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

CHAPTER 2. AVO MODELING 17
2.3.3 Shuey’s approximation
This approximation is modified version of the Aki and Richards’s approximation. But in
this case the equation was rearranged by Shuey (1985) to incorporate the Poisson’s ratio as
parameter instead of the S-wave velocity. The equation has the from
where
R(θi) = Rp + Rg sin 2 θ + 1 ∆α
2 α (tan2 θ − sin 2 θ) (2.29)
Rp = 1
2 (∆α
∆ρ
+
α ρ ),
Rg = [RpA0 + ∆σ
],
(1 − σ) 2
σ =
1
γ2 − 2
2( 1
γ2 − 1) ,
∆σ = σ2 − σ1,
A0 = B − 2(1 + B)
B =
2 ∆α
α
Rp
.
1 − 2σ
1 − σ ,
Note that α, β and ρ has the same definition as in the Aki and Richards’s approximation.
The parameter σ is called the Poisson’s ratio. This approximation is very useful if the
analysis is in terms of the Poisson’s ratios. It is an interesting property of rocks which is
directly related to their elasticity.
2.3.4 Verm and Hilterman’s approximation
Further simplification on Shuey’s equation was by ignoring the last term and setting γ = 4
(Verm and Hilterman, 1995). The simplified two term Rpp has the form,
where
Rpp(θi) = Rp + Gp sin 2 θ, (2.30)
Gp = 9
∆σ − Rp.
4
The first term in equation (2.30), Rp, is called the Intercept and Gp is called the Gradient.
This is often used in AVO analysis and is called Intercept-Gradient analysis.