We have discussed the mathematical and physical development of our theoretical Kirchhoff migration/inversion solution from an intuitive perspective (we hope!).

The Kirchhoff reflectivity estimation is an elastic prestack depth
migration. It is *elastic* because it uses the elastic properties
*V*_{p}, *V*_{s}, , and anelastic *Q* factors
in the background migration (velocity)
model, and backpropagates the recorded wavefield in accordance with
the elastodynamic wave equation, as opposed to the standard but
more approximate acoustic wave equation. It is a *depth* migration
because it uses interval properties (velocities) specified as a function
of *spatial* coordinates, as opposed to time migration and
rms velocities specified as a function of a temporal coordinate. This
implies that the elastic depth migration will tend to be more correct
than acoustic time migration if the elastic interval velocity model
is known.

The migration is *true amplitude* in the sense that it
can recover the correct relative amplitude reflectivity information, without
any special preprocessing of the data, provided the data amplitudes are
reliable and the elastic migration model is sufficiently accurate.
reliable. The migration can estimate the *elastic Zoeppritz
reflectivity coefficients* *as a function of specular
reflection angle*. The subscripts (i, j) indicate that any of the
coefficients , , , etc., can be estimated, although at
present only the implementation is available. The estimates are theoretically correct for *heterogeneous media* (including
lateral velocity variations) that are ray valid (i.e., smoothly
varying velocities within the model and each layer, if any).
The present implementation is for a 2-D geometry, but the theory is
valid for *3-D implementation*, which will be considered in the
near future.
Finally, the technique is well suited to
*flexible acquisition geometries*,
and should be adaptable to VSP, crosshole, and other non-standard acquisition
geometries.

The elastic parameter inversion is *fast* and *robust*.
Several (six at present)
*parameterization choices* are available, as well as *confidence
estimator maps*
which give quantitative measures of the relative confidence in the
elastic parameter inversion depth images.

12/18/1997