Notes on the CREWES TI Explorer Applet

Use of the CREWES TI Explorer Applet

This applet can display reflection coefficients for an interface between two VTI media or between two HTI media. Either or both of the layers may be specified as isotropic as well. Both exact and linearized reflection coefficients may be displayed.

Conventional Thomsen coefficients, γ, δ, and ε, are used to define the anisotropy. Thus the P-wave and S-wave velocities supplied as input refer to the vertical velocities in the VTI case and to the horizontal velocities along the symmetry axis in the HTI case. Velocities in other directions are then derived from these using the Thomsen coefficients. Note that Ruger and Tsvankin (1997) have also defined parameters such as δ(V) and ε(V) which can be related to the Thomsen coefficients and which can be used in connection with HTI. They are not used in the present version of this Explorer, but may be incorporated as options in future versions.

The plot shows how the reflection coefficients change with polar angle of incidence. To see how the coefficients change with polar angle of incidence or with the elastic properties of each medium, use features in the control panel to change these parameters. These may be fixed to particular values in the text fields, or interactively scanned over a range of values using the slider bars. Drop down menus allow you to explore other useful combinations of these variables as well, such as differences and ratios. Note that you are not prevented from selecting unphysical values of parameters.

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 first quantity is plotted with a solid line and the second with a dashed line.

Any of the scales may be adjusted using the control panel. Angles may only be adjusted to integer numbers of degrees, and the incident angles must be between 0° and 90°.

The locations of critical angles are indicated by vertical lines, which are annotated with the value of the critical angle, and the relevant velocity conditions.

Technical Notes

This is the original version and is numbered 1.0. It was first placed on the Internet on November 26, 2007.

Theory References:

• Zoeppritz and Aki-Richards: A standard reference is Aki & Richards (1980) “Quantitative Seismology”, vol. I, sec. 5.2.
• Thomsen coefficients: Thomsen, 1987, "Weak elastic anisotropy", Geophysics, 51, 1954-1966.
• VTI Exact: Daley & Hron, 1977, "Reflection and transmission coefficients for transversely isotropic solids", Bull. Seis. Soc. Am., 67, 661-675.
• VTI Linearized: Ruger, 1997, "P-wave reflection coefficients for transversely isotropic models with vertical and horizontal axis of symmetry", Geophysics, 62, 713-722.
• HTI Exact: Schoenberg & Protazio, 1992, "'Zoeppritz' rationalized and generalized to anisotropy", Journal of Seismic Exploration, 1, 125-144.
• HTI Linearized: Ruger and Tsvankin, 1997, "Using AVO for fracture detection: Analytic basis and practical solutions", The Leading Edge, October, 713-722.

The current implementation of the exact HTI solution suffers from two shortcomings. One is that values past the critical angle are not currently calculated. The other is that it experiences numerical instability near angles of 0° and 90°. Fixing these defects will be an objective of future versions.

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 may be found at www.crewes.org/Explorers/.