Seismic Interevent Time: A Spatial Scaling and Multifractality

The optimal scaling problem for the time t(L × L) between two successive events in a seismogenic cell of size L is considered. The quantity t(L × L) is defined for a random cell of a grid covering a seismic region G. We solve that problem in terms of a multifractal characteristic of epicenters in G known as the tau-function or generalized fractal dimensions; the solution depends on the type of cell randomization. Our theoretical deductions are corroborated by California seismicity with magnitude M ≥ 2. In other words, the population of waiting time distributions for L = 10–100 km provides positive information on the multifractal nature of seismicity, which impedes the population to be converted into a unified law by scaling. This study is a follow-up of our analysis of power/unified laws for seismicity (see Pure and Applied Geophysics 162 (2005), 1135 and GJI 162 (2005), 899).     

Sensitivity analysis of the parameters of earthquake recurrence time power law scaling

The stability of the power law scaling of earthquake recurrence time distribution in a given space–time window is investigated, taking into account the magnitude of completeness and the effective starting time of aftershock sequences in earthquake catalogs from Southern California and Japan. A new method is introduced for sampling at different distances from a network of target events. This method allows the recurrence times to be sampled many times on the same area. Two power laws with unknown exponents are assumed to govern short- and long-recurrence-time ranges. This assumption is developed analytically and shown to imply simple correlation between these power laws. In practice, the results show that this correlation structure is not satisfied for short magnitude cutoffs (m _c = 2.5, 3.5, 4.5), and hence the recurrence time distribution departs from the power law scaling. The scaling parameters obtained from the stack of the distributions corresponding to different magnitude thresholds are quite different for different regions of study. It is also found that significantly different scaling parameters adjust the distribution for different magnitude thresholds. In particular, the power law exponents decrease when the magnitude cutoff increases, resulting in a slower decrease of the recurrence time distribution, especially for short time ranges. For example, in the case of Japan, the exponent p₂ of the power law scaling at large recurrence times follows roughly the relation: , where m _c is the magnitude cutoff. In case of Southern California, it is shown that Weibull distribution provides a better alternative fit to the data for moderate and large time scales.     

A mixed model for earthquake interevent times

A mixed model is proposed to fit earthquake interevent time distribution. In this model, the whole distribution is constructed by mixing the distribution of clustered seismicity, with a suitable distribution of background seismicity. Namely, the fit is tested assuming a clustered seismicity component modeled by a non-homogeneous Poisson process and a background component modeled using different hypothetical models (exponential, gamma and Weibull). For southern California, Japan, and Turkey, the best fit is found when a Weibull distribution is implemented as a model for background seismicity. Our study uses earthquake random sampling method we introduced recently. It is performed here to account for space–time clustering of earthquakes at different distances from a given source and to increase the number of samples used to estimate earthquake interevent time distribution and its power law scaling. For Japan, the contribution of clustered pairs of events to the whole distribution is analyzed for different magnitude cutoffs, m _c, and different time periods. The results show that power laws are mainly produced by the dominance of correlated pairs at small and long time ranges. In particular, both power laws, observed at short and long time ranges, can be attributed to time–space clustering revealed by the standard Gardner and Knopoff’s declustering windows.     

A role of the altitude gradient of the thermospheric wind velocity in ionospheric <Emphasis Type="Italic">F</Emphasis>-region dynamics

By processing the data of vertical ionospheric sounding in Almaty for 2000–2009, we obtained the distributions of the heights of the maximum (h _m F) and bottom (h _bot F) of the F2-layer, incremental changes in its semi-thickness (δh), the characteristic time of losses (τ), and the vertical displacement velocity of the node of the thermospheric wind (V) during the transitional time of the day during nighttime increases in the electron concentration at the layer maximum. The comparison of the measured V and modeled V _m velocities showed a certain discrepancy. The influence of the altitude gradient of the meridional thermospheric wind velocity on the behaviors of h _m F, h _bot F, δh, and τ during nighttime increases in the electron concentration is studied.     

