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Using the dispersion relation of equation 17, we can derive an acoustic wave equation for
orthorhombic media as shown in Appendix A.
The resultant wave equation is sixthorder in time given by,
 

 
 (18) 
This equation is two timederivative orders higher than its VTI equivalent
and 4 orders higher than the conventional isotropic acoustic wave equation. Setting , v_{1}=v_{2}, and
, the conditions necessary for the medium to be VTI, equation (18) reduces
to
 

 (19) 
Substituting gives us the acoustic wave equation for
VTI media derived by Alkhalifah (1997a),
 
(20) 
Thus, as expected, equation (18) reduces to the exact VTI form when VTI model conditions
are used, and subsequently it will reduce to the exact isotropic form (the conventional secondorder
acoustic wave equation) when isotropic model parameters are used (i.e., , and
v_{1}=v_{2}=v_{v}).
Equation (18) can be solved numerically using finite difference methods.
However, such solutions require complicated numerical evaluations based on sixthorder
approximation of derivative in time. Substituting into equation (18)
yields,
 

 
 (21) 
which is a fourth order equation in derivates of t. In addition, substituting
into equation (21) yields
 

 
 (22) 
which is now second order in derivates of t. Equation (22) also clearly displays
the various levels of parameter influence on the wave equation. For example, if and v_{1}=v_{2} the last term
in equation (22) drops.
Next: Analytical solutions of the
Up: Alkhalifah: Wave equation for
Previous: Accuracy tests
Stanford Exploration Project
8/21/1998