Mar. 18, 1998 Two Columbia University scientists have devised a computer model that shows how during geologic deformation, earthquakes in a specific location and time period will occur in a pyramidal distribution, with one larger quake and a specific number of smaller ones in no particular order.
While the new work will not help seismologists predict earthquakes with any greater accuracy, it does help them estimate the likelihood of future damage, since an area with many smaller quakes can be tagged as one likely to experience a larger quake.
The research, by Christopher Scholz, professor of earth and environmental science, and Chrysanthe Spyropoulos, a graduate student in applied physics, will be presented March 16 at the Seismological Society of America conference in Boulder, Colo. The work was conducted at Lamont-Doherty Earth Observatory, Columbia's earth sciences campus in Palisades, N.Y.
In 1935, American seismologist Charles Francis Richter defined earthquake magnitude as the log of the seismic moment, or the mean slip -- the distance apart the two sides of a fault have moved -- multiplied by the affected surface area. This magnitude, which came to be known as the Richter scale, means that when earth movement increases arithmetically, an earthquake's destructive power increases exponentially. The law that produced this insight has become known as the Gutenberg-Richter law, after Richter and his mentor at Caltech, Beno Gutenberg.
Though seismologists can't yet pinpoint when and where a quake will occur, they can record many small earthquakes in an area and use that information to estimate how often much larger -- but much less frequent -- events will occur. In the case of large quakes, the Gutenberg-Richter law says that in a given period, if there were a thousand earthquakes of a certain energy, there will be a hundred earthquakes ten times larger, 10 a hundred times larger and one a thousand times larger. If seismologists find that there are 100 earthquakes of a given magnitude in 10 years, they can estimate that in any 100-year period there should be a single earthquake with a magnitude a thousand times greater, on average.
"This is important because the earthquakes at the smaller measured magnitudes do not cause damage, but the ones a thousand times larger are very damaging," Professor Scholz said. "Those are the ones we want to know about, but we don't have good information on them because they are rare."
Tectonic earthquakes occur within the brittle part of the Earth's crust, from the surface to a depth of about 12 miles. Larger earthquakes occur at depths greater than 12 miles. Smaller, tectonic earthquakes, at about magnitude 6, obey a slightly different distribution, with eight small quakes needed to expect a larger one. The researchers show that faults, cracks in the crust, also obey these two kinds of distribution and that the underlying cause of the earthquake distribution is the distribution of these faults.
A computer model devised by Ms. Spyropoulos explains the distributions as the growth of a population of cracks. Her statistical model of the earth's upper layers assumes a brittle sheet of rock overlying a more pliable substrate. She generated random numbers to represent the different strengths of different areas of the model, just like different rocks on the surface have different strengths.
She then strained the model just as motion between tectonic plates strains rocks. "The top can handle only so much force before it goes pow!" Ms. Spyropoulos said. "In the beginning, we get lots of small faults, and then we see that the model organizes itself into this kind of pyramidal distribution."
Professor Scholz put the puzzle together when he saw that the model's fault propagation spontaneously exhibited the same pyramidal size distribution as earthquakes do. Geological observations show that faults and fault segments have pyramidal distributions that look just like those for earthquakes, with eight fault segments organizing into larger faults. Ms. Spyropoulos's model shows how that happens. "We didn't tell the model to do this," Professor Scholz said. "The cracks organize themselves through their stress fields, and that explains the size distribution."
Professor Scholz says this relationship is a hallmark of a new branch of physics related to chaos, called complexity science. The relationship is fractal because earthquakes (or faults) of different sizes look exactly the same, a property known as self-similarity. This results in the observed distribution, because the interactions between faults are complex on all scales. "This is one of the most important examples that involve nonlinear dynamics," Professor Scholz said. "It shows there is a class of problems that can be solved only by using computer models."
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