Improved geometric-spreading approximation in layered transversely isotropic media

Stovas, Alexey; Ursin, Bjørn

doi:10.1190/1.3158051

Cited by 28 publications

(15 citation statements)

References 16 publications

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“…The DRA for the relative geometrical spreading (

L_{N}

) is given by Stovas and Ursin ()

{scriptL}_{N (DRA)} = {scriptL}_{0} (1 + A_{2} {\hat{x}}^{2} + \frac{A_{4} {\hat{x}}^{4}}{1 + B_{2} {\hat{x}}^{2}}),

where the zero‐offset term

{scriptL}_{0} = t_{0} {bold-italicv}_{n}^{2}

, A 2 and A 4 are the second‐ and fourth‐order coefficients in Taylor series computed at the zero‐offset,

A_{2} = 1 + 8 η, A_{4} = - 9 η (1 + 4 η),

and the coefficient B 2 can be obtained from the slope of

L_{N}

at the infinite offset limit given by

B_{2} = \frac{9 η false(1 + 4 η false) \sqrt{1 + 2 η}}{false(1 + 8 η false) \sqrt{1 + 2 η} - 1} .

…”

Section: Methodsmentioning

confidence: 99%

“…Note that the coefficients C 2 and C 4 in the approximation are only defined by fitting the subsequent term for the asymptotic behaviours at the infinite offset limit for the relative geometrical spreading. By setting

C_{4} = 2 C_{2}

, the GMA form approximation (equation ) is reduced to the ration form (equation ) defined by Stovas and Ursin (). When setting

η = 0

for elliptical anisotropic model, all approximations (rational and GMA form) result in the same expression with the reference one regardless of direct or indirect type shown by

{scriptL}_{N} = {scriptL}_{0} false(1 + {\hat{x}}^{2} false) .

…”

Section: Methodsmentioning

confidence: 99%

“…Substituting equation into equation gives (Stovas and Ursin )

{scriptL}_{N} = \frac{t_{0} {bold-italicv}_{n m o}^{2} \sqrt{1 + 4 η p^{2} {bold-italicv}_{n m o}^{2} - 6 η (1 + 2 η) p^{4} {bold-italicv}_{n m o}^{4}}}{{(1 - 2 η p^{2} {bold-italicv}_{n m o}^{2})}^{2} [1 - (1 + 2 η) p^{2} {bold-italicv}_{n m o}^{2}]} .

…”

Section: Relative Geometrical Spreading In An Acoustic Transversely Imentioning

confidence: 99%

“…The DRA for the relative geometrical spreading (L N ) is given by Stovas and Ursin (2009) where the zero-offset term L 0 = t 0 v 2 n , A 2 and A 4 are the second-and fourth-order coefficients in Taylor series computed at the zero-offset,…”

Section: Direct Rational Approximation For the Relative Geometrical Smentioning

confidence: 99%

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Generalized non‐hyperbolic approximation for qP‐wave relative geometrical spreading in a layered transversely isotropic medium with a vertical symmetry axis

Stovas

2018

Geophysical Prospecting

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Compensation for geometrical spreading along the ray‐path is important in amplitude variation with offset analysis especially for not strongly attenuative media since it contributes to the seismic amplitude preservation. The P‐wave geometrical spreading factor is described by a non‐hyperbolic moveout approximation using the traveltime parameters that can be estimated from the velocity analysis. We extend the P‐wave relative geometrical spreading approximation from the rational form to the generalized non‐hyperbolic form in a transversely isotropic medium with a vertical symmetry axis. The acoustic approximation is used to reduce the number of parameters. The proposed generalized non‐hyperbolic approximation is developed with parameters defined by two rays: vertical and a reference rays. For numerical examples, we consider two choices for parameter selection by using two specific orientations for reference ray. We observe from the numerical tests that the proposed generalized non‐hyperbolic approximation gives more accurate results in both homogeneous and multi‐layered models than the rational counterpart.

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“…The DRA for the relative geometrical spreading (

L_{N}

) is given by Stovas and Ursin ()

{scriptL}_{N (DRA)} = {scriptL}_{0} (1 + A_{2} {\hat{x}}^{2} + \frac{A_{4} {\hat{x}}^{4}}{1 + B_{2} {\hat{x}}^{2}}),

where the zero‐offset term

{scriptL}_{0} = t_{0} {bold-italicv}_{n}^{2}

, A 2 and A 4 are the second‐ and fourth‐order coefficients in Taylor series computed at the zero‐offset,

A_{2} = 1 + 8 η, A_{4} = - 9 η (1 + 4 η),

and the coefficient B 2 can be obtained from the slope of

L_{N}

at the infinite offset limit given by

B_{2} = \frac{9 η false(1 + 4 η false) \sqrt{1 + 2 η}}{false(1 + 8 η false) \sqrt{1 + 2 η} - 1} .

