Seismic Refraction - The Wavefront Approach

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
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Seismic refraction leads to some puzzling results. If layers are dipping, travel time graphs run in opposite directions have different shapes and crossover distances. Branches of the travel time graph can be flat, as if the velocity of the waves was infinite, or even negative - signals get to distant locations sooner than nearby ones.

In the example above, we can see that profiles run in opposite directions are not equivalent. In the top example, the bed gets shallower with distance, so a large increase in distance leads to a small increase in travel time. In the lower diagram, the bed gets steeper with distance, so an increase in distance leads to a large increase in travel time and a steeper second leg of the travel time graph. Even so, it's pretty unsatisfying intuitively. Fortunately, there's a better way.

The top diagram, using the simple ray approach, is problematic for two reasons. First, how does the signal "know" to come up to a receiver? And second, if the signal is leaking energy all the time as shown, where does the energy come from? If the boundary has zero thickness, doesn't that imply an infinite energy density?

There are two ways to visualize signals: as rays or as wave fronts. At bottom is the wave front visualization. When a signal from the shot point (red x) enters the lower layer, it travels faster. Where the wave front in the lower layer intersects the layer boundary, it disturbs it. The disturbance is traveling with the velocity in the second layer, so it continually outruns its own wave front in the first layer and creates a "bow wave" as shown.

At the crossover distance (blue, at right), the bow wave has outrun the direct wave and signals from the second layer arrive first.

Here's the same situation, visualized as wave fronts. It becomes trivially obvious the two graphs should not be the same. The top case, with the signal traveling updip, will take longer to achieve crossover, but the gently sloping bow wave will hit the surface nearly simultaneously along its whole length. So the graph of travel time in the second layer will be nearly flat.

In the bottom case, with the signal traveling downdip, the steep bow wave will achieve crossover sooner, but its progress across the surface will be slower and the graph of travel times in the second layer will be steeper.

The "Bow Wave"

If a source of waves (light, sound, seismic, water) is stationary, it emits wave fronts that are concentric circles as shown at left, above. If the source is moving slower than the waves, as at right, above, it partially overtakes waves traveling in the direction of motion. The source also moves away from waves behind it. Waves in the forward direction are compressed, those behind are stretched. This results in the familiar drop in pitch as an ambulance passes by, and the red- and blue- shifting of light in astronomy.

However, if the source is traveling faster than the waves, it outruns its own wave fronts. Each successive wave is emitted beyond the last wavefront. The resulting wave is a V-shaped wave tangent to all the emitted wave fronts. If the signals are light, this is impossible for objects in a vacuum, but is possible for objects in a medium like water. The velocity of light in water is only about 3/4 that in a vacuum. It is easy for particles to travel faster than light in water, for example, particles emitted from a nuclear reactor. These particles emit a conical wave front called Cerenkov Radiation, which is seen as a blue glow. Boats traveling faster than the speed of waves in water create the familiar wake, and aircraft traveling faster than sound trail a conical shock wave which is heard as a sonic boom.

In the case of a seismic signal, if the velocity in the top layer is Vw, and the velocity in the bottom layer is Vs, a disturbance races along the layer boundary with velocity Vs, trailing a bow wave into the upper layer.

Now it becomes easy to see how the apparently impossible case of a flat travel time curve can come about. If the bow wave is parallel to the surface, it will hit the surface everywhere simultaneously (top). If the dip is sufficiently steep and the velocity disparity large enough, the wave front may even slope downdip, in which case the leading part of the wave will hit the surface first. In this case we will actually have a travel time curve with a negative slope (bottom).


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Created 19 December 2000, Last Update 07 March 2012
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