# Error Calibration of Cross Magnetic Gradient Tensor System with Total Least-Squares Method

Authors: Cheng Chi; Dan Wang; Ronghua Tao; Zhentao Yu
Journal: Mathematical Problems in Engineering
 (2023)

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## Abstract

The magnetic gradient tensor system configured by fluxgate magnetometers is subjected to different scale factors, three-axis nonorthogonality, bias, and misalignment errors. All those errors above will influence the measurement precision directly, so the magnetic gradient tensor system must be calibrated before use. In this paper, an error calibration method of the magnetic gradient tensor system is proposed. The procedure of the proposed method is as follows. Firstly, the error calibration model of the single fluxgate magnetometer is established and generated an ellipsoid mathematical expression. To simplify the calculation, the ellipsoid mathematical expression is transformed into a linear model of intermediate variables, and then, error parameters are estimated by the total least-squares method. Secondly, the orthogonal procrustes problem is adopted to calibrate misalignment errors. Finally, simulations and experiments with a cross magnetic gradient tensor system are carried out for verification of the proposed error calibration method. Results show that compared with the original least-squares method, the proposed method can increase the measurement accuracy of the cross magnetic gradient tensor system greatly.


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## Body

## 1. Introduction

Magnetic gradient tensor measurements [1–3] have many theoretical advantages over conventional magnetic surveys (i.e., vector and total magnetic intensity measurements). For example, tensor elements are true potential fields with desirable mathematical properties, magnetic gradient tensor measurements are relatively insensitive to orientation errors and geomagnetic gradients, and gradient tensor elements can provide valuable additional information and have better resolution. Based on those merits, magnetic gradient tensor measurements are widely used in magnetic target detection, demining, geological survey, and navigation. Over the past decades, there has been growing interest in developing magnetic gradient tensor systems, such as the hexahedron magnetic gradient tensor system designed by the Naval Surface Warfare Center [4] and the tetrahedron magnetic gradient tensor system designed by the DSO National Laboratories [5, 6].Generally, superconducting quantum interference devices (SQUID) [7] or fluxgate magnetometers are two major sensors to construct a magnetic gradient tensor system. SQUID sensors have the merits of high sensitivity and precision, but the SQUID system needs to be liquid He-cooled or liquid nitrogen-cooled, which results in the SQUID system being complicated and expensive. Compared with SQUID sensors, fluxgate magnetometers have the advantages of small size, low power with relatively high sensitivity, and low cost [8], so there are many kinds of magnetic gradient tensor systems configured by fluxgate magnetometers. However, being restricted by technological limitations and manufacturing crafts, the magnetic gradient tensor system is subjected to different scale factors, three-axis nonorthogonality, bias, and misalignment errors [9]. All those errors above will influence the measurement precision directly, so the magnetic gradient tensor system must be calibrated before use.Magnetic gradient tensor system is always configured by several vector magnetometers, so the error calibration process can be divided into two steps: the first step is to calibrate the single vector magnetometer, and the second step is to calibrate misalignment errors between magnetometers. Many research papers have a focus on the calibration of single vector magnetometer, and the calibration method can be divided into vector calibration and scalar calibration.Vector calibration [10, 11] needs a rigorous calibration magnetic field and a high-precision tri-axial nonmagnetic platform, the calibration equipment is very expensive, and the calibration procedure is complicated, so vector calibration is not suitable for practical applications. However, on the other hand, scalar calibration [12–14] is usually described as the “poor man’s” calibration method, because scalar calibration only needs a high-precision proton magnetometer to monitor the background geomagnetic field, and the calibration procedure is simple and easy to perform. Yu et al. [12, 13] proposed a calibration method for a magnetic vector gradiometer, but two or higher-order small quantities were omitted in this method, and the precision of the corrected outs was reduced. Gang et al. [14] calibrated the magnetic gradient tensor system using a linear calibration method, but two nonlinear conversions were used to linearize the nonlinear mathematical model of single vector magnetometer calibration, and the calculation process was complicated. Zhen-tao et al. [15] achieved calibration of the tetrahedron magnetic gradiometer with the restriction of traceless and symmetric properties. Calibration matrixes were estimated with a genetic algorithm; however, the optimization algorithm was easily trapped into local optimization, and the convergent speed was slow.In this paper, an error calibration method of the cross magnetic gradient tensor system is proposed, and the calibration process is divided into two steps. Firstly, considering the scale factors, bias, and three-axis nonorthogonal errors, an error calibration model of the single fluxgate magnetometer is established, the nonlinear calibration model is transformed into a linear model of intermediate variables, and least-squares approximation method is a traditional method to solve an over-determined system of equations, which is called the original method in this paper; however, the least-squares method is asymmetry, and motivated by this, the total least-squares method is introduced to calculate the error parameters in this paper, which is called the proposed method. Secondly, the orthogonal procrustes problem is adopted to calibrate misalignment errors. Simulations and experiments are carried out, and the results show that the proposed error calibration method can improve the measurement accuracy of the cross magnetic gradient tensor system.


