Gutenberg-Richter law
Encyclopedia
In seismology
Seismology
Seismology is the scientific study of earthquakes and the propagation of elastic waves through the Earth or through other planet-like bodies. The field also includes studies of earthquake effects, such as tsunamis as well as diverse seismic sources such as volcanic, tectonic, oceanic,...

, the Gutenberg–Richter law (GR law) expresses the relationship between the magnitude
Richter magnitude scale
The expression Richter magnitude scale refers to a number of ways to assign a single number to quantify the energy contained in an earthquake....

 and total number of earthquake
Earthquake
An earthquake is the result of a sudden release of energy in the Earth's crust that creates seismic waves. The seismicity, seismism or seismic activity of an area refers to the frequency, type and size of earthquakes experienced over a period of time...

s in any given region and time period of at least that magnitude.

or


Where:
  • is the number of events having a magnitude
  • and are constants


The relationship was first proposed by Charles Francis Richter
Charles Francis Richter
Charles Francis Richter , was an American seismologist and physicist. Richter is most famous as the creator of the Richter magnitude scale which, until the development of the moment magnitude scale in 1979, quantified the size of earthquakes...

 and Beno Gutenberg
Beno Gutenberg
Beno Gutenberg was a German-American seismologist who made several important contributions to the science...

. The relationship is surprisingly robust and does not vary significantly from region to region or over time.

The constant b is typically equal to 1.0 in seismically active regions. This means that for every magnitude 4.0 event there will be 10 magnitude 3.0 quakes and 100 magnitude 2.0 quakes. There is some variation with b-values in the range 0.5 to 1.5 depending on the tectonic environment of the region. A notable exception is during earthquake swarm
Earthquake swarm
Earthquake swarms are events where a local area experiences sequences of many earthquakes striking in a relatively short period of time. The length of time used to define the swarm itself varies, but the United States Geological Survey points out that an event may be on the order of days, weeks, or...

s when the b-value can become as high as 2.5 indicating an even larger proportion of small quakes to large ones. A b-value significantly different from 1.0 may suggest a problem with the data set; e.g. it is incomplete or contains errors in calculating magnitude.

There is a tendency for the b-value to decrease for smaller magnitude events. This effect is described as "roll-off" of the b-value, a description due to the plot of the logarithmic version of the GR law becoming flatter at the low magnitude end of the plot. Data which is perfectly following the GR law plots to a straight line. Formerly, this was taken as an indicator of incompleteness of the data set. That is, it was assumed that many low-magnitude earthquakes are missed because fewer stations detect and record them. However, some modern models of earthquake dynamics have roll-off as a natural consequence of the model without the need for the feature to be inserted arbitrarily.

The a-value is of less scientific interest and simply indicates the total seismicity rate of the region. This is more easily seen when the GR law is expressed in terms of the total number of events:

where,, the total number of events.


Modern attempts to understand the law involve theories of self-organized criticality
Self-organized criticality
In physics, self-organized criticality is a property of dynamical systems which have a critical point as an attractor. Their macroscopic behaviour thus displays the spatial and/or temporal scale-invariance characteristic of the critical point of a phase transition, but without the need to tune...

 or self similarity.
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