Intermittent behaviour of volcanic tremor at Mt. Etna

Harmonic tremor is widely studied and modelled in a very narrow frequency band (1–5 Hz) which represents the eigenfrequencies of a resonator assumed as the source of the phenomenon. Minimal effort was dedicated towards understanding its behaviour in larger temporal scales. Here we characterise the dynamic behaviour of volcanic tremor while evaluating the complete spectrum of the generalised dimension of the phase space. The starting time series constitutes the tremor amplitude picked every 10 minutes. The choice of this lag time is made on the basis of a qualitative analysis of the properties of the tremor. The results show intermittent behaviour of the dynamics which requires an 8-dimensional map to be completely described. An interesting result is that the maximum clustering of point density in phase space occurs in a monodimensional space which implies a periodicity sometimes observed experimentally. An appropriate predictive model needs more constraints on the nature of the eight variables involved in the process.     

A NEW APPROACH TO VIBROSEIS DECONVOLUTION*

A method is proposed to obviate the shortcomings of conventional deconvolution approaches applied to vibroseis data. The vibroseis wavelet reduces the time domain resolution of the earth's impulse response by restricting its passband. The spectrum of the wavelet is assumed to be a “low quefrency”phenomenon, and hence it can be estimated by low cut cepstral filtering. The wavelet's amplitude spectrum can then be removed by spectral division. By using an approach which is consistent with the principle of maximum entropy, the undetermined portions of the seismogram's Fourier transform can be filled in by autoregressive prediction. The process of initially deconvolving in a restricted passband reduces the enhancement of noise contaminated parts of the spectrum, and the spectral extension scheme increases the time domain resolution of the process.     

An application of the autoregressive extrapolation technique to enhance deconvolution results: a 2D marine data example

Interpreting a post‐stack seismic section is difficult due to the band‐limited nature of the seismic data even post deconvolution. Deconvolution is a process that is universally applied to extend the bandwidth of seismic data. However, deconvolution falls short of this task as low and high frequencies of the deconvolved data are either still missing or contaminated by noise. In this paper we use the autoregressive extrapolation technique to recover these missing frequencies, using the high signal‐to‐noise ratio (S/N) portions of the spectrum of deconvolved data. I introduce here an algorithm to extend the bandwidth of deconvolved data. This is achieved via an autoregressive extrapolation technique, which has been widely used to replace missing or corrupted samples of data in signal processing. This method is performed in the spectral domain. The spectral band to be extrapolated using autoregressive prediction filters is first selected from the part of the spectrum that has a high signal‐to‐noise ratio (S/N) and is then extended. As there can be more than one zone of good S/N in the spectrum, the results of prediction filter design and extrapolation from three different bands are averaged. When the spectrum of deconvolved data is extended in this way, the results show higher vertical resolution to a degree that the final seismic data closely resemble what is considered to be a reflectivity sequence of the layered medium. This helps to obtain acoustic impedance with inversion by stable integration. The results show that autoregressive spectral extrapolation highly increases vertical resolution and improves horizon tracking to determine continuities and faults. This increase in coherence ultimately yields a more interpretable seismic section.     

Statistical analyses and characteristics of volcanic tremor on Stromboli Volcano (Italy)

 A study of volcanic tremor on Stromboli is carried out on the basis of data recorded daily between 1993 and 1995 by a permanent seismic station (STR) located 1.8 km away from the active craters. We also consider the signal of a second station (TF1), which operated for a shorter time span. Changes in the spectral tremor characteristics can be related to modifications in volcanic activity, particularly to lava effusions and explosive sequences. Statistical analyses were carried out on a set of spectra calculated daily from seismic signals where explosion quakes were present or excluded. Principal component analysis and cluster analysis were applied to identify different classes of spectra. Three clusters of spectra are associated with two different states of volcanic activity. One cluster corresponds to a state of low to moderate activity, whereas the two other clusters are present during phases with a high magma column as inferred from the occurrence of lava fountains or effusions. We therefore conclude that variations in volcanic activity at Stromboli are usually linked to changes in the spectral characteristics of volcanic tremor. Site effects are evident when comparing the spectra calculated from signals synchronously recorded at STR and TF1. However, some major spectral peaks at both stations may reflect source properties. Statistical considerations and polarization analysis are in favor of a prevailing presence of P-waves in the tremor signal along with a position of the source northwest of the craters and at shallow depth. Received: 15 December 1996 / Accepted: 31 March 1998     

Depth Estimation from Gravity Data Using the Maximum Entropy Method (MEM) and the Multi Taper Method (MTM)

