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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* = (*β*_{1}+*β* _{2})/(*α*_{1}+*α* _{2}),
*I*_{P} (the acoustic
impedance), *I*_{S} (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*).
Δ*F* 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., Δ*ρ*/*ρ*, where
Δ*ρ* = *ρ*_{2} - *ρ*_{1} and *ρ* = (*ρ*_{1} + *ρ*_{2})/2, and similarly for
Δ*α*/*α* and Δ*β*/*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 *I*_{P}) is equal to *ρα*, while
the **S-wave impedance** (denoted *J* or *I*_{S}) is equal to *ρβ*.
Thus elastic properties for a single layer can also be expressed using
( *ρ*,*I*_{P},*I*_{S} ).
Some other quantities in common use are the **bulk modulus**
(*K* = *ρ* [*α*^{2}-(4/3)*β*^{2}]),
the **shear modulus**
( *μ* = *ρβ*^{2} ),
the **Lamé constant**
(*λ* = *ρ*[*α*^{2}-2*β*^{2}]),
**Young's modulus**
(*E* = *ρβ*^{2}[3*α*^{2}-4*β*^{2}] / [*α*^{2}-*β*^{2}]),
and **Poisson's ratio**
(*σ* = [*α*^{2}/2-*β*^{2}] / [*α*^{2}-*β*^{2}]).
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

Δ*μ*/*μ* =
(*μ*_{2}-*μ*_{1}) / [(*μ*_{1}+*μ*_{2}) / 2] =
(*ρ*_{2}*β*_{2}^{2}-*ρ*_{1}*β*_{1}^{2}) /
[(*ρ*_{1}*β*_{1}^{2}+*ρ*_{2}*β*_{2}^{2}) / 2].

It can be shown that this is equal to

[Δ*ρ*/*ρ* + 2Δ*β*/*β* + (Δ*ρ*/*ρ*)(Δ*β*/*β*)^{2}/4] /
[1 + (Δ*β*/*β*)^{2} / 4 + (Δ*ρ*/*ρ*)(Δ*β*/*β*) / 2].

Thus to linear order we can write

(Δ*μ*/*μ*)^{linear} =
Δ*ρ*/*ρ* + 2Δ*β*/*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:

(Δ*I*_{P}/*I*_{P})^{linear} =
Δ*ρ*/*ρ* + Δ*α*/*α*.

(Δ*I*_{S}/*I*_{S})^{linear} =
Δ*ρ*/*ρ* + Δ*β*/*β*.

(Δ*K*/*K*)^{linear} =
Δ*ρ*/*ρ* + [2 / (1-(4/3)(*β*/*α*)^{2})] Δ*α*/*α* + [2 / (1-(3/4)(*α*/*β* )^{2})] Δ*β*/*β*.

(Δ*λ*/*λ*)^{linear} =
Δ*ρ*/*ρ* + [2 / (1-2(*β*/*α*)^{2})] Δ*α*/*α* + [2 / (1-(1/2)(*α*/*β* )^{2})] Δ*β*/*β*.

(Δ*E*/*E*)^{linear} =
Δ*ρ*/*ρ* + 2Δ*β*/*β* + 2 / [(3(*α*/*β* )^{2}-4)(1-(*β*/*α*)^{2})] (Δ*α*/*α* - Δ*β*/*β* ) .

Fatti et al. (1994) have shown effective methods for extracting impedance contrasts.
Grey et al. (1999) have shown how to obtain
Δ*K*/*K*, Δ*λ*/*λ*, and Δ*μ*/*μ*.

Certain specific quantities have also become recognized as useful indicators in AVO analysis. The **intercept** and **gradient**
obtained from plotting reflection coefficients against sin^{2}*θ* are useful because P-P reflection
coefficients vary with incidence angle as

*A* + *B* sin^{2}*θ* + *
O*(sin^{4}*θ* )

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* = (Δ*α*/*α* + Δ*ρ*/*ρ*) / 2

*Bp* =
(1/2) Δ*α*/*α* + 2 (*β*/*α*)^{2}(2Δ*β*/*β* + Δ*ρ*/*ρ*) .

The converted-wave P-S reflection coefficients vary as

*A*sin*θ* + *B* sin^{3}*θ* + *
O*(sin^{5}*θ* ).

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*)Δ*β*/*β*
-[(1/2)+(*β*/*α*)]Δ*ρ*/*ρ*

*Bs* =
[2(*β*/*α*)^{2}+*β*/*α*]Δ*β*/*β* + [(3/4)(*β*/*α*)^{2}+(1/2)*β*/*α*]Δ*ρ*/*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 Δ*σ*/*σ* and its linearization,

Δ*σ*/*σ* ^{linear} =
2 (*β*/*α*)^{2} / [(1-(*β*/*α*)^{2})(1-2(*β*/*α*)^{2})] (Δ*α*/*α* - Δ*β*/*β*)

are calculated in the velocity section of the calculator. The quantity
Δ*σ*/(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,

Δ*σ*/(1-*σ*)^{2} ^{linear} =
4 (*β*/*α*)^{2} (Δ*α*/*α* - Δ*β*/*β*).

We note that the Poisson ratio is determined by the ratio *β*/*α*, whose linearized contrast,

Δ(*β*/*α*) /(*β*/*α*) ^{linear} =
-(Δ*α*/*α* - Δ*β*/*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 Δ*α* = 1.16 Δ*β*, and thus the fluid
indicator, defined as

Δ*F* =
Δ*α*/*α* - 1.16 (*β*/*α*) Δ*β*/*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.

- Aki, K. and Richards, P. G., 1980, "Quantitative Seismology", vol. I, sec. 5.2, Freeman & Co.
- Castagna, J.P., Batzle, M.L., and Eastwood, R.L., 1985, Relationship between compressional-wave and shear-wave velocities in clastic
silicate rocks: Geophysics,
**50**, 571-581. - Fatti, J.L., Smith, G.C., Vail, P.J., Strauss, P.J., and Levitt, P.R., 1994, Detection of gas in sandstone reservoirs
using AVO analysis: A 3-D seismic case history using the Geostack technique: Geophysics,
**59**, 1362-1376. - Richards, P. G. and Frasier, C. W., 1976, Scattering of elastic waves from depth-dependent inhomogeneities:
Geophysics,
**41**, 441-458. - Goodway, W., Chen, T., and Downton, J., 1997, Improved AVO fluid detection and lithology discrimination using Lamé petrophysical parameters; "lr", "mr", & "l/m fluid stack", from P and S inversions: SEG Expanded Abstracts, 183-186.
- Gray F. D., Goodway, W. and Chen, T., 1999, Bridging the Gap: Using AVO to detect changes in fundamental elastic constants: SEG Expanded Abstracts.
- Koefoed, O., 1955, On the effect of Poisson's ratios of rock strata on the reflection coefficients of plane waves: Geophys. Prosp.,
**3**, 381-387. - Ramos, A. C. B. and Castagna, J. P., 2001, Useful approximations for converted-wave AVO: Geophysics,
**66**, 1721-1734. - Shuey, R.T., 1985, A simplification of the Zoeppritz Equations: Geophysics,
**50**, 609-614. - Smith, G.C. and Gidlow, P.M. ,1987, Weighted stacking for rock property estimation and detection of gas: Geophys. Prosp.,
**35**, 993-1014.

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