时频峰值滤波在地震勘探资料中随机噪声压制的研究
|
摘要
传统的地学随机噪声压制技术多是基于有效纵波在空间上相关,而随机干扰在空间上不相关的性质来进行噪声压制的。这些方法有时受到假设条件的限制使处理效果变差,而这种假设往往并不符合地震勘探的实际情况。
时频峰值滤波算法是一种新颖的基于时频分析的信号增强算法,能够有效的消除噪声,恢复有效波信息。本文将这种时频分析算法用于消减地震勘探资料中的随机噪声。无需假设前提和已知条件如视速度等,从而更符合地震勘探领域的实际应用条件。
文中根据地震记录即地表检波器接收到的地震信号构成的特点,得到合成地震记录的数学模型,并讨论了双曲时距方程的精度和Ricker子波频率的问题。然后通过合成地震记录来验证算法,将时频峰值滤波算法分别应用于具有单个同相轴,多个同相轴的地震资料和一些复杂的地震资料中,而恢复出来的共炮点记录可以清楚地表现原始记录同相轴的位置,并且将湮没在随机噪声下的Ricker子波基本上恢复出来。经过滤波前后的子波形态、Wigner-Ville分布、傅立叶谱等的比较,验证了该方法可以有效地消减地震资料中的噪声。
实验结果表明,在无需任何假设条件情况下,有用信息被基本无损伤地保留下来。
IN practice, signals are often corrupted by noise, and this has the effect ofhindering the recovery of important information encoded in the signal. All signalprocessing fields including radar, sonar, communications, seismology, andbiomedicine suffer from this problem. The performance of signal enhancementalgorithms generally depends on the signal-to-noise ratio (SNR). Many signalprocessing algorithms work well for high SNR situations, but most perform poorlywhen SNR decreases below a given threshold. In this case, if no adequatealternative algorithm is available, preprocessing or filtering is required to improvethe SNR.
Adaptive and fixed methods have been developed for signal enhancement ofnonstationary signals in noise. Although both approaches have merits, adaptivetechniques are generally superior in performance to fixed methods when the signalstatistics are nonstationary and unknown. Some of the well-known adaptivefiltering techniques are based on the least-mean-squares (LMS) algorithm, therecursive least-squares algorithm, and the Kalman filter (KF).
Adaptive filters could not perform very well in certain conditions, however.An example of this is filtering of a nonstationary signal whose spectral contentchanges quickly with time. The filter designed using LMS approach, for instance,may not adapt quickly enough to track the rapidly changing signal. This is due tothe delayed convergence of the algorithm, which is a function of the signalautocorrelation matrix eigenvalues. Further, in many signal processing applicationssuch as electroencephalogram (EEG) signal, the structure of the underlying signalis often unknown and too complicated to model accurately. An attempt to provide amodel framework could probably lead to suboptimal results.
Recently, efforts have been made to develop Time-frequency analysis, whichdescribes the frequency varying with time. This method uses the jointtime-frequency domain to analysis signal and maps the one-dimensional signal totwo-dimensional time-frequency plane. It shows the information of the signal in
both time domain and frequency domain. Time-frequency analysis can be used toexhibit significant energy concentration around the signal's IF and develop thetime-frequency filtering.In 2004, Time-frequency peak filtering (TFPF) have recently been used byB.Boashash to enhance signals, which may be represented as a sum of bandlimitednonstationary processes in additive white Gaussian noise(WGN). The noisy signalis encoded as the IF of a unit amplitude frequency modulated (FM) analytic signal.The instantaneous frequency (IF) of the analytic signal is then estimated usingstandard time-frequency peak detection methods based WVD to obtain an estimateof the underlying deterministic signal. TFPF which is a noncoherent method hasalready been applied to Newborn EEG data.The aim of this paper is to use TFPF to suppress random noise in a reflectionseismic data. The experiments show that a very clean recovery of seismic reflectiondata can be accomplished. Therefore it indicates the efficiency of the algorithm as anoise-eliminated method for seismic data.Seismic data is the important information resource of geological prospect andexploration. Random noise in seismic reflection data can be introduced by varioussources and is often a problem in geophysical data visualization because it obscuresfine details and complicates identification of image features.In present, there are basically two kinds of methods to reduce the noise inseismic data: signal enhancement algorithms and noise suppression algorithms.These methods including f ? x