Out‐of‐plane geometrical spreading in anisotropic media

Ettrich, N.; Sollid, A.; Ursin, Bjørn

doi:10.1046/j.1365-2478.2002.00317.x

Cited by 11 publications

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“…The integrand V 2 / p 2 tends to a finite value as both V 2 and p 2 tend to zero (in accordance with the 2.5‐D approach, where p 2 = 0 and V 2 = 0 at x 2 = 0). Closed‐form expressions for the integrand for isotropic, transversely isotropic and orthorhombic media can be found in Ettrich et al (2002), , respectively. Note that the argument in for the splitting of the in‐plane and out‐of‐plane spreading is not dependent on the particular constant value of p 2 =∂ T ( x , x s )/∂ x 2 = p s 2 .…”

Section: Modellingmentioning

confidence: 99%

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Linearized 2.5-dimensional parameter imaging inversion in anisotropic elastic media

Foss¹,

Hoop

Ursin³

2005

Geophysical Journal International

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S U M M A R YIn this paper we derive 2.5-D short-period seismic modelling and imaging-inversion formulae in the Born approximation for anisotropic elastic media. The 2.5-D approach encompasses 3-D wave scattering measured in a common-azimuth acquisition geometry subject to 2-D computations under appropriate assumptions. The lowest possible symmetry of the medium in this approach, in principle, is monoclinic, while the medium must be translationally invariant in the normal direction to the associated symmetry plane. In the presence of caustics, artefacts may be generated by the imaging-inversion procedures. We show that in the 2.5-D approach the analysis of artefacts in the 2-D symmetry plane implies the corresponding analysis in 3-D in the framework of the common-azimuth acquisition geometry. An interesting aspect of our results is the occurrence of out-of-plane geometrical spreading in the least-squares removal of the contrast source radiation patterns on the data. We finally introduce the 2.5-D generalized Radon transform that generates common-image-point gathers. The reconsideration of 2.5-D scattering theory is motivated by the increase in use of ocean-bottom acquisition technology. It is not uncommon for ocean bottom cable (OBC) seismic data to be collected along a single line, and the question arises of how to make optimal use of these data. We show the effect of the factors making up the amplitude in the 2.5-D generalized Radon transform on OBC field data from the North Sea.

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Section: Modellingmentioning

confidence: 99%

“…Hence the lowest possible symmetry is transversely isotropic with a symmetry axis in the plane. Then, because of the rotational symmetry of the medium, parameters needed in out‐of‐plane amplitude calculations can be found from in‐plane propagation ( Ettrich et al 2002 ; ).…”

Section: Modellingmentioning

confidence: 99%

Linearized 2.5-dimensional parameter imaging inversion in anisotropic elastic media

Foss¹,

Hoop

Ursin³

2005

Geophysical Journal International

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“…This restricts the rays to remaining in this plane. The kinematics of the wave propagation is therefore 2D (Ettrich, Sollid and Ursin 2002). The waves still travel in a 3D medium and exhibit in‐plane and out‐of‐plane geometrical spreading (Wang and Houseman 1994; Wang 2003).…”

Section: Introductionmentioning

confidence: 99%

“…In the first section of the paper, the notation used is introduced and we show that, for 2.5D, the geometrical spreading of the wave propagation can be split into an in‐plane part and an out‐of‐plane part, as shown by Ettrich et al (2002). Starting with the single‐scattering 3D Born modelling formula (Ursin and Tygel 1997), we proceed as in Bleistein (1986) to approximate the integral in the out‐of‐plane direction by the method of stationary phase.…”

