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希尔伯特—黄变换（HHT）与地震动时程的希尔伯特谱
Hilbert-Huang Transform and Hilbert Spectrum of Earthquake Ground Motion

【作者】张郁山

【导师】胡聿贤; 梁建文

【作者基本信息】中国地震局地球物理研究所，固体地球物理学，2003年，博士

【副题名】方法与应用研究

【摘要】 地震动时程是非平稳的，而且蕴涵着土层等介质的非线性信息。传统的地震动时程的频谱分析方法，如Fourier谱分析方法，在单独的时间域或单独的频率域中描述地震动时程，因此，其很难直观、明确地把握非平稳地震动时程的某些时变特征。基于Fourier变换的加窗Fourier谱分析或短时Fourier谱分析、以及小波分析方法是时频分析方法，它们能够在联合的时间-频率域中描述地震动时程，因此它们能够在一定程度上把握地震动时程的非平稳特性。但是，一方面，与Fourier变换一样，这两种时频分析方法的基函数(即简谐波函数与母波函数)都是预先设定好的，当需要处理的波形与规则的简谐波或母小波相比发生扭曲时，为了从数学上拟合原始波形，Fourier变换与小波变换不得不引入大量的“伪”谐波分量，而这种“伪”谐波分量是不具备明确的物理意义的；另一方面，短时Fourier变换所得到谱图与小波变换所得到小波谱都存在着固有的分辨率方面的问题。希尔伯特-黄变换(Hilbert-Huang Transform，简称HHT)是一种新的非平稳信号的处理技术，它是由美国宇航局的Norden E．Huang教授于1998年在经典的...更多Hilbert变换的基础上提出的。该方法由经验模态分解(Empirical Mode Decomposition，简称EMD)与Hilbert谱分析(Hilbert SpectralAnalysis，简称HSA)两部分组成：任意的非平稳信号首先经过EMD方法处理后被分解为若干个本征模态函数(Intrinsic Mode Function，简称IMF)；然后对每个IMF分量进行Hilbert谱分析得到相应分量的Hilbert谱；最后汇总所有IMF分量Hilbert谱就得到了原始非平稳信号的Hilbert谱。按照这种方法得到的Hilbert谱在联合的时间-频率域中描述非平稳信号，其具有非常高的时频分辨率，而且EMD方法分解所得到的IMF分量也具备明确的物理意义。本文即是围绕着HHT方法，对其方法本身及其在地震工程中的应用展开研究。在HHT方法本身的研究方面，针对EMD方法中的边界问题，本文利用自回归模型(AR模型)，提出了一种“边筛分、边延拓”的算法。该算法利用AR模型(或称为线性预测模型)分别在信号的两端预测出两个附加的极大值点和两个附加的极小值点，然后利用三次样条插值将信号本身的极值点与估计出的附加极值点连结起来形成信号的上下包络线，最后通过“筛分”处理将原始信号分解成一系列IMF分量。该算法充分利用了自回归线性预测模型计算效率高、有限几步内预测精度高、以及对低频信号的预测精度高的优点，可以比较好地抑制信号的边界效应对EMD分解结果的影响。在HHT方法的应用研究方面，本文首先应用HHT方法分析了基本的结构动力学问题，揭示了HHT方法得到的IMF分量与Hilbert谱所蕴涵的物理意义。这方面的研究包括两个主要部分：线性SDOF体系动力响应的分析与双线性SDOF体系动力响应的分析。(1) 线性SDOF体系动力响应的分析这一部分主要研究了线性时不变SDOF体系与两种线性时变SDOF体系(即刚度渐变SDOF体系与刚度突变SDOF体系)的强迫动力响应的IMF分量与Hilbert谱所蕴涵的物理意义。其研究结果表明，在规则的输入下(如体系的输入为简谐波、线性调频波、正弦调频波等)，HHT以体系动力响应的不同本征振动模态之间的波间组合机制来描述体系线性的力学行为，即Hilbert谱能够将体系动力响应中描述输入特性的稳态反应部分与描述体系特性的瞬态反应部分识别出来:利用表示稳态反应部分的IMF分量可以获得有关输入的信息;而利用表示瞬态反应部分的IMF分量，则可以获得有关体系自振特性(如自振频率与阻尼等)方面的信息，而且体系的自振特性可以是时变的(对于时变SDOF体系来说)。如果体系的输入为地震波，由于输入运动的复杂性，从体系动力响应的Hilbert谱中无法将上述稳态反应部分与瞬态反应部分区分开来;尽管如此，研究结果表明，Hilbert谱依然能够如实地描述体系在地震动输入下动力响应的能量在时频平面内的分布。此外，这一部分也对体系在不同规则输入下的共振现象进行了探讨。 (2)双线性SDOF体系动力响应的分析这一部分主要研究了双线性SDOF体系动力响应的Hilbert谱与Hilbert边缘谱所蕴涵的物理意义、以及不同的因素(如输入单分量简谐波的幅值、频率及体系的阻尼等)对这两种谱的特征的影响。该部分的研究结果表明，HHT以蕴涵在双线性SDOF体系动力响应的某个本征振动模态中的波内调节机制来描述体系非线性的力学行为，这种波内调节机制必然使得相应IMF分量的瞬时频率产生波动，本文通过双线性体系在振动过程中的屈服时程论述了该瞬时频率波动所蕴涵的物理意义;上述由体系非线性力学行为引起的某个】N『分量瞬时频率的波动必然会导致体系动力响应的Hi】bert边缘中某个频率附近出现宽频带，这一宽频带是体系动力响应的非线性特征和体系非线性力学行为的重要标志，相比Fourier谱中的“伪”谐波分量，该宽频带更具各明确的物理意义。这一部分除了系统地研究了不同因素对单分量谐波输入下双线性SDOF体系动力响应的频谱特征的影响之外，还对地震动输入和多分量简谐波输入下体系动力响应的Fourier谱、Morlet小波谱及其边缘谱、Hilbert谱  还原[page]分页标题[/page]

