Magnetic Gradient Tensor Data Research Articles

Joint Gramian inversion of gravity, magnetic and gravity gradient tensor data for the modelling of Mesoproterozoic iron oxide deposits, southeast Missouri, USA

SUMMARY Within the iron metallogenic province of southeast Missouri, USA, there are several mines that contain not only economic iron resources, magnetite and/or hematite, but also contain rare earth elements, copper and gold. An area including three major deposits, Pea Ridge, Bourbon and Kratz Spring, was selected for detailed modelling for the upper crustal magnetic susceptibility and density structures. For the study area, ground gravity and high-resolution airborne magnetic and gravity gradiometry data sets are available. An efficient and novel joint inversion algorithm for the simultaneous inversion of these multiple data sets is presented. The Gramian coupling constraint is used to correlate the reconstructed density and magnetic susceptibility models. The implementation relies on the structures of the sensitivity matrices and an efficient minimization algorithm to achieve significant reductions in the memory requirements and computational costs. Consequently, it is feasible to use a laptop computer for the inversion of multiple data sets, each containing thousands of data points, for the recovery of models on the study area, each including approximately one million model parameters. This is the first time that these multiple data sets have been simultaneously inverted for this area. The L1-norm stabilizer is used to provide compact and focused images of the ore deposits. For contrast, independent inversions of each data set are also discussed. In general, our results provide new insights about the concealed ore deposits in the Mesoproterozoic basement rocks of southeast Missouri. Both short- and long-wavelength anomalies exist in the recovered models; these provide a high-resolution image of the subsurface. The geometry and physical properties of the known deposits are determined very well. Additionally, some unknown concealed deposits are revealed; these could be economically valuable and should be considered in future geophysical and geological investigations.

Detection, Location, and Classification of Multiple Dipole-like Magnetic Sources Based on L2 Norm of the Vertical Magnetic Gradient Tensor Data.

In recent years, there has been a growing interest in the detection, location, and classification (DLC) of multiple dipole-like magnetic sources based on magnetic gradient tensor (MGT) data. In these applications, the tilt angle is usually used to detect the number of sources. We found that the tilt angle is only suitable for the scenario where the positive and negative signs of the magnetic sources' inclination are the same. Therefore, we map the L2 norm of the vertical magnetic gradient tensor on the arctan function, denoted as the VMGT2 angle, to detect the number of sources. Then we use the normalized source strength (NSS) to narrow the parameters' search space and combine the differential evolution (DE) algorithm with the Levenberg-Marquardt (LM) algorithm to solve the sources' locations and magnetic moments. Simulation experiments and a field demonstration show that the VMGT2 angle is insensitive to the sign of inclination and more accurate in detecting the number of magnetic sources than the tilt angle. Meanwhile, our method can quickly locate and classify magnetic sources with high precision.

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.

Estimating Locations and Moments of Multiple Dipole-Like Magnetic Sources From Magnetic Gradient Tensor Data Using Differential Evolution

In this article, we present a new method for the detection and localization of multiple dipole-like magnetic sources using magnetic gradient tensor data. We use the tilt angle that is the ratio of the vertical magnetic component to the horizontal field to determine the number of the magnetic sources and use the rotational-invariant normalized source strength (NSS) to estimate approximate horizontal coordinates of the magnetic sources. The differential evolution (DE) algorithm is used to estimate the locations and moments of the magnetic sources. Three synthetic examples, with and without random noise, are designed to validate the proposed method. The locations and magnetic moments of the magnetic sources can be well determined. The proposed method has also been applied to a field test with multiple magnetic sources and the result shows that the location of the magnetic source can be estimated well.

