where the RMS velocity is
Justification [Continuum derivation of the hyperbolic moveout approximation]: The 2-way traveltime from the surface at z=0 to depth z and back at offset x is related to the solution of the eikonal equation with point source at z=x=0 by
Differentiate this twice with respect to x and use the vanishing of odd-order x derivatives at x=0 (implied by symmetry) to conclude that the second x derivative
Introduce temporarily a new depth coordinate
Then in terms of , q satisfies the Ricatti equation
The solution which is singular at , i.e. z=0, is
Since , you can also write this as
the above can be rewritten
which reveals that the hyperbolic moveout approximation is just the second order Taylor expansion of T2 in x, which should be good for ``small'' x.
This report adopts the hyperbolic moveout approximation, i.e. truncate the Taylor expansion above and take
This amounts to assuming that all events in the data have precisely hyperbolic moveout. Of course this assumption is not entirely consistent with geometric optics. It has been suggested that the deviation of actual two-way time from the hyperbolic moveout approximation may be mistaken for evidence of anisotropy in some cases. In any case the error caused by replacing actual two way time by its hyperbolic moveout approximation is not an asymptotic error in the sense of the last section, so I will treat it as a component of data noise.
The reciprocal square RMS velocity, or RMS square slowness is the primary expression of velocity in the above formula. It occurs so often as to warrant its own notation:
The conditions defining the mute can be restated: since
the quantity on the right hand side of this equation must be bounded away from zero. Since v generally increases with depth, hence u decreases, such a lower bound will only be possible for t0 exceeding a threshold for each x, which is the mute boundary mentioned before. In the data, i.e. (t,x), coordinates, the stretch factor condition becomes
and as before the mute must be supported in the set specified by this condition.
The upper and lower velocity envelopes implied by membership of the velocity in imply corresponding envelope mean square slownesses ( corresponding to and vis-versa) so that .
It is usually reasonable to assume the lower velocity bound to be constant (independent of t0) - for example, equal to sound velocity in water, or close to it. Then is also constant, so you can explicitly estimate a lower bound for T0:
The velocity bounds also imply a bound on the derivative of u:
The bounds on v, the known value of v at the surface, and the maximum two way time imply bounds on the slope
whence a bound on the derivative of u follows immediately.
Since both the lower bound on T0 and the uppoer bound on the derivative of u are uniform over , a -uniform bound on the stretch factor follows:
From this you can derive a -uniform mute boundary. Therefore assume henceforth that is a -uniform mute.