The estimation of spectra from time‐frequency transforms for use in attenuation studies

We introduce the signal dependent time–frequency distribution, which is a time–frequency distribution that allows the user to optimize the tradeoff between joint time–frequency resolution and suppression of transform artefacts. The signal‐dependent time–frequency distribution, as well as the short‐time Fourier transform, Stockwell transform, and the Fourier transform are analysed for their ability to estimate the spectrum of a known wavelet used in a tuning wedge model. Next, the signal‐dependent time–frequency distribution, and fixed‐ and variable‐window transforms are used to estimate spectra from a zero‐offset synthetic seismogram. Attenuation is estimated from the associated spectral ratio curves, and the accuracy of the results is compared. The synthetic consisted of six pairs of strong reflections, based on real well‐log data, with a modeled intrinsic attenuation value of 1000/Q = 20. The signal‐dependent time–frequency distribution was the only time–frequency transform found to produce spectra that estimated consistent attenuation values, with an average of 1000/Q = 26±2; results from the fixed‐ and variable‐window transforms were 24±17 and 39±10, respectively. Finally, all three time–frequency transforms were used in a pre‐stack attenuation estimation method (the pre‐stack Q inversion algorithm) applied to a gather from a North Sea seismic dataset, to estimate attenuation between nine different strong reflections. In this case, the signal‐dependent time‐frequency distribution produced spectra more consistent with the constant‐Q model of attenuation assumed in the pre‐stack attenuation estimation algorithm: the average L1 residuals of the spectral ratio surfaces from the theoretical constant‐Q expectation for the signal‐dependent time‐frequency distribution, short‐time Fourier transform, and Stockwell transform were 0.12, 0.21, and 0.33, respectively. Based on the results shown, the signal‐dependent time‐frequency distribution is a time–frequency distribution that can provide more accurate and precise estimations of the amplitude spectrum of a reflection, due to a higher attainable time–frequency resolution.     

Wavelet‐based cepstrum decomposition of seismic data and its application in hydrocarbon detection

How to use cepstrum analysis for reservoir characterization and hydrocarbon detection is an initial question of great interest to exploration seismologists. In this paper, wavelet‐based cepstrum decomposition is proposed as a valid technology for enhancing geophysical responses in specific frequency bands, in the same way as traditional spectrum decomposition methods do. The calculation of wavelet‐based cepstrum decomposition, which decomposes the original seismic volume into a series of common quefrency volumes, employs a sliding window to move over each seismic trace sample by sample. The key factor in wavelet‐based cepstrum decomposition is the selection of the sliding‐window length as it limits the frequency ranges of the common quefrency section. Comparison of the wavelet‐based cepstrum decomposition with traditional spectrum decomposition methods, such as short‐time Fourier transform and wavelet transform, is conducted to demonstrate the effectiveness of the wavelet‐based cepstrum decomposition and the relation between these two technologies. In hydrocarbon detection, seismic amplitude anomalies are detected using wavelet‐based cepstrum decomposition by utilizing the first and second common quefrency sections. This reduces the burden of needing dozens of seismic volumes to represent the response to different mono‐frequency sections in the interpretation of spectrum decomposition in conventional spectrum decomposition methods. The model test and the application of real data acquired from the Sulige gas field in the Ordos Basin, China, confirm the effectiveness of the seismic amplitude anomaly section using wavelet‐based cepstrum decomposition for discerning the strong amplitude anomalies at a particular quefrency buried in the broadband seismic response. Wavelet‐based cepstrum decomposition provides a new method for measuring the instantaneous cepstrum properties of a reservoir and offers a new field of processing and interpretation of seismic reflection data.     

Logarithm of short‐time Fourier transform for extending the seismic bandwidth

Improving seismic resolution is essential for obtaining more detailed structural and stratigraphic information. We present a new algorithm to increase seismic resolution with a minimum of user‐defined parameters. The algorithm inherits useful properties of both the short‐time Fourier transform and the cepstrum to smooth and broaden the frequency spectrum at each translation of the spectral decomposing window. The key idea is to replace the amplitude spectrum with its logarithm in each window of the short‐time Fourier transform. We describe the mathematical formulation of the algorithm and its testing on synthetic and real seismic data to obtain broader frequency spectra and thus enhance the seismic resolution.     

