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

CHAPTER 4. TWO-TERM AVO INVERSION 41
or
where Lf = WGf and
d = Lf mf + n, (4.5)
mf =
4.2.1 Likelihood function of the data
rα
rβ
. (4.6)
Theoretically, the uncertainty between the observed data and synthetic data (data generated
using convolution model) is the noise in the observed data. Assuming the noise terms are in-
dependent and Gaussian, the noise can be modeled using Multivariate Gaussian probability
distribution given by
P (n|Cd) = exp{− 1
2 nT C −1
d n}, (4.7)
where Cd is the data covariance matrix having the form
⎛
σ
⎜
Cd = ⎜
⎝
2 d1
0
0
σ
. . . 0
2 .
.
.
d2
.
.
.
.
.
.
0
.
.
.
0 0 . . . σ2 dMN
⎞
⎟ . (4.8)
⎟
⎠
Using equation (4.5) and (4.7), the likelihood function of the data can be expressed as
where,
P (d|mf ) = Po exp{− 1
2 (Υ(d − Lf mf )) T Cd −1 (Υ(d − Lf mf ))}, (4.9)
Po =
1
(2π) (NM)/2 . (4.10)
|Cd| 1/2
A diagonal matrix, Υ, is also introduced for muting i.e to protect the algorithm from trying
to fit with the input data with zero entries.