The moment magnitude, M_w, is a measure of the total energy released by an earthquake. Unlike other methods of earthquake magnitude, it is directly related to the size of rupture and the amount of slip on the rupture surface, and not based solely on empirical studies of the observed instrumental recordings.

The concept of defining a magnitude related to the energy release, Er, of an earthquake was developed by Hiroo Kanamori in the 1970's at the Seismo Lab at Caltech. He defined the Energy Magnitude, M_w, as:

M_w =[log10(Er) - C] / 1.5

Er is a difficult parameter to estimate, but in general it is approximately linearly related to the Moment, M_o, of an earthquake, which is directly proportional to area of rupture during an earthquake, A, and the average slip across the rupture length, D.

             M_o =A D    (= shear modulus of the rock)

 M_w can then be determined from an estimate of the seismic moment:

M_w = [log10(Mo) - 16.1]/1.5

(M_o in dyne.cm; dyne.cm = 10E-7Nm,)

 At the same time, Hanks found a similar relation using estimated source ruptures. A compromise was made where the moment magnitude, M is defined as:

M = [log10(Mo) - 16.05]/1.5

 The symbols M and M_w are now interchangeable used for the moment magnitude in the literature. There is rarely a distinction between moment magnitude and energy magnitude.

 As M_w is physically dependent on the rupture, and not some narrow frequency band of observations like other magnitudes (M_L, M_b) it is the most reliable magnitude scale for large earthquakes, as it does not saturate. The Richter Magnitude, or the local magnitude, M_L, typically saturates at about M_w 6.5 as the amplitude of the waves generated in the frequency band measured by Ml does not increase once earthquakes become larger than M_w6.5. Various different techniques are available for M_w estimation. At Caltech, the M_w is determined using a complete broadband inversion (10s-100s) from various stations, a code developed by Doug Dreger at Berkeley Seismology Laboratory. This technique also determines the moment tensor of the Earthquake. This method is automatically run by the Northern California Seismic Network (NCSN) URL, and by NIED in Japan URL. The Caltech solution has both Mw and the moment tensor available within 12 minutes of an event trigger. Solutions are assigned a Quality value dependent on the goodness of fit between synthetic and observed waveforms, and numbers of station used in the inversion. Solutions are automatically distributed if they reach acceptable quality.

It has been found that though this technique is useful in determining M_w down to M_w 3.5, a complete catalogue of M_w of a reliable nature is only available for M_w >4.0. The method used at Caltech assumes a point-source representation for the earthquake. During very large events, with magnitude in excess of Mw8.0, the rupture length becomes a significant fraction of the distance to the seismic station used in the inversion, M_w can also saturate. Figures 1 and 2 present the moment tensor and M_w solutions for the SCSN catalogue, for events with Ml>3.0, occurring from 2000-2004.

The Richter Magnitude, Ml, is a logarithmic increase of 1 magnitude unit represents a factor of ten times in amplitude. The seismic waves of a magnitude 6 earthquake are 10 times greater in amplitude than those of a magnitude 5 earthquake. However, in terms of energy release, a magnitude 6 earthquake is about 31 times greater than a magnitude 5.

A magnitude 6.2 releases four times the energy than a magnitude 6.0 does. A magnitude 6.4 releases eight times the energy than a magnitude 6.0 does. A magnitude 6.8 releases sixteen times the energy as does a magnitude 6.0. A magnitude 7.0 releases thirty-two times the energy than a magnitude 6.0 earthquake. A magnitude 8.0 releases 1064 (32x32 or 32²)times the energy than a magnitude 6.0 does. Finally, a magnitude 9.0 releases more than 32,000 times the energy (32x32x32 or 32³) than a magnitude 6.0 does (this means we would need to have more than 32,000 M_w6.0 earthquakes to release the energy of one M_w9.0!).

M_L - Local Magnitude

Local magnitude, as practiced today, is derived directly from Charles Richter's original magnitude scale: the magnitude is the logarithm (base 10) of the amplitude in microns on a Wood-Anderson torsion seismometer located 100 km from the earthquake (in other words, if the amplitude is 1.0 mm at this distance, the quake is a 3.0).  Magnitude estimate is computed for each available horizontal component, and the mean or median of the values is taken as the earthquake magnitude.  We generally require readings from components from at least two different stations for a valid magnitude.

We currently do not operate any Wood-Anderson torsions, so we use synthetic Wood-Anderson records generated by computer from the broadband instruments.  The Wood-Anderson static magnification used in the computation is 2080 (based on empirical calibration), rather than the theoretical value of 2800.

 Another change from Richter's original work is that we use an attenuation curve based on thousands of earthquakes, as opposed to Richter's table based only on a few dozen events.  Using Richter's attenuation table underestimated M_L for earthquakes close to the station (less than about 50 km) and overestimated M_L for earthquakes beyond about 300 km.

For earthquakes within our current dense network, we are able to get M_L estimates from multiple stations for most earthquakes above about M_L1.8.  M_L saturates above about M_L6.5 because most of the energy for larger earthquakes falls outside of the response band of the Wood-Anderson instrument.  This means that most earthquakes larger than that have M_Ls that underestimate their true size, and M_L is useless.  M_w is preferable for the large events.

M_c Coda Magnitude

 Coda magnitudes are based on the fit of an exponential decay curve to the S-wave coda.  The decay rate is assumed to be the same for all earthquakes, and the station correction, representing the gain of the instrument and the site geology, was determined empirically using a set of earthquakes with known M_L.

 M_c was used primarily from 1977 through 2000, under the CUSP (Caltech-USGS Seismic Processing) software.  M_c's are subject to overestimates due to noise on the signals and multiple earthquakes close together in time.

 M_d Duration Magnitude

 M_d is also based on the fit of an exponential decay curve to the S-wave coda, with particular emphasis on the time duration that the earthquake signal is above predetermined amplitude.  M_d was used with Develocorder (microfilm) recorded data in the 1970's and is still being used by the Northern California Seismic Network.  We are in the process of calibrating the SCSN stations for it.

 M_h Hand-Determined Magnitude

M_h, the hand-determined magnitude, is the last resort when no other estimate is available.  Unfortunately, a hodge-podge of different methods fall into this category.  For the data from 1932 through the demise of the photographic and visible (ink) records in the 1990's, M_h was most often similar to M_L, except that in addition to the few Wood-Andersons with good readings, the short-period vertical amplitudes were used as well. At later dates, M_h is most often an M_d that was determined with someone's ruler on the computer screen, rather than a computer algorithm.  Unfortunately, until the M_d calibration is operational, we are currently using this form of M_h for earthquakes too small for M_L to be determined.