A third-order magnetic gradient tensor optimization algorithm based on the second-order improved central difference method

An improved method based on the second order improved central difference method is proposed to improve the third-order magnetic gradient tensor, which is affected by noise and characterized by poor stability. The third-order magnetic gradient tensor at the center point is calculated by using the three-component magnetic field at four points along a single route; this approach effectively improves the stability of the third-order magnetic gradient tensor data. In this paper, the errors of the traditional method and the new method in calculating the third-order magnetic gradient tensor are given by the Taylor series. Based on the traditional third-order magnetic gradient tensor system, a new third-order magnetic gradient tensor system and the corresponding measurement method are presented. Through simulations, in the case of noise, it is verified that the proposed method exhibits better stability by comparing the components of the third-order magnetic gradient tensor with the positioning results. In experiments, the root-mean-square error is used to compare the positioning results before and after noise addition. The traditional method changes the root-mean-square error of the positioning results from (1.3626, 3.6573, 0.5766 m) to (3.3661, 9.5966, 1.9465 m), and the root-mean-square error obtained with the new method displays no obvious change.

Magnetic object positioning method based on tensor spacial invariant relations

We address the problem of extracting the position and the Euler angles of an underground object from measurements of the magnetic field above that object. The traditional Euler inversion positioning method requires us to make use of the high-order magnetic gradient tensor system and extract the second-order tensor data, but at the same time it will increase the sensitivity to errors and measurement noise, which reduces the positioning accuracy. To solve this problem, we propose a method that uses only first-order magnetic gradient tensor data and supplements the tensor spacial invariant relations to achieve a higher-precision positioning performance for magnetic object. We analyze the tensor invariants and the eigenvalues of the tensor matrix under a magnetic dipole source field, and derive two tensor spacial invariant relations: (1) the angle between magnetic moment and position vector is constant and can be represented by the tensor eigenvalues, and (2) the eigenvector of the absolute minimum eigenvalue is perpendicular to the magnetic moment and position vector, as well as the eigenvectors of remaining eigenvalues are coplanar to the magnetic moment and position vector. Hence, we compute a total four possible solutions of the position vector with respect to the four quadrants of a plane over the measuring center, and finally determine the true one based on the actual detection direction and measured magnetic field data. The results show that the proposed method has significantly higher detection accuracy and larger range under a same level noise condition than the Euler inversion positioning method. When positioning small-scale magnets (e.g. about 5 cm in diameter and 0.5 cm in thickness), we control the tri-axial coordinates accuracy with just the first-order magnetic gradient tensor system within 1 cm root mean square error.

Three-directional analytic signal analysis and interpretation of magnetic gradient tensor

Compared to conventional magnetic data, magnetic gradient tensor data contain more high-frequency signal components, which can better describe the features of geological bodies. The directional analytic signal of the magnetic gradient tensor is not easily interfered from the tilting magnetization, but it can infer the range of the field source more accurately. However, the analytic signal strength decays faster with depth, making it difficult to identify deep field sources. Balanced-boundary recognition can effectively overcome this disadvantage. We present here a balanced-boundary identification technique based on the normalization of three-directional analytic signals from aeromagnetic gradient tensor data. This method can effectively prevent the fast attenuation of analytic signals. We also derive an Euler inversion algorithm of three-directional analytic signal derivative. By combining magnetic-anomaly model testing with the traditional magnetic anomaly interpretation method, we show that the boundary-recognition technology based on a magnetic gradient tensor analytic signal has a greater advantage in identifying the boundaries of the geological body and can better reflect shallow anomalies. The characteristics of the Euler equation based on the magnetic anomaly direction to resolve the signal derivative have better convergence, and the obtained solution is more concentrated, which can obtain the depth and horizontal range information of the geological body more accurately. Applying the above method to the measured magnetic-anomaly gradient data from Baoding area, more accurate field source information is obtained, which shows the feasibility of applying this method to geological interpretations.

