Simulation Analysis of Magnetic Gradient Full-Tensor Measurement System

Abstract

The magnetic gradient full-tensor measurement system is diverse, and the magnetometer array structure is complex. Aimed at the problem, seven magnetic gradient full-tensor measurement system models are studied in detail. The full-tensor measurement theories of the tensor measurement arrays are analyzed. Under the same baseline distance, the magnetic dipole model is used to simulate the measurement system. Based on different measurement systems, the paper quantitatively compares and analyzes the error of the structure. A more optimized magnetic gradient full-tensor measurement system is suggested. The simulation results show that the measurement accuracy of the planar measurement system is slightly higher than that of the stereo measurement system. Among them, the cross-shaped and square measurement systems have relatively smaller structural errors and higher measurement accuracy.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The ferromagnetic object magnetized by the geomagnetic field will produce a magnetic field, which will affect the constant distribution of the geomagnetic field. This phenomenon that causes the abnormality of the geomagnetic field is called magnetic anomaly [1]. Magnetic anomaly detection refers to the technology of obtaining several characteristics of the magnetic target with the process of acquiring and processing the magnetic anomaly information of a magnetic target. Magnetic anomaly detection technology has many advantages such as strong anti-interference ability, rich information, and good concealment performance [2–4]. It is widely used in underwater or underground target detection, aerial magnetic survey, navigation and positioning, mineral exploration, and other fields [5–9]. At present, it can be roughly divided into four phases: total magnetic field detection, total magnetic field gradient detection, the magnetic component field detection, and the magnetic gradient full-tensor field detection phase [10]. The phase of magnetic gradient full-tensor detection is the latest development stage of magnetic anomaly detection. It has unique advantages such as being less affected by system attitude changes and less environmental interference. It can get much richer target information and be not affected by the total magnetic background field basically. This technology has gradually developed into a research hotspot in the fields of Earth resource exploration, aerial exploration, and military reconnaissance, and its application prospects are very broad [11–14].

It is impossible to directly measure the magnetic gradient full tensor. In actual measurement, the magnetic gradient full-tensor measurement system is generally used to measure the components of the magnetic field vector field; and the difference method is used to obtain the full-tensor field of the magnetic gradient approximately. Presently, the magnetic gradient full-tensor measurement system mainly includes two types: superconducting magnetometer and fluxgate magnetometer [15, 16]. The former has high sensitivity, good stability, and strong reliability, but the preparation process is complicated and costly. The latter uses several fluxgates to measure the magnetic vector component to obtain the spatial change rate, which is portable, economical, and practical.

At present, the magnetic gradient full-tensor measurement systems constructed by the fluxgate magnetometer can be divided into two types. One is the two-dimensional plane shape including cross, equilateral triangle, right triangle, and square; the other is three-dimensional, which includes regular tetrahedron, right-angle tetrahedron, and regular tetrahedron. Various systems can achieve the measurement purpose under specific environmental conditions. However, under the same measurement environment, the selection of the measurement system is still unclear, and there is very little analysis in the structure of different systems. But the configuration of the fluxgate magnetometer will have a great impact on the measurement of the magnetic gradient full tensor. It even directly affects the feasibility of measurement. Therefore, it is necessary to analyze and compare the magnetic gradient full-tensor measurement system to seek a more optimized system. Aiming at this problem, this paper establishes the structural models of various magnetic gradient full-tensor measurement systems in detail. Using magnetic dipole models, simulation experiments are performed to analyze various magnetic gradient full-tensor measurement systems.

