# Correction of Three Sensors in a Magnetic Field Gradiometer

Authors: Li Yang; Shuting Xiao; Xiaoyu Tang; Hui Chen; Song Zhang; Caihong Li
Journal: Journal of Sensors
 (2022)

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## Abstract

A magnetic sensor array is usually used for weak magnetic target detection. This paper presents a vector second-order gradiometer that is arranged in a vertical structure. The measurement accuracy of the gradiometer is affected by the error of each three-axis magnetometer (TAM) and the misalignment error between TAMs. By analyzing the error mechanism of the magnetic gradiometer test system, the corresponding error model for the correction of the gradient of the component is established. The error parameters of the gradiometer were estimated by the proposed method, and the gradient error of the magnetic field and the component gradient error of the magnetic field were corrected. After correction, the gradiometer is used to localize the magnetic target. Simulation results illustrate that the localization error of magnetic targets is reduced by two orders. Thus, this proposed correction method can effectively improve the locating performance of gradiometer.


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## Body

## 1. Introduction

The magnetic sensor array is composed of magnetic vector sensors following a certain spatial distribution. It can detect magnetic anomalies by measuring the spatial change rates of the three components of a magnetic field along the orthogonal coordinate system and has been widely applied in the fields of aviation geophysical exploration [1–4], magnetic dipole localization [5–10], geophysical surveys [11–13], biomagnetic [14–16], underwater unmanned aerial vehicle navigation, and geomagnetic navigation [17–22]. To achieve high sensitivity detection of weak magnetic targets, it is necessary to improve its data quality [23]. However, the quality can be affected by factors such as interference, unsatisfactory performance of the sensors, and installation of the sensor array. Due to the limitations of machining and installation levels, a three-axis magnetometer (TAM) cannot be absolutely orthogonal [24]. There are also zero bias, temperature drift, and scale factor from each magnetometer [25–27]. In addition, external magnetic distortion affects the measurement accuracy. Most importantly, the misalignment between magnetic sensors in the sensor array cause measurement errors [28, 29]. Therefore, it is particularly important to correct instrument errors in a magnetic sensor array.The vector correction and scalar correction are commonly used in magnetometer calibration. For the vector correction method, a three-dimensional Helmholtz coil is needed to provide a reference magnetic field to correct its output and obtain error parameters. Lee et al. [30] designed a uniaxial square loop coil system, which can generate the pure second-order field gradient and can be used for calibration SQUID gradiometers. The vector correction needs to know the attitude of the TAM, and the procedures are very complicated, especially for TAM array. For the scalar correction method, a homogeneous magnetic field is required. Xuan [31] transformed the objective function into a recursive form, used the central approximate estimation method to obtain the initial value, and then used the extended Kalman filter to estimate the sensor correction parameters. This method is easily influenced by the initial value and can diverge. Merayo et al. [32] established an intrinsic reference systems and use linear least squares estimator to calibrate one vector magnetometers, and the absolute precision of magnetic magnitude decreases to 0.5 nT. Pang et al. [33] presented an integrated vector calibration model for the cross-magnetometer array, in which magnetometer error, misalignment error, and magnetic distortion fields are integrated as one error matrix. Weiss et al. [34] used an approximate algorithm to determine an initial value and then used Newton iteration to solve the bias of the magnetic sensor. Yu et al. [35] proposed a calibration algorithm for two TAMs by applying nonlinear mapping between the difference of geomagnetic vector norm and error sources of two TAMs; however, it only corrects magnitude gradient of the magnetic field, but not the component gradient of the magnetic field.In this study, a magnetic sensor array with three TAMs for detecting weak magnetic anomaly target is developed. By analyzing the error mechanism of the system, an error model related to the system is established. The sensor array for correction is rotated as a whole together by a dedicated nonmagnetic turntable and does not need to strictly control the rotation angle. The performance of the proposed method is demonstrated through both simulation and experimentation. Both the scalar and vector performances of the magnetic gradiometer have been greatly improved. After correction, the localization accuracy of gradiometer is improved. Thus, the proposed correction method is beneficial for magnetic target localization.


