Inversion methods which use the directional dependence of the reflection
coefficient to estimate the elastic parameters of the medium (AVO inversion)
use in general an angular functionality to express such a dependence.
This choice is not the more convenient because the angle estimation
is strongly dependent on the macro model that was used in the estimation
process. Moreover, the propagation angles depend on the elastic perturbations
that it is been used to estimate. A more appropriate choice for expressing the
directional dependence of the reflection coefficient is the *local
Snell parameter*, which is defined as the component of the slowness
parallel to the ``reflector plane" at each position of the underground.
Evidently, not all points of the subsurface can be considered
as a reflector, but at all points of interest; that is, where the
the upcoming wavefronts intercepts the downgoing wavefront, a ``reflector
plane can be defined". As defined, the local Snell parameter is
conserved for any perturbation in the local elastic parameters. Although
its estimated value will be still dependent on the macro model, it will
be much less sensitive to errors in the model than the angle.

Figure shows a descending (incident) wavefield crossing an ascending (reflected) wavefield at a given time step of the backward propagation part of the scheme. The crossing point defines the point of the interface that was imaged at that time, and the angles and i are measured, respectively, from the tangent and from the normal to the interface at that point.

reflect
The points where the ascending and descending wavefronts overlap
define the location of the reflector. The reflection angle can
be determined by the gradients of the two wavefields at the reflection
point, at the time when the reflection occurred.
Figure 1 |

From the figure we get the following relation,

where the(3) |

The unit vectors and are estimated from the particle velocity field, by using the following equations:

(4) |

12/18/1997