The depth determination from the gravity data in frequency domain is carried out using the classical fast Fourier transform (FFT) method utilizing scaling properties of ensemble of anomalous source. The problem of calculating power spectrum from the FFT is well described in the literature. Here, the application of other high-resolution methods of power spectrum calculation, such as maximum entropy method (MEM) and multi-taper method (MTM) are explored to estimate depth to anomalous sources. At the outset, the FFT, the MEM and the MTM are tested on synthetic gravity data, generated for different types of synthetic models and then all these methods are applied to the field gravity data of the Bengal basin. The MTM with scaling is found to be superior for providing the detailed subsurface information rather than the MEM and the FFT methods in the case of synthetic as well as field examples.     

Stochastic considerations in seismic analysis of structures

A method is presented for stochastic modelling of a design earthquake by a power spectral density function for seismic analysis of structures. The method can be adopted with information currently available in the form of design response spectra for earthquake motion. Accurate seismic responses of structures can be easily obtained using such stochastic models. The methods for accurate response analysis of structures with closely spaced modes and for generation of floor response spectra of a building using a prescribed ground response spectrum directly are also presented. The hypothesis that a design earthquake can be modelled by a power spectral density function is used only implicitly in developing these methods.     

Non‐stationary seismic response of tanks with soil interaction by wavelets

A wavelet‐based random vibration theory has been developed for the non‐stationary seismic response of liquid storage tanks including soil interaction. The ground motion process has been characterized via estimates of statistical functionals of wavelet coefficients obtained from a single time history of ground accelerations. The tank–liquid–soil system has been modelled as a two‐degree‐of‐freedom (2‐DOF) system. The wavelet domain equations have been formulated and the wavelet coefficients of the required response state are obtained by solving two linear simultaneous algebraic equations. The explicit expression for the instantaneous power spectral density function (PSDF) in terms of the functionals of the input wavelet coefficients has been obtained. The moments of this PSDF are used to estimate the expected pseudo‐spectral acceleration (PSA) response of the tank. Parametric variations are carried out to study the effects of tank height, foundation natural frequency, shear wave velocity of soil and ratio of the mass of tank (including liquid) to the mass of foundation on the PSA responses of tanks. Copyright © 2001 John Wiley & Sons, Ltd.     

IMPROVED LOW-FREQUENCY DECAY ESTIMATION USING THE MULTITAPER SPECTRAL ANALYSIS METHOD1

Spectral analysis is one of the most ubiquitous signal processing tools used in exploration geophysics. Among many applications, it is used simply to look at the frequency content of seismic traces, to find notches, to estimate wavelets under the minimum-phase assumption, and to match broadband synthetic seismograms to seismic data. Seismic spectra exhibit very large dynamic ranges, particularly at low frequencies. Estimation of low-frequency decay is very important for accurate modelling. However, when using traditional spectral estimates incorporating smoothing windows, too much sidelobe energy leaks from high power into low power areas, spoiling our ability to estimate low-frequency spectral decay. The multitaper method of spectral analysis due to D. Thomson does not employ just a single window, but rather a set of orthogonal data tapers. It is possible to have much less sidelobe contamination, while maintaining a stable estimate. The trace is tapered by each of a subset of the orthogonal tapers, and a raw spectral estimate produced in each case. These are combined to produce a final spectral estimate. The technique can be made adaptive by applying different weights to the different raw spectra at different frequencies. A comparison of seismic spectral estimation using this multitaper technique with a traditional approach having the same analysis bandwidth and stability demonstrates the very different estimates of spectral decay in the areas of high dynamic range. The multitaper approach provides estimates with much reduced sidelobe leakage, and hence is a very appealing method for reflection seismology.     

A note on autoregressive spectral estimates for frequency-wavenumber analysis of strong-motion array data

Frequency-wavenumber (f-k) spectra of seismic strong-motion array data are useful in estimating back-azimuth and apparent propagation velocity of seismic waves arriving at the array. Such estimates are required to model wave passage effects while studying spatial variability of strong ground motion. Although periodogram-based spectral estimates are commonly used, practical applications based on them encounter limitations, such as, lack of objective criteria for selecting a proper smoothing window and its associated bandwidth, and relatively large variance of the estimated spectral quantities. We present an alternative spectral estimate based on parametric time series modelling approach. The well-known autoregressive (AR) time series model is used in a system-based approach to estimate the spectral matrix of auto- and cross-spectral densities. Such spectral estimates are found to be smoother than the windowed periodogram estimates, and can directly be used in f-k spectral analysis. We present an example application of the proposed technique using strong-motion data recorded by the SMART-1 array in Taiwan during the January 29 1981 $M_{L}$ 6.3 earthquake. Our results, in terms of back azimuth and apparent propagation velocity, are found to be in excellent agreement with those reported in the literature.     

