(1) Memory effect in large earthquakes
The memory effect for the M≥7 earthquakes occurred in Taiwan during 1906–2006 and the M≥6 earthquakes occurred in the South-North Seismic Belt, Mainland China during 1901–2008 are studied using the fluctuation analysis technique. Calculated results show that the exponents of scaling law of fluctuation versus window length are less than 0.5 for the sequences of earthquake magnitude and inter-event time. The migration of earthquakes in study is taken to discuss the possible correction between events. The phase portraits of two sequent magnitudes and two sequent inter-event times are also applied to explore if large (or small) earthquakes are followed by large (or small) events. Together with all kinds of given information, we conclude that the earthquakes in study is short-term corrected and thus the short-term memory effect would be operative.
(2) The energy-magnitude scaling law for Ms<5.5 earthquakes
The scaling law of seismic radiation energy, Es, versus surface-wave magnitude, Ms, proposed by Gutenberg and Richter (1956) was originally based on earthquakes with Ms>5.5. A comparison of the data points of log(Es) versus Ms with Gutenberg and Richter’s law leads to a conclusion that the law is still valid for earthquakes with 0<Ms≤5.5.
(3) Observations and mechanism of b-values prior to a mainshock
The temporal variation in b-values prior to a mainshock shows that the b-value starts to increase from the normal value at time t1, reaches its peak one at time t2, then begins to decrease from the peak one at t2, and returns to the normal one at time t3. As t>t3, the b-value varies around the normal one or rightly decreases with time until the occurrence of the forthcoming mainshock at time t4. The precursor time, T=t4-t1, is related to the magnitude, M, of the event in a form: log(T)=q+rM where q and r are two constants. Wang (1995) found a power-law correlation between b and s, where the parameter s is the ratio of the spring constant (K) between two sliders to that (L) between a slider and the moving plate. The power-law correlations are b~s-2/3 for the cumulative frequency and b~s-1/2 for the discrete frequency. Since L of a source area is almost constant for a long time period, b directly relates to K. Lower K results in a higher b-value. Wang (2012) found K=rAvp2, where rA and vp are, respectively, the areal density and P-wave velocity of a fault zone. Experimental results show that vp is strongly influenced by the water saturation in rocks. The water saturation in the source area varies with time, thus leading to a temporal variation in vp as well as K. This results in the temporal variation in b-values prior to a mainshock. The modeled result is consistent with the observation.
(4) Frictional effect on scaling of earthquake source displacement spectra
The scaling of earthquake source displacement spectra or seismic spectra is analytically studied based on the continuous form of 1-D spring-slider model with either linearly slip-weakening friction or linearly velocity-weakening friction. The main parameters of the model are the natural angular frequency, wo, and the decreasing rate, D, of friction with slip for slip-weakening friction as well as the decreasing rate, u, of friction with velocity for velocity-weakening friction. The analytic solutions show that slip-weakening friction cannot produce the power-law seismic spectra. includes the complementary and particular parts. For velocity-weakening friction with u>0.5, the log-log plot of spectral amplitude versus w exhibits almost w0 scaling when w is lower than the corner angular frequency, wc. When w>wc, the spectral amplitude exhibits a w-1 scaling at high w.
(5) Frictional and viscous effects on the nucleation phase of an earthquake
The frictional and viscous effects are specified by the characteristic displacement, Uc, and viscosity coefficient, h, respectively. Simulation results show that Uc and h can both lengthen the natural period of the system and viscosity increases the duration time of motion of the slider. Higher h causes a smaller amplitude of lower velocity motion than lower h. A change of either Uc (under large h) or h (under large Uc) from a large value (Uch for Uc and hh for h) to a small one (Ucl for Uc and hl for h) in two stages during sliding can result in a clear nucleation phase prior to the P wave. The differences dUc=Uch-Ucl and dh=hh-hl are two important factors in producing a nucleation phase. The difference between the nucleation phase and the P wave increases with either dUc or dh. Like seismic observations, the peak amplitude of P wave, which is associated with the earthquake magnitude, is independent upon the duration time of nucleation phase.
(6) Multi-stable slip in a one-body spring-slider model with friction and viscosity
This study is focused on multistable slip of earthquakes based on a one-body slider-slider model in the presence of thermal-pressurized slip-weakening friction and viscosity by using the normalized equation of motion of the model. The major model parameters are the normalized characteristic displacement, Uc, of the friction law and the normalized viscosity coefficient, h, between the slider and background plate. Analytic results at small slip suggest that there is a solution regime for h and g (=1/Uc) to make the slider slip steadily. Numerical simulations exhibit that the time variation in normalized velocity, V/Vmax (Vmax is the maximum velocity), obviously depends on Uc and h. The effect on the amplitude is stronger due to h than due to Uc. In the phase portrait of V/Vmax versus the normalized displacement, U/Umax (Umax is the maximum displacement), there are two fixed points. The one at large V/Vmax and large U/Umax is not an attractor; while that at small V/Vmax and small U/Umax can be an attractor for some values of h and Uc. When Uc<0.55, unstable slip does not exist. When Uc≥0.55, Uc and h divide the solution domain into three regimes: stable, intermittent, and unstable (or chaotic) regimes. For a certain Uc, the three regimes are controlled by a lower bound, hl, and an upper bound, hu, of h. The values of hl, hu, and hu-hl all decrease with increasing Uc, thus suggesting that it is easier to yield unstable slip for larger Uc than for smaller Uc or for larger h than for smaller h. When Uc<1, the Fourier spectra calculated from simulation velocity waveforms exhibit several peaks, thus suggesting the existence of nonlinear behavior of the system. When Uc>1, the related Fourier spectra show only one peak, thus suggesting linear behavior of the system.