【Abstract】 The ground motion time history is non-stationary, as well as contains the non-linear information of media, such as the surface soil layers. Traditional spectral analysis method in the analysis of ground motion time histories, such as the Fourier spectral analysis, describes the ground motion time histories either in the time domain singly or in the frequency domain singly, thus, it has some difficulties in grasping directly and definitely some time-variant characteristics of non-stationary ground motion time history. The Fourier-transform-based windowed Fourier spectral analysis, or in other words, the short-time Fourier spectral analysis, and the wavelet analysis are time-frequency analysis methods; they describe the ground motion time histories in the joint time-frequency domain, so they can grasp the non-stationary properties of ground motion to some extent. However, on the one hand, just like the Fourier transform, the bases (the harmonics and the mother wavelet functions) of these...更多 two time-frequency analysis methods are both set up beforehand, as a result, when the waveform to be processed is deformed as compared with the regular harmonic waveform or the regular mother wavelet waveform, in order to fit the original waveform in mathematical sense, the Fourier transform and the wavelet transform have to introduce numerous spurious harmonics, which have not distinct physical meanings; on the other hand, the spectrogram obtained by short-time Fourier transform and the wavelet spectrum obtained by wavelet transform both suffer from the inherent issues of resolution.Hilbert-Huang transform (HHT) is a new technology for the analysis of the non-stationary signals, which was introduced by Prof. Norden E. Huang of NASA in 1998 on the foundation of classic Hilbert transform. This method consists of two successive parts, i.e., the empirical mode decomposition (EMD) and the Hilbert spectral analysis (HSA): firstly, an arbitrary non-stationary signal is decomposed into a number of intrinsic mode functions (IMF) by EMD; then, HSA is performed on each IMF, and the Hilbert spectrum of the corresponding IMF is obtained; at last, the Hilbert spectra of all IMFs are grouped to get the Hilbert spectrum of the original signal. The Hilbert spectrum obtained by this way describes the non-stationary signal in the joint time-frequency domain, and possesses high time-frequency resolution. Furthermore, the IMF components decomposed by EMD method have distinct physical senses.Centering on the Hilbert-Huang transform, this dissertation is devoted to the study of the method itself as well as the study of its application in the field of earthquake engineering.In the aspect of the study of HHT method, aiming at the boundary problem in EMD, a new algorithm called "sifting and extending at the same time" is introduced in this dissertation by the aide of auto-regressive (AR) model. This algorithm uses the AR model, or in other words the linear prediction model, to predict the two additional maxima and the two additional minima at both ends of the signal respectively, and then all of the signal's original extrema and the estimated extrema are connected by cubic spline interpolation method to construct the upper and the lower envelopes of the signal, at last the original signal is decomposed into a series of IMF components by the sifting process. The algorithm utilizes fully the advantages of auto-regressive linear prediction model in the aspects of its high computation efficiency, its high prediction precision in finite steps, and its high prediction precision for low-frequency signal, and thus it can control well the influences of boundary effects on the results of EMD method.In the aspect of the study of the applications of HHT method, this dissertation analyzes the basic issues in structural dynamics by HHT, which reveals the physical meanings of the IMF components and the Hilbert spectrum obtained by HHT. This study is composed by two parts, the analysis of dynamic response of linear SDOF system and the analysis of dynamic respons[page]分页标题[/page]

【关键词】 地展动时程; 非平称; 非线性; 频谱分析; 时频分析; 瞬时频率; 希尔伯特-黄变换(HHT); 经验模态分解(EMD); 本征模态函数(IMF); Hilbert谱; Fourier变换; Fourier谱; 小波变换; 小波谱; 自回归模型; 线性SDOF体系; 双线性SDOF体系; 场地液化
【Key words】 ground motion time history; non-stationarity; non-linearity; spectra analysis; time-frequency analysis; instantaneous frequency; Hilbert-Huang transform (HHT); empirical mode decomposition (EMD); intrinsic mode function (IMF); Hilbert spectrnm; Fourier transform; Fourier spectrum; wavelet transform; wavelet spectrum; auto-regressive model; linear SDOF system; bilinear SDOF system; site liquefaction.

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