Anisotropic complex Padé hybrid finite-difference depth migration Daniela Amazonas 1 , Rafael Aleixo 4 , Jörg Schleicher 2,3 , and Jessé C. Costa 1,3 ABSTRACT Standard real-valued finite-difference FD and Fourier fi- nite-difference FFD migrations cannot handle evanescent waves correctly, which can lead to numerical instabilities in the presence of strong velocity variations. A possible solution to these problems is the complex Padé approximation, which avoids problems with evanescent waves by rotating the branch cut of the complex square root. We have applied this approximation to the acoustic wave equation for vertical transversely isotropic media to derive more stable FD and hy- brid FD/FFD migrations for such media. Our analysis of the dispersion relation of the new method indicates that it should provide more stable migration results with fewer artifacts and higher accuracy at steep dips. Our studies lead to the conclu- sion that the rotation angle of the branch cut that should yield the most stable image is 60° for FD migration, as confirmed by numerical impulse responses and work with synthetic data. INTRODUCTION When ray-based and wave-equation-based migration methods are compared, the latter are generally considered to show superior per- formance see, e.g., Bevc and Biondi, 2005. However, the draw- back is their general difficulty to image steep dips Mulder and Plessix, 2004; Xu and Jin, 2007. This fact is owed to the use of one- way wave equations, which need an approximation of the involved square root of a differential operator. This becomes rather obvious in finite-difference FD migration Claerbout, 1971, which is based on 15° or 45° approximations to the wave equation. More sophisti- cated techniques such as Fourier finite-difference FFD migration Ristow and Rühl, 1994 reach higher propagation angles yet fail for near-vertical reflectors. Moreover, standard real-valued FD and FFD migrations cannot handle evanescent waves correctly Millinazzo et al., 1997. Consequently, FFD algorithms tend to become numerically un- stable in the presence of strong velocity variations Biondi, 2002 and the real Padé approximation is unsatisfactory in anelliptic media for large propagation angles with respect to the vertical direction Bakker, 2009. To overcome these limitations, Claerbout 1985 proposes a numerical trick that introduces artificial damping of the evanescent waves. The disadvantage of his method is that it intro- duces dispersive behavior. In a more recent paper, Biondi 2002 proposes using FFD plus interpolation, an unconditionally stable ex- tension of the FFD algorithm. Prior to Biondi’s idea, Millinazzo et al. 1997 proposed a differ- ent approach to overcoming these difficulties with incorrect treat- ment of the evanescent modes in ocean acoustic applications. They introduced a complex extension of the Padé approximation. It con- sists of a rotation of the branch cut of the square-root operator from the negative axis into the complex plane. The complex Padé expan- sion has already found use in applied geophysics. For example, Zhang et al. 2003 use the method in finite-difference migration. However, their implementation is not optimized for wide angles. Zhang et al. 2004, 2007 propose an FFD migration based on a dif- ferent realization of the complex Padé approximation. Amazonas et al. 2007 derive FD and FFD algorithms using the complex Padé ap- proximation for isotropic media to handle evanescent waves. They demonstrate that this procedure stabilizes FD and FFD migration without requiring special treatment for the migration-domain boundaries and enables an accurate migration up to higher dips. All of these methods are based on approximations to the acoustic wave equation. However, the acoustic wave equation can only be generalized to include elliptic anisotropy. More complex anisotropic phenomena cannot be described by a physically meaningful scalar wave equation. Thus, Alkhalifah 1998 uses the dispersion relation for vertical transversely isotropic VTI elastic media to derive an approximate acoustic wave equation for P-waves in VTI media. Manuscript received by the Editor 27 April 2009; revised manuscript received 14 August 2009; published online 1 April 2010. 1 Federal University of Pará, Faculty of Geophysics, Belém, Brazil. E-mail: daniela.amazonas@gmail.com, jesse@ufpa.br. 2 University of Campinas, Department of Applied Mathematics/Institute of Mathematics, Statistics, and Scientific Computing DMA/IMECC, Campinas, Brazil. E-mail: aleixo@ime.unicamp.br, js@ime.unicamp.br. 3 National Institute of Petroleum Geophysics INCT-GP, Brazil. 4 Formerly University of Campinas, presently CGGVeritas, Houston. E-mail: rafael.aleixo@cggveritas.com © 2010 Society of Exploration Geophysicists. All rights reserved. GEOPHYSICS, VOL. 75, NO. 2 MARCH-APRIL 2010; P. S51–S59, 11 FIGS. 10.1190/1.3337317 S51