2. Magnetic Gradient Full-Tensor Measurement Elements

The magnetic field is a vector field, and the spatial rate of change in each direction of the component is the tensor of magnetic gradient [17], which includes 9 elements in total. The magnetic gradient full tensor ($G$) of the magnetic field vector ($\stackrel{⟶}{B}$) can be shown as$$\begin{array}{c}\text{(1)}& G=\begin{array}{ccc}\frac{{\partial }^{2}U}{\partial x^2}& \frac{{\partial }^{2}U}{\partial x\partial y}& \frac{{\partial }^{2}U}{\partial x\partial z}\\ \frac{{\partial }^{2}U}{\partial y\partial x}& \frac{{\partial }^{2}U}{\partial y^2}& \frac{{\partial }^{2}U}{\partial y\partial z}\\ \frac{{\partial }^{2}U}{\partial z\partial x}& \frac{{\partial }^{2}U}{\partial z\partial y}& \frac{{\partial }^{2}U}{\partial z^2}\end{array},=\begin{array}{ccc}\frac{\partial B_x}{\partial x}& \frac{\partial B_x}{\partial y}& \frac{\partial B_x}{\partial z}\\ \frac{\partial B_y}{\partial x}& \frac{\partial B_y}{\partial y}& \frac{\partial B_y}{\partial z}\\ \frac{\partial B_z}{\partial x}& \frac{\partial B_z}{\partial y}& \frac{\partial B_z}{\partial z}\end{array},\\ =\begin{array}{ccc}{B}_{xx}& B_{xy}& B_{xz}\\ B_{yx}& B_{yy}& B_{yz}\\ B_{zx}& B_{zy}& B_{zz}\end{array}.\end{array}$$

In equation (1), $U$ is the magnetic scalar potential; $B_x$, $B_y$, and $B_z$ are the three components of the magnetic field at any point in space $x,y,z$; $B_{ij}i,j=x,y,z$ is the component of the magnetic gradient full tensor. According to Maxwell’s equations, in a passive ambient static magnetic field, the divergence and curl of the magnetic field are both zero.$$\begin{array}{c}\text{(2)}& \text{div}\stackrel{⟶}{B}=\nabla ·\stackrel{⟶}{B}=0,\text{(3)}& \text{rot}\stackrel{⟶}{B}=\nabla \times \stackrel{⟶}{B},=\begin{array}{ccc}i& j& k\\ \frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\\ B_x& B_y& B_z\end{array}=0.\end{array}$$

According to equations (2) and (3),$$\begin{array}{}\text{(4)}& \begin{array}{c}\frac{\partial B_x}{\partial x}+\frac{\partial B_y}{\partial y}+\frac{\partial B_z}{\partial z}=0,\\ \frac{\partial B_x}{\partial y}=\frac{\partial B_y}{\partial x},\\ \frac{\partial B_x}{\partial z}=\frac{\partial B_z}{\partial x},\\ \frac{\partial B_y}{\partial z}=\frac{\partial B_z}{\partial y}.\end{array}\end{array}$$

Therefore, in equation (1), only five elements, $B_{xx}$, $B_{xy}$, $B_{xz}$, $B_{yy}$, and $B_{yz}$, are required to obtain the magnetic gradient full tensor of the magnetic field.$$\begin{array}{}\text{(5)}& G=\begin{array}{ccc}{B}_{xx}& B_{xy}& B_{xz}\\ B_{xy}& B_{yy}& B_{yz}\\ B_{xz}& B_{yz}& -B_{xx}-B_{yy}\end{array}.\end{array}$$

The essence of the magnetic gradient full tensor is the second-order partial differential of the magnetic scalar potential U and the first-order derivative of the magnetic field vector, which cannot be directly measured in actual operation. Generally, the ratio of the magnetic field to the distance is used to approximately obtain each component of the magnetic gradient full tensor. The difference calculation method is as follows:$$\begin{array}{}\text{(6)}& B_{ij}=\frac{\partial B_i}{\partial j}\approx \frac{\Delta B_i}{\Delta d_j}.\end{array}$$

In equation (6), $\mathrm{\Delta }{B}_i$ is the difference between the magnetic field components measured by two adjacent vector fluxgates, and $\mathrm{\Delta }{d}_j$ is the distance between the two vector fluxgates in the direction of $j$, which is defined as the baseline of the magnetic gradient full-tensor measurement system distance.

