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.

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.

THE SEISMIC WAVE ENERGY REFLECTED FROM VARIOUS TYPES OF STRATIFIED HORIZONS

Geophysics ◽  
1940 ◽  
Vol 5 (2) ◽  
pp. 149-155 ◽  
Author(s):  
M. Muskat ◽  
M. W. Meres
Keyword(s):  
Seismic Wave ◽  
Wave Energy ◽  
Incident Energy ◽  
Preceding Paper ◽  
Reflection And Transmission Coefficients ◽  
Angle Of Incidence ◽  
Reflection And Transmission ◽  
Transmission Coefficients

Two applications are made of the reflection and transmission coefficients reported in the preceding paper. These concern the effect of the angle of incidence upon the fraction of incident energy returning to the surface, and the effect of velocity stratification upon the energy return.

Amplitude‐variation‐with‐angle behavior of self‐similar interfaces

Geophysics ◽  
1999 ◽  
Vol 64 (6) ◽  
pp. 1928-1938 ◽  
Author(s):  
Kees Wapenaar
Keyword(s):  
Normal Incidence ◽  
Reflection And Transmission Coefficients ◽  
Well Logs ◽  
Amplitude Variation ◽  
Reflection And Transmission ◽  
Velocity Function ◽  
Transmission Coefficients ◽  
Step Functions ◽  
Self Similar ◽  
Amplitude Variation With Angle

Amplitude‐variation‐with‐angle (AVA) analysis is generally based on the assumption that the medium parameters behave as step functions of the depth coordinate z, at least in a finite region around the interface. However, outliers observed in well logs often behave quite differently from step functions. In this paper, outliers in the acoustic propagation velocity are parameterized by functions of the form [Formula: see text]. The wavelet transform of this function reveals properties similar to those of several outliers in real well logs. Moreover, this function is self‐similar, according to [Formula: see text], for β > 0. Analytical expressions are derived for the acoustic normal incidence reflection and transmission coefficients for this type of velocity function. For oblique incidence, no explicit solutions are available. However, by exploiting the self‐similarity property of the velocity function, it turns out that the acoustic angle‐dependent and frequency‐dependent reflection and transmission coefficients are self‐similar as well. To be more specific, these coefficients appear to be constant along curves described by [Formula: see text], where p is the raypath parameter and ω the angular frequency. The singularity exponent α that is reflected in these curves may prove to be a useful indicator in seismic characterization.

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).

Transmission and Reflection Coefficients for Damage Identification in 1D Elements

2009 ◽  
Vol 413-414 ◽  
pp. 95-100 ◽  
Author(s):  
Marek Krawczuk ◽  
Magdalena Palacz ◽  
Arkadiusz Zak ◽  
Wiesław M. Ostachowicz
Keyword(s):  
Damage Identification ◽  
Reflection And Transmission Coefficients ◽  
Reflection Coefficients ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
Transmission And Reflection Coefficients ◽  
Structural Discontinuity ◽  
Spectral Finite Element ◽  
Processing Techniques ◽  
Transmission And Reflection

According to the latest research results presented in the literature changes in propagating waves are one of the most promising parameters for damage identification algorithms. Numerous publications describe methods of damage identification based on the analysis of signals reflected from damage. They also include complicated signal processing techniques. Such methods work well for damage localisation, but it is rather difficult to use them in order to estimate the size of damage. It is natural that propagating wave reflects from any structural discontinuity. The bigger the disturbance the bigger part of a propagating wave reflects from it. The amount of energy reflected and transmitted through any discontinuity can expressed as reflection and transmission coefficients. In the literature different application for these coefficients may be found – the most often cited application is connected with localising changes in the geometry of structures. Changes in the coefficients due to cross section variations in rods and beams or due to existence of stiffeners in plates are well documented. However there are no application of using the reflection and transmission coefficients for damage size identification. For this reason the analysis presented in this paper has been carried out. The article presents a method of damage identification in 1D elements based on the wave propagation phenomenon and changes in reflection and transmission coefficients. The changes in transmission and reflection coefficients for waves propagating in isotropic rods with different types of damage have been analysed. The rods have been modelled with the elementary, two and three mode theories or rods. For numerical modelling the Spectral Finite Element Method has been used. Several examples are given in the paper.

