A new time-series analysis called "wavelet transform" is applied to measure the group velocity of surface waves as compared with that obtained by the conventional Fourier transform. We use vertical-component Rayleigh waves for both synthetic seismograms and GDSN long-period data of oceanic paths. The results of this study are summarized as follows: for synthetic seismograms, moving-window analysis using the Fourier transform can measure the group velocity of the fundamental mode correctly, while the group velocity of the first-higher mode is systematically larger than the correct value. In contrast, the wavelet transform measures the group velocity of both modes precisely although the resolution in frequency may not be sufficiently high. For GDSN data propagating along the Pacific Ocean, both methods provide stable results for the group velocity of the fundamental mode in the period range of 20 to 100 s. Using the Fourier transform, we obtain the group velocities of the first-higher mode between 20 and 40 s although these values seem unreliable. In contrast, the wavelet transform can measure both modes precisely in the period range of 20 to 100 s for non-shallow events and even for shallow events with relatively small noise in the data. Another advantage of the wavelet analysis is that we can specify resolving power in group velocity measurement rigorously.

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The generalized fractal dimensions are measured for the time series based on two complete earthquake catalogues: one with M≥6 earthquakes occurring in the north-south seismic belt of mainland China during the 1900-1990 period published by Ma et al. (1992) and the other with M≥5.5 earthquakes occurring in southern California, USA during the 1915-1994 period compiled by Press and Allen (1995). The log-log plot of C_q versus t, where C_q(t) is the generalized correlation integral and t is the interoccurrence time in years between two events, at positive q shows a linear istribution when t<t_c. D_q is the slope of this linear portion. The value of t_c decreases from 50.1 to 39.8 years for Chinese earthquakes and from 50.1 to 31.6 years for southern California events as q is increased from 0 to 15. For M≥6 Chinese earthquakes, the well-distributed, monotonically decreasing function of D_q, with increasing q would imply that such earthquakes have formed a multifractal time series. In contrast, the M≥5.5 southern California earthquakes might have not yet formed a complete multifractal time series or the number of these events is too small to accurately estimate the multifractal dimensions, especially for large qs. Different degrees of complexity of fault distributions in the two seismic regions might also be a factor in causing the difference in the D_q-q relations. In addition, the results also suggest that a D_q-q relation is better than the first three commonly-used values of D_q to completely represent a multifractal time series.

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A two-dimensional reflection/transmission problem for SH-waves at a corrugated interface between homogeneous transversely isotropic half-spaces is considered. Rayleigh's method is adopted and expressions for reflection and transmission coefficients are obtained in closed form for the first-order approximation of the corrugation. Numerical computations for a particular model have been performed.

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A one-dimensional BK dynamical lattice model (Burridge and Knopoff, 1967) is applied to simulate earthquakes for the study of the scaling relation between frequency and rupture length of earthquakes. Velocity-dependent friction controls the motion of mass elements. The distribution of the breaking strengths (i.e., static friction) is considered to be a fractal function. Simulation results show that the fractal dimension of the distribution of the breaking strengths is a minor factor in affecting the scaling of frequency versus rupture length. A fast velocity-weakening process from static friction to dynamic friction and a slow velocity-hardening one from dynamic friction to static friction are appropriate for interpreting the scaling of the frequency-rupture length (FL) relation. The frictional drop rather than the level of the breaking strength affects the FL scaling. Hence, the friction drop ratio (g) which determines the minimum value of the dynamic frictional force, is an important factor in influencing the FL relation. Smaller g (which a large friction drop) leads to a smaller scaling exponent value in the regime of localized events than larger g (with a smaller friction drop). The stiffness ratio, which is defined as the ratio of the stiffness of the coil spring to that of the leaf spring of the model, is also a significant parameter affecting the FL distribution. Nevertheless, simulation results show that small s is unable to produce a ower-law FL relation.

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