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

CHAPTER 2. AVO MODELING 12
and strain components. According to the usual geometry depicted in Figure 2.1.
σij = λ∆δij + 2µeij, (2.6)
where σij is the stress component, eij is the strain component and ∆ is the dilation which
is the sum of the normal strain components. the indices ij refers to the various combination
x, y and z directions. the constants λ and µ (shear modulus) are the Lamé parameters.
The following boundary conditions holds at z = 0.
Boundary Condition 1: The tangential displacement component is continuous across the
boundary,
ux1 = ux2. (2.7)
Using the first two equations of equation (2.5) and setting z=0, we can easily see that
ω
A0 cos θi − ω
A1 cos θi + ω
B1 sin ϕr = ω
A2 cos θt − ω
B2 sin ϕt. (2.8)
α1
α1
β1
Boundary Condition 2: The normal displacement component is continuous across the
boundary,
α2
uz1 = uz2. (2.9)
Using the third and the fourth equations of equation (2.5) and setting z=0, we have
ω
A0 sin θi + ω
A1 sin θr + ω
B1 cos ϕr = ω
A2 sin θt + ω
B2 cos ϕt. (2.10)
α1
α1
β1
Boundary Condition 3: The normal stress component is continuous across the boundary.
According to equation (2.6), the normal stress component can be written as
where
We can rewrite equation (2.11) as
α2
σzz = λ(exx + ezz) + 2µezz, (2.11)
exx = ∂ux
∂x ,
ezz = ∂uz
. (2.12)
∂z
σzz = (λ + 2µ) ∂uz
∂z
β2
β2
+ λ∂ux . (2.13)
∂x