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Zoeppritz equation

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Journal Article

The ℓ 1 − 2 -norm-regularized basis pursuit seismic inversion based on the exact Zoeppritz equation

Guangtan Huang, Shuying Wei, Davide Gei, Tongtao Wang
Journal: Geophysics
Published: 27 March 2024
Geophysics (2024) 89 (3): R247–R260.
DOI: https://doi.org/10.1190/geo2022-0336.1
... constraints. For the estimation of elastic parameters, a high-resolution and reliable seismic basis pursuit inversion is developed based on the exact Zoeppritz equation. Furthermore, the ℓ 1 − 2 norm is developed as a constraint, wherein a regularized function is minimized with the alternating direction...
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Book Chapter

The Zoeppritz equations

Author(s)
Dries Gisolf, Eric Verschuur
Book: The Principles of Quantitative Acoustical Imaging
Publisher: EAGE
Published: 01 January 2010
EISBN: 9789462820135
... The Zoeppritz equations listed below are different from the ones derived in Aki and Richards (2002) , because they are based on the potentials as defined in Eq. (9.4) , and not on displacement. In presenting the equations we will follow the approach adopted by Aki Richards and first define...
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Zoeppritz Equations and their Approximations

Author(s)
Satinder Chopra, John P. Castagna
Book: AVO
Series: Society of Exploration Geophysicists Geophysics Reprint Series
Publisher: Society of Exploration Geophysicists
Published: 01 January 2014
EISBN: 9781560803201
... of incidence (hence, of source-receiver offset) and of the media’s elastic properties, which comprise densities and bulk and shear moduli. The Zoeppritz equations (Chapter 1) give the reflection and transmission coefficients for plane waves, as a function of the angle of incidence and as a function...
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Abstract When an incident P-wave strikes the boundary (or interface) between two media obliquely, the wave is split into reflected and refracted P-wave components and reflected and refracted S-wave components. The reflection and transmission coefficients vary as a function of the angle of incidence (hence, of source-receiver offset) and of the media’s elastic properties, which comprise densities and bulk and shear moduli. The Zoeppritz equations (Chapter 1) give the reflection and transmission coefficients for plane waves, as a function of the angle of incidence and as a function of the three independent elastic parameters on each side of the reflecting interface. If the reflection amplitude is observed as a function of the angle of incidence, the variation of that parameter can be used to make inferences about the elastic parameters.

Journal Article

A simplification of the Zoeppritz equations

R. T. Shuey
Journal: Geophysics
Published: 01 April 1985
Geophysics (1985) 50 (4): 609–614.
DOI: https://doi.org/10.1190/1.1441936
...R. T. Shuey Abstract The compressional wave reflection coefficient R(theta ) given by the Zoeppritz equations is simplified to the following:EquationThe first term gives the amplitude at normal incidence (theta = 0), the second term characterizes R(theta ) at intermediate angles, and the third term...
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Journal Article

Approximations to the Zoeppritz equations and their use in AVO analysis

Yanghua Wang
Journal: Geophysics
Published: 01 December 1999
Geophysics (1999) 64 (6): 1920–1927.
DOI: https://doi.org/10.1190/1.1444698
...Yanghua Wang Abstract To efficiently invert seismic amplitudes for elastic parameters, pseudoquartic approximations to the Zoeppritz equations are derived to calculate P-P-wave reflection and transmission coefficients as a function of the ray parameter p. These explicit expressions have a compact...
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Journal Article

Probabilistic estimation of density and shear information from Zoeppritz's equation

Ivan Lehocki, Per Avseth, Vedad Hadžiavdić
Journal: The Leading Edge
Published: 01 September 2015
The Leading Edge (2015) 34 (9): 1036–1047.
DOI: https://doi.org/10.1190/tle34091036.1
... uncertainties in the elastic parameters. Zoeppritz's equation for calculation of PP reflection coefficients (RCs) is used to create 2D conditional probability density functions (CPDFs) of model parameters which are constructed by systematic grid search over the entire space of the model parameters. The method...
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Journal Article

