小波消噪与分解对结构地震反应的影响研究
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摘要
小波分析及小波变换是80年代中期发展起来的一门新兴的数学理论和方法，它被认为是傅里叶分析方法的突破性进展，它具有许多优良的特性。小波变换的基本思想类似于傅里叶变换就是用信号在一簇基函数张成的空间上的投影表征该信号。经典的傅里叶变换把信号按三角正、余弦基展开，将任意函数表示为具有不同频率的谐波函数的线性迭加，能较好的刻画信号的频率特性，但它在时空域上无任何分辨，不能做局部分析，这在理论和应用上都带来了许多不便。小波分析优于傅里叶之处在于，小波分析在时域和频域同时具有良好的局部化性质，因为小波函数是紧支集，而三角正、余弦的区间是无穷区间，所以小波变换可以对高频成分采用逐渐精细的时域或空间域取代步长，从而可以聚焦到对象的任意细节。因此，小波变换被誉为分析信号的显微镜，傅里叶分析发展史上的一个新的里程碑。小波分析是一个新的数学分支，它是泛函分析、傅里叶分析、数值分析的最完美结晶；在应用领域，特别是在信号处理、图象处理、语音分析、模式识别、量子物理、生物医学工程、计算机视觉、故障诊断及众多非线性科学领域都有广泛的应用。本文主要内容如下：
第一章简单介绍了小波发展的历史以及MATLAB软件的一些特点。
第二章介绍了小波分析理论以及小波分析中一些重要的概念。
第三章介绍了小波变换与时-频分析。
第四章介绍了小波变换模极大值和奇异性的定义。
第五章介绍了噪声以及小波消噪的目的、原理、方法并给出了小波消噪的例子。
第六章应用小波分析对地震波进行消噪，并比较消噪前后储罐的地震响应。应用小波分析对地震信号进行多层分解，提取我们想要得地震波数据进行储罐地震响应分析，得出这种方法是可行的。
Wavelet analysis is a novel mathematical theory and method proposed In 1980s. It is regarded as a breakthrough of Fourier analysis because It has many wonderful characteristics. Fourier analysis consists of breaking up a signal Into slue waves of various frequencies. Similarly, wavelet analysis is the breaking-up a signal into shifted and scaled versions of the original wavelet. Lacking of space locality in time domain, Fourier analysis can only make certain of the Integral singularity of a function or signal. As a result; It Is difficult to detect the spatial position and distribution of broken signal by Fourier analysis. Wavelet analysis has the characteristic of spatial locality, and its wideness in both windows of the time and the frequency can be adjusted, so it can analyze the details of a signal. Therefore Wavelet analysis is called a microscope of signal analysis and a milestone of Fourier analysis. As a new embranchment of mathematics. Wavelet analysis is the most perfect combination of the function
al analysis. Fourier analysis and numerical analysis Wavelet analysis has been widely used in signal processing, image manipulation, voice analysis, pattern recognition, quanta physics, biomedicine engineering, computer vision, fault diagnosis, some nonlinear fields and so on. The dissertation is outlined below.
In chapter 1 we will briefly introduce the history of Wavelet analysis's development, and also introduce some useful feature of the MATLAB software.
In chapter 2 we introduce you the Wavelet analysis theory, and you can also find some important concept used in Wavelet analysis.
In chapter 3 we introduce you the Wavelet analysis and time-frequency analysis.
In chapter4 we will introduce the modulus maximum of Wavelet transform and the definition of singularity.
In chapter 5 we will introduce the different noise, and you will know why and how we should transient signal denoise , you will also find some examples of transient signal denoise.
Chapter 6 is devoted to use Wavelet analysis to transient the signal of ELCENTRO denoise, and to compare the Seismic response of tank to both signal. We also use the method of multiple-level decomposition to get what we want of the signal, and do some compare about the Seismic response of tank to this signal and the original signal. We find this method is practicable.
