Inversion ``confidence'' estimation

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 $\rho$.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.

 

invnoise
Figure 11 Noisy marine synthetics impedance inversion: I_p, I_s, and $\rho$.

 

conf
Figure 12 Impedance inversion ``confidence'' map for noisy synthetics.

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.