Constrained nonlinear amplitude variation with offset inversion using Zoeppritz equations

Ali Gholami, Hossein S. Aghamiry, Mostafa Abbasi
Journal: Geophysics
Published: 24 April 2018
Geophysics (2018) 83 (3): R245–R255.
DOI: https://doi.org/10.1190/geo2017-0543.1
... equations, but traditional inversion techniques are based on various linear or quasilinear approximations to these nonlinear equations. We have developed an efficient algorithm for nonlinear AVO inversion of precritical reflections using the exact Zoeppritz equations in multichannel and multi-interface form...
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Journal Article

Amplitude variation with angle inversion using the exact Zoeppritz equations — Theory and methodology

Lixia Zhi, Shuangquan Chen, Xiang-yang Li
Journal: Geophysics
Published: 18 February 2016
Geophysics (2016) 81 (2): N1–N15.
DOI: https://doi.org/10.1190/geo2014-0582.1
... the exact Zoeppritz equations is preferred by several authors. Liu et al. (2011) derive the analytical Fréchet derivatives for P- and S-wave velocities and density using the exact Zoeppritz equations.  Wei and Chen (2011) present a practical method for the joint inversion of PP and PS data by expressing...
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Journal Article

Joint PP and PS AVA seismic inversion using exact Zoeppritz equations

Jun Lu, Zhen Yang, Yun Wang, Ying Shi
Journal: Geophysics
Published: 29 July 2015
Geophysics (2015) 80 (5): R239–R250.
DOI: https://doi.org/10.1190/geo2014-0490.1
...Jun Lu; Zhen Yang; Yun Wang; Ying Shi ABSTRACT The Zoeppritz equations describe the relationship between seismic properties that are useful for the interpretation of lithology and fluid properties. Various approximations to the Zoeppritz equations are often used in linear amplitude variation...
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Journal Article

Bayesian Time-lapse Difference Inversion Based on the exact Zoeppritz Equations with Blockiness Constraint

Lin Zhou, Jianping Liao, Jingye Li, Xiaohong Chen, Tianchun Yang, Andrew Hursthouse
Published: 27 May 2020
Journal of Environmental and Engineering Geophysics (2020) 25 (1): 89–100.
DOI: https://doi.org/10.2113/JEEG19-045
.... Previously numerous time-lapse seismic inversion methods based on the approximate formulas of exact Zoeppritz equations or wave equations have been used to estimate these changes. However the low accuracy of calculations using approximate formulas and the significant calculation effort for the wave equations...
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Journal Article

Generally applicable solutions of Zoeppritz' amplitude equations

Keith McCamy, Robert P. Meyer, Thomas J. Smith
Published: 01 October 1962
Bulletin of the Seismological Society of America (1962) 52 (4): 923–955.
DOI: https://doi.org/10.1785/BSSA0520040923
... of the amplitude of an incident wave front at an interface. The Zoeppritz equations describe this partition for a plane wave and while they are simple simultaneous linear equations, their solution by hand methods is tedious and receptive to errors. Solutions of these equations have been obtained by numerical...
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The accuracy comparison of nonlinear <span class="search-highlight">equation</span>  1 with <span class="search-highlight">Zoeppritz</span> <span class="search-highlight">equation</span> a...

The accuracy comparison of nonlinear equation  1 with Zoeppritz equation a...

Published: 13 June 2017
Figure 1. The accuracy comparison of nonlinear equation  1 with Zoeppritz equation and Aki-Richards approximation.
Image
Comparison of exact <span class="search-highlight">Zoeppritz</span> <span class="search-highlight">equation</span>, Aki-Richard approximate <span class="search-highlight">equation</span>, Y...

Comparison of exact Zoeppritz equation, Aki-Richard approximate equation, Y...