第一章简单介绍了小波发展的历史以及MATLAB软件的一些特点。
第二章介绍了小波分析理论以及小波分析中一些重要的概念。
第三章介绍了小波变换与时-频分析。
第四章介绍了小波变换模极大值和奇异性的定义。
第五章介绍了噪声以及小波消噪的目的、原理、方法并给出了小波消噪的例子。
第六章应用小波分析对地震波进行消噪，并比较消噪前后储罐的地震响应。应用小波分析对地震信号进行多层分解，提取我们想要得地震波数据进行储罐地震响应分析，得出这种方法是可行的。
Wavelet analysis is a novel mathematical theory and method proposed In 1980s. It is regarded as a breakthrough of Fourier analysis because It has many wonderful characteristics. Fourier analysis consists of breaking up a signal Into slue waves of various frequencies. Similarly, wavelet analysis is the breaking-up a signal into shifted and scaled versions of the original wavelet. Lacking of space locality in time domain, Fourier analysis can only make certain of the Integral singularity of a function or signal. As a result; It Is difficult to detect the spatial position and distribution of broken signal by Fourier analysis. Wavelet analysis has the characteristic of spatial locality, and its wideness in both windows of the time and the frequency can be adjusted, so it can analyze the details of a signal. Therefore Wavelet analysis is called a microscope of signal analysis and a milestone of Fourier analysis. As a new embranchment of mathematics. Wavelet analysis is the most perfect combination of the function
al analysis. Fourier analysis and numerical analysis Wavelet analysis has been widely used in signal processing, image manipulation, voice analysis, pattern recognition, quanta physics, biomedicine engineering, computer vision, fault diagnosis, some nonlinear fields and so on. The dissertation is outlined below.
In chapter 1 we will briefly introduce the history of Wavelet analysis's development, and also introduce some useful feature of the MATLAB software.
In chapter 2 we introduce you the Wavelet analysis theory, and you can also find some important concept used in Wavelet analysis.
In chapter 3 we introduce you the Wavelet analysis and time-frequency analysis.
In chapter4 we will introduce the modulus maximum of Wavelet transform and the definition of singularity.
In chapter 5 we will introduce the different noise, and you will know why and how we should transient signal denoise , you will also find some examples of transient signal denoise.
Chapter 6 is devoted to use Wavelet analysis to transient the signal of ELCENTRO denoise, and to compare the Seismic response of tank to both signal. We also use the method of multiple-level decomposition to get what we want of the signal, and do some compare about the Seismic response of tank to this signal and the original signal. We find this method is practicable.
引文
[1] 张贤达．现代信号处理．北京：清华大学出版社，1995
[2] 吕启霞，胡维礼．小波网络在控制系统中的应用．信息与控制，2000，29(6)：532～539
[3] 程正兴．小波分析算法与应用．西安：西安交通大学出版社，1998
[4] 焦李成．神经网络的应用与实现．西安：西安电子科技大学出版社，1995
[5] 闻新，周露，王丹等．MATLAB神经网络应用设计．北京：科学出版社，2000
[6] 胡昌华，张军波，夏军，张伟．基于MATLAB的系统分析与设计．西安：西安电子科技大学出版社，2000
[7] 邓东皋，彭立中．小波分析．数学进展，1991，20(3)：294～310
[8] 刘贵忠，邸双亮．小波分析及其应用．西安：西安电子科技大学出版社，1993．1～6，159～164
[9] 王建中．小波理论及其在物理和工程中的应用．数学进展，1992，21(3)：290～310