## 2. The Configuration and Error Model of Cross Magnetic Gradient Tensor System

### 2.1. The Configuration of Cross Magnetic Gradient Tensor System

The magnetic gradient tensor is defined as the vector gradient of the magnetic flux density in three orthogonal directions, and the magnetic gradient tensor can be represented by the following3×3 matrix:(1)G=∇B=∂Bx∂x∂Bx∂y∂Bx∂z∂By∂x∂By∂y∂By∂z∂Bz∂x∂Bz∂y∂Bz∂z=G11G12G13G21G22G23G31G32G33.There are nine components in the matrixG. In an area that does not contain conduction currents, both the divergence and the curl of the magnetic flux density are zero, so the tensor is traceless and symmetric [16].(2)∇⋅B=∂Bx∂x+∂By∂y+∂Bz∂z=0,∇×B=ijk∂∂x∂∂y∂∂zBxByBz=0.According to equations (1) and (2), the number of independent components is reduced to 5, so the magnetic gradient tensor matrix can be obtained by measuring the independent 5 components.The magnetic gradient tensor components cannot be measured intrinsically in an actual measurement application, so the tensor components are approximated by the difference between two measurement readings of the magnetic field at different locations. However, the real measured value is slightly different compared to the theoretical value, and this difference is commonly called structure error. Structure error is usually affected by different configurations of fluxgate magnetometers. The structure error of different configurations is analyzed in reference [17], and research results show that the cross magnetic gradient tensor system has minimal structure error. Based on the above results, a cross magnetic gradient tensor system is designed in this paper, and the configuration of the cross magnetic gradient tensor system is shown in Figure 1.Figure 1
The configuration of the cross magnetic gradient tensor system.As shown in Figure1, a right-handed coordinate system is established, and the cross magnetic gradient tensor system comprises four fluxgate magnetometers (1 to 4). Four magnetometers lie on the plane xoy, and the baseline distance between two magnetometers along the same axes is d. The magnetic gradient tensor of the point o can be estimated by the measurement readings of four fluxgate magnetometers:(3)G=1dB1x−B3xB2x−B4xB1z−B3zB1y−B3yB2y−B4yB2z−B4zB1z−B3zB2z−B4zB4y−B2y+B3x−B1x,where Bij (i=1,2,3,4, j=x,y,z) is the j component of the magnetometer i.


### 2.2. Error Calibration Model for Single Fluxgate Magnetometer

Due to manufacturing imperfections, the fluxgate magnetometer is subjected to nonorthogonal error, different scale factors, and bias. Nonorthogonal error is that the three actual axes of the fluxgate magnetometer may not coincide with the three-orthogonal axis. The geometrical relationship of the nonorthogonal angles is shown in Figure2, suppose O−XYZ is the actual measurement coordinate system of the fluxgate magnetometer, O−X0Y0Z0 is an ideal three-axis orthogonal coordinate system, and suppose the measurement axis OZ is completely aligned with the ideal coordinate axis OZ0. The plane YOZ is coplanar with the plane Y0OZ0. The angle between the axis OY and OY0 is ψ. OX′ is the projection of the measurement axis OX in the plane X0OY0, the angle between the axis OX and OX′ is θ, and the angle between the axis OX0 and OX′ is φ.Figure 2
Geometrical relationship of the nonorthogonal angles.Each measurement axis of the fluxgate magnetometer has different biases and sensitivities, so we defineb=bx,by,bzT as the biases and kx,ky,kz as the scale factors (sensitivities) for OX,OY,OZ axes. Suppose B are the theoretical outputs under the ideal three-axis orthogonal coordinate system O−X0Y0Z0, Bm are the actual outputs of the fluxgate magnetometer, and then, the relationship between Bm and B can be written as follows:(4)Bm=kx000ky000kzcosθcosφcosθsinφsinθ0cosψsinψ001B+b+ε=SCNOB+b+ε,where S is the scale factor matrix, CNO is the nonorthogonal error coefficient matrix, and ε is the measurement noise of the magnetometer. Equation (4) is the error model of the fluxgate magnetometer.Define the combined error parameter matrixC=SCNOI+K, and then, the error model of the fluxgate magnetometer can be described by(5)Bm=CB+b+ε.Compared with the nonorthogonal error, different scale factors, and bias, the measurement noise is relatively small, and hence, it is negligible, and then, the error calibration model for the fluxgate magnetometer can be written as follows:(6)B=C−1Bm−b.According to equation (6), we know that the error calibration for a single fluxgate magnetometer is to estimate the calibration matrices C and b, and then, the corrected output B can be obtained by equation (6).


### 2.3. Misalignment Error Model

Due to technical limitations, the four fluxgate magnetometers cannot be completely aligned with each other, and then, the misalignment error is brought out. We calibrate the misalignment error using one of the sensors as a reference. We use magnetometer 1 as a reference, and the actual outputs of magnetometer 2–4 are transformed into the reference orthogonal coordinate system of magnetometer 1. As shown in Figure3, OX1Y1Z1 is the reference orthogonal coordinate system, and OXiYiZi is the orthogonal coordinate system of the magnetometer i. OX1Y1Z1 can be converted to OXiYiZi by rotating around three orthogonal axes. As shown in Figure 3, the procedure is as follows: firstly, rotate OX1Y1Z1 through angle α about Z1 axes to OX'Y'Z', then rotate OX'Y'Z' through angle β about X′ axes to OX″Y″Z″, and at last, rotate OX″Y″Z″ through angle γ about Y″ axes to OXiYiZi. α,β,γ are misalignment angles.Figure 3
Misalignment angles of the magnetometeri.The mathematical relationship between the output of the magnetometeri and the output of the reference magnetometer is described as follows:(7)Bi=R1R2R3B1,where R1=cosγ0−sinγ010sinγ0cosγ, R2=1000cosβsinβ0−sinβcosβ, and R3=cosαsinα0−sinαcosα0001 are three coordinate-transformation matrices. Equation (7) is the misalignment error model of the magnetometer i, and once the coordinate-transformation matrices are obtained, the actual outputs of the magnetometer i can be transformed into the reference orthogonal coordinate system.As discussed in Section2.1, the tensor components are estimated using the difference between two measurement values of the magnetic field at different locations. Nonorthogonal error, different scale factors, bias, and misalignment errors of the magnetometers will influence the measurement precision directly, so the magnetic gradient tensor system must be calibrated before use.


