Theoretical Pseudo-gravity Anomalies, Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109480

ARTICLE IN PRESS

JID: CHAOS

[m5G;October 18, 2019;11:6]

J. Li, J. Li and W. Li et al. / Chaos, Solitons and Fractals xxx (xxxx) xxx

7

Table 1 Model parameters. Density Difference (g/cm3 ) Magnetization (10−3 A/m) Magnetic Declination (°) Magnetic Inclination (°)

Model Center Coordinates (m) Geometry(m) Length Width Height 1 2 3

(650, 650, 500) (200, 200, 50) (650, 650, 50)

300 50 50

300 50 50

100 50 50

0.4 0.5 0.5

80 100 100

0° 0° 0°

45° 45° 45°

Fig. 10. Apparent magnetization anomaly maps at different depths from aeromagnetic inversion of the Xisha Trough.

model 3 in Fig. 1 (Fig. 4a). The large-scale anomaly is extended downward by 450 m to obtain the pseudo-gravity anomaly data of the model 1 (Fig. 4b). Then, the apparent density inversion calculations of the top interface of pseudo-gravity anomaly are performed at the above two scales to obtain the apparent density distribution of the magnetic

source bodies (Fig. 5). It can be seen from the map that the apparent density of the inversion of large and small scale anomalies can basically reflect the real density of the model. This is due to that the downward continuation depth of the iterative method is consistent with the depth of the top interface of the model magnetic source. However, the large-scale apparent density still con-

Please cite this article as: J. Li, J. Li and W. Li et al., Multi-scale Apparent Magnetization Inversion Imaging Method Based on the Theoretical Pseudo-gravity Anomalies, Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109480

JID: CHAOS 8

ARTICLE IN PRESS

[m5G;October 18, 2019;11:6]

J. Li, J. Li and W. Li et al. / Chaos, Solitons and Fractals xxx (xxxx) xxx

tains some small-scale anomalies, which is attributed to two reasons that the small-scale error of separation is amplified with the increase of depth and the cutting radius of large-scale anomaly separation is relatively small. Further, according to the relationship between the magnetic susceptibility and the density (Eq. 14) and the magnetization equation (Eq. 15), the apparent magnetization conversion is performed on the deep apparent density, and the apparent magnetization distribution of the two-scale magnetic source bodies can be obtained (Fig. 6). Based on the results, it can be seen that the apparent density and apparent magnetization obtained from the inversion are well

coincided with the model parameters by using the apparent intensity imaging method of pseudo-gravity in this paper (Fig. 7). Among them, the inversion magnetization of the large-scale model is 0.075 A/m, which is close to the theoretical data of 0.08 A/m. The magnetization inversion of the small-scale model is 0.1 A/m, which is in agreement with the theoretical value of 0.1 A/m. This shows that when the continuation depth is close to the actual depth, the apparent parameters of the inversion are close to the real parameters. Of course, in practical applications, since the depth of the geological body is not determined and the cutting radius of the anomaly separation is uncertain, so the separated multi-scale residual anomalies cannot truly reflect the anomalies

Fig. 11. Apparent density anomaly maps at different depths through the gravity inversion of the Xisha Trough.

Please cite this article as: J. Li, J. Li and W. Li et al., Multi-scale Apparent Magnetization Inversion Imaging Method Based on the Theoretical Pseudo-gravity Anomalies, Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109480

JID: CHAOS

ARTICLE IN PRESS

[m5G;October 18, 2019;11:6]

