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

CHAPTER 1. INTRODUCTION 8
are explained. The inversion algorithms for different possible scenarios are discussed. These
include ordinary least squares solution (OLS), weighted least squares solution (WLS), it-
erative re-weighted least squares solution (IRLS), and the target oriented or window by
window AVO inversion via IRLS. In the same section, two possible ways of selecting a reg-
ularization parameter (or hyper-parameter), the chi-square test and trade-off-diagram, are
also discussed.
In Chapter 4, a two-term AVO inversion using the Fatti’s approximation on Zoeppritz equa-
tion as forward modeling is investigated. Three prior distributions are used independently
to formulate the inversion via Bayesian approach. The first prior distribution is the Gaus-
sian probability distribution which leads to a quadratic regularization term. The other
two are family of the Multivariate t distribution, Univariate Cauchy and Bivariate Cauchy
distributions. The later two prior distribution lead to non-linear regularizations i.e. regular-
ization term which depend on the model parameters. Finally, using the methods of solving
the inversion given in chapter 3, the three regularizations are demonstrated by synthetic
examples.
In Chapter 5, a three-term AVO inversion using the Aki and Richards’s approximation as
forward modeling is investigated. Three prior distributions are independently used to formu-
late the inversion via Bayesian approach. These include Multivariate Gaussian, Univariate
Cauchy, and Trivariate Cauchy probability distributions. Similar to two-term AVO, regular-
ization via Multivariate Gaussian is the inverse of parameter covariance matrix and the other
two result non-linear regularizations. In the last section of this chapter, the regularizations
are demonstrated by synthetic and real data examples.
In the last chapter, the summary of all chapters are briefly described. It focuses on the aim
and the findings of this research.