The space-time domain: Theory and modelling for anisotropic media

In transversely isotropic media with a vertical axis of symmetry (VTI media), we represent the image in vertical time, as opposed to depth, thus eliminating the inherent ambiguity of resolving the vertical P-wave velocity from surface seismic data. In this new (χ-τ)-domain, the ray tracing and eikonal equations are completely independent of the vertical P-wave velocity, with the condition that the ratio of the vertical to normal-moveout (NMO) P-wave velocity (α) is laterally invariant. Moderate size departures of x from lateral homogeneity affect traveltimes only slightly. As a result, for all practical purposes, the VTI equations in the (x-τ)-domain become dependent on only two parameters in laterally inhomogeneous media: the NMO velocity for a horizontal reflector, and an anisotropy parameter, η. An acoustic wave equation in the (x-τ)-domain is also independent of the vertical P-wave velocity. It includes an asymmetric Laplacian operator to accommodate the unbalanced axis units in this new domain. In summary, we have established the basis for a full inhomogeneous time-processing scheme in VTI media that is dependent on only ν and η, and independent of the vertical P-wave velocity.