The electromagnetic response of an azimuthally anisotropic half‐space

Geophysics ◽  
1991 ◽  
Vol 56 (9) ◽  
pp. 1462-1473 ◽  
Author(s):  
Xiaobo Li ◽  
Laust B. Pedersen
Keyword(s):  
Half Space ◽  
Transfer Functions ◽  
Hankel Transform ◽  
Electromagnetic Response ◽  
Dipole Source ◽  
Impedance Tensor ◽  
Fracture Zones ◽  
Transform Method ◽  
Homogeneous Half Space ◽  
Horizontal Electric Dipole

Fracture zones are pervasive in crystalline areas. When the earth is seen over a sufficiently large volume fracture zones may be too thin to be individually identified. If they have preferred directions in that volume, the volume can be considered to be an azimuthally anisotropic medium. We have formulated the electromagnetic fields induced by a horizontal electric dipole on the surface of a homogeneous half‐space with azimuthally anisotropic conductivity. The field components are expressed by the two‐dimensional Fourier transform which can be computed by a fast Hankel transform method. The impedance tensor and tipper functions of controlled source tensor magnetotellurics are derived by exciting the dipole source in two different directions. We show the behavior of impedance tensor, tipper functions and their derived quantities: rotational invariants and Parkinson vectors. All transfer functions clearly show anisotropic characteristics. Contours of rotationally invariant apparent resistivities and phases for fixed frequencies are elongated in the direction of maximum conductivity, and Parkinson’s vectors tend to point in the same direction.

Controlled source tensor magnetotellurics

Geophysics ◽  
1991 ◽  
Vol 56 (9) ◽  
pp. 1456-1461 ◽  
Author(s):  
Xiaobo Li ◽  
Laust B. Pedersen
Keyword(s):  
Half Space ◽  
Transfer Functions ◽  
Unit Vector ◽  
Impedance Tensor ◽  
Wave Excitation ◽  
Controlled Source ◽  
Practice Analysis ◽  
Homogeneous Half Space ◽  
Dipole Sources ◽  
Layered Half Space

Impedance tensor and tipper vectors, known to connect the electromagnetic surface components for plane‐wave excitation, are shown to be uniquely defined for horizontal electric or horizontal magnetic dipole sources. Two independent source polarizations are needed for their estimation in practice. Analysis of impedance tensors and tipper vectors for a layered half‐space shows that the impedance tensor can be antidiagonalized by rotating the measurement system so that one of the measurement directions coincides with the direction to the transmitter dipole. The tipper vector points towards the transmitter dipole. In the static limit, all transfer functions are real, and simple analytic results for a homogeneous half‐space show that impedance elements are proportional to the inverse of the product of conductivity and distance between source and receiver, while the tipper vector is a unit vector pointing towards the transmitter dipole.

Electromagnetic induction by a finite electric dipole source over a 2-D earth

Geophysics ◽  
1993 ◽  
Vol 58 (2) ◽  
pp. 198-214 ◽  
Author(s):  
Martyn J. Unsworth ◽  
Bryan J. Travis ◽  
Alan D. Chave
Keyword(s):  
Finite Element ◽  
Electromagnetic Fields ◽  
Electric Dipole ◽  
Numerical Solutions ◽  
Current Source ◽  
Three Dimensional ◽  
Iterative Solution ◽  
Electromagnetic Response ◽  
Dipole Source ◽  
Horizontal Electric Dipole

A numerical solution for the frequency domain electromagnetic response of a two‐dimensional (2-D) conductivity structure to excitation by a three‐dimensional (3-D) current source has been developed. The fields are Fourier transformed in the invariant conductivity direction and then expressed in a variational form. At each of a set of discrete spatial wavenumbers a finite‐element method is used to obtain a solution for the secondary electromagnetic fields. The finite element uses exponential elements to efficiently model the fields in the far‐field. In combination with an iterative solution for the along‐strike electromagnetic fields, this produces a considerable reduction in computation costs. The numerical solutions for a horizontal electric dipole are computed and shown to agree with closed form expressions and to converge with respect to the parameterization. Finally some simple examples of the electromagnetic fields produced by horizontal electric dipole sources at both the seafloor and air‐earth interface are presented to illustrate the usefulness of the code.