Statistical features of the seismic noise-field

We present a preliminary study of the dependence of the statistical features of the soil motion due to seismic noise on the near-surface geology in the frequency range from 1 Hz to ∼ 40 Hz. In detail, we have investigated the 3D average squared soil displacement 〈r²〉 and the distribution function of the displacement flctuations at different geological sites. The anomalous scaling of the average squared soil displacement 〈r²(τ)〉~τ_α, and the Gaussian shape of the probability distribution function of its fluctuations suggest that the soil motion under the influence of the seismic noise is consistent with a persistent fractional Brownian motion (fBm) characterized by a scaling exponent 1.5 < α < 2. Therefore, the seismic noise-field, thought as a stochastic process, shows a markovian character with a memory longer than a pure Brownian motion (α = 1/2). Moreover, a dependence of such persistent behavior of the noise-field dynamics on the near-surface local geology has been found and it is discussed.     

Variability of parameters of the <Emphasis Type="Italic">F</Emphasis>2-layer maximum in the quiet midlatitude ionosphere under low solar activity: 1. Statistical properties

Results of statistical analysis of the properties of variability of F2-layer maximum parameters (critical frequency foF2 and the height hmF2) in quiet midlatitude ionosphere under low solar activity in the daytime (1000–1500 LT) and nighttime (2200–0300 LT) hours are presented on the basis of Irkutsk station data for 2007–2008. It is found that the distribution density of δfoF2 could be presented as consisting of two distinctly different normal laws of this distribution, one of which corresponds to weak (|δfoF2| < 10%) fluctuations in foF2 and the other corresponds to strong (30% > |δfoF2| > 10%) fluctuations. Weak fluctuations in foF2 to a substantial degree are related to ionospheric variability at times less of than 1–3 h and determine the δfoF2 variability in the daytime hours. Strong fluctuations in foF2 are mainly related to day-to-day variability of the ionosphere at a fixed local time, the variability increasing by approximately a factor of 3 during the transition from day to night and determining the δfoF2 variability in the nighttime hours. The distribution density of ΔhmF2 is close to the normal distribution law. An interpretation of the different character of the distribution densities of δfoF2 and ΔhmF2 is given.     

Possible precursory accelerated Benioff strain in the region of Sistan Suture Zone of Eastern Iran

We investigated whether accelerated seismic strain release precedes large earthquakes occurring in and around the Sistan Suture Zone, Eastern Iran. Online catalogs of teleseismic events occurring post-1960 within the region 27.0°–37.0°N, 55.0°–65.0°E, report five M _w > 7.0 earthquakes, namely, 1968 Dasht-e-Bayaz, 1978 Tabas, 1979 Khuli-Buniabad, 1981 Sirch and 1997 Zirkuh-e-Q’aenat events. We defined four earthquake test episodes, 1968–1978, 1978–1981, 1979–1981, and 1981–1997, with all catalogued intermediate events having magnitudes within 2.0 units that of the final large event. Using the 1968 event as the starting point, we investigated possible increased moderate earthquake activity patterns prior to the large events of 1978, 1981 and 1997 by examining if the cumulative Benioff strain released from such preceding events followed a power law time-to-failure. Our investigation seem to suggest that the 1978, 1981 and 1997 events (i) followed a period of accelerated moderate earthquake activity and (ii) the radius of their optimal critical region, R, scaled with their magnitude, M, according to the scaling law log R ∝ 0.36 M. Our suggestions conform to those proposed by similar investigations in varied seismotectonic regimes.     

