Compared with the elastic wave equation, the acoustic wave equation has two features: it is simpler, and thus, more efficient to use, and it does not yield Shear waves, and as a result, it can be used for zero-offset modeling of P-waves. Though in anisotropic media, an acoustic wave equation does not, physically, exit, Alkhalifah (1997b) derived such an acoustic wave equation for transversely isotropic media with a vertical symmetry axis (VTI media). If we ignore the physical aspects of the problem, an acoustic equation for TI media can be extracted by simply setting the shear wave velocity to zero. Kinematically such equations yield good approximations to the elastic ones.
Orthorhombic anisotropic media are more complicated than VTI ones. They represent models where we can have vertical fractures along with the general VTI preference (i.e. horizontal thin layering). Orthorhombic media have three mutually orthogonal planes of mirror symmetry; for the model with a single system of vertical cracks in a VTI background, the symmetry planes are determined by the crack orientation.
Numerous equations have been derived lately for orthorhombic media including the normal moveout (NMO) equation for horizontal and dipping reflectors derived by Grechka and Tsvankin (1997). These equations are generally complicated and are often solved numerically. The complexity of these equations stem from the complexity of the phase velocity and the dispersion relation for orthorhombic media. However, the phase velocity and dispersion relation, as we will see later, can be simplified considerably using the acoustic medium assumption. Thus, practical analytical solutions for the NMO equation in orthorhombic media is possible.
In this paper, I derive a dispersion relation for orthorhombic anisotropic media based on the acoustic approximation. I then look into the accuracy of such an equation, and subsequently use it to derive an acoustic wave equation for orthorhombic anisotropic media. Finally, I solve the acoustic wave equations analytically. This is a preliminary study of the subject and a follow up paper will include details and experiments left out of this paper.