SUMMARY

We propose a new method for wave equation depth migration applied in the space-temporal frequency domain. The method uses as base the constant density variable velocity wave equation. The method also uses a complex Padé aproximation of an exponential term in the solution of the wave equation. This method improves the inaccuracy and instabilities due to evanescent waves and produce high quality imagens of complex media with fewer numerical artifacts than those obtained with a real Padé aproximantion of the exponential operator. Tests on zero-offset salt data from the SEG/EAEG show that the method can handle strong lateral variation and also has a good steep dip response.

INTRODUCTION

Wave equation migration have a better performance than ray-based migration when the velocity model has strong lateral velocity variation. The majority of wave equation methods are based on solving the one-way wave acoustic equation. There are many different methods to numerically solve the one-wave equation. These methods can be grouped into three classes: Fourier methods (Stoffa et al.,1990; Gazdag and Sguazerro, 1984) solved in the wavenumber domain; finite difference methods (Claerbout, 1985; Hale, 1991) and mixed methods that are a linear combination of spectral and finite-difference methods (FFD) (Ristow and Rulh, 1994). In both FD e FFD migration methods, the purely real Padé approximation or a simples Taylor expansion is usually used to approximate the square-root operator. Real root operators can not handle evanescent waves (Millinazzo et at.,1997). To deal with this problem Millenazzo et al. (1997) proposed to use a complex Padé approximation. The complex Padé expantion was applied previously in applied geophysics. Zhang et al. (2003) used the method in finite-difference migration and its implementation were not efficient to wide-angle. Zhang et al. (2004) derived a split-step complex Padé -Fourier solution for one-way equation using both [1/1] and [2/2] Padé approximantion. Recently, Amazonas et al. (2007) use the complex Padé complex for wide-angle FD and FFD prestack depth migration. In this paper, we present a new FD migration method that is quite different from the above mentioned methods. The method is not based on one-wave equation or on perturbations to constant velocity solution to the wave equation. It uses the exact acoustic wave equation for the actual subsurface velocity. The propagation of the solution from one depth level to the next is carried out by the complex Padé approximation of an exponential term of the one way evolution operator. This approch produces a tridiagonal matrix system that is solved to obain the wavefield from the previous wavefield. The complex Padé approximation is used to mitigate the inaccuaracy and instability due the evanescent waves. In the following sections, we derive the FD complex Padé solution for the wave equation using [1/1] Padé approximantion. Then we present zero-offset migration results for the SEG/EAGE salt model. A comparison with the standard (real-valued) FD e FFD, split-step and PSPI and our method is given. The test results show that the our FD migration method is capable of imaging structures with complicated velocity variation.

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