Notes on the CREWES Reflectivity Explorer Applet
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. The slider bars generate ratios between 0 and 2, but other values can be accessed through the text fields. 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. Angles may only be adjusted to integer numbers of degrees, and the incident angle 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.
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.
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).
Related applications: Users of this application may also be interested in the CREWES Zoeppritz Explorer, which displays exact solutions and Aki-Richards and Bortfeld approximations for all possible reflection and transmission coefficients as a function of incident angle.
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) and shear waves (Rss) as well. 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 or Rss) 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, Rps and Rss 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].
class=small>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. 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.
style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto'> 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, 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. Rss, like Rpp, has an even-power expansion and can be expressed by a 2-term approximation.
Another approach to simplifying the Aki-Richards approximation for Rpp is given by Smith and Gidlow, 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, and the approximation is equally valid at all angles of incidence.
A different way of removing the fractional change in density is by the method of Fatti et al. 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.
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.
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