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

CHAPTER 3. BAYESIAN INVERSION APPROACH AND ALGORITHMS 27
where P (m|d) is the posterior distribution, P (d|m) is the likelihood of the observed data,
P (d) is a non-vanishing marginal probability of the observed data, and P (m) is the prior
distribution which is a knowledge one has before observation. This theorem describes how
the conditional probability of m given d is related to the converse probability of d given m.
Since the marginal probability of d is a constant, equation (3.1) is equivalent to
P (m|d) ∝ P (d|m)P (m). (3.2)
This equation indicates that Bayesian inference mainly depends on constructing or modeling
the prior distribution of the problem. Therefore, a reasonable choice of the probability
distributions is mandatory such that the results of inversion using this prior is agreeable.
3.3 Prior distributions
From Bayesian point of view, the additional information which can be added to constraint
a given problem is called a priori. The choice of the prior distributions is the most contro-
versial part of the bayesian analysis (Scales and Tenorio, 2001; Ulrych et al., 2001). This
is because incorrect choice of the a priori leads to wrong inferences about the problem
under consideration. It should be chosen on the basis of the type of problem, the target
and kind of system under consideration. When prior distributions are used to formulate
inverse problems, different prior distributions lead to different regularizations. For instance,
regularizations are incorporated for stability, smoothing via Gaussian or l2-norm (Tikhonov
and Arsenin, 1987), sparsity via Cauchy norm (Sacchi and Ulrych, 1995), and edge preserv-
ing and image de-noising via total variation regularization (Vogel and Oman, 1998). The
later two regularizations are non-quadratic regularizations which help to reinforce relatively
larger parameters and suppress small fluctuations (noise). Discussing all the existing reg-
ularization methods is beyond the scope of this thesis. The focus of this thesis is to use
Multivariate Gaussian and Multivariate t distribution to regularize AVO inversion. In the
next sections, the behavior of these distributions are discussed.
Multivariate Gaussian distribution
Gaussian distribution function is a smooth and continuous function which tells us clusters
of a given data around the average or mean(Johnson and Kotz, 1972). The Multivariate
Gaussian distribution for n model parameters, x, is given by
P (x|Σ) =
1
(2π) n/2
exp −
|Σ| 1/2 1
2 (x − µ)Σ−1
(x − µ) , (3.3)