Stochastic characterization of earthquakes through their response spectrum

If earthquakes are modelled by a stochastic process, it is possible to interpret the associated response spectrum in terms of the statistics of extreme values of oscillator response to the process. For a stationary earthquake model this interpretation leads to a relationship between the power spectral density function of the process and the response spectrum. This relationship is examined in this paper and forms the basis for two methods presented to obtain the power spectrum of the earthquake process from its response spectrum. One of these methods is approximate but leads to an explicit representation of the power spectral density function in terms of the response spectrum. The other method is exact wherein an iterative scheme for the solution of the problem is established. An example problem is solved to illustrate the use of the two methods and it is shown that for small values of damping, the approximate derivation yields a fairly accurate solution.     

Observations of volcanic tremor during the January–February 2005 eruption of Mt. Veniaminof,Alaska

Mt. Veniaminof, Alaska Peninsula, is a stratovolcano with a summit ice-filled caldera containing a small intracaldera cone and active vent. From January 2 to February 21, 2005, Mt. Veniaminof erupted. The eruption was characterized by numerous small ash emissions (VEI 0 to 1) and accompanied by low-frequency earthquake activity and volcanic tremor. We have performed spectral analyses of the seismic signals in order to characterize them and to constrain their source. Continuous tremor has durations of minutes to hours with dominant energy in the band 0.5–4.0 Hz, and spectra characterized by narrow peaks either irregularly (non-harmonic tremor) or regularly spaced (harmonic tremor). The spectra of non-harmonic tremor resemble those of low-frequency events recorded simultaneously with surface ash explosions, suggesting that the source mechanisms might be similar or related. We propose that non-harmonic tremor at Mt. Veniaminof results from the coalescence of gas bubbles while low-frequency events are related to the disruption of large gas pockets within the conduit. Harmonic tremor, characterized by regular and quasi-sinusoidal waveforms, has duration of hours. Spectra containing up to five harmonics suggest the presence of a resonating source volume that vibrates in a longitudinal acoustic mode. An interesting feature of harmonic tremor is that frequency is observed to change over time; spectral lines move towards higher or lower values while the harmonic nature of the spectra is maintained. Factors controlling the variable characteristics of harmonic tremor include changes in acoustic velocity at the source and variations of the effective size of the resonator.     

The objective determination of the instantaneous predominant frequency of seismic signals and inferences onQ of coda waves

A technique to detect spectrum variations versus time along seismic signals is applied to coda waves of local earthquakes (Friuli, Northern Italy). The technique consists of an autoregressive modeling and utilizes nonlinear spectral analysis where the spectrum of stochastic processes is estimated as the transfer function of the filter that whitens the process under analysis. This approach appears to be particularly well suited to those investigations where automatic measurements of the instantaneous frequency have to be carried out on digital data. The detection of variations of the instantaneous frequency along the coda allows computation of seismic-Q in the lithosphere and its frequency dependence: the result obtained is $$Q = 100f^{0.4} $$ which appears to be strongly consistent with that, based on the estimate of the coda amplitude decay in the band including the most significant frequencies of the signals under analysis.     

A time varying AR coefficient model for modelling and simulating earthquake ground motion

A smoothness priors-time varying autoregressive (AR) coefficient model method for the modelling of earthquake ground motion is shown. The method yields the instantaneous smoothed values of the AR coefficients and the instantaneous smoothed values of the innovations variance. These results in turn yield estimates of the instantaneous spectral density, the time varying covariance function and a simulation model for the ground motion data. An example of the application of the method to the analysis of an accelogram from the February 1971 San Fernando, California earthquake is shown.     

俯冲带板内地震竖向加速度谱阻尼修正系数模型研究

基于日本K-NET和KiK-net台网的4695条俯冲带板内地震记录,采用最小二乘法对阻尼修正系数(DMF)的几何均值进行关于阻尼比和谱周期的回归拟合,分场地类别建立了考虑阻尼比和谱周期的竖向加速度谱DMF模型.为探究震源、路径和场地效应对该模型残差分布的影响,采用随机效应模型将残差分离得到各类残差及相应的残差标准差,...     