## 2.1. The Configuration of Cross Magnetic Gradient Tensor System

The magnetic gradient tensor is defined as the vector gradient of the magnetic flux density in three orthogonal directions, and the magnetic gradient tensor can be represented by the following3×3 matrix:(1)G=∇B=∂Bx∂x∂Bx∂y∂Bx∂z∂By∂x∂By∂y∂By∂z∂Bz∂x∂Bz∂y∂Bz∂z=G11G12G13G21G22G23G31G32G33.There are nine components in the matrixG. In an area that does not contain conduction currents, both the divergence and the curl of the magnetic flux density are zero, so the tensor is traceless and symmetric [16].(2)∇⋅B=∂Bx∂x+∂By∂y+∂Bz∂z=0,∇×B=ijk∂∂x∂∂y∂∂zBxByBz=0.According to equations (1) and (2), the number of independent components is reduced to 5, so the magnetic gradient tensor matrix can be obtained by measuring the independent 5 components.The magnetic gradient tensor components cannot be measured intrinsically in an actual measurement application, so the tensor components are approximated by the difference between two measurement readings of the magnetic field at different locations. However, the real measured value is slightly different compared to the theoretical value, and this difference is commonly called structure error. Structure error is usually affected by different configurations of fluxgate magnetometers. The structure error of different configurations is analyzed in reference [17], and research results show that the cross magnetic gradient tensor system has minimal structure error. Based on the above results, a cross magnetic gradient tensor system is designed in this paper, and the configuration of the cross magnetic gradient tensor system is shown in Figure 1.Figure 1
The configuration of the cross magnetic gradient tensor system.As shown in Figure1, a right-handed coordinate system is established, and the cross magnetic gradient tensor system comprises four fluxgate magnetometers (1 to 4). Four magnetometers lie on the plane xoy, and the baseline distance between two magnetometers along the same axes is d. The magnetic gradient tensor of the point o can be estimated by the measurement readings of four fluxgate magnetometers:(3)G=1dB1x−B3xB2x−B4xB1z−B3zB1y−B3yB2y−B4yB2z−B4zB1z−B3zB2z−B4zB4y−B2y+B3x−B1x,where Bij (i=1,2,3,4, j=x,y,z) is the j component of the magnetometer i.


## 2.2. Error Calibration Model for Single Fluxgate Magnetometer

Due to manufacturing imperfections, the fluxgate magnetometer is subjected to nonorthogonal error, different scale factors, and bias. Nonorthogonal error is that the three actual axes of the fluxgate magnetometer may not coincide with the three-orthogonal axis. The geometrical relationship of the nonorthogonal angles is shown in Figure2, suppose O−XYZ is the actual measurement coordinate system of the fluxgate magnetometer, O−X0Y0Z0 is an ideal three-axis orthogonal coordinate system, and suppose the measurement axis OZ is completely aligned with the ideal coordinate axis OZ0. The plane YOZ is coplanar with the plane Y0OZ0. The angle between the axis OY and OY0 is ψ. OX′ is the projection of the measurement axis OX in the plane X0OY0, the angle between the axis OX and OX′ is θ, and the angle between the axis OX0 and OX′ is φ.Figure 2
Geometrical relationship of the nonorthogonal angles.Each measurement axis of the fluxgate magnetometer has different biases and sensitivities, so we defineb=bx,by,bzT as the biases and kx,ky,kz as the scale factors (sensitivities) for OX,OY,OZ axes. Suppose B are the theoretical outputs under the ideal three-axis orthogonal coordinate system O−X0Y0Z0, Bm are the actual outputs of the fluxgate magnetometer, and then, the relationship between Bm and B can be written as follows:(4)Bm=kx000ky000kzcosθcosφcosθsinφsinθ0cosψsinψ001B+b+ε=SCNOB+b+ε,where S is the scale factor matrix, CNO is the nonorthogonal error coefficient matrix, and ε is the measurement noise of the magnetometer. Equation (4) is the error model of the fluxgate magnetometer.Define the combined error parameter matrixC=SCNOI+K, and then, the error model of the fluxgate magnetometer can be described by(5)Bm=CB+b+ε.Compared with the nonorthogonal error, different scale factors, and bias, the measurement noise is relatively small, and hence, it is negligible, and then, the error calibration model for the fluxgate magnetometer can be written as follows:(6)B=C−1Bm−b.According to equation (6), we know that the error calibration for a single fluxgate magnetometer is to estimate the calibration matrices C and b, and then, the corrected output B can be obtained by equation (6).


## 2.3. Misalignment Error Model

Due to technical limitations, the four fluxgate magnetometers cannot be completely aligned with each other, and then, the misalignment error is brought out. We calibrate the misalignment error using one of the sensors as a reference. We use magnetometer 1 as a reference, and the actual outputs of magnetometer 2–4 are transformed into the reference orthogonal coordinate system of magnetometer 1. As shown in Figure3, OX1Y1Z1 is the reference orthogonal coordinate system, and OXiYiZi is the orthogonal coordinate system of the magnetometer i. OX1Y1Z1 can be converted to OXiYiZi by rotating around three orthogonal axes. As shown in Figure 3, the procedure is as follows: firstly, rotate OX1Y1Z1 through angle α about Z1 axes to OX'Y'Z', then rotate OX'Y'Z' through angle β about X′ axes to OX″Y″Z″, and at last, rotate OX″Y″Z″ through angle γ about Y″ axes to OXiYiZi. α,β,γ are misalignment angles.Figure 3
Misalignment angles of the magnetometeri.The mathematical relationship between the output of the magnetometeri and the output of the reference magnetometer is described as follows:(7)Bi=R1R2R3B1,where R1=cosγ0−sinγ010sinγ0cosγ, R2=1000cosβsinβ0−sinβcosβ, and R3=cosαsinα0−sinαcosα0001 are three coordinate-transformation matrices. Equation (7) is the misalignment error model of the magnetometer i, and once the coordinate-transformation matrices are obtained, the actual outputs of the magnetometer i can be transformed into the reference orthogonal coordinate system.As discussed in Section2.1, the tensor components are estimated using the difference between two measurement values of the magnetic field at different locations. Nonorthogonal error, different scale factors, bias, and misalignment errors of the magnetometers will influence the measurement precision directly, so the magnetic gradient tensor system must be calibrated before use.


