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

CHAPTER 2. AVO MODELING 15
The resulting equation is a specific case in which we have a down-going P-wave (incident),
reflected P-wave, reflected S-wave, transmitted P-wave and transmitted S-wave. Following
a similar procedure given above, it is also possible to generalize for all possible incidence,
reflected and transmitted waves. For AVO analysis, the Rpp and Rps are the most exploited
reflection coefficients i.e in typical seismic experiment using compressional wave sources ,
P-waves, and receiving the P-wave component and/ or S-wave component by receivers. In
this thesis, only the Rpp of equation (2.23) part is used.
2.3 Approximation of the Zoeppritz Equations
Solving for Rpp from Zoeppritz equation, the exact P P wave reflection coefficients in terms
of the ray parameter p, velocity and density of the media can be expressed as
where
Rpp =
(b cos θi
α1
cos θt
cos θi cos θt
− c )F − (a + d α2 α1 α2 )Hp2
D
a = ρ2(1 − 2β2p 2 ) − ρ1(1 − 2β1p 2 ),
b = ρ2(1 − 2β2p 2 ) + 2ρ1β1p 2 ,
c = ρ1(1 − 2β1p 2 ) + 2ρ2β2p 2 ,
d = 2(ρ2β2 − ρ1β 2 1),
cos ϕr cos ϕt
F = b + c ,
β1
β2
(2.24)
H =
cos θt cos ϕr
a − d ,
α2 β1
D =
det M
. (2.25)
α1α2β1β2
Although the above equation shows the exact transformation of the reflection coefficient,
it is non-linear and complicated with respect to the the model parameters of the medium.
For this reason, under the assumption of small property contrasts, expanding and retaining
only linear terms, a number of papers has been published on linearized approximations
which can be directly used for AVO inversion. The next section is devoted in revisiting the
approximations