## 1731 Magnitude

An individual earthquake is a unique release of strain energy — quantification of this energy has formed the basis for measuring the earthquake event. Richter (1935) was the first to define earthquake magnitude, as

where ML is the local magnitude (which Richter only defined for Southern California), A is the maximum trace amplitude in micrometers recorded on a standard Wood-Anderson short-period torsion seismometer,3 at a site 100 km from the epicenter, and log A0 is a standard value as a function of distance for instruments located at distances other than 100 km and less than 600 km. Subsequently, a number of other magnitudes have been defined, the most important of which are surface wave magnitude MS, body wave magnitude mb, and moment magnitude MW. Due to the fact that Ml was only locally defined for California (i.e., for events within about 600 km of the observing stations), surface wave magnitude MS was defined analogously to ML, using teleseismic observations of surface waves of 20 s period (Richter 1935). Magnitude, which is defined on the basis of the amplitude of ground displacements, can be related to the total energy in the expanding wave front generated by an earthquake, and thus to the total energy release — an empirical relation by Richter is log10 Es = 11.8 + 1.5Ms (17.2)

where ES is the total energy in ergs. 4 Note that 10L5 = 31.6, so that an increase of one magnitude unit is equivalent to 31.6 times more energy release, two magnitude units increase equivalent to 998.6 ffi 1000 times more energy, etc. Subsequently, due to the observation that deep-focus earthquakes commonly do not register measurable surface waves with periods near 20 s, a body wave magnitude mb was defined (Gutenberg and Richter 1954), which can be related to MS (Darragh et al. 1994):

Body wave magnitudes are more commonly used in eastern North America, due to the deeper earthquakes there. A number of other magnitude scales have been developed, most of which tend to saturate — that is, asymptote to an upper bound due to larger earthquakes radiating significant amounts of energy at periods longer than used for determining the magnitude (e.g., for MS, defined by measuring 20 s surface waves, saturation occurs at about MS > 7.5). More recently, seismic moment has been employed to define a moment magnitude MW (Hanks and Kanamori 1979; also denoted as boldface M), which is finding increased and widespread use log M0 = 1.5MW + 16.0 (17.4)

where seismic moment M0 (dyne cm) is defined as (Lomnitz 1974)

where p is the material shear modulus, A is the area of fault plane rupture, and U is the mean relative displacement between the two sides of the fault (the averaged fault slip). Comparatively, MW and MS are

3The instrument has a natural period of 0.8 s, critical damping ration 0.8, magnification 2800.

4Richter (1958) gives 11.4 for the constant term, rather than 11.8, which is based on subsequent work — the uncertainty in the data makes this difference inconsequential.

23456789 10 Moment magnitude

FIGURE 17.4 Relationship between moment magnitude and various magnitude scales (Campbell, K.W. 1985).

23456789 10 Moment magnitude

FIGURE 17.4 Relationship between moment magnitude and various magnitude scales (Campbell, K.W. 1985).

numerically almost identical up to magnitude 7.5. Figure 17.4 indicates the relationship between moment magnitude and various magnitude scales.

For lay communications, it is sometimes customary to speak of great earthquakes, large earthquakes, etc. There is no standard definition for these, but the following is an approximate categorization:

Earthquake Micro Small Moderate Large Great

a Not specifically defined.

From the foregoing discussion, it can be seen that magnitude and energy are related to fault rupture length and slip. Slemmons (1977) and Bonilla et al. (1984) have determined statistical relations between these parameters for worldwide and regional data sets, aggregated and segregated by type of faulting (normal, reverse, strike-slip). Bonilla et al.'s worldwide results for all types of faults are which indicates, for example, that, for MS = 7, the average fault rupture length is about 36 km (and the average displacement is about 1.86 m), and s indicates standard deviation. Conversely, a fault of 100 km length is capable of about an MS = 7.5 event5. More recently, Wells and Coppersmith (1994) have performed an extensive analysis of a dataset of 421 earthquakes — their results are presented in Table 17.1.

Note that L = g(MS) should not be inverted to solve for MS = f(L), as a regression for y = f (x) is different than a regression for x = g (y).

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