Magnetic Gradient Tensor Research Articles

Fast 3D forward modeling of a potential field based on spherical symmetry of gravitational potential

A novel approach is developed to implement the forward modeling of a potential field caused by prismatic grids on gridded observations with higher efficiency and less memory. Based on the spherical symmetry of gravitational potential, we summarize the parity, symmetry, and interconversion relations from derivatives and higher derivatives of gravitational potential, which reveal that there are vast repetitive calculation and storage redundancies in conventional forward modeling of a potential field for prismatic grids, especially for cubic grids. These properties help to reduce not only the size of a single kernel matrix but also the number of necessary kernels in multicomponent forward modeling of a potential field, such as a gradient tensor field or joint inversion of a potential field. Several experiments on the synthetic and the realistic Bishop models are presented to illustrate how these properties are used in the forward modeling of gravity and magnetic gradient tensors. Based on the proposed properties, a 2D fast Fourier transform is performed to accelerate the discrete deconvolution of the kernel and the model parameters. With the new method, one single kernel can be reduced to half the number of model parameters, and the memory requirement and time cost of our method can be reduced by 1/8 and 1/2, respectively, compared with previous works. In forward modeling of six gravity gradient tensor components, only two kernels need to be computed and stored, which are 1/3 of the conventional method. These examples indicate that, with high precision, our method requires less memory and less time than conventional methods. It shows potential in the fast large-scale inversion of gravity, magnetic, and gradient tensors with limited hardware resources.

Statistics of geometrical invariants of magnetic field gradient tensors in the turbulent magnetosheath based on magnetospheric multiscale mission

Geometrical invariants of magnetic field gradient tensors are used to classify the topological structures of a magnetic field. This study presents a statistical analysis on the geometrical invariants of magnetic field gradient based on high-quality data measured by magnetospheric multiscale mission in turbulent magnetosheath. The method for the classification of velocity field topologies cannot be applied to magnetic field with strong intensity directly because the magnetic field cannot be transformed to zero by selecting a co-moving reference frame in which the velocity is zero. During a strong magnetic field, flux ropes and tubes are the most possible magnetic structures. Statistics in the plane formed by geometrical invariants show that about 23% are force-free structures comprised of 20.5% flux tubes and 79.5% flux ropes. The remaining actively evolved structures are comprised of 30% flux tubes and 70% flux ropes. Moreover, the conditional average of current density and Lorentz force decomposition in geometrical invariants plane are investigated. The results show that flux ropes carried more current density than flux tubes for the same geometrical invariants, and flux ropes tend to associate with magnetic pressure force and flux tubes tend to associate with magnetic tension.

High-precision 3D reconstruction of multiple magnetic targets based on center weighting method

A high-precision three-dimensional reconstruction method of multiple magnetic targets is studied in this paper. The inversion space is divided into cubic grid cells, and various complex shapes of magnetic targets are simulated by the combination of grid cells. The forward calculation of the magnetic gradient tensor in the spatial domain is performed. Then, the correlation imaging method is used to calculate cross correlation between the observed magnetic gradient tensor and the theoretical magnetic gradient at each grid cell tensor due to a magnetic dipole. The spatial distribution and center position of magnetic targets are preliminarily estimated by the resultant correlation coefficients, and use them to construct the center weighting function as a constraint to the inversion equations. The solution is completed by using the conjugate gradient method to estimate the magnetic susceptibility of each grid cell, and the 3D outline of the magnetic target is depicted by using the visualization technique. The experimental results show that the proposed method can achieve the 3D reconstruction of multiple simple magnetic targets with high precision.

Regularity criteria for 3D MHD equations via mixed velocity-magnetic gradient tensors

A well-known regularity criterion in He and Xin (2005 J. Differ. Equ. 213 235–54) shows that if the velocity gradient tensor belongs to with , and , then the corresponding weak solution to the 3D magnetohydrodynamic (MHD) equations is smooth on . In this paper, we prove that the role of the velocity gradient tensor can be replaced by the combination of the diagonal part of the velocity gradient tensor and the non-diagonal part of the magnetic gradient tensor or by the combination of the diagonal part of the magnetic gradient tensor and the non-diagonal part of the velocity gradient tensor. The main interest among others is that the diagonal part of a gradient tensor is related to the deformation while the non-diagonal part is related to the rotation. Thus, our theorems may provide us with a new viewpoint to understand the potential singularity of the 3D MHD flow.

