Notes on the CREWES Reflectivity Explorer Applet

Notes on the CREWES Reflectivity Explorer applet

Use of the CREWES Reflectivity Explorer Applet

Technical Notes.

Rationale and use of the reflectivity approximations.

AVO and inversion

One common application of P-P reflectivity (RPP) expressions is in Amplitude-Variation-with-Offset (AVO) studies in reservoir seismology. More recent work has applied AVO to converted waves (RPS). The general objective in AVO is to collect information on how seismic amplitudes vary with incident angle, and to use this along with RPP (or RPS) expressions to obtain information on properties of earth layers. Since these coefficients are essentially expressions of amplitude in terms of earth properties (densities and velocities), such a process involves a mathematical inversion.

The Zoeppritz Equations (Aki & Richards, 1980) give exact expressions for RPP and RPS for elastic plane waves at a non-slip horizontal boundary between two semi-infinite isotropic elastic media. Using these solutions for inversion may be termed a full Zoeppritz Inversion. Since there are six earth variables [density (ρ), P-wave velocity (VP) and S-wave velocity (VS) for each of the two layers] it can be difficult to obtain accurate results. It is helpful to recognize that only four of the six variables are independent [see for instance Lavaud et al. (1999) J. Seis. Expl. 8, 279-302 and Pate, A.J. (1996) AAPG Bulletin 80, 978].

Linear Approximations

The Aki-Richards (1980) and Bortfeld (1961) approximations are linear in property differences across the boundary. In other words, they are accurate when there is only a small change in density and velocities between the two media. The Bortfeld expression depends explicitly on individual properties of the two media, while the Aki-Richards is given in terms of average properties and property differences. Thus an advantage of the Aki-Richards approximation is that it simplifies the inversion procedure by changing the independent variables to Δρ/ρ, ΔVP/VP, ΔVS/VS, and VS/VP. The first three are fractional changes and may be referred to as density contrast, VP contrast, etc. VS/VP is often approximated by some reasonable value, effectively reducing to three the number of quantities sought in the inversion.

The Aki-Richards approximation can be written in terms of the ray parameter, p, or in terms of the average angles of reflection and transmission, q and j. Shuey (1985) also suggested the possibility of replacing the average P-wave angle by the angle of incidence. These three possibilities have each been implemented in this current version of the Explorer to allow comparison of their behaviour. For a discussion of this in relation to post-critical behaviour see Downton and Ursenbach (2005).

Approximations to the Aki-Richards approximation

Shuey (1985) has rewritten the Aki-Richards RPP in a useful form consisting of three terms. The first is the zero-offset reflection coefficient, the second depends on the square of the sine of the average P-wave angle, and the third essentially depends on fourth and higher even-order powers of the sine. The full three-term expression is equivalent to Aki-Richards (to linear order), but the 2-term approximation is commonly used and is normally considered accurate up to an angle of about 30 degrees. This allows one to invert to only two variables, the two variables being combinations of the four Aki-Richards variables. In the case of RPS one obtains an odd-power series of sine functions and RPS is sometimes approximated as a function linear in the sine. Ramos and Castagna (2001) obtained the expression appropriate to RPS.

Another approach to simplifying the Aki-Richards approximation for RPP is given by Smith and Gidlow (1987), who use Gardner's approximate relation (which states that density is proportional to the fourth root of VP) to justify expressing the density contrast in terms of the VP contrast. This reduces by one the number of variables sought in an inversion. Stewart (1990) has developed a similar theory for inversions involving converted-wave data.

A different way of removing the fractional change in density is by the method of Fatti et al (1994). They replace the velocity contrasts in the Aki-Richards RPP with corresponding impedance contrasts, or reflectivities. The impedance contrast includes a large portion of the density contrast, and the remaining density contrast term, being relatively small, is dropped. This approximation is most accurate for small density contrasts and low angles. A similar approach can be carried out for RPS using and expression by Larsen (1999).

In general there is a trade-off between the higher accuracy of more rigorous expressions, and the lower number of variables and easier interpretability of simpler approximations.

A different approach is taken by Connolly who has derived the concept of elastic impedance starting from Shuey's form of the Aki-Richards RPP approximation. The elastic impedance is a quantity that can be calculated for each layer as a function of angle of incidence. It is defined such that the elastic impedance contrast approximates the angle-dependent reflection coefficient. It is exact (and equal to the acoustic impedance) at zero offset. For this method to be accurate one must assume a constant VP/VS ratio throughout the entire system. An analogous expression for RPS is given by Duffaut et al. (2000).

Nonlinear approximations

The advantages of employing linear approximations include a simplified interpretation picture and the ability to invert AVO data directly with a closed solution. The principle advantage of nonlinear methods is that they allow for greater accuracy.

The quadratic shear approximation is unique among nonlinear theories because of two key properties:

  1. It can be inverted non-iteratively. The solution is more complex than for linear inversion because one must solve a cubic polynomial, but it is still a closed solution.
  2. It incorporates the principle corrections to the linear inversion.
This method is implemented in the current explorer. For completeness, the analogous theory for RPS (i.e., the Aki-Richards approximation, but with an additional (ΔJ/J)2 term) is included also [cf. Xu, Y. and Bancroft, J. C., Joint AVO analysis of PP and PS seismic data: CREWES 1997 Research Report].

Copyright 2001-2005, CREWES, University of Calgary, All Rights Reserved