Dependence of the energy released during earthquake on ambient shear stress

Starting with dislocation model, using the result of the fracture mechanics: the slip displacement at the crack tip is proportional to the length of the crack and the applied ambient shear stressτ ₀ ² , we consider the dislocation in the earthquake to be the slip displacement at the crack tip and have obtained the analysis expresses of displacement and velocity pulse for the circular crack and have calculated the seismic wave energy radiated by earthquake. The seismic wave energyE ∞M ₀ τ ₀ ² f(v) _r , i. e.E is proportional to the seismic momentM ₀ and the square of the ambient shear stressτ ₀ ² and increases with the rupture velocityv _r . In frequency domain, integrating the square of source velocity spectrum derived from our the scaling law model, we have also obtained the seismic wave energyE released by earthquake and earthquake radiated effficiencyη.E ∞M ₀ τ ₀ ² also. If takingτ ₀ = 10.0 MPa, E=4.79M ₀. This result is consistent with the estimate by Vassiliou and Kanamori (1982). Theη=5.26%. The distribution of the seismic wave energy is that most of the energy contains in the frequency range between the first corner frequencyf _c1 and thirdf _c3, amount to 92.3% the energy in the rangef<f _c1 is about 3.85% and 3.85% whenf>f _c3. Thef _c3 is about 8Hz forM ⩾ 6, thus most of radiated energy is below 2Hz. This phenomenon had been verified by Vassiliou Kanamori. Previous results show the energy radiated by earthquake to be strongly dependent on ambient shear stress. The Chinese version of this paper appeared in the Chinese edition ofActa Seismologica Sinica,15, 146–152, 1993. This work was supported by the Deutsche Forschungsgemeinschaft, Bonn, F. R. Germany. The support is grateful acknowledged. The authors are also grateful to Professor Klussmann and Mr. Hasthoff for their lots of help.     

Characteristics of ambient stress values for micro－earthquake sequences in Western Yunnan Earthquake Prediction Experimental Field

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A probabilistic model of seismicity: Kamchatka earthquakes

The catalog of Kamchatka earthquakes is represented as a probability space of three objects {Ω, $ \tilde F $ \tilde F P}. Each earthquake is treated as an outcome ω _i in the space of elementary events Ω whose cardinality for the period under consideration is given by the number of events. In turn, ω _i is characterized by a system of random variables, viz., energy class k_i, latitude φ _i , longitude λ _i , and depth h _i . The time of an outcome has been eliminated from this system in this study. The random variables make up subsets in the set $ \tilde F $ \tilde F and are defined by multivariate distributions, either by the distribution function $ \tilde F $ \tilde F (φ, λ, h, k) or by the probability density f(φ, λ, h, k) based on the earthquake catalog in hand. The probabilities P are treated in the frequency interpretation. Taking the example of a recurrence relation (RR) written down in the form of a power law for probability density f(k), where the initial value of the distribution function f(k ₀) is the basic data [Bogdanov, 2006] rather than the seismic activity A ₀, we proceed to show that for different intervals of coordinates and time the distribution f _elim(k) of an earthquake catalog with the aftershocks eliminated is identical to the distribution f _full(k), which corresponds to the full catalog. It follows from our calculations that f ₀(k) takes on nearly identical numeral values for different initial values of energy class k ₀ (8 ≤ k ₀ ≤ 12) f(k ₀). The difference decreases with an increasing number of events. We put forward the hypothesis that the values of f(k ₀) tend to cluster around the value 2/3 as the number of events increases. The Kolmogorov test is used to test the hypothesis that statistical recurrence laws are consistent with the analytical form of the probabilistic RR based on a distribution function with the initial value f(k ₀) = 2/3. We discuss statistical distributions of earthquake hypocenters over depth and the epicenters over various areas for several periods     

An evidence for earthquake occurrence time related with geologic structure — Relations between local mean lunar times of earthquake occurrence and their fault trends

