基于小波变换的地震波时变功率谱估计
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摘要
为进一步认识地震动的非平稳特性,基于小波变换提出一种估计时变功率谱的方法.利用小波变换的带通滤波特性,结合时变功率谱的物理意义,从能量的角度推导各频带上小波系数与时变功率谱密度函数的关系,从而通过地震动信号的小波系数估计时变功率谱.利用基于小波变换的估计时变功率谱方法,针对实测和模拟地震动信号进行时变功率谱估计,并与基于傅里叶变换估计的功率谱进行了比较验证.结果表明:该方法正确、可行,它将原有的频率幅值的二维认识提高到频率时间幅值的三维认识,解决了傅里叶变换估计功率谱缺乏时间信息的问题.
In order to further understand the nonstationary characteristic of ground motion,a new method to estimate the time-dependent power spectral density (PSD) was proposed based on wavelet transform. With utilizing the band-pass filtering characteristics of wavelet transform and combining the physical meaning of time-dependent PSD,the relations between the wavelet coefficient and time-dependent PSD on different frequency bands were deduced in terms of energy. And then the time-dependent PSD was estimated through the wavelet coefficient of ground motion signals. Using the proposed method,the time-dependent PSD for both measured and simulated ground motion signals was respectively estimated and compared with the PSD based on Fourier transform. The results show that the present method is feasible and correct. The method can extend the original two-dimentional relation of frequency-amplitude to three-dimentional relation of time-frequency-amplitude,and avoid the lack of time information in the traditional PSD estimation based on Fourier transform.
In order to further understand the nonstationary characteristic of ground motion,a new method to estimate the time-dependent power spectral density (PSD) was proposed based on wavelet transform. With utilizing the band-pass filtering characteristics of wavelet transform and combining the physical meaning of time-dependent PSD,the relations between the wavelet coefficient and time-dependent PSD on different frequency bands were deduced in terms of energy. And then the time-dependent PSD was estimated through the wavelet coefficient of ground motion signals. Using the proposed method,the time-dependent PSD for both measured and simulated ground motion signals was respectively estimated and compared with the PSD based on Fourier transform. The results show that the present method is feasible and correct. The method can extend the original two-dimentional relation of frequency-amplitude to three-dimentional relation of time-frequency-amplitude,and avoid the lack of time information in the traditional PSD estimation based on Fourier transform.
引文
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[3]Mukherjee S,Gupta V K.Wavelet-based characteriza-tion of design ground motions[J].Earthquake Engi-neering and Structural Dynamics,2002,31:1173-1190
[4]Basu B,Gupta V K.Non-stationary seismic response of MDOF systems by wavelet transform[J].Earth-quake Engineering and Structural Dynamics,1997,26:1243-1258.
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[12]徐科军,李永三.基于连续小波变换的功率谱估计方法[J].应用科学学报,2003,21(2):157-160.(XU Ke-jun,LI Yong-san.The power spectrum esti-mation method based on continuous wavelet transfor-mation[J].Journal of Applied Sciences,2003,21(2):157-160.)
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[14]曹晖,赖明,白绍良.地震地面运动局部谱密度的小波变换估计[J].工程力学,2004,21(5):109-115.(CAO Hui,LAI Ming,BAI Shao-liang.Estimation of local spectral density of earthquake ground motion based on wavelet transform[J].Engineering Mecha-nics,2004,21(5):109-115.)
[15]曹晖,赖明,白绍良.地震地面运动的局部谱密度描述及其估计方法[J],世界地震工程,2004,20(2):43-49.(CAO Hui,LAI Ming,BAI Shao-liang.The descrip-tion of earthquake ground motion based on local spec-tral density and its estimation methods[J].World Earthquake Engineering,2004,20(2):43-49.)
[16]应怀樵.波形和频谱分析与随机数据处理[M].北京:中国铁道出版社,1983.(YING Huai-qiao.Waveform and frequency spectrum analyze and random data processing[M].Beijing:China Railway Press,1983.)
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[18]梁建文.非平稳地震动过程模拟方法(Ⅰ)[J].地震学报,2005,27(2):213-224.(LIANG Jian-wen.Simulation of non-stationary ground motion processes(Ⅰ)[J].Acta Seismologica Sinica,2005,27(2):213-224.)
[19]曹资,薛素铎.空间结构抗震理论与设计[M].北京:科学出版社,2005.(CAO Zi,XUE Su-duo.Seismic analysis and design of spatial structures[M].Beijing:Science Press,2005.)
