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Optimized implicit finite-difference migration for TTI media



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Stanford Exploration Project Report 125 January 16 2007 pages 217 Optimized implicit finite difference migration for TTI media Guojian Shan ABSTRACT I develop an implicit finite difference migration algorithm for tilted transversely isotropic TTI media I approximate the dispersion relation of TTI media with a rational function series whose coefficients are estimated by least squares optimization The dispersion relation of TTI media is not a symmetric function so an odd rational function series is required in addition to the even one These coefficients are functions of Thomsen anisotropy parameters They are calculated and stored in a table before the wavefield extrapolation Similar to the isotropic and VTI media in 3D a phase correction filter is applied after the finite difference operator to eliminate the numerical error caused by two way splitting I generate impulse responses for this algorithm and compare them to those generated using the phase shift method INTRODUCTION Anisotropy is becoming increasingly important in seismic imaging A vertical transversely isotropic VTI medium is one of the simplest and most practical approximations for anisotropic media However the VTI approximation is only valid for simple geologic formations where the bedding plane is horizontal In an area where the sediments are steeply dipping such as anticline structures and thrust sheets the symmetry axis of the medium is not vertical and the medium cannot be simply approximated as VTI medium In these area it is usually better to consider them as tilted transversely isotropic TTI media For VTI media to image steeply dipping reflector using one way wave equation Shan and Biondi 2004 rotate the coordinates In the new coordinates the medium becomes TTI media For both cases we need to design wavefield extrapolation operators for TTI media Compared to those of isotropic and VTI media the dispersion relation of TTI media is much more complicated The dispersion relation of an isotropic medium is



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