[10] 李世雄，刘家琦．小波理论和反演数学基础．北京：地质出版社，1994．99～118
[11] J Morlet. Seismic Tomorrow: Interferometry and Quantum Mechanics, Presented at the 45th Annual International SEG meeting, Oct. 15, 1972
[12] 崔锦泰著．小波分析导论．程正兴译．西安：西安交通大学出版社，1994．1～28
[13] 冉启文．小波分析与分数傅里叶变换理论及应用．哈尔滨：哈尔滨工业出版社，2001．18～19
[14] Young R. An introduction to nonlinear Fourier series. New York: Academic Press, 1980. 1～150
[15] 李建平．小波分析与信号处理—理论、应用及软件实现，重庆：重庆出版社，1997
[16] Morler J. Wave propation and sampling theory and complex waves. Geophysics, 1982, 47 (2): 222～236
[17] Daubechies I. Orthonormal bases of compactly supported wavelets. Comm. On Pure and Appl. Math., 1988, 41 (7): 909～996
[18] Burt P J, Adelson E. The Laplacian Pyramid as a compact images code. IEEE Trans on Communications, 1983, 31 (4): 532～540
[19] Mallat S. A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans. On PAMI, 1989, 11 (7): 674～693
[20] Mallat S. Multifrequency channel decompositions of images and wavelet models. IEEE Trans. on Acoustics, Speech, and Signal Processing, 1989, 37 (12): 2091～2110
[21] Rioul O, Vetterti M. IEEE SP Mag., 1991. 14～28
[22] 陈规明，张明照，戚红雨等．应用MATLAB语言处理数字信号与数字图像．北京：科学出版社，2001
[23] L.科恩著．时—频分析：理论与应用．白居宪译．西安：西安交通大学出版社，1998．3
[24] 秦前清，杨宗凯．实用小波分析．西安：西安电子科技大学出版社，2001
[25] 温熙森，胡鸢庆，邱静．模式识别与状态监控．长沙：国防科技大学出版社，1997
[26] Daubechies I. Ten lectures on wavelets．New York: SIAM, 1992.1～213
[27] 刘贵忠，冯铁岭，宗涛．样条多分辨分析．电子学报，1996，24(7)72～77
[28] 崔锦泰．小波分析导论．程正兴译．西安：西安交通大学出版社，1995．1～354
[29] Schumaker L L. Basic theory of spline functions. New York: Academic Press,1981
[30] Zwart P. Multivariate splines with nondegenerate partitions. SIAM J. Numer. Anal. 1979,10: 665～673
[31] Chui C K. Multivariate splines: theory and applications. New York: SIAM, 1987
[32] 徐佩霞，孙功宪．小波分析与实用实例．合肥：中国科学技术大学出版社，1996
[33] D Gabor. Theory of Communication. J. Inst. Elect. Eng. (London), 1946, 93(Ⅲ): 492-457
[34] M V Wickerhauser. Acoustic Signal Compression with Wavelet Packets. Wavelet Analysis and its Application. Chui C K (et.). Academic Press, Inc, 1992: 679～700
[35] Y Meyer. Wavelets: Algorithms and Applications. SIAM, Philadelphia, 1993: 95～99
[36] 崔锦泰 小波分析导论 西安 西安交通大学出版社1992
[37] D Gabor. theory of Communication. J. Inst.. Elect. Eng.(London), 1946, 93(Ⅲ): 429～457
[38] 雷斌、杨宗凯等，线性时频变换在弱信号检测中的应用，电子科学杂志．Vol．3 1993
[39] 黄正强等．基于小波包的信号检测算法．华中理工大学学报，1994，22(11)：32～35
[40] 冯玉珉，邵玉明，张星编译．数据图像压缩编码．北京：中国铁道出版社，1993
[41] Daubechies I. The wavelet transform,. time-frequency localization and signal analysis. IEEE Trans Inform Theory, 1990, 36: 961-1005
[42] Daubechies I and Grossmann A. Frames in the Bargmann space of entire functions. Comm Pure and Appl Math, 1988, 41: 151-164
[43] Mallat S and Zhong S. Characterization of signals from multiscale edges. IEEE Trans Part Anal and Math Intell, 1992, 14: 710-732
[44] 梁学章，金光日．二元样条函数与小波．天津计算技术学年会文集，1991
[45] 侯遵泽，杨文采．小波分析应用研究．物探化探计算技术，1995，17：1～9
[46] Mallat S and Hwang W L. Singularity detection and processing with wavelet. IEEE Trans. on Information Theory, 1192, 38(2): 617～643
