The Omori formula n(t)=K(t+c)^-1 and its modified form n(t)=K(t+c)^-P have been successfully applied to many aftershock sequences since the former was proposed just 100 years ago. This paper summarizes studies using these formulae. The problems of fitting these formulae and related point process models to observational data are discussed mainly. Studies published during the last 1/3 century confirmed that the modified Omori formula generally provides an appropriate representation of the temporal variation of aftershock activity. Although no systematic dependence of the index p has been found on the magnitude of the main shock and on the lowest limit of magnitude above which aftershocks are counted, this index (usually p = 0.9-1.5) differs from sequence to. sequence. This variability may be related to the tectonic condition of the region such as structural heterogeneity, stress, and temperature, but it is not clear which factor is most significant in controlling the p value. The constant c is a controversial quantity. It is strongly influenced by incomplete detection of small aftershocks in the early stage of sequence. Careful analyses indicate that c is positive at least for some sequences. Point process models for the temporal pattern of shallow seismicity must include the existence of aftershocks, most suitably expressed by the modified Omori law. Among such models, the ETAS model seems to best represent the main features of seismicity with only five parameters. An anomalous decrease in aftershock activity below the level predicted by the modified Omori formula sometimes precedes a large aftershock. An anomalous decrease in seismic activity of a region below the level predicted by the ETAS model is sometimes followed by a large earthquake in the same or in a neighboring region.

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We have studied magma solidification during volcanic events on the basis of conductive interchange in systems that undergo phase changes within a wide temperature range. Assuming this conductive hypothesis, we approach the heat equation via an enthalpic path, which takes account of the phase changes, where the thermodynamic parameters are a function of system temperature. This result is obtained by solving a nonlinear partial differential equation. Applying the finite group theory, we have obtained an analytic solution to the problem in different geometries.

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Seismic sources are described by the 'traction tensors' which are expressed as the surface integral of traction around the sources.

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The three-dimensional (3-D) velocity structure of the upper crust in the central part of the Hida region, central Honshu, Japan, has been investigated by simultaneous inversion of travel time data for velocity and hypocentral parameters using Thurber's method (1983). The data used for this purpose were 2, 231 P-wave arrival times from 204 local earthquakes observed at 16 high-sensitivity seismograph stations. The iterative damped least-squares inversion used here provided reliable results with the diagonal elements of resolution matrix well exceeding 0.90 and standard errors less than 3% for the central area. The central part of the Hida region covering an extremely low seismicity area has high velocities probably extending down to a mid-crust. The high-velocity area composed of hard metamorphic and granitic complex extends from the western flank of the Hida mountains to the western section of the seismically active Atotsugawa-Ushikubi faults. The axial part of the Hida mountains and its southwestward area, on the other hand, is covered by remarkable low-velocities. The low velocities may be associated with a high thermal state due to active volcanism beneath the mountains involving volcanoes Mts. Tateyama, Yake, Norikura, and Ontake. The southwestern part of the Hida region is also partially covered by low-velocities beneath the Ryohaku mountains, which may also be related to volcanism of Mt. Hakusan. The obtained 3-D velocity structure appears to be qualitatively consistent with the Bouguer gravity anomalies and also with the large-scale 3-D structure so far obtained.

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A new approach which uses the full-space Green's function is proposed to eliminate the influence of the finitude of the segmented free surface in the indirect formulation of the Boundary Element Method. The absence of a contribution from the far parts of the free surface that are neglected in the computation is compensated for, assuming that the wave field along flat parts of the surface far from the localized irregularity can be substituted by the reference solution, i.e., the wave field in the half space with a completely flat free surface which is given analytically. The nonphysical waves radiated at the two ends of the segmented model of the free surface are eliminated efficiently by this approach. This reduces the amount of computer main memory required and the CPU time consumed for calculation. Moreover, physical consideration of the relation to existing approaches shows clearly that our approach gives a more precise approximation, in particular for high frequency problems. The capacity of the indirect formulation of the Boundary Element Method therefore is bettered by this approach.

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