Velocity Analysis for Transversely Isotropic Media

Velocity Analysis for Transversely Isotropic Media

August 1994 | Tariq Alkhalifah and Ilya Tsvankin
This paper presents a method for velocity analysis in transversely isotropic (TI) media, focusing on recovering anisotropic velocity fields from surface reflection data. The key idea is to invert the dependence of P-wave moveout velocities on the ray parameter using an exact analytic equation for the normal-moveout (NMO) velocity for dipping reflectors in anisotropic media, derived by Tsvankin (1994). The method involves determining two parameters: the zero-dip NMO velocity $ V_{\mathrm{nmo}}(0) $ and a new effective parameter $ \eta $, which reduces to the difference between Thomsen parameters $ \epsilon $ and $ \delta $ in the limit of weak anisotropy. The inversion procedure allows the reconstruction of NMO velocity as a function of ray parameter using moveout velocities for two different dips. $ V_{\mathrm{nmo}}(0) $ and $ \eta $ determine not only the NMO velocity but also long-spread (nonhyperbolic) P-wave moveout for horizontal reflectors and time-migration impulse response. This means that inversion of dip-moveout information allows one to perform all time-processing steps in TI media using only surface P-wave data. Isotropic time-processing methods remain valid for elliptical anisotropy ($ \epsilon = \delta $). The method is tested on synthetic data and field data from offshore Africa. Accurate time-to-depth conversion requires resolving the vertical velocity $ V_{P0} $ independently. If $ V_{P0} $ is known, $ \epsilon $ and $ \delta $ can be found by inverting two P-wave NMO velocities corresponding to a horizontal and a dipping reflector. If no well information is available, all three parameters ($ V_{P0} $, $ \epsilon $, and $ \delta $) can be obtained by combining inversion results with shear-wave information. The paper also generalizes Tsvankin's (1994) single-layer NMO equation for layered anisotropic media with a dipping reflector, providing a basis for extending anisotropic velocity analysis to vertically inhomogeneous media. A Dix-type differentiation procedure is used to strip the influence of a stratified overburden on moveout velocity. The paper concludes with an application of the method to a marine data set from offshore Africa. The analysis shows that the NMO velocity for dipping reflectors in TI media is primarily controlled by the difference between anisotropies $ \epsilon $ and $ \delta $. The inversion procedure is shown to be stable and effective for weak anisotropy, with the NMO velocity containing only two combinations of the anisotropies $ V_{\mathrm{nmo}}(0) $ and $ \epsilon - \delta $. The paper also discusses the properties of the family of equivalent solutions (ES), showing that they yield the same long-sThis paper presents a method for velocity analysis in transversely isotropic (TI) media, focusing on recovering anisotropic velocity fields from surface reflection data. The key idea is to invert the dependence of P-wave moveout velocities on the ray parameter using an exact analytic equation for the normal-moveout (NMO) velocity for dipping reflectors in anisotropic media, derived by Tsvankin (1994). The method involves determining two parameters: the zero-dip NMO velocity $ V_{\mathrm{nmo}}(0) $ and a new effective parameter $ \eta $, which reduces to the difference between Thomsen parameters $ \epsilon $ and $ \delta $ in the limit of weak anisotropy. The inversion procedure allows the reconstruction of NMO velocity as a function of ray parameter using moveout velocities for two different dips. $ V_{\mathrm{nmo}}(0) $ and $ \eta $ determine not only the NMO velocity but also long-spread (nonhyperbolic) P-wave moveout for horizontal reflectors and time-migration impulse response. This means that inversion of dip-moveout information allows one to perform all time-processing steps in TI media using only surface P-wave data. Isotropic time-processing methods remain valid for elliptical anisotropy ($ \epsilon = \delta $). The method is tested on synthetic data and field data from offshore Africa. Accurate time-to-depth conversion requires resolving the vertical velocity $ V_{P0} $ independently. If $ V_{P0} $ is known, $ \epsilon $ and $ \delta $ can be found by inverting two P-wave NMO velocities corresponding to a horizontal and a dipping reflector. If no well information is available, all three parameters ($ V_{P0} $, $ \epsilon $, and $ \delta $) can be obtained by combining inversion results with shear-wave information. The paper also generalizes Tsvankin's (1994) single-layer NMO equation for layered anisotropic media with a dipping reflector, providing a basis for extending anisotropic velocity analysis to vertically inhomogeneous media. A Dix-type differentiation procedure is used to strip the influence of a stratified overburden on moveout velocity. The paper concludes with an application of the method to a marine data set from offshore Africa. The analysis shows that the NMO velocity for dipping reflectors in TI media is primarily controlled by the difference between anisotropies $ \epsilon $ and $ \delta $. The inversion procedure is shown to be stable and effective for weak anisotropy, with the NMO velocity containing only two combinations of the anisotropies $ V_{\mathrm{nmo}}(0) $ and $ \epsilon - \delta $. The paper also discusses the properties of the family of equivalent solutions (ES), showing that they yield the same long-s
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