J. Li, J. Li and W. Li et al. / Chaos, Solitons and Fractals xxx (xxxx) xxx

of the actual magnetic body. The downward continuation depth of the iterative method for each scale anomaly may not coincide with the depth of the real magnetic body, so that the inverted intensity parameters are different from the real parameters to some extent. Therefore, the inverted magnetization can only be regarded as the apparent magnetization of the geological body of that depth. However, the apparent magnetization parameters obtained by inversion do not affect the magnetic qualitative comparison of multiple magnetic bodies in transverse direction. Therefore, the parameters inverted in this paper are called apparent magnetization imaging inversion. 4. Application to a real case In previous sections we have illustrated the advantages of multi-scale apparent magnetization imaging method based on pseudo-gravity anomaly theory with test examples. Now the method will be applied to the real case of an aeromagnetic data set in the conjunction of observed gravity data from the Xisha Trough, South China Sea. The South China Sea located in Southeast Asia is surrounded by three plates: the Pacific to the east, the Eurasian to the north and Indian-Australian plate to the west and south (Fig. 8). It is the products of the closure of Neo-Tethys Ocean. Due to its special position, geological complexity and prolific petroleum, it has become the focus of tectonics, geophysics and petroleum geology [4,10,11,14,16–18,24]. Research results indicate the South China Sea experiences multi-phase rifting and seafloor spreading [4,11]. However, its tectonic evolution and geodynamics are still in debate. The Xisha Trough between Hainan and Xisha Islands in the northwestern part of the South China Sea provides an ideal case which helps understand the rifting process and the tectonic evolution of the South China Sea. The tectonic features of the South China Sea rest largely on the interpretation of gravity and magnetic data. In this paper, the multi-scale cutting method for anomaly separation and iterative method with downward continuation techniques are used for the multi-scale anomaly separation of the Bouguer gravity anomaly (Fig. 9a) and aeromagnetic anomaly (Fig. 9b) in the Xisha Trough to obtain the information of gravity anomaly and pseudo-gravity anomaly at different continuation positions and to perform the apparent magnetization imaging inversion on gravity and magnetic anomaly. Apparent magnetization maps (Fig. 10) and apparent density maps (Fig. 11) with the depths of 8–18 km in the middle and lower crust in the study area are selected. The inversion results show that the apparent magnetization imaging of pseudo-gravity anomaly is consistent with the apparent density imaging of Bouguer gravity anomaly. The Xisha Trough has an east-west striking symmetrical anomalies zone along 18° N latitude, reflecting the feature of gravity and magnetic homology in the center of the trough. The anomaly belt becomes wider with increasing depth. Those results indicate that the magnetic signature is more clearly illustrated after data processing by this method. It is a typical rift structure. The formation may be related to the expansion of the northwest sub-basin in the eastern trough. The gravity and magnetic imaging strongly supports that the Xisha Trough is a residual rift during the process of the northwest subbasin expansion. 5. Conclusion The apparent magnetization inversion proposed in this paper, without considering the direction of remnant magnetization, transforms the observed magnetic anomalies into unipolar gravity anomalies. Then, the gravity anomalies are separated by interpolation cutting method at multi-scale, and continued downward by

9

iteration method to the level of close to the top of each magnet source so that the high resolution magnetic anomaly near the top of each source can be obtained. Therefore, the method is essentially a high resolution apparent magnetization imaging inversion algorithm. In comparison, the traditional magnetization vector inversion needs to invert the magnitude and direction of the magnetization, and the multiplicity of solutions obviously exist. The traditional magnetic anomaly inversion must search for the regularization factors by repeatedly solving non-linear problems. This process is very time-consuming. The imaging inversion algorithm in this paper is performed in the frequency domain and the inversion is only performed on the data at the continuation depth. The depth parameter is regarded as a constant. Therefore, the method is a linear inversion with the characteristics of fast calculation speed, small computational memory consumption and small amount of parameter inversion calculation. Multi-scale anomaly separation and inversion can be performed for anomalies of different depths, especially with the effect of iterative downward continuation, which further improves the resolution of anomalies. It is a large-scale, fast and high-resolution inversion method of three-dimensional apparent magnetization imaging, which can be widely applied to the mineral exploration and deep structure. Declaration of Competing Interest None.

