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ADI plus interpolation: accurate finite‐difference solution to 3D paraxial wave equation
Author(s) -
Wang Yanghua
Publication year - 2001
Publication title -
geophysical prospecting
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.735
H-Index - 79
eISSN - 1365-2478
pISSN - 0016-8025
DOI - 10.1046/j.1365-2478.2001.00278.x
Subject(s) - paraxial approximation , extrapolation , interpolation (computer graphics) , azimuth , tridiagonal matrix , wave equation , seismic migration , finite difference , alternating direction implicit method , mathematical analysis , finite difference method , mathematics , richardson extrapolation , wave propagation , physics , geometry , optics , geophysics , classical mechanics , quantum mechanics , motion (physics) , beam (structure) , eigenvalues and eigenvectors
An accurate finite‐difference solution is developed for the paraxial wave equation in 3D seismic migration. The conventional alternating‐direction‐implicit (ADI) scheme used in migration causes errors, because the variables in the migration problem are complex‐valued, not real‐valued, and the imaginary part of the higher‐order spatial derivatives cannot be ignored. The accuracy of the 3D paraxial extrapolator is preserved by (i) retaining these higher‐order terms so that it does not produce the apparent azimuthal anisotropy in conventional migration, and (ii) filtering the non‐physical evanescent waves during the downward extrapolation. The implementation of the accurate solution consists of two steps: firstly, the application of ADI to solve two tridiagonal systems sequentially, and secondly, an interpolation between the extrapolated wavefields of successive extrapolation levels. The method is computationally efficient as it uses the ADI scheme and, in addition, couples the correction for azimuthal anisotropy and the suppression of evanescent waves into a single operation, the interpolation step.