Global correlation imaging of magnetic total field gradients

At a survey area, we take a coordinate system with the (x-, y-) planes at sea level and the z-axis positive downward. Suppose that an arbitrary magnetic dipole is presented at a point q(x_q, y_q, z_q) in the subsurface, and its magnetic moment is M_q. The inclination and declination of the geomagnetic field are I₀ and A'₀, respectively, while those of the total magnetization of the dipole are I and A' separately. The theoretical magnetic total field α-gradient (α = x, y, or z) at an arbitrary station (x_i, y_i, z_i) on the observational surface caused by the dipole can be calculated by

$$\begin{equation} \frac{{\partial \Delta T_q (x_i ,y_i ,z_i )}}{{\partial \alpha }} = \frac{{\mu _0 M_q }}{{4\pi }} \cdot B_{\alpha ,q} (x_i ,y_i ,z_i ), \end{equation} \tag{ 1 }$$

where, ΔT_q(x_i, y_i, z_i) is the theoretical magnetic total field anomaly, μ₀ is the vacuum magnetic permeability, and B_{α, q}(x_i, y_i, z_i) is the geometrical function of the dipole for the magnetic total field α-gradient at the station,

$$\begin{eqnarray} &&\fl B_{x,q} (x_i ,y_i ,z_i ) = \frac{{5 \cdot (x_q - x_i )}}{{r_i^2 }}B_q (x_i ,y_i ,z_i ) \nonumber \\ &&- \frac{1}{{r_i^5 }}\big[ {2A \cdot (x_q - x_i ) + E \cdot (y_q - y_i ) + C \cdot (z_q - z_i )} \big], \end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray} &&\fl B_{y,q} (x_i ,y_i ,z_i ) = \frac{{5 \cdot (y_q - y_i )}}{{r_i^2 }}B_q (x_i ,y_i ,z_i )\nonumber \\ &&- \frac{1}{{r_i^5 }}\big[ {E \cdot (x_q - x_i ) + 2B \cdot (y_q - y_i ) + D \cdot (z_q - z_i )} \big], \end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray} &&\fl B_{z,q} (x_i ,y_i ,z_i ) = \frac{{5 \cdot (z_q - z_i )}}{{r_i^2 }}B_q (x_i ,y_i ,z_i ) \nonumber \\ &&- \frac{1}{{r_i^5 }}\big[ {C \cdot (x_q - x_i ) + D \cdot (y_q - y_i ) + 2F \cdot (z_q - z_i )} \big]. \end{eqnarray} \tag{ 4 }$$

where $r_i = \sqrt {(x_q - x_i )^2 + (y_q - y_i )^2 + (z_q - z_i )^2 }$, l = cos I₀cos A'₀, m = cos I₀sin A'₀, n = sin I₀, L = cos Icos A' M = cos Isin A' N = sin I' A = 2 · l · L − n · N − m · M, B = 2 · m · M − l · L − n · N, C = 3 · (l · N + n · L), D = 3 · (m · N + n · M), E = 3 · (l · M + m · L), and F = 2 · n · N − m · M − l · L.

The correlation coefficient between the observed magnetic total field α gradient and the theoretical magnetic total field α gradient caused by the dipole is defined as

$$\begin{equation} C_{\alpha ,q} = \frac{{\sum_{i = 1}^{N_s } {\frac{{\partial \Delta T(x_i ,y_i ,z_i )}}{{\partial \alpha }}\frac{{\partial \Delta T_q (x_i ,y_i ,z_i )}}{{\partial \alpha }}} }}{{\sqrt {\sum_{i = 1}^{N_s } {\big( {\frac{{\partial \Delta T(x_i ,y_i ,z_i )}}{{\partial \alpha }}} \big)^2 } \sum_{i = 1}^{N_s } {\big( {\frac{{\partial \Delta T_q (x_i ,y_i ,z_i )}}{{\partial \alpha }}} \big)^2 } } }}, \end{equation} \tag{ 5 }$$

where $\frac{{\partial \Delta T(x_i ,y_i ,z_i )}}{{\partial \alpha }}$ is the observed magnetic total field α gradient at the station (x_i, y_i, z_i), and N_s is the total number of the observed stations.

Assuming M_q > 0, then substituting equation (1) into equation (5), gives

$$\begin{equation} C_{\alpha ,q} = \frac{{\sum_{i = 1}^{N_s } {\frac{{\partial \Delta T(x_i ,y_i ,z_i )}}{{\partial \alpha }}B_{\alpha ,q} (x_i ,y_i ,z_i )} }}{{\sqrt {\sum_{i = 1}^{N_s } {\big( {\frac{{\partial \Delta T(x_i ,y_i ,z_i )}}{{\partial \alpha }}} \big)^2 } \sum_{i = 1}^{N_s } {B_{\alpha ,q}^2 (x_i ,y_i ,z_i )} } }}, \end{equation} \tag{ 6 }$$

According to the Cauchy inequality, the correlation coefficient C_{α, q} in equation (6) satisfies the condition −1 ⩽ C_{α, q} ⩽ +1.

