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Suppose that the data *S*(*t*,*x*) is the sum of model-consistent data and another
field, regarded as noise or error:
*S*(*t*,*x*)=*S*^{*}(*t*,*x*)+*E*(*t*,*x*)

where ``model-consistent'' means as before
and *E*(*t*,*x*) is arbitrary (but finite ``energy'' = mean square).
Since there are several data running around in this part of the discussion,
include the name of the data in the notation for the differential semblance
objective:

etc. Then
*J*_{0}[*v*,*S*]=*J*_{0}[*v*.*S*^{*}]+*J*_{0}[*v*,*E*]+*K*[*v*,*S*^{*},*E*]

where
satisfies
Here and in the following, *C* will stand for a constant uniform over
(though the precise value may vary from display to
display).
Likewise,

Similarly, the gradients *with respect to RMS square
slowness* *u* satisfy
and
Suppose that *u* (or its corresponding *v*) is a stationary point of
*J*_{0}[*v*,*S*], i.e. . Then

If you presume that the data error is less than 100%, i.e. ,which seems reasonable (or pick any other fixed percentage, if 100% seems
wrong to you - just absorbs in *C*), then this becomes
That is,
**Theorem:** At a stationary point of the differential semblance
objective, its value is bounded by a -uniform multiple of the
distance of the data to the set of model-consistent data.

Thus for a family of data converging to model-consistent data, any set
of corresponding stationary points of *J*_{0} must have *J*_{0} values which
converge to zero, modulo asymptotic errors.

This result may well *not* imply that stationary points for noisy
data are global minima. Indeed, substitute the ``target'' velocity
*v*^{*} in the expression for *J*_{0}[*v*,*S*]: from the expansion and estimates
above you easily see that

Certainly one hopes that the asymptotic error is no worse than other
errors, in particular than the data error *E*, so this inequality
effectively implies that the global minimum value of is
proportional to for near consistent data, whereas the
theorem shows only that the stationary values are proportional
to , so presumeably larger at least in some cases.
In the next section I will show that when the differential semblance
minimization is supplemented with proper constraints on the velocity
model, in addition to those already imposed, the error in the
*RMS square slowness* is proportional to the error in the
data. It then follows from the estimates above that stationary
values conforming to these constraints are indeed proportional
to the square of the error level, hence essentially global
minima. It would be interesting to know whether relaxing these
constraints actually permits anomalously large stationary values.

** Next:** Data driven model parametrization
** Up:** Symes: Differential semblance
** Previous:** Global Analysis of Stationary
Stanford Exploration Project

4/20/1999