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

CHAPTER 4. TWO-TERM AVO INVERSION 42
4.2.2 Multivariate Gaussian prior
In this section, Gaussian probability distribution is used a priori for the model parameters.
It is given by
P (mf ) =
1
πN exp {−1
|Cf | 1/2 2 mTf C −1
f mf }, (4.11)
where Cf is parameter covariance matrix which shows the correlation between the two model
parameters; P-wave impedance and S-wave impedance. Assuming the model parameters are
correlated at a given time sample but independent from one time sample to another, the
covariance matrix for 2N model parameters can be expressed as
⎛
⎜
Cf = ⎜
⎝
σ 2 rα
σrβrα
. ..
. ..
σ 2 rα
σrβrα
σ rαrβ
σ 2 rβ
. ..
. ..
σrαrβ
σ 2 rβ
⎞
⎟
⎠
(4.12)
which is (2N) × (2N) symmetric matrix. In this matrix, the diagonal elements are the
variances of each of the model parameter elements and the off-diagonal matrices shows the
degree of correlation of various possible combination between a pair of model elements. In
real data inversion, this matrix can be constructed from available bore-hole data (Downton
and Lines, 2004).
Objective function
The objective function of the problem can be constructed by combining the likelihood func-
tion and the prior model using Bayes’ theorem. Thus, using equations (3.2), (4.9) and
(4.11), we have
P (mf |d) ∝ exp{− 1
2 (d − Lf mf ) T Υ T C −1
d Υ(d − Lf mf ) − 1
2 mT f C −1
f mf )}. (4.13)
Maximizing the posterior distribution, equation (4.13), is equivalent to minimizing the fol-
lowing objective function
J mg (mf ) = 1
2 (d − Lf mf ) T Υ T C −1
d Υ(d − Lf mf ) + 1
2 mT f C −1
f mf )}. (4.14)
Differentiating (4.14) with respect to mf and setting the resulting expression to zero, we