Notes on the CREWES Spherical-Wave Reflectivity Explorer

This Explorer is modeled after the CREWES Zoeppritz Explorer. In this application, however, reflection coefficients are calculated for both plane waves and spherical waves. It demonstrates the importance of including spherical-wave effects in long-offset, post-critical AVO studies. Spherical effects are important near the critical angle, even at considerable depth (see Haase, 2004 CSEG & SEG Abstracts), so that spherical-wave reflection coefficients are valuable when seeking to extract subsurface density parameters from AVO.

Unlike plane-wave reflections, spherical-wave reflection coefficients depend on depth of the reflector (Z), and P-wave speed (α1) and these can be specified within the control panel. Spherical-wave reflectivity is dependent on frequency as well, and so it is important to specify the wavelet employed. The principle assumption in this calculation is that the wavelet is restricted to the form of a Rayleigh wavelet, namely, w( f ) = f n exp(-nf /f0), where n is a positive integer and f0 is the maximum-amplitude frequency (see Hubral and Tygel, 1989). This wavelet, which makes possible the efficient calculations of this code, is displayed as a red curve in each of the two panels at the top of the applet. The parameters n and f0 defining the curve may be entered in text boxes in the control panel. (Press enter for changes to take effect.)

The Rayleigh wavelet is less commonly used at present than some other wavelets, such as the Ricker wavelet, defined by w Ricker( f ) = f 2 exp[-( f /f0)2], and the Ormsby wavelet, defined by the corner frequencies, f1, f2, f3, and f4. While such wavelets cannot be used in the method of this applet, their effect can be approximated by an appropriate choice of n and f0 for the Rayleigh wavelet. The upper panels display comparisons with the Ricker and Ormsby wavelets, and their parameters can also be adjusted in the control panel. It can be shown (Ursenbach et al., 2006) that the greatest similarity in reflectivity curves for two different wavelets occurs when the wavelets share the same average frequency, given by (1+1/n) f0 for the Rayleigh wavelet, (2/√π) f0 for the Ricker wavelet, and

( f1 + f2 + f3 + f4) / 4 - (1/12) [ ( f4 - f3)2 - ( f2 - f1)2 ] / [ ( f3 + f4) - ( f1 + f2) ]

for the Ormsby wavelet. The choice of n also plays a secondary role in helping to match reflectivity curves (Ursenbach et al., 2006). These observations lead to a method for approximating the reflection coefficients of other wavelets. In order to find an optimal approximation to a reflectivity curve for a Ricker or Ormsby wavelet, do the following:

We emphasize again that the Ricker and Ormsby wavelets are not actually used in the calculation of the spherical-wave reflection coefficient. It uses only the Rayleigh wavelet, whose parameters are chosen to mimic the effect of these other wavelets. It is apparent from the comparisons of spectra that the Ricker wavelet is better represented than the Ormsby wavelet, but tests show that even the latter is surprisingly well represented (Ursenbach et al., 2006).

The spherical-wave reflection coefficient calculation involves a numerical integration, which allows for imperfection in the results. One imperfection is associated with extreme values of α1/(2Zf0), which is a measure of sphericity. It may first be seen near 0 and 90 degree angles of incidence, as an erratic deviation of spherical-wave results from plane-wave results. The region of interest is generally at critical angles in the 30 to 60 degree region, which will generally be affected negligibly. Another imperfection is associated with very large values of n. This introduces oscillations near the critical angle, and a slightly jagged appearance indicates sub-optimal sampling. These imperfections will be further addressed in future versions of this explorer.

References

Hubral, P. and Tygel, M., 1989, "Analysis of the Rayleigh pulse (short note): Geophysics, 54, 654-658

Ursenbach, C.P., Haase, A.B., and Downton, J.E., 2006, "Improved modeling of spherical-wave AVO": CREWES Research Report, 18.

Feedback

Do you have feedback on 1) problems with using this applet, 2) suggestions for improvement, or 3) examples of geological systems where consideration of spherical effects may be important? Feel free to contact Chuck crewesinfo@crewes.org.

Copyright 2006, CREWES, University of Calgary