Partly because of their association with earthquakes, many attempts have been made to define the size or severity of tsunami in terms of one or another magnitude scales. Most of these are derived from measurements of runup, which as noted above is the height that the wave extends up to on steep shorelines, measured above a reference level (the normal height of the sea, corrected to the state of the tide at the time of wave arrival). Where tide gauges are available, the displacements on these are sometimes used (although a whole further set of corrections are required, because tide gauges are designed precisely NOT to record displacements due to short - period waves and they therefore attenuate signals associated even with the relatively long - period tsunami waves).

Two widely used measures are:

Tsunami magnitude m (Iida et al., 1967):

where H is the maximum observed or measured runup (in metres)

Tsunami intensity I (Soloviev & Go, 1974):

The following table shows the range of values of m in relation to runups and effects of tsunamis:

MAXIMUM RUNUP (METRES)	MAGNITUDE m	COMMENTS
0.25	-2	Smallest observable tsunami (except with very sensitive tide gauges)
1	0	Smallest tsunami generally reported by eyewitnesses
2	1	-
4	2	Usual lower limit for onset of significant damage
16	4	Onset of catastrophic damage and large numbers of deaths
64	6	Largest tsunami along open coastlines to be found in the historic record (e.g. Kamchatka, 1737). Severe damage in many parts of ocean affected
256	8	Typical oceanic island collapse - generated tsunami (e.g. Lanai, Canary Islands)? Catastrophic damage on transoceanic scales
1000+	10	Large asteroid impact - generated tsunami?

The basic problem with all of these measurements, however, is that the local effects of wave amplification, refraction, resonance and so on in the immediate vicinity of the measuring point distort the impression given of the true size of the tsunami. Furthermore, the impression given of the magnitude of the tsunami will also depend on the distance between the tsunami source and the nearest measurement point, because no formal allowance is made in the above equations for the attenuation of tsunami waves between source and measuring point, although Iida et al did recommend the use of runup data between 10 km and 250 km from the source (although this begs the question of how to define the location of the source, when the source is an earthquake rupture hundreds of kilometres long). Directivity of tsunami is also a problem in applying these equations.

This situation is at its worst in the case of fjord tsunami where the trapped waves can attain enormous heights relative to the amount of energy actually involved, especially in the immediate vicinity of the place of generation where surge waves ("splashes") can greatly distort the impression given by maximum runup height values (as at Lituya Bay in 1958). In contrast, earthquake magnitude scales derived from seismometers at least have the merit that they are derived from measurements on standardized instruments which have predictable responses to earthquake waves whatever their source and can also be used to locate the position of the source and thus provide a means of distance correction.

These problems should always be borne in mind when interpreting tsunami event and runup catalogues in terms of frequency-magnitude distributions of tsunami in a particular area of interest. Two important attempts have been made to overcome them:

K. Abe (1979) recognized that the near-source runups are perhaps the most unreliable for estimation of the overall magnitude of a tsunami. His approach, which applies specifically to earthquake-generated tsunami in the Pacific, averaged tide-gauge data at distant sites to derive a relationship between wave height H and tsunami magnitude Mt which was then adjusted to be approximately equivalent to the separately known moment magnitudes Mw of a reference set of source earthquakes by incorporation of a constant B, a procedure that normally works at the order of magnitude level for any one earthquake (except for the so-called tsunami earthquakes, which Abe recognized as forming about 10% of the total set of earthquakes considered in his study):

Abe's study was originally intended to form an alternative means of reevaluating earthquake magnitudes, particularly those for which instrumental data was unavailable or unreliable. For the purposes of evaluating tsunami magnitudes, however, his principal contribution was to establish that (subject to errors of the order of a factor of 2 or 3) the value of B varies systematically between certain specific pairs of sources and regions of runup measurement in the Pacific:

SOURCE REGION	OBSERVATION REGION/POINT
	Honolulu	Hilo	California	Japan	Aleutians
Peru/Chile	9.3	8.5	9.3	9.1	9.3
Alaska/Aleutians	9.2	9.1	9.5	9.4	---
Kamchatka/Kuriles	9.1	8.7	9.4	8.9	8.9
Japan	9.1	8.7	9.4	---	9.9

This procedure offers the prospect, in the Pacific at least, of using tide gauge records at specific points (such as Honolulu) to evaluate tsunami magnitudes for earthquake - generated tsunami in the Pacific in such a way as to make systematic corrections for different source regions.

Use of deep water buoy or pressure sensor data.

A more general solution may be to use measurements of tsunami in deep water. Advances in instrumentation (see Warning Systems) now make it possible to detect very long wavelength, very low amplitude waves in the deep ocean. As noted under tsunami propagation, tsunami waves in the deep ocean where water depths are both large (averaging 4-6 km) and generally relatively uniform, are not affected by all the complications that arise in shallow water. Records of the passage of tsunami made by the networks of buoys being installed for warning purposes will in future provide the basis for estimates of tsunami magnitude based on deep water wave height, corrected for distance from the source. This development will be particularly relevant to any impact-generated tsunami that may be observed in future, since the many theoretical models produced to predict the behaviour of these rather unusual tsunami generally produce output data in the form of calculated deep-ocean wave heights at particular distances from source.

Until the buoy and ocean bottom pressure sensor systems have been in place for several decades, however, estimates of the frequency-magnitude distribution of tsunami, particularly in regions of moderate to low tsunami frequency, will have to remain dependent upon the more traditional methods of estimating tsunami magnitude, as applied to historical, archaeological and geological data sets.