%0 Journal Article
%T Three-Dimensional Complex Pad¨¦ FD Migration: Splitting and Corrections
%A D. Mondini
%A J. C. Costa
%A J. Schleicher
%A A. Novais
%J International Journal of Geophysics
%D 2012
%I Hindawi Publishing Corporation
%R 10.1155/2012/479492
%X Three-dimensional wave-equation migration techniques are still quite expensive because of the huge matrices that need to be inverted. Several techniques have been proposed to reduce this cost by splitting the full 3D problem into a sequence of 2D problems. To reduce errors, the Li correction is applied at regular multiples of depth extrapolation increment. We compare the performance of splitting techniques in wave propagation for 3D finite-difference (FD) migration in terms of image quality and computational cost. We study the behaviour of the complex Pad¨¦ approximation in combination with two- and alternating four-way splitting, that is, splitting into the coordinate directions at one depth and the diagonal directions at the next depth level. We also extend the Li correction for use with the complex Pad¨¦ expansion and diagonal directions. From numerical examples in inhomogeneous media, we conclude that alternate four-way splitting is the most cost-effective strategy to reduce numerical anisotropy in complex Pad¨¦ 3D FD migration. 1. Introduction In three dimensions, migration methods based on solving the one-way wave equation, besides facing problems to image dip reflectors and handle evanescent waves, are still computationally expensive. For the problems of imaging dip reflectors and evanescent waves, we use the complex Pad¨¦ approximation. Because the resolution of three-dimensional problem is computationally expensive, over the years various techniques have been developed in order to reduce costs and still maintain the quality of the migration method in use. A commonly used technique is splitting. For the case of splitting in two directions, we face the problem of numerical anisotropy, that is, the migration operator acts differently in different directions, resulting in positioning errors of reflectors in the situation where the direction of the dip reflector is far from the directions of the migration planes. To correct this problem it is common to use the correction of Li [1]. Without changing the basic principle of applying subsequent 2D FD migrations in the and directions, the Li correction is an extrapolation of the residual field by a phase shift. In our extension using complex Pad¨¦, the modified Li correction also includes compensation for evanescent energy propagation. When splitting is applied alternately in four directions (the horizontal coordinates and the diagonals), we may still face problems of numerical anisotropy and, consequently, of positioning errors of steeply dipping reflectors. Therefore, we also tested the application of a Li
%U http://www.hindawi.com/journals/ijge/2012/479492/