A Robust Calibration and Adaptive Multipair of Magnetic Gradient Tensors Localization Method for Magnetic Anomaly Detection

The performance of magnetic anomaly detection (MAD) with magnetic gradient tensor (MGT) measurement system is severely limited by both the measurement errors and accuracy of localization algorithm. However, the robust calibration and localization method against various interferences and noises has not received the deserved attention. This study proposes a joint calibration and localization method with MGT to improve the accuracy of non-cooperative target (NCT) detection with environmental magnetic interferences and noise. First, compared to the previous calibration method requiring a uniform magnetic field environment and precise triaxial nonmagnetic turntable, a new calibration method based on multi-invariant constraints is studied, which is insensitive to the magnetic interferences and movement of array center during the calibration. Then, an adaptive multi-pair of MGTs localization (AMML) method is proposed for the MAD of NCT. On one hand, the corresponding multi-pair of MGTs are chosen adaptively depending on the condition number of MGT, which is far more robust against varying movement directions of NCT compared to previous methods. On the other hand, the AMML method is optimized with the Levenberg–Marquardt (LM) algorithm guided by the prior information of analytical scalar triangulation and ranging (STAR) method, which enhances the efficiency and accuracy of localization method significantly. The experimental results demonstrate that the AMML method reduces the localization error by 39%, 54.3%, and 55.4% compared to the new STAR method (NSM), Lv-STAR method (LSM) and conventional STAR method, respectively.

Topology of Magnetic and Velocity Fields at Kinetic Scales in Incompressible Plasma Turbulence

AbstractThe topology of the magnetic and velocity fields at the kinetic scales are investigated in the context of nearly incompressible magnetosheath plasma turbulence. Using the unprecedented high‐resolution data from the Magnetospheric MultiScale mission, the joint probability distribution functions of geometrical invariants characterizing the magnetic and velocity fields gradient tensor at the kinetic scales are computed. The topological features of the magnetic and velocity field gradient tensors and their symmetric component tensors present axisymmetric distribution patterns, indicating that the structure of the plasma turbulence at the kinetic scales are different from those in hydrodynamic and magnetohydrodynamic turbulence. Moreover, a strong correlation between the straining and rotational parts of the magnetic and velocity field gradient tensors was found, which manifests the dominance of sheet‐like structures at the kinetic‐scales dissipation in incompressible plasma turbulence.

A Magnetic Gradient Tensor Based Method for UXO Detection on Movable Platform

A positioning method based on magnetic gradient tensor was proposed for UXO detection on movable platforms. A multi-resolution linear regression algorithm was developed to select measurement points collected when the instrument was relatively stable. In this algorithm, Pearson Correlation Coefficient (PCC) was adopted as a quality factor for characterizing the stability of instrument. Then, selected measurement points were processed in a modified Euler method for UXO position inversion. The results from the modified Euler method were selected again based on normalized magnetic source strength (NSS) and clustered by the DBSCAN algorithm. Validity of the method was tested with both simulation data and field experiment data collected by a backpack gradiometer. The test result showed that all 4 targets were detected and positioning errors were all lower than 0.15m.

Evaluation Method for Airborne Full Magnetic Gradient Tensor Data Measured Over a Survey Area

Airborne surveys of the full magnetic gradient tensor have been established as a powerful method in environmental, engineering, exploration and lithospheric studies. However, the accuracy limitation of the gradiometer, the inevitable interference from the carrying platform, and the complexity of the flight environment could degrade the quality of the airborne full magnetic gradient tensor (AFMGT) measurements. High-quality measurements are essential to obtain reliable data for subsequent analysis and robust interpretations, but there is currently no method for evaluating accuracy of the AFMGT measurements. Therefore, we propose an evaluation method for AFMGT data measured over a survey area, constrained by the harmonicity property for all tensor components, without the necessity for independent referencing data. Considering that one of the features of the AFMGT measurements is high-order and spatial correlation, we represent them by tensors that are data structures in multilinear algebra. The accuracy and utility of our method have been tested and verified by simulations and field experiments. Our work can serve as a groundwork for instrument manufacturing and quality control.

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

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.

