THE FIRST ANELLIPTIC APPROXIMATION

In cases such as cross-borehole tomography where true vertical and horizontal scale measurements are available, elliptical anisotropy can be useful as a ``first step'' away from isotropy. If only surface kinematic data are available, though, equation () is enough for doing NMO and assuming elliptical anisotropy accomplishes nothing more than making explicit the depth ambiguity that was always there. To add another velocity parameter to this isotropic (really elliptically anisotropic) case, we need to add some sort of an anelliptic parameter.

Note that equations (), (), and () share a common polynomial form:  

$$f = \mbox{\rm z-term}^2 + \mbox{\rm x-term}^2 .$$ (9)

We add an anelliptic parameter to equation () by generalizing it to  

$$f = { \mbox{\rm z-term}^4 + (1 + F) \, \mbox{\rm z-term}^2 \... ...m x-term}^4 \over \mbox{\rm z-term}^2 + \mbox{\rm x-term}^2 } .$$ (10)

Varying the cross term F lets us vary the behavior away from perfect ellipticity in between the coordinate axes. If $F \equiv 1$ the numerator becomes a perfect square and equation () reverts to the original elliptic form in equation () for all propagation directions.

Why this particular generalization? This anelliptic form needs only one more parameter beyond elliptical anisotropy (or isotropy for the surface survey case), retains the properties that make elliptical anisotropy so convenient (in approximation), and makes a good approximation to exact transverse isotropy, especially in the phase-slowness domain. (See Figure .)

Following this template, equation () (the ray equation) becomes  

$$M(\phi_{r}) = {{ \Bigl(M_z C\Bigr)^2 + ( 1 + M_{{\mbox{\rm\s... ...igl(M_x S\Bigr) + \Bigl(M_x S\Bigr)^2 \over M_z C + M_x S }} ,$$ (11)

where $S = \sin^2(\phi_{r})$,$C = \cos^2(\phi_{r})$,and, as before, M indicates slowness squared (M_x = 1/ V_x², $M_{{\mbox{\rm\scriptsize NMO}}} = 1/ V_{{\mbox{\rm\scriptsize NMO}}}^2$,and M_z = 1/ V_z²). Note this equation explicity separates the true horizontal velocity V_x from the paraxial NMO velocity $V_{\mbox{\rm\scriptsize NMO}}$.

We can similarly extend equation () (the dispersion relation) to  

$$W(\phi_{w}) = {{ \Bigl(W_z C\Bigr)^2 + ( 1 + W_{{\mbox{\rm\s... ...igl(W_x S\Bigr) + \Bigl(W_x S \Bigr)^2 \over W_z C + W_x S }} ,$$ (12)

where $S = \sin^2(\phi_{w})$,$C = \cos^2(\phi_{w})$,and as before W indicates velocity squared (W_x = V_x² = 1 / M_x, $W_{{\mbox{\rm\scriptsize NMO}}} = V_{{\mbox{\rm\scriptsize NMO}}}^2 = 1 / W_{{\mbox{\rm\scriptsize NMO}}}$,and W_z = V_z² = 1 / M_z).

Note the symmetry between equations and ; as for elliptical anisotropy, converting from the representation in one domain to the other is as simple as replacing each of the velocity parameters with its reciprocal.