[47] Mallat. S. A theory for multiresolution signal decompositions-the wavelet representation. IEEE Trans Pattern and Machine Intel., 1989, 11: 674～693
[48] 平玉娇．利用小波变换进行图像数据压缩．重庆：重庆大学，1995
[49] 田逢春，金吉成．数字信号二进小波变换的滤波器与边界失真．重庆大学学报，1996，19(5)：23～29
[50] 龙瑞麟．高维小波分析．北京：世界图书出版社，1995．1～362
[51] 李东舸，柳健．小波理论及其在图像处理中的应用．无线电工程，1994，24(4)：19～27
[52] 王玉平，蔡元龙．小波分析在信号处理中的应用．无线电工程，1994，24(3)：11～18
[53] 程俊等．小波变换用于突变检测．通信学报，1995，16(3)96～104
[54] 钟声．图像压缩技术及其应用．电子学报，1995，23(10)：117～123
[55] 柳树田，徐建东，张岩，李淳飞．透镜组合实现分数傅里叶变换．光学学报，1995，10：1404～1408
[56] O Aytur, H M zaktas, Nonorthogonal Domains in phase Space of Quantum Optics and Their Relation to Fractional Fourier Transform. Opt Comm.. 1995, 120: 166～170
[57] S Y Lee, H H Szu. Franctional Fourier Transform, Wavelet Transforms and Adaptive Networks. Optical Engineering. 1994, 33(7): 23～26
[58] Daubechies I. Time-frequency localization operators: a geometric phase space approach. IEEE Trans Inform Theory, 1998, 34: 605～612
[59] 马晶，谭立英，冉启文．光学小波滤波理论初探．中国激光，1999，26(4)：1～4
[60] 孟昭波，杨丽华．地震资料的小波压缩．石油地球物理勘探，1995，30(2)：70～75
[61] S Mallat, S Zhong. Characterization of Signals from Multiscale Edges．第三次中法小波会议
论文集．
[62] 程正兴．国外小波及其应用概况．工程力学学报，1992，9(3)：125～126
[63] Wicherhauser Mladen Victor. Adapted wavelet analysis from theory to software. New York: SIAM, 1994, 329～360
[2] 吕启霞，胡维礼．小波网络在控制系统中的应用．信息与控制，2000，29(6)：532～539
[3] 程正兴．小波分析算法与应用．西安：西安交通大学出版社，1998
[4] 焦李成．神经网络的应用与实现．西安：西安电子科技大学出版社，1995
[5] 闻新，周露，王丹等．MATLAB神经网络应用设计．北京：科学出版社，2000
[6] 胡昌华，张军波，夏军，张伟．基于MATLAB的系统分析与设计．西安：西安电子科技大学出版社，2000
[7] 邓东皋，彭立中．小波分析．数学进展，1991，20(3)：294～310
[8] 刘贵忠，邸双亮．小波分析及其应用．西安：西安电子科技大学出版社，1993．1～6，159～164
[9] 王建中．小波理论及其在物理和工程中的应用．数学进展，1992，21(3)：290～310
[10] 李世雄，刘家琦．小波理论和反演数学基础．北京：地质出版社，1994．99～118
[11] J Morlet. Seismic Tomorrow: Interferometry and Quantum Mechanics, Presented at the 45th Annual International SEG meeting, Oct. 15, 1972
[12] 崔锦泰著．小波分析导论．程正兴译．西安：西安交通大学出版社，1994．1～28
[13] 冉启文．小波分析与分数傅里叶变换理论及应用．哈尔滨：哈尔滨工业出版社，2001．18～19
[14] Young R. An introduction to nonlinear Fourier series. New York: Academic Press, 1980. 1～150
[15] 李建平．小波分析与信号处理—理论、应用及软件实现，重庆：重庆出版社，1997
[16] Morler J. Wave propation and sampling theory and complex waves. Geophysics, 1982, 47 (2): 222～236
[17] Daubechies I. Orthonormal bases of compactly supported wavelets. Comm. On Pure and Appl. Math., 1988, 41 (7): 909～996
[18] Burt P J, Adelson E. The Laplacian Pyramid as a compact images code. IEEE Trans on Communications, 1983, 31 (4): 532～540
[19] Mallat S. A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans. On PAMI, 1989, 11 (7): 674～693
[20] Mallat S. Multifrequency channel decompositions of images and wavelet models. IEEE Trans. on Acoustics, Speech, and Signal Processing, 1989, 37 (12): 2091～2110
[21] Rioul O, Vetterti M. IEEE SP Mag., 1991. 14～28
[22] 陈规明，张明照，戚红雨等．应用MATLAB语言处理数字信号与数字图像．北京：科学出版社，2001
[23] L.科恩著．时—频分析：理论与应用．白居宪译．西安：西安交通大学出版社，1998．3
[24] 秦前清，杨宗凯．实用小波分析．西安：西安电子科技大学出版社，2001