Receiver Functions from Autoregressive Deconvolution

Summary Receiver functions can be estimated by minimizing the square errors of Wiener filter in time-domain or spectrum division in frequency domain. To avoid the direct calculation of auto-correlation and cross-correlation coefficients in Toeplitz equation or of auto-spectrum and cross-spectrum in spectrum division equation as well as empirically choosing a damping parameter, autoregressive deconvolution is presented to isolate receiver function from three-component teleseismic P waveforms. The vertical component of teleseismic P waveform is modeled by an autoregressive model, which can be forward and backward, predicted respectively. The optimum length of the autoregressive model is determined by the Akaike criterion. By minimizing the square errors of forward and backward predicting filters, autoregressive filter coefficients can be recursively solved, and receiver function is also estimated in the similar procedure. Both synthetic and real data tests show that autoregressive deconvolution is an effective method to isolate receiver function from teleseismic P waveforms in time-domain.     

Approximate Stochastic Self-Similarity of Envelopes of High-Frequency Teleseismic P-Waves from Large Earthquakes

A wavetrain of high-frequency (HF) P waves from a large earthquake, when recorded at a distant station, looks like a segment of modulated noise, with its duration close to the duration of rupture. These wavetrains, with their bursts and fadings, look much more intermittent than a segment of common stationary random noise. We try to describe quantitatively this bursty behavior. To this end, variogram and spectral analyses are applied to time histories of P-wave envelopes (squared-amplitude or instant-power signals) in six HF bands of 1-Hz width. Nine M _w = 7.6–9.2 earthquakes were examined, using, in total, 232 records and 992 single-band traces. Variograms of integrated instant power are approximately linear on a log–log scale, indicating that the correlation structure of the instant-power signal is approximately self-similar. Also, estimates of the power spectrum of the instant-power signal look approximately linear on a log–log scale. Log–log slopes of the variograms and spectra deliver estimates of the Hurst exponent H that are mostly in the range 0.6–0.9, markedly above the value H = 0.5 of uncorrelated (white-noise) signals. The preferred estimate over the entire data set is H = 0.83, still, this estimate may include some bias, and must be treated as preliminary. The inter-event scatter of H estimates is about 0.04, reflecting individual event-to-event variations of H. Many of the average log–log spectral plots show slight concavity that perturbs the approximately linear slope; this is a secondary effect that seems to be mostly related to the limited bandwidth of the data. Evidence is given in support of the idea that the observed approximately self-similar correlation structure of the P-wave envelope originates in a similar structure of the body wave instant-power signal radiated by the source, so that the propagation-related distortions can be regarded as limited. The facts presented suggest that the space–time organization of the earthquake rupture process is multiscaled and bears significant fractal features; it deviates from the brittle-crack model with its two well-separated characteristic scales. Phenomenologically, the high-frequency body-wave radiation from an earthquake source can be thought of as a product of stationary noise and the square root of a positive random envelope function with a power-law spectrum. From the viewpoint of applications, the self-similarity of body wave envelopes provides a useful constraint for earthquake source models used to simulate strong ground motions.     

基于AR模型的相干函数有效频段范围的确定

目前人们在进行地震动相干函数模型参数的拟合时,选取的频段范围存在很大差异,而确定有效的频段范围对相干函数模型拟合结果的合理性很重要。本文利用AR模型(Autoregressive Model,自回归模型)计算地震动的功率谱,提出了根据功率谱"信噪比(SNR)"确定相干函数有效频段范围的方法。本文的信噪比中的"噪",专指AR模型中的激励白噪声,用于衡量AR模型中某一频率地震动功率谱与激励白噪声功率谱之间的比例。定义信噪比SNR=0时对应的频率为有效频段的最高频率,大于最高频率的频率成分,被当作是地震波中的随机成分,可以不考虑地震动的相关性。采用这一方法,对SMART-1第45次地震记录的不同台站间距下的地震波进行了功率谱估计,根据信噪比确定出了有效频段范围,并计算了相应的相干函数,结果表明此方法是可行的。     

基于随机介质模型的储层非均质性分析

本文利用随机介质模型对复杂岩性储层进行了非均质性描述.利用模型特征量即非均质纵横比、纵向谱指数、横向谱指数以及扰动标准差等来模拟不同的随机介质.在前人工作的基础上,由某油气田的两口井资料估计储层非均质性能谱,从能谱曲线上提取储层纵向大小尺度非均质谱指数.通过将二维随机介质模拟的合成井记录互相关系数与实际井记录互相关系数进行分析比较,分别得到大小尺度非均质情况下最佳拟合时的横向谱指数和非均质性纵横比.以上求得的各种特征量从不同角度定量分析了储层非均质性的纵横向变化,为储层横向预测提供了依据.