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

CHAPTER 2. AVO MODELING 13
At this point, we need to express the Lamé parameters in terms of density, P-wave and
S-wave velocities in order to express the final result solely in terms of density and velocities
of the media. To this end, the following relationships are useful
α = (
Using these relations in equation (2.13), we have
From which follows,
λ + 2µ
)
ρ
1
2 ,
β = ( µ 1
) 2 . (2.14)
ρ
2 ∂uz
σzz = ρα
∂z
ρ1α 2 ∂uz1
1
∂z + ρ1β 2 ∂ux1
1
∂x
(σzz)1 = (σzz)2
∂ux
+ ρβ2 . (2.15)
∂x
= ρ2α 2 ∂uz2
2
∂z + ρ2β 2 2
∂ux2
. (2.16)
∂x
Substituting equations (2.5) into equation (2.13) and applying differentiation, one can easily
show that
ρ1α1(1 − 2β 2 1p 2 ) ω
A0 + ρ1α1(1 − 2β 2 1p 2 ) ω
α1
= ρ2α2(1 − 2β 2 2p 2 ) ω
ω
A1 − 2ρ1β
α1
2 1p cos ϕr B1
β1
A2 − 2ρ2β
α2
2 2p cos ϕt
β2
ω
B2. (2.17)
Boundary Condition 4: The tangential stress component is continuous across the bound-
ary i.e According to the relation given by equation (2.6), the tangential stress component
can be written as
But exz can be expressed as
σxz = 2µexz. (2.18)
exz = 1
2 (∂uz
∂x
∂ux
+ ). (2.19)
∂z
Using this expression and equation (2.14) in equation (2.18), we can write the boundary
condition as
ρ1β 2 1( ∂uz1
∂x
∂ux1
+
∂z ) = ρ2β 2 2( ∂uz2
∂x
∂ux2
+ . (2.20)
∂z