[2]杨庆山,姜海鹏,陈英俊.基于相位差谱的时频非平稳人造地震动的生成[J].地震工程与工程振动,2001,21(3):10-16.(YANG Qing-shan,JIANG Hai-peng,CHEN Ying-jun.Generation of ground motion non-stationary both in time and frequency domains based on phase diffe-rence spectrum[J].Earthquake Engineering and En-gineering Vibration,2001,21(3):10-16.)
[3]Mukherjee S,Gupta V K.Wavelet-based characteriza-tion of design ground motions[J].Earthquake Engi-neering and Structural Dynamics,2002,31:1173-1190
[4]Basu B,Gupta V K.Non-stationary seismic response of MDOF systems by wavelet transform[J].Earth-quake Engineering and Structural Dynamics,1997,26:1243-1258.
[5]Gurley K,Kareem A.Applications of wavelet trans-forms in earthquake wind and ocean engineering[J].Engineering Structures,1999,21:149-167.
[6]曹晖,林学鹏.地震动非平稳特性对结构非线性响应影响的分析[J].工程力学,2006,23(12):30-35.(CAO Hui,LIN Xue-peng.The effect of nonstationary characteristic of earthquake ground motion on the structural nonlinear response[J].Engineering Me-chanics,2006,23(12):30-35.)
[7]Iyama J,Kuwamura H.Application of wavelets to analysis and simulation of earthquake motions[J].Earthquake Engineering and Structural Dynamics,1999,28:255-272.
[8]Daubechies I.Ten lectures on wavelets[M].Philadel-phia:Society for Industrial and Applied Mathematics,1992.
[9]杨福生.小波变换的工程分析与应用[M].北京:科学出版社,1999.(YANG Fu-sheng.Wavelet transform for project anal-ysis and application[M].Beijing:Science Press,1999.)
[10]彭玉华.小波变换与工程应用[M].北京:科学出版社,1999.(PENG Yu-hua.Wavelet transform and engineering application[M].Beijing:Science Press,1999.)
[11]丁宏,戴逸松,石要武.采用小波变换对短数据信号的谱估计方法[J].电子学报,1997,25(1):11-14.(DING Hong,DAI Yi-song,SHI Yao-wu.Short data signal spectrum estimation based on wavelet transfor-mation[J].Acta Electronica Sinica,1997,25(1):11-14.)
[12]徐科军,李永三.基于连续小波变换的功率谱估计方法[J].应用科学学报,2003,21(2):157-160.(XU Ke-jun,LI Yong-san.The power spectrum esti-mation method based on continuous wavelet transfor-mation[J].Journal of Applied Sciences,2003,21(2):157-160.)
[13]Spanos P D,Failla G.Evolutionary spectra estimation using wavelets[J].Journal of Engineering Mecha-nics,ASCE,2004,130(8):952-960.
[14]曹晖,赖明,白绍良.地震地面运动局部谱密度的小波变换估计[J].工程力学,2004,21(5):109-115.(CAO Hui,LAI Ming,BAI Shao-liang.Estimation of local spectral density of earthquake ground motion based on wavelet transform[J].Engineering Mecha-nics,2004,21(5):109-115.)
[15]曹晖,赖明,白绍良.地震地面运动的局部谱密度描述及其估计方法[J],世界地震工程,2004,20(2):43-49.(CAO Hui,LAI Ming,BAI Shao-liang.The descrip-tion of earthquake ground motion based on local spec-tral density and its estimation methods[J].World Earthquake Engineering,2004,20(2):43-49.)
[16]应怀樵.波形和频谱分析与随机数据处理[M].北京:中国铁道出版社,1983.(YING Huai-qiao.Waveform and frequency spectrum analyze and random data processing[M].Beijing:China Railway Press,1983.)
[17]欧进萍,王光远.结构随机振动[M].北京:高等教育出版社,1998.(OU Jin-ping,WANG Guang-yuan.Random vibration of structures[M].Beijing:Higher Education Press,1998.)
[18]梁建文.非平稳地震动过程模拟方法(Ⅰ)[J].地震学报,2005,27(2):213-224.(LIANG Jian-wen.Simulation of non-stationary ground motion processes(Ⅰ)[J].Acta Seismologica Sinica,2005,27(2):213-224.)
[19]曹资,薛素铎.空间结构抗震理论与设计[M].北京:科学出版社,2005.(CAO Zi,XUE Su-duo.Seismic analysis and design of spatial structures[M].Beijing:Science Press,2005.)
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