Weak-contrast approximation of the elastic scattering matrix in anisotropic media

Ursin, Bjørn; Haugen, Geir

doi:10.1007/bf00874584

Cited by 53 publications

(7 citation statements)

References 16 publications

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“…Considering a discrete subsurface model, the linear approximation of the PP‐wave reflection coefficient for the i th planar interface separating the i th and the i + 1th VTI layers can be expressed as (Ursin and Haugen ; Rüger ; Vavryčuk and Pšenčík ; Stovas and Ursin ; Shaw and Sen ; Zhang and Li )

R_{VTI} false(θ false) = R_{ISO} false(θ false) + \frac{1}{2} \sin^{2} θ Δ δ + \frac{1}{2} \sin^{2} θ \tan^{2} θ Δ ε,

where θ is the incidence phase angle, and can be approximated to the average angle

\bar{θ}

of the incidence phase angle and the transmission phase angle under the assumption of weak impedance contrast and small angle,

\bar{ρ V_{P 0}}

refers to average values of the two adjacent layers across the interface, Δ refers to contrasts in the properties of the two adjacent layers across the interface,

R_{ISO} false(θ false)

is the linear approximation of PP‐wave reflection coefficient for isotropic media (Bortfeld ; Richards and Frasier ; Aki and Richards )

\begin{matrix} true \\ right R_{ISO} (θ) & = & left \frac{1}{2} \frac{normalΔ false(ρ V_{P 0} false)}{\bar{ρ V_{P 0}}} \\ left + \frac{1}{2} \sin^{2} θ ([], \frac{normalΔ V_{P 0}}{\bar{V_{P 0}}} - {(\frac{2 \bar{V_{S 0}}}{\bar{V_{P 0}}})}^{2} \frac{normalΔ false(ρ {V_{S 0}}^{2} false)}{\bar{ρ {V_{S 0}}^{2}}}) \\ left + \frac{1}{...} \end{matrix}

…”

Section: Methodsmentioning

confidence: 99%

“…Exact solutions for reflection coefficients for anisotropic media have been studied by Keith and Crampin (), Daley and Hron (), Graebners () and Schoenberg and Protázio (), which enable amplitude variation with offset (AVO) analysis and modelling of VTI media. Approximations of reflection coefficients in anisotropic media are extensively investigated to provide expressions in simple and closed forms (Banik ; Thomsen ; Ursin and Haugen ; Rüger ; Rüger ; Vavryčuk and Pšenčík ; Zillmer, Gajewski and Kashtan ; Vavryčuk ; Stovas and Ursin ; Shaw and Sen ; Zhang and Li ). In spite of that, the inversion of amplitudes of reflected PP waves in VTI media is a difficult issue.…”

Section: Introductionmentioning

confidence: 99%

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Seismic amplitude inversion for the transversely isotropic media with vertical axis of symmetry

Zhang

2019

Geophysical Prospecting

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Transverse isotropy with a vertical axis of symmetry is a common form of anisotropy in sedimentary basins, and it has a significant influence on the seismic amplitude variation with offset. Although exact solutions and approximations of the PP‐wave reflection coefficient for the transversely isotropic media with vertical axis of symmetry have been explicitly studied, it is difficult to apply these equations to amplitude inversion, because more than three parameters need to be estimated, and such an inverse problem is highly ill‐posed. In this paper, we propose a seismic amplitude inversion method for the transversely isotropic media with a vertical axis of symmetry based on a modified approximation of the reflection coefficient. This new approximation consists of only three model parameters: attribute A, the impedance (vertical phase velocity multiplied by bulk density); attribute B, shear modulus proportional to an anellipticity parameter (Thomsen's parameter ε−δ); and attribute C, the approximate horizontal P‐wave phase velocity, which can be well estimated by using a Bayesian‐framework‐based inversion method. Using numerical tests we show that the derived approximation has similar accuracy to the existing linear approximation and much higher accuracy than isotropic approximations, especially at large angles of incidence and for strong anisotropy. The new inversion method is validated by using both synthetic data and field seismic data. We show that the inverted attributes are robust for shale‐gas reservoir characterization: the shale formation can be discriminated from surrounding formations by using the crossplot of the attributes A and C, and then the gas‐bearing shale can be identified through the combination of the attributes A and B. We then propose a rock‐physics‐based method and a stepwise‐inversion‐based method to estimate the P‐wave anisotropy parameter (Thomsen's parameter ε). The latter is more suitable when subsurface media are strongly heterogeneous. The stepwise inversion produces a stable and accurate Thomsen's parameter ε, which is proved by using both synthetic and field data.

