Generalized form of Snell's law in anisotropic media: A practical approach

Application of Snell’s law in weakly anisotropic media

Geophysics ◽

10.1190/1.3137571 ◽

2009 ◽

Vol 74 (5) ◽

pp. WB147-WB152 ◽

Cited By ~ 5

Author(s):

Claudia Vanelle ◽

Dirk Gajewski

Keyword(s):

Phase Velocity ◽

Anisotropic Media ◽

Transmitted Wave ◽

Sixth Order ◽

Wave Type ◽

Weak Anisotropy ◽

Slowness Vector ◽

Order Polynomial ◽

Snell’S Law ◽

Snell's Law

Snell’s law describes the relationship between phase angles and velocities during the reflection or transmission of waves. It states that horizontal slowness with respect to an interface is preserved during reflection or transmission. Evaluation of this relationship at an interface between two isotropic media is straightforward. For anisotropic media, it is a complicated problem because phase velocity depends on the angle; in the anisotropic reflection/transmission problem, neither is known. Solving Snell’s law in the anisotropic case requires a numerical solution for a sixth-order polynomial. In addition to finding the roots, they must be assigned to the correct reflected or transmitted wave type. We show that if the anisotropy is weak, an approximate solution based on first-order perturbation theory can be obtained. This approach permits the computation of the full slowness vector and, thereby, the phase velocity and angle. In addition to replacing the need for solving the sixth-order polynomial, the resulting expressions allow us to prescribe the desired reflected or transmitted wave type. The method is best implemented iteratively to increase accuracy. The result can be applied to anisotropic media with arbitrary symmetry. It converges toward the weak-anisotropy solution and provides overall good accuracy for media with weak to moderate anisotropy.

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A generalized form of Snell’s law in anisotropic media

Geophysics ◽

10.1190/1.1444759 ◽

2000 ◽

Vol 65 (2) ◽

pp. 632-637 ◽

Cited By ~ 18

Author(s):

Michael A. Slawinski ◽

Raphaël A. Slawinski ◽

R. James Brown ◽

John M. Parkin

Keyword(s):

Standard Form ◽

Anisotropic Media ◽

Practical Application ◽

Amplitude Variation ◽

Sh Waves ◽

Amplitude Variation With Offset ◽

Geometrical Properties ◽

Analytic Expressions ◽

Snell’S Law ◽

Snell's Law

We have reformulated the law governing the refraction of rays at a planar interface separating two anisotropic media in terms of slowness surfaces. Equations connecting ray directions and phase‐slowness angles are derived using geometrical properties of the gradient operator in slowness space. A numerical example shows that, even in weakly anisotropic media, the ray trajectory governed by the anisotropic Snell’s law is significantly different from that obtained using the isotropic form. This could have important implications for such considerations as imaging (e.g., migration) and lithology analysis (e.g., amplitude variation with offset). Expressions are shown specifically for compressional (qP) waves but they can easily be extended to SH waves by equating the anisotropic parameters (i.e., ε = δ ⇒ γ) and to qSV and converted waves by similar means. The analytic expressions presented are more complicated than the standard form of Snell’s law. To facilitate practical application, we include our Mathematica code.

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Estimating SV‐wave stacking velocities for transversely isotropic solids

Geophysics ◽

10.1190/1.1442970 ◽

1991 ◽

Vol 56 (10) ◽

pp. 1596-1602 ◽

Cited By ~ 9

Author(s):

Patricia A. Berge

Keyword(s):

Shear Wave ◽

Isotropic Medium ◽

Transversely Isotropic ◽

Anisotropic Media ◽

Vertical Axis ◽

Transversely Isotropic Medium ◽

Transversely Isotropic Media ◽

Isotropic Media ◽

Snell’S Law ◽

Snell's Law

Conventional seismic experiments can record converted shear waves in anisotropic media, but the shear‐wave stacking velocities pose a problem when processing and interpreting the data. Methods used to find shear‐wave stacking velocities in isotropic media will not always provide good estimates in anisotropic media. Although isotropic methods often can be used to estimate shear‐wave stacking velocities in transversely isotropic media with vertical symmetry axes, the estimations fail for some transversely isotropic media even though the anisotropy is weak. For a given anisotropic medium, the shear‐wave stacking velocity can be estimated using isotropic methods if the isotropic Snell’s law approximates the anisotropic Snell’s law and if the shear wavefront is smooth enough near the vertical axis to be fit with an ellipse. Most of the 15 transversely isotropic media examined in this paper met these conditions for short reflection spreads and small ray angles. Any transversely isotropic medium will meet these conditions if the transverse isotropy is weak and caused by thin subhorizontal layering. For three of the media examined, the anisotropy was weak but the shear wave-fronts were not even approximately elliptical near the vertical axis. Thus, isotropic methods provided poor estimates of the shear‐wave stacking velocities. These results confirm that for any given transversely isotropic medium, it is possible to determine whether or not shear‐wave stacking velocities can be estimated using isotropic velocity analysis.

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Simple Refraction Law for Uniaxial Anisotropic Absorbing Media

Applied Spectroscopy ◽

10.1366/0003702934066695 ◽

1993 ◽

Vol 47 (3) ◽

pp. 338-340 ◽

Cited By ~ 3

Author(s):

Takeshi Hasegawa ◽

Junzo Umemura ◽

Tohru Takenaka

Keyword(s):

Anisotropic Media ◽

Absorbing Media ◽

Snell’S Law ◽

Snell's Law ◽

Uniaxial Media

In anisotropic media, there exist ordinary and extraordinary rays. Until now, there has been no simple refraction law like Snell's law for the extraordinary ray. In this study, with the Fresnel model, we made a simple refraction law applicable to both ordinary and extraordinary rays, though it is limited for uniaxial media. The obtained formula could explain the differences between the two rays and would be very useful in extending the traditional isotropic electromagnetic theories to the uniaxial anisotropic electromagnetic theories.

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