Direct time‐domain calculation of the transient response for a rectangular loop over a two‐layer medium

Geophysics ◽  
1987 ◽  
Vol 52 (7) ◽  
pp. 997-1006 ◽  
Author(s):  
Mark M. Goldman ◽  
David V. Fitterman
Keyword(s):  
Magnetic Field ◽  
Time Domain ◽  
Electric Dipole ◽  
Late Stage ◽  
Apparent Resistivity ◽  
Time Derivative ◽  
Vertical Magnetic Field ◽  
First Case ◽  
Layer Medium ◽  
The Time Domain

The time derivative of the vertical magnetic field due to an electric dipole on the surface of a two‐layer half‐space is computed directly in the time domain by applying the residue theorem to the analytic field expressions. The second layer must be either insulating [Formula: see text] or perfectly conducting [Formula: see text]. The first case can be used to estimate the response of a conductive overburden for mining exploration problems. The second case is useful in explaining the overshoot seen in transient sounding voltage apparent‐resistivity curves when a conductive basement underlies a resistive first layer. In the late stage, the time derivative of the vertical magnetic field decays as [Formula: see text] and the late‐stage apparent resistivity increases as t for [Formula: see text], while for [Formula: see text], these quantities behave as [Formula: see text] and [Formula: see text], respectively, where [Formula: see text], [Formula: see text], is the first‐layer conductivity, [Formula: see text] is the first‐layer thickness, and [Formula: see text]. The electric dipole expressions are integrated to obtain solutions for rectangular loops. Numerical results for a rectangular loop on a layer over an insulating basement (overburden case) show that the overburden response is initially positive inside the loop and negative outside the loop. At later times, the response outside the loop becomes positive. The thinner the overburden layer, the greater the maximum response.

Time‐domain electromagnetic detection of a hidden target

Geophysics ◽  
1988 ◽  
Vol 53 (4) ◽  
pp. 537-545 ◽  
Author(s):  
David C. Bartel ◽  
A. Becker
Keyword(s):  
Magnetic Field ◽  
Time Domain ◽  
Time Window ◽  
Time Derivative ◽  
Step Response ◽  
Time Range ◽  
Geometrical Parameters ◽  
Target Signal ◽  
Time Domain Electromagnetic ◽  
The Time Domain

Numerical modeling of the time‐domain electromagnetic (EM) step response of a vertical tabular target hidden beneath a thin conductive overburden reveals that the target’s presence may be detected only during a well‐defined time window. In a situation where the secondary magnetic field is sensed by an airborne system equipped with horizontal coaxial dipoles, a conductance contrast of about ten between the target and the overburden is needed to ensure target detection. This value will, of course, vary with the size and depth of the target and, to a lesser extent, with the geometry of the system. In general, the time at which the window opens is a function of the geometrical parameters of the target, the height of the system, and the conductance of the overburden. For a given target, its width (defined as the ratio of the time of closure to the time of opening) is only a function of the conductance contrast between the target and the overburden. While the target signal is visible, one observes a maximum value of the target‐to‐overburden response ratio. The time at which this occurs is mainly controlled by the conductance of the target. The presence of the overburden causes the target signal to build up gradually before decaying toward zero. However, once the target signal dominates the overburden response, the signal can be approximated by a simple exponential decay over the time range of interest. The time constant of this decay is determined by the size and conductance of the target. Using this model, it is easy to relate the magnetic field step response calculated here to the response observable with a conventional EM system that transmits a primary field pulse of finite duration and detects the time derivative of the secondary magnetic field.

scholarly journals Electromagnetic induction in an inhomogeneous conductive thin sheet

Geophysics ◽  
1987 ◽  
Vol 52 (12) ◽  
pp. 1677-1688 ◽  
Author(s):  
Richard S. Smith ◽  
G. F. West
Keyword(s):  
Magnetic Field ◽  
Time Domain ◽  
Time Domain Method ◽  
Step Response ◽  
Scale Model ◽  
Frequency Domain Method ◽  
Vertical Magnetic Field ◽  
Zero Crossing ◽  
Domain Method ◽  
The Time Domain