Published: 03 October 2013
Figure 2. Comparison of exact Zoeppritz equation, Aki-Richard approximate equation, YPD equation, and YPD-EI of the positive reflector. (a) Reflection coefficients with exact Zoeppritz equation, Aki-Richard approximate equation, YPD equation, and YPD-EI (b) Difference of reflection coefficients
Image
Comparison of exact <span class="search-highlight">Zoeppritz</span> <span class="search-highlight">equation</span>, Aki-Richard approximate <span class="search-highlight">equation</span>, Y...

Comparison of exact Zoeppritz equation, Aki-Richard approximate equation, Y...

Published: 03 October 2013
Figure 1. Comparison of exact Zoeppritz equation, Aki-Richard approximate equation, YPD equation, and YPD-EI of the negative reflector. (a) Reflection coefficients with exact Zoeppritz equation, Aki-Richard approximate equation, YPD equation, and YPD-EI (b) Difference of reflection coefficients
Image
Synthetic model generated with (a) the <span class="search-highlight">Zoeppritz</span> <span class="search-highlight">equation</span> and (b) <span class="search-highlight">equation</span> ...

Synthetic model generated with (a) the Zoeppritz equation and (b) equation ...

Published: 28 August 2006
Figure 2. Synthetic model generated with (a) the Zoeppritz equation and (b) equation 2 . (c) The difference between (a) and (b).
Journal Article

To: “A Simplification of the Zoeppritz Equations,” by R. T. Shuey, which appeared in GEOPHYSICS, v. 50, no. 4 (April 1985), p. 609–614

Journal: Geophysics
Published: 01 September 1985
Geophysics (1985) 50 (9): 1522.
DOI: https://doi.org/10.1190/1.1442022
... Abstract The acknowledgments should have included the statement that E. M. Haacke originally pointed out to me the possibility of studying the Zoeppritz equations by making a parabolic (Taylor) expansion of the linearized reflection amplitude, equation (1). Society of Exploration Geophysicists...
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Comparison of exact <span class="search-highlight">Zoeppritz</span> <span class="search-highlight">equation</span> (black lines), Aki-Richards approxim...

Comparison of exact Zoeppritz equation (black lines), Aki-Richards approxim...

Published: 22 September 2023
Figure 6. Comparison of exact Zoeppritz equation (black lines), Aki-Richards approximation (pink lines), Gray approximation (blue lines), and the new approximation (equation  14 , red lines) of the negative reflector. (a) The P-wave reflectivity and (b) error analysis.
Image
Comparison of exact <span class="search-highlight">Zoeppritz</span> <span class="search-highlight">equation</span> (black lines), Aki-Richards approxim...

Comparison of exact Zoeppritz equation (black lines), Aki-Richards approxim...

Published: 22 September 2023
Figure 7. Comparison of exact Zoeppritz equation (black lines), Aki-Richards approximation (pink lines), Gray approximation (blue lines), and the new approximation (equation  14 , red lines) of the positive reflector. (a) The P-wave reflectivity and (b) error analysis.
Image
The synthetic PP and PS angle gathers by the exact <span class="search-highlight">Zoeppritz</span> <span class="search-highlight">equation</span>. (a) ...

The synthetic PP and PS angle gathers by the exact Zoeppritz equation. (a) ...

Published: 29 March 2023
Figure 4. The synthetic PP and PS angle gathers by the exact Zoeppritz equation. (a) The PP gather, (b) PS gather in PS time, (c) PS gather in PP time obtained by the DTW algorithm, (d) the perfect PS gather with γ rep = 2.5 , and (e) the perfect PS gather with γ rep = 4
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Comparison of the exact <span class="search-highlight">Zoeppritz</span> <span class="search-highlight">equation</span>, the Aki-Richards approximation ...

Comparison of the exact Zoeppritz equation, the Aki-Richards approximation ...

Published: 09 January 2023
Figure 1. Comparison of the exact Zoeppritz equation, the Aki-Richards approximation equation, and the viscoelastic approximation reflection coefficient equation for the negative reflector at reference frequencies of 20 and 40 Hz.