## 3. Error Calibration of Cross Magnetic Gradient Tensor System

### 3.1. Error Calibration of Single Fluxgate Magnetometer

Within a homogeneous magnetic field, the intensity of the magnetic field is constant. By multiplying both sides of equation (6) with the transpose, we can obtain(8)BTB=Bm−boTCTCBm−bo=const.DefineA=C−1TC−1, and the following equation is obtained:(9)BmTABm−2bTABm+bTAb−BTB=0.Equation (9) is the expression of an ellipsoidal surface. Within a homogeneous magnetic field, the distribution surface of actual outputs is an ellipsoidal surface, and the distribution surface of theoretical outputs is a standard sphere, so the error calibration of the fluxgate magnetometer is a problem of the ellipsoidal surface fitting. Suppose the actual output of the fluxgate magnetometer is x,y,zT, the general mathematical expression of an ellipsoid can be written as follows:(10)ax2+by2+cz2+2fyz+2gxz+2hxy+2px+2qy+2rz+d=0.Both sides of equation (10) are divided by the parameter p, and the following equation is obtained:(11)apx2+bpy2+cpz2+2fpyz+2gpxz+2hpxy+2qpy+2rpz+dp=−2x.A set of actual outputsxi,yi,ziTi=1N is obtained by presenting the magnetometer in different orientations. Best estimates for parameters (a/p,b/p,c/p,f/p,g/p,h/p,q/p,r/p,d/p) in the least-squares sense can be found by restructuring equation (11) because the parameters are constant for all N measurements.(12)XN×9P9×1=WN×1,where P=1/pabcfghqrdT, W=−2x1−2x2⋅⋅⋅−2xNT, X=x12y12z122y1z12x1z12x1y12y12z11x22y22z222y2z22x2z22x2y22y22z21⋮⋮⋮⋮⋮⋮⋮⋮⋮xN2yN2zN22yNzN2xNzN2xNyN2yN2zN1. Since rankX=9≤N, equation (12) has no exact solution. Finally, the least-squares approximation for the parameter matrix P^ can be obtained by(13)P^LS=XXT−1XW.The matrixW is affected by the magnetic noise, and the least-squares approximation is obtained as a solution to the optimization problem [18–20].(14)P^LS,ΔWTLS≔argminP,ΔWΔWFsubjecttoXP=W+ΔW.However, bothX and W are measurement data, and the matrix X is also affected by the magnetic noise. From equation (14), W is corrected while X is not, and then, the least-squares method is asymmetry. Provided that both X and W are affected by the magnetic noise, it is reasonable to treat them symmetrically, and then, the optimization problem becomes that(15)P^TLS,ΔXTLS,ΔWTLS≔argminP,ΔX,ΔWΔXΔWFsubjecttoX+ΔXP=W+ΔW.The estimated solution of equation (15) can be obtained by the total least-squares method [21, 22]. Define the matrix G=XW, and the singular value decomposition of the matrix G is written as follows:(16)C=XW=U∑VT,whereΣ=diagσ1,σ2,…,σ10,whereσ1,σ2,…,σ10 are the singular values of the matrix C, and the right unitary matrix V can be rewritten as follows:(17)91V10×10=V11V12V21V229.1IfV22 is nonsingular, then the total least-squares solution exists and is given by(18)P^TLS=−V12V22−1.P^TLS is a total least-squares solution of equation (12). The relationship between the calibration matrices A,b and the estimated error parameters is as follows:(19)A=a^h^g^h^b^f^g^f^c^,(20)b=−A−1p^q^r^,(21)bTAb−BTB=d.^The measurement is carried out within a homogeneous magnetic field, and the intensity of the magnetic fieldB is measured by a scalar proton magnetometer. The calibration matrices A,b can be obtained by solving equations (19)–(21). Thus, to accomplish the calibration of the single magnetometer, we need to estimate the parameter matrix C−1. Since A is a symmetric positive definite matrix, the Cholesky factorization [19] of A provides a unique upper triangular matrix C−1, such that C−1TC−1=A. Then, the calibrated output B of a single magnetometer can be obtained by equation (6).


### 3.2. Misalignment Error Calibration

According to the misalignment error model of the magnetometeri (2–4), the misalignment error calibration of the magnetometer i (2–4) is to estimate the coordinate-transformation matrices, and the orthogonal procrustes problem [23] is adopted to determine the coordinate-transformation matrices in this paper.Suppose we getNN≥3 groups of measurement data by presenting the cross magnetic gradient tensor system in different attitudes. After the error calibration for a single fluxgate magnetometer, the corrected outputs of magnetometer i are Mi=Bi1⋅⋅⋅BiN, where BiN=BiNx,BiNy,BiNzT, the corrected outputs of the reference magnetometer are M1=B11⋅⋅⋅B1N, suppose M1MiT is nonsingular, the singular value decomposition of the matrix M1MiT is M1MiT=UM∑VMT, where UM,VM are 3×3 unitary matrices, and Σ is a diagonal matrix. The optimal orthogonal matrix R minimizes the transformation from Mi to M1, specifically,(22)minR∑j=1NB1j−RBij2subjecttoRTR=I.The optimal solution of equation (22) is unique and given by R=UMVMT, and then, the corrected outputs of magnetometer i can be transformed into the reference orthogonal coordinate system.


