The (reference or background) velocity *v* includes
the slowly varying components of velocity and perhaps other fields.
The reflectivity *r* is a field (or vector of fields)
encompassing the rapidly varying components of the model. Linearized
scattering treats *r* as a perturbation of *v*. Thus in the forward
modeling operator *F*[*v*]*r* the dependence on *r* is linear,
whereas the dependence on *v* is (quite!) nonlinear.

*Minimal data sets* are those on which the kinematic relations
in the data are bijective.
Minimal data sets include common shot and common offset gathers, and -
for layered models - single traces. For a few models, such as
constant density acoustics, the forward modeling operator *F*[*v*] is
invertible (modulo smoothing operators) on minimal data sets. This
paper deals only with constant density acoustics.

Denote by *G*[*v*] an approximate inverse operator for *F*[*v*] on each
(minimal) data bin (common source, common offset, single trace,...).
Thus *G*[*v*] applied to the data produces a *prestack reflectivity
volume*. Similarly, understand by *F*[*v*] the application of forward
modeling independently for each reflectivity bin.

Each binning scheme also implies a notion of *neighboring bins*:
that is, neighboring source positions, offsets,... Denote by *W* an
operator approximating the derivative or gradient in the bin
parameter(s). Generally the definition of *F*[*v*] necessarily
incorporates a cutoff or mute, as does that of *G*[*v*]. Differentiation
in the bin direction across this mute produces edge artifacts. To
control these, introduce an additional mute slightly more
severe than the mutes built into *F* and *G*. Since the edge effects
are localized, application of this secondary mute eliminates
them.

Differentiation enhances high frequency content. To keep the spectrum
of the differential semblance output comparable to that of the data,
employ a smoothing operator *H*. An appropriate choice is the inverse
square root of the Helmholtz operator in all of the variables on which the data depends, i.e. both
within-bin and cross-bin variables.

With these notations, define differential semblance *J _{0}*[

4/20/1999