Stability Analysis of Third-order Magnetic Gradient Tensor

So far, magnetic gradient tensor has been studied in the field of exploration. The third-order magnetic gradient tensor(MGT) can obtain more target information. But its stability is poor. It will lead to a large error under the condition of noise. Aiming at the above shortcomings, this paper puts forward an improved method. The approximate calculation formula of the third-order MGT is analyzed and improved. It can be got from the simulation that the proposed method is more excellent.

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

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%.

A Small Target Localization Method Based on the Magnetic Gradient Tensor.

Currently, many small target localization methods based on a magnetic gradient tensor have problems, such as complex solution processes, poor stability, and multiple solutions. This paper proposes an optimization method based on the Euler deconvolution localization method to solve these problems. In a simulation, the Euler deconvolution method, an improved method of the Euler deconvolution method and our proposed method are analyzed under noise conditions. These three methods are evaluated in the field with complex magnetic interference in an experiment. The simulations show that the accuracy of the proposed method is higher than that of the improved Euler deconvolution method and is slightly lower for noisy conditions. The experimental results show that the proposed method is more precise and accurate than the Euler deconvolution and enhanced methods.

Magnetic object recognition with magnetic gradient tensor system heading-line surveys based on kernel extreme learning machine and sparrow search algorithm

We found that single heading-line surveys from magnetic gradient tensor system (MGTS) can be used to realize pattern recognition of magnetic objects, such as shape and posture, which can greatly improve the target detection efficiency compared with the two-dimensional grid measurement. Abandoning complex mathematical process, we measure and learn several training routes in advance, and use kernel extreme learning machine (KELM) and sparrow search algorithm (SSA) to recognize the target. The magnetic gradient tensor and its derived variables are analyzed for the sensitivity of the magnetization direction, and two types of characteristic attributes suitable for the target posture and shape categories are summarized. Time-domain waveform feature extraction from continuously sampled signals helps build datasets with corresponding category labels. Principal component analysis (PCA) is used to reduce feature dimensionality and improve classifier efficiency. Both simulation and experiment dataset have achieved 100% accurate recognition of the target posture and shape categories.

Magnetic Anomalies Caused by 3D Polyhedral Structures With Arbitrary Polynomial Magnetization

AbstractMagnetization is a natural property of magnetic materials which is widely used to analyze paramagnetic data, locate unexploded ordnance, explore mineral deposits and gas resources, and image anomalous magnetic structures in the Earth's crust. An essential and challenging task is to compute accurate magnetic anomalies caused by realistic magnetic 3D structures. In this study, for the first time, we derive the analytical expressions of magnetic potential, magnetic field and magnetic gradient tensor for magnetic materials with 3D polyhedral shapes and magnetization vectors that may vary in the space following polynomial trends. The order of polynomials can vary from one to high positive integers in both horizontal and vertical directions. Synthetic and realistic deposit models are used to validate our analytical solutions. We release the open‐source code in C++.

Magnetic anomaly detection via a combination approach of minimum entropy and gradient orthogonal functions

Aiming at the problem of magnetic anomaly detection under low signal-to-noise ratio, a full magnetic gradient detection via minimum entropy and gradient orthonormal basis function is proposed in this paper. Firstly, the mathematical model of magnetic dipole gradient tensor and the experimental method of collecting magnetic gradient tensor signals are introduced. Secondly, a new minimum entropy detector is designed, the basic functions which are processed by new minimum entropy detector can be orthogonalized easily. Thirdly, a group of new standard orthogonal basis functions are constructed according to the characteristics of magnetic gradient anomaly signals. Finally, the detector and orthogonal basis functions are used to detect the magnetic anomaly. Both numerical simulation signals and experiment signals are used to prove the effectiveness of the methods.

Application of Helbig integrals to magnetic gradient tensor multi-target detection

We extended the magnetic gradient tensor Helbig integrals (MGTHI) from classical Helbig integrals (HI), which can be perfectly adapted to the grid measurement of planar-cross magnetic gradient tensor system. With different sizes of windows, we can estimate the MGTHI by using the two-dimensional trapezoidal rule algorithm and calculate the total magnetization direction of grids, where the node with the smallest change in the magnetization direction is target horizontal position. In the targets recognition area determined by the invariant improved tilt angles, the source depth and dipole moment magnitude can be calculated with magnetic vector and tensor field data directly above the source. In experiments, the horizontal position deviation of multiple small magnets in the survey area of 2.1 m × 2.1 m and 1.2 m × 1.2 m is less than 0.08 m, and the buried depth deviation is less than 0.15 m, the positioning accuracy is improved by more than 5 times from the classical HI method.

