GEOPHYSICS, VOL. 68, NO. 4 (JULY-AUGUST2003);P. 1348–1356, 9 FIGS. 10.1190/1.1598128 3D ray+Born migration/inversion—Part 1: Theory Gilles Lambar ´ e ∗ , St ´ ephane Operto ‡ , Pascal Podvin ∗ , and Philippe Thierry ∗ ABSTRACT Prestack ray+Born migration/inversion can be split in two steps : the computation of common image gathers (CIGs) and their weighted stack (the migration stack). The choice of the domain for the CIGs (shot, offset, an- gle, etc.) has a direct impact on the resolution of the mi- gration stack. This resolution can be studied easily in the frame of ray+Born migration/inversion theory result- ing into improved migration/inversion formulas accord- ing to the acquisition geometry. This paper is devoted to this analysis in the cases of a simple 2D acquisition and of a 3D swath acquisition, both corresponding to classical data sets from the SEG/EAGE 3D overthrust experiment. We show that the migration formula orig- inally designed for 3D marine acquisition is not adapt- able to the 3D swath acquisition. Finally, we propose a new formula for this specific acquisition, which im- proves the resolution of the final migrated image. The relevance of this new formula is illustrated in the frame of the SEG/EAGE experiment in the companion paper. INTRODUCTION In the context of 3D seismic imaging, ray-based migrations are particularly appreciated for their CPU efficiency and for their ability to provide a quantitative estimation of the physi- cal properties of the reflectors. Three-dimensional applications of quantitative ray-based migration (migration/inversion) have already been presented as feasability studies (Sevink, 1996; Clochard et al., 1997a,b; Tura et al., 1998; Thierry et al., 1999b). The capacity of the approach, in particular for improving am- plitude variation with offset/amplitude variation with angle (AVO/AVA) attribute estimation on real cases, is currently under examination (Baina et al., 2001; Lenain et al., 2001). Its success depends on accurate control of the preprocessing sequence and refinements of the migration/inversion strategy Manuscript received by the Editor March 11, 2002; revised manuscript received December 12, 2002. ∗ ´ Ecole des Mines de Paris, Centre de Recherche en G´ eophysique, 35 rue Saint Honor´ e, 77 305 Fontainebleau C´ edex, France. E-mail: lambare@ geophy.ensmp.fr; podvin@geophy.ensmp.fr; thierry@geophy.ensmp.fr. ‡Formerly ´ Ecole des Mines de Paris, Centre de Recherche en G´ eophysique, 35 rue Saint Honor´ e, 77 305 Fontainebleau C´ edex, France; presently UMR G ´ eosciences Azur, CNRS-UNSA, 06235 Villefranche-sur-mer, France. E-mail: operto@obs-vlfr.fr. c  2003 Society of Exploration Geophysicists. All rights reserved. to cure potential amplitude artifacts and improve spatial res- olution. This last point is the main focus of this pair of paper and its companion paper (Operto et al., 2003). The theory of ray-based quantitative migration was devel- oped more than a decade ago (Beylkin, 1985; Bleistein, 1987; Beylkin and Burridge, 1990; Jin et al., 1992). It relies either on the ray+Born or on the ray+Kirchhoff linearized approxima- tions. These approximations differ by their description of the “reflecting/diffracting” components of the model (i.e. in terms of impedance and density perturbations or in terms of specular reflectivity, respectively). The associated formulas and numer- ical implementations are very close, and it is not so easy to discriminate between them in the context of seismic imaging (Beydoun and Jin, 1994). Some significant differences remain, however, in their practical algorithms. Ray+Born migration/ inversion provides impedance and density perturbations in a single step, whereas ray+Kirchhoff migration/inversion re- quires some postprocessing of the common image gathers (CIGs) in order to provide an equivalent result. A two-step strategy has many advantages. In particular, it opens ways to mitigate the effects of incomplete illumination or incorrect mi- gration velocities (via migration velocity analysis). Whatever the approach, the proper choice of the domain used to com- pute CIGs has well-known impact on the quality of the results. In the Kirchhoff approach, this amounts to choosing properly minimal subdata sets to improve CIG reliability. CIGs in the offset domain are the most frequently used, but the computa- tion of CIGs in the diffracting/reflecting angle domain seems to exhibit advantages in complex media (Xu et al., 2001). In these papers, we focus on the single-step ray+Born ap- proach, where similar questions arise, but are reformulated in terms of improving the choice of the stacking domain and the design of an optimal weighting of the migration stack with both resolution and amplitude reliability in mind. The question is in- tricate, however; for instance, resolution could be improved by favoring the short offset contributions (less affected by stretch- ing and velocity errors), but this would compromise the signal- to-noise ratio for AVO estimation, where the contribution of large offset is essential. 1348