Magnetic Dipole Moment Determination Using Near-Field Magnetic Gradient Tensor Data

Determination of the magnetic dipole moment of a spacecraft is a key issue for magnetic cleanliness and attitude control of various space scientific spacecraft. Two main methods, namely the multiple magnetic dipole modeling method and near-field analysis, have been used with good results inside a coil facility for prelaunch magnetic testing of a spacecraft. When field-canceling coil sets are unavailable, or coil feedback controls are inadequate to compensate for time-varying geomagnetic fields, determination of the magnetic signature of a spacecraft from measurements of field components is compromised. Therefore, we propose a magnetic dipole moment determination method using magnetic gradient tensor data without any coil facility, which is based on the fact that the time variation of the gradient tensor of the geomagnetic field is very small compared with that of field. We improve the near-field analysis method by expanding the magnetic gradient tensor in a series of spherical harmonics, obtain sets of near-field equations using Fourier analysis, and solve these equations to determine the dipole-moment components. The proposed tensor method can effectively suppress the interference from time variations of ambient magnetic fields. Simulations and experiments have been conducted to prove the validity, robustness, and accuracy of the tensor method.

A level-set algorithm for the inverse problem of full magnetic gradient tensor data

We propose a level-set approach for recovering the susceptibility distribution from five-component full magnetic gradient tensor data. Given a value of the average susceptibility, the level-set inversion inverts for the susceptibility distribution so as to depict the shape of the underlying magnetic source. Numerical examples illustrate that the method is able to recover multiple magnetic sources from noisy data.

Determination of the Direction of Magnetization and Orientation of a Tilted Sheet From Downhole Magnetic Gradient Tensor Data

A tilted sheet model can usefully represent a dipping vein, dyke, sedimentary layer, or metasedimentary unit. Downhole magnetometer measurements provide proximal remote sensing of magnetic properties of magnetized sheets from a different perspective than ground or aeromagnetic surveys. However, a priori geometric information about a sheet is still required to constrain its magnetization, unless the measurements include data acquired in a borehole that penetrates the sheet. In particular, if only external measurements are available, knowledge of the dip of the sheet is required to constrain the direction of magnetization and an estimate of the thickness of the sheet is required to constrain the intensity of magnetization. This article describes a program for determination of both the direction of magnetization and geometry of a sheet from downhole magnetic gradient tensor data. We give expressions for the magnetic gradient tensor on the axis of a vertical borehole within a tilted sheet. We use an invariant of the tensor to determine the location and thickness of the sheet and tensor components to obtain the direction of magnetization and orientation of the sheet. An iterative procedure can improve the accuracy of the estimated parameters. We also discuss the robustness of the proposed program in the presence of variations in the diameter of the borehole, measurement offset from the center of the borehole, and nonuniform dip of the sheet. Experiments verify the proposed program and demonstrate the advantages of downhole magnetic gradient tensor measurements. The method can be extended to inclined boreholes by a straightforward rotation of coordinate axes.

Estimating the location of magnetic sources using magnetic gradient tensor data

In this paper, a new method for the interpretation of magnetic sources is developed using magnetic gradient tensor data. The total magnetisation direction of each magnetic source is estimated, and the magnetic gradient tensor is transformed into tensor data, which would be produced by the same sources with vertical magnetisation. Furthermore, the horizontal edges of the broad steep-sided magnetic sources are estimated from the ratio of the sum of absolute values of the eigenvalues (SAE) and Bzz of the transformed magnetic gradient tensor. The depth estimation of the magnetic sources is obtained using a modified Euler deconvolution method that eliminates the structural index from the equations. The process was tested on both synthetic and measured magnetic data. The results show that this method reduces considerably the effects of total magnetisation and is tolerant to the noise of the data. In addition, this method calculates vertical locations without any prior information about the structural index of the source.

Three-dimensional reconstruction of a small-scale magnetic target from magnetic gradient observations

Abstract A three-dimensional reconstruction method of a small-scale magnetic target is studied in this paper. Firstly, the tilt angle and local wavenumber are calculated from the measured magnetic gradient tensor data, leading to estimates of an approximate horizontal boundary and central depth of the magnetic target. From these, a spatial range of the inversion space and depth weighting function of the magnetic target are determined. Secondly, the inversion space is divided into cuboid cells, and the inversion equations for the magnetic parameters are established. They are regularized subject to the determined spatial range and depth weighting. Finally, a generalized cross validation method is used to obtain optimal regularization parameters, allowing the magnetization intensity of each cuboid to be estimated, and the three-dimensional profile of magnetic target to be delimited. Our simulation and experimental results show that the proposed method can realize, with a high degree of fidelity, the three-dimensional reconstruction of a simple small-scale magnetic target.