Dilatational waves at a microstretch solid/fluid interface

2016 ◽  
Vol 23 (20) ◽  
pp. 3448-3467 ◽  
Author(s):  
Dilbag Singh ◽  
Neela Rani ◽  
Sushil Kumar Tomar
Keyword(s):  
Half Space ◽  
Aluminum Matrix ◽  
Elastic Solid ◽  
Specific Model ◽  
Reflection And Transmission Coefficients ◽  
Angle Of Incidence ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
Special Cases ◽  
Microstretch Fluid

The present work is concerned with the study of reflection and transmission phenomena of dilatational waves at a plane interface between a microstretch elastic solid half-space and a microstretch liquid half-space. Eringen's theory of micro-continuum materials has been employed for addressing the mathematical analysis. Reflection and transmission coefficients, corresponding to various reflected and transmitted waves, have been obtained when a plane dilatational wave strikes obliquely at the interface after propagating through the solid half-space. It is found that the reflection and transmission coefficients are functions of the angle of incidence, the frequency of the incident wave and the elastic properties of the half-spaces. Numerical calculations have been carried out for a specific model by taking an aluminum matrix with randomly distributed epoxy spheres as the microstretch solid medium, while the microstretch fluid is taken arbitrarily with suitably chosen elastic parameters. The computed results obtained have been depicted graphically. The results of earlier studies have been deduced from the present formulation as special cases.

Propagation of Surface Waves Across an Anisotropic Layer

1994 ◽  
Vol 61 (3) ◽  
pp. 596-604 ◽  
Author(s):  
E. N. Its ◽  
J. S. Lee
Keyword(s):  
Surface Waves ◽  
Rayleigh Waves ◽  
Differential Operators ◽  
Interface Layer ◽  
Reflection And Transmission Coefficients ◽  
Function Technique ◽  
Angle Of Incidence ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
Matrix Differential

Propagation of elastic surface waves across a thin anisotropic interface layer between two vertically inhomogeneous isotropic quarter-spaces is considered. The relationship between the surface wave fields at the opposite sides of the layer is obtained in the form of matrix differential operators. Based on the Green’s function technique, an analytical method is developed to calculate reflection and transmission coefficients of Rayleigh waves at the layer. The reflection and transmission coefficients of Rayleigh waves at the interface layer are calculated as a function of the angle of incidence for various models of layers with hexagonal symmetry and results are discussed in some detail. Several isotropic layers of low or high velocity materials are also considered to examine the trade-off between anisotropy and inhomogeneity of the interface layer.

Free Space Microwave Characterization of Silicon Wafers for Microelectronic Applications

2005 ◽  
Vol 2 (2) ◽  
pp. 35
Author(s):  
Zaiki Awang ◽  
Deepak Kumar Ghodgaonkar ◽  
Noor Hasimah Baba
Keyword(s):  
Complex Permittivity ◽  
Free Space ◽  
Normal Incidence ◽  
Silicon Wafers ◽  
Reflection And Transmission Coefficients ◽  
Frequency Range ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
Quarter Wave ◽  
Mode Transitions

A contactless and non-destructive microwave method has been developed to characterize silicon semiconductor wafers from reflection and transmission measurements made at normal incidence using MNDT. The measurement system consists of a pair of spot-focusing horn lens antenna, mode transitions, coaxial cables and a vector network analyzer (VNA). In this method, the free-space reflection and transmission coefficients, S11 and S21 are measured for silicon wafers sandwiched between two Teflon plates of 5mm thickness which act as a quarter-wave transformer at mid-band. The actual reflection and transmission coefficients, S11 and S21 of the silicon wafers are then calculated from the measured S11 and S21 using ABCD matrix transformation in which the complex permittivity and thickness of the Teflon plates are known. From the complex permittivity, the resistivity and conductivity can be obtained. Results for p-type and n-type doped silicon wafers are reported in the frequency range of 11 – 12.5 GHz. The dielectric constant of silicon wafer obtained by this method agrees well with that measured in the same frequency range by other conventional methods.

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