地震勘探信号时频分析方法对比与应用分析

针对地震勘探信号,对非平稳信号时频分析几种适效方法:短时Fourier变换、小波变换、S变换、Wigner分布、平滑伪Wigner分布、锥形核时频分布、AOK(adaptive optimum kernel,自适应最优核函数)分布等进行对比与应用研究.在阐明各种方法基本原理的基础上,进行数值分析与应用研究.首先对非平稳地震勘探模拟信号进行试算及时频属性提取,结合各类方法的信号表示理论,在时频局部化的精度和交叉项抑制等方面对计算结果进行对比分析;进一步应用于实际二维地震数据,提取瞬时频率和瞬时带宽等时频属性,进行比较研究.研究表明:对于地震勘探信号,就本文涉及的几种时频分析方法而言,AOK分布是时频局部化精度最高、交叉项抑制最好、时频匹配最优的方法,值得在地震勘探信号分析和地震属性提取、频谱分解等应用中深入研究和应用.     

小波变换与地震信号特征分析

地震波的瞬时信号,如瞬时振幅、瞬时频率及瞬时相位等,是研究地球介质的重要参数。根据小波定义,对Morlet 小波进行了修正,并对修正后的小波形态进行了深入讨论。理论分析表明,小波变换效果受到整形参数、小波长度、中心频率、频带宽度及小波个数等参数的制约,特别是整形参数与小波中心频率及频带之间关系对小波变换起到决定性作用。在地震波信号的实际处理中,可选取恰当的整形参数,同时采用合适的小波中心频率以避免小波变换对信号产生的遗漏和冗余。文中给出了实际地震记录处理的示例。     

基于振幅和瞬时频率的震相自动识别方法

为了充分反映地震信号振幅增大和频率改变的特征,发展了一种以振幅和瞬时频率比乘积为特征参量的震相自动识别方法。谐波信号和有限实际地震记录的应用表明,本方法具有较高的识别精度。     

A model‐based data‐driven dictionary learning for seismic data representation

Planar waves events recorded in a seismic array can be represented as lines in the Fourier domain. However, in the real world, seismic events usually have curvature or amplitude variability, which means that their Fourier transforms are no longer strictly linear but rather occupy conic regions of the Fourier domain that are narrow at low frequencies but broaden at high frequencies where the effect of curvature becomes more pronounced. One can consider these regions as localised “signal cones”. In this work, we consider a space–time variable signal cone to model the seismic data. The variability of the signal cone is obtained through scaling, slanting, and translation of the kernel for cone‐limited (C‐limited) functions (functions whose Fourier transform lives within a cone) or C‐Gaussian function (a multivariate function whose Fourier transform decays exponentially with respect to slowness and frequency), which constitutes our dictionary. We find a discrete number of scaling, slanting, and translation parameters from a continuum by optimally matching the data. This is a non‐linear optimisation problem, which we address by a fixed‐point method that utilises a variable projection method with ?₁ constraints on the linear parameters and bound constraints on the non‐linear parameters. We observe that slow decay and oscillatory behaviour of the kernel for C‐limited functions constitute bottlenecks for the optimisation problem, which we partially overcome by the C‐Gaussian function. We demonstrate our method through an interpolation example. We present the interpolation result using the estimated parameters obtained from the proposed method and compare it with those obtained using sparsity‐promoting curvelet decomposition, matching pursuit Fourier interpolation, and sparsity‐promoting plane‐wave decomposition methods.     

Enhancement of Signal-to-noise Ratio in Natural-source Transient Magnetotelluric Data with Wavelet Transform

—For audio-frequency magnetotelluric surveys where the signals are lightning-stroke transients, the conventional Fourier transform method often fails to produce a high quality impedance tensor. An alternative approach is to use the wavelet transform method which is capable of localizing target information simultaneously in both the temporal and frequency domains. Unlike Fourier analysis that yields an average amplitude and phase, the wavelet transform produces an instantaneous estimate of the amplitude and phase of a signal. In this paper a complex well-localized wavelet, the Morlet wavelet, has been used to transform and analyze audio-frequency magnetotelluric data. With the Morlet wavelet, the magnetotelluric impedance tensor can be computed directly in the wavelet transform domain. The lightning-stroke transients are easily identified on the dilation-translation plane. Choosing those wavelet transform values where the signals are located, a higher signal-to-noise ratio estimation of the impedance tensor can be obtained. ? In a test using real data, the wavelet transform showed a significant improvement in the signal-to-noise ratio over the conventional Fourier transform.     