## 3.1. Error Calibration of Single Fluxgate Magnetometer

Within a homogeneous magnetic field, the intensity of the magnetic field is constant. By multiplying both sides of equation (6) with the transpose, we can obtain(8)BTB=Bm−boTCTCBm−bo=const.DefineA=C−1TC−1, and the following equation is obtained:(9)BmTABm−2bTABm+bTAb−BTB=0.Equation (9) is the expression of an ellipsoidal surface. Within a homogeneous magnetic field, the distribution surface of actual outputs is an ellipsoidal surface, and the distribution surface of theoretical outputs is a standard sphere, so the error calibration of the fluxgate magnetometer is a problem of the ellipsoidal surface fitting. Suppose the actual output of the fluxgate magnetometer is x,y,zT, the general mathematical expression of an ellipsoid can be written as follows:(10)ax2+by2+cz2+2fyz+2gxz+2hxy+2px+2qy+2rz+d=0.Both sides of equation (10) are divided by the parameter p, and the following equation is obtained:(11)apx2+bpy2+cpz2+2fpyz+2gpxz+2hpxy+2qpy+2rpz+dp=−2x.A set of actual outputsxi,yi,ziTi=1N is obtained by presenting the magnetometer in different orientations. Best estimates for parameters (a/p,b/p,c/p,f/p,g/p,h/p,q/p,r/p,d/p) in the least-squares sense can be found by restructuring equation (11) because the parameters are constant for all N measurements.(12)XN×9P9×1=WN×1,where P=1/pabcfghqrdT, W=−2x1−2x2⋅⋅⋅−2xNT, X=x12y12z122y1z12x1z12x1y12y12z11x22y22z222y2z22x2z22x2y22y22z21⋮⋮⋮⋮⋮⋮⋮⋮⋮xN2yN2zN22yNzN2xNzN2xNyN2yN2zN1. Since rankX=9≤N, equation (12) has no exact solution. Finally, the least-squares approximation for the parameter matrix P^ can be obtained by(13)P^LS=XXT−1XW.The matrixW is affected by the magnetic noise, and the least-squares approximation is obtained as a solution to the optimization problem [18–20].(14)P^LS,ΔWTLS≔argminP,ΔWΔWFsubjecttoXP=W+ΔW.However, bothX and W are measurement data, and the matrix X is also affected by the magnetic noise. From equation (14), W is corrected while X is not, and then, the least-squares method is asymmetry. Provided that both X and W are affected by the magnetic noise, it is reasonable to treat them symmetrically, and then, the optimization problem becomes that(15)P^TLS,ΔXTLS,ΔWTLS≔argminP,ΔX,ΔWΔXΔWFsubjecttoX+ΔXP=W+ΔW.The estimated solution of equation (15) can be obtained by the total least-squares method [21, 22]. Define the matrix G=XW, and the singular value decomposition of the matrix G is written as follows:(16)C=XW=U∑VT,whereΣ=diagσ1,σ2,…,σ10,whereσ1,σ2,…,σ10 are the singular values of the matrix C, and the right unitary matrix V can be rewritten as follows:(17)91V10×10=V11V12V21V229.1IfV22 is nonsingular, then the total least-squares solution exists and is given by(18)P^TLS=−V12V22−1.P^TLS is a total least-squares solution of equation (12). The relationship between the calibration matrices A,b and the estimated error parameters is as follows:(19)A=a^h^g^h^b^f^g^f^c^,(20)b=−A−1p^q^r^,(21)bTAb−BTB=d.^The measurement is carried out within a homogeneous magnetic field, and the intensity of the magnetic fieldB is measured by a scalar proton magnetometer. The calibration matrices A,b can be obtained by solving equations (19)–(21). Thus, to accomplish the calibration of the single magnetometer, we need to estimate the parameter matrix C−1. Since A is a symmetric positive definite matrix, the Cholesky factorization [19] of A provides a unique upper triangular matrix C−1, such that C−1TC−1=A. Then, the calibrated output B of a single magnetometer can be obtained by equation (6).


## 3.2. Misalignment Error Calibration

According to the misalignment error model of the magnetometeri (2–4), the misalignment error calibration of the magnetometer i (2–4) is to estimate the coordinate-transformation matrices, and the orthogonal procrustes problem [23] is adopted to determine the coordinate-transformation matrices in this paper.Suppose we getNN≥3 groups of measurement data by presenting the cross magnetic gradient tensor system in different attitudes. After the error calibration for a single fluxgate magnetometer, the corrected outputs of magnetometer i are Mi=Bi1⋅⋅⋅BiN, where BiN=BiNx,BiNy,BiNzT, the corrected outputs of the reference magnetometer are M1=B11⋅⋅⋅B1N, suppose M1MiT is nonsingular, the singular value decomposition of the matrix M1MiT is M1MiT=UM∑VMT, where UM,VM are 3×3 unitary matrices, and Σ is a diagonal matrix. The optimal orthogonal matrix R minimizes the transformation from Mi to M1, specifically,(22)minR∑j=1NB1j−RBij2subjecttoRTR=I.The optimal solution of equation (22) is unique and given by R=UMVMT, and then, the corrected outputs of magnetometer i can be transformed into the reference orthogonal coordinate system.


