Brief notes on the CREWES Interface Parameter Converter

The Interface Parameter Converter is designed to be a convenient tool for converting data on interfaces between earth layers into various useful formats. For instance, given the density (ρ), compressional velocity (α), and shear velocity (β), for each side of the interface, one can calculate ratios (e.g. β1/a1), or relative conrasts (e.g. Δρ/ρ) that are commonly used in AVO applications. The subscript "1" indicates the upper layer and "2" the lower layer.

Some other symbols employed in this applet 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.

For questions or suggestions regarding this applet, contact Chuck Ursenbach (ursenbach@crewes.org).

Extended notes on the CREWES Interface Parameter Converter

The elastic properties of any homogeneous earth layer can be summarized by three quantities. The choice is not unique, but is commonly given as the density (denoted by ρ), the compressional-wave velocity (denoted by VP or α), and the shear-wave velocity (denoted by VS or β ).

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.

(ΔE/E)linear = 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

ΔF = 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. For questions or suggestions regarding this applet, please contact Chuck Ursenbach (ursenbach@crewes.org).

References

  1. Aki, K. and Richards, P. G., 1980, "Quantitative Seismology", vol. I, sec. 5.2, Freeman & Co.
  2. 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.
  3. 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.
  4. Richards, P. G. and Frasier, C. W., 1976, Scattering of elastic waves from depth-dependent inhomogeneities: Geophysics, 41, 441-458.
  5. 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.
  6. Gray F. D., Goodway, W. and Chen, T., 1999, Bridging the Gap: Using AVO to detect changes in fundamental elastic constants: SEG Expanded Abstracts.
  7. 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.
  8. Ramos, A. C. B. and Castagna, J. P., 2001, Useful approximations for converted-wave AVO: Geophysics, 66, 1721-1734.
  9. Shuey, R.T., 1985, A simplification of the Zoeppritz Equations: Geophysics, 50, 609-614.
  10. 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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