I found however, after several numerical simulations (see Figure ), that under some conditions (on the deviation of the perturbations from the background) a good approximation can be achieved by the consideration of only the primary P energy. A theoretical consideration of this approximation is painful for the elastic case, but its possible to have a good feeling of the limitations involved by the analysis of the simple case of normal incidence. The reflection coefficient when the two layers are in direct contact is

whereWhen the two layers are separated by the background layer whose thickness approaches zero, the reflection coefficient will be (if we consider only primaries)

where
The elastic response from an interface separating two layers
(*I _{1}* and

The ratio between modeled and actual normal incidence reflection coefficient, as a function of the background impedance, for fixed impedance between the two layers.

Figure shows how the ratio (*F*) between the
two reflection coefficients
behaves for fixed *I _{1}* and

The differences between the reflectivities for non-normal
incidence were computed for some specific cases corresponding
to a background P velocity of 2500 m/s and a maximum variation
of 500 m/s around it for the perturbation layers. The results
are shown in Figure . All graphs start at *p ^{1/2}*=0.012
(which corresponds to an angle of 21 degrees in the background
layer) because the errors are too small for smaller values. For
figures

All figures correspond to *v*_{min}=2000m/s, *v*_{max}=3000m/s,
*v*_{back}=2500m/s, g/cc (*v* refers to
*P* velocities, to density, and indexes 1, 2 and *back*,
to media 1, 2 and background respectively). Different curves
correspond to increasing P velocity of medium 1 or 2, according
to the following convention: continuous=2167, dot=2333,
dot-dash=2500, fine-dash=2667, large-dash=2833. *(a)*
, , , ( represents Poisson's ratios), and
*v _{2}* has a different value for each curve (according to previous
convention).

1/13/1998