The relation between the local mean lunar time τ of earthquake occurrence and their fault trends is studied in this paper. The local mean lunar times τ of 53 earthquakes in 24 groups are calculated. Because the tidal generation force arisen by the moon is a cyclic function of about 12 hours 25 minutes in the main, the two tidal generation forces anywhere in the earth arising by the moon are equal in general when the moon lies to the two sites of 180° interval of local mean lunar time. Based on this phenomenon the values Δτ of τ₁–τ₂ or τ₁–τ₂ ± 180° of two earthquakes occurring repetitiously in the same place are also calculated. The calculated results show that if the fault trends of the two earthquakes in the same place is near, the Δτ is usually smaller and if the fault trends of the two ones is not near, the Δτ is usually larger and the distribution of the local mean lunar time τ of earthquakes in different places is dispersive even if fault trends of these earthquakes are near, and the τ does not concentrate on the lower and upper transit of the moon. The above phenomena clear up that the triggering earthquake of earth solid tide arisen by the moon is relative with the fault trends of earthquakes and we ought to think over the difference of environmental conditions of earthquake preparation of each seismogenic zone and can not make statistics to earthquakes in different places when we study the relation between the solid earth tide arisen by the moon and earthquakes.     

Alignment silkworms as seismic animal anomalous behavior (SAAB) and electromagnetic model of a fault: a theory and laboratory experiment

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尾波持续时间比τH/τV 在强震短临预报中的应用

本文根据Aki等人提出的尾波理论,导出了地方震尾波水平分量与垂直分量的持续时间比的具体表达式: τ_H/τ_V=I_H/I_V(Q_H/Q_V)~(1/4)·B_H/B_V。该式表明,τ_H/τ_V的变化主要反映了震源体一定范围内,由于介质的各向异性而引起的尾波在不同方向上的激发及衰减能力的差异。本文还讨论了地震前τ_H/τ_V短临异常的物理机制,认为异常的产生与孕震期间介质内裂隙的出现和闭合有关。1986年门源6.4级地震和1975年海城7.3级地震前,τ_H/τ_V都有不同程度的短期低值异常及临震高值突跳。门源地震前后门源台记录的直达S波的最大振幅比A_(mH)/A_(mv)也有与尾波持续时间比类似的异常。最后通过对一些震例的分析,初步得到震级与异常时间的关系为M=0.657lnT+3.44。     

Ambient tectonic shear stress field in Southern California and seismic hazard regions

According to the fracture mechanics rupture model of earthquakes put forward by us, several equations to compute tectonic ambient shear stress value τ₀ have been derived [equations (1), (2), (3), (5)]. τ₀ values for intermediate and small earthquakes occurred in Chinese mainland and Southern California have been calculated by use of these equations. The results demonstrate that the level and distribution of τ₀ are closely related to the location where large earthquakes will occur, i.e. the region with higher level of τ₀ will be prone to occur large earthquakes and the region with lower level will usually occur small earthquakes. According to the spatial distribution of τ₀, the seismic hazard regions or the potential earthquake source regions can in some degree be determined. According to the variation of τ₀ with time, the large earthquake occurrence time can be roughly estimated. According to the distribution of τ₀ in Southern California and variation with time, three high stress level regions are determined, one (Goldfield area) of them is the present seismic hazard region. Contribution No. 98A02023, Institute of Geophysics, State Seismological Bureau, China     

Measuring nonlinear dependence in hydrologic time series

It has been a common practice to employ the correlation dimension method to investigate the presence of nonlinearity and chaos in hydrologic processes. Although the method is generally reliable, potential limitations that exist in its applications to hydrologic data cannot be dismissed altogether. As for these limitations, two issues have dominated the discussions thus far: small data size and presence of noise. Another issue that is equally important, but less discussed in the literature, is the selection of delay time (τ _d ) for reconstruction of the phase-space, which is an essential first step in the correlation dimension method, or any other chaos identification and prediction method for that matter. It has also been increasingly recognized that fixing the delay time window (τ _w ) rather than just the delay time itself could be more appropriate, since the delay time window is the one that is of actual interest at the end to represent the dynamics. To this effect, Kim et al. (1998a) [Phys Rev E 58(5):5676–5682] developed a procedure for fixing the delay time window and demonstrated its effectiveness on three artificial chaotic series, and followed it up with the development of the C–C method to estimate both the delay time and the delay time window. The purpose of the present study is to test this procedure on real hydrologic time series and, hence, to assess their nonlinear deterministic characteristics. Three hydrologic time series are studied: (1) daily streamflow series from St. Johns near Cocoa, FL, USA; (2) biweekly volume time series from the Great Salt Lake, UT, USA; and (3) daily rainfall series from Seoul, South Korea. The results are also compared with those obtained using the conventional autocorrelation function (ACF) method.     