[25] 温熙森，胡鸢庆，邱静．模式识别与状态监控．长沙：国防科技大学出版社，1997
[26] Daubechies I. Ten lectures on wavelets．New York: SIAM, 1992.1～213
[27] 刘贵忠，冯铁岭，宗涛．样条多分辨分析．电子学报，1996，24(7)72～77
[28] 崔锦泰．小波分析导论．程正兴译．西安：西安交通大学出版社，1995．1～354
[29] Schumaker L L. Basic theory of spline functions. New York: Academic Press,1981
[30] Zwart P. Multivariate splines with nondegenerate partitions. SIAM J. Numer. Anal. 1979,10: 665～673
[31] Chui C K. Multivariate splines: theory and applications. New York: SIAM, 1987
[32] 徐佩霞，孙功宪．小波分析与实用实例．合肥：中国科学技术大学出版社，1996
[33] D Gabor. Theory of Communication. J. Inst. Elect. Eng. (London), 1946, 93(Ⅲ): 492-457
[34] M V Wickerhauser. Acoustic Signal Compression with Wavelet Packets. Wavelet Analysis and its Application. Chui C K (et.). Academic Press, Inc, 1992: 679～700
[35] Y Meyer. Wavelets: Algorithms and Applications. SIAM, Philadelphia, 1993: 95～99
[36] 崔锦泰 小波分析导论 西安 西安交通大学出版社1992
[37] D Gabor. theory of Communication. J. Inst.. Elect. Eng.(London), 1946, 93(Ⅲ): 429～457
[38] 雷斌、杨宗凯等，线性时频变换在弱信号检测中的应用，电子科学杂志．Vol．3 1993
[39] 黄正强等．基于小波包的信号检测算法．华中理工大学学报，1994，22(11)：32～35
[40] 冯玉珉，邵玉明，张星编译．数据图像压缩编码．北京：中国铁道出版社，1993
[41] Daubechies I. The wavelet transform,. time-frequency localization and signal analysis. IEEE Trans Inform Theory, 1990, 36: 961-1005
[42] Daubechies I and Grossmann A. Frames in the Bargmann space of entire functions. Comm Pure and Appl Math, 1988, 41: 151-164
[43] Mallat S and Zhong S. Characterization of signals from multiscale edges. IEEE Trans Part Anal and Math Intell, 1992, 14: 710-732
[44] 梁学章，金光日．二元样条函数与小波．天津计算技术学年会文集，1991
[45] 侯遵泽，杨文采．小波分析应用研究．物探化探计算技术，1995，17：1～9
[46] Mallat S and Hwang W L. Singularity detection and processing with wavelet. IEEE Trans. on Information Theory, 1192, 38(2): 617～643
[47] Mallat. S. A theory for multiresolution signal decompositions-the wavelet representation. IEEE Trans Pattern and Machine Intel., 1989, 11: 674～693
[48] 平玉娇．利用小波变换进行图像数据压缩．重庆：重庆大学，1995
[49] 田逢春，金吉成．数字信号二进小波变换的滤波器与边界失真．重庆大学学报，1996，19(5)：23～29
[50] 龙瑞麟．高维小波分析．北京：世界图书出版社，1995．1～362
[51] 李东舸，柳健．小波理论及其在图像处理中的应用．无线电工程，1994，24(4)：19～27
[52] 王玉平，蔡元龙．小波分析在信号处理中的应用．无线电工程，1994，24(3)：11～18
[53] 程俊等．小波变换用于突变检测．通信学报，1995，16(3)96～104
[54] 钟声．图像压缩技术及其应用．电子学报，1995，23(10)：117～123
[55] 柳树田，徐建东，张岩，李淳飞．透镜组合实现分数傅里叶变换．光学学报，1995，10：1404～1408
[56] O Aytur, H M zaktas, Nonorthogonal Domains in phase Space of Quantum Optics and Their Relation to Fractional Fourier Transform. Opt Comm.. 1995, 120: 166～170
[57] S Y Lee, H H Szu. Franctional Fourier Transform, Wavelet Transforms and Adaptive Networks. Optical Engineering. 1994, 33(7): 23～26
[58] Daubechies I. Time-frequency localization operators: a geometric phase space approach. IEEE Trans Inform Theory, 1998, 34: 605～612
[59] 马晶，谭立英，冉启文．光学小波滤波理论初探．中国激光，1999，26(4)：1～4
[60] 孟昭波，杨丽华．地震资料的小波压缩．石油地球物理勘探，1995，30(2)：70～75
[61] S Mallat, S Zhong. Characterization of Signals from Multiscale Edges．第三次中法小波会议
论文集．
[62] 程正兴．国外小波及其应用概况．工程力学学报，1992，9(3)：125～126
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