Magnetotelluric response on vertically inhomogeneous earth having conductivity varying exponentially with depth

Geophysics ◽  
1982 ◽  
Vol 47 (1) ◽  
pp. 89-99 ◽  
Author(s):  
D. Kao
Keyword(s):  
Half Space ◽  
Apparent Resistivity ◽  
Bessel Functions ◽  
Homogeneous Layer ◽  
Modified Bessel Functions ◽  
Layer Model ◽  
Transitional Layer ◽  
Number Of Layers ◽  
Inhomogeneous Models ◽  
Electric And Magnetic Fields

Magnetotelluric (MT) response is studied for a vertically inhomogeneous earth, where conductivity (or resistivity) varies exponentially with depth as [Formula: see text]. Horizontal electric and magnetic fields in such an inhomogeneous medium are given in terms of modified Bessel functions. Impedance and apparent resistivity are calculated for (1) an inhomogeneous half‐space having conductivity varying exponentially with depth, (2) an inhomogeneous half‐space overlain by a homogeneous layer, and (3) a three‐layer model with the second layer as an inhomogeneous or transitional layer. Results are presented graphically and are compared with those of homogeneous multilayer models. In the case of resistivity increasing exponentially with depth, the results of the above inhomogeneous models are equivalent to those of Cagniard two‐layer models, with [Formula: see text]. In the case of resistivity decreasing exponentially with depth, the homogeneous multilayer approximation depends upon the number of layers and the layer parameters chosen; |Z/ωμ| as a function of frequency is more useful than the apparent resistivity in determining the values of p and [Formula: see text].

Asymptotic expansions and converging factors V. Lommel, Struve, modified Struve, Anger and Weber functions, and integrals of ordinary and modified Bessel functions

1959 ◽  
Vol 249 (1257) ◽  
pp. 284-292 ◽  
Keyword(s):  
Asymptotic Expansions ◽  
Bessel Functions ◽  
Modified Bessel Functions ◽  
Lommel Functions ◽  
Special Cases

A theory of Lommel functions is developed, based upon the methods described in the first four papers (I to IV) of this series for replacing the divergent parts of asymptotic expansions by easily calculable series involving one or other of the four ‘basic converging factors’ which were investigated and tabulated in I. This theory is then illustrated by application to the special cases of Struve, modified Struve, Anger and Weber functions, and integrals of ordinary and modified Bessel functions.

(p, q)-Extended Bessel and Modified Bessel Functions of the First Kind

2017 ◽  
Vol 72 (1-2) ◽  
pp. 617-632 ◽  
Author(s):  
Dragana Jankov Maširević ◽  
Rakesh K. Parmar ◽  
Tibor K. Pogány
Keyword(s):  
Bessel Functions ◽  
Modified Bessel Functions

Airborne resistivity and susceptibility mapping in magnetically polarizable areas

Geophysics ◽  
2000 ◽  
Vol 65 (2) ◽  
pp. 502-511 ◽  
Author(s):  
Haoping Huang ◽  
Douglas C. Fraser
Keyword(s):  
Half Space ◽  
Magnetic Permeability ◽  
Apparent Resistivity ◽  
Dynamic Range ◽  
Single Frequency ◽  
Measured Signal ◽  
Apparent Permeability ◽  
Quadrature Component ◽  
New Methods ◽  
Quadrature Response

The apparent resistivity technique using half‐space models has been employed in helicopter‐borne resistivity mapping for twenty years. These resistivity algorithms yield the apparent resistivity from the measured in‐phase and quadrature response arising from the flow of electrical conduction currents for a given frequency. However, these algorithms, which assume free‐space magnetic permeability, do not yield a reliable value for the apparent resistivity in highly magnetic areas. This is because magnetic polarization also occurs, which modifies the electromagnetic (EM) response, causing the computed resistivity to be erroneously high. Conversely, the susceptibility of a magnetic half‐space can be computed from the measured EM response, assuming an absence of conduction currents. However, the presence of conduction currents will cause the computed susceptibility to be erroneously low. New methods for computing the apparent resistivity and apparent magnetic permeability have been developed for the magnetic conductive half‐space. The in‐phase and quadrature responses at the lowest frequency are first used to estimate the apparent magnetic permeability. The lowest frequency should be used to calculate the permeability because this minimizes the contribution to the measured signal from conduction currents. Knowing the apparent magnetic permeability then allows the apparent resistivity to be computed for all frequencies. The resistivity can be computed using different methods. Because the EM response of magnetic permeability is much greater for the in‐phase component than for the quadrature component, it may be better in highly magnetic environments to derive the resistivity using the quadrature component at two frequencies (the quad‐quad algorithm) rather than using the in‐phase and quadrature response at a single frequency (the in‐phase‐quad algorithm). However, the in‐phase‐quad algorithm has the advantage of dynamic range, and it gives credible resistivity results when the apparent permeability has been obtained correctly.