Acknowledgments This paper owes its debt to the support of the National Key Research and Development Program (2017YFC0601504) and Major Research Projects of the National Natural Science Foundation of China (no. 91755215). References [1] Baranov V. A new method for interpretation of aeromagnetic maps: pseudo– gravimetric anomalies. Geophysics 1957;22(2):359–83. [2] Bott MHP, Smith RA, Stacey RA. Estimation of the direction of magnetization of a body causing a magnetic anomaly using a pseudo-gravity transformation. Geophysics 1966;31(4):803–11. [3] Cheng FD, Liu DJ, Yao RX. A study on the identification of regional and local graviy fields. Comput Tech Geophys Geochem Explor (in Chinese), 1987;9(3):1–9. [4] Cullen A, Reemst P, Henstra G, Gozzard S, Ray A. Rifting of the South China Sea: new perspectives. Pet Geosci 2010;16(3):273–82. [5] Gerovska D, Araúzo-Bravo Marcos J, Stavrev P. Estimating the magnetization direction of sources from south-east bulgaria through correlation between reduced-to-the-pole and total magnitude anomalies. Geophys Prospect 2008;57(4):491–505. [6] Green WR. Inversion of gravity profiles by use of a backus-gilbert approach. Geophysics 1975;40(5):763–72. [7] Lelièvre PG, Oldenburg DW. A 3D total magnetization inversion applicable when significant, complicated remanence is present. Geophysics 2009;74(3):21–30. [8] Li SL, Li YG. Inversion of magnetic anomaly on rugged observation surface in the presence of strong remanent magnetization. Geophysics 2014;79(2):11–19. [9] Li YG, Oldenburg D, Douglas W. 3-D inversion of magnetic data. Geophysics 1996;61(2):394–408. [10] Lu BL, Wang WY, Zhao ZG, Feng XL, Zhang GC, Luo XG, Yao P, Ji XL. Characteristics of deep structure in the South China Sea and geological implications: Insights from gravity and magnetic inversion. Chin J Geophys (in Chinese), 2018;61(10):4231–41. doi:10.6038/cjg2018L0564. [11] Morley CK. A tectonic model for the tertiary evolution of strike-slip faults and rift basins in SE Asia. Tectonophysics 2002;347(4):189–215. [12] Mottl J, Mottlová L. Solution of the inverse gravimetric problem with the aid of integer linear programming. Geoexploration 1972;10(1):53–62. [13] Pilkington M. 3-D Magnetic Imaging Using Conjugate Gradients. Geophysics 1997;62(4):1132–42. [14] Qiu X, Ye S, Wu S, Shi X, Zhou D, Xia K, Flueh ER. Crustal structure across the Xisha trough, northwestern south china sea. Tectonophysics 2001;341(1):179–93.

Please cite this article as: J. Li, J. Li and W. Li et al., Multi-scale Apparent Magnetization Inversion Imaging Method Based on the Theoretical Pseudo-gravity Anomalies, Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109480

JID: CHAOS 10

ARTICLE IN PRESS

[m5G;October 18, 2019;11:6]

J. Li, J. Li and W. Li et al. / Chaos, Solitons and Fractals xxx (xxxx) xxx

[15] Roest WR, Pilkington M. Identifying remanent magnetization effects in magnetic data. Geophysics 1993;58(5):653–9. [16] Sun Q, Wu S, Cartwright J, Wang S, Lu Y, Chen D, Dong D. Neogene igneous intrusions in the northern South China Sea: Evidence from high-resolution three dimensional seismic data. Mar Pet Geol 2014;54:83–95. [17] Taylor B, Hayes DE. Origin and history of the South China Sea basin. The tectonic and geologic evolution of Southeast Asian Seas and Islands (Part 2). Geophys Monogr, 27. American Geophysical Union; 1983. [18] Wan K, Xia S, Cao J, Sun J, Xu H. Deep seismic structure of the northeastern South China Sea: Origin of a high-velocity layer in the lower crust. J Geophys Res 2017;122(4):2831–58. [19] Wang BZ, Xu SZ, Liu BH, Yan HJ, Zhang JS, et al. An example of aeromagnetic anomaly separation using multi-interpolation division (in Chinese). Oil Geophys Prospect 1997;32(3):431–8. [20] Wen BH, Cheng FD. A New interpolating cut method for identifying regional and local fields of magnetic anomaly. J Cent South Inst Min Metall (in Chinese), 1990;21(3):229–35.

[21] Xie J. Calculation method of pseudo-gravity anomaly and its derivatives (in Chinese). J Changchun Geol. Inst. 1978;8(3):62–78. [22] Xu SZ. A compansion of effccts between the iteration method and FFT for downward continuation of potential fields. Chin J Geophys (in Chinese), 2007;50(1):285–9. [23] Xu SZ, Zhang Y, Wen BH, Yang H, Yang CF. The application of cutting method to interpretation of magnetic anomaly in region of Ludong. Geophys Prospect Pet (in Chinese), 2006;45(3):316–18. [24] Yao BC, Zeng WJ, Chen YZ, Zhang YL, Hayes DE, Diebold JBuhl, P Spangler. Xisha Trough of South China Sea-An ancient suture. Mar Geol Quat Geol 1994;14(1):1–10. [25] Yao CL, Hao TY, Guan ZN. Restrictions in gravity and magnetic inversions and technical strategy of 3D properties inversion. Geophys Geochem Explor (in Chinese), 2002;26(4):253–7. [26] Zhu ZQ, Huang GX. Gravity and magnetic inversion by using artificial neural networks and its application in Southern Hunan. J Ent-South Inst Min Metall (in Chinese), 1994;25(3):288–93.

Please cite this article as: J. Li, J. Li and W. Li et al., Multi-scale Apparent Magnetization Inversion Imaging Method Based on the Theoretical Pseudo-gravity Anomalies, Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109480