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

CHAPTER 2. AVO MODELING 16
2.3.1 Bortfeld’s approximation
The first attempt in making an approximation on Zoeppritz equation to make it more useful
for AVO analysis was done by Bortfeld (1962). This equation is given by
Rpp(θi) = 1
2 ln α2ρ2 cos θi
α1ρ1 cos θt
+ sin2 θi
α2 (β
1
2 1 − β 2 2)(2 +
ln ρ2
ρ1
ln α2 α2β1
− ln α1 α1β2
). (2.26)
This approximation is easier to understand how the reflection coefficient is related to the
angles and the earth model parameters than the exact equation (2.24). But it still non-linear
with respect to the Earth model parameters which makes it not suitable for AVO inversion.
2.3.2 Aki and Richards’s approximation
The Bortfield’s approximation was revisited by Richards and Frasier (1976) and later re-
defined by Aki and Richards (1980) which is linear in three model parameters, the P-wave
reflectivity, S-wave reflectivity, and density reflectivity. This equation is given by,
where
RP P (θi) = 1
2 [1 + tan2 (θ)] ∆α
α − 4γ2 sin 2 (θ) ∆β 1
+
β 2 [1 − 4γ2 sin 2 (θ)] ∆ρ
, (2.27)
ρ
∆α = α2 − α1,
∆β = β2 − β1,
∆ρ = ρ2 − ρ1,
α = 1
2 (α2 + α1),
β = 1
2 (β2 + β1),
ρ = 1
2 (ρ2 + ρ1),
γ = β
α ,
θt = sin −1 ( α2
α1
sin(θi)),
θ = 1
2 (θi + θt). (2.28)
This is a very elegant approximation in the sense that it is expressed in terms of three model
parameters independently. It is even the starting equation for other approximations which
are described below. The parameters defined in the Aki and Richards’s approximation are
used in the next sections.