The Convolutional Model for Laterally Homogenous Acoustics

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 $r=\frac{\delta v}{v}$ 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

$$t_0 = 2\int_0^z \frac{dz}{v}$$

rather than depth: thus v=v(t₀), r=r(t₀,x).

With these conventions, the forward modeling operator is

F[v]r(t,x)= a(t,x)r(T₀(t,x),x)

where a is the geometric amplitude and T₀(t,x) is the inverse function of the two-way traveltime function T(t₀,x).