History
Richter scale: the original measure of earthquake magnitude
At the beginning of the twentieth century, very little was known about how earthquakes happen, how seismic waves are generated and propagate through the earth's crust, and what information they carry about the earthquake rupture process; the first magnitude scales were therefore empirical. The initial step in determining earthquake magnitudes empirically came in 1931 when the Japanese seismologist Kiyoo Wadati showed that the maximum amplitude of an earthquake's seismic waves diminished with distance at a certain rate.Single couple or double couple
The study of earthquakes is challenging as the source events cannot be observed directly, and it took many years to develop the mathematics for understanding what the seismic waves from an earthquake can tell us about the source event. An early step was to determine how different systems of forces might generate seismic waves equivalent to those observed from earthquakes. The simplest force system is a single force acting on an object. If it has sufficient strength to overcome any resistance it will cause the object to move ("translate"). A pair of forces, acting on the same "line of action" but in opposite directions, will cancel; if they cancel (balance) exactly there will be no net translation, though the object will experience stress, either tension or compression. If the pair of forces are offset, acting along parallel but separate lines of action, the object experiences a rotational force, or torque. In mechanics (the branch of physics concerned with the interactions of forces) this model is called a ''Dislocation theory
A double couple model suffices to explain an earthquake's far-field pattern of seismic radiation, but tells us very little about the nature of an earthquake's source mechanism or its physical features. While slippage along a fault was theorized as the cause of earthquakes (other theories included movement of magma, or sudden changes of volume due to phase changes), observing this at depth was not possible, and understanding what could be learned about the source mechanism from the seismic waves requires an understanding of the source mechanism. Modeling the physical process by which an earthquake generates seismic waves required much theoretical development of dislocation theory, first formulated by the Italian Vito Volterra in 1907, with further developments by E. H. Love in 1927. More generally applied to problems of stress in materials, an extension by F. Nabarro in 1951 was recognized by the Russian geophysicist A. V. Vvedenskaya as applicable to earthquake faulting. In a series of papers starting in 1956 she and other colleagues used dislocation theory to determine part of an earthquake's focal mechanism, and to show that a dislocation – a rupture accompanied by slipping — was indeed equivalent to a double couple. In a pair of papers in 1958, J. A. Steketee worked out how to relate dislocation theory to geophysical features. Numerous other researchers worked out other details, culminating in a general solution in 1964 by Burridge and Knopoff, which established the relationship between double couples and the theory of elastic rebound, and provided the basis for relating an earthquake's physical features to seismic moment.Seismic moment
'' Seismic moment'' – symbol – is a measure of the fault slip and area involved in the earthquake. Its value is the torque of each of the two force couples that form the earthquake's equivalent double-couple. (More precisely, it is the scalar magnitude of the second-order moment tensor that describes the force components of the double-couple.) Seismic moment is measured in units of Newton meters (N·m) or Joules, or (in the older CGS system) dyne-centimeters (dyn-cm). The first calculation of an earthquake's seismic moment from its seismic waves was byIntroduction of an energy-motivated magnitude ''M''w
Most earthquake magnitude scales suffered from the fact that they only provided a comparison of the amplitude of waves produced at a standard distance and frequency band; it was difficult to relate these magnitudes to a physical property of the earthquake. Gutenberg and Richter suggested that radiated energy Es could be estimated as : (in Joules). Unfortunately, the duration of many very large earthquakes was longer than 20 seconds, the period of the surface waves used in the measurement of . This meant that giant earthquakes such as the 1960 Chilean earthquake (M 9.5) were only assigned an . Caltech seismologist Hiroo Kanamori recognized this deficiency and took the simple but important step of defining a magnitude based on estimates of radiated energy, , where the "w" stood for work (energy): : Kanamori recognized that measurement of radiated energy is technically difficult since it involves the integration of wave energy over the entire frequency band. To simplify this calculation, he noted that the lowest frequency parts of the spectrum can often be used to estimate the rest of the spectrum. The lowest frequency asymptote of a seismic spectrum is characterized by the seismic moment, . Using an approximate relation between radiated energy and seismic moment (which assumes stress drop is complete and ignores fracture energy), : (where E is in Joules and is in Nm), Kanamori approximated by :Moment magnitude scale
The formula above made it much easier to estimate the energy-based magnitude , but it changed the fundamental nature of the scale into a moment magnitude scale. USGS seismologist Thomas C. Hanks noted that Kanamori's scale was very similar to a relationship between and that was reported by : combined their work to define a new magnitude scale based on estimates of seismic moment : where is defined in newton meters (N·m).Current use
