Fourier amplitude spectra of strong motion acceleration: Extension to high and low frequencies

It is shown how the empirical equations for scaling the Fourier amplitude spectra in the frequency band from ～0.1 to 25 Hz can be extended to describe the strong motion amplitudes in a much broader frequency range. At long periods, the proposed equations are in excellent agreement with (1) the seismological and field estimates of permanent ground displacement (near field) and (2) the independent estimates of seismic moment (far field). At high frequencies, f ≥ 25 Hz, the spectral amplitudes can be described by exp (? πkf), where k ranges from 0·02 (near source) to about 0·06 at an epicentral distance of about 200 km. It is also shown how amplification by local soil and geological site conditions can be defined to apply in the same broad frequency range.     

<Emphasis Type="Italic">Q</Emphasis>-type bursts in magnetic oscillations of ELF range as a phenomenon reflecting the properties of the global Earth-ionosphere resonator

The observation results of Q-type bursts in the measurements of the horizontal component of the noise magnetic field in the range of the first Schumann resonance in polar regions (Lovozero high-latitude observatory) are presented. Automatic selection of Q-type bursts from the experimental data series is implemented on the basis of a waveform recognition algorithm. The resonant nature of Q-bursts is shown. The possibility of selecting such events in magnetic excitation conditions is highlighted. The global resonator quality upon decreasing the selected waveform amplitudes is estimated. The data obtained by this method are compared with estimates on the basis of Fourier analysis and values known from the world literature. The possible reasons for disagreements of the estimates are analyzed, including the problem of selecting the function approximating the spectrum, the problem of accounting for the background, and the possible irregularity of the spectrum. It is shown that Q-type bursts, besides the quality, allow estimating the resonant frequency of the first oscillation mode; however, the accuracy of such estimation is lower as compared to the results of Fourier analysis methods.     

Q and high-frequency strong motion spectra

Empirical scaling equations for Fourier amplitude spectra of strong ground motion are used to describe A₀ and τ in the assumed (high-frequency) shape of strong motion amplitudes: FS(_φ) = A₀e-^πτφ. The res of computed A₀ and τ with other related estimates of spectral amplitudes; (2) smooth decay of strong motion spectral amplitudes up to φ = 25 Hz, without an abrupt low-pass filtering of high frequecies; and (3) good agreement with other estimates of the regionally specific attenuation of high-frequncy seismic waves.As the recorded strong earthquake shaking in the western United States typically samples only the shallow (10 km) and local (100km) characteristics of wave attenuation, the processed strong motion accelerograms can be used as the most direct means of describing the nature of the high-frequency attenuation of the entire strong motion signal for use in earthquake engineering applications. Seismological body wave, Lg and coda wave estimates of Q sample different volumes of the crust surrounding the station, and involve different paths of the waves. These differences must be carefully documented and understood before the results can be used in earthquake engineering characterization of strong motion amplitudes.     

DEPTH ESTIMATION OF MAGNETIC SOURCES BY MEANS OF FOURIER AMPLITUDE SPECTRA*

The Fourier transform of a square-shaped section of a magnetic survey, digitized in a square grid, forms a rectangular matrix of coefficients which can be condensed to a series of average amplitudes dependent only on their frequency and no longer on the direction of the respective partial waves. These average amplitudes together represent a spectrum which–when plotted in a semilogarithmic coordinate system (log amplitude versus frequency)–often shows straight segments which decrease with increasing frequency. By continuing the given field downwards these straight segments become horizontal at a certain depth, the so-called “white depth”. This white depth may be used as a first estimate for the depth of magnetic sources producing the respective part of the field. It is shown that the sources which correspond to such use of the white depth can be expected to be “randomly distributed with some positive autocorrelation”. As an example for such a depth estimation the interpretation of the aeromagnetic survey of NW-Germany by a relief in 8–16 km depth is given. The relief divides the subsurface in an upper nonmagnetic layer and a lower layer with magnetization M= 2 Am^?1.     