## 2. Error Mathematical Model

The magnetic array was composed of three TAMs (Figure1). The target location using magnetic field gradient was measured by gradiometer, and the magnetic field gradient measurement was realized based on the principle of difference equivalent. Then, the error of gradiometer is mainly composed of the nonorthogonal, zero deviation, and scale factor errors of each TAM, which is referred to as the magnetometer error and the misalignment error between TAMs.Figure 1
Structure design of magnetic gradiometer.
(a)(b)For magnetometer error correction, the same calibration model is applied to each TAM in the magnetic gradiometer. As shown in Figure2, OiXi′Yi′Zi′i=1,2,3 and OiXiYiZii=1,2,3 are the orthogonal and actual coordinates of the three axes of the i-th sensor, respectively.Figure 2
Schematic of the three-axis nonorthogonal angle of the TAM.The calibration experiment was conducted in the zero magnetic field laboratory which can produce homogeneous magnetic field. Because of the homogeneous magnetic field, if thethree axes of the i-th sensor are orthogonal, the total magnetic field does not change with the sensor rotation. In order to simplify the mathematical model of nonorthogonal, we build an orthogonal coordinate system Oi,Xi,Yi,Zi, for each TAM as follows: let OZi be one axis of the orthogonal coordinate and the second orthonormal axis (Oi,Yi,) of orthogonal coordinate system lies in the plane OiYiZi. The latter is coplanar with the former ϕi1 is the angle between axes OiYi′ and OiYi, φi1 is the angle between OiXi and plane Oi,Xi,Yi,, and φi2 is the angle between Oi,Xi, and the projection of plane OiXi in plane Oi,Xi,Yi,. Then, the true magnetic field of the three TAMs can be expressed as follows:
(1)BmxiBmyiBmzi=cosφi1cosφi200cosφi1sinφi2cosϕi10sinφi1sinϕi11kxi000kyi000kziBtxiBtyiBtzi+++B0xiB0yiB0zi,where Btxi, Btyi, and Btzi are the true magnetic fields and Bmxi, Bmyi, and Bmzii=1,2,3 are measured magnetic fields of the i-th TAM. kxi,kyi,kzii=1,2,3 are scale factor, and B0xi, B0yi, and B0zii=1,2,3 are offset of the i-th TAM.As shown in Figure3(a), if there is no misalignment error between each TAM, the three components of magnetic field after magnetometer error correction should be identical in the homogeneous magnetic field, as follows:
(2)Btx1Bty1Btz1=BtxjBtyjBtzjj=2,3,Figure 3
(a) Three components of magnetic field after error correction in the homogeneous magnetic field. (b) Schematic of misalignment angle between TAMs.
(a)(b)whereBtxj, Btyj, and Btzjj=2,3 represent the magnetic field values of TAM2 and TAM3 after the magnetometer correction. However, there is misalignment error between each TAM, and considering the misalignment errors between each TAM, the relationship between three components of magnetic field of each TAM should be expressed as follows:
(3)Btx1Bty1Btz1=cosα1jcosβ1j−sinα1jcosγ1j+cosα1jsinβ1jsinγ1jsinα1jsinγ1j+cosα1jsinβ1jcosγ1jsinα1jcosβ1jcosα1jcosγ1j+sinα1jsinβ1jsinγ1j−cosα1jsinγ1j+sinα1jsinβ1jcosγ1j−sinβ1jcosβ1jsinγ1jcosβ1jcosγ1jBtxjBtyjBtzj,whereα1j, β1j, and γ1jj=2,3 are Euler’s angles, and the misalignment error model of the gradiometer is established by Euler’s angles. The misalignment correction rotates the α1j, β1j, γ1jj=2,3 angles around the axes, and the order is z-y-x. The rotation process is shown in Figure 3(b), in which Xi, Yi, and Zii=1,2,3 are the three axes after nonorthogonal correction of the i-th sensor, respectively. In order to express the misalignment error conveniently, the misalignment error can be conveniently expressed in the following matrix form.
(4)Bt1=M12Bt2,Bt1=M13Bt3,whereM1j (j=2,3) are misalignment error matrix between TAM1 and j-th TAM.