deconvolution, polynomial fitting, radialpredictive deconvolution, belong to the signal enhancement algorithms, and thevector decomposition belongs to the noise suppression algorithms. These methodswere often used in practice. However, some assumptions of seismic data for themethods above will lead to suboptimal results about the signal. Generally speaking,the assumptions are that effective primary wave is correlated in space, whereasrandom noise is not correlated in space. One-dimensional noise reduction methodswere studied very little.In this paper, Time-frequency peak filtering is used to reconstruct signals from
reflection seismic data corrupted by random noise. This method which is aone-dimensional signal enhancement algorithm doesn't need any assumptions toseismic data.Firstly, this paper discussed the basic principle and algorithm realization ofTFPF. Secondly get the signal model according to the characteristic of seismic data.This signal model showed the signal composition in reason. Thirdly, simulationwas conducted on synthetic seismic data with an event or multi-event andcomplicated seismic data. Simulations have demonstrated that TFPF could reducerandom noise effectively. In comparison with other methods, TFPF used thediscarding seismic recordings, so it could save economic resources. Finally, weanalyzed the effect of each parameter on TFPF to assist the simulation. There arethree factors: window length, sample frequency and the noise. When implementingthe experiments, we should pay attention to these factors according to the seismicdata.The different experiments conducted demonstrated that this method of signalenhancement could clearly show the position of events and the Ricker wavelets.We compared seismic data corrupted in noise with the filtering result from thewavelets, Wigner-Ville distribution and Fourier spectrum. It was concluded thatreducing random noise of seismic data by TFPF is feasible and has research value.
时频峰值滤波算法是一种新颖的基于时频分析的信号增强算法,能够有效的消除噪声,恢复有效波信息。本文将这种时频分析算法用于消减地震勘探资料中的随机噪声。无需假设前提和已知条件如视速度等,从而更符合地震勘探领域的实际应用条件。
文中根据地震记录即地表检波器接收到的地震信号构成的特点,得到合成地震记录的数学模型,并讨论了双曲时距方程的精度和Ricker子波频率的问题。然后通过合成地震记录来验证算法,将时频峰值滤波算法分别应用于具有单个同相轴,多个同相轴的地震资料和一些复杂的地震资料中,而恢复出来的共炮点记录可以清楚地表现原始记录同相轴的位置,并且将湮没在随机噪声下的Ricker子波基本上恢复出来。经过滤波前后的子波形态、Wigner-Ville分布、傅立叶谱等的比较,验证了该方法可以有效地消减地震资料中的噪声。
实验结果表明,在无需任何假设条件情况下,有用信息被基本无损伤地保留下来。
IN practice, signals are often corrupted by noise, and this has the effect ofhindering the recovery of important information encoded in the signal. All signalprocessing fields including radar, sonar, communications, seismology, andbiomedicine suffer from this problem. The performance of signal enhancementalgorithms generally depends on the signal-to-noise ratio (SNR). Many signalprocessing algorithms work well for high SNR situations, but most perform poorlywhen SNR decreases below a given threshold. In this case, if no adequatealternative algorithm is available, preprocessing or filtering is required to improvethe SNR.
Adaptive and fixed methods have been developed for signal enhancement ofnonstationary signals in noise. Although both approaches have merits, adaptivetechniques are generally superior in performance to fixed methods when the signalstatistics are nonstationary and unknown. Some of the well-known adaptivefiltering techniques are based on the least-mean-squares (LMS) algorithm, therecursive least-squares algorithm, and the Kalman filter (KF).
Adaptive filters could not perform very well in certain conditions, however.An example of this is filtering of a nonstationary signal whose spectral contentchanges quickly with time. The filter designed using LMS approach, for instance,may not adapt quickly enough to track the rapidly changing signal. This is due tothe delayed convergence of the algorithm, which is a function of the signalautocorrelation matrix eigenvalues. Further, in many signal processing applicationssuch as electroencephalogram (EEG) signal, the structure of the underlying signalis often unknown and too complicated to model accurately. An attempt to provide amodel framework could probably lead to suboptimal results.