VELOCITY-STACK PROCESSING1

A conventional velocity-stack gather consists of constant-velocity CMP-stacked traces. It emphasizes the energy associated with the events that follow hyperbolic traveltime trajectories in the CMP gather. Amplitudes along a hyperbola on a CMP gather ideally map onto a point on a velocity-stack gather. Because a CMP gather only includes a cable-length portion of a hyperbolic traveltime trajectory, this mapping is not exact. The finite cable length, discrete sampling along the offset axis and the closeness of hyperbolic summation paths at near-offsets cause smearing of the stacked amplitudes along the velocity axis. Unless this smearing is removed, inverse mapping from velocity space (the plane of stacking velocity versus two-way zero-offset time) back to offset space (the plane of offset versus two-way traveltime) does not reproduce the amplitudes in the original CMP gather. The gather resulting from the inverse mapping can be considered as the model CMP gather that contains only the hyperbolic events from the actual CMP gather. A least-squares minimization of the energy contained in the difference between the actual CMP gather and the model CMP gather removes smearing of amplitudes on the velocity-stack gather and increases velocity resolution. A practical application of this procedure is in separation of multiples from primaries. A method is described to obtain proper velocity-stack gathers with reduced amplitude smearing. The method involves a t²-stretching in the offset space. This stretching maps reflection amplitudes along hyperbolic moveout curves to those along parabolic moveout curves. The CMP gather is Fourier transformed along the stretched axis. Each Fourier component is then used in the least-squares minimization to compute the corresponding Fourier component of the proper velocity-stack gather. Finally, inverse transforming and undoing the stretching yield the proper velocity-stack gather, which can then be inverse mapped back to the offset space. During this inverse mapping, multiples, primaries or all of the hyperbolic events can be modelled. An application of velocity-stack processing to multiple suppression is demonstrated with a field data example.     

Preliminary empirical model for scaling fourier amplitude spectra of strong ground acceleration in terms of modified mercalli intensity and recording site conditions

This paper presents an empirical model for scaling Fourier amplitude spectra of ground acceleration during strong earthquake shaking in terms of the reported Modified Mercalli Intensity (MMI) and the simplified characteristics of the geologic environment at the recording station. This analysis shows that (i) for the intermediate and high-frequency motions the spectral amplitudes approximately double for every level of the MMI; that (ii) the uncertainties associated with estimation of Fourier spectral amplitudes in terms of MMI are not greater than the uncertainties associated with similar estimation in terms of earthquake magnitude and epicentral distance; that (iii) the high frequency spectral amplitudes tend to be greater on basement rock sites relative to alluvium sites, with this trend being reversed for the low-frequency spectral amplitudes; and that (iv) the spectral amplitudes of very high-frequency vertical shaking are equal to or higher than the corresponding spectral amplitudes for horizontal shaking.     

两相饱和介质中的集中力点源位移场解与应用

通过两相饱和介质Ｂｉｏｔ方程变换,利用势场分解较好地解决了饱和两相介质中Ｐ１和Ｐ２波的耦合问题．再由Ｐｏｉｓｓｏｎ方程和Ｈｅｌｍｈｏｔｚ方程的特性,求解得到两相饱和介质在集中力点源作用下的位移场Ｇｒｅｅｎ函数,进而通过阶跃函数和Ｆｏｕｒｉｅｒ逆变换,求得两相介质在集中力．作用下的波场．由上述结果,根据扩容进水模型,合理地解释了孕震前各主要阶段在地震记录中垂直向振幅与振幅比统计值的变化．最后结合工作实例,利用两相介质波场理论求得由一般弹性介质理论不易求得的骨架固体力学参数．     

Spectral amplitudes of strong earthquake accelerations recorded in buildings

Fourier spectrum amplitudes of horizontal and vertical earthquake accelerations recorded at the foundation levels of 57 buildings in the Los Angeles metropolitan area have been used to study the dependence of spectral amplitudes on the building foundation sizes. Comparison of these amplitudes with those predicted by empirical models for scaling ‘free field’ Fourier amplitude spectra does not indicate any significant dependence of the spectral amplitudes on the size of the foundation. Third degree polynomials have been employed to smooth the spectra of the accelerations recorded inside the buildings and their coefficients have been examined as functions of the foundation plan dimensions. These results also indicate no significant dependence of the spectral amplitudes on the foundation dimensions. A qualitative analysis of the spectral amplitudes for possible effects caused by the phenomena associated with soil-structure interaction indicates that the Fourier spectra of the recorded accelerations may experience some amplification as the relative ‘density’ of the foundation-structure system increases.     

