Linearization of the acoustic model for a layered fluid and
application of high frequency asymtotics leads to the *
convolutional model* of primaries-only reflection seismograms. The
convolutional model of offset traces is one of the simplest models of
the reflection process within which to pose the velocity analysis
problem. A similar model for plane wave traces is almost equally
simple, and was the subject of earlier work on differential semblance
Minkoff and Symes (1997); Symes and Carazzone (1991). However synthesis of accurate plane wave
traces is a nontrivial task. Accordingly the version of the model
developed here uses offset domain data.

A natural binning scheme for this model is the common midpoint
gather. Since all midpoint gathers are in principle the same for a
layered model, the data consists of a single CMP. The bins contain
single traces, parameterized by offset *x*.

The velocity parameter is simply the interval velocity *v*(*z*), whereas the
reflectivity is and is regarded as bin-dependent,
i.e. *r*=*r*(*z*,*x*); this section plays
the role of a common image gather, as every trace represents reflectivity
below the same midpoint. Thus successful velocity estimation will produce a
``flat'' (x-independent) *r*(*z*,*x*).

The simple version of DS presented here will assume that source signature deconvolution has been applied to the data, so that it is essentially impulsive.

It will be convenient to parametrize velocity
and reflectivity by *vertical two-way time *

With these conventions, the forward modeling operator is

*F*[*v*]*r*(*t*,*x*)= *a*(*t*,*x*)*r*(*T _{0}*(

4/20/1999