Notes on the CREWES Reflectivity Explorer Applet

- Use of the CREWES Reflectivity Explorer Applet
- Technical Notes
- Rationale and use of the reflectivity approximations

- Select the type of reflection coefficient you wish to display [P-wave (
*R*_{PP}) and/or converted wave (*R*_{PS})]. This may be changed at any time. - Select the density and velocity properties of both upper and lower media. These may be fixed to particular values in the text fields, or interactively scanned over a range of values using the slider bars. Only four of these six variables are independent, one for the densities and three for the velocities. Accordingly one can use the drop down menus to select up to four density and velocity ratios as well. Note that you are not prevented from selecting properties corresponding to a negative Poisson's ratio.
- Select which approximations to the reflection coefficient you wish to have plotted. These may also be changed at any time. These approximations are normally used in the subcritical region, but have been extended beyond the critical point for this application. Notes on the approximations are given below.
- The results are plotted in modified polar form. The magnitude is shown as positive or negative in order that the phase will always be zero below the first critical angle, and as continuous as possible beyond that. (The phase below the first critical angle is not plotted in this routine. It is always zero or pi in this region for standard polar form [when magnitudes are always positive].) Either magnitude or phase may be deselected for plotting using checkboxes at the bottom of the control panel.
- The magnitudes are plotted with solid lines, and the phases with dashed lines. The color code for the various approximations is given on the
control panel. Any of the scales may be adjusted using the control panel. The incident angles must be between 0 and 90. One can also choose
to display the results versus half-offset over depth. Non-negative integers are required for the limits of this variable.
Two other abcissa options are the sine and sine-squared of the incident angle. (
*R*_{PP}and*R*_{PS}are commonly expressed as expansions in sin*q*). - The location of critical angles is indicated by vertical lines, which are annotated with the value of the critical angle or critical offset, and the relevant velocity conditions.

- References:
__Exact solution__: Aki & Richards "Quantitative Seismology", vol. I, sec. 5.2.__Aki-Richards approximation__: Aki & Richards "Quantitative Seismology", vol. I, sec. 5.2.__Alternate forms of the Aki-Richards approximation__: Downton, J.E. and Ursenbach, C. (2005) Linearized AVO inversion with supercritical angles: CSEG / SEG Expanded Abstract.__Smith & Gidlow approximation__: Smith, G.C. and Gidlow, P.M. (1987) Weighted stacking for rock property estimation and detection of gas: Geophys. Prosp., 35, 993-1014.__Stewart approximation__: Stewart, R.R. (1990) Joint P and P-SV inversion: CREWES 1990 Research Report, chapter 9.__Fatti approximation__: 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.__PS analogue of Fatti approximation__: see equation 2.4, with D*r*/*r*set to zero, in Larsen, J. A. (1999) "AVO Inversion by Simultaneous P-P and P-S Inversion": M.Sc. Thesis, CREWES, University of Calgary__Shuey approximation__: Shuey, R.T. (1985) A simplification of the Zoeppritz Equations: Geophysics,**50**, 609-614.__Ramos & Castagna approximation__: Ramos, A.C.B. and Castagna, J.P. (2001) Useful approximations for converted-wave AVO: Geophysics,**66**, 1721–1734.__Elastic Impedance__: Connolly, P. (1999) Elastic Impedance: Leading Edge,**18**, 438-452.__Converted-wave Elastic Impedance__: Duffaut, K., Alsos, T., Landrø, M., Rognø, and H., Al-Najjar, N.F. (2000) Shear-wave elastic impedance: Leading Edge,**19**, 1222.__Bortfeld approximation__: Bortfeld, R. (1961) Approximations to the reflection and transmission coefficients of plane longitudinal and transverse waves: Geophysical Prospecting,**9**, 485-502.__Quadratic shear approximation__: Ursenbach, C. (2004) Nonlinear estimation of*R*_{J}from AVO intercept and gradient: SEG Expanded Abstract.__Full Offset/Large R__: Ursenbach, C., Stewart, R. R. (2001) Extending AVO inversion techniques: CREWES Research Report, vol. 13._{ρ}__Corrected Slope__: Ursenbach, C. (200?) A simple way to improve AVO approximations: CREWES Research Report, vol. 17.

- The source code for this applet is available to sponsors of CREWES (Consortium for Research in Elastic Wave Exploration Seismology, located at the University of Calgary).
- Version history:
- The original Reflectivity Explorer was placed on the Internet on July 14, 2001.
- Updated August 9, 2001
*R*_{PS}and*R*_{SS}options added

- Updated September 2, 2005
- An error in the Bortfeld
*R*_{PS}approximation calculation was corrected - The
*R*_{SS}option retired (legacy code will be kept available) - Alternate versions of the Aki-Richards expression given
- Converted-wave analogues of Smith-Gidlow, Fatti, and Elastic Impedance approximations added
- Quadratic-shear approximation added
- Facility to plot
*R*_{PP}and*R*_{PS}simultaneously added - Facility to plot versus sin
*q*and sin^{2}*q*added - Several features added in keeping with current CREWES Zoeppritz Explorer

- An error in the Bortfeld
- Further updates will be recorded at www.crewes.org.

- Users employing Microsoft's Internet Explorer may occasionally experience difficulties in running the applet. This is an error associated with the Internet Explorer browser and not with this applet. The user should ensure that their browser is properly Java enabled. The most rigorous way to do this is to download the free Java Runtime Environment (JRE) from java.sun.com and install according to instructions. This should solve any difficulties and allow other web-based Java applications to run more effectively as well. One can also use an alternate browser, such as Netscape or Mozilla, as these are not subject to the same problems as Internet Explorer.
__Related applications__: Users of this application may also be interested in other CREWES Explorers found at www.crewes.org/Explorers.

One common application of P-P reflectivity (*R*_{PP}) expressions is in Amplitude-Variation-with-Offset (AVO) studies in reservoir
seismology. More recent work has applied AVO to converted waves (*R*_{PS}). The general objective in AVO is to collect
information on how seismic amplitudes vary with incident angle, and to use this along with *R*_{PP} (or *R*_{PS}) 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 *R*_{PP} and *R*_{PS} 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 (*V*_{P}) and S-wave velocity (*V*_{S}) 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].

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, Δ*V*_{P}/*V*_{P}, Δ*V*_{S}/*V*_{S}, and *V*_{S}/*V*_{P}. The first
three are fractional changes and may be referred to as density contrast, *V*_{P} contrast, etc. *V*_{S}/*V*_{P} 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).

Shuey (1985) has rewritten the Aki-Richards *R*_{PP} 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 *R*_{PS} one obtains an odd-power series of sine functions and *R*_{PS} is sometimes approximated as a function linear in the sine.
Ramos and Castagna (2001) obtained the expression appropriate to *R*_{PS}.

Another approach to simplifying the Aki-Richards approximation for *R*_{PP} is given by Smith and Gidlow (1987), who use Gardner's approximate relation (which
states that density is proportional to the fourth root of *V*_{P}) to justify expressing the density contrast in terms of the *V*_{P} 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
*R*_{PP} 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 *R*_{PS} 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 *R*_{PP}
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 *V*_{P}/*V*_{S} ratio throughout the entire system.
An analogous expression for *R*_{PS} is given by Duffaut et al. (2000).

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:

- 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.
- It incorporates the principle corrections to the linear inversion.