# Earthquake scaling and statistics The scaling of slip

Earthquake scaling and statistics The scaling of slip with length Stress drop Seismic moment Earthquake magnitude Magnitude statistics Fault statistics The scaling of fault length and slip Normalized slip profiles of normal faults of different length.

From Dawers et al., 1993 The scaling of fault length and slip Displacement versus fault length What emerges from this data set is a linear scaling between displacement and fault length. Figure from: Schlische et al, 1996 The seismic moment The seismic moment is a physical quantity (as opposed to

earthquake magnitude) that measures the strength of an earthquake. It is equal to: moment = G A D , where: G is the shear modulus A = LxW is the rupture area D is the average co-seismic slip (It may be calculated from the amplitude spectra of the seismic waves.)

The scaling of seismic moment with rupture length What emerges from this is that co-seismic stress drop is constant over a wide range of sizes. The constancy of the stress drop implies linear scaling between co-seismic slip and rupture length. slope=3 Figure from: Schlische et al, 1996

Earthquake magnitude Richter noticed that the vertical offset between every two curves is independent of the distance. Thus, one can measure the magnitude of a given event with respect to the magnitude of a reference event as: M L = log10 A() log10 A0 (),

log(a) event1 event2 event3 where A0 is the amplitude of the reference event and is the epicentral distance. distance Earthquake magnitude

Richter arbitrarily chose a magnitude 0 event to be an earthquake that would show a maximum combined horizontal displacement of 1 micrometer on a seismogram recorded using a Wood-Anderson torsion seismometer 100 km from the earthquake epicenter. Problems with Richters magnitude scale: The Wood-Anderson seismograph is no longer in use and cannot record magnitudes greater than 6.8. Local scale for South California, and therefore difficult to compare with other regions. Earthquake magnitude

Several magnitude scales have been defined, but the most commonly used are: Local magnitude (ML), commonly referred to as "Richter magnitude". Surface-wave magnitude (MS). Body-wave magnitude (mb). Moment magnitude (Mw). Earthquake magnitude Both surface-wave and body-waves magnitudes are a function of the ratio between the displacement amplitude, A, and the dominant period, T, and are given by:

M S or mb = log10 (A /T) + distance correction. The moment magnitude is a function of the seismic moment, M 0, asfollows: 2 MW = log10 (M 0 ) 10.7 . 3 where M0 is in dyne-cm. Earthquake magnitude The diagrams to the right

show slip distribution inferred for several well studied quakes. It is interesting to compare the rupture area of a magnitude 7.3 (top) with that of a magnitude 5.6 (smallest one near the bottom). Earthquake magnitude Magnitude classification (from the USGS): 0.0-3.0 :

3.0-3.9 : 4.0-4.9 : 5.0-5.9 : 6.0-6.9 : 7.0-7.9 : 8.0 and greater : micro minor light moderate strong major

great Intensity scale The intensity scale, often referred to as the Mercalli scale, quantifies the effects of an earthquake on the Earths surface, humans, objects of nature, and man-made structures on a scale of 1 through 12. (from Wikipedia) I V VIII XII

shaking is felt by a few people shaking is felt by almost everyone cause great damage to poorly built structures total destruction The Gutenberg-Richter statistics Fortunately, there are many more small quakes than large ones. The figure below shows the frequency of earthquakes as a function of their magnitude for a world-wide catalog during the year of 1995. This distribution may be fitted with:

log N(> M) = a bM , Figure from simscience.org where n is the number of earthquakes whose magnitude is greater than M. This result is known as the GutenbergRichter relation. The Gutenberg-Richter statistics While the a-value is a measure of earthquake productivity, the bvalue is indicative of the ratio between large and small quakes. Both a and b are, therefore, important parameters in hazard

analysis. Usually b is close to a unity. Note that the G-R relation describes a power-law distribution. 1. log N(> MW ) = a bMW . Recall that : 2 2. MW = log10 M 0 10.7 . 3 Replacing 1 in 2 gives : 3a. log N(> MW ) = a b log M 0 , which is equivalent to : b 3b. N(> MW ) = aM 0 .

The Gutenberg-Richter distribution versus characteristic distribution G-R distribution characteristic distribution Two end-member models can explain the G-R statistics: Each fault exhibits its own G-R distribution of earthquakes. There is a power-law distribution of fault lengths, with each fault exhibiting a characteristic distribution. Fault distribution and earthquake statistics

Cumulative length distribution of subfaults of the San Andreas fault. Scholz, 1998 Fault distribution and earthquake statistics Loma Prieta Fault distribution and earthquake statistics In conclusion: For a statistically meaningful population of faults, the distribution is often consistent with the G-R relation.

For a single fault, on the other hand, the size distribution is often characteristic. Note that the extrapolation of the b-value inferred for small earthquakes may result in under-estimation of the actual hazard, if earthquake size-distribution is characteristic rather than powerlaw. Question: what gives rise to the drop-off in the small magnitude with respect to the G-R distribution? The controls on rupture final dimensions Seismological observations show that: 1. Co-seismic slip is very heterogeneous. 2. Slip duration (rise time) at any given point is much shorter than the total rupture duration

Example from the 2004 Northern Sumatra giant earthquake Preliminary result by Yagi. Uploaded from: www.ineter.gob.ni/geofisica/tsunami/com/20041226-indonesia/rupture.htm The controls on rupture final dimensions Barriers are areas of little slip in a single earthquake (Das and Aki, 1977). Asperities are areas of large slip during a single earthquake (Kanamori and Stewart, 1978). The origin and behavior with time of barriers and asperities:

1. Fault geometry - fixed in time and space? 2. Stress heterogeneities - variable in time and space? 3. Both? The controls on rupture final dimensions According to the barrier model (Aki, 1984) maximum slip scales with barrier interval. If this was true, fault

maps could be used to predict maximum earthquake magnitude in a given region. The controls on rupture final dimensions But quite often barriers fail to stop the rupture The 1992 Mw7.3 Landers (CA): The 2002 Mw7.9 Denali (Alaska): Figure from: pubs.usgs.gov Figure from: www.cisn.org

The controls on rupture final dimensions While in the barrier model ruptures stop on barriers and the bigger the rupture gets the bigger the barrier that is needed in order for it to stop, according to the asperity model (Kanamori and Steawart, 1978) earthquakes nucleate on asperities and big ruptures are those that nucleate on strong big asperities. That many ruptures nucleate far from areas of maximum slip is somewhat inconsistent with the asperity model. The controls on rupture final dimensions In the context of rate-state friction: Asperities are areas of a-b<0.

Barriers are areas of a-b>0. Further reading: Scholz, C. H., The mechanics of earthquakes and faulting, NewYork: Cambridge Univ. Press., 439 p., 1990. Aki, K., Asperities, barriers and characteristics of earthquakes, J. Geophys. Res., 89, 5867-5872, 1994.

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