Reverse time migration (RTM) images reflectors by using time-extrapolation modeling codes to synthesize source and receiver wavefields in the subsurface. Asymptotic analysis of wave propagation in transversely isotropic (TI) media yields a dispersion relation describing coupled P- and SV-wave modes. This dispersion relation can be converted into a fourth-order scalar partial differential equation (PDE). Increased computational efficiency can be achieved using equivalent coupled second-order PDEs. Analysis of the corresponding dispersion relations as matrix eigenvalue systems allows one to characterize all possible coupled linear second-order systems equivalent to a given linear fourth-order PDE and to determine which ones yield optimally efficient finite-difference implementations. Setting the shear velocity along the axis of symmetry to zero yields a simpler approximate TI wave equation that is more efficient to implement. This simpler approximation, however, can become unstable for some plausible combinations of anisotropic parameters. The same eigensystem analysis can be applied using finite vertical shear velocity to obtain solutions that avoid these instability problems.