Notes on the CREWES Zoeppritz Explorer Applet

- To use the applet, select whether the incident wave is P or S, and whether it originates in the upper or lower level. These choices may be changed at any time.
- Also select which coefficients you wish to have plotted. These may also be changed at any time. Only those for one type of incident wave may be displayed simultaneously.
- The plot shows how the reflection coefficients change with angle of incidence. To see how the coefficients change with properties of each medium, use the six scroll bars in the control panel to change the density and velocities of each layer. 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.
- 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 location of critical angles is indicated by vertical lines, which are annotated with the value of the critical angle, and the relevant velocity conditions.
- The magnitudes are plotted with solid lines, and the phases with dashed lines. The color code for the various coefficients 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.
- The Aki-Richards and Bortfeld approximations can be accessed using the appropriate checkbox on the panel. These were originally defined only for sub-critical angles, but are extended beyond the critical point in this applet.

- A standard reference for the Zoeppritz equations and for the
Aki-Richards approximation is Aki & Richards (1980) "Quantitative
Seismology", vol. I, sec. 5.2.
Information on the Bortfeld approximation is found in “Bortfeld, R.
(1961) Approximations to the reflection and transmission coefficients of
plane longitudinal and transverse waves. Geophys. Prosp.,
**9**, 485-502”. - 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).
- The Java applet was adapted by Chuck Ursenbach from Gary Margrave's zoeppritz.m routine in the CREWES Matlab toolbox, which in turn was translated from a Fortran routine written by Ed Krebes.
- Version history:
- The original Zoeppritz Explorer was placed on the Internet on June 4, 2001.
- Updated July 2001
- menus to allow specification of density and velocity ratios rather than densities and velocities themselves
- checkboxes to deselect plotting of magnitude or phase
- option for the Bortfeld approximation

- Updated January 2002
- correct treatment of liquid interfaces, i.e., when
*V*_{S }= 0 - Inclusion of the SH-wave reflection and transmission

- correct treatment of liquid interfaces, i.e., when
- Updated August 2003
- added more choices of slider variable, e.g.,
*V*_{S}/*V*_{P}, Δ*V*_{P}/*V*_{P}, Δ*V*_{S}/*V*_{S}, and Δ*ρ*/*ρ* - Allowed option of displaying either S.I. or Imperial units
- Allowed one 'enter' to effect several changes, instead of requiring users to press 'enter' in each text box changed

- added more choices of slider variable, e.g.,
- Updated September 2, 2005
- corrected errors in calculation of the Bortfeld approximations to
*R*_{PS},*R*_{SP},*T*_{PS}, and*T*_{SP}(note that there is also an error in equation 22 of Bortfeld (1961) which affected previous versions)

- corrected errors in calculation of the Bortfeld approximations to
- Further updates will be recorded at www.crewes.org.

Sound waves travel through solids just as
they do through the air. When a sound wave going through a solid reaches a
boundary with another solid of a different density, it can either reflect off
of the boundary at some angle, or pass through it, but have a different
direction after the boundary than before it. Most people are familiar with
similar phenomena for light rays at a boundary between air and water. In some
conditions the water surface acts like a mirror, reflecting light rays back
into the air. Other times light rays pass into the water but are bent, so that
an object such as a pencil placed in a glass of water appears to be bent
itself. The Zoeppritz Equations describe how likely a sound wave traveling
through the earth is to be reflected at a boundary between two different layers
of earth, or to be bent when passing from one layer of earth to another. *[To
be more precise, the Zoeppritz Equations give angle-dependent reflection and
transmission coefficients for elastic plane waves at a non-slip horizontal
boundary between two semi-infinite isotropic elastic media.] *

Actually, in addition to compressional sound waves (or P-waves) the equations also describe shear waves (S-waves). P-waves travel by the medium being alternately squashed and stretched, such as a disturbance traveling through a Slinky toy. S-waves travel by the medium alternately moving up and down, such as a wave traveling along a rope, the end of which is being jiggled up and down. When either a P-wave or S-wave is traveling through the earth (such as happens in an earthquake) it will run into boundaries between different layers of earth. At this point it may be reflected as either a P-wave or S-wave, or it may be transmitted through the boundary as a P- or S-wave, but with its direction changed. Normally it is split up into a combination of these kinds of outgoing waves, and the magnitudes of the coefficients calculated from the Zoeppritz Equations (which are plotted in the applet above) show the relative amplitudes of the waves produced by each of these four possibilities.

The notation used here for each coefficient
is a three-letter symbol such as "*R*_{PP}" or "*T*_{SP}".
The first letter indicates whether it is a reflection or transmission
coefficient, the second letter indicates whether the incident wave is P or S,
and the third letter indicates whether the outgoing wave is P or S. The sizes
of the four coefficients *R*_{PP}, *R*_{PS}, *T*_{PP}, and *T*_{PS},
for example, are related to how the energy of a P-wave is distributed when it
reaches an interface.The
coefficients *R*_{SS} and *T*_{SS} are also appended with (v) or
(h). This is because an S-wave can
oscillate either in a plane containing a vertical line (v) or one containing a
horizontal line (h).Only the former
can generate or be derived from P-waves.

The values of the various reflection coefficients
are determined by the angle of incidence, and by the density and wave
velocities in each layer. The plot shows first of all how the coefficients
change with angle of incidence, all the way from an angle of zero, where the
wave is traveling perpendicular to the boundary, right up to 90 degrees, where
the wave is parallel (grazing incidence). One can also use the applet to see
how the coefficients vary with density (often denoted *ρ*)
and with *V*_{P} and *V*_{S}. (*V*_{P}
is the velocity at which P-waves travel, and *V*_{S} is the velocity at
which S-waves travel.)

The coefficients are all real numbers for small enough angles, but for some combinations of velocities, the coefficients become complex past a certain critical angle. Physically this corresponds to the angle of the reflected or transmitted wave reaching 90 degrees, so that the wave travels along the boundary itself, dissipating exponentially away from it. To accommodate this behavior, the results are plotted with both a magnitude and a phase past the critical angle.

The Zoeppritz Equations are somewhat complicated, and the Aki-Richards approximation has proven to be a very useful tool in many applications, and can be displayed here as well. This approximation gives the linear dependence of the coefficients on property differences between the two layers (such as the difference in density) so it is most accurate when the two layers are very similar. An earlier linear approximation to the Zoeppritz equations was given by Bortfeld.

The Zoeppritz Equations are very useful in exploration geophysics. Seismology is a way of studying the interior of the earth by creating sound waves, and recording them after they have traveled through the earth. For instance, a blast of dynamite will create a sound wave (P-wave) that will travel through the earth, be reflected off of various layers, and return to the surface where it can be detected. The time it takes to return is related to the depth of the various layers. By detecting the sound waves at various points on the surface, one can see how the reflections change with angle of incidence. One can then use this information along with the Zoeppritz Equations to learn more about the density and velocity of each layer. This is helpful for instance in locating underground reservoirs and deposits.

Copyright 2001-2005, CREWES, University of Calgary, All Rights Reserved