Application of wavelet theory to identify yielding in seismic response of bi‐linear structures

The time–frequency and the time‐scale analysis methods are used in this paper to identify the dynamic characteristics of non‐linear seismic response of structural systems with single degree of freedom (SDOF) and multiple degrees of freedom (MDOF). Based on the floor acceleration response time histories of bi‐linear SDOF and MDOF structures, the current study compares the results of system identification using the short‐time Fourier transform (STFT), continuous wavelet transform (CWT) and discrete wavelet transform (DWT) methods. The aim is to identify the frequency variations and the time at on‐set of yielding and unloading of a bi‐linear structural system during seismic response. The results demonstrate that the CWT method is better than the STFT method in both time and frequency resolutions, and that the DWT method is the best at detecting the time at on‐set of yielding and unloading. Combining the results of CWT and DWT methods therefore provides accurate information of both frequency variations and yielding time in non‐linear seismic response. To alleviate the problems associated with noise‐contaminated signals, e.g. seismic response data recorded on site, the study suggests that low‐pass filtering be carried out before applying the DWT method to decompose the signals into multiple levels of details. Copyright © 2001 John Wiley & Sons, Ltd.     

Surface wave attenuation based polarization attributes in time-frequency domain for multicomponent seismic data

In the paper, we propose a surface wave suppression method in time-frequency domain based on the wavelet transform, considering the characteristic difference of polarization attributes, amplitude energy and apparent velocity between the effective signals and strong surface waves. First, we use the proposed method to obtain time–frequency spectra of seismic signals by using the wavelet transform and calculate the instantaneous polarizability at each point based on instantaneous polarization analysis. Then, we separate the surface wave area from the signal area based on the surface-wave apparent velocity and the average energy of the signal. Finally, we combine the polarizability, energy, and frequency characteristic to identify and suppress the signal noise. Model and field data are used to test the proposed filtering method.     

信号的瞬时参数与正交小波基

对于可表成ｕ（ｔ）＝ａ（ｔ）ｃｏｓθ（ｔ）的信号,称ａ（ｔ）为其瞬时振幅,θ（ｔ）为瞬时位相,为瞬时频率．在许多实际问题中,瞬时振幅、位相、频率等瞬时参数的提取有重要意义．但已知的一些瞬时参数计算方法均各有其局限性．本文构造了一种频谱支集紧且有解析表达式的小波,利用它得到一种计算信号瞬时参数的能抗噪声干扰的快速、局部算法．理论分析和数值模拟的结果表明,该方法是实用而有效的,可望能满足许多实际问题的需要．     

基于双谱幅值和相位重构的地震子波提取

利用地震记录双谱中包含子波的幅值和相位信息,以及其超强的抗噪声干扰能力,采用一种基于双谱幅值和相位重构的地震子波提取方法,首先提取出子波幅值及相位信息,进而通过傅立叶反变换,使子波得以完全恢复.本文针对双谱相位重构递推公式,提出一种新的初值选取方法,使地震子波估计的稳定性得到了提高.仿真实验证实了该方法的可行性.     

Wavelet‐based characterization of design ground motions

With the recent emergence of wavelet‐based procedures for stochastic analyses of linear and non‐linear structural systems subjected to earthquake ground motions, it has become necessary that seismic ground motion processes are characterized through statistical functionals of wavelet coefficients. While direct characterization in terms of earthquake and site parameters may have to wait for a few more years due to the complexity of the problem, this study attempts such characterization through commonly available Fourier and response spectra for design earthquake motions. Two approaches have been proposed for obtaining the spectrum‐compatible wavelet functionals, one for input Fourier spectrum and another for input response spectrum, such that the total number of input data points are 30–35% of those required for a time‐history analysis. The proposed methods provide for simulating ‘desired non‐stationary characteristics’ consistent with those in a recorded accelerogram. Numerical studies have been performed to illustrate the proposed approaches. Further, the wavelet functionals compatible with a USNRC spectrum in the case of 35 recorded motions of similar strong motion durations have been used to obtain the strength reduction factor spectra for elasto‐plastic oscillators and to show that about ±20% variation may be assumed from mean to 5 and 95% confidence levels due to uncertainty in the non‐stationary characteristics of the ground motion process. Copyright © 2002 John Wiley & Sons, Ltd.     