3. Magnetic Gradient Full-Tensor Measurement System

3.1. Cross Measurement System

The model of the cross-shaped magnetic gradient full-tensor measurement system is shown in Figure 1. The system is composed of four fluxgate magnetometers. The center point is taken as the axis center ($O$) of the system coordinate axis. The fluxgates are marked as 1, 2, 3, and 4, respectively. The centers of No. 1 and No. 3 fluxgates are parallel to the x-axis, and their baseline distance is b. No. 2 and No. 4 fluxgates are parallel to the y-axis, and their baseline distance is also b [18].

[figure omitted; refer to PDF]

The three-component magnetic fields measured by the four fluxgates are as follows:$$\begin{array}{c}\text{(7)}& B_1=\begin{array}{c}{B}_{1x}\\ B_{1y}\\ B_{1z}\end{array},B_2=\begin{array}{c}{B}_{2x}\\ B_{2y}\\ B_{2z}\end{array},\\ B_3=\begin{array}{c}{B}_{3x}\\ B_{3y}\\ B_{3z}\end{array},\\ B_4=\begin{array}{c}{B}_{4x}\\ B_{4y}\\ B_{4z}\end{array}.\end{array}$$

In equation (7), $B_{i}i=\mathrm{1,2,3,4}$ is the magnetic field vector of the fluxgate No. $i$, and $B_{ij}i=\mathrm{1,2,3,4};j=x,y,z$ is the component of the fluxgate No. $i$ in the $j$ direction. The expression of the magnetic gradient full tensor at the axis $O$ of the system is as follows:$$\begin{array}{}\text{(8)}& G=\frac{1}{d}\begin{array}{ccc}{B}_{1x}-B_{3x}& B_{1y}-B_{3y}& B_{1z}-B_{3z}\\ B_{2x}-B_{4x}& B_{2y}-B_{4y}& B_{2z}-B_{4z}\\ B_{1z}-B_{3z}& B_{2z}-B_{4z}& B_{3x}+B_{4y}-B_{1x}+B_{2y}\end{array}.\end{array}$$

3.2. Equilateral Triangle Measurement System

The triangular magnetic gradient full-tensor measurement system is composed of three fluxgate magnetometers. No. 1, No. 2, and No. 3 fluxgates are, respectively, located at the apex of the triangle. The side length of the equilateral triangle is the baseline distance (d) of the system. The geometric center of the triangle is taken as the axis center (O) of the system coordinate axis. The system model is shown in Figure 2.

[figure omitted; refer to PDF]

Similar to the cross-shaped measurement system, according to the three components of the magnetic field vector measured by the No. 1, No. 2, and No. 3 fluxgates, the full tensor of the magnetic gradient of the axis can be obtained:$$\begin{array}{}\text{(9)}& G=\frac{1}{\sqrt{3}d}\begin{array}{ccc}\sqrt{3}{B}_{1x}-B_{3x}& \sqrt{3}{B}_{1y}-B_{3y}& \sqrt{3}{B}_{1z}-B_{3z}\\ 2B_{2x}-B_{1x}+B_{3x}& 2B_{2y}-B_{1y}+B_{3y}& 2B_{2z}-B_{1z}+B_{3z}\\ \sqrt{3}{B}_{1z}-B_{3z}& 2B_{2z}-B_{1z}+B_{3z}& B_{1y}+B_{3y}-2B_{2y}-\sqrt{3}{B}_{1x}-B_{3x}\end{array}.\end{array}$$

3.3. Right Triangle Measurement System

The model of the right triangle magnetic gradient full-tensor measurement system is shown in Figure 3. The difference from the equilateral triangle measurement system is that the connection between the centers of the three fluxgate magnetometers is isosceles right triangle. The position where the No. 3 fluxgate is located is the axis center ($O$) of the system coordinate axis, and the right-angle side length $d$ is the baseline distance. The full tensor of the magnetic gradient of the axis $O$ is shown as follows:$$\begin{array}{}\text{(10)}& G=\frac{1}{d}\begin{array}{ccc}{B}_{1x}-B_{3x}& B_{1y}-B_{3y}& B_{1z}-B_{3z}\\ B_{2x}-B_{3x}& B_{2y}-B_{3y}& B_{2z}-B_{3z}\\ B_{1z}-B_{3z}& B_{2z}-B_{3z}& B_{3x}+B_{3y}-B_{1x}+B_{2y}\end{array}.\end{array}$$

[figure omitted; refer to PDF]

The structure comparison of seven kinds of full-tensor measurement systems is shown in Table 1.