The value of C_{α, q} reflects the cross correlation degree between the observed magnetic total field α gradient and the theoretical magnetic total field α gradient due to the dipole. It implies the probability of finding a magnetic dipole with a total magnetized inclination I and declination A' at point q responsible for the observed α gradient. The lower the absolute value of C_{α, q}, the lower the probability of the occurrence of a magnetic dipole at point q, while the higher the value of positive C_{α, q}, a higher probability of the occurrence of a magnetic dipole with a total magnetized inclination I and declination A'. The higher the value of a negative C_{α, q}, the higher the probability of the occurrence of a magnetic dipole but with the total magnetized inclination and declination different from I and A', respectively.

The approach is called correlation imaging because it derives the subsurface equivalent magnetic dipole distribution in a probability sense by calculating the cross correlation between the observed magnetic total field α gradient and the theoretical magnetic total field α gradient due to the magnetic dipole.

To apply the correlation imaging procedure for the observed magnetic total field α gradient, we first divided the subsurface space into a 2D or 3D regular grid. Then by using equations (6) and (2)–(4), we calculate the correlation coefficient of each node of the 2D or 3D regular grid from the top to bottom, in turn. This yields a correlation coefficient dataset. By visualizing the dataset in 2D or 3D form, we can describe the equivalent magnetic dipole distribution according to the correlation coefficients.

Different-directional magnetic gradients emphasize structures in particular orientations (Schmidt and Clark 2006). Therefore, to provide a global interpretation of the most likely distribution of magnetic sources, we define a global correlation coefficient function by integrating the three results of the correlation imaging of magnetic total field gradients.

The global correlation coefficient function mainly considers the role of the correlation imaging of the magnetic z-gradient, but also considers the role of the x-gradient and y-gradient. The function takes the maximum of the three correlation coefficients of x, y, and z-gradients when the correlation coefficient of the z-gradient is positive, while it is set to zero when the correlation coefficient of the z-gradient is not positive, because we are mainly interested in the probability of the occurrence of a magnetic dipole with a given total magnetized inclination I and declination A'. Hence, the global correlation imaging function is defined as

$$\begin{equation} C_{g,q} = \left\{ \begin{array}{@{}lc@{}} {\max (C_{x,q} ,C_{y,q} ,C_{z,q} ),} & {C_{z,q} > 0}, \\ {0,} & C_{z,q} \le 0. \end{array} \right. \end{equation} \tag{ 7 }$$

The global correlation coefficient C_{g, q} has no actual physical meaning, because it is obtained by means of a particular composition of the correlation coefficient functions of the three gradients C_{α, q}(α = x, y, or z). It can be used to comprehensively describe the probability of the occurrence of a magnetic dipole at point q. The value of C_{g, q} satisfies the condition 0 ⩽ C_{g, q} ⩽ 1. The closer the value of C_{g, q} is to 1, the higher the probability of the occurrence of a magnetic dipole with a total magnetized inclination I and declination A' at point q in a global sense.

The test model consists of six rectangular prisms with various sizes (see figure 1), among which the bigger prisms A1, A2 and A3 are at deeper depths and the smaller prisms B1, B2 and B3 are in the shallow subsurface. Prisms A1 and B3 have no obvious strike, while prisms A3 and B1 are eastward striking and prisms A2 and B2 are northward striking. The geometric parameters of each prism are listed in table 1. The total magnetization of each prism is 1 A m⁻¹, and the total magnetized inclination and declination are, respectively, 80° and −50°. The inclination and declination of the geomagnetic field are, respectively, 45° and −5°. We did the forward modelling of the model for the magnetic total field x, y and z-gradients on a topographic surface (figure 2(a)). The observed geometry is a 201 × 201 regular grid with a grid spacing of 0.04 km along both easting and northing. Then the three synthetic magnetic total field gradients were all contaminated by Gaussian noise (figures 2(b)–(d)).

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Then we tested the correlation imaging approach on the synthetic magnetic total field x-, y- and z-gradients, respectively, with a depth extent of −1.2 km∼0 km and a depth step of 0.04 km. Here the total magnetized inclination 80° and declination −50° of magnetic dipole are used in equations (2)–(4).

Figure 3 shows the images of the correlation coefficients derived from the correlation imaging of the synthetic magnetic total field x-gradient along profiles A–A', B–B', C–C' and D–D' (shown with black lines in figure 2(a)). The outline of the true prisms was drawn by black lines in each image section. Figure 4 displays those of the synthetic y-gradient, and figure 5 displays those of the synthetic z-gradient. The high positive correlation coefficients shown by the bright colour portions in figures 3–5 indicate a high probability of the occurrence of a magnetic dipole with the total magnetized inclination 80° and declination −50°.