…”

Section: Methodsmentioning

confidence: 99%

C_{4} = 2 C_{2}

, the GMA form approximation (equation ) is reduced to the ration form (equation ) defined by Stovas and Ursin (). When setting

η = 0

for elliptical anisotropic model, all approximations (rational and GMA form) result in the same expression with the reference one regardless of direct or indirect type shown by

{scriptL}_{N} = {scriptL}_{0} false(1 + {\hat{x}}^{2} false) .

…”

Section: Methodsmentioning

confidence: 99%

“…Substituting equation into equation gives (Stovas and Ursin )

{scriptL}_{N} = \frac{t_{0} {bold-italicv}_{n m o}^{2} \sqrt{1 + 4 η p^{2} {bold-italicv}_{n m o}^{2} - 6 η (1 + 2 η) p^{4} {bold-italicv}_{n m o}^{4}}}{{(1 - 2 η p^{2} {bold-italicv}_{n m o}^{2})}^{2} [1 - (1 + 2 η) p^{2} {bold-italicv}_{n m o}^{2}]} .

…”

Section: Relative Geometrical Spreading In An Acoustic Transversely Imentioning

confidence: 99%

Section: Direct Rational Approximation For the Relative Geometrical Smentioning

confidence: 99%

See 2 more Smart Citations

Generalized non‐hyperbolic approximation for qP‐wave relative geometrical spreading in a layered transversely isotropic medium with a vertical symmetry axis

Stovas

2018

Geophysical Prospecting

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show abstract

“…Different from the indirect type approximation, which is approximating the traveltime for geometric spreading approximation, the direct-type approximation is computed by approximating the geometric spreading term directly from the exact parametric equations obtained from dynamic ray tracing. The first example of this comparison between indirect and direct type approximation is done by Stovas and Ursin (2009), who developed the rational type of approximation in direct form. They showed that the direct rational approximation is simpler and more accurate than the indirect counterpart for a homogeneous and multilayered VTI model.…”

Section: Introductionmentioning

confidence: 99%

An anelliptic approximation for geometrical spreading in transversely isotropic and orthorhombic media

Stovas

Sripanich

2017

SEG Technical Program Expanded Abstracts 2017

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The relative geometric spreading along the raypath contributes to the amplitude decay of the seismic wave propagation that needs to be considered for amplitude variation with offset or other seismic data processing methods that require the true amplitude processing. Expressing the P-wave geometric spreading factor in terms of the offset-traveltime-based parameters is a more practical and convenient way because these parameters can be estimated from the nonhyperbolic velocity analysis. We have developed an anelliptic approximation for the relative geometric spreading of P-wave in a homogeneous transversely isotropic medium with vertical symmetry axis (VTI) and an orthorhombic (ORT) medium under the acoustic anisotropy assumption. The coefficients in our approximation are only defined within the symmetry planes and computed from fitting with the exact parametric expression. For an ORT model, due to the symmetric behavior in different symmetry planes, the other coefficients in the approximation can be easily obtained by making corresponding changes in indices from the computed coefficients in one symmetry plane. From the numerical examples, we found that for a homogeneous VTI model, the anelliptic approximation is more accurate than the generalized nonhyperbolic moveout approximation form for larger offset. For a homogeneous ORT model, our anelliptic approximation is more accurate than its traveltime-based counterparts. Using the Dix-type equations for the effective parameters, our anelliptic form approximation is extended to a multilayered VTI and ORT models and has accurate results in both models.

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Reflection and transmission coefficient approximation at weak-contrast interfaces for strong VTI media

Yin,

Yang,

Liang

et al. 2024

Sci. China Earth Sci.

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Improved geometric-spreading approximation in layered transversely isotropic media

Cited by 28 publications

References 16 publications

Generalized non‐hyperbolic approximation for qP‐wave relative geometrical spreading in a layered transversely isotropic medium with a vertical symmetry axis

Generalized non‐hyperbolic approximation for qP‐wave relative geometrical spreading in a layered transversely isotropic medium with a vertical symmetry axis

An anelliptic approximation for geometrical spreading in transversely isotropic and orthorhombic media

Reflection and transmission coefficient approximation at weak-contrast interfaces for strong VTI media

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