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R_{VTI} false(θ false) = R_{ISO} false(θ false) + \frac{1}{2} \sin^{2} θ Δ δ + \frac{1}{2} \sin^{2} θ \tan^{2} θ Δ ε,

where θ is the incidence phase angle, and can be approximated to the average angle

\bar{θ}

of the incidence phase angle and the transmission phase angle under the assumption of weak impedance contrast and small angle,

\bar{ρ V_{P 0}}

refers to average values of the two adjacent layers across the interface, Δ refers to contrasts in the properties of the two adjacent layers across the interface,

R_{ISO} false(θ false)

is the linear approximation of PP‐wave reflection coefficient for isotropic media (Bortfeld ; Richards and Frasier ; Aki and Richards )

\begin{matrix} true \\ right R_{ISO} (θ) & = & left \frac{1}{2} \frac{normalΔ false(ρ V_{P 0} false)}{\bar{ρ V_{P 0}}} \\ left + \frac{1}{2} \sin^{2} θ ([], \frac{normalΔ V_{P 0}}{\bar{V_{P 0}}} - {(\frac{2 \bar{V_{S 0}}}{\bar{V_{P 0}}})}^{2} \frac{normalΔ false(ρ {V_{S 0}}^{2} false)}{\bar{ρ {V_{S 0}}^{2}}}) \\ left + \frac{1}{...} \end{matrix}

…”

Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Seismic amplitude inversion for the transversely isotropic media with vertical axis of symmetry

Zhang

2019

Geophysical Prospecting

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“…He finds the relations for the linearized reflection and transmission coefficients corresponding to an incident P-wave, including the P-S converted waves. Ursin and Haugen (1996) abandon the assumption of weak anisotropy and allow for a transversely isotropic background with a vertical axis of symmetry. They assume a horizontal weak-contrast interface (coinciding with a coordinate plane) separating two transversely isotropic media with a vertical (perpendicular to the interface) axis of symmetry and find the explicit expressions for the corresponding linearized reflectiontransmission coefficients.…”

Section: Introductionmentioning

confidence: 99%

Weak‐contrast reflection–transmission coefficients in a generally anisotropic background

Klimeš

2003

GEOPHYSICS

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Explicit equations for approximate linearized reflection-transmission coefficients at a generally oriented weak-contrast interface separating two generally and independently anisotropic media are presented. The equations are derived also for all singular directions and are thus valid in degenerate cases (e.g., in an isotropic background). The equations are expressed in general Cartesian coordinates, with arbitrary orientation of the interface. The explicit equations for linearized reflection-transmission coefficients have a very simple form-much simpler than the equations published previously. The equations for all reflection-transmission coefficients, with the exception of the unconverted transmitted wave, have a common form. The form of the equations is very suitable for inversion and for analyzing the sensitivity of seismic data to discontinuities in individual elastic moduli. The factors of proportionality of the contrasts of elastic moduli and density are expressed in terms of the slowness and polarization vectors of the corresponding generated wave and incident wave.

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“…The analytical expression for the P‐wave reflection coefficient even in the case of isotropic media is too complicated to provide insight into the influence of medium parameters. In order to overcome this problem, various approximations are developed in the assumption of weak contrast at the reflection boundary (Thomsen ; Blangy ; Ursin and Haugen ; Rüger ; Golikov and Stovas ) and weak anisotropy (Thomsen ). Weak‐contrast PP ‐reflection and transmission coefficients in weakly anisotropic media of an arbitrary symmetry were presented by Ps̆enc̆ík and Vavryc̆uk (), Vavryc̆uk and Ps̆enc̆ík (), Vavryc̆uk (), and Ps̆enc̆ík and Martins ().…”

Section: Introductionmentioning

confidence: 99%

Weak‐anisotropy approximation for P‐wave reflection coefficient at the boundary between two tilted transversely isotropic media

Ivanov

Stovas

2016

Geophysical Prospecting

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Existing and commonly used in industry nowadays, closed‐form approximations for a P‐wave reflection coefficient in transversely isotropic media are restricted to cases of a vertical and a horizontal transverse isotropy. However, field observations confirm the widespread presence of rock beds and fracture sets tilted with respect to a reflection boundary. These situations can be described by means of the transverse isotropy with an arbitrary orientation of the symmetry axis, known as tilted transversely isotropic media. In order to study the influence of the anisotropy parameters and the orientation of the symmetry axis on P‐wave reflection amplitudes, a linearised 3D P‐wave reflection coefficient at a planar weak‐contrast interface separating two weakly anisotropic tilted tranversely isotropic half‐spaces is derived. The approximation is a function of the incidence phase angle, the anisotropy parameters, and symmetry axes tilt and azimuth angles in both media above and below the interface. The expression takes the form of the well‐known amplitude‐versus‐offset “Shuey‐type” equation and confirms that the influence of the tilt and the azimuth of the symmetry axis on the P‐wave reflection coefficient even for a weakly anisotropic medium is strong and cannot be neglected. There are no assumptions made on the symmetry‐axis orientation angles in both half‐spaces above and below the interface. The proposed approximation can be used for inversion for the model parameters, including the orientation of the symmetry axes. Obtained amplitude‐versus‐offset attributes converge to well‐known approximations for vertical and horizontal transverse isotropic media derived by Rüger in corresponding limits. Comparison with numerical solution demonstrates good accuracy.

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Weak-contrast approximation of the elastic scattering matrix in anisotropic media

Cited by 53 publications

References 16 publications

Seismic amplitude inversion for the transversely isotropic media with vertical axis of symmetry

Seismic amplitude inversion for the transversely isotropic media with vertical axis of symmetry

Weak‐contrast reflection–transmission coefficients in a generally anisotropic background

Weak‐anisotropy approximation for P‐wave reflection coefficient at the boundary between two tilted transversely isotropic media

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