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

CHAPTER 5. THREE-TERM AVO INVERSION 64
The stability and the uncertainty in the inversion using the various probability distribution
functions are also analyzed via Monte Carlo simulation for various S/N. Increasing the noise
level corresponds to large uncertainty in the inverted results. Especially, the instability of
the result using the Univariate Cauchy regularization is more clearly observed. This can
be explained by the absence of the correlation information in this regularization. Adding
only the correlation information does not grant the accuracy of the result as seen by the
Gaussian prior. The density reflectivity usually has very small magnitude as compared to
the P-wave and S-wave reflectivity. Therefore, one has to expect smaller root mean square
error as compared to the P-wave and S-wave reflectivity. But the smaller is the magnitude
the more likely appears to be dominated by the noise. This characteristic is also seen in
the simulation. The relative mean square error in the density reflectivity is larger than
the P-wave reflectivity and S-wave reflectivity. Nevertheless, using the Trivariate Cauchy
regularization has the least error than the other two for all the three parameters.
5.4 Real data examples
The real data set is provided by Petrobras, Argentina. The data consist of 61 NMO corrected
CDP gathers offset ranges from 240 m - 3210 m in a difference of 240 m. The time ranges
from 0 to 1.502 s with sampling interval of 2 ms. The wavelet is estimated from each trace
and an average wavelet is used for each CDP. The inversion is done only in a time window
of 0.75 s -1.35 s which is subdivided into two window each with an overlapping window of
0.146 s (the time length of the wavelet) below and above each window. Separate inversion is
done using the target oriented inversion algorithm for each window and the results are put
together. The inversion is constrained by including a correlation information matrix which
is obtained from well-log data. Understanding the ineffectiveness of the Univariate Cauchy
for correlated parameters, the real data inversion is done for Gaussian and Trivariate Cauchy
only. Figure 5.14 and 5.15 are the inversion results using the Gaussian and Trivariate Cauchy
probability distributions as prior respectively. Figure 5.16a shows the stacked real data.
Figures 5.16 b and c shows the data predicted from the results of the inverted reflectivities
from each CDP gather and then stacked using the Multivariate Gaussian and Trivariate
Cauchy priors respectively.
The real data results show similar behaviors like the synthetic case. The same trend is
observed both in the Gaussian and Trivariate Cauchy prior distributions which reflects they
do similar job incorporating the well-log information thereby stabilizing the inversion. As
in the synthetic case, the resolution enhancement is evident in the Trivariate Cauchy prior.
The stacked sections (Figures 5.16 (b) and (c)) are quite similar to the original stacked data
(Figure 5.16 a). This implies that the inversion honors the original input data.