Spotting an unfortunate spot

If for no other reason, 2011 will be remembered for its tragic earthquakes: first Burma, then China and Christchurch, and then the truly disastrous earthquake in Japan.

Earthquakes affect very large regions. To pinpoint the exact source of an earthquake, seismologists employ a world-wide network of seismographs, which are continuously registering ground movement. And, there is some very clever mathematics.

Above is a typical seismographic record of an earthquake. More details can be found here, but in brief the seismograph has recorded two types of waves emitted from the source, or focus, of the earthquake: there are fast-moving primary waves and slower secondary waves. The seismograph records the amplitude of the waves and the times that the waves are detected.

Seismologists also know the speeds of the primary and secondary waves. So, the time interval between the detection of these waves can be used to determine the distance of the seismograph from the earthquake’s focus.

Now, if your seismograph indicates that there has been an earthquake 100 km away, this means that the focus of the earthquake is located somewhere on a circle of radius 100 km, with your seismograph at the centre. However, to determine exactly where the focus is on this circle, you’ll also require the circles obtained from other seismographs.

Two circles corresponding to the same earthquake will intersect at either one or two points. In either case, the circle from a third seismograph will be able to locate the focus of the earthquake. 

Of course, this description oversimplifies the many details. For one thing, earthquakes do not always occur close to the Earth’s surface. This means that instead of drawing circles around each seismograph, we should be imagining spheres. 

Still, the same ideas work. Two spheres will either intersect in a single point, which must then be the focus, or in a circle. In the latter case, a suitable third sphere will intersect the circle in two points. One of these points may be above ground level, which is an unlikely spot for an earthquake, and so the other point must be the focus. In any case, a suitable fourth sphere will always suffice to locate the focus of the earthquake. 

Similar calculations play a role when a location has to be determined from the distances from certain measuring sites. For example, the worldwide GPS system relies on a system of satellites orbiting the Earth. The satellites carry synchronised clocks that are constantly sending out signals, indicating the time when the signal is sent, and the coordinates of the satellite at that moment. Your GPS device then combines the information carried by the signals from (at least) four different satellites, to calculate your position. 

By the way, news reports on earthquakes typically contain three mathematical pieces of information: the epicenter, which is the point on the Earth’s surface directly above the focus; the depth of the focus; and the magnitude of the earthquake. The epicenter and the depth of the focus are easily calculated once we know the location of the focus. Finally, the magnitude can be calculated from the amplitude of the seismograph output, together with the distance of the quake from the seismograph.

 

Puzzle to ponder: Let’s consider an earthquake with focus on the Earth’s surface. So, we can stick to drawing circles rather than spheres. Can you think of how four seismographs might be located so that the focus of this earthquake cannot be pinpointed? 

Burkard Polster teaches mathematics at Monash and is the university's resident mathemagician, mathematical juggler, origami expert, bubble-master, shoelace charmer, and Count von Count impersonator. 

Marty Ross is a mathematical nomad. His hobby is smashing calculators with a hammer.

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