Difference between revisions of "Dictionary:Polar anisotropy"

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[[File:Segt13.jpg|thumb|FIG. T-13. <b>Transverse isotropy</b>. <b>(a)</b> Wavefront with vertical axis of symmetry (TIV); <b>(b)</b> with horizontal symmetry axis (TIH) leading to azimuthal anisotropy; <b>(c)</b> phase (wavefront) angle &#x03B8; and group (ray) angle '' for transverse isotropy; <b>(d)</b> elliptical wavefront where &#x03B5;=&#x2013;&#x03B4;; in this case ''V''<sub>NMO</sub>&#x003E;''V''<sub>vertical</sub>; <b>(e)</b> anisotropic wavefront where &#x03B5;=&#x03B4;; in this case ''V''<sub>NMO</sub>&#x003C;''V''<sub>vertical</sub>; <b>(f)</b> wavefront for tilted symmetry axis; <b>(g)</b> wave equations for transverse isotropy.]]
 
[[File:Segt13.jpg|thumb|FIG. T-13. <b>Transverse isotropy</b>. <b>(a)</b> Wavefront with vertical axis of symmetry (TIV); <b>(b)</b> with horizontal symmetry axis (TIH) leading to azimuthal anisotropy; <b>(c)</b> phase (wavefront) angle &#x03B8; and group (ray) angle '' for transverse isotropy; <b>(d)</b> elliptical wavefront where &#x03B5;=&#x2013;&#x03B4;; in this case ''V''<sub>NMO</sub>&#x003E;''V''<sub>vertical</sub>; <b>(e)</b> anisotropic wavefront where &#x03B5;=&#x03B4;; in this case ''V''<sub>NMO</sub>&#x003C;''V''<sub>vertical</sub>; <b>(f)</b> wavefront for tilted symmetry axis; <b>(g)</b> wave equations for transverse isotropy.]]
  
<b>Transverse isotropy</b>. It involves elastic properties that are the same in any direction perpendicular to a symmetry axis but different parallel to the axis, and it has five independent elastic constants; see [[Dictionary:Thomsen_anisotropic_parameters_(tom&#x2019;_s&#x2202;n)|''Thomsen anisotropic parameters'']] and see Figure [[Dictionary:Fig_T-13|T-13]].
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<b>Transverse isotropy</b>. It involves elastic properties that are the same in any direction perpendicular to a symmetry axis but different parallel to the axis, and it has five independent elastic constants; see [[Special:MyLanguage/Dictionary:Thomsen_anisotropic_parameters_(tom&#x2019;_s&#x2202;n)|''Thomsen anisotropic parameters'']] and see Figure [[Special:MyLanguage/Dictionary:Fig_T-13|T-13]].
  
This symmetry is like a crystal having hexagonal symmetry; see Figure [[Dictionary:Fig_S-29|S-29]].
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This symmetry is like a crystal having hexagonal symmetry; see Figure [[Special:MyLanguage/Dictionary:Fig_S-29|S-29]].
  
A sequence of generally horizontal, isotropic layers (such as sedimentary bedding) tends to produce <b>layering anisotropy</b> (sometimes called <b>periodic thin-layer anisotropy</b>, although the layering need not be periodic) for wavelengths that are appreciably larger than the layer thickness. The axis of symmetry is generally perpendicular to the bedding, more-or-less vertical, with the velocities of P-waves parallel to the bedding and S-waves that are polarized parallel to the bedding being larger than for those perpendicular to the bedding. Parallel isotropic layering, where there are more than eight or so layers per wavelength, behaves as a polar anisotropic medium. Roughly horizontal layering is also called <b>transverse isotropy</b> because properties are the same in any transverse direction with a vertical axis of symmetry (TIV). See also [[Dictionary:azimuthal_asymmetry|''azimuthal asymmetry'']].
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A sequence of generally horizontal, isotropic layers (such as sedimentary bedding) tends to produce <b>layering anisotropy</b> (sometimes called <b>periodic thin-layer anisotropy</b>, although the layering need not be periodic) for wavelengths that are appreciably larger than the layer thickness. The axis of symmetry is generally perpendicular to the bedding, more-or-less vertical, with the velocities of P-waves parallel to the bedding and S-waves that are polarized parallel to the bedding being larger than for those perpendicular to the bedding. Parallel isotropic layering, where there are more than eight or so layers per wavelength, behaves as a polar anisotropic medium. Roughly horizontal layering is also called <b>transverse isotropy</b> because properties are the same in any transverse direction with a vertical axis of symmetry (TIV). See also [[Special:MyLanguage/Dictionary:azimuthal_asymmetry|''azimuthal asymmetry'']].
  
 
[[File:Sega14.jpg|thumb|left|FIG. A-14. <b>Anisotropy</b>. (<b>a</b>) Application of Huygens&#x2019; principle to anisotropic velocity illustrates why phase and ray velocities may differ in both direction and magnitude. (<b>b</b>) The application of Fermat&#x2019;s principle to anisotropic velocity illustrates why the angle of incidence for a reflection for a coincident source and receiver may not make a right angle with the reflector. (<b>c</b>) SH-wavefronts in transversely isotropic media are elliptical but P- and SV-wavefronts are not.]]
 