Distinguishing between the electromagnetic (EM) response of a subsurface conductor and the EM response of an overburden whose conductivity and/or thickness varies laterally requires a capability to calculate the EM response of both types of conductor. While methods for calculating the response of some simple subsurface conductors such as dipping rectangular sheets are already available, methods for computing the response of an irregular overburden are not common. Using Price’s analysis, we have formulated two numerical techniques for calculating the response of a laterally varying overburden which is thin and flat, and which lies on a perfectly resistive subspace. The first technique is a frequency‐domain method in which a large matrix equation is solved to find the horizontal‐wavenumber components of the secondary vertical magnetic field. The method is best suited to calculating the response of the overburden when the EM source and receiver are located above the sheet, such as in airborne EM systems. Helicopter EM profiles calculated using this technique have been checked against a simple scale model. The second method calculates the time‐domain step response of the overburden by time‐stepping the vertical component of the magnetic field. The method is suitable for calculating the response of the overburden when the EM source is a large transmitter loop close to the overburden. Using the time‐domain method to investigate the response of simple conductance structures illustrates that the zero crossing of the vertical magnetic field moves more slowly across conductive regions than across resistive regions. This is because the rate of decay of the vertical field in a region varies in proportion to the resistance of the region. A response profile from a UTEM survey shows a response that could be interpreted as due to a dipping subsurface conductor. This response has been modeled using the time‐domain method, and a geologically acceptable pattern of lateral variations in the overburden conductance yields a response close to the measured EM response. Thus, a subsurface conductor need not lie below the profile line to explain the response.

On: “Time‐Dependent Electromagnetic Fields of an Infinite, Conducting Cylinder Excited by a Long Current‐Carrying Cable” by S. K. Verma (GEOPHYSICS, April 1973, p. 369–379)

Geophysics ◽  
1974 ◽  
Vol 39 (3) ◽  
pp. 355-355
Author(s):  
Shri Krishna Singh
Keyword(s):  
Magnetic Field ◽  
Frequency Domain ◽  
Electric Current ◽  
Electromagnetic Fields ◽  
Time Domain ◽  
Static Solution ◽  
Transverse Magnetic ◽  
Time Dependent ◽  
Axially Symmetric ◽  
The Time Domain

In this paper Verma obtains a time‐domain solution by inverting the frequency‐domain solution given by Wait (1952). However, it has been recently pointed out by Singh (1973a) (see also Wait, 1973) that there is an error in the quasi‐static solution of Wait. Wait neglected the axially symmetric inducted electric current in the cylinder giving rise to a secondary transverse magnetic field outside (the n=0 term in the scattered wavefield). Singh (1973a) has shown that this term dominates. [It should be noted that Wait in his other works on the cylinder retains this term (e.g., Wait, 1959).] It is clear that this term would be dominant in the time‐domain also. This has been shown by Singh (1972, 1973b). Since the theoretical solution given by Verma in the paper under discussion is incomplete, his interpretation schemes are meaningless.

Symmetric Arc Solutions of ζ¨ = ζn

1968 ◽  
Vol 35 (3) ◽  
pp. 565-570
Author(s):  
C. P. Atkinson ◽  
B. L. Dhoopar
Keyword(s):  
Differential Equation ◽  
Time Domain ◽  
Complex Variable ◽  
Digital Computer ◽  
Nonlinear Differential Equations ◽  
Time Derivative ◽  
Approximate Solutions ◽  
Complex Variables ◽  
Complex Modes ◽  
The Time Domain

This paper, “Symmetric Arc Solutions of ζ¨ = ζn,” presents periodic solutions of this differential equation relating the complex variable ζ(t) = u(t) + iv(t) and its second time derivative ζ¨ The solutions are called symmetric arc solutions since they form such arcs on the ζ = u + iv-plane. The solutions, ζ(t), are “complex modes” of coupled nonlinear differential equations in the complex variables z1 and z2. Symmetric arc solutions are presented for a range of n from n = 3 to n = 101. Approximate solutions are presented and compared with solutions generated by digital computer. Solutions are presented on the ζ-plane and in the time domain as u(t) and v(t).

