Unification of Euler and Werner deconvolution in three dimensions via the generalized Hilbert transform

Geophysics ◽  
2001 ◽  
Vol 66 (6) ◽  
pp. 1805-1810 ◽  
Author(s):  
Misac N. Nabighian ◽  
R. O. Hansen
Keyword(s):  
Magnetic Dipole ◽  
Hilbert Transform ◽  
Natural Generalization ◽  
Three Dimensions ◽  
Explicit Calculation ◽  
Euler Deconvolution ◽  
Hilbert Transforms ◽  
Vertical Magnetic Dipole ◽  
Generalized Hilbert Transform ◽  
Practical Level

The extended Euler deconvolution algorithm is shown to be a generalization and unification of 2‐D Euler deconvolution and Werner deconvolution. After recasting the extended Euler algorithm in a way that suggests a natural generalization to three dimensions, we show that the 3‐D extension can be realized using generalized Hilbert transforms. The resulting algorithm is both a generalization of extended Euler deconvolution to three dimensions and a 3‐D extension of Werner deconvolution. At a practical level, the new algorithm helps stabilize the Euler algorithm by providing at each point three equations rather than one. We illustrate the algorithm by explicit calculation for the potential of a vertical magnetic dipole.

Toward a three‐dimensional automatic interpretation of potential field data via generalized Hilbert transforms: Fundamental relations

Geophysics ◽  
1984 ◽  
Vol 49 (6) ◽  
pp. 780-786 ◽  
Author(s):  
Misac N. Nabighian
Keyword(s):  
Hilbert Transform ◽  
Potential Field ◽  
Field Data ◽  
Three Dimensional ◽  
Analytic Signal ◽  
Three Dimensions ◽  
Hilbert Transforms ◽  
Potential Field Data ◽  
Automatic Interpretation ◽  
The Hilbert Transform

The paper extends to three dimensions (3-D) the two‐dimensional (2-D) Hilbert transform relations between potential field components. For the 3-D case, it is shown that the Hilbert transform is composed of two parts, with one part acting on the X component and one part on the Y component. As for the previously developed 2-D case, it is shown that in 3-D the vertical and horizontal derivatives are the Hilbert transforms of each other. The 2-D Cauchy‐Riemann relations between a potential function and its Hilbert transform are generalized for the 3-D case. Finally, the previously developed concept of analytic signal in 2-D can be extended to 3-D as a first step toward the development of an automatic interpretation technique for potential field data.

Electroseismic waves excited by vertical magnetic dipole in borenole

2011 ◽  
Vol 57 (5) ◽  
pp. 610-614 ◽  
Author(s):  
Zhiwen Cui ◽  
Jinxia Liu ◽  
Guijin Yao ◽  
Kexie Wang
Keyword(s):  
Magnetic Dipole ◽  
Vertical Magnetic Dipole

scholarly journals Quasi-Static Solution for ELF/SLF near-Zone Field of a Vertical Magnetic Dipole near the Surface of a Lossy Half-Space

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Lina He ◽  
Tong He ◽  
Kai Li
Keyword(s):  
Magnetic Dipole ◽  
Numerical Solutions ◽  
Low Frequency ◽  
Analytical Techniques ◽  
Static Solution ◽  
Static Approximation ◽  
Near Zone ◽  
Vertical Magnetic Dipole ◽  
Validity Limit ◽  
Zone Field

Dipole antennas over the boundary between two different media have been widely used in the fields of geophysics exploration, oceanography, and submerged communication. In this paper, an analytical method is proposed to analyse the near-zone field at the extremely low frequency (ELF)/super low frequency (SLF) range due to a vertical magnetic dipole (VMD). For the lack of feasible analytical techniques to derive the components exactly, two reasonable assumptions are introduced depending on the quasi-static definition and the equivalent infinitesimal theory. Final expressions of the electromagnetic field components are in terms of exponential functions. By comparisons with direct numerical solutions and exact results in a special case, the correctness and effectiveness of the proposed quasi-static approximation are demonstrated. Simulations show that the smallest validity limit always occurs for component H2z, and the value of k2ρ should be no greater than 0.6 in order to keep a good consistency.

Response above a plane-conducting earth to a pulsed vertical magnetic dipole at the surface

2000 ◽  
Vol 78 (9) ◽  
pp. 833-844 ◽  
Author(s):  
O M Abo-Seida ◽  
S T Bishay
Keyword(s):  
Wave Equation ◽  
Magnetic Dipole ◽  
Fourier Transformation ◽  
Laplace Transformation ◽  
Theoretical Study ◽  
Initial Boundary ◽  
Frequency Space ◽  
Hertz Vector ◽  
Vertical Magnetic Dipole ◽  
Pulsed Electromagnetic

A theoretical study of the pulsed electromagnetic radiation from a vertical magnetic dipole placed on a plane-conducting earth is presented. The application of a Laplace transformation in time and a Fourier transformation in the two orthogonal, horizontal, spatial components leads, under consideration of initial, boundary, and transition conditions, to an integral representation of the solution of the wave equation in frequency space. A modified Cagniard method is then used to derive closed-form expressions for the magnetic Hertz vector anywhere above the conducting earth. The method is used to perform numeric calculations of the magnetic Hertz vector, for different source-receiver distances, as well as different values of the earth's conductivity and permittivity. PACS Nos.: 41.20Jb, 42.25Bs, 42.25Gy, 44.05+e

Theory of 2-D complex seismic trace analysis

Geophysics ◽  
1996 ◽  
Vol 61 (1) ◽  
pp. 264-272 ◽  
Author(s):  
Arthur E. Barnes
Keyword(s):  
Phase Velocity ◽  
Group Velocity ◽  
Hilbert Transform ◽  
Instantaneous Frequency ◽  
Trace Analysis ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Instantaneous Amplitude ◽  
Dip Angle

The ideas of 1-D complex seismic trace analysis extend readily to two dimensions. Two‐dimensional instantaneous amplitude and phase are scalars, and 2-D instantaneous frequency and bandwidth are vectors perpendicular to local wavefronts, each defined by a magnitude and a dip angle. The two independent measures of instantaneous dip correspond to instantaneous apparent phase velocity and group velocity. Instantaneous phase dips are aliased for steep reflection dips following the same rule that governs the aliasing of 2-D sinusoids in f-k space. Two‐dimensional frequency and bandwidth are appropriate for migrated data, whereas 1-D frequency and bandwidth are appropriate for unmigrated data. The 2-D Hilbert transform and 2-D complex trace attributes can be efficiently computed with little more effort than their 1-D counterparts. In three dimensions, amplitude and phase remain scalars, but frequency and bandwidth are 3-D vectors with magnitude, dip angle, and azimuth.

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