## 4. Simulations and Experiments

### 4.1. Simulations

Simulations are performed to verify the performance of the proposed calibration method with Matlab. We set the average total intensity of the geomagnetic field as 50000 nT, the declination angle as−7∘, and the inclination angle as 55∘. Four simulative magnetometers are used to configure the simulated cross magnetic gradient tensor system, the baseline distance is 0.5 m, and error parameters of the four simulative magnetometers are shown in Table 1. Put the cross magnetic gradient tensor system under the uniform geomagnetic field environment, and the sampling data are obtained by presenting the cross magnetic gradient tensor system in different orientations. 1000 groups of measurement data under different system orientations are recorded.Table 1
Error parameters of the simulated cross magnetic gradient tensor system.
Sensor 1Sensor 2Sensor 3Sensor 4kx1.0451.0531.0650.973ky0.9811.0410.9851.033kz0.9750.9711.0491.042bx/nT129−838593by/nT88769371bz/nT−74131−65−89ψ/rad0.0510.026−0.0440.047φ/rad0.037−0.0330.0350.042θ/rad−0.0290.0490.039−0.031α/rad00.0320.043−0.035β/rad00.0420.0360.031γ/rad00.0350.0370.045To simulate the magnetic noise of the measurement, we add independent Gaussian white noises with a mean of 0 nT and a variance of6nT in each axis of the different magnetometers. With 1000 groups of measurement data, error calibration parameters are estimated by the total least-squares method, and error calibration matrices calculated by the proposed method are shown in Table 2. The raw data are calibrated using the error calibration matrices as shown in Table 2. To compare the calibration performance, the corrected outputs of the original calibration method and the proposed method are both shown in Figures 4 and 5.Table 2
Error calibration matrices of the proposed method.
C−1bRSensor 10.958−0.03780.031701.0207−0.0524001.0257129.37486.4032−73.8938Sensor 20.95140.0317−0.051400.961−0.0268001.0299−83.223676.9102130.80630.9988−0.0320.03630.03340.9986−0.0408−0.0350.0420.9985Sensor 30.9403−0.0356−0.038701.01630.042000.953385.77393.115−64.94060.9983−0.0430.03850.04430.9984−0.0343−0.0370.0360.9987Sensor 41.0292−0.04070.031700.9692−0.0451000.959793.340970.9987−88.89170.99840.0350.0439−0.03360.9989−0.0325−0.0450.0310.9985Figure 4
Comparison of total magnetic intensity before and after calibration.
(a)(b)(c)(d)Figure 5
Comparison of tensor components before and after calibration.
(a)(b)(c)(d)(e)(f)(g)According to the simulation results, the fluctuations of the magnetic field intensity and magnetic gradient tensor components are large before calibration, and the fluctuations are mainly caused by the measurement of the system under different orientations. After calibration, the fluctuations of the corrected outputs decrease obviously. As shown in Figure4, before calibration, the max deviation of the magnetic field intensity is 3916 nT; however, the max deviation of the corrected data is 14 nT.As shown in Figure5, before calibration, the max deviation of the tensor components is 12290 nT/m, and after calibration, the max deviation is reduced to 34.4 nT/m, so we can see that both the original calibration method and the proposed method can calibrate the magnetic gradient tensor system effectively.RMS (root-mean-square) errors are used to represent the error calibration performance, and the formula is written as follows:(23)ERMS=1N∑i=1Nei2,where ei is the calibration error between corrected data and ideal data, and N is the number of measurement orientations. Then, the RMS errors of total magnetic intensity before and after calibration are listed in Table 3. RMS errors of tensor components before and after calibration are listed in Table 4.Table 3
RMS errors of total magnetic intensity before and after calibration.
Reference sensor (nT)Sensor 2 (nT)Sensor 3 (nT)Sensor 4 (nT)Raw data1112.71576.12073.51799.8Corrected data by the original method3.9144.0373.8734.035Corrected data by the proposed method3.8923.9893.8293.993Table 4
RMS errors of tensor components before and after calibration.
BxxnT/mBxynT/mBxznT/mByxnT/mByynT/mByznT/mBzznT/mRaw data3549.969585765.34455.62920.45036.65113.1Corrected data by the original method6.2429.9177.0697.4997.217.1159.55Corrected data by the proposed method6.1426.7857.0677.2996.9537.1139.468According to the RMS errors of total magnetic intensity shown in Table3, the background geomagnetic intensity is uniform, and both the original calibration method and the proposed method can achieve the accurate calibration of the single magnetometer, so the RMS errors of corrected data are in the Gaussian noise error range; however, the performance of the original method is slightly worse. According to the RMS errors of tensor components before and after calibration shown in Table 4, RMS errors of all the components calibrated by the proposed method are smaller than the original method, and the RMS errors of Bxy can be reduced by 31.6% of errors of the original method.