Simulations of non‐stationary frequency content and its importance to seismic assessment of structures

To verify the importance of the non‐stationary frequency characteristic of seismic ground motion, a joint time–frequency analysis technique of time signals, called chirplet‐based signal approximation, is developed to extract the non‐stationary frequency information from the recorded data. The chirplet‐based signal approximation is clear in concept, similar to Fourier Transform in mathematical expressions but with different base functions. Case studies show that the chirplet‐based signal approximation can represent the joint time–frequency variation of seismic ground motion quite well. Both the random models of uniform modulating process and evolutionary process are employed to generate artificial seismic waves. The joint time–frequency modulating function in the random model of evolutionary process is determined by chirplet‐based signal approximation. Finally, non‐linear response analysis of a SODF system and a frame structure is performed based on the generated artificial seismic waves. The results show that the non‐stationary frequency characteristic of seismic ground motion can significantly change the non‐linear response characteristics of structures, particularly when a structure goes into collapse phase under seismic action. It is concluded that non‐stationary frequency characteristic of seismic ground motion should be considered for the assessment of seismic capacity of structures. Copyright © 2002 John Wiley & Sons, Ltd.     

地震信号的复地震道分析及应用

复地震道分析又称三瞬分析,该分析方法可将反映地震信号局部变化情况的地震波的瞬时振幅、瞬时相位和瞬时频率等信息分离开.本文应用Hilbert变换求解虚地震记录,用复地震道分析方法求取"三瞬"信息,并用该方法计算了理论合成地震记录的瞬时振幅、瞬时相位和瞬时频率,获得了较好的效果.同时,本文也利用该方法对某区块实际地震资料进行了处理,结果表明,复地震道分析方法获得的"三瞬"信息可反映地震信号的局部变化,有助于进行地震薄互层分析,并能提高数据的解释精度.     

MCZT在气枪震源信号频谱分析中的应用

气枪震源信号是短时非平稳信号,采用频谱细化算法能提高频谱分析的准确性。首先进行了改进线性调频Z变换(MCZT)和FFT两种频谱分析算法的误差仿真计算,然后进行气枪震源的水下子波信号和地震波信号的对比计算。结果表明MCZT计算误差较小、计算时间较少,能有效提高气枪震源信号频率和幅度特征提取的准确性,是气枪震源信号频谱特征分析的一个有效方法。     

Ground roll attenuation using the S and x-f-k transforms

Ground roll, which is characterized by low frequency and high amplitude, is an old seismic data processing problem in land‐based seismic acquisition. Common techniques for ground roll attenuation are frequency filtering, f‐k or velocity filtering and a type of f‐k filtering based on the time‐offset windowed Fourier transform. These techniques assume that the seismic signal is stationary. In this study we utilized the S, x‐f‐k and t‐f‐k transforms as alternative methods to the Fourier transform. The S transform is a type of time‐frequency transform that provides frequency‐dependent resolution while maintaining a direct relationship with the Fourier spectrum. Application of a filter based on the S transform to land seismic shot records attenuates ground roll in a time‐frequency domain. The t‐f‐k and x‐f‐k transforms are approaches to localize the apparent velocity panel of a seismic record in time and offset domains, respectively. These transforms provide a convenient way to define offset or time‐varying reject zones on the separate f‐k panel at different offsets or times.     

Hilbert-Huang变换在提取地震信号动力特性中的应用

H ilbert-Huang变换(HHT)是一种处理非线性、非平稳信号的新方法。它通过经验模态分解将信号分解为有限个固有模态函数,并对每个固有模态函数进行H ilbert变换得到H ilbert谱。本文将这种方法应用于地震信号动力特性的提取,有效地获得了信号能量的时频分布,量化提取了中心频率、瞬时相位、瞬时能量、H ilbert能量、最大振幅对应的时频分布等动力特性,并与Fourier变换、小波变换等进行了比较,显示了HHT的优势以及对于进一步实现结构分析和控制的重要意义。     

基于保角变换的双谱域相位谱估计方法及其在地震子波估计中的应用(英文)

地震子波估计是地震资料处理与解释中的重要环节,它的准确与否直接关系到反褶积及反演等结果的好坏。高阶谱(双谱和三谱)地震子波估计方法是一类重要的、新兴的子波估计方法,然而基于高阶谱的地震子波估计往往因为高阶相位谱卷绕的原因,导致子波相位谱求解产生偏差,进而影响了混合相位子波估计的效果。针对这一问题,本文在双谱域提出了一种基于保角变换的相位谱求解方法。通过缩小傅里叶相位谱的取值范围,有效避免了双谱相位发生卷绕的情况,从而消除了原相位谱估计中双谱相位卷绕的影响。该方法与最小二乘法相位谱估计相结合,构成了基于保角变换的最小二乘地震子波相位谱估计方法,并与最小二乘地震子波振幅谱估计方法一起,应用到了地震资料混合相位子波估计中。理论模型和实际资料验证了该方法的有效性。同时本文将双谱域地震子波相位谱估计中保角变换的思想推广到三谱域地震子波相位谱估计中。