This section describes the inversion of a single *R*_{pp} and
offset gather by SVD to produce a single depth trace each of P-impedance
(*I*_{p}), S-impedance (*I*_{s}), and density (). Recall that these
parameters are of the form , where is the
*change* in a given property value, and is the
*average* property value, i.e. -- *relative changes* in elastic
material properties.

Figure shows the three elastic parameter traces *I*_{p}(*z*;*x*),
*I*_{s}(*z*;*x*) and (as a function of depth *z*, for a fixed
surface position *x*).
The first depth trace is the P-impedance trace
*I*_{p}. The correct model values (in %) from Table are plotted in
parentheses, adjacent to the m/i estimated peak values. The *I*_{p}
estimates are seen to be reasonably accurate. Again the first event
at 1960 m is weak due to the mute pattern in the shot records affecting
the *R*_{pp} estimation, and the events below the gas (2960 m) are weak
due to attenuation through the low *Q* gas layer. The *I*_{s}
estimates are also seen to be quite reasonable, although perhaps
less robust than *I*_{p}. Also, although the *I*_{s} relative
change of 0% has been correctly estimated at the gas sand, there
are wavelet sidelobe artifacts which would make it difficult to resolve
in field data situations. A similar situation applies to the impedance
change at 3100 m where two closely spaced reflections overlap into
one apparent reflection. This is analogous to a thin bed resolution
problem and linearized amplitude tuning effects. Although the
*I*_{s} estimate is correct at 3100 m (somewhat fortuitously),
the overlap of two
distinct wavelets erroneously suggests a single impedance change of a large
negative polarity which is clearly incorrect. Lastly, the density variation
is completely unreliable, except for perhaps the 1960 m event which has
the correct polarity and roughly approximate amplitude due to its
large specular angle coverage (). This will be explained
further in the next section under the topic of choice of elastic
parameterization for the inversion.

Figure 6

In summary, with accurate synthetic data and a realistic recording geometry
and background model, we are able to recover very reasonable *I*_{p} estimates,
somewhat less robust *I*_{s} estimates, and very little (if any) information
about density variations.

12/18/1997