A simplification of the Zoeppritz equations

Geophysics ◽  
1985 ◽  
Vol 50 (4) ◽  
pp. 609-614 ◽  
Author(s):  
R. T. Shuey
Keyword(s):  
Reflection Coefficient ◽  
Elastic Properties ◽  
Critical Angle ◽  
Wave Reflection ◽  
Normal Incidence ◽  
Compressional Wave ◽  
Zoeppritz Equations ◽  
The Third ◽  
Reflection Data ◽  
Simplified Formula

The compressional wave reflection coefficient R(θ) given by the Zoeppritz equations is simplified to the following: [Formula: see text] The first term gives the amplitude at normal incidence (θ = 0), the second term characterizes R(θ) at intermediate angles, and the third term describes the approach to critical angle. The coefficient of the second term is that combination of elastic properties which can be determined by analyzing the offset dependence of event amplitude in conventional multichannel reflection data. If the event amplitude is normalized to its value for normal incidence, then the quantity determined is [Formula: see text] [Formula: see text] specifies the normal, gradual decrease of amplitude with offset; its value is constrained well enough that the main information conveyed is [Formula: see text] where [Formula: see text] is the contrast in Poisson’s ratio at the reflecting interface and [Formula: see text] is the amplitude at normal incidence. This simplified formula for R(θ) accounts for all of the relations between R(θ) and elastic properties first described by Koefoed in 1955.

Waveform‐based AVO inversion and AVO prediction‐error

Geophysics ◽  
1996 ◽  
Vol 61 (6) ◽  
pp. 1575-1588 ◽  
Author(s):  
James L. Simmons ◽  
Milo M. Backus
Keyword(s):  
Reflection Coefficient ◽  
Thin Layer ◽  
Shear Wave ◽  
Prediction Error ◽  
Wave Reflection ◽  
A Priori ◽  
Normal Incidence ◽  
Strong Increase ◽  
Time Range ◽  
Zoeppritz Equations

A practical approach to linear prestack seismic inversion in the context of a locally 1-D earth is employed to use amplitude variation with offset (AVO) information for the direct detection in hydrocarbons. The inversion is based on the three‐term linearized approximation to the Zoeppritz equations. The normal‐incidence compressional‐wave reflection coefficient [Formula: see text] models the background reflectivity in the absence of hydrocarbons and incorporates the mudrock curve and Gardner’s equation. Prediction‐error parameters, [Formula: see text] and [Formula: see text], represent perturbations in the normal‐incidence shear‐wave reflection coefficient and the density contribution to the normal incidence reflectivity, respectively, from that predicted by the mudrock curve and Gardner’s equation. This prediction‐error approach can detect hydrocarbons in the absence of an overall increase in AVO, and in the absence of bright spots, as expected in theory. Linear inversion is applied to a portion of a young, Tertiary, shallow‐marine data set that contains known hydrocarbon accumulations. Prestack data are in the form of angle stack, or constant offset‐to‐depth ratio, gathers. Prestack synthetic seismograms are obtained by primaries‐only ray tracing using the linearized approximation to the Zoeppritz equations to model the reflection amplitudes. Where the a priori assumptions hold, the data are reproduced with a single parameter [Formula: see text]. Hydrocarbons are detected as low impedance relative to the surrounding shales and the downdip brine‐filled reservoir on [Formula: see text], also as positive perturbations (opposite polarity relative to [Formula: see text]) on [Formula: see text] and [Formula: see text]. The maximum perturbation in [Formula: see text] from the normal‐incidence shear‐wave reflection coefficient predicted by the a priori assumptions is 0.08. Hydrocarbon detection is achieved, although the overall seismic response of a gas‐filled thin layer shows a decrease in amplitude with offset (angle). The angle‐stack data (70 prestack ensembles, 0.504–1.936 s time range) are reproduced with a data residual that is 7 dB down. Reflectivity‐based prestack seismograms properly model a gas/water contact as a strong increase in AVO and a gas‐filled thin layer as a decrease in AVO.

Linearized amplitude variation with offset (AVO) inversion with supercritical angles

Geophysics ◽  
2006 ◽  
Vol 71 (5) ◽  
pp. E49-E55 ◽  
Author(s):  
Jonathan E. Downton ◽  
Charles Ursenbach
Keyword(s):  
Reflection Coefficient ◽  
Critical Angle ◽  
Waveform Inversion ◽  
Phase Variation ◽  
Reflection Coefficients ◽  
Amplitude Variation ◽  
Amplitude Variation With Offset ◽  
Zoeppritz Equations ◽  
Avo Inversion ◽  
Angle Data

Contrary to popular belief, a linearized approximation of the Zoeppritz equations may be used to estimate the reflection coefficient for angles of incidence up to and beyond the critical angle. These supercritical reflection coefficients are complex, implying a phase variation with offset in addition to amplitude variation with offset (AVO). This linearized approximation is then used as the basis for an AVO waveform inversion. By incorporating this new approximation, wider offset and angle data may be incorporated in the AVO inversion, helping to stabilize the problem and leading to more accurate estimates of reflectivity, including density reflectivity.

