Modeling of pseudo-acoustic P-waves in orthorhombic media with a lowrank approximation |

In transversely isotropic (TI) media,
the model is fully characterized by five elastic parameters and density.
In orthorhombic media, nine elastic parameters and density are needed to describe the elastic model.
The stiffness tensor for an orthorhombic model can be represented,
using the compressed two-index Voigt notation, as follows:

Instead of strictly adhering to the orthorhombic media used by Tsvankin (2005,1997), Alkhalifah (2003) slightly changed the notations and used the following nine parameters determined from the above stiffness tensor:

The Christoffel equation in 3D anisotropic media takes the
following general form (Chapman, 2004):

Alkhalifah (1998) pointed out that setting the S-wave velocity to zero
does not compromise accuracy in traveltime computations for TI media.
This conclusion can be applied to orthorhombic media as well (Tsvankin, 1997).
Alkhalifah (2003) showed that the kinematics of wave propagation
is well described by acoustic approximation.

In orthorhombic media, the Christoffel equation 9 reduces to the following form if , , and are set to zero:

We evaluate the determinant of matrix 10 and set it to zero.
After replacing with
,
with
,
and with
,
we obtain a cubic polynomial in as follows:

One of the roots of the cubic polynomial corresponds to P-waves in acoustic media and is given by the following expression:

This root reduces to the isotropic -wave solution
when we set ,
, and ,
in which in expression 12 is then given by , which is the same dispersion relation in isotropic media as that is shown in equation 6.
In VTI media:
,
, and ,
in expression 12 reduces to

Modeling of pseudo-acoustic P-waves in orthorhombic media with a lowrank approximation |

2013-07-26