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

• Geophysics and Geophysical Exploration
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• v.25 no.1
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• pp.44-49
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• 2022

Closed-form expressions of vector gravity and gravity gradient tensor based on a line segment are derived. If a cylindrical object with axial symmetry is observed from a distance, it is possible to approximate it as a line segment; therefore, it is necessary to compute the gravity and the gravity gradient tensor due to a line source by using closed-form expressions. The gravitational potential for a line segment is defined as a one-dimensional integral, and this integral is differentiated with respect to the Cartesian coordinate system to derive the vector gravity. The expressions of the gravity gradient tensor are derived by differentiating the vector gravity once more in the same coordinate system.

• Geophysics and Geophysical Exploration
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• v.24 no.1
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• pp.1-5
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• 2021

The closed-form expressions of the vector gravity and gravity gradient tensor due to a circular disk are derived. The gravity potential due to a circular disk with a constant density is defined for a cylindrical system. Then, the vector gravity is derived by differentiating the gravity potential with respect to cylindrical coordinates. The radial component of the vector gravity in the cylindrical system is converted into horizontal gravity components in the Cartesian system. Finally, the gravity gradient tensor due to a circular disk is obtained by differentiating the vector gravity with respect to the Cartesian coordinates.

• Journal of the Korean earth science society
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• v.35 no.7
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• pp.529-536
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• 2014

In this paper, the automatic determination algorithm of strike and dip of a line source using gravity gradient on a single profile is proposed. In general, the gravity gradient tensor due to a line source has only two independent components because of its 2-Dimensional (2-D) characteristics. However, if the line source has the strike and dip regarding the observation profile, it comes to have five independent components. The proposed algorithm of the determination both strike and dip is based on the rotational transform that converts full gravity gradient tensor to reduced 2-D gravity gradient tensor. The least-square method is applied in order to find optimum rotational angles that make one of the row components minimalized simultaneously. The two synthetic cases of a line source are represented; one has strike only and the other has both strike and dip. This study finds that the automatic determination method using gravity gradient tensor can find directions of a line source in each case.

• Geophysics and Geophysical Exploration
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• v.26 no.1
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• pp.1-7
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• 2023

This study derives the expressions of vector gravity and gravity gradient tensor due to an elliptical cylinder. The vector gravity for an arbitrary three-dimensional (3D) body is obtained by differentiating the gravitational potential, including the triple integral, according to the shape of the body in each axis direction. The vector gravity of the 3D body with axial symmetry is integrated along the axial direction and reduced to a double integral. The complex Green's theorem using complex conjugates subsequently converts the double integral into a one-dimensional (1D) closed-line integral. Finally, the vector gravity due to the elliptical cylinder is derived using 1D numerical integration by parameterizing a boundary of the elliptical cross-section as a closed line. Similarly, the gravity gradient tensor due to the elliptical cylinder is second-order differentiated from the gravitational potential, including the triple integral, and integrated along the vertical axis direction reducing it to a double integral. Consequently, all the components of the gravity gradient tensor due to an elliptical cylinder are derived using complex Green's theorem as used in the case of vector gravity.

• Geophysics and Geophysical Exploration
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• v.27 no.1
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• pp.51-56
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• 2024

In this paper, the vector gravity and gravity gradient tensor of an elliptical disk are derived. The vector gravity of an elliptical disk is defined by differentiating the gravitational potential due to the elliptical disk expressed by a double integral with respect to each axial direction. The vector gravity defined by the double integral is then transformed into a line integral of a closed curve along the elliptical disk boundary using the complex Green's theorem. Finally, vector gravity due to the elliptical disk is derived by 1D parametric numerical integration along the elliptical disk boundary. The xz, yz, zz components of the gravity gradient tensor due to the elliptical disk are obtained by differentiating the vector gravity with respect to vertical direction. The xx, yy, xy components are derived by differentiating the horizontal components of the vector gravity in the form of a double integral with respect to horizontal directions and then using the complex Green's theorem.

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

• Geophysics and Geophysical Exploration
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• v.25 no.1
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• pp.38-43
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• 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
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• v.25 no.2
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• pp.85-92
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• 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.

• Journal of the Korean earth science society
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• v.38 no.4
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• pp.263-268
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• 2017

The determination algorithm of the location of a line source with strike and dip using the gravity gradient tensor on a single profile is proposed. We already proposed the determination of strike and dip in the previous paper and then, now we improved the algorithm to locate a line source after determining strike and dip. The strike and dip of the line source can be determined by rotating the gravity gradient tensor matrix as reducing 2 independent components. Using the ratio of remaining 2 components, the location can be determined by the least square manner of the pointing vectors on each observation point. A synthetic model is tested for proving the usefulness of the proposed algorithm.