Frequency-dependent spherical-wave reflection coefficient inversion in acoustic media: Theory to practice

Geophysics ◽  
2020 ◽  
Vol 85 (4) ◽  
pp. R425-R435
Author(s):  
Binpeng Yan ◽  
Shangxu Wang ◽  
Yongzhen Ji ◽  
Xingguo Huang ◽  
Nuno V. da Silva
Keyword(s):  
Reflection Coefficient ◽  
Wave Reflection ◽  
Incident Angle ◽  
Spherical Wave ◽  
Frequency Dependent ◽  
Amplitude Variation With Offset ◽  
Reflection Data ◽  
Alternative Approach ◽  
Acoustic Media ◽  
The Time Domain

As an approximation of the spherical-wave reflection coefficient (SRC), the plane-wave reflection coefficient does not fully describe the reflection phenomenon of a seismic wave generated by a point source. The applications of SRC to improve analyses of seismic data have also been studied. However, most of the studies focus on the time-domain SRC and its benefit to using the long-offset information instead of the dependency of SRC on frequency. Consequently, we have investigated and accounted for the frequency-dependent spherical-wave reflection coefficient (FSRC) and analyzed the feasibility of this type of inversion. Our inversion strategy requires a single incident angle using reflection data for inverting the density and velocity ratios, which is distinctly different from conventional inversion methods using amplitude variation with offset. Hence, this investigation provides an alternative approach for estimating media properties in some contexts, especially when the range of aperture of the reflection angles is limited. We apply the FSRC theory to the inversion of noisy synthetic and field data using a heuristic algorithm. The multirealization results of the inversion strategy are consistent with the feasibility analysis and demonstrate the potential of the outlined method for practical application.

Impedance‐type approximations of the P–P elastic reflection coefficient: Modeling and AVO inversion

Geophysics ◽  
2004 ◽  
Vol 69 (2) ◽  
pp. 592-598 ◽  
Author(s):  
Lúcio T. Santos ◽  
Martin Tygel
Keyword(s):  
Reflection Coefficient ◽  
Normal Incidence ◽  
Simple Expression ◽  
P Wave ◽  
Reflection Coefficients ◽  
Seismic Reflection Data ◽  
Acoustic Reflection ◽  
Reflection Data ◽  
Avo Inversion ◽  
Elastic Impedance

The normal‐incidence elastic compressional reflection coefficient admits an exact, simple expression in terms of the acoustic impedance, namely the product of the P‐wave velocity and density, at both sides of an interface. With slight modifications a similar expression can, also exactly, express the oblique‐incidence acoustic reflection coefficient. A severe limitation on the use of these two reflection coefficients in analyzing seismic reflection data is that they provide no information on shear‐wave velocities that refer to the interface. We address the natural question of whether a suitable impedance concept can be introduced for which arbitrary P–P reflection coefficients can be expressed in a form analogous to their acoustic counterparts. Although no closed‐form exact solution exists, our analysis provides a general framework for which, under suitable restrictions of the medium parameters, possible impedance functions can be derived. In particular, the well‐established concept of elastic impedance and the recently introduced concept of reflection impedance can be better understood. Concerning these two impedances, we examine their potential for modeling and for estimating the AVO indicators of intercept and gradient. For typical synthetic examples, we show that the reflection impedance formulation provides consistently better results than those obtained using the elastic impedance.

ESTIMATION FORMULAS FOR WAVE REFLECTION COEFFICIENT OF LONG-PERIOD WAVE ABSORBING MOUND STRUCTURES

2019 ◽  
Vol 75 (2) ◽  
pp. I_787-I_792
Author(s):  
Jun MITSUI ◽  
Iwao HASEGAWA ◽  
Masashi TANAKA ◽  
Akira MATSUMOTO
Keyword(s):  
Reflection Coefficient ◽  
Wave Reflection ◽  
Long Period ◽  
Period Wave

scholarly journals The application to real data of a technique for measuring the plane‐wave reflection coefficient of the ocean bottom

1982 ◽  
Vol 72 (S1) ◽  
pp. S97-S97
Author(s):  
George V. Frisk ◽  
Douglas R. Mook ◽  
James A. Doutt ◽  
Earl E. Hays ◽  
Alan V. Oppenheim
Keyword(s):  
Reflection Coefficient ◽  
Plane Wave ◽  
Wave Reflection ◽  
Real Data ◽  
Ocean Bottom

Effect of caffeine on aortic elastic properties and wave reflection

2003 ◽  
Vol 21 (3) ◽  
pp. 563-570 ◽  
Author(s):  
Charalambos Vlachopoulos ◽  
Kozo Hirata ◽  
Michael F O'Rourke
Keyword(s):  
Elastic Properties ◽  
Wave Reflection ◽  
Aortic Elastic Properties
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