## 3. Calibration Method

The calibration process is shown in Figure4. BE is the true magnitude of the external magnetic field, which is homogeneous magnetic field; f1., f2. and f3. are the transfer functions of the three TAMs; Bmii=1,2,3 are the measured value of the three TAMs. C1., C2., and C3. are the error correctors of the three TAMs, which are used to correct the magnetometer errors. M12. and M13. are the misalignment correctors between TAM1 and TAM2 and TAM1 and TAM3, respectively. Then, Bt1, Bt2, and Bt3 must be the true magnitude of the external magnetic field. For the i-th TAM, according to the true magnetic field of the three TAMs of Equation (1), the relationship between the measured value and the ideal value can be rewritten as
(5)BtxiBtyiBtzi=kxi000kyi000kzi−1cosφi1cosφi200cosφi1sinφi2cosϕi10sinφi1sinϕi11−1BmxiBmyiBmzi−B0xiB0yiB0zi.Figure 4
Schematic diagram of calibration process.For ease of description, Equation (5) can be rewritten as
(6)BtxiBtyiBtzi=c11c12c130c22c2300c33BmxiBmyiBmzi−B0xiB0yiB0zi,where matrix C is c11=1kxicosφi1cosφi2, c12=1kxicosφi2cosφi1, c13=1−cosφi1sinφi2sinφi1/kxicosφi1cosφi2cosϕi1, c22=1kyicosφi1, c23=−sinϕi1/kyicosϕi1, and c33=1/kzi. And the norm of Bti can be expressed as follows:
(7)Bti2=c11i2Bmxi2+c12i2+c22i2Bmyi2+c13i2+c23i2+c33i2Bmzi2+2c11ic12iBmxiBmyi+2c11ic13iBmxiBmzi+2c12iM13i+2c22iM23iBmyiBmzi−2c11iBmxiB0xi−2c12iB0xi+2c12iB0yiBmyi−2c13iB0xi+2c23iB0yi+2c33iB0ziBmzi+B0xi2+B0yi2+B0zi2.Consideringx0=1, x1i=Bmxi2, x2i=Bmyi2, x3i=Bmzi2, x4i=BmxiBmyi, x5i=BmyiBmzi, x6i=BmxiBmzi, x7i=Bmxi, x8i=Bmyi, x9i=Bmzi, θ0i=B0xi2+B0yi2+B0zi2, θ1i=c11i2, θ2i=c12i2+c22i2, θ3i=c13i2+c23i2+c33i2, θ4i=2c11ic12i, θ5i=2c11ic13i, θ6i=2c12ic13i+2c22ic23i, θ7i=−2c11iB0xi, θ8i=2c12iB0xi+2c12iB0yi, and θ9i=2c13iB0xi+2c33iB0zi, (7) is rewritten as follows:
(8)Bti2=hθXi=∑n=09θnXni,and the function is defined as follows:
(9)Jθ=12∑q=1mhθXiq−Yq2,where Xiq are input variables consisting of the measurement data for the i-th (i=1,2,3) TAM and Y is the target variable and the ideal value of the magnetic field. An updated algorithm [36] is used to find θthat minimizes Jθ, which can be expressed as
(10)θj≔θj−α∂∂θjJθ,where α is the learning factor. Through the definition of θ in the front, estimated coefficients of the error matrix C are determined.After the magnetometer error calibration, the estimated value of the three TAMs isBt1, Bt2, and Bt3, respectively. Since the three TAMs have been calibrated by the same method, their corrected data are all in the orthogonal coordinate system, and the misalignment errors of the three TAMs are mainly reflected in the different component values of the three axes. It is obvious that Btj (j=2,3) can be transformed to the coordinate system of sensor 1.
(11)Btj′=M1jBtj,where Btj′ (j=2,3) is the true magnitude of the external magnetic field and M1j is the rotation matrix between TAM1 and jj=2,3. Considering the gradiometer will be calibrated in the homogeneous magnetic field, Btj′should be equal to Bt1, and the relationship between them can be described as follows:
(12)Bt1−M1jBtj=0.The coefficients in the misalignment matrix are solved by the same optimization algorithm as the magnetometer error correction which is mentioned above, and the misalignment angle is finally obtained.


