Scalar generalized-screen algorithms in transversely isotropic media with a vertical symmetry axis

The scalar generalized-screen method in isotropic media is extended here to trans-versely isotropic media with a vertical symmetry axis (VTI). Although wave propaga-tion in a transversely isotropic medium is essentially elastic, we introduce an equiv-alent ‘acoustic’ system of equations for the qP-waves which we prove to be accuratefor both the dispersion relation and the polarization angle, in the case of ‘mild’ an-isotropy. The enhanced accuracy of the generalized-screen method as compared tothe phase-screen and the split-step Fourier methods allows the extension to VTI me-dia. The generalized-screen expansion of the one-way propagator follows closely themethod used in the isotropic case. The medium is defined in terms of a background anda perturbation. The generalized-screen expansion of the vertical slowness is based uponan expansion of the medium parameters simultaneously into magnitude and smooth-ness of variation. We cast the theory into numerical algorithms. We assess the accuracyof the generalized-screen method in a particular VTI medium with complex structure,viz. the BP Amoco Valhall model, in which multi-pathing is significant.IntroductionIn realistic geological models, heterogeneity in medium proper-ties is such that the phenomenon of multiple scattering is signif-icant. We distinguish two classes of multiple scattering: one inwhich the multiples are identified with respect to the projectionof their propagation paths onto the vertical direction (depth),and one where the multiples are identified with respect to theprojection of their propagation paths onto the horizontal plane.In the asymptotic framework of wavefront analysis, paths arerays. The first class of multiple scattering is associated with‘turning rays’ and ‘internal multiples’ as well as ‘surface mul-tiples’, the second, possibly combined with the first class ofmultiple scattering, is associated with ‘multi-pathing’.Wave extrapolation methods are able to account for multi-pathing (second class of multiple scattering), with no need tofollow the formation of caustics explicitly. However, their com-putational complexity is significant and hence fast, approxi-mate, algorithms are of interest, in particular in 3D. Methodssuch as the phase-screen (Ratcliffe, 1956) and the closely re-lated split-step Fourier (Stoffa et al., 1990) methods yield fast3D algorithms. They are, however, limited in their capacity topredict large-angle propagation where significant lateral het-erogeneities are present. Because of their attractive properties(3D, multi-pathing), De Hoop et al. (1999) and Le Rousseauand De Hoop (1999) generalized this latter family of algo-rithms, enhancing their accuracy. With the generalized-screen(GS) approach, the accuracy of the phase-screen method is ex-tended to larger-contrast, wider-angle, and back-scattering. Wepropose here to extend the GS method further to anisotropicmedia, in particular to transversely isotropic media with a ver-tical symmetry axis (VTI media). The enhanced accuracy ac-complished by the GS approach becomes a necessity in the ap-plication to VTI media.Our approach accounts for the first class of multi-ple scattering through use of the generalized Bremmer se-ries (De Hoop, 1996), and for the second class of multi-pathing by means of the GS propagation (De Hoop et al., 1999;Le Rousseau & De Hoop, 1999). Examples of implementationof the Bremmer series with the one-way wave operator approx-imated with the GS method can be found in Le Rousseau andDe Hoop (1999). Here, we only consider the first term of theBremmer series associated with the one-way wave operator thatmodels transverse scattering.The propagator that generates the Bremmer series can berepresented by a Hamiltonian path integral (De Witte-Moretteet al., 1979; Fishman & McCoy, 1984a; Fishman & McCoy,1984b; De Hoop, 1996) that accounts for not only the energytraveling along the ray but also for the transport along non-stationary paths. These path integrals contain all possible multi-pathing. In the path integral, ‘time’ is identified with depth, and‘momenta’ are identified with the horizontal wave slownesses 64 J.H. LeRousseau & M.V. de Hoopwhich, in the ray-theoretic limit, coincide with the horizontalcomponents of the gradient of travel time. The (square-root)Hamiltonian, appearing in the phase of the path integral, isidentified with vertical wave slowness. The GS approach yieldsa fast algorithm for the path integrals.We analyze the accuracy of the GS method in complexVTI structures using the synthetic BP Amoco VTI Valhallmodel. This model exhibits significant multi-pathing and is rep-resentative of a North Sea geology. In the Valhall model, a so-called ‘gas cloud’ in the overburden creates a low velocity zonefor qP-waves. This geologic situation yields poor imaging be-low the ‘gas cloud’ with standard single-path methods usingthe qP-qP energy. With the help of the GS propagator, whichwe prove to be accurate in these situations, we shall illustratethat the origin of this problem is possibly associated with multi-pathing.Wave propagation in VTI media is essentially elastic. Yet,in various applications the propagation of qP-waves only isconsidered as if the medium were acoustic. Following the workof Schoenberg and De Hoop (1999) we introduce in Appen-dix A an equivalent ‘acoustic’ system of equations for VTIqP-wave propagation. We show the accuracy of this equiva-lent ‘acoustic’ system for the qP-wave propagation for boththe dispersion relation, i.e. the wavefront set, and the polar-ization angle. In Appendix B, we follow the procedure intro-duced by De Hoop (1996) to decompose the wavefield into up-and downgoing components. Doing so, we introduce the verti-cal slowness operator for the equivalent ‘acoustic’ medium andgive the general form for the one-way wave propagator in sucha medium.We first present the dispersion relation for qP-waves inVTI media and the approximate, yet accurate, simplificationintroduced by Schoenberg and De Hoop (1999). Starting fromthat simplified dispersion relation, we derive the GS represen-tation of the thin-slab propagator in VTI media. The results arethen cast into a numerical algorithm. We carry out our accuracyanalysis through modeling and therefore indirectly analyze themigration operator before stacking that is performed in the pro-cess of imaging. We focus on multi-pathing and second-arrivalenergy. The scalar generalized-screen propagator intransversely isotropic media with a verticalsymmetry axisFor transversely isotropic (TI) media, the dispersion relationassociated with qP-wave propagation is not quite as simple asin the isotropic case (Le Rousseau & De Hoop, 1999, equation(9)). Because the phase velocity is a function of angle, the slow-ness surface is not a sphere. Nevertheless, to apply a GS-typeexpansion (De Hoop et al., 1999; Le Rousseau & De Hoop,1999) as in the isotropic case, one would like to have a disper-sion relation as close as possible to the one of the isotropic case.This is accomplished here using the approximate, yet accurate,dispersion relation in TI media developed by Schoenberg andDe Hoop (1999). Transversely isotropic media with a vertical symmetry axisThroughout the paper, we shall treat the qP-wave as a scalarwave. In the scalar approximation, we neglect any type of modeconversion. We consider the case of TI media with a verticalsymmetry axis (VTI), so without loss of generality one can con-fine attention to a single vertical plane.An elastic medium is defined by its stiffness tensor. With the so-called Voigt notation (Thomsen, 1986),one can represent the medium by a matrix in accordancewith