Preprocessed Method and Application of Magnetic Gradient Tensor Data

The edge detection method is significant in the interpretation of magnetic data, and can effectively identify the edge of the magnetic source. In this paper, a new edge detection method of magnetic sources is proposed under the oblique magnetization condition. First, the data of the magnetic field component is transformed into the total magnitude anomaly and its direction cosinesaccording to the inter-relationship between different magnetic field components. To eliminate noise, we use the K-Singular Value Decomposition denoising method based on the mean signal improvement to reconstruct the magnetic signal. Then, the three components of the magnetic field data is converted into the data of three components from the same magnetic source under the vertical magnetization condition by using the calculated magnetization direction of the magnetic source. Finally, the edge of the magnetic source is calculated through the ratio of the magnetic gradient tensor invariant to the B <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">zz</sub> component data under the vertical magnetization condition. This method is verified by simulation and experiment. The simulation and experimental results obtained show that this method considerably reduces the effects of the oblique magnetization. Besides, the method in this paper has high noise immunity.

Three‐dimensional inversion of full magnetic gradient tensor data based on hybrid regularization method

ABSTRACTRapid developments in SQUID‐based technology make it possible for geophysical exploration to direct measure, inverse and interpret magnetic gradient tensor data. This contribution introduces a novel three‐dimensional hybrid regularization method for inversion of magnetic gradient tensor data, which is based on the minimum support functional and total variation functional. Compared to the existing stabilizers, for example, the minimum support stabilizer, the minimum gradient support stabilizer or the total variation stabilizer, our proposed hybrid stabilizer, in association with boundary penalization, improves the revision result greatly, including higher spatial and depth resolution, more clear boundaries, more highlighted images and more evident structure depiction. Moreover, suitable selection of model parameter λ will further improve the image quality of the recovered model. We verify our proposed hybrid method with various synthetic magnetic models. Experiment results prove that this method gives more accurate results, exhibiting advantages of less computational costs even when less prior information of magnetic sources are provided. Comparison of results with different types of magnetic data with and without remanence indicates that our inversion algorithm can obtain more detailed information on the source structure based on rational estimation of total magnetization direction. Finally, we present a case study for inverting SQUID‐based magnetic tensor data acquired at Da Hinggan Mountains area, inner Mongolia, China. The result also certifies that the method is reliable and efficient for real cases.

Full Tensor Eigenvector Analysis on Air-Borne Magnetic Gradiometer Data for the Detection of Dipole-Like Magnetic Sources.

The detection of dipole-like sources, such as unexploded ordnances (UXO) and other metallic objects, based on a magnetic gradiometer system, has been increasingly applied in recent years. In this paper, a novel dipole-like source detection algorithm, based on eigenvector analysis with magnetic gradient tensor data interpretation is presented. Firstly, the theoretical basis of the eigenvector decomposition of magnetic gradient tensor is analyzed. Then, a detection algorithm is proposed by using the properties of the tensor eigenvector decomposition to locate dipole-like magnetic sources. The algorithm can automatically detect magnetic dipole-like sources without estimating the magnetic moment direction. It performs well for locating weak, anomalous dipole-like sources in air-borne magnetic data through quantitative interpretation. The effectiveness of the proposed algorithm has been demonstrated in the designed synthetic experiment. Finally, an air-borne magnetic field data taken at high altitude with exact source position information is used to validate the practicality of the proposed algorithm. All of the experiments prove that the proposed algorithm is suitable for magnetic dipole-like source detecting and air-borne magnetic gradiometer data interpretation.