Numerical seismograms obtained using ω-κ integrals, for a point P source in a vertically inhomogeneous anelastic model

1996 ◽  
Vol 86 (3) ◽  
pp. 750-760
Author(s):  
F. Abramovici ◽  
L. H. T. Le ◽  
E. R. Kanasewich
Keyword(s):  
Half Space ◽  
Homogeneous Layer ◽  
Variable Step Size ◽  
Step Size ◽  
Adaptive Process ◽  
Cpu Time ◽  
Homogeneous Half Space ◽  
Linear Layer ◽  
Variable Step ◽  
Viscoelastic Layers

Abstract This article presents some numerical experiments in using a computer program for calculating the displacements due to a P source in a vertically inhomogeneous structure, based on the Fourier-Bessel representation. The structure may contain homogeneous, inhomogeneous, elastic, or viscoelastic layers. The source may act in any type of sublayer or in the half-space. Synthetic results for the simple case of a homogeneous layer overlaying a homogeneous half-space compare favorably with computations based on the Cagniard method. Numerical seismograms for an elastic layer having velocities and density varying linearly with depth were computed by integrating numerically the governing differential systems and compared with results based on the Haskell model of splitting the linear layer in homogeneous sublayers. Even an adaptive process with a variable step size based on the Haskell model has a poorer performance on the accuracy-cpu time scale than numerical integration.

Seismic waves from a horizontal stress discontinuity in a layered solid

1982 ◽  
Vol 72 (5) ◽  
pp. 1483-1498
Author(s):  
F. Abramovici ◽  
E. R. Kanasewich ◽  
P. G. Kelamis
Keyword(s):  
Half Space ◽  
Seismic Waves ◽  
Horizontal Stress ◽  
Radial Component ◽  
Elastic Half Space ◽  
Homogeneous Layer ◽  
Approximate Methods ◽  
Crustal Layer ◽  
Finite Fault ◽  
Stress Discontinuity

abstract The displacement components for a horizontal stress discontinuity along a buried finite fault in an elastic homogeneous layer on top of an elastic half-space are given analytically in terms of generalized rays. For a particular case of a concentrated horizontal force pointing in an arbitrary direction, detailed time-dependent expressions are given. For a simple model of a “crustal” layer over a “mantle” half-space, the numerical seismograms in the near- and intermediate-field show some interesting features. These include a prominent group of compressional waves whose radial component is substantial at distances four times the crustal thickness. All the dominant shear arrivals (s, SS, and sSS) are important and show large variations of amplitude as the source depth and receiver distance are varied. Some of the prominent individual generalized rays are shown, and it is found that they can be grouped naturally into families based on the number of interactions with the boundaries. The subdivision into individual generalized rays is useful for analysis and for checks on the numerical stability of the synthetic seismograms. Since the solution is analytic and the numerical evaluation is complete up to any desired time, the results are useful in comparing other approximate methods for the computation of seismograms.

Approximated Fundamental Frequency for Thin Circular Plates Clamped or Pinned at the Edge

2007 ◽  
Author(s):  
George Weiss
Keyword(s):  
Circular Plate ◽  
Fundamental Frequency ◽  
Bessel Functions ◽  
Unit Area ◽  
Continuous System ◽  
Modified Bessel Functions ◽  
Circular Plates ◽  
Numerical Parameter ◽  
Elliptical Plate ◽  
Distributed Load

Calculating the exact solution to the differential equations that describe the motion of a circular plate clamped or pinned at the edge, is laborious. The calculations include the Bessel functions and modified Bessel functions. In this paper, we present a brief method for calculating with approximation, the fundamental frequency of a circular plate clamped or pinned at the edge. We’ll use the Dunkerley’s estimate to determine the fundamental frequency of the plates. A plate is a continuous system and will assume it is loaded with a uniform distributed load, including the weight of the plate itself. Considering the mass per unit area of the plate, and substituting it in Dunkerley’s equation rearranged, we obtain a numerical parameter K02, related to the fundamental frequency of the plate, which has to be evaluated for each particular case. In this paper, have been evaluated the values of K02 for thin circular plates clamped or pinned at edge. An elliptical plate clamped at edge is also presented for several ratios of the semi–axes, one of which is identical with a circular plate.

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