P‐wave reflection coefficients for transversely isotropic models with vertical and horizontal axis of symmetry

Geophysics ◽  
1997 ◽  
Vol 62 (3) ◽  
pp. 713-722 ◽  
Author(s):  
Andreas Rüger
Keyword(s):  
Reflection Coefficient ◽  
Wave Reflection ◽  
Numerical Solutions ◽  
Anisotropy Parameter ◽  
Transversely Isotropic ◽  
Horizontal Axis ◽  
P Wave ◽  
Reflection Coefficients ◽  
Gradient Term ◽  
Axis Of Symmetry

The study of P‐wave reflection coefficients in anisotropic media is important for amplitude variation with offset (AVO) analysis. While numerical evaluation of the reflection coefficient is straightforward, numerical solutions do not provide analytic insight into the influence of anisotropy on the AVO signature. To overcome this difficulty, I present an improved approximation for P‐wave reflection coefficients at a horizontal boundary in transversely isotropic media with vertical axis of symmetry (VTI media). This solution has the same AVO‐gradient term describing the low‐order angular variation of the reflection coefficient as the equations published previously, but is more accurate for large incidence angles. The refined approximation is then extended to transverse isotropy with a horizontal axis of symmetry (HTI), which is caused typically by a system of vertical cracks. Comparison of the approximate reflection coefficients for P‐waves incident in the two vertical symmetry planes of HTI media indicates that the azimuthal variation of the AVO gradient is a function of the shear‐wave splitting parameter γ, and the anisotropy parameter describing P‐wave anisotropy for nearvertical propagation in the vertical plane containing the symmetry axis.

PP-wave reflection coefficients in weakly anisotropic elastic media

Geophysics ◽  
1998 ◽  
Vol 63 (6) ◽  
pp. 2129-2141 ◽  
Author(s):  
Václav Vavryuk ◽  
Ivan Peník
Keyword(s):  
Reflection Coefficient ◽  
Approximate Formula ◽  
Wave Reflection ◽  
Transversely Isotropic ◽  
Reflection Coefficients ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Pp Wave ◽  
Axis Of Symmetry ◽  
Amplitude Versus

Approximate PP-wave reflection coefficients for weak contrast interfaces separating elastic, weakly transversely isotropic media have been derived recently by several authors. Application of these coefficients is limited because the axis of symmetry of transversely isotropic media must be either perpendicular or parallel to the reflector. In this paper, we remove this limitation by deriving a formula for the PP-wave reflection coefficient for weak contrast interfaces separating two weakly but arbitrarily anisotropic media. The formula is obtained by applying the first‐order perturbation theory. The approximate coefficient consists of a sum of the PP-wave reflection coefficient for a weak contrast interface separating two background isotropic half‐spaces and a perturbation attributable to the deviation of anisotropic half‐spaces from their isotropic backgrounds. The coefficient depends linearly on differences of weak anisotropy parameters across the interface. This simplifies studies of sensitivity of such coefficients to the parameters of the surrounding structure, which represent a basic part of the amplitude‐versus‐offset (AVO) or amplitude‐versus‐azimuth (AVA) analysis. The reflection coefficient is reciprocal. In the same way, the formula for the PP-wave transmission coefficient can be derived. The generalization of the procedure presented for the derivation of coefficients of converted waves is also possible although slightly more complicated. Dependence of the reflection coefficient on the angle of incidence is expressed in terms of three factors, as in isotropic media. The first factor alone describes normal incidence reflection. The second yields the low‐order angular variations. All three factors describe the coefficient in the whole region, in which the approximate formula is valid. In symmetry planes of weakly anisotropic media of higher symmetry, the approximate formula reduces to the formulas presented by other authors. The accuracy of the approximate formula for the PP reflection coefficient is illustrated on the model with an interface separating an isotropic half‐space from a half‐space filled by a transversely isotropic material with a horizontal axis of symmetry. The results show a very good fit with results of the exact formula, even in cases of strong anisotropy and strong velocity contrast.

Variation of P-wave reflectivity with offset and azimuth in anisotropic media

Geophysics ◽  
1998 ◽  
Vol 63 (3) ◽  
pp. 935-947 ◽  
Author(s):  
Andreas Rüger
Keyword(s):  
Reflection Coefficient ◽  
Shear Wave ◽  
Wave Reflection ◽  
Symmetry Plane ◽  
Anisotropic Media ◽  
Shear Wave Splitting ◽  
P Wave ◽  
Reflection Coefficients ◽  
Wave Splitting ◽  
Splitting Parameter