Computation of Green’s tensor integrals for three‐dimensional electromagnetic problems using fast Hankel transforms

Geophysics ◽  
1984 ◽  
Vol 49 (10) ◽  
pp. 1754-1759 ◽  
Author(s):  
Walter L. Anderson
Keyword(s):  
Half Space ◽  
Three Dimensional ◽  
Hankel Transform ◽  
Electromagnetic Modeling ◽  
Electromagnetic Problems ◽  
Hankel Transforms ◽  
Homogeneous Half Space ◽  
Green's Tensor ◽  
Green’S Tensor ◽  
The Many

A new method is presented that rapidly evaluates the many Green’s tensor integrals encountered in three‐dimensional electromagnetic modeling using an integral equation. Application of a fast Hankel transform (FHT) algorithm (Anderson, 1982) is the basis for the new solution, where efficient and accurate computation of Hankel transforms are obtained by related and lagged convolutions (linear digital filtering). The FHT algorithm is briefly reviewed and compared to earlier convolution algorithms written by the author. The homogeneous and layered half‐space cases for the Green’s tensor integrals are presented in a form so that the FHT can be easily applied in practice. Computer timing runs comparing the FHT to conventional direct convolution methods are discussed, where the FHT’s performance was about 6 times faster for a homogeneous half‐space, and about 108 times faster for a five‐layer half‐space. Subsequent interpolation after the FHT is called is required to compute specific values of the tensor integrals at selected transform arguments; however, due to the relatively small lagged convolution interval used (same as the digital filter’s), a simple and fast interpolation is sufficient (e.g., by cubic splines).

Three-dimensional Land FD-CSEM Regularized Inversion Based on Edge Finite-element Method

2018 ◽  
Vol 23 (2) ◽  
pp. 211-222
Author(s):  
Jianxin Liu ◽  
Pengmao Liu ◽  
Xiaozhong Tong
Keyword(s):  
Finite Element ◽  
Electric Fields ◽  
Three Dimensional ◽  
Dipole Source ◽  
Unique Problem ◽  
Three Dimensional Modeling ◽  
Homogeneous Half Space ◽  
Electromagnetic Responses ◽  
Horizontal Electric Dipole ◽  
The Galerkin Method

There is a desire to obtain rapid and stable inversion results and clearly reconstruct subsurface resistivity structure in frequency domain (FD) electromagnetics. Three-dimensional modeling of land FD controlled-source electromagnetic (CSEM) data is vital to improve the understanding of electromagnetic responses collected in increasingly complex geologic settings. Three-dimensional inversion of land FD-CSEM data is a mathematically non-unique problem with instability, due to the noise contained in the data and its inherent incompleteness. The main difference between our method and those from previous work is that the edge finite-element approach is applied to solve the three-dimensional FD-CSEM generated by a horizontal electric dipole source. Firstly, we formulate the edge finite-element equation through the Galerkin method, based on the Helmholtz equation of the electric fields. Secondly, in order to check the validity of the modeling code, we compare the numerical results with the analytical solutions for a homogeneous half-space model. For further tests, we calculate the electromagnetic responses for another two models with more practical structures. Finally, the three-dimensional inversion is carried out based on a regularization method with smoothness-constraints to obtain stable solutions.

Laser-Induced Heating of a Multilayered Medium Resting on a Half-Space: Part I—Stationary Source

1988 ◽  
Vol 55 (1) ◽  
pp. 93-97 ◽  
Author(s):  
R. Kant
Keyword(s):  
Laplace Transform ◽  
Half Space ◽  
Hankel Transform ◽  
Optical Recording ◽  
Time Intervals ◽  
Homogeneous Half Space ◽  
Multilayered Medium ◽  
Rule Application ◽  
The Laplace Transform ◽  
The Time Domain

Laser induced heating of a multilayered medium resting on a homogeneous half-space is considered. The transient heat transfer equation is solved by employing the Laplace transform in the time domain and the Hankel transform in the space domain (r direction). Numerical inversion of the Laplace transform is obtained by using a technique developed by Crump. For the time intervals of interest, inversion of the Hankel transform is obtained by the Simpson rule. Application to magneto-optical recording is discussed.

