TTIdip
A transversely isotropic medium with a tilted symmetry axis normal to the reflector
The computational tools for imaging in transversely isotropic media with tilted axes of symmetry (TTI) are complex and in most cases do not have an explicit closed-form representation. As discussed in this paper, developing such tools for a TTI medium with tilt constrained to be normal to the reflector dip (DTI) reduces their complexity and allows for closed-form representations. We show that, for the homogeneous case, zero-offset migration in such a medium can be performed using an isotropic operator scaled by the velocity of the medium in the tilt direction. \geouline{We also show that, for the nonzero-offset case, the reflection angle is always equal to the incidence angle, and thus, the velocities for the source and receiver waves at the reflection point are equal and explicitly dependent on the reflection angle. This fact allows us to develop explicit representations for angle decomposition as well as moveout formulas for analysis of extended images obtained by wave-equation migration. Although setting the tilt normal to the reflector dip may not be valid everywhere (i.e., salt flanks), it can be used in the process of velocity model building where such constrains are useful and typically used.
Below is a schematic plot of how rays may behave in a DTI model.
Below is a plot of the vertical wavenumber, kz, as a function of midpoint wavenumber and angle gather for a DTI model with v=2.0 km/s, vT=1.8 km/s (in the tilt direction), and eta=0.3 (left) and the difference in kz between the DTI model and an isotropic model with v=1.8 km/s (right). The wave numbers are given in units of 1/km.