Moment magnitude is now the most common measure of earthquake size for medium to large earthquake magnitudes, but in practice, seismic moment (), the seismological parameter it is based on, is not measured routinely for smaller quakes. For example, the United States Geological Survey does not use this scale for earthquakes with a magnitude of less than 3.5, which includes the great majority of quakes. Popular press reports most often deal with significant earthquakes larger than . For these events, the preferred magnitude is the moment magnitude , not Richter's local magnitude .Definition
The symbol for the moment magnitude scale is , with the subscript "w" meaning mechanical work accomplished. The moment magnitude is a dimensionless value defined by Hiroo Kanamori as : where is the seismic moment in dyne⋅cm (10−7 N⋅m). The constant values in the equation are chosen to achieve consistency with the magnitude values produced by earlier scales, such as the local magnitude and the surface wave magnitude. Thus, a magnitude zeroRelations between seismic moment, potential energy released and radiated energy
Seismic moment is not a direct measure of energy changes during an earthquake. The relations between seismic moment and the energies involved in an earthquake depend on parameters that have large uncertainties and that may vary between earthquakes. Potential energy is stored in the crust in the form of elastic energy due to built-up stress and gravitational energy. During an earthquake, a portion of this stored energy is transformed into * energy dissipated in frictional weakening and inelastic deformation in rocks by processes such as the creation of cracks * heat * radiated seismic energy The potential energy drop caused by an earthquake is related approximately to its seismic moment by : where is the average of the ''absolute'' shear stresses on the fault before and after the earthquake (e.g., equation 3 of ) and is the average of the shear moduli of the rocks that constitute the fault. Currently, there is no technology to measure absolute stresses at all depths of interest, nor method to estimate it accurately, and is thus poorly known. It could vary highly from one earthquake to another. Two earthquakes with identical but different would have released different . The radiated energy caused by an earthquake is approximately related to seismic moment by : where is radiated efficiency and is the static stress drop, i.e., the difference between shear stresses on the fault before and after the earthquake (e.g., from equation 1 of ). These two quantities are far from being constants. For instance, depends on rupture speed; it is close to 1 for regular earthquakes but much smaller for slower earthquakes such as tsunami earthquakes and slow earthquakes. Two earthquakes with identical but different or would have radiated different . Because and are fundamentally independent properties of an earthquake source, and since can now be computed more directly and robustly than in the 1970s, introducing a separate magnitude associated to radiated energy was warranted. Choy and Boatwright defined in 1995 the ''energy magnitude'' : where is in J (N·m).Comparative energy released by two earthquakes
Assuming the values of are the same for all earthquakes, one can consider as a measure of the potential energy change Δ''W'' caused by earthquakes. Similarly, if one assumes is the same for all earthquakes, one can consider as a measure of the energy ''E''s radiated by earthquakes. Under these assumptions, the following formula, obtained by solving for the equation defining , allows one to assess the ratio of energy release (potential or radiated) between two earthquakes of different moment magnitudes, and : : As with the Richter scale, an increase of one step on the logarithmic scale of moment magnitude corresponds to a 101.5 ≈ 32 times increase in the amount of energy released, and an increase of two steps corresponds to a 103 = 1000 times increase in energy. Thus, an earthquake of of 7.0 contains 1000 times as much energy as one of 5.0 and about 32 times that of 6.0.Comparison with TNT equivalents
To make the significance of the magnitude value plausible, the seismic energy released during the earthquake is sometimes compared to the effect of the conventional chemical explosive TNT. The seismic energy results from the above mentioned formula according to Gutenberg and Richter to : or converted into Hiroshima bombs: : For comparison of seismic energy (in joules) with the corresponding explosion energy, a value of 4.2 - 109 joules per ton of TNT applies. The table''FAQs – Measuring Earthquakes: How much energy is released in an earthquake?''Subtypes of Mw
Various ways of determining moment magnitude have been developed, and several subtypes of the scale can be used to indicate the basis used. * – Based on ''moment tensor inversion'' of long-period (~10 – 100 s) body-waves. * – From a ''moment tensor inversion'' of complete waveforms at regional distances (~ 1,000 miles). Sometimes called RMT. * – Derived from a ''centroid moment tensor inversion'' of intermediate- and long-period body- and surface-waves. * – Derived from a ''centroid moment tensor inversion'' of the W-phase. * () – Developed by Seiji Tsuboi for quick estimation of the tsunami potential of large near-coastal earthquakes from measurements of the P-waves, and later extended to teleseismic earthquakes in general. * – A duration-amplitude procedure which takes into account the duration of the rupture, providing a fuller picture of the energy released by longer lasting ("slow") ruptures than seen with ., pp. 137–128.See also
* Earthquake engineering * Lists of earthquakes * Seismic magnitude scalesNotes
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