## 4. Results

This section may be divided by subheadings. It should provide a concise and precise description of the experimental results and their interpretation as well as the experimental conclusions that can be drawn.


### 4.1. Simulations

The performance of the proposed correction method was verified through MATLAB simulations. By simulating the free rotation of a gradiometer in a uniform magnetic field space, 32 sets of three-component magnetic field data are obtained. Error parameters are added to these three-component data to obtain simulated three-component magnetic field data; the preset values for the error parameters of TAM1, TAM2, and TAM3 are shown in Table1.Table 1
Comparison between presetting error parameter and estimating error parameter.
Sensor 1Sensor 2Sensor 3Preset parameterk1=1.000570000.999520000.99981N1=14.3633×10−62.2689×10−6016.8068×10−6001B°1=−130.5645.8420.04τk2=1.000410000.999710000.99935N2=15.9341×10−61.9199×10−6014.5379×10−6001B°2=−125.14−40.5226.42το12=−0.00458840.0015903−0.0066582k3=1.000250000.999770000.99982N3=12.2689×10−63.6652×10−6016.2832×10−6001B°3=75.83−70.75−86.0το13=−0.01560.02540.0324Estimate parameterk1=1.0005660000.9995160000.999806N1=1−4.3609×10−62.2677×10−6016.8101×10−6001B°1=130.56145.84020.040τk2=1.0004060000.9997060000.999346B°2=125.14040.52026.420το12=−0.004588260.00159032−0.00665814k3=1.0002460000.9997660000.999816N3=12.2684×10−62.2684×10−6016.2847×10−6001B°3=75.830−70.750−86.00το13=−0.015600.025400.03240kii=1,2,3 are scale factor, Nii=1,2,3 are nonorthogonal error, boii=1,2,3 are offset error, and O1jj=2,3 are misalignment error. The error parameters of the three sensors were estimated by using the simulated data. The estimated results are shown in Table 1. The simulated magnetic field data was used for error correction, and the comparison of magnetic magnitude before and after correction is shown in Figure 5. From Figure 5, we can know that the maximum fluctuations of the magnetic magnitude about TAM1 at 32 different attitudes are 254.99 nT before correction, while the maximum fluctuations are 0.013 nT after correction. It is obvious that the magnetic magnitude becomes smooth at 32 different attitudes after correction. It can be seen that the magnetic magnitude fluctuation of TAM2 and TAM3 also decreases from 269.28 and 258.07 nT before correction to 0.01639 and 0.001295 nT after correction at different attitudes.Figure 5
Comparison of simulated total magnetic field before and after correction.
(a)(b)