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From figures 3–5, we can see that all of the correlation imaging of the three gradients produces high positive correlation coefficients inside the outline of prisms A1 and B3, which have an approximate equi-axed shape. This means that they all can reveal causative sources with an approximate equi-axed shape. Both the correlation imaging of the x-gradient and the z-gradient produce high positive correlation coefficients inside the outline of prisms A2 and B2 (see figures 3 and 5), which have a northward strike. This indicates that they both can reveal causative sources with a northward strike. Both the correlation imaging of the y-gradient and the z-gradient produce high positive correlation coefficients inside the outline of prisms A3 and B1 (see figures 4 and 5), which have an eastward strike. This implies that they both can reveal causative sources with an eastward strike. Therefore, the correlation imaging of different-directional magnetic gradients can reveal magnetic sources with different strike. We used global correlation imaging to obtain a global interpretation of the most likely distribution of magnetic sources.

We then calculated the global correlation coefficient function of the three gradients at each node in the subsurface regular grid by using equation (7). Figures 6(a)–(d) show the images of the global correlation coefficients along profiles A–A', B–B', C–C' and D–D', respectively. High positive correlation coefficients occur clearly inside the outline of both the deep prisms A1–A3 and the shallow prisms B1–B3 in figure 6, which indicates that the global correlation coefficient function can reveal causative sources with almost all kinds of strike.

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The real magnetic total field anomaly data (figure 7) is taken from a metallic deposit area in the middle-lower reaches of the Yangtze River in China. The data were observed on a fairly flat surface with an average elevation of 15 m, and was gridded on a 132 × 101 regular grid with a grid spacing of 0.05 km along both easting and northing. The inclination and declination of the geomagnetic field are 46.65° and −4.4°, respectively. None of the magnetic total field gradients were observed directly.

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The deposit is located in the effusive rock basin formed during the faults and magma activities in the late Yanshan period (Mesozoic age). The metallic ore includes magnetite, specularite and chalcopyrite with strong magnetism, and mainly occurs in faults or fracture zones, intrusive contact zones and depression-uplift structures of the effusive rock basin. The background terrane is effusive rock with medium magnetism and sedimentary rock with almost no magnetism. The magnetic source distribution in the subsurface remains unknown due to no existing borehole and little a priori geological information for this area. Here, we tried to use the correlation imaging approach to obtain a rough imaging of the subsurface.

We firstly calculated the x-, y- and z-gradients of the real magnetic total field anomaly by the conventional Fourier transformation method. Then we tested the correlation imaging approach, on the calculated x-, y- and z-gradients, respectively, with a depth extent of −1.5–0 km and a depth step of 0.05 km. In this case, assuming that the effects of remnant magnetization and demagnetization can be neglected, the inclination and declination of total magnetization of the magnetic dipole used in equations (2)–(4) were the same as those of the geomagnetic field. Then we calculated the global correlation coefficient function of the three gradients at each node of the subsurface regular grid according to equation (7). Figures 8(a)–(c) show the images of the correlation coefficients derived from the correlation imaging of the x-, y- and z-gradients, respectively, along profile A–A' (white line in figure 7), and figure 8(d) displays the image of the global correlation coefficients.

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High positive values of correlation coefficients alternate with low positive or negative values above a depth of −0.5 km in each image section of figure 8. This indicates a laterally heterogeneous magnetism distribution in the shallow depths. A large zone of high positive correlation coefficients occurs at a distance range of 0.6–1.4 km and below a depth of −0.5 km as shown in figures 8(a)–(c), which implies a high probability of the occurrence of a magnetic source with an approximate equi-axed shape. Another large zone of high positive correlation coefficients appears at a distance range of 4.7–5.3 km and below a depth of −0.5 km, which is shown in figures 8(a) and (c) but disappears in figure 8(b). This implies a high probability of the occurrence of a magnetic source with an approximately northward strike. These two large zones of magnetic source distribution are delineated more directly and clearly in the global correlation coefficients image section (figure 8(d)).

Based on the above results and the preliminary geological evaluation, one borehole was drilled at a distance of 5.1 km in profile A–A' (the black downward arrow in each image section of figure 8). The borehole was drilled to a depth of −1.3 km. The rock core is andesite above a depth of −1.17 km while it is syenite below this depth. The metallic ores of specularite, magnetite and chalcopyrite were found among the crack and fracture zones of the andesite terrane. Specularite of 2.5–4 m thickness was found at depths of −0.243, −0.356 and −0.710 km, respectively. Magnetite of 2–3 m thickness was found at depths of −0.818 and −0.871 km, respectively. Chalcopyrite of 3–9 m thickness was found at depths of −0.340, −0.954, −0.979 and −1.101 km, respectively.