ANELLIPTIC PARAMETERS FOR TI MEDIA

The anelliptic approximations are useful in their own right for fitting measured phase- and group-velocity surfaces, but it is also useful to know how to relate these approximations to a real anisotropic symmetry system. Because the approximations are two-dimensional, the appropriate symmetry system is transverse isotropy (TI). The equations for TI media are most easily written in terms of phase-velocity squared, suggesting we follow the notation of equation ().

Let $W(\phi_w)$ be the phase-velocity squared measured at phase angle $\phi_{w}$ from the vertical, and let $S = \sin^2(\phi_{w})$ and $(1-S) = \cos^2(\phi_{w})$.Express the elastic constants in units of phase-velocity squared as well, so $W_{ij} = C_{ij} / \rho$.Then we have for the TI SH wavetype  

$$W(\phi_w) = W_{44} (1-S) + W_{66} S \ \ .$$ (17)

Equation () is linear because SH waves in TI media are exactly elliptically anisotropic; by matching coefficients in equations () and () we find the required anelliptic parameters:  

$$W_{{\mbox{\rm\scriptsize SH}},x} = W_{{\mbox{\rm\scriptsize SH}},x{\mbox{\rm\scriptsize NMO}}} = W_{66} \ \ ;$$ (18)

$$W_{{\mbox{\rm\scriptsize SH}},z} = W_{{\mbox{\rm\scriptsize SH}},z{\mbox{\rm\scriptsize NMO}}} = W_{44} \ \ .$$

The TI qP-qSV wavetype is rather more complicated:

$$W(\phi_w) = {1 \over 2} \biggl( (W_{33} + W_{55}) (1-S) + (W_{11} + W_{55}) S$$

 

$$\pm \sqrt{( (W_{33} - W_{55}) (1-S) - (W_{11} - W_{55}) S )^2 + 4 (W_{13} + W_{55})^2 (1-S) S } \; \biggr) ,$$ (19)

where the + sign is for qP, - for qSV. We can find W_z or W_x by substituting S=0 or S=1, respectively, into equation (). To find $W_{x{\mbox{\rm\scriptsize NMO}}}$ or $W_{z{\mbox{\rm\scriptsize NMO}}}$, however, we first have to fit a paraxial ellipse about the z or x axes, respectively. We have already seen that elliptical anisotropy is linear in these coordinates. To find the equation for the paraxial ellipse about the z axis we therefore linearize about S=0, obtaining  

$$W_{{\mbox{\rm\scriptsize z-paraxial}}}(S) = W\biggl\vert_{S=0} + {d W \over d S}\biggl\vert_{S=0} \; S \ \ .$$ (20)

Next, remembering that $W_{x{\mbox{\rm\scriptsize NMO}}}$is the horizontal velocity for the paraxial approximation about the z axis, we set S=1 (horizontal propagation) in equation (), obtaining  

$$W_{x{\mbox{\rm\scriptsize NMO}}} = W\biggl\vert_{S=0} + {d W \over d S}\biggl\vert_{S=0} \ \ .$$ (21)

Similarly,

$$W_{z{\mbox{\rm\scriptsize NMO}}} = W\biggl\vert_{S=1} - \; {d W \over d S}\biggl\vert_{S=1} \ \ .$$ (22)

For the TI qP wavetype we obtain:  

$$W_{{\mbox{\rm\scriptsize P}},x} = W_{11}$$ (23)

$$W_{{\mbox{\rm\scriptsize P}},x{\mbox{\rm\scriptsize NMO}}} = W_{55} + {(W_{13} + W_{55})^2 \over W_{33} - W_{55} }$$

$$W_{{\mbox{\rm\scriptsize P}},z} = W_{33}$$

$$W_{{\mbox{\rm\scriptsize P}},z{\mbox{\rm\scriptsize NMO}}} = W_{55} + {(W_{13} + W_{55})^2 \over W_{11} - W_{55} } .$$

For the TI qSV wavetype we obtain:  

$$W_{{\mbox{\rm\scriptsize SV}},x} = W_{{\mbox{\rm\scriptsize SV}},z} = W_{55}$$ (24)

$$W_{{\mbox{\rm\scriptsize SV}},x{\mbox{\rm\scriptsize NMO}}} = W_{11} - {(W_{13} + W_{55})^2 \over W_{33} - W_{55} }$$

$$W_{{\mbox{\rm\scriptsize SV}},z{\mbox{\rm\scriptsize NMO}}} = W_{33} - {(W_{13} + W_{55})^2 \over W_{11} - W_{55} } .$$