### 4.2. Experimental Test and Result Analysis

The input module consists of three TAMs, a positioning chip, and an attitude sensor. The data communication processing module included a signal acquisition module with 16-bit resolution and a calculation unit. Three Mag648 fluxgate sensors were used, which is produced by Bartington Instruments Ltd. With the help of the high-precision scalar magnetometer (potassium optical pumped magnetometer) developed by us, the magnetic magnitude which can be used as the expected value of the true magnitude was measured. The attitude of the TAM was changed by the customized high-precision nonmagnetic turntable, and its three-dimensional rotation angle was accurate to 0.015 degrees. The calibration experiment was conducted in the zero magnetic field laboratory, and there is a Helmholtz coil with size of16m∗16m∗16m which can produce homogeneous magnetic field with a diameter of 3 meters. As shown in Figure 6, the gradiometer was placed on the nonmagnetic turntable and rotated around the x-axis, y-axis, and z-axis, respectively, and 32 groups of magnetic field three-component data under different attitudes were collected. The error parameters of the gradiometer were estimated, as shown in Table 2.Figure 6
The nonmagnetic turntable.Table 2
Calculated error parameters.
Sensor 1Sensor 2Sensor 3Estimate parameterk1=1.00270001.00290001.0030N1=10.62×10−30.33×10−301−0.76×10−4001B°1=−23.31−44.73−170.94τk2=1.00310000.99960001.0008N2=10.36×10−30.01010.47×10−3001B°2=8.14−100.1126.13το12=0.97−0.130.160.140.980.030.15−0.060.98k3=0.99960000.99990000.9998N3=1−0.25×10−3−0.62×10−3010.53×10−4001B°3=8.14−100.1126.13Tο13=0.99−0.03−0.13×10−20.030.99−0.11×10−20.13×10−20.10×10−20.99The comparison of magnetic magnitude error before and after correction is shown in Figure7. The errors in measured magnetic magnitude using three TAMs are, respectively, represented by ∆Bt1, ∆Bt2, and ∆Bt3. It can be observed that the errors in magnetic magnitude decrease clearly through correction, and the means of ∆Bt1, ∆Bt2, and ∆Bt3 at different attitudes decrease 131.80, 60.90, and 21. 33 nT to 0.31, 0.10, and 0.59 nT, respectively. Figure 8 shows the gradient error of the total magnetic field between TAM1 and TAM2 and TAM1 and TAM3; the maximum of magnitude gradient errors of the magnetic field between TAM1 and TAM2 and TAM1 and TAM3 at different attitudes are reduced from 2105.97 and 1299.78 nT/m before correction to 28.02 and 18.72 nT/m after correction. Misalignment errors were calibrated using the estimated misalignment error parameters shown in Table 2, and the calibration results between TAM1 and TAM2 are shown in Figure 9.Figure 7
Comparison of magnetic magnitude error before and after correction: (a) error of the first TAM; (b) error of the second TAM; (c) error of the third TAM.
(a)(b)(c)Figure 8
Comparison of magnetic gradient error before and after correction: (a) error between TAM1 and TAM2; (b) error between TAM1 and TAM3; (c) error between TAM1 and TAM2 after correction; (d) error between TAM1 and TAM3 after correction.
(a)(b)(c)(d)Figure 9
Comparison of component gradient error before and after correction.
(a)(b)(c)(d)(e)(f)Figures9(a), 9(c), and 9(e) represent the component gradient error of the magnetic field before misalignment calibration, and Figures 9(b), 9(d), and 9(f) represent the component gradient errors of the magnetic field after calibration using estimated misalignment parameters. The component gradient errors on the three axes were greatly reduced at different attitudes after correction, and the error maximum of the three components has decreased from 54845, 35102, and 40151 nT/m before correction to 104.17, 59.68, and 40.07 nT/m after correction. The results of the misalignment correction between TAM1 and TAM3 are also greatly reduced from 8525, 7765, and -867 nT/m to 32.86, 102.81, and 42.99 nT/m. Based on these results, it was determined that this method has a very good correction effect on the magnetic and misalignment errors.In order to investigate the effect of the proposed method on positioning error, the simulation is conducted as follows: the gradiometer is used to localize the magnetic object. As shown in Figure10(a), the survey lines are distributed in the space of 4m×4m. A magnetic target with known magnetic moment mx=0.34A∙m2,my=0.25A∙m2,mz=0.91A∙m2 is placed at different positions, and the localization accuracy is analyzed. Position 1 and position 2 are marked in red point and green point, respectively. As shown in Figure 10(b), the plane of the second sensor in the gradiometer is taken as the detection plane, then the coordinate position of second TAM is x,y,z, and the coordinate position of the first and third TAM is x,y,z+L and x,y,z−L, respectively, where L=20cm is the baseline length of the gradiometer. On the basis of Euler equation, the vertical gradient information is used to locate the target.
(13)x−x0Bxz+y−y0Byz+z−z0Bzz=−NBz−bz,Figure 10
The schematic of the simulation experiment.
(a)(b)wherex,y,z and x0,y0,z0 are the coordinates of measuring point and the magnetic objection, respectively, Bz and bz are the vertical component of the magnetic field and background magnetic field at the measuring point, respectively, and N is structure indexes. A sliding rectangular calculation window is set up in the survey lines area, and the overdetermined equations are established by using the data of grid points in each calculation window; using the least square method resolve overdetermined equation to acquire the location of the magnetic object. The localization errors of the magnetic objection are expressed as follows:
(14)errri=riprset−riestimatedi=x,y,z,whereriprset and riestimated are the preset and estimated coordinate position of magnetic object, respectively. The localization errors of the target are calculated and are shown in Table 3.Table 3
Calculated error parameters.
Point 1Point 2errrx (m)errry (m)errrz (m)errrx (m)errry (m)errrz (m)Before correction0.592-0.956-0.9740.673-0.3140.307After correction0.004-0.00020.0060.010-0.007-0.123As shown in Table3, at position 1, the localization errors of x-, y-, and z-axis are 0.592 m, -0.956 m, and 0.974 m before correction. It is obvious that the localization resolution without correction is unreliable. After correction with the proposed method, the localization errors reduce to 0.004 m, -0.0002 m, and 0.006 m. It means that the localization errors largely declined after correction by the proposed method. At position 2, it shows that the localization errors of the x-, y-, and z-axes are reduced from 0.673 m, -0.314 m, and 0.307 m before correction to 0.010 m, -0.007 m, and 0.123 m after correction. The simulation results indicate that the positioning accuracy is further improved by correction.


