INTRODUCTION

All the models that we have analyzed so far have consisted of layers of uniform velocity. We also have discussed many of the factors that influence velocity: pressure, temperature, cementation, jointing, weathering and so on. In many cases these factors are negligible; in others they are very important. Thus we need to derive formulae to account for them.

Strong velocity gradients occur in sediments, sedimentary rocks, ocean water, glacial ice, the mantle and the lunar regolith. In some instances the velocity changes linearly with depth but in others the change is nonlinear. We’ll only derive travel-time and ray path equations for the linear case and a flat earth.

FLAT EARTH CASE

The first part of the derivations deals with any change of velocity with depth. Basically we apply Snell’s Law of Refraction to find equations for travel-time and the horizontal coordinate of the ray path, both in terms of integrals over depth (the independent variable). Both integrals involve velocity as a function of depth and the "ray parameter" (p). The ray parameter is the inverse of the apparent velocity of a wave sweeping across the ground surface. Thus p equals the seismic velocity at the ground surface divided by the sine of the angle of incidence of the ray at the ground surface. The ray parameter, also called "horizontal slowness", is constant for any given ray.

In general we have to evaluate the two integrals numerically. If the velocity changes linearly with depth (constant velocity gradient) we can solve the integrals analytically. Many real-life cases can at least be approximated by a constant gradient over some depth of interest.

The resulting expression for the ray path reveals that it is just a circle. The radius equals 1/(kp) where k is the velocity gradient. For a positive gradient, the circular ray path is centered at a height of Vo/k above the ground. With this information we can easily draw the ray path with a compass. Alternatively we can use equation of the ray path.

The ray path is sharply curved for large k and almost straight for small k. For k=0 the ray path is perfectly straight as in all the previous cases we’ve studied. If the initial angle of incidence is small the ray descends steeply to a large depth and returns to the surface far away. If the angle is large the ray stays shallow and returns to the surface in a small distance.

We can also carry out the integration for the travel-time. The result is that the time equals 2/k multiplied by the "inverse hyperbolic sine" of (kR)/(2Vo). We’ll explain this function in class. Alternatively we can express the time in terms of the gradient and initial angle of incidence using logarithms and cosines.

If the gradient is zero (constant velocity) the travel-time curve is straight as in the previous cases we have studied. Otherwise the travel-tile curve flattens with distance. At the origin the slope equals 1/Vo. At any other point the slope equals the inverse of the velocity at the "turning point" or maximum depth of the ray path.

MORE COMPLEX CASES

We can build more complex models using the results for a constant gradient as the basic module. Rather than deriving a lot of formulae we’ll just think about these cases and draw the ray paths and travel time curves in a qualitative manner.

Our first complex model consists of a positive gradient layer (increase ov velocity with depth) overlying a lower-velocity layer. That is, the velocity sharply decreases below the interface. Geologically this model might represent the upper mantle. Increasing pressure causes the velocity to increase with depth until rising temperature takes over and slows the waves.

Ray paths in the gradient layers are circular arcs and the travel-time curve flattens with distance. Rays that penetrate too deeply (small angle of incidence) refract downward and don’t return to the surface, at least not in our oversimplified model. Thus we have a "shadow zone" where the rays don’t go and were the travel-time curve suddenly is terminated. This kind of travel-time curve was our first seismic hint for a mantle low-velocity layer or asthenosphere.

The second complex model has a higher velocity half-space below the gradient layer. Up to a point the ray paths are curved as in the previous case. The travel-tile curve is the same. The ray that just grazes the interface is refracted back to the surface as a head wave. The travel-time plot becomes a straight line. This model could approximate the continental crust and the uppermost mantle.

The third case is very strange. It consists of a low-gradient layer overlying a high-gradient half-space. The rays are all composed of circular arcs but the radius is much smaller in the lower medium. Thus, in contrast to all the other models, the ray paths actually cross in places. As the angle of incidence decreases the rays may actually decrease in both distance and time. In effect the travel-time graph "goes backwards"! This behaviour leads to a "triplication" where there are three arrival times for the same distance.

The "SOFAR" channel in the ocean presents a fourth interesting example. Here temperature and pressure combine to yield a negative velocity gradient near the surface and a positive one below. Taken together they form a sound channel in which the rays are refracted back and forth such that the energy is confined and the waves travel great distances with little loss. Seismic "T phases" are waves from earthquakes or submarine volcanic eruptions. They travel at about 1500 m/s and can be readily detected by hydrophones installed at the depth of minimum velocity.

REFERENCE

Dobrin, M. B. and C. H. Savit, 1988, "Introduction to Geophysical Prospecting", Fourth edition, McGraw-Hill, 867 pp.