A note on distribution of amplitudes of peaks in structural response including uncertainties of the exciting ground motion and of the structural model

This note is an extension of earlier works that presented probability distribution functions for amplitudes of the peaks (the highest, the second highest … the m-th highest) in response of deterministic single degree-of-freedom (SDOF) and multi degree-of-freedom (MDOF) structures to ground motion, with deterministic Fourier spectrum and duration. It shows how these probability distribution functions can be evaluated if the Fourier spectrum and duration of the excitation are random variables specified via distribution functions. Two cases are considered: (l) when the structural model is deterministic, and (2) when the modal frequencies are random variables. The procedure presented here approximates the transfer function of the structural response by Dirac delta functions at the modal frequencies, and is applicable to multi-storey buildings with small modal damping, and with natural frequencies that are not too close. The resulting probability distribution functions are needed in seismic hazard calculations of peak response amplitudes of SDOF and MDOF structures that will not be exceeded with given confidence during the service time of the structure from any earthquake at all known faults within certain distance from the structure.     

Reconstruction of the harmonic component of the magnetic field modulus anomalies

An algorithm is proposed for calculating a harmonic function equal to the projection of the anomalous magnetic field vector onto the normal field direction from in situ measurements of the anomalous magnetic field modulus (the scalar magnetic anomaly) ΔT, which is a nonharmonic function and is nonlinearly related to the magnetization of anomaly sources. It is shown that the inferred estimates tend to the desired harmonic function if the iterative algorithm converges. The convergence conditions and stability of the process are studied numerically in a wide range of amplitudes of the anomalous field. The results of the modeling simulation demonstrate the efficiency of the proposed algorithm in solving magnetic field interpretation problems often encountered in practice.     

MAKING AVO SECTIONS MORE ROBUST1

When large quantities of seismic data are involved it is impossible to examine all gathers by eye for AVO anomalies. The standard approach is to compute, for each amplitude profile (at a specific time) on each gather, the intercept and gradient of a straight-line fit to seismic amplitudes. These intercepts and gradients are each plotted as a sort of seismic section - an intercept section, and a gradient section. Estimation of the intercept and gradient for a straight-line fit to each amplitude profile proceeds traditionally via least-squares. Two undesirable features can be hidden from the user by the fitting procedure, namely (i) the effect of outlying or uncharacteristic amplitudes on the intercept and gradient estimates, and (ii) complete breakdown of the straight-line model for the amplitudes, thus rendering meaningless the intercept and gradient estimates. It should be remembered that least-squares can always fit any sequence of numbers to any other sequence of numbers; checks are needed to show that the result is meaningful. It is shown that statistically robust estimation methods can greatly limit the damage done by outlying amplitudes, and that a simple test on the model, the runs-statistic, is capable of detecting breakdown of the straight-line assumption. It is demonstrated using two seismic data sets that these two techniques, used in tandem, facilitate much better quality control of AVO intercept and gradient calculations.     

Yield Estimation from Surface-wave Amplitudes

— Surface-wave amplitudes from explosion sources show less variation for a given event han body wave amplitudes, so it is natural to expect that yield estimates derived from surface waves will be more accurate than yield estimates derived from body waves. However, yield estimation from surface waves is complicated by the presence of tectonic strain release, which acts like one or more earthquake sources superimposed on top of the explosion. Moment-tensor inversion can be used to remove the tectonic component of the surface waves, however moment-tensor inversion for shallow sources is inherently non-unique so the explosion isotropic moment cannot be determined with the necessary accuracy by this means. Explosions on an island or near a mountain slope can exhibit anomalous surface waves similar to those caused by tectonic strain release. These complications cause yield estimates derived from surface waves to be less accurate than yield estimates from body waves recorded on a well-calibrated network with good coverage. Surface-wave amplitudes can be expressed as a surface-wave magnitude M _s , which is defined as the logarithm of the amplitude plus a distance correction, or as a path corrected spectral magnitude, log $M^{\prime}_0$ , which is derived from the surface-wave spectrum. We derive relations for M _s vs. yield and log $M^{\prime}_0$ vs. yield for a large data set and estimate the accuracy of these estimates.     