P-wave amplitudes may be sensitive even to relatively weak anisotropy of rock mass. Recent results on symmetry‐plane P-wave reflection coefficients in azimuthally anisotropic media are extended to observations at arbitrary azimuth, large incidence angles, and lower symmetry systems. The approximate P-wave reflection coefficient in transversely isotropic media with a horizontal axis of symmetry (HTI) (typical for a system of parallel vertical cracks embedded in an isotropic matrix) shows that the amplitude versus offset (AVO) gradient varies as a function of the squared cosine of the azimuthal angle. This change can be inverted for the symmetry‐plane directions and a combination of the shear‐wave splitting parameter γ and the anisotropy coefficient [Formula: see text]. The reflection coefficient study is also extended to media of orthorhombic symmetry that are believed to be more realistic models of fractured reservoirs. The study shows the orthorhombic and HTI reflection coefficients are very similar and the azimuthal variation in the orthorhombic P-wave reflection response is a function of the shear‐wave splitting parameter γ and two anisotropy parameters describing P-wave anisotropy for near‐vertical propagation in the symmetry planes. The simple relationships between the reflection amplitudes and anisotropic coefficients given here can be regarded as helpful rules of thumb in quickly evaluating the importance of anisotropy in a particular play, integrating results of NMO and shear‐wave‐splitting analyses, planning data acquisition, and guiding more advanced numerical amplitude‐inversion procedures.

scholarly journals Elastic-Impedance-Based Fluid/Porosity Term and Fracture Weaknesses Inversion in Transversely Isotropic Media with a Tilted Axis of Symmetry

Geofluids ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-17
Author(s):  
Xinpeng Pan ◽  
Lin Li ◽  
Guangzhi Zhang ◽  
Yian Cui
Keyword(s):  
Reflection Coefficient ◽  
Seismic Data ◽  
Wave Reflection ◽  
Transversely Isotropic ◽  
Unknown Parameters ◽  
Fluid Identification ◽  
Pp Wave ◽  
Axis Of Symmetry ◽  
Dip Angle ◽  
Elastic Impedance

The rock containing a set of tilted fractures is equivalent to a transversely isotropic (TTI) medium with a tilted axis of symmetry. To implement fluid identification and tilted fracture detection, we propose an inversion approach of utilizing seismic data to simultaneously estimate parameters that are sensitive to fluids and tilted fractures. We first derive a PP-wave reflection coefficient and elastic impedance (EI) in terms of the dip angle, fluid/porosity term, shear modulus, density, and fracture weaknesses, and we present numerical examples to demonstrate how the PP-wave reflection coefficient and EI vary with the dip angle. Based on the information of dip angle of fractures provided by geologic and well data, we propose a two-step inversion approach of utilizing azimuthal seismic data to estimate unknown parameters involving the fluid/porosity term and fracture weaknesses: (1) the constrained sparse spike inversion (CSSI) for azimuthally anisotropic EI data and (2) the estimation of unknown parameters with the low-frequency constrained regularization term. Synthetic and real data demonstrate that fluid and fracture parameters are reasonably estimated, which may help fluid identification and fracture characterization.

scholarly journals Body‐wave radiation patterns and AVO in transversely isotropic media

Geophysics ◽  
1995 ◽  
Vol 60 (5) ◽  
pp. 1409-1425 ◽  
Author(s):  
Ilya Tsvankin
Keyword(s):  
Reflection Coefficient ◽  
Radiation Pattern ◽  
Transversely Isotropic ◽  
P Wave ◽  
Wave Radiation ◽  
Reflection Coefficients ◽  
Radiation Patterns ◽  
Avo Analysis ◽  
Transversely Isotropic Media ◽  
Isotropic Media

The angular dependence of reflection coefficients may be significantly distorted in the presence of elastic anisotropy. However, the influence of anisotropy on amplitude variation with offset (AVO) analysis is not limited to reflection coefficients. AVO signatures (e.g., AVO gradient) in anisotropic media are also distorted by the redistribution of energy along the wavefront of the wave traveling down to the reflector and back up to the surface. Significant anisotropy above the target horizon may be rather typical of sand‐shale sequences commonly encountered in AVO analysis. Here, I examine the influence of P‐ and S‐wave radiation patterns on AVO in the most common anisotropic model—transversely isotropic media. A concise analytic solution, obtained in the weak‐anisotropy approximation, provides a convenient way to estimate the impact of the distortions of the radiation patterns on AVO results. It is shown that the shape of the P‐wave radiation pattern in the range of angles most important to AVO analysis (0–40°) is primarily dependent on the difference between Thomsen parameters ε and δ. For media with ε − δ > 0 (the most common case), the P‐wave amplitude may drop substantially over the first 25–40° from vertical. There is no simple correlation between the strength of velocity anisotropy and angular amplitude variations. For instance, for models with a fixed positive ε − δ the amplitude distortions are less pronounced for larger values of ε and δ. The distortions of the SV‐wave radiation pattern are usually much more significant than those for the P‐wave. The anisotropic directivity factor for the incident wave may be of equal or greater importance for AVO than the influence of anisotropy on the reflection coefficient. Therefore, interpretation of AVO anomalies in the presence of anisotropy requires an integrated approach that takes into account not only the reflection coefficient but also the wave propagation above the reflector.

