Seismic reflection is simple in principle but the interpretation leads to unexpected subtleties. The simplest variation of seismic reflection is sonar, which reveals not just sea floor topography but often structure beneath. However, what you see is not always what there is. Consider the seamount below. Not only do we get echoes from directly above it, but also from off to the side as well.
On the left, we see the formula relating distance x, depth d and travel time t. The velocity of sound in water is denoted by v. Now on a sonar plot (or any seismic reflection profile) we do not actually graph vertical distance. All we see is time; specifically, round-trip travel time from the source to the receiver. We can rewrite the equation for travel time as shown at right - it is the equation for a hyperbola. What we actually see on the sonar plot is a hyperbola, shown in purple. A strongly-reflecting source shows up on a sonar or seismic-reflection plot as a hyperbola. The vertex of the hyperbola is the actual location of the point.
Areas of complex sea-floor topography show up as overlapping hyperbolas, rather reminiscent of classical Chinese art. Any strongly reflecting points on the flanks of the seamounts will also show up as hyperbolas.
The hyperbolas show up elsewhere as well. Suppose there is a fault offsetting a strongly reflecting layer (left, yellow). The layer itself will show up as a horizontal line on the reflection profile because the vertical reflection is strongest and because any oblique signals will be reflected away from the receiver. But where the layer is offset, the end will reflect oblique signals, so each end of the layer on the reflection profile will grade into a half hyperbola. (For clarity the ray paths for the layer right of the fault are not shown.) The horizontal reflectors are real but the hyperbolas are artifacts.
Since reflection profiling shows things that aren't there and sometimes shows features in the wrong place, we need a way to remove artifacts and restore features to the correct location. This technique is called migration.
We have to figure out where the real source of a signal is, and the beginning step is to figure out where it could be. In a medium with uniform velocity, if the source and receiver are close together, then any given signal must originate on a circle whose radius corresponds to the signal travel time. We say the locus of possible signal locations is a circle. If the source and receiver are far apart, the locus is an ellipse with the source and receiver at the foci.
Consider the three point sources shown below. The travel times from source to receiver are tabulated at right.
|Travel Times (Seconds)
1 - A 0.93
1 - B 1.36
1 - C 2.32
2 - A 1.03
2 - B 0.79
2 - C 1.82
3 - A 1.45
3 - B 0.59
3 - C 1.52
4 - A 2.05
4 - B 1.05
4 - C 1.50
5 - A 2.68
5 - B 1.67
5 - C 1.78
|Now, when we run a reflection profile over this area, we have no way of knowing where those sources are. All we know is the times when signals reach the receivers. That data looks as shown at left.|
|If we naively assume the reflections represent continuous
structures, we'll get something like at left. We might guess we had an
anticline cut by a thrust fault.
However, anyone familiar with reflection profiling would look at this data and notice the hyperbolic shape.
By the way, the colors are for visualization purposes only. In real life there would be no way to distinguish the red, green, and blue points.
|One way to analyze the data is to draw circles through all
the time marks as shown. Any place numerous circles intersect is obviously
the location of a signal.
The circles are colored here to aid in visualizing the process, but in reality there is no way to identify any circle with any given source. The two nearly identical circles that pass through the red and green points could not be linked to one source or the other.
|Here's how it's done in practice. The profile is divided
into an array and the signal from a given source is added to all the other
array elements along a circular arc. Since there are negative oscillations
as well, random noise sums to zero but actual signals add up.
Migration doesn't eliminate all artifacts (and creates some of its own) but it does go a long way toward removing them and relocating signals to their true location.