Experimental results on the tidal resonance effect of the liquid core

Abstract

Molodenskii's theory on the resonance effect of the liquid core predicts anomalous amplitudes in the diurnal spectrum for earth tide registrations of gravity and tilt. A two years recording of an Askania gravimeter at Kiel University has been analysed using three different numerical methods: harmonic analysis (least squares method), gliding harmonic analysis, and Fourier spectral analysis. Because of difficulties in resolving tidal waves near the resonance frequency, efforts were made towards improving the Fourier spectral analysis by successive subtraction of high amplitudes in the Fourier spectrum. The resonance of the liquid core has been observed with a high degree of confidence thereby supporting Molodenskii's model calculations.     

Sq指数的周期变化分析

利用1960-1980年中国北京和广州的地磁场X分量小时值数据,根据徐文耀(1992)提出的用来描述每日Sq变化幅度的地磁活动指数的方法,计算并分析了Asq指数的周期变化特征.结果表明:Asq指数具有11年、年和半年变化等主要周期成分,与F107指数傅氏谱主要周期成分存在对应关系,表明Asq指数的周期变化与太阳辐射密切相关.然而互相关分析表明,日变幅dSq与F107的相关关系略强,这是由于Asq指数计算中每月平均ΔS_q(t)携带了部分Sq场的周期变化信息.     

Geomagnetic field effects of the Chelyabinsk meteoroid

An analysis was conducted of time variations in geomagnetic field components on the day of the Chelyabinsk meteorite event (February 15, 2013) and on control days (February 12 and 16, 2013). The analysis uses the data collected by magnetic observatories in Novosibirsk, Almaty, Kyiv, and Lviv. The distance R from the explosion site to the observatories varies in the range 1200–2700 km. The flyby and explosion of the Chelyabinsk cosmic body is found to have been accompanied by variations mainly in the horizontal component of the geomagnetic field. The variations are quasi-periodic with a period of 30–40 min, an amplitude of 0.5–2 nT for R ≈ 2700?1200 km, respectively, and a duration of 2–3 h. The horizontal velocity of the geomagnetic field disturbances is close to 260–370 m/s. A theoretical model of wave disturbances is proposed. According to the model, wave disturbances in the geomagnetic field are caused (a) by the motion of the gravity wave generated in the atmosphere by the falling space body and (b) by traveling ionospheric disturbances, which modulate the ionospheric current at dynamo altitudes. The calculated amplitudes of the wave disturbances are 0.6–1.8 nT for R ≈ 2700?1200 km, respectively. The estimates are in good agreement with the observational data. Disturbances in the geomagnetic field level (geomagnetic pulsations) in the period range 1–1000 s are negligible (less than 1 nT).     

Simulation of digital earthquake accelerograms using the inverse discrete Fourier transform

Acceleration time histories of horizontal earthquake ground motion are obtained by inverting the discrete Fourier transform, which is defined by modelling the probability distribution of the Fourier phase differences conditional on the Fourier amplitude. The Fourier amplitude spectrum is modelled as a scaled, lognormal probability density function. Three parameters are necessary to define the Fourier amplitude spectrum. They are the total energy of the accelerogram, the central frequency, and the spectral bandwidth. The Fourier phase differences are simulated conditional on the Fourier amplitudes. The amplitudes are classified into three categories: small, intermediate and large. For each amplitude category, a beta distribution or a combination of a beta distribution and a uniform distribution are defined for the phase differences. Seven parameters are needed to completely define the phase difference distributions: two for each of the three beta distributions, and the weight of the uniform distribution for phase differences corresponding to small Fourier amplitudes. Approximately 300 uniformly processed horizontal ground motion records from recent California earthquakes are used to develop prediction formulas for the model parameters, as well as to validate the simulation model. The moment magnitude of the earthquakes ranges from 5.8 to 7.3. The source to site distance for all the records is less than 100 km. Copyright © 2002 John Wiley & Sons, Ltd.     

Duration of earthquake fault motion in California

Duration of high frequency (5–25 Hz) radiation of energy from earthquake sources in California is consistent with the estimates of fault length and with dislocation velocity estimates of 2–3 km/sec. This duration can be described by an exponential function of magnitude for 2·5 < M < 7·5. It is related to the times it takes the dislocation to spread over the fault width (1/f₂), and the fault length (～ 1/f₁), and to reach its ultimate amplitude (T₀). The results in this paper can be used to estimate the range of amplitudes and the duration of long period pulses of strong ground motion near faults, as these long period pulses can be related to the properties of high-frequency radiation from the source. Such pulses must be considered in the analyses of yielding structures, when the average peak acceleration of the pulse exceeds the yield resistance seismic coefficient of the structure.     