Table 1

Comparison table of measurement system structure.

Element system	Dimension	Number of magnetometers	Number of independent components	Observation points
Cross shape	2	4	6	Plane center
Equilateral triangle shape	2	3	6	Plane center
Right triangle shape	2	3	6	Plane center
Square shape	2	4	6	Plane center
Regular tetrahedron shape	3	4	9	Three-dimensional center
Right-angle tetrahedron shape	3	4	9	Three-dimensional center
Hexahedron shape	3	8	9	Three-dimensional center

It can be seen from Table 1 that the measurement system can be divided into two types: two-dimensional plane and three-dimensional stereo according to the number of dimensions, which also determines the type of observation point. Among all measurement systems, the number of magnetometers used in triangle is the smallest, which is 3; the number of magnetometers used in the regular hexahedron is the largest, which is 8. The three-dimensional shape measurement system can measure 9 independent components individually, while the two-dimensional plane can measure 6 independent components.

4. Simulation Experiment of Magnetic Gradient Full-Tensor Measurement System

4.1. Magnetic Dipole Model

When the distance between the magnetic target point and the observation point is more than 2.5 times the diameter or length of the magnetic target point, the magnetic field generated by the magnetic target point can be equivalent to a magnetic dipole [19–21]. In the process of measuring the magnetic gradient full tensor, the magnetic gradient full-tensor measurement system is often far away from the magnetic target. According to actual experience, a magnetic dipole model can be used to replace the magnetic target for modeling.

As shown in Figure 8, the magnetic dipole model is established. In the Cartesian coordinate system, the position of the magnetic target is $x_0,y_0,z_0$, and the position of the observation point $x,y,z$ is the axis of the magnetic gradient full-tensor measurement system. The projection components at the observation point of the magnetic anomaly field generated are represented by $B_x$, $B_y$, and $B_z$, which are the three-component magnetic anomaly.

[figure omitted; refer to PDF]

The magnetic anomaly vector field generated by the magnetic target at the observation point is$$\begin{array}{}\text{(15)}& \stackrel{⟶}{B}=\frac{{\mu }_0}{4\pi r^3}\frac{3\stackrel{⟶}{m}·\stackrel{⟶}{r}\stackrel{⟶}{r}}{r^2}-\stackrel{⟶}{m}.\end{array}$$

In equation (15), $\stackrel{⟶}{B}$ is the magnetic field vector; ${\mu }_0=4\times {10}^{-7}\text{H}/\text{m}$ is the magnetic permeability of the vacuum; $\stackrel{⟶}{r}=x-x_0,y-y_0,z-z_0$ is the position vector of the observation point relative to the target point; $r$ is the modulus of the vector $\stackrel{⟶}{r}$; $\stackrel{⟶}{m}=m_x,m_y,m_z$ is the magnetic moment vector of the magnetic dipole. Therefore, the coordinate equation of the magnetic dipole is$$\begin{array}{c}\text{(16)}& \stackrel{⟶}{B}=\begin{array}{c}{B}_x\\ B_y\\ B_z\end{array},=\frac{{\mu }_0}{4\pi r^5}\begin{array}{ccc}3x^2-r^2& 3xy& 3xz\\ 3xy& 3y^2-r^2& 3yz\\ 3xz& 3yz& 3z^2-r^2\end{array}\begin{array}{c}{m}_x\\ m_y\\ m_z\end{array}.\end{array}$$

