• Geophysics and Geophysical Exploration
• /
• v.23 no.1
• /
• pp.55-60
• /
• 2020

The closed-form expressions of gravity, magnetic, gravity gradient tensor, and magnetic gradient tensor due to a rectangular prism are derived. The vertical gravity is derived via triple integration of a rectangular prism in Cartesian coordinates, and the two horizontal components of vector gravity are then derived via cycle permutation of the axis variables of vertical gravity through the axial symmetry of the rectangular prism. The gravity gradient tensor is obtained by differentiating the vector gravity with respect to each coordinate. Using Poisson's relation, a vector magnetic field with constant magnetic direction can be obtained from the gravity gradient tensor. Finally, the magnetic gradient tensor is derived by differentiating the vector magnetic with respect to appropriate coordinates.

• Journal of the Korean earth science society
• /
• v.32 no.6
• /
• pp.595-601
• /
• 2011

In this paper, I propose the algorithm that the location of a magnetic dipole can be detected from the magnetic gradient tensor. I induce the location vector of a vertically magnetizated dipole from the magnetic gradient tensor. Deficit of magnetic moment of magnetic dipole makes the induced location information incomplete. However, if the observation of magnetic gradient tensor would be collected on more points, the algorithm is able to catch the location of the magnetic dipole by clustering the solution of the proposed algorithm. For example, I show that the synthetic case of borehole observation of magnetic gradient tensor can find the source location successively by picking common solution area.

• Geophysics and Geophysical Exploration
• /
• v.25 no.1
• /
• pp.38-43
• /
• 2022

In case axial symmetrical bodies with varying cross sections such as volcanic conduits and unexploded ordnance (UXO), it is efficient to approximate them by adding the response of thin disks perpendicular to the axis of symmetry. To compute the vector magnetic and magnetic gradient tensor respones by such bodies, it is necessary to derive an analytical expression of the circular disk. Therefore, in this study, we drive closed-form expressions of the vector magnetic and magnetic gradient tensor due to a circular disk. First, the vector magnetic field is obtained from the existing gravity gradient tensor using Poisson's relation where the gravity gradient tensor due to the same disk with a constant density can be transformed into a magnetic field. Then, the magnetic gradient tensor is derived by differentiating the vector magnetic field with respect to the cylindrical coordinates converted from the Cartesian coordinate system. Finally, both the vector magnetic and magnetic gradient tensors are derived using Lipschitz-Hankel type integrals based on the axial symmetry of the circular disk.

• Geophysics and Geophysical Exploration
• /
• v.25 no.2
• /
• pp.85-92
• /
• 2022

An elongated object in one direction can be approximated as a line segment. Here, the closed-form expressions of a line segment's vector magnetic and magnetic gradient tensor are required to interpret responses by a line segment. Therefore, the analytical expressions of the vector magnetic and magnetic gradient tensor are derived. The vector magnetic is converted from the existing gravity gradient tensor using Poisson's relation where the gravity gradient tensor caused by a line segment can be transformed into a vector magnetic. Then, the magnetic gradient tensor is derived by differentiating the vector magnetic with respect to each axis in the Cartesian coordinate system. The synthetic total magnetic data simulated by an iron pile on boreholes are inverted by a nonlinear inversion process so that the physical parameters of the iron pile, including the beginning point, the length, orientation, and magnetization vector are successfully estimated.

• Geophysics and Geophysical Exploration
• /
• v.23 no.2
• /
• pp.67-71
• /
• 2020

In this study, we derive closed-form expressions of magnetic gradient tensor due to a circular cylinder. Because the expression for magnetic field has been derived in a previously conducted study, expressions are developed for the magnetic gradient tensor based on the derivatives of the expressions of magnetic field with respect to the variables of the Cartesian coordinates. Furthermore, expressions are derived for the magnetic gradient tensor based on the relations between the Cartesian and cylindrical coordinates in the derivative because the expression for magnetic field contains variables of cylindrical coordinates owing to its axial symmetry.

• Geophysics and Geophysical Exploration
• /
• v.26 no.2
• /
• pp.77-83
• /
• 2023

In this study, the expressions of magnetic vector and magnetic gradient tensor due to an elliptical cylinder were derived. Igneous intrusions and kimberlite structures are often shaped like elliptical cylinders with axial symmetry and different radii in the strike and perpendicular directions. The expressions of magnetic fields due to this elliptical cylinder were derived from the Poisson relation, which includes the direction of magnetization in the gravity gradient tensor. The magnetic gradient tensor due to an elliptical cylinder is derived by differentiating the magnetic fields. This method involves obtaining a total of 10 triple derivative functions acquired by differentiating the gravitational potential of the elliptical cylinder three times in each axis direction. As the order of differentiation and integration can be exchanged, the magnetic gradient tensor was derived by differentiating the gravitational potential of the elliptical cylinder three times in each direction, followed by integration in the depth direction. The remaining double integration was converted to a complex line integral along the closed boundary curve of the elliptical cylinder in the complex plane. The expressions of the magnetic field and magnetic gradient tensor derived from the complex line integral in the complex plane were shown to be perfectly consistent with those of the circular cylinder derived by the Lipschitz-Hankel integral.

