First-Order Derivatives of Principal and Main Invariants of Magnetic Gradient Tensor of a Uniformly Magnetized Tesseroid and Spherical Shell

Deng, Xiao-Le; Shen, Wenbin; Yang, Meng; Kühn, Michael; Ran, Jiangjun

doi:10.1007/s10712-022-09697-1

“…We can see that there are nine components in the MGT matrix. Essentially, the components of the MGT matrix are the first-order partial derivative of the magnetic field vector components [31]. In practical applications, it is unfeasible to measure the derivation directly.…”

Section: Design Of Sensor Array Structurementioning

confidence: 99%

Structural Design and Parameter Optimization of Magnetic Gradient Tensor Measurement System

Liu,

Zhang,

Liu

2024

Sensors

1

0

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Magnetic anomaly detection (MAD) technology based on the magnetic gradient tensor (MGT) has broad application prospects in fields such as unexploded ordnance detection and mineral exploration. The difference approximation method currently employed in the MGT measurement system introduces measurement errors. Designing reasonable geometric structures and configuring optimal structural parameters can effectively reduce measurement errors. Based on research into differential MGT measurement, this paper proposes three simplified planar MGT measurement structures and provides the differential measurement matrix. The factors that affect the design of the baseline distance of the MGT measurement system are also theoretically analyzed. Then, using the magnetic dipole model, the error analysis of the MGT measurement structures is carried out. The results demonstrate that the planar cross-shaped structure is optimal, with the smallest measurement error, only 3.15 × 10−10 T/m. Furthermore, employing the control variable method, the impact of sensor resolution constraints, noise level, target magnetic moment, and detection distance on the design of the optimal baseline distance of the MGT measurement system is simulated and verified. The results indicate that the smaller the target magnetic moment, the farther the detection distance, the lower the magnetometer resolution, the greater the noise, and the greater the baseline distance required. These conclusions provide reference and guidance for the construction of the MGT measurement system based on triaxial magnetometers.

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“…The tensor invariants derived from the magnetic gradient tensor do not vary with the change in the coordinate system and exhibit great potential in reducing approximation errors [31]. They have appeared in many inversion methods, including the aforementioned eigenvalues and eigenvectors method, and the STAR method [32].…”

Section: Introductionmentioning

confidence: 99%

A New Magnetic Target Localization Method Based on Two-Point Magnetic Gradient Tensor

Liu

¹

,

Zhang

²

,

Wang

³

et al. 2022

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The existing magnetic target localization methods are greatly affected by the geomagnetic field and exist approximation errors. In this paper, a two-point magnetic gradient tensor localization model is established by using the spatial relation between the magnetic target and the observation points derived from magnetic gradient tensor and tensor invariants. Based on the model, the equations relating to the position vector of magnetic target are constructed. Solving the equations, a new magnetic target localization method using only a two-point magnetic gradient tensor and no approximation errors is achieved. To accurately evaluate the localization accuracy of the method, a circular trajectory that varies in all three directions is proposed. Simulation results show that the proposed method is almost error-free in the absence of noise. After adding noise, the maximum relative error percentage is reduced by 28.4% and 2.21% compared with the single-point method and the other two-point method, respectively. Furthermore, the proposed method is not affected by the variation in the distance between two observation points. At a detection distance of 20 m, the maximum localization error is 1.86 m. In addition, the experiments also verify that the new method can avoid the influence of the geomagnetic field and the variation in the distance, and achieve high localization accuracy. The average relative error percentage in the y-direction is as low as 3.78%.

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Analytical Solutions for Gravitational Potential up to Its Third-order Derivatives of a Tesseroid, Spherical Zonal Band, and Spherical Shell

Deng

¹

,

Sneeuw

²

2023

Surv Geophys

2

0

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The spherical shell and spherical zonal band are two elemental geometries that are often used as benchmarks for gravity field modeling. When applying the spherical shell and spherical zonal band discretized into tesseroids, the errors may be reduced or cancelled for the superposition of the tesseroids due to the spherical symmetry of the spherical shell and spherical zonal band. In previous studies, this superposition error elimination effect (SEEE) of the spherical shell and spherical zonal band has not been taken seriously, and it needs to be investigated carefully. In this contribution, the analytical formulas of the signal of derivatives of the gravitational potential up to third order (e.g., V, $$V_{z}$$ V z , $$V_{zz}$$ V zz , $$V_{xx}$$ V xx , $$V_{yy}$$ V yy , $$V_{zzz}$$ V zzz , $$V_{xxz}$$ V xxz , and $$V_{yyz}$$ V yyz ) of a tesseroid are derived when the computation point is situated on the polar axis. In comparison with prior research, simpler analytical expressions of the gravitational effects of a spherical zonal band are derived from these novel expressions of a tesseroid. In the numerical experiments, the relative errors of the gravitational effects of the individual tesseroid are compared to those of the spherical zonal band and spherical shell not only with different 3D Gauss–Legendre quadrature orders ranging from (1,1,1) to (7,7,7) but also with different grid sizes (i.e., $$5^{\circ }\times 5^{\circ }$$ 5 ∘ × 5 ∘ , $$2^{\circ }\times 2^{\circ }$$ 2 ∘ × 2 ∘ , $$1^{\circ }\times 1^{\circ }$$ 1 ∘ × 1 ∘ , $$30^{\prime }\times 30^{\prime }$$ 30 ′ × 30 ′ , and $$15^{\prime }\times 15^{\prime }$$ 15 ′ × 15 ′ ) at a satellite altitude of 260 km. Numerical results reveal that the SEEE does not occur for the gravitational components V, $$V_{z}$$ V z , $$V_{zz}$$ V zz , and $$V_{zzz}$$ V zzz of a spherical zonal band discretized into tesseroids. The SEEE can be found for the $$V_{xx}$$ V xx and $$V_{yy}$$ V yy , whereas the superposition error effect exists for the $$V_{xxz}$$ V xxz and $$V_{yyz}$$ V yyz of a spherical zonal band discretized into tesseroids on the overall average. In most instances, the SEEE occurs for a spherical shell discretized into tesseroids. In summary, numerical experiments demonstrate the existence of the SEEE of a spherical zonal band and a spherical shell, and the analytical solutions for a tesseroid can benefit the investigation of the SEEE. The single tesseroid benchmark can be proposed in comparison to the spherical shell and spherical zonal band benchmarks in gravity field modeling based on these new analytical formulas of a tesseroid.

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First-Order Derivatives of Principal and Main Invariants of Magnetic Gradient Tensor of a Uniformly Magnetized Tesseroid and Spherical Shell

Cited by 5 publications

References 99 publications

Structural Design and Parameter Optimization of Magnetic Gradient Tensor Measurement System

Structural Design and Parameter Optimization of Magnetic Gradient Tensor Measurement System

A New Magnetic Target Localization Method Based on Two-Point Magnetic Gradient Tensor

Analytical Solutions for Gravitational Potential up to Its Third-order Derivatives of a Tesseroid, Spherical Zonal Band, and Spherical Shell

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