The *forward map* *F*[*v*] is a linear operator depending on a
velocity function *v*. The velocity function depends on
all or part of the subsurface coordinates; the examples presented below
use depth-dependent velocity. *F*[*v*] is a *prestack* forward
modeling operator; it takes an image volume or bin-dependent
reflectivity as input, and outputs a seismic
data volume. In the examples presented below, the data will be a
common midpoint gather, each bin will contain a single trace, and the
bin parameter is offset. Thus the input reflectivity also has the
appearance of a common midpoint gather, and can be identified with the
image gather in this setting.

The *inverse map* *G*[*v*] is an approximate inverse to *F*[*v*]. That
is, if data *d* and reflectivity *r* satisfy *d*=*F*[*v*]*r*, then . For multioffset data and
multidimensional models, Beylkin (1985) showed how to build such
operators as weighted diffraction sums. For layered modeling, *G*[*v*]
is essentially moveout correction, after compensation for amplitude
and wavelet deconvolution; *F*[*v*] inverts these steps.

Differential semblance measures nonflatness by comparing neighboring
image bins. That is, if the image is *r* = *G*[*v*]*d*, and the bin index is
*i*, then the differential semblance is the mean square power of . A convenient notation for root mean square of a
field, say *r*, is . The dot product of two fields (viewed as
vectors of samples), say *r _{1}* and

The dual regularization objective function is then

Note that the differential semblance objective explored in the above cited references is the special case of this one with . In general, a Lagrange multiplier exists for which the solution
Besides accurate numerical
implementations of the operators described above, we require
methods for solving the system of normal and secular equations. For
the latter, we use the Moré-Hebden algorithm Björk (1997)
with the linear systems occuring in this method solved approximately
by conjugate gradient iteration.
The best estimate of *v* results from gradient-based optimization (a
quasi-Newton method) applied to . In the experiments
reported here, we have used the Limited Memory
Broyden-Fletcher-Goldfarb-Shanno method as developed in
Nocedal (1980).

4/20/1999