Recently, efforts have been made to develop Time-frequency analysis, whichdescribes the frequency varying with time. This method uses the jointtime-frequency domain to analysis signal and maps the one-dimensional signal totwo-dimensional time-frequency plane. It shows the information of the signal in
both time domain and frequency domain. Time-frequency analysis can be used toexhibit significant energy concentration around the signal's IF and develop thetime-frequency filtering.In 2004, Time-frequency peak filtering (TFPF) have recently been used byB.Boashash to enhance signals, which may be represented as a sum of bandlimitednonstationary processes in additive white Gaussian noise(WGN). The noisy signalis encoded as the IF of a unit amplitude frequency modulated (FM) analytic signal.The instantaneous frequency (IF) of the analytic signal is then estimated usingstandard time-frequency peak detection methods based WVD to obtain an estimateof the underlying deterministic signal. TFPF which is a noncoherent method hasalready been applied to Newborn EEG data.The aim of this paper is to use TFPF to suppress random noise in a reflectionseismic data. The experiments show that a very clean recovery of seismic reflectiondata can be accomplished. Therefore it indicates the efficiency of the algorithm as anoise-eliminated method for seismic data.Seismic data is the important information resource of geological prospect andexploration. Random noise in seismic reflection data can be introduced by varioussources and is often a problem in geophysical data visualization because it obscuresfine details and complicates identification of image features.In present, there are basically two kinds of methods to reduce the noise inseismic data: signal enhancement algorithms and noise suppression algorithms.These methods including f ? x deconvolution, polynomial fitting, radialpredictive deconvolution, belong to the signal enhancement algorithms, and thevector decomposition belongs to the noise suppression algorithms. These methodswere often used in practice. However, some assumptions of seismic data for themethods above will lead to suboptimal results about the signal. Generally speaking,the assumptions are that effective primary wave is correlated in space, whereasrandom noise is not correlated in space. One-dimensional noise reduction methodswere studied very little.In this paper, Time-frequency peak filtering is used to reconstruct signals from
reflection seismic data corrupted by random noise. This method which is aone-dimensional signal enhancement algorithm doesn't need any assumptions toseismic data.Firstly, this paper discussed the basic principle and algorithm realization ofTFPF. Secondly get the signal model according to the characteristic of seismic data.This signal model showed the signal composition in reason. Thirdly, simulationwas conducted on synthetic seismic data with an event or multi-event andcomplicated seismic data. Simulations have demonstrated that TFPF could reducerandom noise effectively. In comparison with other methods, TFPF used thediscarding seismic recordings, so it could save economic resources. Finally, weanalyzed the effect of each parameter on TFPF to assist the simulation. There arethree factors: window length, sample frequency and the noise. When implementingthe experiments, we should pay attention to these factors according to the seismicdata.The different experiments conducted demonstrated that this method of signalenhancement could clearly show the position of events and the Ricker wavelets.We compared seismic data corrupted in noise with the filtering result from thewavelets, Wigner-Ville distribution and Fourier spectrum. It was concluded thatreducing random noise of seismic data by TFPF is feasible and has research value.
引文
[1] Cohen L, Time-frequency analysis, Prentice-Hall,1995
[2] 王宏禹,非平稳随机信号分析与处理,国防工业出版社,1999
[3] 张贤达,保铮,非平稳信号分析与处理.国防工业出版社,1998
[4] Koening R,Dunn H K, Lacy L Y, The sound spectrograph. J.Acoust. Soc.Amcr.,1946,18:19~49
[5] Wigner E P, On the quantum correction for thermodynamic equilibrium, Phys.Rev,1932,40
[6] Zhenyu Guo, Louis-Gilles, Durand, Howard C. Lee, The Time-Frequency distributions of. Nonstationary Signals Based on a Bessel Kernel, IEEE Trans. on Sig. Proc., 1993, 41(4):1589~1602
[7] Choi H I,Williams W J, Improved time-frequency representation of multicomponent signals using exponential kernels, IEEE Trans. Acoust., Speech, Signal Processing, 1989.37:862~871
[8] Cohen L, Generalized phase-space distribution functions,Math.Phys,1966
[9] Baraniuk T G, Jones D L, A signal-dependent time-frequency analysis using a radially Gaussian kernel, Signal Processing, 1993,32(3):263~284
[10] Daubechies O, The wavelet transform, time frequency localization and signal analysis, IEEE Trans IT, 1990,36(5):961～1005
[11] 保铮,孙晓兵.时频平面分析非平稳信号的研究进展.电子科技导报 1996,4:2~4
[12] Barkat B and Boashash B, Design of higher order polynomial Wigner–Ville distributions IEEE Trans. Signal Processing, 1999,47(9):2608~2611.
