Once I have estimated ,
an inverse problem for three isotropic elastic
parameters can be posed. Under the assumption that *relative contrasts*
in material properties are small at reflecting boundaries, and the reflection
angles are well within the pre-critical region (Aki and Richards, 1980),
a linearization of the
Zoeppritz plane wave reflection coefficients can be made at every subsurface
point :

(15) |

where are the relative contrasts in *P* impedance,
*S* impedance and
density at the reflecting boundary, and are known basis
functions which are analytical in . The three basis functions
are plotted in Figure , with *c _{1}* at the top,

(16) |

where is the shear to compressional velocity ratio *v*_{s}/*v*_{p}.
I used a constant value of in this data example, but
it could be specified as a function if the appropriate
*v*_{p} and *v*_{s} information is available. In fact, the basis functions
*c*_{i} are not too sensitive to reasonable ranges of values.

In principle, any three elastic parameters can be chosen that
span the space. I choose the elastic *impedance* parameterization,
because of its robust inversion properties for surface seismic geometries
when a narrow (e.g., 5-35) reflection illumination aperture is only
available in the data. I explored this issue more completely in
Lumley and Beydoun (1991), and it has recently become an active area of
discussion in AVO inversion (e.g., De Nicolao et al., 1991).

In particular, I invert (15) at every subsurface location by a least-squares method which bootstraps with offset and angle.
The logic behind my approach is based on the properties of the basis functions
, as plotted in Figure .
I first find a least-squares estimate for *I*_{p} using only values
for which . Next, I find a least-squares estimate for
*I*_{s} using the data in the range
and using the estimate of *I*_{p} as a *constraint* on the system.
Finally, if there are angles in the data greater than 35, I perform
a least-squares estimate for the density parameter using the *I*_{p} and
*I*_{s} values as constraints. I have found this method to be a very robust
procedure for estimating *I*_{p} and *I*_{s} (e.g., better than damped SVD),
and also for demonstrating that little or
no *independent* information on the contribution of density
contrasts to a reflection in typical surface seismic geometries is
invertible.

Figure 2

11/17/1997