We derive an approximate expression for far-field spectral amplitude of acceleration radiated by circular cracks. The crack tip velocity is assumed to make abrupt changes, which can be the sources of high-frequency radiation, during the propagation of crack tip. This crack model will be usable as a source model for the study of high-frequency radiation. The expression for the spectral amplitude of acceleration is obtained in the following way. In the high-frequency range its expression is derived, with the aid of geometrical theory of diffraction, by extending the two-dimensional results. In the low-frequency range it is derived on the assumption that the source can be regarded as a point. Some plausible assumptions are made for its behavior in the intermediate-frequency range. Theoretical expressions for the root-mean-square and peak accelerations are derived by use of the spectral amplitude of acceleration obtained in the above way. Theoretically calculated accelerations are compared with observed ones.
The observations are shown to be well explained by our source model if suitable stress-drop and crack tip velocity are assumed. Using Brune's model as an earthquake source model, Hanks and McGuire showed that the seismic accelerations are well predicted by a stress-drop which is higher than the statically determined stress-drop. However, their conclusion seems less reliable since Brune's source model cannot be applied to the study of high-frequency radiation. According to our results, the seismic accelerations can be explained by a lower stress-drop, even by the same value of static stress-drop, if there are more abrupt changes in the crack tip velocity, the magnitude of its change is larger, or the crack tip velocity averaged over the crack surface is higher. Precise information about crack tip velocity is necessary to estimate the stress-drop associated with high-frequency radiation.

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The modified first and third asymptotic functions of the extreme value theory are applied to compute return periods and maximum potential earthquakes in densely populated regions such as the Tokyo, Guatemalan, Managuan, and Los Angeles areas.
The plot of the modified first asymptotic distribution follows an upward convex curve on Gumbel probability paper and is almost identical to that of the third asymptotic distribution. An application of the third asymptotic function shows too large maximum potential earthquakes to be realistic in view of the finiteness of the seismic source, whereas the modifled first asymptotic function gives more reasonable maximum potential earthquakes and more optimum evaluation in seismic risk than the third one.

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