## 4.1. Simulations

The performance of the proposed correction method was verified through MATLAB simulations. By simulating the free rotation of a gradiometer in a uniform magnetic field space, 32 sets of three-component magnetic field data are obtained. Error parameters are added to these three-component data to obtain simulated three-component magnetic field data; the preset values for the error parameters of TAM1, TAM2, and TAM3 are shown in Table1.Table 1
Comparison between presetting error parameter and estimating error parameter.
Sensor 1Sensor 2Sensor 3Preset parameterk1=1.000570000.999520000.99981N1=14.3633×10−62.2689×10−6016.8068×10−6001B°1=−130.5645.8420.04τk2=1.000410000.999710000.99935N2=15.9341×10−61.9199×10−6014.5379×10−6001B°2=−125.14−40.5226.42το12=−0.00458840.0015903−0.0066582k3=1.000250000.999770000.99982N3=12.2689×10−63.6652×10−6016.2832×10−6001B°3=75.83−70.75−86.0το13=−0.01560.02540.0324Estimate parameterk1=1.0005660000.9995160000.999806N1=1−4.3609×10−62.2677×10−6016.8101×10−6001B°1=130.56145.84020.040τk2=1.0004060000.9997060000.999346B°2=125.14040.52026.420το12=−0.004588260.00159032−0.00665814k3=1.0002460000.9997660000.999816N3=12.2684×10−62.2684×10−6016.2847×10−6001B°3=75.830−70.750−86.00το13=−0.015600.025400.03240kii=1,2,3 are scale factor, Nii=1,2,3 are nonorthogonal error, boii=1,2,3 are offset error, and O1jj=2,3 are misalignment error. The error parameters of the three sensors were estimated by using the simulated data. The estimated results are shown in Table 1. The simulated magnetic field data was used for error correction, and the comparison of magnetic magnitude before and after correction is shown in Figure 5. From Figure 5, we can know that the maximum fluctuations of the magnetic magnitude about TAM1 at 32 different attitudes are 254.99 nT before correction, while the maximum fluctuations are 0.013 nT after correction. It is obvious that the magnetic magnitude becomes smooth at 32 different attitudes after correction. It can be seen that the magnetic magnitude fluctuation of TAM2 and TAM3 also decreases from 269.28 and 258.07 nT before correction to 0.01639 and 0.001295 nT after correction at different attitudes.Figure 5
Comparison of simulated total magnetic field before and after correction.
(a)(b)


