Notes on the CREWES Zoeppritz Explorer Applet
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 "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 applet 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 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.
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