The previous section has illuminated the fact that the accuracy of inversion results can vary depending on parameter choice, data noise levels, and incomplete coverage of a target zone constrained by acquisition geometry. In this section we briefly address the issue of estimating a quantitative ``confidence'' criterion to display alongside elastic parameter inversions. We believe our confidence maps can be of great aid to guiding interpretations of inversion results. This work is a currently ongoing part of our active research, and is in preparation for journal publication.

Figure is the impedance inversion of the noisy gather
(Figure ),
tabulated in Table , and should be compared to the noise-free inversion
in Figure . It is immediately obvious from
Figure that *I*_{p} is the
most robust parameter to noise, followed by *I*_{s}, followed by .This agrees with our intuitive interpretation of the Impedance radiation
curves (Figure ), and the values tabulated in Table .
It is evident that
there are many false impedance changes in the inversion which are caused
by noise. We have developed a confidence measure which helps to discriminate
real from false impedance changes, and an overall quantitative measure
of the inversion's stability at each subsurface point.

Figure 11

Figure 12

Our inversion confidence estimates for the results of Figure are
displayed in Figure . Immediately, one can recognize from the
confidence maps that only four
events on the inversion traces are of reliable merit. These correspond
to four of the five correct events in the synthetic model. The fifth
event at 1960 m has low confidence because the random noise has
severely contaminated its AVO response as shown in Figure .
Comparison of
the lateral coherency and amplitude of events in Figure qualitatively
reinforces
the relative quantitative confidence estimates in Figure . However,
the peak confidence value is about 30%, which indicates that on the
whole, the inversion is having trouble with the noise level and
coherency. Furthermore,
it is evident that the confidence in *I*_{p} is about six times greater than
the confidence in *I*_{s}, and there is little or no confidence in any of
the density variations. Finally, a false event due to coherent noise
at 3650 m is given some nonzero confidence because of its coherency in
the *R*_{pp} gather as shown in Figure . Hence, our confidence estimates
can greatly aid in the appraisal of inversion results (evaluating their
worth and reliability from one location to another), but are not immune
to noise which is coherently correlated with reflection events.

We now briefly describe the confidence criteria from a general perspective.
Readers interested in more detail can request a paper preprint from
the authors, or stay tuned for subsequent SEP reports.
The confidence measure consists of
objective numerical criteria calculated directly from the *R*_{pp} data
and the inversion step. From the *R*_{pp} data, we calculate quantitative
measures of
event coherency and average coherent amplitude, as a function of offset.
During the inversion step, we calculate the ``goodness of fit'' of the
parameter estimates to the *R*_{pp} values, and the eigenvalue spectrum
stability of the matrix system, at each subsurface location. The
confidence estimate is a function of these measures. A confidence value
of 100% means implies maximum offset coherency and amplitude in the
CDP gather, as well as a perfect fit of the parameter estimate to the
data, and maximum stability in the singular value decomposition.

12/18/1997