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Summary We present a practical implementation of the one-way wave equation based 3-D prestack implicit finite-difference depth migration(IFDM). Conventional aspects of IFDM, including the Crank-Nicholson method, the continued-fraction expansion of the square-root operator, the compact implicit finite difference stencil, the x-y operator split, and Zhiming Li's error compensation, are combined to produce a downward continuation process with accurate kinematic and dynamic response up to 90°, even when the velocity field varies laterally by a factor of 2 or more. This downward continuation method is combined with appropriate aperture and imaging conditions to produce, from shot-gathered full-azimuth prestack data, depth-migrated image gathers in the conventional offset domain. A computationally highly efficient parallel implementation has been achieved without any sacrifice in accuracy. Introduction Even sophisticated Kirchhoff prestack depth migration techniques are faced with many theoretical and practical difficulties. These include the limitations of the high frequency and single arrival approximations, as well as the problems with amplitude control based on simplified geometrical or statistical considerations. It has long been known that the solution to these problems is a wave equation based method. Due to the difficulties associated with imaging with the full acoustic wave equation(Claerbout), most attention has been focused on the one-way or paraxial wave equation. However, because of its high computational cost, even full solutions to the one-way equation have been considered impractical when 3-D prestack migrated image gathers are desired. This situation has forced researchers to seek various additional approximations to the one-way equation that can significantly increase computational efficiency. One class of such approaches limits accurate wave propagation angle to 75° from vertical, when lateral velocity variation is severe. Another approach is the reduction of dimensionality through the common azimuth approximation (Biondi). Even after such approximations, the wave equation based methods are still generally capable of producing images better than those from Kirchhoff migrations(Vaillant). Basing our technique on the Salvo software(Ober), we have developed a practical one-way wave-equation migration system that has wide angle (up to 90°) accuracy for any realistic velocity contrast and can handle full-azimuth wave propagation and data acquisition. While this method employs a series of approximations to the one-way equation, the error introduced by each approximation, as well as any numerical error inherent in finite differences, are collected and compensated for. The result is a solution to the one-way equation that has no error of significance in the context of seismic imaging. Compared to both common Kirchhoff migrations and double- square-root(DSR) based wave-equation methods, finite difference(FD) shot migrations require less memory, disk, and network resources, which in modern computing environments often overshadow raw CPU cost. It can scale to massive, distributed-memory environments as well as large shared- memory computers, with no loss in efficiency. This method also does not require data regularization, velocity model smoothing, or other quality compromising preprocessing steps. More importantly, our implementation proves to be efficient enough to produce offset-domain 3-D depth-migrated image gathers at a computational cost that is competitive with high- end Kirchhoff migration. Additionally, it offers potential savings in the cost of data acquisition and transmission, which is often geared toward Kirchhoff migrations, and therefore retains more data redundancy than necessary to generate high quality images. |