Section: Introductionmentioning

confidence: 99%

2.5D modelling, inversion and angle migration in anisotropic elastic media

Foss¹,

Ursin

2004

Geophysical Prospecting

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2.5D modelling approximates 3D wave propagation in the dip‐direction of a 2D geological model. Attention is restricted to raypaths for waves propagating in a plane. In this way, fast inversion or migration can be performed. For velocity analysis, this reduction of the problem is particularly useful. We review 2.5D modelling for Born volume scattering and Born–Helmholtz surface scattering. The amplitudes are corrected for 3D wave propagation, taking into account both in‐plane and out‐of‐plane geometrical spreading. We also derive some new inversion/migration results. An AVA‐compensated migration routine is presented that is simplified compared with earlier results. This formula can be used to create common‐image gathers for use in velocity analysis by studying the residual moveout. We also give a migration formula for the energy‐flux‐normalized plane‐wave reflection coefficient that models large contrast in the medium parameters not treated by the Born and the Born–Helmholtz equation results. All results are derived using the generalized Radon transform (GRT) directly in the natural coordinate system characterized by scattering angle and migration dip. Consequently, no Jacobians are needed in their calculation. Inversion and migration in an orthorhombic medium or a transversely isotropic (TI) medium with tilted symmetry axis are the lowest symmetries for practical purposes (symmetry axis is in the plane). We give an analysis, using derived methods, of the parameters for these two types of media used in velocity analysis, inversion and migration. The kinematics of the two media involve the same parameters, hence there is no distinction when carrying out velocity analysis. The in‐plane scattering coefficient, used in the inversion and migration, also depends on the same parameters for both media. The out‐of‐plane geometrical spreading, necessary for amplitude‐preserving computations, for the TI medium is dependent on the same parameters that govern in‐plane kinematics. For orthorhombic media, information on additional parameters is required that is not needed for in‐plane kinematics and the scattering coefficients. Resolution analysis of the scattering coefficient suggests that direct inversion by GRT yields unreliable parameter estimates. A more practical approach to inversion is amplitude‐preserving migration followed by AVA analysis. SYMBOLS AND NOTATION A list of symbols and notation is given in .

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“…They do not, however, separate the geometrical spreading into in-plane and out-of-plane factors. Ettrich et al (2002) consider the out-ofplane contribution to the geometrical spreading for wave propagation in a symmetry plane of transversely isotropic (TI) or orthorhombic media with arbitrary orientation of the other symmetry planes.…”

Section: Introductionmentioning

confidence: 99%

Geometrical spreading in a layered transversely isotropic medium with vertical symmetry axis

Ursin

Hokstad

2003

GEOPHYSICS

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Compensation for geometrical spreading is important in prestack Kirchhoff migration and in amplitude versus offset/amplitude versus angle (AVO/AVA) analysis of seismic data. We present equations for the relative geometrical spreading of reflected and transmitted P-and S-wave in horizontally layered transversely isotropic media with vertical symmetry axis (VTI). We show that relatively simple expressions are obtained when the geometrical spreading is expressed in terms of group velocities. In weakly anisotropic media, we obtain simple expressions also in terms of phase velocities. Also, we derive analytical equations for geometrical spreading based on the nonhyperbolic traveltime formula of Tsvankin and Thomsen, such that the geometrical spreading can be expressed in terms of the parameters used in time processing of seismic data.Comparison with numerical ray tracing demonstrates that the weak anisotropy approximation to geometrical spreading is accurate for P-waves. It is less accurate for SV-waves, but has qualitatively the correct form. For P waves, the nonhyperbolic equation for geometrical spreading compares favorably with ray-tracing results for offset-depth ratios less than five. For SV-waves, the analytical approximation is accurate only at small offsets, and breaks down at offset-depth ratios less than unity. The numerical results are in agreement with the range of validity for the nonhyperbolic traveltime equations.

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Out‐of‐plane geometrical spreading in anisotropic media

Cited by 11 publications

References 0 publications

Linearized 2.5-dimensional parameter imaging inversion in anisotropic elastic media

Linearized 2.5-dimensional parameter imaging inversion in anisotropic elastic media

2.5D modelling, inversion and angle migration in anisotropic elastic media

Geometrical spreading in a layered transversely isotropic medium with vertical symmetry axis

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