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

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 Appendix D.