Regularization of the AVO inverse problem by means of a ...

CHAPTER 6. CONCLUSIONS 78
Therefore, IRLS algorithm is used to solve the AVO inverse problem.
The results of the inversion in both synthetic and real data examples indicated that the corre-
lation information is very important to stabilize the inverse problem. This was demonstrated
by the Multivariate Gaussian and Multivariate Cauchy regularizations. This especially is
evident in the three-term AVO inversion where we have density term. In the two-term case,
even Univariate Cauchy may perform good as the inversion is less ill-conditioned than the
three-term AVO inversion. But it still shows a little discrepancy due to the treatment of
the parameters as independent. In the three-term AVO inversion, Monte Carlo Simulation
for inverted results was done by calculating the root mean square errors for a number of
realizations at each signal to noise ratio. This helps to understand which part of the model
parameters can be resolved reliably as the noise level increases. We have seen that increas-
ing the noise level affects the density reflectivity as compared to the other two parameters
(P-wave reflectivity and S-wave reflectivity). Furthermore, for all the three parameters,
the Trivariate Cauchy has least uncertainty as compared to the Univariate Cauchy and
Multivariate Gaussian regularizations. The over all investigation of the prior distributions
used to constrain the inverse problem showed that the Multivariate Cauchy distribution are
promising for better prediction of subsurface physical parameters.
Finally, solving inverse problems in general involves an hyper-parameter selection. Large
data sets should have an automatic hyper-parameter selection method that can be imple-
mented in the inversion algorithm. More work is required in order to select hyper-parameters
automatically.