Including lateral velocity variations into true-amplitude common-shot wave-equation migration Daniela Amazonas 1 , Rafael Aleixo 2 , Gabriela Melo 3 , Jörg Schleicher 4,5 , Amélia Novais 4,5 , and Jessé C. Costa 1,5 ABSTRACT In heterogeneous media, standard one-way wave equa- tions describe only the kinematic part of one-way wave prop- agation correctly. For a correct description of amplitudes, the one-way wave equations must be modified. In media with vertical velocity variations only, the resulting true-amplitude one-way wave equations can be solved analytically. In media with lateral velocity variations, these equations are much harder to solve and require sophisticated numerical tech- niques. We present an approach to circumvent these prob- lems by implementing approximate solutions based on the one-dimensional analytic amplitude modifications. We use these approximations to show how to modify conventional migration methods such as split-step and Fourier finite-dif- ference migrations in such a way that they more accurately handle migration amplitudes. Simple synthetic data exam- ples in media with a constant vertical gradient demonstrate that the correction achieves the recovery of true migration amplitudes. Applications to the SEG/EAGE salt model and the Marmousi data show that the technique improves ampli- tude recovery in the migrated images in more realistic situations. INTRODUCTION Many seismic migration methods, particularly those directly based on the one-way wave equation, address only the kinematic as- pects of the imaging problem i.e., the position and structure of the seismic reflectors but incorrectly treat the dynamics amplitudes, related to the energy carried by the seismic wavefield. However, as postmigration amplitude variation with offset AVO and angle AVA studies are becoming increasingly important, the correct treatment of migration amplitudes is necessary. Most practical wave-equation migration techniques are based on so-called one-way wave equations, which describe wave propaga- tion in one preferential direction, only Claerbout, 1971. The one- way wave operators permit the separation of the full wavefield into its directional components of propagation. Generally, the factoriza- tion is used to split the wavefield into its up- and downward propa- gating parts. In this form, the one-way wave equations are useful in modeling and, principally, in migration. In homogeneous media, the product of the two one-way operators yields the full wave-equation operator. Then, the traveltimes and amplitudes of the one-way waves i.e., the solutions of the one-way wave equations are identical to those of the solution of the full wave equation. However, in heterogeneous media, the use of the same one-way wave equation leads to incorrect amplitudes. Since the original work of Leontovich and Fock 1946 on one- way wave equations, much research has been dedicated to the deri- vation of better approximations to the correct one-way wave equa- tions in laterally varying media. One of the first attempts can be at- tributed to Fishman et al. 1997, who provide an extensive treat- ment of asymptotic approximations of the symbol of the one-way operator using a path integral approach. They achieve the construc- tion of several explicit, uniform asymptotic approximations of the square-root Helmholtz operator. Next in line are de Hoop et al. 2000 and Rousseau and de Hoop 2001, who propose a normaliz- ing operator to obtain approximately restored amplitude behavior but do not discuss its numerical implementation. Their work has been extended by de Hoop et al. 2003, who introduce a sequence of approximate representations that, in a certain limit, converges to the “true” one. Each member of the sequence corresponds to a finite cas- cade of thin-slab propagators. They show that the limit of the se- Manuscript received by the Editor 5 August 2009; revised manuscript received 22 March 2010; published online 5 October 2010. 1 Federal University of Pará, Faculty of Geophysics, Belém, Pará, Brazil. E-mail: daniela.amazonas@gmail.com, jesse@ufpa.br. 2 CGGVeritas, Houston,Texas, U.S.A. E-mail: aleixocarvalho@gmail.com. 3 Massachusetts Institute of Technology, Department of Earth, Atmospheric, and Planetary Sciences, Cambridge, Massachusetts, U.S.A. E-mail: gmelo @mit.edu. 4 University of Campinas, DMA/IMECC, Campinas, São Paulo, Brazil. E-mail: js@ime.unicamp.br, amelia@ime.unicamp.br. 5 National Institute of Petroleum Geophysics INCT-GP, Brazil. © 2010 Society of Exploration Geophysicists. All rights reserved. GEOPHYSICS, VOL. 75, NO. 5 SEPTEMBER-OCTOBER 2010; P. S175–S186, 14 FIGS., 1 TABLE. 10.1190/1.3481469 S175