The wave equation is the central ingredient in defining and constraining wave propagation in a given medium. No other constraint, such as the eikonal or raytracing equations, is as conclusive and elaborate (includes all traveltime and amplitude aspects) as the wave equation. Having such an equation in a simple form for transversely isotropic media with a vertical symmetry axis will help us get a better grip on wave propagation in such media.
Wave propagation in the Earth's subsurface is well simulated by solving the elastic wave equation. The wavefield, in this case, is described by a three-component displacement vector that includes two types of waves, compressional and shear. Probably, the only drawback of the elastic assumption is that it ignores the inelastic nature of the Earth. Since the wavefield in elastic media is described by a vector, simulating wave propagation using finite-difference techniques requires the dynamic computation of all three components of the wavefield at once, resulting in a process that is computationally expensive. Often, geophysicists have resorted to acoustic media approximations to simulate P-wave propagation. The wavefield in acoustic media is described by a scalar quantity rather than a vector. Kinematically, for P-waves in the far field, the acoustic and elastic wave equations are similar; they both yield the same eikonal equation in isotropic media. The reflection and transmission behavior of waves differs considerably for each of the two media. In elastic media, when encountering an interface, some of the P-wave energy transforms to S-wave energy and vice versa, whereas in acoustic media, all the P-wave energy is conserved.
The importance of the acoustic wave equation in transversely isotropic media with a vertical symmetry axis (VTI media) is not strictly computational. The acoustic wave equation does not yield shear waves, and as a result it can be used for zero-offset modeling of P-waves. No splitting filters Dellinger and Etgen (1990) are needed to separate P-waves from S-waves when using this acoustic equation.
In anisotropic media, the acoustic wave equation does not describe a physical phenomenon. This is because acoustic media cannot be anisotropic. If the shear wave velocity equals zero, the medium is rendered isotropic. However, if we ignore the physical aspects of the problem, an acoustic equation for VTI media can be extracted by simply setting the shear wave velocity to zero. Though physically impossible, kinematically the equations resulting from setting the shear wave velocity to zero yield good approximations of the elastic equations.
Numerous attempts through various approximations have been made to simplify the anisotropic equations; some approximations, such as the weak-anisotropy approximation Cohen (1996); Thomsen (1986), elliptical approximations Dellinger and Muir (1988); Helbig (1983), and small dip-angle approximations Cohen (1996) do indeed simplify these anisotropic equations considerably. These approximations, nevertheless, have certain limitations that are in some cases unacceptable for use in practice. The approximation considered in this paper (mainly the acoustic assumption of setting the S-wave velocity, VS0, to zero in the new parameter representation) is far more accurate than the weak-anisotropy or small-propagation-angle approximations, as well as yielding simplified equations.
In an earlier paper 1997, I have derived a simple dispersion equation that relates the vertical slowness to the horizontal one in transversely isotropic media. The simplicity of this acoustic equation is a direct result of setting the shear wave velocity to zero. Although the equation results in an approximation, the accuracy is far within the typical accuracies expected in practical geophysical applications. Simply stated, the equation is exact within the confines of seismic error tolerance. This equation served as the starting point for the development of an acoustic wave equation that describes P-wave propagation in VTI media.
In this paper, I first derive the acoustic wave equation for VTI media using the dispersion relation. The acoustic equation is then used to extract the eikonal and transport equations necessary to describe the ray-theoretical aspects of wave propagation in VTI media. Numerical simulations of wave propagation using finite difference techniques demonstrate the accuracy and efficiency of the VTI acoustic wave equation, especially in comparaison with the elastic wave equation.