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

APPENDIX A. MULTIVARIATE T DISTRIBUTION AND REGULARIZATIONS 84
and the model X is now has p times by N elements which can be put in a vector form as
X = [x 1 1, x 2 1, ..., x N 1 , x 1 2, x 2 2, ..., x N 2 , ..., x 1 p, x 2 p, ..., x N p ] T .
For simplicity, we define p by (pN) matrix D i which selects elements in vector X at a given
sample i
x i = D i X, i = 1, 2, 3, .., N. (A.3)
After substituting equation (A.3) into equation (A.2), we obtain
P (X|µ, Ψ, ν) = Po
N
i=0
1
[1 + 1
ν (DiX − µ) T Ψ −1 (DiX − µ)] (ν+p)
2
We can rewrite equation (A.4) in exponential form as follows
(ν + p)
P (X|µ, Ψ, ν) = Po exp −
2
N
i=0
. (A.4)
ln [1 + 1
ν (DiX − µ) T Ψ −1 (D i
X − µ)] . (A.5)
This equation was used in Chapters 3 and 4 to derive the prior distributions of parameters
for our AVO problem.
A.2 Differentiation of the Regularization R(m)
The regularization given by equation (5.28) is derived from the joint probability distribution,
equation (A.5), under the assumption that the center for the model parameters is zero
(µ = 0). Furthermore, p = 3 and ν = 1 which leads to Trivariate Cauchy distribution. In
order to minimize the objective function, we need to differentiate the regularization with
respect to m. Thus,
∂R tc (m)
∂m
=
∂R t c(m)
∂m1
... ∂Rtc (m)
... ∂mk
∂Rtc (m)
∂m3N
T
. (A.6)
Taking the derivative of R tc (m) with respect to mk where k = 1, 2, 3, ..., 3N, we have
∂Rtc N
(m)
= 2
∂mk
i=1
1
1 + m T Φ i m
∂
∂mk
m T Φ i
m . (A.7)