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

APPENDIX B
EM Algorithm
B.1 EM-Algorithm to estimate the scale matrix Ψ
EM algorithm can be used to estimate the scale matrix, (Ψ), and the location parameter,
(µ l). If the center (the location parameter) for the model parameters to be zero, we only
need to estimate scale matrix (a matrix that contains the correlation information about
the model parameter). Taking the natural log of equation (5.23), the prior distribution for
model parameters m, we have
where
L(Ψ) =
N
i=1
[− 1
2 ln |Ψ| − 2 ln(π) − 2 ln(1 + ∆T i Ψ −1 ∆i), (B.1)
∆i = D i m. (B.2)
Differentiating equation (B.1) with respect to Ψ −1 keeping the other parametes constant,
we have
∂L
∂(Ψ −1 )
N
= NΨ − diag{Ψ} −
2
N
4
1 + ∆
i=1
T i Ψ−1 ∆i∆
∆i
T i − 1
2 diag{∆i∆ T i }] (B.3)
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