### 4.2. Experiments

Experiments were carried out in Heikuang Mountain, Yantai, China, where the external magnetic interference was very low. A cross magnetic gradient tensor system consisting of four homemade fluxgate magnetometers is shown in Figure6. To avoid additional magnetic interference, the holder is made of aluminum. The baseline distance of the system is 0.26 m. To find a relatively steady background geomagnetic field, a proton magnetometer is used, the accuracy of the proton magnetometer is 1 nT, and the resolution is 0.1 nT. The average value of the measured background geomagnetic intensity was 52368.3 nT. 80 groups of measurement data were recorded by presenting the cross magnetic gradient tensor system in different orientations. Then, we used the measurement data to calibrate the cross magnetic gradient tensor system, and the results were shown in Figures 7 and 8. As we knew that in a uniform geomagnetic field, the corrected output of the total magnetic intensity should be close to the background geomagnetic intensity, and the corrected output of the magnetic gradient tensor components should be close to 0 nT/m. As shown in Figure 7, the fluctuations of raw data are large, and the max fluctuation of the raw data is up to 3880 nT, and after calibration, fluctuation of total magnetic intensity is descended evidently, the corrected outputs are close to 52368.3 nT, the corresponding RMS errors of the four fluxgate magnetometers are presented in Table 5, and both the original method and the proposed method can calibrate the single fluxgate magnetometer’s error effectively. As shown in Figure 8, because of the single fluxgate magnetometer errors and misalignment errors, the actual outputs of the magnetic gradient tensor components have big deviations, and after calibration, the corrected outputs are close to 0 nT/m. The corresponding RMS errors of tensor components are presented in Table 6, the RMS errors of the proposed method are smaller than the original method, and the RMSE of Bxy can be reduced to 31.316 nT/m by the original method; however, the RMSE of Bxy calibrated by the proposed method is 14.158 nT/m, and 54.79% of errors can be reduced by the proposed method, so the results of the experiments show that the proposed method is more accurate and reliable than the original method.Figure 6
Cross magnetic gradient tensor system.Figure 7
Comparison of total magnetic intensity before and after calibration.Figure 8
Comparison of tensor components before and after calibration.Table 5
RMS errors of total magnetic intensity before and after calibration.
Reference sensor (nT)Sensor 2 (nT)Sensor 3 (nT)Sensor 4 (nT)Raw data1147.16914.771080.621055.16Corrected data by the original method4.1993.7774.7724.725Corrected data by the proposed method4.1633.7034.6544.534Table 6
RMS errors of tensor components before and after calibration.
BxxnT/mBxynT/mBxznT/mByxnT/mByynT/mByznT/mBzznT/mRaw data3317.252723.872936.668606.444405.691752.097213.74Corrected data by the original method51.73631.31617.54529.71120.69113.22665.439Corrected data by the proposed method28.30214.15817.19618.65516.15313.04732.214


## 4.1. Simulations

Simulations are performed to verify the performance of the proposed calibration method with Matlab. We set the average total intensity of the geomagnetic field as 50000 nT, the declination angle as−7∘, and the inclination angle as 55∘. Four simulative magnetometers are used to configure the simulated cross magnetic gradient tensor system, the baseline distance is 0.5 m, and error parameters of the four simulative magnetometers are shown in Table 1. Put the cross magnetic gradient tensor system under the uniform geomagnetic field environment, and the sampling data are obtained by presenting the cross magnetic gradient tensor system in different orientations. 1000 groups of measurement data under different system orientations are recorded.Table 1
Error parameters of the simulated cross magnetic gradient tensor system.
Sensor 1Sensor 2Sensor 3Sensor 4kx1.0451.0531.0650.973ky0.9811.0410.9851.033kz0.9750.9711.0491.042bx/nT129−838593by/nT88769371bz/nT−74131−65−89ψ/rad0.0510.026−0.0440.047φ/rad0.037−0.0330.0350.042θ/rad−0.0290.0490.039−0.031α/rad00.0320.043−0.035β/rad00.0420.0360.031γ/rad00.0350.0370.045To simulate the magnetic noise of the measurement, we add independent Gaussian white noises with a mean of 0 nT and a variance of6nT in each axis of the different magnetometers. With 1000 groups of measurement data, error calibration parameters are estimated by the total least-squares method, and error calibration matrices calculated by the proposed method are shown in Table 2. The raw data are calibrated using the error calibration matrices as shown in Table 2. To compare the calibration performance, the corrected outputs of the original calibration method and the proposed method are both shown in Figures 4 and 5.Table 2
Error calibration matrices of the proposed method.
C−1bRSensor 10.958−0.03780.031701.0207−0.0524001.0257129.37486.4032−73.8938Sensor 20.95140.0317−0.051400.961−0.0268001.0299−83.223676.9102130.80630.9988−0.0320.03630.03340.9986−0.0408−0.0350.0420.9985Sensor 30.9403−0.0356−0.038701.01630.042000.953385.77393.115−64.94060.9983−0.0430.03850.04430.9984−0.0343−0.0370.0360.9987Sensor 41.0292−0.04070.031700.9692−0.0451000.959793.340970.9987−88.89170.99840.0350.0439−0.03360.9989−0.0325−0.0450.0310.9985Figure 4
Comparison of total magnetic intensity before and after calibration.
(a)(b)(c)(d)Figure 5
Comparison of tensor components before and after calibration.
(a)(b)(c)(d)(e)(f)(g)According to the simulation results, the fluctuations of the magnetic field intensity and magnetic gradient tensor components are large before calibration, and the fluctuations are mainly caused by the measurement of the system under different orientations. After calibration, the fluctuations of the corrected outputs decrease obviously. As shown in Figure4, before calibration, the max deviation of the magnetic field intensity is 3916 nT; however, the max deviation of the corrected data is 14 nT.As shown in Figure5, before calibration, the max deviation of the tensor components is 12290 nT/m, and after calibration, the max deviation is reduced to 34.4 nT/m, so we can see that both the original calibration method and the proposed method can calibrate the magnetic gradient tensor system effectively.RMS (root-mean-square) errors are used to represent the error calibration performance, and the formula is written as follows:(23)ERMS=1N∑i=1Nei2,where ei is the calibration error between corrected data and ideal data, and N is the number of measurement orientations. Then, the RMS errors of total magnetic intensity before and after calibration are listed in Table 3. RMS errors of tensor components before and after calibration are listed in Table 4.Table 3
RMS errors of total magnetic intensity before and after calibration.
Reference sensor (nT)Sensor 2 (nT)Sensor 3 (nT)Sensor 4 (nT)Raw data1112.71576.12073.51799.8Corrected data by the original method3.9144.0373.8734.035Corrected data by the proposed method3.8923.9893.8293.993Table 4
RMS errors of tensor components before and after calibration.
BxxnT/mBxynT/mBxznT/mByxnT/mByynT/mByznT/mBzznT/mRaw data3549.969585765.34455.62920.45036.65113.1Corrected data by the original method6.2429.9177.0697.4997.217.1159.55Corrected data by the proposed method6.1426.7857.0677.2996.9537.1139.468According to the RMS errors of total magnetic intensity shown in Table3, the background geomagnetic intensity is uniform, and both the original calibration method and the proposed method can achieve the accurate calibration of the single magnetometer, so the RMS errors of corrected data are in the Gaussian noise error range; however, the performance of the original method is slightly worse. According to the RMS errors of tensor components before and after calibration shown in Table 4, RMS errors of all the components calibrated by the proposed method are smaller than the original method, and the RMS errors of Bxy can be reduced by 31.6% of errors of the original method.