## 4.2. Experimental Test and Result Analysis

The input module consists of three TAMs, a positioning chip, and an attitude sensor. The data communication processing module included a signal acquisition module with 16-bit resolution and a calculation unit. Three Mag648 fluxgate sensors were used, which is produced by Bartington Instruments Ltd. With the help of the high-precision scalar magnetometer (potassium optical pumped magnetometer) developed by us, the magnetic magnitude which can be used as the expected value of the true magnitude was measured. The attitude of the TAM was changed by the customized high-precision nonmagnetic turntable, and its three-dimensional rotation angle was accurate to 0.015 degrees. The calibration experiment was conducted in the zero magnetic field laboratory, and there is a Helmholtz coil with size of16m∗16m∗16m which can produce homogeneous magnetic field with a diameter of 3 meters. As shown in Figure 6, the gradiometer was placed on the nonmagnetic turntable and rotated around the x-axis, y-axis, and z-axis, respectively, and 32 groups of magnetic field three-component data under different attitudes were collected. The error parameters of the gradiometer were estimated, as shown in Table 2.Figure 6
The nonmagnetic turntable.Table 2
Calculated error parameters.
Sensor 1Sensor 2Sensor 3Estimate parameterk1=1.00270001.00290001.0030N1=10.62×10−30.33×10−301−0.76×10−4001B°1=−23.31−44.73−170.94τk2=1.00310000.99960001.0008N2=10.36×10−30.01010.47×10−3001B°2=8.14−100.1126.13το12=0.97−0.130.160.140.980.030.15−0.060.98k3=0.99960000.99990000.9998N3=1−0.25×10−3−0.62×10−3010.53×10−4001B°3=8.14−100.1126.13Tο13=0.99−0.03−0.13×10−20.030.99−0.11×10−20.13×10−20.10×10−20.99The comparison of magnetic magnitude error before and after correction is shown in Figure7. The errors in measured magnetic magnitude using three TAMs are, respectively, represented by ∆Bt1, ∆Bt2, and ∆Bt3. It can be observed that the errors in magnetic magnitude decrease clearly through correction, and the means of ∆Bt1, ∆Bt2, and ∆Bt3 at different attitudes decrease 131.80, 60.90, and 21. 33 nT to 0.31, 0.10, and 0.59 nT, respectively. Figure 8 shows the gradient error of the total magnetic field between TAM1 and TAM2 and TAM1 and TAM3; the maximum of magnitude gradient errors of the magnetic field between TAM1 and TAM2 and TAM1 and TAM3 at different attitudes are reduced from 2105.97 and 1299.78 nT/m before correction to 28.02 and 18.72 nT/m after correction. Misalignment errors were calibrated using the estimated misalignment error parameters shown in Table 2, and the calibration results between TAM1 and TAM2 are shown in Figure 9.Figure 7
Comparison of magnetic magnitude error before and after correction: (a) error of the first TAM; (b) error of the second TAM; (c) error of the third TAM.
(a)(b)(c)Figure 8
Comparison of magnetic gradient error before and after correction: (a) error between TAM1 and TAM2; (b) error between TAM1 and TAM3; (c) error between TAM1 and TAM2 after correction; (d) error between TAM1 and TAM3 after correction.
(a)(b)(c)(d)Figure 9
Comparison of component gradient error before and after correction.
(a)(b)(c)(d)(e)(f)Figures9(a), 9(c), and 9(e) represent the component gradient error of the magnetic field before misalignment calibration, and Figures 9(b), 9(d), and 9(f) represent the component gradient errors of the magnetic field after calibration using estimated misalignment parameters. The component gradient errors on the three axes were greatly reduced at different attitudes after correction, and the error maximum of the three components has decreased from 54845, 35102, and 40151 nT/m before correction to 104.17, 59.68, and 40.07 nT/m after correction. The results of the misalignment correction between TAM1 