Ray Paths in Layered Media

Reflections and refractions at a plane interface

Consider a P-wave which is incident at an  angle 1 measured with respect to the normal of the interface. There will be a reflected wave and a transmitted wave but the directions of the waves are given by the diagram to the right. Law of reflection: The angle of reflection equals the angle of incidence. So r = 1 . Law of refraction: The angle of refraction 2  is determined through Snell's Law, which is  If the wave travels from a low velocity medium to a high velocity medium the wave gets refracted away from the normal. Conversely, it gets refracted toward the normal if the wave goes from a high velocity to a low velocity medium.

"Critical refraction" is an important concept in refraction seismics. The maximum value of 2 is 90o. If this is the case, then refracted waves travel horizontally in the second medium. The incident angle that causes this, known as the critical angle, c is found using Snell's law as follows:

When the wave in the second medium is critically refracted, it travels parallel to the interface at a speed of V₂. As it travels, it radiates energy into the upper medium with the associated ray path making an angle _c with the normal. This critically refracted wave is also called a "head wave". It is somewhat analagous to the bow wave of a moving boat.

Mode Conversion A P-wave incident upon a boundary can produce reflected and transmitted P-waves, but reflected and transmitted S-waves can also be produced. Analogous conversions occur when there is an incident S wave on a plane boundary. The mode conversions (P -> S, or S -> P) can complicate interpretation, but S-waves are always slower than P-waves, so first arrivals will always be P-waves unless a special S-wave energy source is used. A valuable benefit of using shear waves is that they provide important information about the rigidity of the material.

Waves for a layer over a halfspace

We do seismic refraction surveys in order to learn about the geometry of geologic layers and velocities (ie types of) materials. To do this we must build relations relating what we know and can measure to the things we want. In other words we must build equations that relate what we want (depths and velocities) to what we measure (surface distance and total travel times).

Consider a layer of thickness h and velocity V1 overlying a uniform halfspace of velocity V2. A source is detonated at time t=0. We are interested in the waves and arrival times of those waves at a receiver which is located a distance x from the source at position D in the figure below.

There are three principle waves that will travel through the earth and arrive at position D.   i) direct waves, ii) reflected waves, and  iii) critically refracted waves.

The travel time curves for these ray paths are shown to the right, and expressions for the ray paths and important parameters of these travel time curves are as follows: xcrit  is the critical distance at which the refracted arrival first arrives. xcross  is the crossover distance. Beyond this distance the refracted arrival is the first arrival on the record. Travel times of visible arrivals are related to distance between geophone and source (x), thickness of the layer (h) and the velocities of signals within the two layers (V1 and V2). Three times are of interest: tdir is travel time of direct arrivals, tref is arrival time of reflections and tcr is the refraction travel time (see figure above). These parameters are all related as follows:

The first and third of these are important for interpretation of seismic refraction data, and the next page explains how they arise, and how they are used with refraction data on a T-X plot to obtain useful geological information.