Calculation of the magnetic gradient tensor from total magnetic anomaly field based on regularized method in frequency domain

To obtain accurate magnetic gradient tensor data, a fast and robust calculation method based on regularized method in frequency domain was proposed. Using the potential field theory, the transform formula in frequency domain was deduced in order to calculate the magnetic gradient tensor from the pre-existing total magnetic anomaly data. By analyzing the filter characteristics of the Vertical vector transform operator (VVTO) and Gradient tensor transform operator (GTTO), we proved that the conventional transform process was unstable which would zoom in the high-frequency part of the data in which measuring noise locate. Due to the existing unstable problem that led to a low signal-to-noise (SNR) for the calculated result, we introduced regularized method in this paper. By selecting the optimum regularization parameters of different transform phases using the C-norm approach, the high frequency noise was restrained and the SNR was improved effectively. Numerical analysis demonstrates that most value and characteristics of the calculated data by the proposed method compare favorably with reference magnetic gradient tensor data. In addition, calculated magnetic gradient tensor components form real aeromagnetic survey provided better resolution of the magnetic sources and original profile.

Detection, localization and classification of multiple dipole-like magnetic sources using magnetic gradient tensor data

We have developed a method for automatic detection, localization and classification (DLC) of multiple dipole sources using magnetic gradient tensor data. First, we define modified tilt angles to estimate the approximate horizontal locations of the multiple dipole-like magnetic sources simultaneously and detect the number of magnetic sources using a fixed threshold. Secondly, based on the isotropy of the normalized source strength (NSS) response of a dipole, we obtain accurate horizontal locations of the dipoles. Then the vertical locations are calculated using magnitude magnetic transforms of magnetic gradient tensor data. Finally, we invert for the magnetic moments of the sources using the measured magnetic gradient tensor data and forward model. Synthetic and field data sets demonstrate effectiveness and practicality of the proposed method.

Corrigendum to: New methods for interpretation of magnetic vector and gradient tensor data I: eigenvector analysis and the normalised source strength

Acquisition of magnetic gradient tensor data is likely to become routine in the near future. New methods for inverting gradient tensor surveys to obtain source parameters have been developed for several elementary, but useful, models. These include point dipole (sphere), vertical line of dipoles (narrow vertical pipe), line of dipoles (horizontal cylinder), thin dipping sheet, and contact models. A key simplification is the use of eigenvalues and associated eigenvectors of the tensor. The normalised source strength (NSS), calculated from the eigenvalues, is a particularly useful rotational invariant that peaks directly over 3D compact sources, 2D compact sources, thin sheets and contacts, and is independent of magnetisation direction. In combination the NSS and its vector gradient determine source locations uniquely. NSS analysis can be extended to other useful models, such as vertical pipes, by calculating eigenvalues of the vertical derivative of the gradient tensor. Inversion based on the vector gradient of the NSS over the Tallawang magnetite deposit obtained good agreement between the inferred geometry of the tabular magnetite skarn body and drill hole intersections. Besides the geological applications, the algorithms for the dipole model are readily applicable to the detection, location and characterisation (DLC) of magnetic objects, such as naval mines, unexploded ordnance, shipwrecks, archaeological artefacts, and buried drums.

Kalman Filter Processing Applications for Motion Noise

In geophysics exploration, using gradient tensor instead of the full magnetic field gradient has many advantages, which magnetic gradient tensor data to better describe small anomalies. However, the measurement of magnetic gradiometer contains a very complex motion noise, separating the motion noise from the signal component is a large challenge. In this paper, we show the expression for the magnetic gradient tensor, and then through model tests proved the Kalman filter good filtering effect.

Magnetic dipole localization based on magnetic gradient tensor data at a single point

Magnetic dipole localization methods that rely on measurement of the magnetic field vector are compromised by the relatively strong background geomagnetic field. A localization method that uses only magnetic gradient tensor data is proposed. The localization equations are established by transforming Euler’s equation of degree −3 into degree −4 and using the orthogonality of the intermediate eigenvector of the magnetic gradient tensor that is produced by a magnetic dipole and the source-sensor displacement vector. To measure the quantities required in the localization equations, we designed a magnetic gradient tensor system in which finite differences are used to approximate the first- and second-order spatial gradients of magnetic field components. Numerical simulations show that the proposed method can accurately and uniquely solve for the location of a magnetic dipole in the presence of the geomagnetic field, and the experimental results show the superiority and the practicability of the proposed method.