• Geophysics and Geophysical Exploration
• /
• v.27 no.2
• /
• pp.108-118
• /
• 2024

In this study, expressions for the magnetic field and magnetic gradient tensor due to an elliptical disk were derived. Igneous intrusions and kimberlite structures often have elliptical cylinders with axial symmetry and elliptical cross sections. An elliptical cylinder with varying cross-sectional areas was approximated using stacks of elliptical disks. The magnetic fields of elliptical disks were derived using the Poisson relation, which includes the direction of magnetization in the gravity gradient tensor, as described in a previous study (Rim, 2024). The magnetic gradient tensor due to an elliptical disk is derived by differentiating the magnetic fields, which is equivalent to obtaining ten triple-derivative functions acquired by differentiating the gravitational potential of the elliptical disk three times in each axis direction. Because it is possible to exchange the order of differentiation, the magnetic gradient tensor is derived by differentiating the gravitational potential of the elliptical disk three times, which is then converted into a complex line integral along the closed boundary curve of the elliptical disk in the complex plane. The expressions for the magnetic field and magnetic gradient tensor derived from a complex line integral in complex plane are perfectly consistent with those of the circular disk derived from the Lipschitz-Hankel integral.

• Geophysics and Geophysical Exploration
• /
• v.23 no.1
• /
• pp.50-54
• /
• 2020

Herein, the closed-form expressions of the magnetic field due to an axially symmetric body such as a right cylinder, are derived. The magnetic field due to a right cylinder is converted from the gravity gradient tensor using Poisson's relation; the magnetic field induced by a constant magnetization can be obtained from the gravity gradient tensor with a constant density. Because of the axial symmetry of the cylinder, the expressions of gravity gradient tensor are derived in cylindrical coordinate and then transformed into Cartesian coordinates for the three components of the magnetic field using an arbitrary magnetization direction.

• Journal of the Korean Magnetic Resonance Society
• /
• v.7 no.1
• /
• pp.16-24
• /
• 2003

Nuclei with the spin quantum number not smaller than unity have not only the nuclear magnetic moment but also the electric quadrupole moment. The quadrupole moment couples with the electric field gradient (EFG) to produce the nuclear quadrupole interaction. It is well known that two independent parameters, i.e. the quadrupole coupling constant (QCC) and the asymmetry parameter ($\eta$) together with the principal axis directions can fully describe the interaction and are very sensitive to the local symmetry and structure of the solid. In order to obtain quantitative estimates of the EFG tensor for various simple ionic configurations surrounding the nucleus under consideration, we employ the simple point charge approximation and apply the calculated results to some real crystals. General agreement is rather satisfactory.

• Journal of Biomedical Engineering Research
• /
• v.28 no.6
• /
• pp.817-824
• /
• 2007

Tractography using Diffusion Tensor Magnetic Resonance Imaging (DT-MRI) is a method to determine the architecture of axonal fibers in the central nervous system by computing the direction of the principal eigenvector in the white matter of the brain. However, the fiber tracking methods suffer from the noise included in the diffusion tensor images that affects the determination of the principal eigenvector. As the fiber tracking progresses, the accumulated error creates a large deviation between the calculated fiber and the real fiber. This problem of the DT-MRI tractography is known mathematically as the ill-posed problem which means that tractography is very sensitive to perturbations by noise. To reduce the noise in DT-MRI measurements, a tensor-valued median filter which is reported to be denoising and structure-preserving in fiber tracking, is applied in the tractography. In this paper, we proposed the modified gradient descent method which converges fast and accurately to the optimal tensor-valued median filter by changing the step size. In addition, the performance of the modified gradient descent method is compared with others. We used the synthetic image which consists of 45 degree principal eigenvectors and the corticospinal tract. For the synthetic image, the proposed method achieved 4.66%, 16.66% and 15.08% less error than the conventional gradient descent method for error measures AE, AAE, AFA respectively. For the corticospinal tract, at iteration number ten the proposed method achieved 3.78%, 25.71 % and 11.54% less error than the conventional gradient descent method for error measures AE, AAE, AFA respectively.