[[File:Sega14.jpg|thumb|left|FIG. A-14. <b>Anisotropy</b>. (<b>a</b>) Application of Huygens&#x2019; principle to anisotropic velocity illustrates why phase and ray velocities may differ in both direction and magnitude. (<b>b</b>) The application of Fermat&#x2019;s principle to anisotropic velocity illustrates why the angle of incidence for a reflection for a coincident source and receiver may not make a right angle with the reflector. (<b>c</b>) SH-wavefronts in transversely isotropic media are elliptical but P- and SV-wavefronts are not.]]
With a vertical symmetry axis, pure P- and S-waves may exist only in certain directions. SH-wavefronts are ellipsoidal in shape (see Figure [[Dictionary:Fig_A-14|A-14]]c) and SV- and P-modes of propagation are coupled with nonelliptical wavefronts that in general are not othogonal to the directions of wave propagation. <b>Phase velocity</b> (<b>wavefront velocity</b>) perpendicular to the wavefront surface of constant phase and <b>ray velocity</b> in the direction of energy transport (also called <b>group velocity</b>) are generally not in the same direction (see Figure [[Dictionary:Fig_A-14|A-14]]a). The reciprocal of phase velocity (also a vector quantity) is called <b>slowness</b>. SV-wavefronts may have cusps. See Thomsen, 2002<ref name="Thomsen 2002">{{cite book | last=Thomsen | first=Leon | title=Understanding Seismic Anisotropy in Exploration and Exploitation | publisher=Society of Exploration Geophysicists | year=2002 | doi=10.1190/1.9781560801986}}</ref>.
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With a vertical symmetry axis, pure P- and S-waves may exist only in certain directions. SH-wavefronts are ellipsoidal in shape (see Figure [[Special:MyLanguage/Dictionary:Fig_A-14|A-14]]c) and SV- and P-modes of propagation are coupled with nonelliptical wavefronts that in general are not othogonal to the directions of wave propagation. <b>Phase velocity</b> (<b>wavefront velocity</b>) perpendicular to the wavefront surface of constant phase and <b>ray velocity</b> in the direction of energy transport (also called <b>group velocity</b>) are generally not in the same direction (see Figure [[Special:MyLanguage/Dictionary:Fig_A-14|A-14]]a). The reciprocal of phase velocity (also a vector quantity) is called <b>slowness</b>. SV-wavefronts may have cusps. See Thomsen, 2002<ref name="Thomsen 2002">{{cite book | last=Thomsen | first=Leon | title=Understanding Seismic Anisotropy in Exploration and Exploitation | publisher=Society of Exploration Geophysicists | year=2002 | doi=10.1190/1.9781560801986}}</ref>.
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== References ==
 
== References ==
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Revision as of 16:22, 14 February 2019

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FIG. T-13. Transverse isotropy. (a) Wavefront with vertical axis of symmetry (TIV); (b) with horizontal symmetry axis (TIH) leading to azimuthal anisotropy; (c) phase (wavefront) angle θ and group (ray) angle for transverse isotropy; (d) elliptical wavefront where ε=–δ; in this case VNMO>Vvertical; (e) anisotropic wavefront where ε=δ; in this case VNMO<Vvertical; (f) wavefront for tilted symmetry axis; (g) wave equations for transverse isotropy.

Transverse isotropy. It involves elastic properties that are the same in any direction perpendicular to a symmetry axis but different parallel to the axis, and it has five independent elastic constants; see Thomsen anisotropic parameters and see Figure T-13.

This symmetry is like a crystal having hexagonal symmetry; see Figure S-29.

A sequence of generally horizontal, isotropic layers (such as sedimentary bedding) tends to produce layering anisotropy (sometimes called periodic thin-layer anisotropy, although the layering need not be periodic) for wavelengths that are appreciably larger than the layer thickness. The axis of symmetry is generally perpendicular to the bedding, more-or-less vertical, with the velocities of P-waves parallel to the bedding and S-waves that are polarized parallel to the bedding being larger than for those perpendicular to the bedding. Parallel isotropic layering, where there are more than eight or so layers per wavelength, behaves as a polar anisotropic medium. Roughly horizontal layering is also called transverse isotropy because properties are the same in any transverse direction with a vertical axis of symmetry (TIV). See also azimuthal asymmetry.

FIG. A-14. Anisotropy. (a) Application of Huygens’ principle to anisotropic velocity illustrates why phase and ray velocities may differ in both direction and magnitude. (b) The application of Fermat’s principle to anisotropic velocity illustrates why the angle of incidence for a reflection for a coincident source and receiver may not make a right angle with the reflector. (c) SH-wavefronts in transversely isotropic media are elliptical but P- and SV-wavefronts are not.

With a vertical symmetry axis, pure P- and S-waves may exist only in certain directions. SH-wavefronts are ellipsoidal in shape (see Figure A-14c) and SV- and P-modes of propagation are coupled with nonelliptical wavefronts that in general are not othogonal to the directions of wave propagation. Phase velocity (wavefront velocity) perpendicular to the wavefront surface of constant phase and ray velocity in the direction of energy transport (also called group velocity) are generally not in the same direction (see Figure A-14a). The reciprocal of phase velocity (also a vector quantity) is called slowness. SV-wavefronts may have cusps. See Thomsen, 2002[1].


References

  1. Thomsen, Leon (2002). Understanding Seismic Anisotropy in Exploration and Exploitation. Society of Exploration Geophysicists. doi:10.1190/1.9781560801986.