Reflection moveout approximations for P-waves in a moderately anisotropic homogeneous tilted transverse isotropy layer

Geophysics ◽  
2017 ◽  
Vol 82 (5) ◽  
pp. C175-C185 ◽  
Author(s):  
Ivan Pšenčík ◽  
Véronique Farra
Keyword(s):  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
P Wave ◽  
Weak Anisotropy ◽  
P Waves ◽  
3D Space ◽  
Axis Of Symmetry ◽  
Relative Errors ◽  
Symmetry Tests ◽  
Ti Media

We have developed approximate nonhyperbolic P-wave moveout formulas applicable to weakly or moderately anisotropic media of arbitrary anisotropy symmetry and orientation. Instead of the commonly used Taylor expansion of the square of the reflection traveltime in terms of the square of the offset, we expand the square of the reflection traveltime in terms of weak-anisotropy (WA) parameters. No acoustic approximation is used. We specify the formulas designed for anisotropy of arbitrary symmetry for the transversely isotropic (TI) media with the axis of symmetry oriented arbitrarily in the 3D space. Resulting formulas depend on three P-wave WA parameters specifying the TI symmetry and two angles specifying the orientation of the axis of symmetry. Tests of the accuracy of the more accurate of the approximate formulas indicate that maximum relative errors do not exceed 0.3% or 2.5% for weak or moderate P-wave anisotropy, respectively.

Two-step joint PP- and PS-wave three-term amplitude-variation with offset inversion

2016 ◽  
Vol 4 (4) ◽  
pp. T613-T625 ◽  
Author(s):  
Qizhen Du ◽  
Bo Zhang ◽  
Xianjun Meng ◽  
Chengfeng Guo ◽  
Gang Chen ◽  
...  
Keyword(s):  
Reflection Coefficient ◽  
Wave Reflection ◽  
Coefficient Matrix ◽  
P Wave ◽  
Inversion Method ◽  
S Wave ◽  
Amplitude Variation ◽  
Amplitude Variation With Offset ◽  
Avo Inversion ◽  
Pp Wave

Three-term amplitude-variation with offset (AVO) inversion generally suffers from instability when there is limited prior geologic or petrophysical constraints. Two-term AVO inversion shows higher instability compared with three-term AVO inversion. However, density, which is important in the fluid-type estimation, cannot be recovered from two-term AVO inversion. To reliably predict the P- and S-waves and density, we have developed a robust two-step joint PP- and PS-wave three-term AVO-inversion method. Our inversion workflow consists of two steps. The first step is to estimate the P- and S-wave reflectivities using Stewart’s joint two-term PP- and PS-AVO inversion. The second step is to treat the P-wave reflectivity obtained from the first step as the prior constraint to remove the P-wave velocity related-term from the three-term Aki-Richards PP-wave approximated reflection coefficient equation, and then the reduced PP-wave reflection coefficient equation is combined with the PS-wave reflection coefficient equation to estimate the S-wave and density reflectivities. We determined the effectiveness of our method by first applying it to synthetic models and then to field data. We also analyzed the condition number of the coefficient matrix to illustrate the stability of the proposed method. The estimated results using proposed method are superior to those obtained from three-term AVO inversion.

scholarly journals Analysis of fluid substitution in a porous and fractured medium

Geophysics ◽  
2011 ◽  
Vol 76 (3) ◽  
pp. WA157-WA166 ◽  
Author(s):  
Samik Sil ◽  
Mrinal K. Sen ◽  
Boris Gurevich
Keyword(s):  
Water Saturation ◽  
Transversely Isotropic ◽  
P Wave ◽  
Reflection Coefficients ◽  
Fluid Substitution ◽  
Fracture Density ◽  
Maximum Increase ◽  
Amplitude Variation With Offset ◽  
Fractured Medium ◽  
Medium Change

To improve quantitative interpretation of seismic data, we analyze the effect of fluid substitution in a porous and fractured medium on elastic properties and reflection coefficients. This analysis uses closed-form expressions suitable for fluid substitution in transversely isotropic media with a horizontal symmetry axis (HTI). For the HTI medium, the effect of changing porosity and water saturation on (1) P-wave moduli, (2) horizontal and vertical velocities, (3) anisotropic parameters, and (4) reflection coefficients are examined. The effects of fracture density on these four parameters are also studied. For the model used in this study, a 35% increase in porosity lowers the value of P-wave moduli by maximum of 45%. Consistent with the reduction in P-wave moduli, P-wave velocities also decrease by maximum of 17% with a similar increment in porosity. The reduction is always larger for the horizontal P-wave modulus than for the vertical one and is nearly independent of fracture density. The magnitude of the anisotropic parameters of the fractured medium also changes with increased porosity depending on the changes in the value of P-wave moduli. The reflection coefficients at an interface of the fractured medium with an isotropic medium change in accordance with the above observations and lead to an increase in anisotropic amplitude variation with offset (AVO) gradient with porosity. Additionally, we observe a maximum increase in P-wave modulus and velocity by 30% and 8%, respectively, with a 100% increase in water saturation. Water saturation also changes the anisotropic parameters and reflection coefficients. Increase in water saturation considerably increases the magnitude of the anisotropic AVO gradient irrespective of fracture density. From this study, we conclude that porosity and water saturation have a significant impact on the four studied parameters and the impacts are seismically detectable.

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