The observation point in Figure 8 is the center point O in Figure 1 to Figure 7. The magnetic field value of point O can be obtained according to the coordinate equation (16). In order to obtain the full tensor of the magnetic gradient at point O, it is necessary to determine the location based on the structural parameters of the measurement system. The magnetic field value of each point can be determined by using the magnetic dipole equation several times. Take Figure 1 as an example to calculate the coordinates of No. 1 to No. 4 fluxgates, as shown in the following equation:$$\begin{array}{c}\text{(17)}& r_1=x+\frac{d}{2},y,z,r_2=x,y+\frac{d}{2},z,r_3=x-\frac{d}{2},y,z,r_4=x,y-\frac{d}{2},z.\end{array}$$

In equation (17), $r_1,r_2,r_3,r_4$ are the position coordinates of No. 1 to No. 4, with point O as the reference point. The magnetic field value of each magnetometer can be obtained, and then the full tensor of the magnetic gradient can be gotten.

4.2. Simulation Experiment

Assuming that the magnetic dipole is located in a vacuum, the position of the magnetic source is $\mathrm{3,4,5}$. The magnetic declination angle $D=15\mathrm{°}$, the magnetic inclination angle $I={30}^{°}$, the magnetic moment modulus $m=200A·m^2$, $m_x=m\cos I\cos d$, $m_y=m\cos I\sin d$, $m_z=\sin I$, and the grid area is $20\text{m}\times 60\text{m}$. During the sampling process, the attitude of the tensor measurement system is unchanged, and it moves unidirectionally along the x-axis. The sampling interval is set to 1 m; the baseline distance is set to 0.3 m. Taking the cross-shaped magnetic gradient full-tensor measurement system as an example, the comparison between distribution value and theoretical value of tensor is shown in Figure 9. The full-tensor distribution value of the seven structures is shown in Figure 10. In order to highlight the impact of the structural error in measurement system on the tensor measurement, the output values of the three-axis fluxgate are ideal during the simulation.

[figure omitted; refer to PDF][figure omitted; refer to PDF]

4.3. Simulation Results and Analysis

It can be seen from Figure 10 that each measurement system can measure the full tensor within the allowable error range, but the error between the tensor value and theoretical value obtained from different structures is different. Among them, the errors of cross, equilateral triangle, right triangle, and square are relatively small. In order to describe the measurement error of tensor more clearly, equation (18) is used to calculate the relative error (RE) of each point in the simulation. In Figure 11, the measurement errors of each point of the measurement system are compared and shown.$$\begin{array}{}\text{(18)}& \text{RE}i=fi-yii=\mathrm{1,2,3},\dots .\end{array}$$

[figures omitted; refer to PDF]

In equation (18), $fi$ is the measured value of each point, and $yi$ is the theoretical value of each point.

It can be seen from Figure 11(a) that the Bxx measured by the square magnetic gradient full-tensor measurement system has the highest accuracy. Within the range of ±30 nT/m, the maximum measurement error does not exceed 0.02 nT/m. The accuracy of right-angle tetrahedron is the worst, exceeding 2 nT/m. As can be seen from Figure 11(b), the Bxy measured by the cross-shaped magnetic gradient full-tensor measurement system has the highest accuracy. Within the range of ±30 nT/m, the maximum measurement error is not about 0.03 nT/m. The measurement accuracy of the right-angled tetrahedron is the worst, reaching 1.7 nT/m. It can be seen in Figure 11(c) that the Bxz measured by the cross-shaped magnetic gradient full-tensor measurement system has the highest accuracy. In the range of ±30 nT/m, the maximum measurement error does not exceed 0.02 nT/m. The measurement accuracy of the right-angled tetrahedron is the worst, exceeding 2 nT/m. It can be seen in Figure 11(d) that Byy measured by the cross-shaped magnetic gradient full-tensor measurement system has the highest accuracy. Within the range of ±30 nT/m, the maximum measurement error does not exceed 0.02 nT/m, and the measurement accuracy of the regular hexahedron is the worst, reaching 30 nT/m. It can be seen in Figure 11(e) that the Byz accuracy measured by the cross-shaped magnetic gradient full-tensor measurement system is the highest. Within the range of ±30 nT/m, the maximum measurement error does not exceed 0.01 nT/m, and the measurement accuracy of the regular tetrahedron is the worst, reaching 15 nT/m. From Figure 11, it is basically possible to analyze the pros and cons of the seven measurement systems for each component, and the order from the best to the worst is shown in Table 2.