scholarly journals Axisymmetric stationary heat conduction problem for half-space with temperature-dependent properties

2020 ◽  
Vol 24 (3 Part B) ◽  
pp. 2137-2150
Author(s):  
Dariusz Perkowski ◽  
Piotr Sebestianiuk ◽  
Jakub Augustyniak
Keyword(s):  
Heat Conduction ◽  
Half Space ◽  
Heat Conduction Problem ◽  
Hankel Transform ◽  
Heating Zone ◽  
Temperature Dependent ◽  
Transform Method ◽  
Special Cases ◽  
Temperature Dependent Properties ◽  
Method Model

The study examines problems of heat conduction in a half-space with a thermal conductivity coefficient that is dependent on temperature. A boundary plane is heated locally in a circle zone at a given temperature as a function of radius. A solution is obtained for any function that describes temperature in the heating zone. Two special cases are investigated in detail, namely Case 1 with given constant temperature in the circle zone and Case 2 with temperature given as a function of radius, r. The temperature of the boundary on the exterior of the heating zone is assumed as zero. The Hankel transform method is applied to obtain a solution for the formulated problem. The effect of thermal properties on temperature distributions in the considered body is investigated. The obtained results were compared with finite element method model.

Electromagnetic Response for Horizontal Electric Dipole Source Considering Conduct & Displacement Current

2014 ◽  
Vol 513-517 ◽  
pp. 3340-3344
Author(s):  
Jia Bin Yan ◽  
Xiang Yu Huang ◽  
Peng Yu Wu
Keyword(s):  
Electric Dipole ◽  
Electromagnetic Response ◽  
Dipole Source ◽  
Displacement Current ◽  
Geophysical Application ◽  
Horizontal Electric Dipole ◽  
Em Field ◽  
Transform Zone ◽  
Displacement Currents ◽  
And Diffusion

Electromagnetic (EM) field is often referred to diffusion (quasi-static assumption) that displacement currents are neglected during data processing in geophysical application, while the ratio of conduct currents to displacement currents is higher than 10, that is ,we think EM field is diffusion dominated and wave dominate for the ratio less than 0.1. Our simulating with Horizontal Electric Dipole Field indicated that frequency range of wave dominated is that the ratio is less than 0.007, the magnitude curves of EM component and impedance referring to diffusion and wave are different from those of diffusion. And in transform zone () the curves are different from those of wave and diffusion.

Inversion of slingram electromagnetic induction data using a Born approximation

Geophysics ◽  
2013 ◽  
Vol 78 (4) ◽  
pp. E201-E212 ◽  
Author(s):  
Jochen Kamm ◽  
Michael Becken ◽  
Laust B. Pedersen
Keyword(s):  
Half Space ◽  
Born Approximation ◽  
High Efficiency ◽  
Three Dimensions ◽  
Background Model ◽  
Desktop Computer ◽  
Near Surface ◽  
Homogeneous Half Space ◽  
Integral Equation Formulation ◽  
Forward Operator

We present an efficient approximate inversion scheme for near-surface loop-loop EM induction data (slingram) that can be applied to obtain 2D or 3D models on a normal desktop computer. Our approach is derived from a volume integral equation formulation with an arbitrarily conductive homogeneous half-space as a background model. The measurements are not required to fulfill the low induction number condition (low frequency and conductivity). The high efficiency of the method is achieved by invoking the Born approximation around a half-space background. The Born approximation renders the forward operator linear. The choice of a homogeneous half-space yields closed form expressions for the required electromagnetic normal fields. It also yields a translationally invariant forward operator, i.e., a highly redundant Jacobian. In connection with the application of a matrix-free conjugate gradient method, this allows for very low memory requirements during the inversion, even in three dimensions. As a consequence of the Born approximation, strong conductive deviations from the background model are underestimated. Highly resistive anomalies are in principle overestimated, but at the same time difficult to resolve with induction methods. In the case of extreme contrasts, our forward model may fail in simultaneously explaining all the data collected. We applied the method to EM34 data from a profile that has been extensively studied with other electromagnetic methods and compare the results. Then, we invert three conductivity maps from the same area in a 3D inversion.

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