Reflection and transmission coefficients for seismic waves in ellipsoidally anisotropic media

Geophysics ◽  
1979 ◽  
Vol 44 (1) ◽  
pp. 27-38 ◽  
Author(s):  
P. F. Daley ◽  
F. Hron
Keyword(s):  
Wave Propagation ◽  
Anisotropic Medium ◽  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Reflection And Transmission Coefficients ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
The Earth ◽  
Special Case

The deficiency of an isotropic model of the earth in the explanation of observed traveltime phenomena has led to the mathematical investigation of elastic wave propagation in anisotropic media. A type of anisotropy dealt with in the literature (Potsma, 1955; Cerveny and Psencik, 1972; and Vlaar, 1968) is uniaxial anisotropy or transverse isotropy. A special case of transverse isotropy which assumes the wavefronts to be ellipsoids of revolution has been used by Cholet and Richard (1954) and Richards (1960) in accounting for the observed traveltimes at Berraine in the Sahara and in the foothills of Western Canada. The kinematics of this problem have been treated in a number of papers, the most notable being Gassmann (1964). However, to appreciate fully the effect of anisotropy, the dynamics of the problem must be explored. Assuming a model of the earth consisting of plane transversely isotropic layers with the axes of anisotropy perpendicular to the interfaces, a prime requisite for obtaining amplitude distance curves or synthetic seismograms is the calculation of reflection and transmission coefficients at the interfaces. In this paper the special case of ellipsoidal anisotropy will be considered. That the quasi‐shear SV wavefront is forced to be spherical by this assumption is unfortunate, but it is instructive to investigate this simple anisotropic model as it incorporates many features inherent to wave propagation in a more general anisotropic medium. A brief outline of the theory for wave propagation in an ellipsoidally anisotropic medium is given and the analytic expressions for the reflection and transmission coefficients are listed. A complete derivation of reflection and transmission coefficients in transversely isotropic media can be found in Daley and Hron (1977). Finally, all 24 possible reflection and transmission coefficients and surface conversion coefficients are displayed for varying degrees of anisotropy.

scholarly journals Approximate reflection coefficients for a thin VTI layer

Geophysics ◽  
2018 ◽  
Vol 83 (1) ◽  
pp. C1-C11 ◽  
Author(s):  
Qi Hao ◽  
Alexey Stovas
Keyword(s):  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
Normal Incidence ◽  
Reflection And Transmission Coefficients ◽  
Reflection Coefficients ◽  
Angle Of Incidence ◽  
Isotropic Layer ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
P And Sv Waves

We have developed an approximate method to derive simple expressions for the reflection coefficients of P- and SV-waves for a thin transversely isotropic layer with a vertical symmetry axis (VTI) embedded in a homogeneous VTI background. The layer thickness is assumed to be much smaller than the wavelengths of P- and SV-waves inside. The exact reflection and transmission coefficients are derived by the propagator matrix method. In the case of normal incidence, the exact reflection and transmission coefficients are expressed in terms of the impedances of vertically propagating P- and S-waves. For subcritical incidence, the approximate reflection coefficients are expressed in terms of the contrast in the VTI parameters between the layer and the background. Numerical examples are designed to analyze the reflection coefficients at normal and oblique incidence and investigate the influence of transverse isotropy on the reflection coefficients. Despite giving numerical errors, the approximate formulas are sufficiently simple to qualitatively analyze the variation of the reflection coefficients with the angle of incidence.

Generalization of von Neumann analysis for a model of two discrete half-spaces: The acoustic case

Geophysics ◽  
2007 ◽  
Vol 72 (5) ◽  
pp. SM35-SM46 ◽  
Author(s):  
Matthew M. Haney
Keyword(s):  
Wave Propagation ◽  
Numerical Model ◽  
Finite Difference ◽  
Stability Criterion ◽  
Reflection And Transmission Coefficients ◽  
Reflection And Transmission ◽  
Von Neumann ◽  
Transmission Coefficients ◽  
Strong Material ◽  
Von Neumann Analysis

