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**Claim:** To good
approximation, for ``small'' offsets,
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

Thus
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
satisfies
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
Thus
Since
the above can be rewritten
which reveals that the hyperbolic moveout approximation is just the second order
Taylor expansion of *T*^{2} 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 *t*_{0}
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 *t*_{0}) - 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 *T*_{0}:

so
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 *T*_{0} 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.

** Next:** Global Analysis of Stationary
** Up:** Symes: Differential semblance
** Previous:** Noise Free Data
Stanford Exploration Project

4/20/1999