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In the text we derived an acoustic dispersion relation [equation 17] for orthorhombic anisotropy.
This relation can be used to derive the acoustic wave equation following the same steps I took
in deriving such an equation for VTI media Alkhalifah (1997b).
First, we cast the dispersion relation in a polynomial form in terms of the slownesses and
substitute these slownesses with wavenumbers as follows,
 

 
 (27) 
where , , and .As a reminder, V_{1} and V_{2} are the horizontal velocities along the xaxis and
the yaxis, respectively.
Multiplying both sides of equation (27) by the wavefield
in the Fourier domain, ,as well as using inverse Fourier transform on k_{x}, k_{y}, and k_{z}
(, , and
) yields a wave equation in the spacefrequency domain,
given by
 

 
 (28) 
Finally, applying inverse Fourier transform on (),
the acoustic wave equation for
VTI media is given by
 

 
 (29) 
Unlike the acoustic wave equation for
VTI media Alkhalifah (1997b) which is fourth order in time,
equation (29) is sixth order in time, and thus can provide us with
6 independent solutions.
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Stanford Exploration Project
8/21/1998