Zoeppritz Explorer

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You will need to have the Java runtime environment (JRE) installed on your computer in order to use this software. As of Java Version 7 Update 51, you will also need to add https://www.crewes.org to the Java Exception Site List. This has been tested in Windows 10 using Microsoft Internet Explorer, Microsoft Edge and Mozilla Firefox.

To start the Zoeppritz Explorer click on the 'Launch' button above, then 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 color code for the various coefficients is given on the control panel.

Version History

The Java code was adapted by Chuck Ursenbach from Gary Margrave’s “zoeppritz” routine in the CREWES Matlab toolbox, which in turn was translated from a Fortran routine written by Ed Krebes. The original Zoeppritz Explorer was placed on the Internet on June 4, 2001.

Technical Notes

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.

Reflection coefficients are complex quantities in general. In this Explorer they can be plotted in any of three ways:

  1. Real and Imaginary components (Cartesian form): The imaginary component is zero below the first critical angle.
  2. Magnitude and Phase (polar form): The phase is 0° or 180° below the first critical angle.
  3. Signed Magnitude and Continuous Phase (modified polar form): The phase is chosen to be zero below the first critical point and continuous thereafter. The magnitude is then made positive or negative to accommodate the phase constraints.

Any of these three pairs may be selected in the control panel, and either of the two quantities in each pair may be deselected for plotting. 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.

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 Explorer.

Meaning of the Zoeppritz Equations

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 Explorer) 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 "RPP" or "TSP". 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 RPP, RPS, TPP, and TPS, for example, are related to how the energy of a P-wave is distributed when it reaches an interface. The coefficients RSS and TSS 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 Explorer to see how the coefficients vary with density (often denoted ρ and with VP and VS. (VP is the velocity at which P-waves travel, and VS 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°, 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.


  1. Aki & Richards, 1980, “Quantitative Seismology”, vol. I, sec. 5.2.
  2. Bortfeld, R., 1961, “Approximations to the reflection and transmission coefficients of plane longitudinal and transverse waves”, Geophys. Prosp., 9, 485-502.

Note that there is an error in equation 22 of Bortfeld (1961) which affected early versions of the Zoeppritz Explorer.

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