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Zoeppritz equations

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Journal Article
Published: 27 May 2020
Journal of Environmental and Engineering Geophysics (2020) 25 (1): 89–100.
.... 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
Journal: Geophysics
Published: 24 April 2018
Geophysics (2018) 83 (3): R245–R255.
... 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
Journal: Geophysics
Published: 18 February 2016
Geophysics (2016) 81 (2): N1–N15.
... 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
Journal: Geophysics
Published: 29 July 2015
Geophysics (2015) 80 (5): R239–R250.
...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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Book: AVO
Series: Society of Exploration Geophysicists Geophysics Reprint Series
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...
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...
Journal Article
Journal: Geophysics
Published: 01 December 1999
Geophysics (1999) 64 (6): 1920–1927.
...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...
Journal Article
Journal: Geophysics
Published: 01 September 1985
Geophysics (1985) 50 (9): 1522.
... 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...
Journal Article
Journal: Geophysics
Published: 01 April 1985
Geophysics (1985) 50 (4): 609–614.
...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...
Journal Article
Published: 01 October 1962
Bulletin of the Seismological Society of America (1962) 52 (4): 923–955.
... 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...
Journal Article
Published: 01 February 1974
Bulletin of the Seismological Society of America (1974) 64 (1): 285.
.... Smith T. J. (1962) . Generally applicable solutions of Zoeppritz' amplitude equations , Bull. Seism. Soc. Am. 52 , 923 - 955 . Richter C. F. (1958) . Elementary Seismology , W. H. Freeman...
Journal Article
Published: 01 February 1970
Bulletin of the Seismological Society of America (1970) 60 (1): 277–280.
.... Smith T. J. (1962) . Generally applicable solutions of Zoeppritz' amplitude equations , Bull. Seism. Soc. Am. 52 , 923 - 955 . Richter C. F. (1958) . Elementary Seismology , W. H. Freeman...
Journal Article
Journal: The Leading Edge
Published: 01 September 2015
The Leading Edge (2015) 34 (9): 1036–1047.
... 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
Journal: Geophysics
Published: 27 March 2024
Geophysics (2024) 89 (3): R247–R260.
... 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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Image
Comparison of the exact <span class="search-highlight">Zoeppritz</span> <span class="search-highlight">equations</span> (the solid line), FMR-<span class="search-highlight">Zoeppritz</span>...
Published: 23 June 2021
Figure 3. Comparison of the exact Zoeppritz equations (the solid line), FMR-Zoeppritz equations ( γ dry 2 = 2.0 , the circles), FMR-Zoeppritz equations ( γ dry 2 = 2.333 , the stars), and FMR-Zoeppritz equations ( γ dry 2 = 2.5 , the triangles) for model 1
Image
Comparison of the exact <span class="search-highlight">Zoeppritz</span> <span class="search-highlight">equations</span> (the solid line), FMR-<span class="search-highlight">Zoeppritz</span>...
Published: 23 June 2021
Figure 2. Comparison of the exact Zoeppritz equations (the solid line), FMR-Zoeppritz equations ( γ dry 2 = 2.333 , circles), Aki-Richards approximation equations (stars), and Russell approximation equations (triangles) for three models. (a-c) The P-wave reflection coefficients and (d
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The AVO responses of the PP-reflections by solving the <span class="search-highlight">Zoeppritz</span> <span class="search-highlight">equations</span>:...
Published: 12 October 2021
Figure 10. The AVO responses of the PP-reflections by solving the Zoeppritz equations: (a) the amplitude and (b) the phase angle.
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Angle gathers from forward modeling using <span class="search-highlight">Zoeppritz</span> <span class="search-highlight">equations</span>. (a) PP- and ...
Published: 17 October 2016
Figure 4. Angle gathers from forward modeling using Zoeppritz equations. (a) PP- and (b) PS-reflectivity angle gathers.
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A comparison between the full <span class="search-highlight">Zoeppritz</span> <span class="search-highlight">equations</span> (see Appendix  A ) and th...
Published: 20 March 2014
Figure 9. A comparison between the full Zoeppritz equations (see Appendix  A ) and the two forms of the Aki-Richards equation computed over the top of a gas sand, where (a) shows the intercept-gradient-curvature approximation of equation  12 and (b) shows the Fatti et al. (1994) approximation
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Exact solutions for the <span class="search-highlight">Zoeppritz</span> <span class="search-highlight">equations</span> for (a) brine-saturated SACROC ...
Published: 01 February 2010
Figure 5. Exact solutions for the Zoeppritz equations for (a) brine-saturated SACROC material, and (b) CO 2 -saturated SACROC material. Results obtained from CREWES Zoeppritz Explorer 2.2. The critical angle for (a) is 56°; (b) is 73°.