THE SECOND ANELLIPTIC APPROXIMATION

The first anelliptic approximation has a certain lack of symmetry. There are two vertical control parameters, V_z and $V_{\mbox{\rm\scriptsize NMO}}$,but only a single horizontal control parameter V_x. If our data includes sources and receivers separated both horizontally and vertically, it makes sense to use an approximation that is symmetric between x and z. We can do this by generalizing to the following template:  

$$f = { \mbox{\rm z-term}^6 + (2 + F_x) \, \mbox{\rm z-term}^4... ...rm}^6 \over ( \mbox{\rm z-term}^2 + \mbox{\rm x-term}^2 )^2 } .$$ (14)

If $F_z \equiv F_x$ equation () reduces to the first anelliptic form given by equation (); if $F_z \equiv F_x \equiv 1$ equation () reduces to the original elliptical form given by equation ().

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

$$M(\phi_{r}) =$$ (15)

$${ \Bigl(M_z C\Bigr)^3 + ( 2 + M_{x{\mbox{\rm\scriptsize NMO}... ... S\Bigr)^2 + \Bigl(M_x S\Bigr)^3 \over ( M_z C + M_x S )^2 } ,$$

where $S = \sin^2(\phi_{r})$,$C = \cos^2(\phi_{r})$,and as before M indicates slowness squared (M_x = 1/ V_x², $M_{x{\mbox{\rm\scriptsize NMO}}} = 1/ V_{x{\mbox{\rm\scriptsize NMO}}}^2$,$M_{z{\mbox{\rm\scriptsize NMO}}} = 1/ V_{z{\mbox{\rm\scriptsize NMO}}}^2$,and M_z = 1/ V_z²).

Similarly, equation () (the dispersion relation) becomes  

$$W(\phi_{w}) =$$ (16)

$${ \Bigl(W_z C\Bigr)^3 + ( 2 + W_{x{\mbox{\rm\scriptsize NMO}... ... S\Bigr)^2 + \Bigl(W_x S\Bigr)^3 \over ( W_z C + W_x S )^2 } ,$$

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

Note that the ${x{\mbox{\rm\scriptsize NMO}}}$ subscript indicates moveout velocity measured for near-vertical propagation; i.e., $W_{x{\mbox{\rm\scriptsize NMO}}}$ is the square NMO velocity we use every day in surface-to-surface data processing. (If an x subscript seems confusing for a paraxial measurement about the vertical, remember that in an elliptic world it is a horizontal velocity that surface moveout measures.) The ${z{\mbox{\rm\scriptsize NMO}}}$ subscript indicates moveout velocity measured for near-horizontal propagation, such as might be found in a cross-borehole experiment.