## 4.2. Experiments

Experiments were carried out in Heikuang Mountain, Yantai, China, where the external magnetic interference was very low. A cross magnetic gradient tensor system consisting of four homemade fluxgate magnetometers is shown in Figure6. To avoid additional magnetic interference, the holder is made of aluminum. The baseline distance of the system is 0.26 m. To find a relatively steady background geomagnetic field, a proton magnetometer is used, the accuracy of the proton magnetometer is 1 nT, and the resolution is 0.1 nT. The average value of the measured background geomagnetic intensity was 52368.3 nT. 80 groups of measurement data were recorded by presenting the cross magnetic gradient tensor system in different orientations. Then, we used the measurement data to calibrate the cross magnetic gradient tensor system, and the results were shown in Figures 7 and 8. As we knew that in a uniform geomagnetic field, the corrected output of the total magnetic intensity should be close to the background geomagnetic intensity, and the corrected output of the magnetic gradient tensor components should be close to 0 nT/m. As shown in Figure 7, the fluctuations of raw data are large, and the max fluctuation of the raw data is up to 3880 nT, and after calibration, fluctuation of total magnetic intensity is descended evidently, the corrected outputs are close to 52368.3 nT, the corresponding RMS errors of the four fluxgate magnetometers are presented in Table 5, and both the original method and the proposed method can calibrate the single fluxgate magnetometer’s error effectively. As shown in Figure 8, because of the single fluxgate magnetometer errors and misalignment errors, the actual outputs of the magnetic gradient tensor components have big deviations, and after calibration, the corrected outputs are close to 0 nT/m. The corresponding RMS errors of tensor components are presented in Table 6, the RMS errors of the proposed method are smaller than the original method, and the RMSE of Bxy can be reduced to 31.316 nT/m by the original method; however, the RMSE of Bxy calibrated by the proposed method is 14.158 nT/m, and 54.79% of errors can be reduced by the proposed method, so the results of the experiments show that the proposed method is more accurate and reliable than the original method.Figure 6
Cross magnetic gradient tensor system.Figure 7
Comparison of total magnetic intensity before and after calibration.Figure 8
Comparison of tensor components before and after calibration.Table 5
RMS errors of total magnetic intensity before and after calibration.
Reference sensor (nT)Sensor 2 (nT)Sensor 3 (nT)Sensor 4 (nT)Raw data1147.16914.771080.621055.16Corrected data by the original method4.1993.7774.7724.725Corrected data by the proposed method4.1633.7034.6544.534Table 6
RMS errors of tensor components before and after calibration.
BxxnT/mBxynT/mBxznT/mByxnT/mByynT/mByznT/mBzznT/mRaw data3317.252723.872936.668606.444405.691752.097213.74Corrected data by the original method51.73631.31617.54529.71120.69113.22665.439Corrected data by the proposed method28.30214.15817.19618.65516.15313.04732.214


## 5. Conclusions

Considering different scale factors, three-axis nonorthogonality, bias, and misalignment errors, the error calibration method of the cross magnetic gradient tensor system is proposed. Firstly, the error calibration model is established, and then, the calibrated parameters are estimated with the total least-squares method. Simulations and experiments with a cross magnetic gradient tensor system are carried out for verification of the proposed error calibration method. The simulation results show that before calibration, the max deviation of the tensor components is 12290 nT/m, and after calibration, the max deviation is reduced to 34.4 nT/m, so the proposed calibration method can calibrate the tensor system effectively. The experiment results show that the RMS errors of the proposed method are smaller than the original method, and the RMSE ofBxy can be reduced to 31.316 nT/m by the original method, and however, the RMSE of Bxy calibrated by the proposed method is 14.158 nT/m, and 54.79% of errors can be reduced by the proposed method. The proposed method can increase the calibrated accuracy compared with the original least-squares method, so the proposed error calibration method has a high value for practical application. Comparing the two methods, the traditional method (the least-squares method) has much lower accuracy than the proposed method (the total least-squares method). However, the least-squares method has less amount of calculations than the total least-squares method. The singular value decomposition of the matrix is needed in the calculation of the total least-squares method.In this paper, an error calibration method with the total least-squares method is proposed, and the corrected output by the proposed method is comparatively accurate. However, the magnetic interference of the platform is not considered in this paper, and in the future, an integrated error calibration method should be proposed to calibrate both the magnetic interference and the vector magnetometer errors.



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*Source: 6974834-2023-01-04.xml*