Earthquake magnitude is expressed in various ways, most of which are strictly speaking measures of the response of seismometers to the earthquakes. The most fundamental measure of earthquake size is however the Moment Magnitude MW, which is a function of the seismic moment M0, or the energy released in the earthquake as motion of the moving rock masses. In order to estimate this accurately it is necessary to measure seismic energy over a wide frequency range, down to 0.01 Hz (corresponding to seismic waves with a period of 100 seconds), or else carry out detailed geophysical, geodetic and geological studies to independently estimate the earthquake parameters on which it depends.

Mo varies as a function of the average amount of movement on the fault surface in the earthquake, u, the area of fault surface on which movement occurs, A, and the shear modulus of elasticity µ , a measure of how much force is required to elastically deform the moving rock on either side of the fault:

The shear modulus of elasticity varies only by a little more than a factor of two for most rocks in the crust and upper mantle where earthquakes occur. This means that seismic moment varies mostly as a function of the amount of movement and the area over which it occurs. Furthermore, as the relationship between these two is governed primarily by the "stiffness" of the rock (which is expressed largely in terms of the shear modulus of elasticity), and thus the permitted gradients in displacement along the fault surface, a relatively simple relationship exists between seismic moment and the area and amount of fault slip. It is therefore to be anticipated that a simple relationship will also exist between the seismic moment of an earthquake and the size of tsunami that it can directly produce, provided that due allowance is made for geometrical and depth. As general rules, only earthquakes with MW ³ 6.5 - 7 produce significant tsunami directly (although smaller earthquakes may trigger submarine landslides, as discussed below) and the largest earthquakes (MW » 9), as they almost invariably occur at the margins of oceans for reasons discussed below, normally produce major tsunami that cause significant damage over wide regions.

Unfortunately, this does not mean that the tsunami-generating potential of an earthquake can be quickly and simply evaluated from near-real-time seismic records, even though rapid processing of records from a number of seismometers allows estimates of the geometrical and (to a lesser extent) depth effects. The reason for this is that most seismometers are relatively short-period instruments and only measure the higher-frequency components of the seismic energy radiated by an earthquake. In reality, a large part of the seismic energy of earthquakes is released at frequencies of 0.01 Hz or less. As a result the various instrumental estimates of earthquake magnitude such as surface- and body- wave magnitudes, although providing good measures of expected intensities of ground shaking, do not provide good estimates of the values of A and u, and thus the tsunamigenic potential of earthquakes. About 1 in 10 submarine earthquakes produce tsunami that are 10 to 100 times larger than expected from their instrumentally-estimated magnitudes. Notable examples include the 1 April 1946 earthquake off Alaska, which produced a tsunami that devastated large parts of Hilo, Hawaii despite having a surface-wave magnitude MS of only 7.4. Conversely, other large earthquakes produce much smaller tsunami than expected, leading to "false alarm" problems for tsunami warning systems.

Some of these "tsunami earthquakes" are in fact likely to have been relatively small earthquakes which triggered submarine landslides, but many others are likely to have been "slow" or "silent" earthquakes. These were first proposed, in the context of unusually large tsunamis, by H. Kanamori in 1972 (Ref.). He proposed that comparatively slow propagation of the rupture along the fault surface led to only a tiny fraction of the seismic energy being released at high frequencies whilst all the rest was emitted at low frequencies that cannot be detected by normal seismometers. Recent studies (Bilek & Lay, 1999) suggest that this type of earthquake tends to occur at shallow depths in subduction zones where thrust faults cut weak, partly ductile sediments in which fault ruptures do not propagate rapidly but where the low stiffness allows for unusually large slips in relation to the size of the fault rupture: they may therefore have a specific regional distribution. The complex relationship between instrumental estimates of earthquake size, moment magnitudes of earthquakes, and their tsunamigenic potential, makes it difficult to use conventional earthquake catalogues to accurately predict tsunami hazard levels, and has led to efforts to do the reverse, i.e. to make estimates of earthquake magnitude using tsunami records.

The maximum size of earthquakes and earthquake-generated tsunami

The largest observed earthquakes (MW » 9 to 9.5) typically involve average slips of about 10 m (with local maxima up to 20 m) over fault slip areas of about 100 000 square km (typically 700 - 1000 km long by 100 - 150 km wide, since the depth of the seismogenic zone in the earth limits the width). The limits imposed by the structure of the earth and the lengths of individual faults mean that it is unlikely that an earthquake much larger than these will ever occur. Furthermore, these earthquakes occur at subduction zones, where the seismogenic layer is thickest, and therefore typically have a thrust type geometry that is an efficient generator of tsunami waves. These very large earthquakes have rupture zones that extend deep into the earth and therefore are not subject to the slow rupture velocities that result in "tsunami earthquakes": the largest earthquakes of the latter type have moment magnitudes of about Mw = 8. It is therefore likely that the tsunami generated by the largest observed earthquakes, such as those produced in the 1964 Alaska earthquake, are about as large as will ever be produced directly by an earthquake.