Dynamics of Infinite Beams Described by Fractional Time Derivatives

2003 ◽  
Author(s):  
Peter Ruge ◽  
Carolin Trinks
Keyword(s):  
Frequency Domain ◽  
Time Domain ◽  
Dynamic Stiffness ◽  
Time Derivative ◽  
Low Frequency ◽  
Internal Variables ◽  
Closed Form Solutions ◽  
Rational Polynomial ◽  
The Time Domain ◽  
Fractional Time Derivative

Closed-form solutions of infinite Bernoulli-Euler beams on a viscoelastic foundation are available for harmonic excitations with frequency Ω. For more general time-dependent loadings and beam-systems with local perturbations, for example caused by non-linear effects an overall treatment of the system in the time-domain is highly appropriated. Here the analytical dynamic stiffness of the infinite beam in the frequency-domain is approximated by a rational polynomial in the low frequency-domain and by an irrational part representing the asymptotic behaviour for Ω tending towards infinity. Thus, the corresponding description in the time-domain contains a fractional time derivative part and additonal internal variables due to splitting the rational polynomial into a linear system with respect to Ω.

scholarly journals Normalized Study of Three-Parameter System in the Time Domain and Frequency Domain

2017 ◽  
Vol 2017 ◽  
pp. 1-21
Author(s):  
Xiao-Lei Jiao ◽  
Yang Zhao ◽  
Wen-Lai Ma
Keyword(s):  
Frequency Domain ◽  
Dynamic Behavior ◽  
Time Domain ◽  
Phase Margin ◽  
Stiffness Ratio ◽  
Parameter System ◽  
Regulatory Factor ◽  
First Case ◽  
The Time Domain ◽  
Isolation Performance

Three-parameter isolation system can be used to isolate microvibration for control moment gyroscopes. Normalized analytical model for three-parameter system in the time domain and frequency domain is proposed by using analytical method. Dynamic behavior of three-parameter system in the time domain and frequency domain is studied. Response in the time domain under different types of excitations is analyzed. In this paper, a regulatory factor is defined in order to analyze dynamic behavior in the frequency domain. For harmonic excitation, a comparison study is made on isolation performance between the case when the system has optimal damping and the case when regulatory factor is 1. Besides, phase margin of three-parameter system is obtained. Results show that dynamic behavior in the time domain and frequency domain changes with regulatory factor. Phase margin has the largest value when the value of regulatory factor is 1. System under impulse excitation and step excitation has the shortest settling time for the response in the time domain when the value of regulatory factor is 1. When stiffness ratio is small, isolation performances of two cases are nearly the same; when system has a large stiffness ratio, isolation performance of the first case is better.

Three‐dimensional transient electromagnetic responses for a grounded source

Geophysics ◽  
1986 ◽  
Vol 51 (11) ◽  
pp. 2117-2130 ◽  
Author(s):  
Brian M. Gunderson ◽  
Gregory A. Newman ◽  
Gerald W. Hohmann
Keyword(s):  
Magnetic Field ◽  
Half Space ◽  
Time Derivative ◽  
Three Dimensional ◽  
The Body ◽  
Vertical Magnetic Field ◽  
Horizontal Magnetic Field ◽  
One Dimensional ◽  
Transient Electromagnetic ◽  
Electromagnetic Responses

