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

Abstract

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 .

Linearized 2.5-dimensional parameter imaging inversion in anisotropic elastic media

“…In this section, we illustrate the relative significance (magnitude) of the factors making up the amplitude in 2.5‐D imaging inversion according to . We use ocean bottom cable (OBC) data (a single cable) from the North Sea, the same data we used in another paper, in which we developed and carried out reflection tomography in 2.5‐D (Foss et al 2004). We apply the transform in both for PP and PSV reflection data and assume a transversely isotropic background medium with a vertical axis of symmetry.…”

“…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. Also, 2.5-D scattering theory can be applied in reflection tomography or migration velocity analysis (Foss et al 2004). Such an application leads to fast, though approximate, algorithms that carry out the reflection tomography slicewise ( Fig.…”

A practical approach to PP seismic angle tomography

“…A similar notation is used for the receiver, namely, p r ( y ) = ( p r 1 , p r 3 ) and p ( x r ). Observe that the out‐of‐plane slowness p 2 is zero in 2.5D (Bleistein 1986; Foss and Ursin 2004). In an isotropic medium, the P‐wave polarization vectors are the unit vectors of the slowness vectors, denoted by h ( x s ) = ( h 1 ( x s ), h 3 ( x s )) and h ( x r ) = ( h 1 ( x r ), h 3 ( x r )) for the polarization vectors at the source and receiver, respectively.…”

“…Here, we present the expressions for the creation of PP common‐image‐point gathers assuming an isotropic medium. A complete description of the 2.5D formulae for anisotropic media and other acquisition geometries can be found in the review by Foss and Ursin (2004). Formulae for the 3D case can be found in the review by Ursin (2004).…”

Full-azimuth subsurface angle domain wavefield decomposition and imaging Part I: Directional and reflection image gathers

“…Many papers have been published which study and emphasize the importance of generating common-image-angle gathers directly at the subsurface points rather than the universally used surface-offset image gathers, especially in complex geological areas where the wavefield includes multipathing ͑e.g., ten Kroode et al, 1994;Nolan and Symes, 1996;Brandsberg-Dahl et al, 1999;Rousseau et al, 2000;Xu et al, 2001;Audebert et al, 2002;Koren et al, 2002;Rickett and Sava, 2002;Brandsberg-Dahl et al, 2003;Foss and Ursin, 2004;Sollid and Ursin, 2003;Soubaras, 2003;Bleistein et al, 2005aBleistein et al, , 2005bWu and Chen, 2006;Biondi, 2007a Manuscript received by the Editor 28 December 2009; revised manuscript received 13 July 2010; published online 4 January 2011. 1 Although the theory of angle-domain imaging is well established, its implementation, especially for large-scale 3D models or for highresolution reservoir imaging, remains extremely challenging.…”