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

CHAPTER 6. CONCLUSIONS 77
two terms and some of them contain three terms. These approximations were compared by
plotting the reflection coefficient versus the angle of incidence in two-layered models with
small and large layer contrasts. For a given two-layered model, it was demonstrated that as
the angle of incidence increases, the deviation from the exact reflection coefficient increases.
Comparing two two-layered models which have small and large contrasts, the deviation be-
comes much more significant in the model with large layer contrast. Nevertheless, all those
approximations are useful for AVO analysis. One has to fix the maximum angle to be used
for a given approximation equation to decrease discrepancy which may arise due to the
inexactness of the model.
Setting a relationship between observed data and model parameters using any of those ap-
proximations leads to a linear equation. In order to retrieve those parameters, the Bayesian
approach was used to formulate the inverse problem. It makes use of the famous statistical
theorem (Bayes’ theorem) which is the base for the formulation of the Bayesian inversion.
This theorem has three probability distribution to deal with: the posterior distribution, the
likelihood function, and a prior distribution. In most cases, the likelihood function is built
from a Gaussian probability distribution function by assuming Gaussian noise. Therefore,
the main concern using Bayesian inversion is to choose a prior distribution. For AVO inver-
sion, two potential prior distributions were investigated in detail: the Multivariate Gaussian
and the Multivariate t distributions. It has been shown that the Multivariate t distributions
has a long tails. Another interesting characteristics for the two prior distribution is that as
the degree of freedom increases, the Multivariate t distributions approaches the Multivariate
Gaussian distribution. Therefore, modeling the model parameters such as P-wave velocity,
S-wave velocity, and density or their attributes using the Multivariate t distributions is still
realistic. In this thesis, two prior distribution which belongs to the Multivariate t distri-
bution family are proposed. These include the Bivariate Cauchy and Trivariate Cauchy
probability distributions which were used to derive regularization terms for two-term and
three-term AVO inversion respectively. The main advantage of this kind of regularization
is to answer the problems I mentioned above. In other words, the two proposed regulariza-
tions allows to incorporate correlation information of the model parameters to stabilize the
inversion and thereby increase stability. Furthermore, they also promote sparse solutions
(high resolution).
Two other prior distribution were also used to constrain the inversion: the Multivariate
Gaussian and Univariate Cauchy distributions. The Multivariate Gaussian prior gives a
quadratic regularization. The AVO inverse problem with quadratic regularizations is solved
by weighted least squares method. When Univariate Cauchy is used as prior distribution for
AVO parameters, it means that the parameters are treated as independent (or uncorrelated).
But any of the Cauchy prior distributions (Univariate or Multivariate) result in a non-
quadratic regularization term i.e the regularization that depend on the model parameters.