The space and time distribution characteristics of the shear stress field for the sequence of the Wuding earthquake

Follow Chen and Duda's model of spectral fall-off of (3, the dependence of peak parameters of ground motion, peak displacement dm, peak velocity vm and peak acceleration am, upon the environment stress (0-values are studied using near source seismic digital recordings for the sequence of the Wuding, Yunnan, M = 6.5 earthquake, in which, as a new thought, the peak parameters are assumed to be related to the medium Q-value. Three formulae for estimating the environment stress (0-values by the peak parameters of three types of ground motions are derived. Using these formulae, the environment stress (0-values are calculated for the sequence of the Wuding earthquake. The result show that (0-values calculated by the three formulae are constant largely, the averages of (0 are in the range of 5.0～35 MPa for most earthquakes. It belongs to the high-stress earthquakes sequence: the high-stress values are restricted to the relatively small area closely near to the epicenter of the main shock. The fine distribution structure for the contours of the environment stress (0-values is related closely to the strong aftershocks. The analysis of spatial and temporal feature of (0-values suggests that the earthquakes sequence in a rupture process generated at the specific intersection zone of seismo-tectonics under high-stress background.     

Magnitude scaling relationships from the first 3 s of P-wave arrivals in South Korea

Two empirical magnitude scaling relationships, predominant period (t_p^max{\tau _p^{\max}}) and peak ground displacement (Pd) magnitudes, were investigated for the first 3 s after P-wave arrivals using 1,412 vertical waveforms recorded by the Korea National Seismic Network (KNSN) between 2001 and 2007. To evaluate the accuracy of the derived magnitude relationships, we simulated off-line ElarmS tests using 65 events occurring inside the KNSN. While the average magnitude error was ∼0.70 magnitude units when using only the closest station to the epicentre, the error dropped to ∼0.62 and ∼0.42 magnitude units when using the closest two and closest four stations, respectively. For events M _L ≥ 3.0, the average magnitude error was ∼0.33 and showed stable values when the closest four stations were available. Our magnitude scaling relationships may be useful for initial work in developing an earthquake early warning system in South Korea.     

1983–2003 decaying rate of deflation at Askja caldera: Pressure decrease in an extensive magma plumbing system at a spreading plate boundary

New deformation data from the Askja volcano, Iceland, show that the volcano's caldera has been deflating continuously for over 20 years, and confirm that the rate of subsidence is slowing down. The decay in subsidence rate can be fitted with a function of the form e ^{− t / τ} , where τ is 39 years. Reanalysis of GPS data from 1993–1998 show that these data can be fitted with a model calling for two Mogi point sources, one shallow, and another one much deeper (16.2 km depth). Pressure decrease occurs in both sources. The deeper source is responsible for observed horizontal contraction towards Askja at distances that cannot be explained by the shallower source. Plate spreading of 19 mm/year distributed evenly over about 100-km-wide zone is also favoured by the data.     

Effective relaxation time and viscosity of the earth inferred from the postseismic vertical deformations of the 1990 M S=7.0 Gonghe earthquake

The postseismic vertical deformation rates of the 1990 Gonghe M _S=7.0 earthquake appears to have decreased exponentially. Based on Okada’s coseismic surface displacement solution caused by a uniform fault slip in an elastic homogeneous half space, we derived its postseismic surface displacement by using a single-layer standard linear solid model, and further derived a simplified formula for determining the effective relaxation time and viscosity of the earth, which is independent of the dislocation parameters of the causative fault. From the postseismic vertical deformation of the 1990 Gonghe earthquake, we inferred that the effective relaxation time defined by τ=η/μ is 2.6 years, and the effective viscosity η is about 10¹⁸ Pa · s. This work was supported by Chinese Joint Seismological Science Foundation under the grants 92088 and 196098.