Lagrangian evolution of field gradient tensor invariants in magneto-hydrodynamic theory

In 1982 in a series of works Vielliefosse [1, 2] discussed a nonlinear homogeneous evolution equation for the velocity gradient tensor in fluid dynamics. Later Cantwell [3] extended this formalism to the non-homogeneous case including the effects of viscous diffusion and cross derivatives of pressure field. Here, we derive the evolution equations of the geometrical invariants of the magnetic and velocity field gradient tensors in the case of magneto-hydrodynamics for both non-homogeneous and homogeneous cases, i.e., considering or neglecting viscous effects and source terms. The inclusion of dissipation effects and higher-order gradient terms introduces a non trivial evolution of invariants, which can be treated as a stochastic evolution equation. Conversely, in the homogeneous case, the magnetic field invariants do not evolve, i.e., the magnetic field line topology is conserved, while the corresponding velocity invariants are affected by magnetic forces. By writing the equations of the velocity field invariants as a dynamical system we can identify the role of the different terms in the evolution equations. In detail, in the homogenous case we show that the term associated with the current density drives transitions between hyperbolic and elliptical structures. Evolution equations are also discussed in the perspective of an application to the analysis of magneto-hydrodynamic turbulence.

3D Inversion of Magnetic Gradient Tensor Data Based on Convolutional Neural Networks

High-precision vector magnetic field detection has been widely used in the fields of celestial magnetic field detection, aeromagnetic detection, marine magnetic field detection and geomagnetic navigation. Due to the large amount of data, the 3D inversion of high-precision magnetic gradient vector data often involves a large number of computational requirements and is very time-consuming. In this paper, a 3D magnetic gradient tensor (MGT) inversion method is developed, based on using a convolutional neural network (CNN) to automatically predict physical parameters from the 2D images of MGT. The information of geometry, depth and parameters such as magnetic inclination (I), magnetic declination (D) and magnetization susceptibility of magnetic anomalies is extracted, and a 3D model is obtained by comprehensive analysis. The method first obtains sufficient MGT data samples by forward modeling of different magnetic anomalies. Then, we use an improved CNN with shear layers to achieve the prediction of each magnetic parameter. The reliability of the algorithm is verified by numerical simulations of synthetic models of multiple magnetic anomalies. MGT data of the Tallawang magnetite diorite deposit in Australia are also predicted by using this method to obtain a slab model that matches the known geological information. The effects of sample size and noise level on the prediction accuracy are discussed. Compared with single-component prediction, the results of multi-component joint prediction are more reliable. From the numerical model study and the field data validation, we demonstrate the capability of using CNNs for inversing MGT data.

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

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

Fast Localization and Characterization of Underground Targets with a Towed Transient Electromagnetic Array System.

A fast inversion algorithm combined with the transient electromagnetic (TEM) detection system has important significance for improving the detection efficiency of unexploded ordnance. The traditional algorithms, such as differential evolution or Gauss–Newton algorithms, usually require tens to thousands of iterations to locate the underground target. A new algorithm with a magnetic gradient tensor and singular value decomposition (SVD) to estimate the target position and characterization quickly and accurately is proposed in this paper. Two modes of magnetic gradient tensor are constructed to accurately locate shallow and deep targets, respectively. The SVD algorithm is applied to the responses to estimate the electromagnetic characteristics of the target quickly and accurately. To verify the performance of the proposed algorithm, a towed TEM sensor is designed, which is constructed with three transmitting coils and nine three-component receiving coils arranged in a 3 × 3 array. Field experiments in survey and cued modes were taken to verify the performance of the proposed algorithm and the towed system. Results show that the magnetic gradient tensor algorithm proposed in this paper can accurately locate a single target within 2.0 m depth, and the error of depth is no more than 8 cm. Even for overlapping response of multi targets, the error of depth is no more than 12 cm. The underground target can be accurately characterized by the SVD algorithm. For targets with depths over 2.0 m, the signal-to-noise ratio of characteristic response estimated by SVD is higher than that of the traditional method. The proposed method needs approximately 40 ms, only 1% of the traditional one, considerably improving detection efficiency and laying a theoretical and experimental foundation for real-time data processing.