When the current in a grounded wire is terminated abruptly, currents immediately flow in the Earth to preserve the magnetic field. Initially the current is concentrated near the wire, with a broad zone of return currents below. The electric field maximum broadens and moves downward with time. Currents are channeled into a conductive three‐dimensional body, resulting in anomalous magnetic fields. At early times, when the return currents are channeled into the body, the vertical magnetic field is less than the half‐space field on the far side of the body but is greater than the half‐space field between the source and the body. Later the current in the body reverses; the vertical field is enhanced on the far side of the body and decreased between the source and the body. The horizontal magnetic field has a well‐defined maximum directly over the body at late times, and is a better indicator of the position of the body. The vertical magnetic field and its time derivative change sign with time at receiver locations near the source if a three‐dimensional body is present. These sign reversals present serious problems for one‐dimensional inversion, because decay curves for a layered earth do not change sign. At positions away from the source, the decay curves exhibit no sign reversals—only decreases and enhancements relative to one‐dimensional decay curves. In such cases one‐dimensional inversions may provide useful information, but they are likely to result in fictitious layers and erroneous interpretations.

scholarly journals Electromagnetic Pulse of a Vertical Electric Dipole in the Presence of Three-Layered Region

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
D. Cheng ◽  
T. T. Gu ◽  
P. Cao ◽  
T. He ◽  
K. Li
Keyword(s):  
Time Domain ◽  
Electric Dipole ◽  
Delta Function ◽  
Observation Point ◽  
Perfect Conductor ◽  
Transient Field ◽  
Planar Surface ◽  
Vertical Electric Dipole ◽  
Trapped Surface ◽  
The Time Domain

Approximate formulas are obtained for the electromagnetic pulses due to a delta-function current in a vertical electric dipole on the planar surface of a perfect conductor coated by a dielectric layer. The new approximated formulas for the electromagnetic field in time domain are retreated analytically and some new results are obtained. Computations and discussions are carried out for the time-domain field components radiated by a vertical electric dipole in the presence of three-layered region. It is shown that the trapped-surface-wave terms should be included in the total transient field when both the vertical electric dipole and the observation point are on or near the planar surface of the dielectric-coated earth.

Numerical Study of the Ship Motion in Waves Using the Three-Dimensional Time-Domain Forward-Speed Free-Surface Green Function and a Second-Order Boundary Element Method

2008 ◽  
Author(s):  
D. C. Hong ◽  
S. Y. Hong ◽  
H. G. Sung
Keyword(s):  
Integral Equation ◽  
Free Surface ◽  
Green Function ◽  
Time Domain ◽  
Time Derivative ◽  
Three Dimensional ◽  
Forward Speed ◽  
Surface Green Function ◽  
The Green Function ◽  
The Time Domain

The radiation and diffraction potentials of a ship advancing in waves are calculated in the time-domain using the three-dimensional time-domain forward-speed free-surface Green function and the Green integral equation on the basis of the Neumann-Kelvin linear wave hypothesis. The Green function approximated by Newman for large time is used together with the Green function by Lamb for small time. The time-domain diffraction problem is solved for the time derivative of the potential by using the time derivative of the impulsive incident wave potential represented by using the complementary complex error function. The integral equation for the potential is discretized according to a second-order boundary element method where the collocation points are located inside the panel. It makes it possible to take account of the line integral along the waterline in a rigorous manner. The six-degree-of-freedom motion and memory functions as well as the diffraction impulse response functions of a hemisphere and the Wigley seakeeping model are presented for various Froude numbers. Comparisons of the wave damping and exciting force and moment coefficients for zero forward speed, calculated by using the Fourier transforms of the time-domain results and the frequency-domain coefficients calculated by using the improved Green integral equation which is free of the irregular frequencies, have been shown to be satisfactory. The wave damping coefficients for non-zero forward speed, calculated by using Fourier transforming of the present time-domain results have also been compared to the experimental results and agreement between them has been shown to be good. A simulation of coupled heave-pitch motion of the Wigley seakeeping model advancing in regular head waves of unit amplitude has been carried out.

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