Unlike plane-wave reflections, spherical-wave reflection coefficients depend on depth
of the reflector (*Z*), and P-wave speed (*α*_{1}) 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(-*n**f* /*f*_{0}),
where *n* is a positive integer and *f*_{0} 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 *f*_{0} 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* /*f*_{0})^{2}],
and the Ormsby wavelet, defined by
the corner frequencies, *f*_{1}, *f*_{2},
*f*_{3}, and *f*_{4}.
While such wavelets
cannot be used in the method of this applet, their effect can be approximated by
an appropriate choice of *n* and *f*_{0} 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*) *f*_{0} for
the Rayleigh wavelet, (2/√π) *f*_{0} for the Ricker wavelet, and

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:

- Enter parameters for the desired Ricker (
*f*_{0}) or Ormsby (*f*_{1},*f*_{2},*f*_{3}, and*f*_{4}) wavelet. - Select "Ricker" or "Ormsby" in the control panel
- The values of
*n*and*f*_{0}for the Rayleigh wavelet will be automatically adjusted so that the reflectivity curve optimally represents that of the desired wavelet. - If it is desired to choose a different set of Rayleigh parameters to represent the displayed Ricker or Ormsby wavelet, select "Rayleigh" in the control panel and adjust parameters as desired.

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 α_{1}/(2*Z**f*_{0}), 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.