Figure 1

Group velocity (= impulse response), group slowness, phase velocity,
and phase slowness (= dispersion relation) plots for the *q*SV
mode of Greenhorn Shale (thick solid line) and an elliptically anisotropic
paraxial approximation to it (thin dashed line). The ``''
and ``'' symbols indicate the recurring transformations
linking the four representations.

These vertical and horizontal velocities have simple geophysical
interpretations. The vertical velocity
of the paraxial elliptic approximation
(equals the true vertical velocity)
is what we need to do time-to-depth
conversion.
The horizontal velocity
of the paraxial elliptic approximation
(*not* the same thing as the true horizontal velocity)
is what is required for NMO.
(In a surface survey the sources and receivers are laid out along a
horizontal line; if the vertical velocity were changed,
it would effectively change the vertical scale of the survey, but
the traveltime field recorded along the horizontal surface would
remain unchanged.
So it is horizontal velocity that matters. But it isn't the true
horizontal velocity, because we're only considering NMO for near-vertical
propagation. It is the horizontal velocity of the vertical paraxial
elliptic approximation.)

Figure 2

Three different approximations (dashed curves) to the
*q*SV impulse-response surface of Greenhorn Shale (bold curves).
On the left is the standard vertical paraxial elliptic approximation. In the
center is the horizontal paraxial elliptic approximation.
On the right is Muir's double-elliptic approximation.

Figure 3

Three different approximations (dashed curves) to the
*q*SV dispersion-relation surface of Greenhorn Shale (bold curves).
On the left is the standard vertical paraxial elliptic approximation. In the
center is the horizontal paraxial elliptic approximation.
On the right is Muir's double-elliptic approximation;
the fit is so close the dashed curve is hard to see.

12/18/1997