A simple model of forest fires could help explain the distribution of the sizes of earthquakes and their aftershocks, a theoretical physicist says. Geoscientists say they have reservations about the accuracy of the bare-bones model, but they welcome the effort to account for aftershocks.
"The basic approach has merit," says Donald Turcotte, a geophysicist at the University of California, Davis. "I’m not aware of anybody else who has done aftershocks."
Earthquakes show a striking statistical regularity. Larger ones occur less often than smaller ones do, and for more than half a century scientists have known that earthquakes of magnitude 2 occur roughly one-tenth as often as those of magnitude 1; those of magnitude 3 occur at one-tenth the rate of those of magnitude 2; and so on. That Gutenberg-Richter relation can be mathematically recast to show that the frequency of earthquakes decreases with increasing size or seismic moment—a different quantity from their magnitude—in proportion to that size raised to an exponent or power. Data show that that exponent is between -1.8 and -1.5.
But where does that number come from? Earlier statistical models of earthquake faults couldn’t explain it, says Eduardo Jagla, a theorist with Argentina's National Atomic Energy Commission in Bariloche. Instead, researchers adjusted parameters to get the right exponent, he says. Moreover, those models left out the aftershocks that follow a big jolt—a major omission, he says.
Now, Jagla says he can both incorporate aftershocks and explain the value of the exponent by tweaking a model of how forest fires spread. In the so-called Drössel-Schwabl model, trees sprout at random on a square grid like a vast checkerboard. Once the forest gets dense enough, lightning sets a random tree on fire, and fire spreads instantaneously among trees that occupy adjacent squares. The conflagration continues until there are no more neighbors to jump to. Then, the process starts all over again.
In Jagla’s model, the “forest” is the plane of a fault cutting through Earth’s crust, divided into a 10,000-by-10,000 grid. Sprouting trees correspond to the buildup of stress along the fault; burning areas, to the part of the fault that moves during a quake.
In this basic model, the size of the fires obeys a power law with the exponent -1.2—significantly higher than the number for real earthquakes. But then Jagla puts in a twist, as he reports in a paper published on 3 December in Physical Review Letters. He assumes that the trees come in two types: more common “A trees" that burn instantly and much rarer "B trees" that burn slowly and light their neighbors only after a delay. In the forest-fire model, the fire pauses when it has to pass through a single B tree.
The result is that the forest fire breaks into a "cluster" of smaller fires slightly separated in time, reducing the frequency of really big fires. Now, the size distribution of individual fires has an exponent of -1.8, just as in the observed distribution of earthquake sizes. "What would have been a single earthquake is now fragmented into a lot of smaller earthquakes that give the correct exponent," Jagla says. The fragments can be interpreted as aftershocks and the B trees as their epicenters on the fault, he says.
The key point, Jagla says, is that merely introducing an internal timescale—the delay in the B trees—drives the system to the correct exponent. As long as the delay is much longer than it takes the A trees to burn but much shorter than the time between lightning strikes, the exponent will be the same, because the B trees' effect is to break up the fire spatially, he says. In terms of earthquakes, the aftershocks must come slowly compared with the duration of each shock but quickly compared with the buildup of stress.
"I like the intuitive simplicity and clarity of his approach, and I admire that he is borrowing from the fire community to provide insights into ours," says Ross Stein, a geologist with the U.S. Geological Survey in Menlo Park, California. "But," he cautions, "many phenomena can explain the Gutenberg-Richter power-law … so it's difficult to be sure that the slow-fire hypothesis is a key element." Turcotte says that many other researchers also claim to have explained the value of the exponent in the relation.
In his paper, Jagla suggests a way to test his scheme. In his model, the exponent for the main shocks and their first few aftershocks should start out relatively high—closer to the -1.2 of the original fire model with only one kind of tree. Then its value should decrease as more and more aftershocks occur. Jagla says data taken in southern California over 20 years shows that main quakes and aftershocks that occur within 30 minutes show an exponent of -1.35, whereas those for aftershocks coming 30 to 90 minutes after the main shock have an exponent of -1.7—just as predicted.
But both Stein and Turcotte note that the catalog of aftershocks in the first 30 minutes after a big quake is notoriously incomplete. So whether this is an earth-shaking advance remains to be seen.