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| [[File:Reflection at an interface.png|thumb|Diagram showing the mode conversions that occur when a P-wave reflects off an interface at non-normal incidence]]
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| In [[geophysics]] and [[reflection seismology]], the '''Zoeppritz equations''' are a set of equations that describe the partitioning of [[seismic wave]] energy at an interface, typically a boundary between two different layers of rock. They are named after their author, the German [[geophysicist]] [[Karl Bernhard Zoeppritz]], who died before they were published in 1919.<ref>Zoeppritz, Karl (1919). Erdbebenwellen VII. VIIb. Über Reflexion und Durchgang seismischer Wellen durch Unstetigkeitsflächen. Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-physikalische Klasse, 66-84.</ref>
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| The equations are important in geophysics because they relate the amplitude of [[P-wave]], incident upon a plane interface, and the amplitude of [[Reflection (physics)|reflected]] and [[refraction|refracted]] P- and [[S-waves]] to the [[angle of incidence]].<ref name="SheriffGeldart">Sheriff, R. E., Geldart, L. P., (1995), 2nd Edition. Exploration Seismology. Cambridge University Press.</ref> They are the basis for investigating the factors affecting the amplitude of a returning seismic wave when the angle of incidence is altered - also known as [[amplitude versus offset]] analysis - which is a helpful technique in the detection of [[petroleum reservoir]]s.
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| The Zoeppritz equations were not the first to describe the amplitudes of reflected and refracted waves at a plane interface. [[Cargill Gilston Knott]] used an approach in terms of potentials almost 30 years earlier, in 1899, to derive [[Knott's equations]]. Both approaches are valid and Zoeppritz's approach is more easily understood.<ref name=SheriffGeldart />
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| ==Equations==
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| There are 4 equations with 4 unknowns and although they can be solved, they do not give an intuitive understanding for how the reflection amplitudes vary with the rock properties involved ([[density]], velocity etc.).<ref name="Shuey-1985 ">{{cite journal | url=http://library.seg.org/getabs/servlet/GetabsServlet?prog=normal&id=GPYSA7000050000004000609000001&idtype=cvips&gifs=yes&ref=no | title=A simplification of the Zoeppritz equations | author=Shuey, R. T. | journal=Geophysics |date=April 1985 | volume=50 | issue=9 | pages=609–614 | doi=10.1190/1.1441936|bibcode = 1985Geop...50..609S }}</ref> Several attempts have been made to develop approximations to the Zoeppritz Equations, such as Bortfeld’s (1961) and Aki & Richards’ (1980),<ref>Aki, K. and Richards, P. G., 1980, Quantitative seismology: Theory and methods, v.1 : W.H. Freeman and Co.</ref> but the most successful of these is the Shuey's, which assumes [[Poisson's ratio]] to be the elastic property most directly related to the angular dependence of the reflection coefficient.
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| ===Shuey Equation===
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| The 3-term Shuey Equation can be written a number of ways, the following is a common form:<ref>Avesth, P, T Mukerji and G Mavko (2005). Quantitative seismic interpretation. Cambridge University Press, Cambridge, UK</ref>
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| :<math>R(\theta ) = R(0) + G \sin^2 \theta + F ( \tan^2 \theta - \sin^2 \theta )</math>
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| where
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| :<math>R(0) = \frac{1}{2} \left ( \frac{\Delta V_\mathrm{P}}{V_\mathrm{P}} + \frac{\Delta \rho}{\rho} \right ) </math> | |
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| :<math>G = \frac{1}{2} \frac{\Delta V_\mathrm{P}}{V_\mathrm{P}} - 2 \frac{V^2_\mathrm{S}}{V^2_\mathrm{P}} \left ( \frac{\Delta \rho}{\rho} + 2 \frac{\Delta V_\mathrm{S}}{V_\mathrm{S}} \right ) </math> ; <math>F = \frac{1}{2}\frac{\Delta V_\mathrm{P}}{V_\mathrm{P}} </math> | |
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| where <math>{\theta}</math>=angle of incidence; <math>{V_p}</math> = P-wave velocity in medium; <math>{{\Delta}V_p}</math> = P-wave velocity contrast across interface;<math>{V_s}</math> = S-wave velocity in medium; <math>{{\Delta}V_s}</math> = S-wave velocity contrast across interface; <math>{{\rho}}</math> = density in medium; <math>{{\Delta}{\rho}}</math> = density contrast across interface;
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| A proposed better approximation of Zoeppritz equations:
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| :<math>R(\theta ) = R(0) - A \sin^2 \theta </math>
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| and
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| :<math>A = 2 \frac{V^2_\mathrm{S}}{V^2_\mathrm{P}} \left ( \frac{\Delta \rho}{\rho} + 2 \frac{\Delta V_\mathrm{S}}{V_\mathrm{S}} \right ) </math>
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| In the Shuey Equation, R(0) is the reflection coefficient at normal incidence and is controlled by the contrast in acoustic impedances. G, often referred to as the AVO gradient, describes the variation of reflection amplitudes at intermediate offsets and the third term, F, describes the behaviour at large angles/far offsets that are close to the critical angle.
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| This equation can be further simplified by assuming that the angle of incidence is less than 30 degrees (i.e. the offset is relatively small), so the third term will tend to zero. This is the case in most seismic surveys and gives the “Shuey Approximation”:
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| :<math>R(\theta ) = R(0) + G \sin^2 \theta </math>
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| ==See also==
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| *[[Amplitude versus offset]], a practical application of the phenomenon described by these equations.
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| *{{Ill|de|Karl Zöppritz (Geographer)|Karl Zöppritz (Geograph)}}
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| ==Further reading==
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| A full derivation of these equations can be found in most [[exploration geophysics]] text books, such as:
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| *Sheriff, R. E., Geldart, L. P., (1995), 2nd Edition. Exploration Seismology. Cambridge University Press.
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| ==References==
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| {{reflist}}
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| ==External links==
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| * [http://www.crewes.org/ResearchLinks/ExplorerPrograms/ZE/ZEcrewes.html crewes.org]
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| [[Category:Seismology measurement]]
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| [[Category:Petroleum geology]]
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Oscar is how he's called and he totally loves this title. To collect badges is what her family members and her appreciate. Her spouse and her reside in Puerto Rico but she will have to move one working day or an additional. For years I've been operating as a payroll clerk.
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