引力场球谐综合的Walsh变换方法

提出直接在序率域内用Walsh变换实现引力场球谐综合的问题。给出球谐函数展开式的Walsh变换及快速算法,讨论了Walsh变换和Walsh-Fourier变换、Fourier变换之间的差异,分析了用地球重力场模型OSU81的位系数作出的Walsh变换和Fourier变换的结果。研究表明:Walsh变换与Walsh-Fourier变换、Fourier变换对应向量在数量方面的差值通常都小于士10~(-5);对于给定的阶数和飞行高度,3种方法求得的球谐综合值总是完全一致的;1°×1°等网格数据和Walsh函数形状相近。在重力场研究中Walsh级数会比Fourier级数收敛得更快;Walsh变换在计算速度、计算准确度、数据储存、收敛速度和方法简单方面都好于Fourier变换。     

Intra-seasonal monthly oscillations in stratospheric NCEP data and model results

Intra-seasonal oscillations (ISO) are observed in the zonal-mean of mesospheric wind and temperature measurements—and the numerical spectral model (NSM) generates such oscillations. Relatively large temperature ISO are evident also in stratospheric CPC (NCEP) data at high latitudes, where the NSM produces amplitudes around 3 K at 30 km. Analyzing the NCEP data for the years 1996–2006, we find in Fourier spectra signatures of oscillations with periods between 1.7 and 3 months. With statistical confidence levels exceeding 70%, the spectral features are induced by nonlinear interactions involving the annual and semi-annual variations. The synthesized data show for the 10-year average that the temperature ISO peak in winter, having amplitudes close to 4 K. The synthesized complete spectrum for periods around 2 months produces oscillations, varying from year to year, which can reach peak amplitudes of 15 and 5 K respectively at northern and southern polar latitudes.     

Order statistics of functionals of strong ground motion for a class of renewal processes

In this paper, probability distribution functions are derived for the order statistics of various functionals of strong ground motion at a site. These functionals can be: Modified Mercalli Intensity (MMI), peak ground acceleration (PGA), Fourier spectral amplitudes of acceleration, response spectrum amplitudes (spectral displacement, pseudo-spectral velocity and pseudo-spectral acceleration), and amplitudes of the peaks (local maxima and local minima) in the time historyof the response of SDOF and MDOF structures at the site. Three parameters of the response of a structure are considered: displacement, shear force and bending moment at each level (storey) of the structure. The earthquake sources contributing to the risk of ground motion at the site are a number of point, area or volume sources, each with defined frequency of occurence-magnitude relationship. The magnitudes of the possible events at these sources are discretized, and the occurrence of events of different magnitudes are assumed to be statistically independent. For each magnitude, it is assumed that the eartquakes occur in a Poissonian sequence or in a renewal process which is a generalization of the Poissonian. For these assumptions, the probability distribution functions are presented for the number of earthquakes, n, during which a given level of site or structural response is exceeded during the exposure time, and for the return period of the exceedances. For example, for single-degree- of-freedom: (SDOF) or multi-degree-of-freedom structures, (MDOF) n can be the number of earthquakes during which the response of a storey will exceed a given level at least m times(m = 1, 2, 3,…) during the exposure time. These probability distribution functions can be used to extend the concept of uniform probability functionals to more than one exceedance. A more important application is to generalize the uniform probability functionals method of site response (uniform probability Fourier or response spectra) to uniform probability envelopes of displacement, shears and bending moments of a given structure. The uniform probability envelopes can be for exceedance at least once during at least one earthquake, or, in general, for exceedance at least m times per earthquake (m = 1, 2,…) during at least n earthquakes. In other words, during at least n earthquakes at least m peaks in the response can be higher than the specified level. Such uniform probability envelopes can be used (1) to define new design guidelines for building codes based on cost-benefit analysis; (2) to construct more refined probability distribution functions for the damage and total economic losses caused by earthquakes; and (3) to develop planning and decision strategies on strengthening and retrofitting existing buildings.