and TAM3 are also greatly reduced from 8525, 7765, and -867 nT/m to 32.86, 102.81, and 42.99 nT/m. Based on these results, it was determined that this method has a very good correction effect on the magnetic and misalignment errors.In order to investigate the effect of the proposed method on positioning error, the simulation is conducted as follows: the gradiometer is used to localize the magnetic object. As shown in Figure10(a), the survey lines are distributed in the space of 4m×4m. A magnetic target with known magnetic moment mx=0.34A∙m2,my=0.25A∙m2,mz=0.91A∙m2 is placed at different positions, and the localization accuracy is analyzed. Position 1 and position 2 are marked in red point and green point, respectively. As shown in Figure 10(b), the plane of the second sensor in the gradiometer is taken as the detection plane, then the coordinate position of second TAM is x,y,z, and the coordinate position of the first and third TAM is x,y,z+L and x,y,z−L, respectively, where L=20cm is the baseline length of the gradiometer. On the basis of Euler equation, the vertical gradient information is used to locate the target.
(13)x−x0Bxz+y−y0Byz+z−z0Bzz=−NBz−bz,Figure 10
The schematic of the simulation experiment.
(a)(b)wherex,y,z and x0,y0,z0 are the coordinates of measuring point and the magnetic objection, respectively, Bz and bz are the vertical component of the magnetic field and background magnetic field at the measuring point, respectively, and N is structure indexes. A sliding rectangular calculation window is set up in the survey lines area, and the overdetermined equations are established by using the data of grid points in each calculation window; using the least square method resolve overdetermined equation to acquire the location of the magnetic object. The localization errors of the magnetic objection are expressed as follows:
(14)errri=riprset−riestimatedi=x,y,z,whereriprset and riestimated are the preset and estimated coordinate position of magnetic object, respectively. The localization errors of the target are calculated and are shown in Table 3.Table 3
Calculated error parameters.
Point 1Point 2errrx (m)errry (m)errrz (m)errrx (m)errry (m)errrz (m)Before correction0.592-0.956-0.9740.673-0.3140.307After correction0.004-0.00020.0060.010-0.007-0.123As shown in Table3, at position 1, the localization errors of x-, y-, and z-axis are 0.592 m, -0.956 m, and 0.974 m before correction. It is obvious that the localization resolution without correction is unreliable. After correction with the proposed method, the localization errors reduce to 0.004 m, -0.0002 m, and 0.006 m. It means that the localization errors largely declined after correction by the proposed method. At position 2, it shows that the localization errors of the x-, y-, and z-axes are reduced from 0.673 m, -0.314 m, and 0.307 m before correction to 0.010 m, -0.007 m, and 0.123 m after correction. The simulation results indicate that the positioning accuracy is further improved by correction.


## 5. Conclusion

To detect the buried weak magnetic field target, a magnetic gradiometer composed of three sensors was constructed. The error mechanism was analyzed to improve the performance of the magnetic gradiometer, the corresponding error model was established, and the measured scalar and vector outputs of the magnetic gradiometer were corrected. The results showed that both the gradient error of the total magnetic field and the gradient error of the magnetic field component decreased to a great extent. The data quality of the gradiometer was improved, and the measurement accuracy of scalar and vector magnetic fields was greatly improved, which has a strong practical value.



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*Source: 1145960-2022-04-06.xml*