Table 2

Arrangement of magnetic gradient full-tensor measurement system.

Tensors	Pros and cons
Bxx (nT/m)	Square shape	Cross shape	Hexahedron shape	Regular tetrahedron shape	Equilateral triangle shape	Right triangle shape	Right-angle tetrahedron shape
Bxy (nT/m)	Cross shape	Square shape	Hexahedron shape	Regular tetrahedron shape	Equilateral triangle shape	Right triangle shape	Right-angle tetrahedron shape
Bxz (nT/m)	Cross shape	Square shape	Hexahedron shape	Regular tetrahedron shape	Equilateral triangle shape	Right triangle shape	Right-angle tetrahedron shape
Byy (nT/m)	Cross shape	Square shape	Right triangle shape	Right-angle tetrahedron shape	Equilateral triangle shape	Regular tetrahedron shape	Hexahedron shape
Byz (nT/m)	Cross shape	Right triangle shape	Square shape	Equilateral triangle shape	Right triangle shape	Right-angle tetrahedron shape	Regular tetrahedron shape

In order to further test the overall measurement accuracy of each system, equation (19) is used to calculate the root mean square error (RMSE) [22] to quantitatively analyze the overall deviation of the measurement system. The calculated values are shown in Table 3.$$\begin{array}{}\text{(19)}& \text{RMSE}=\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^n{f_i-y_i}^2}}\hspace{1em}i=\mathrm{1,2,3},\mathrm{....},n.\end{array}$$

Table 3

RMSE comparison of measured values of seven kinds of magnetic gradient full-tensor system.

Tensor	Structures
	Cross shape	Equilateral triangle shape	Right triangle shape	Square shape	Regular tetrahedron shape	Right-angle tetrahedron shape	Right triangle shape
Bxx (nT/m)	0.0093	0.2468	0.5652	0.0011	0.0565	0.5652	0.0187
Bxy (nT/m)	0.0066	0.1220	0.4182	0.0101	0.1036	0.4182	0.0133
Bxz (nT/m)	0.0080	0.2923	0.4977	0.0086	0.1366	0.4977	0.0161
Byy (nT/m)	0.0090	0.1217	0.3929	0.0164	2.9141	0.3929	7.8728
Byz (nT/m)	0.0019	0.2613	0.4736	0.0207	3.5512	0.4736	0.0039

In equation (19), $f_i$ is the measured value of each point, and $y_i$ is the theoretical value of each point.

It can be seen from Tables 2 and 3 that, under the same measurement conditions, the cross-shaped and square measurement systems have relatively high accuracy, while the right-angled triangle and right-angled tetrahedral measurement systems have relatively great errors. Among them, the overall measurement accuracy of the cross shape is the highest in the analyzed measurement accuracy.

5. Conclusion

In this paper, seven magnetic gradient full-tensor measurement system models are studied in detail, and the full-tensor measurement theory of the tensor measurement array is analyzed. The magnetic dipole model is used to simulate the measurement system. Different measurement systems are quantitatively compared and analyzed. The analysis shows that, under the same measurement conditions, the cross-shaped and square measurement systems have smaller structural errors and higher measurement accuracy. In addition, the cross-shaped measurement system has the best overall performance. Of course, in engineering practice, the planar magnetic gradient full-tensor measurement system is easy to install, and the center point of the structure is easier to find. The three-dimensional magnetic gradient full-tensor measurement system can obtain many independent components and can directly obtain the vertical component. The magnetic gradient full-tensor measurement system should be selected to use according to actual needs. The research results in this paper provide a theoretical reference for the construction and application development of the magnetic gradient full-tensor system.

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