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To start the Interface Parameter Calculator click on the 'Launch' button above.
Some other symbols employed in this calculator are σ (the Poisson ratio), β/α = (β1+β 2)/(α1+α 2), IP (the acoustic impedance), IS (the shear impedance), K (the bulk modulus), λ (the Lamé constant), μ (the shear modulus), E (Young's modulus), Ap and Bp (the coefficients in Shuey's two-term equation), and As and Bs (the converted-wave analogues of Ap and Bp). ΔF refers to the fluid reflectivity of Smith and Gidlow.
Ratios of these quantities are also important. This is because reflection data for the interface can be inverted to obtain information on up to four ratios, one for the densities and three for the velocities. The calculator allows one to calculate or input some of these ratios. Of particular interest in AVO (amplitude variation with offset) studies are the relative contrast ratios, defined as the difference across the interface divided by the average across the interface, i.e., Δρ/ρ, where Δρ = ρ2 - ρ1 and ρ = (ρ1 + ρ2)/2, and similarly for Δα/α and Δβ/β. The reflection and transmission coefficients for the interface may be usefully linearized in terms of these three quantities (Richards & Frasier, 1976; Aki & Richards, 1980).
The information contained in the two velocities can be equivalently expressed by other quantities in common use, such as impedances. The P-wave impedance (denoted I or IP) is equal to ρα, while the S-wave impedance (denoted J or IS) is equal to ρβ. Thus elastic properties for a single layer can also be expressed using ( ρ,IP,IS ). Some other quantities in common use are the bulk modulus (K = ρ [α2-(4/3)β2]), the shear modulus ( μ = ρβ2 ), the Lamé constant (λ = ρ[α2-2β2]), Young's modulus (E = ρβ2[3α2-4β2] / [α2-β2]), and Poisson's ratio (σ = [α2/2-β2] / [α2-β2]). These quantities are commonly used to specify the elastic properties of a homogeneous medium in combinations such as (ρ,K,μ), (ρ,λ,μ), and (ρ,E,σ).
Because they are of interest in AVO studies, the calculator has been designed to also calculate the relative contrasts of impedances and moduli across the interface. This is done in two ways. The exact relative contrast for, say, μ, would be given as
Δμ/μ = (μ2-μ1) / [(μ1+μ2) / 2] = (ρ2β22-ρ1β12) / [(ρ1β12+ρ2β22) / 2].
It can be shown that this is equal to
[Δρ/ρ + 2Δβ/β + (Δρ/ρ)(Δβ/β)2/4] / [1 + (Δβ/β)2 / 4 + (Δρ/ρ)(Δβ/β) / 2].
Thus to linear order we can write
(Δμ/μ)linear = Δρ/ρ + 2Δβ/β.
The difference between the exact and linear contrasts is generally small for interfaces typical of exploration seismology. Both are calculated to allow for comparison. The expressions for other linearized contrasts are given below:
(ΔIP/IP)linear = Δρ/ρ + Δα/α.
(ΔIS/IS)linear = Δρ/ρ + Δβ/β.
(ΔK/K)linear = Δρ/ρ + [2 / (1-(4/3)(β/α)2)] Δα/α + [2 / (1-(3/4)(α/β )2)] Δβ/β.
(Δλ/λ)linear = Δρ/ρ + [2 / (1-2(β/α)2)] Δα/α + [2 / (1-(1/2)(α/β )2)] Δβ/β.
Δρ/ρ + 2Δβ/β + 2 / [(3(α/β )2-4)(1-(β/α)2)] (Δα/α - Δβ/β ) .
Fatti et al. (1994) have shown effective methods for extracting impedance contrasts. Grey et al. (1999) have shown how to obtain ΔK/K, Δλ/λ, and Δμ/μ.
Certain specific quantities have also become recognized as useful indicators in AVO analysis. The intercept and gradient obtained from plotting reflection coefficients against sin2θ are useful because P-P reflection coefficients vary with incidence angle as
A + B sin2θ + O(sin4θ )
as discussed by Shuey (1985). We denote the linearized values of A and B as Ap and Bp and note that they are equal to
Ap = (Δα/α + Δρ/ρ) / 2
Bp = (1/2) Δα/α + 2 (β/α)2(2Δβ/β + Δρ/ρ) .
The converted-wave P-S reflection coefficients vary as
Asinθ + B sin3θ + O(sin5θ ).
Ramos & Castagna (2001) have shown that the linearized values of these latter coefficients, denoted as As and Bs, have the values
As = - 2(β/α)Δβ/β -[(1/2)+(β/α)]Δρ/ρ
Bs = [2(β/α)2+β/α]Δβ/β + [(3/4)(β/α)2+(1/2)β/α]Δρ/ρ .
Goodway et al. (1997) introduced the quantities λρ and μρ as key quantities in LMR (lamda-mu-rho) analysis. These, along with their quotient (λ/μ) and difference (λρ-μρ), have been shown to be useful AVO indicators. They are calculated for the layers above and below the interface.
Koefed (1955), Shuey (1985), and Verm & Hilterman (1996) have emphasized the significance of Poisson ratio contrasts across the interface. The relative contrast Δσ/σ and its linearization,
Δσ/σ linear =
2 (β/α)2 / [(1-(β/α)2)(1-2(β/α)2)] (Δα/α - Δβ/β)
are calculated in the velocity section of the calculator. The quantity Δσ/(1-σ)2 also arises naturally in linearized theories (Shuey, 1985; Verm & Hilterman, 1996) and has been dubbed the Poisson Reflectivity. It is calculated here as an AVO indicator along with its linearization,
Δσ/(1-σ)2 linear =
4 (β/α)2 (Δα/α - Δβ/β).
We note that the Poisson ratio is determined by the ratio β/α, whose linearized contrast,
Δ(β/α) /(β/α) linear =
-(Δα/α - Δβ/β),
is closely related to the Poisson ratio contrasts.
Another important AVO indicator is the fluid indicator introduced by Smith & Gidlow (1987). This is based on the empirical mudrock relation of Castagna et al. (1985),
α = 1.36 km/s + 1.16 β.
In differential form this is Δα = 1.16 Δβ, and thus the fluid indicator, defined as
Δα/α - 1.16 (β/α) Δβ/β,
is zero whenever the mudrock relation is satisfied across the interface. Non-zero values indicate anomalies. The fluid indicator is calculated above, both with the exact β/α and with β/α = 1/2.
The above is intended as a brief introduction to various quantities of interest in AVO analysis. For further information consult the references below.
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