Let us outline the mathematical and physical components of the true-amplitude elastic migration reflectivity theory. We refer the reader to the paper of Stolt and Weglein (1985) for a comprehensive overview of seismic migration and inversion techniques.

We begin by considering a small *local* neighborhood about a single
subsurface location . In general, at we can define an incident
vector displacement wavefield and a scattered vector displacement
wavefield . The wavefield is due to the seismic source
(dynamite, Vibroseis, airgun array, etc.), and contains the scattered
reflections and diffractions caused by the source energy bouncing off
of rapid changes in subsurface elastic properties, such as reflectors,
diffractors and fault zones. The wavefields are vectors since they contain
the full three-component directional information of elastic particle
displacement, as
opposed to directionless measurements of scalar quantities like
acoustic pressure.
We can define a 3x3 scattering matrix which relates the
incident source wavefield to the scattered (reflected or diffracted) wavefield.
At reflectors, the elements of the scattering matrix contain the elastic
Zoeppritz reflection coefficients , , , etc. Hence, if we
can determine any of these coefficients, they will in turn reveal information
about the changes in elastic parameters which define them. It is evident that
determining these coefficients depends on our ability to reconstruct the
source wavefield and the scattered wavefield *locally*
at each subsurface point .

To reconstruct the incident and scattered wavefields, we mathematically
solve the elastodynamic wave equation which describes the propagation of
elastic waves through heterogeneous media (Aki and Richards, 1980).
The mathematical technique used
is the Kirchhoff (Love) integral solution (Wu, 1989),
which gives a way to
reconstruct the wavefields in terms of integrals over the seismic source
and the recorded seismic trace data at the surface. To evaluate the
integrals, we need to calculate the traveltimes and amplitudes of the waves
from the source to each subsurface point, and *A*_{s}, as well as the
traveltimes and amplitudes from each receiver to each subsurface point,
and *A*_{r} (Keho and Beydoun, 1988).
We calculate these traveltimes and amplitudes
numerically in heterogeneous media by raytracing techniques (Cervený
*et al.*, 1977, Beydoun and Keho, 1987).

The final solutions are integrated in a constant offset implementation to
preserve the reflection coefficient estimates as a function of fixed
specular angles. This is equivalent to presorting the recorded data
into constant offset sections, and then migrating each section separately.
In practice we do the sorting internal to the migration computer algorithm,
so the input data can originate from arbitrary acquisition geometries.
Numerically, the integrals become computer summations over all midpoint
coordinates for a fixed offset . The midpoint sum
for the *R*_{pp} coefficient, for a single offset , is

(1) |

or,

(2) |

where is a migration weighting function depending on the raytraced
source and receiver amplitudes *A*_{s} and *A*_{r}, and is
the recorded trace data which has been deconvolved and filtered.
The total factor can be interpreted as a constant
offset scattering function, and by summing along the traveltime hyperbolas
given by , the summed scatterers make a constant offset
reflection.

The specular angles can be determined by a first moment
summation similar to the *R*_{pp} sum,

(3) |

where *F* is some known function like , and and
are the scattering angles made by the incident ray and scattered
ray at each subsurface location. The *R*_{pp} and results
are somewhat similar in spirit to the results of Parsons (1986), which in
turn are based on the work of Beylkin (1985) and Bleistein (1987). The
main difference is that our method is completely based on the
solution to the elastodynamic wave equation in heterogeneous
media, as opposed to their
acoustic formulations. Several additional theoretical and
implementational details further distinguish our work.

So, by implementing the above summations, we obtain estimates of and , which can be combined to obtain ; in other words, we can estimate the elastic reflection coefficient as a function of specular reflection angle at every point in the subsurface.

12/18/1997