Evaluating the performance of finite-difference algorithms typically uses a technique known as von Neumann analysis. For a given algorithm, application of the technique yields both a dispersion relation valid for the discrete time-space grid and a mathematical condition for stability. In practice, a major shortcoming of conventional von Neumann analysis is that it can be applied only to an idealized numerical model — that of an infinite, homogeneous whole space. Experience has shown that numerical instabilities often arise in finite-difference simulations of wave propagation at interfaces with strong material contrasts. These interface instabilities occur even though the conventional von Neumann stability criterion may be satisfied at each point of the numerical model. To address this issue, I generalize von Neumann analysis for a model of two half-spaces. I perform the analysis for the case of acoustic wave propagation using a standard staggered-grid finite-difference numerical scheme. By deriving expressions for the discrete reflection and transmission coefficients, I study under what conditions the discrete reflection and transmission coefficients become unbounded. I find that instabilities encountered in numerical modeling near interfaces with strong material contrasts are linked to these cases and develop a modified stability criterion that takes into account the resulting instabilities. I test and verify the stability criterion by executing a finite-difference algorithm under conditions predicted to be stable and unstable.

scholarly journals Wave propagation in a non-uniform string

2011 ◽  
Vol 33 (4) ◽  
pp. 4306-4306
Author(s):  
Fernando Fuzinatto Dall'Agnol
Keyword(s):  
Wave Propagation ◽  
Pulse Width ◽  
Mass Density ◽  
Reflection And Transmission Coefficients ◽  
Antireflection Coatings ◽  
Reflection And Transmission ◽  
Physical Systems ◽  
Transmission Coefficients

In this article I present the behavior of the reflection and transmission coefficients of a pulse at a joint between two strings with mass densities μ1and μ2. The joint is made of a string segment with mass density varying linearly from μ1 to μ2. It will be shown that the reflection of the pulse at the joint depends largely on the ratio between the pulse width and the length of the joint. Analogies with other physical systems such as antireflection coatings and tsunamis will be considered briefly.

Body waves in a “real Earth”. Part I

1973 ◽  
Vol 63 (1) ◽  
pp. 145-156 ◽  
Author(s):  
A. Cisternas ◽  
O. Betancourt ◽  
A. Leiva
Keyword(s):  
Exact Solution ◽  
Theoretical Analysis ◽  
Power Series ◽  
Arbitrary Number ◽  
Body Waves ◽  
Reflection And Transmission Coefficients ◽  
Earth Model ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
The Earth

abstract A theoretical analysis of body waves in a “real Earth” is presented. The earth model consists of an arbitrary number of spherical liquid and solid layers. The algebraic part of the analysis deals with the way to obtain generalized rays out of the exact solution. It is shown that the Rayleigh matrix, and not the Rayleigh determinant, should be used to expand the solution into a power series of modified reflection and transmission coefficients in order to obtain rays.

Plane‐wave reflection and transmission coefficients for a transversely isotropic solid

Geophysics ◽  
1992 ◽  
Vol 57 (11) ◽  
pp. 1512-1519 ◽  
Author(s):  
Mark Graebner
Keyword(s):  
Plane Wave ◽  
Wave Reflection ◽  
Transversely Isotropic ◽  
Reflection And Transmission Coefficients ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
Isotropic Solid

Numerous investigators have studied the P-SV reflection and transmission coefficients of an isotropic solid (Zoeppritz, 1919; Nafe, 1957; Frasier, 1970; Young and Braile, 1976; Kind, 1976; Aki and Richards, 1980).

THE REFLECTION OF WAVES IN A GENERALIZED EPSTEIN PROFILE

1965 ◽  
Vol 43 (5) ◽  
pp. 921-934 ◽  
Author(s):  
R. Burman ◽  
R. N. Gould
Keyword(s):  
Refractive Index ◽  
Wave Equation ◽  
Reflection Coefficient ◽  
Approximate Formula ◽  
Reflection And Transmission Coefficients ◽  
Stratified Medium ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
Special Case ◽  
Reflection Of Waves

Epstein (1930) and Rawer (1939) studied the reflection of waves in a stratified medium by transforming the hypergeometric equation into the wave equation. A particular case of the Epstein profile is a symmetrical layer. Considerable attention has been given in the literature to this case as well as to symmetrical layers with certain other profiles of the refractive index. In the present paper a generalized Epstein profile is considered and the reflection and transmission coefficients are obtained. The special case in which the refractive index tends to the same constant value on either side of a layer is then discussed. The symmetrical Epstein profile is a special case of this layer which, in general, is asymmetrical. Particular attention is given to a layer differing only slightly from the symmetrical Epstein layer, a simple approximate formula for the reflection coefficient being derived.

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