[13] Ristic B and Boashash B The relationship between polynomial and the high order Wigner-Ville distributions IEEE Signal Processing letter. 1995, 2(12):227~229
[14] Prabhu K M M and Giridhar J Detection performance of an adaptive MTD withWVD as a Doppler filter bank ELSEVIER Signal Processing, 2001, 81:693~698
[15] 李国发,李观寿,张立勤. 利用时空域预测实现地震资料随机噪声衰减[J]. 石油地球物理勘探,1998,33(1):122～128
[16]Canales L. Random noise reduction. the 54th Annual Meeting of SEG, 1984
[17] Yu S, Cai X, Su Y. Seismic signal enhancement by polynomial fitting[J]. Applied Geophysics, 1989, 1:57～65
[18] 张军华, 吕宁, 田连玉, 陆文志, 钟磊. 地震资料去噪方法、技术综合评述[J]. 地球物理学进展,2005,20(4):1083~1091
[19] Christian B?nnemann,Burkhard Buttkus. Estimation of seismic velocities from deep reflection in the τ -pdomain. Pure and Applied Geophysics, 1999,156:279~301
[20]Alan R.Michell, Panos G.Kelamis. Efficient tau-p hyperbolic velocity filtering. Geophysics, 1990, 55(5):619~625
[21] Theophanis S, Queen J. Color display of the localized spectrum. Geophysics, 2000, 65(4):1330~1340
[22] 夏洪瑞,朱勇,周开明.小波变换及其在去噪中的应用.石油地球物理勘探,1994,29(3):274～285.
[23] 王振国,周熙襄,李晶.基于小波变换的最小光滑滤波去噪[J].石油地球物理勘探,2002,37(6):594～600.
[24] 邹文,陈爱萍,顾汉明. 联合时频分析在地震勘探中的应用[J]. 勘探地球物理进展,2004,27(4),246～250
[25] 吴正国,夏立,尹为民,现代信号处理技术,武汉大学出版社,2003
[26] 邹红星,周小波,李衍达,时频分析:回搠与前瞻,电子学报,2000,28(9):78～84
[27] Haykin S, Neural Networks Expand SP's Horizons, IEEE Signal Processing Magazine,1996,24~46
[28] Auger F, Flandrin P, Improving the readability of time-frequency and time-scale representations by the reassignment method, IEEE Trans. Signal Processing, 1995, 43:1068~1089
[29] Haykin S, Analog and Digital Communications, New York: Wiley, 1989.
[30] Boashash B, Estimating and interpreting the instantaneous frequency of a signal—Part 1: fundamentals;part 2: algorithms and applications, Proc. IEEE, 1992,80:520~538
[31] Amin M G, Time-frequency spectrum analysis and estimation for nonstationary random processes, in Time-Frequency Signal Analysis, B.Boualem Boashash, Ed. London, U.K.: Longman Cheshire, 1992,208~232.
[32] Boashash B and O'Shea P J, Polynomial Wigner–Ville distributions and their relationship to time-varying higher order spectra, IEEE Trans.Signal Processing, 1992, 42:216~220.
[33] Wilbur J and Bono J, Wigner/cycle spectrum analysis of spread spectrum and diversity transmissions, IEEE J. Ocean. Eng.,1991, 16:98~105
[34] Roessgen M, A comparative study of spectral estimation techniques for noisy nonstationary EEG data, in Proc. Workshop Signal Process, 1993,191~198
[35] Arnold M J, Roessgen M, and Boashash B, Signal filtering using frequency encoding and the time-frequency plane, in Proc. Asilomar Conf., Pacific Grove, CA,1993,1459~1463.
[36] Arnold M J, Roessgen M, and Boashash B, Filtering real signals through frequency modulation and peak detection in the time-frequency plane, in Proc. ICASSP, Adelaide, Australia, 1994, 345~348.
[37] Claasen T A C M. and Mecklenbrauker , W F G. , The Wigner distribution—A tool for time-frequency signal analysis, Philips J. Res., pt. 1, 1981, 35: 217~250 (Part 1), 276~300 (Part 2), 372~389 (Part 3)
[38] Boashash B, Time-frequency signal analysis, in Advances in Spectral Estimation and Array Processing, S. Haykin, Ed. Englewood Cliffs, NJ: Prentice-Hall, 1990, 1:418~517.
[39] Papoulis A, Random modulation: a review, IEEE Trans. Acoust., Speech, Signal Processing, 1983, vol. ASSP-31:96~105
[40] Stankovic′ L J and Katkovnik V, The Wigner distribution of noisy signals with adaptive time-frequency varying window, IEEE Trans. Signal Processing ,1999, 47(4):1099~1108
[41] Boashash B, Signal enhancement by time-frequency peak Filtering, IEEE Trans. Signal Processing,2004, 52(4):929~937
[42] Ristau J P, Wooil M. Moon, Adaptive filtering of random noise in 2-D geophysical data, Geophysics, 2001, 66:342～349.
[43] Erick Baziw, Iain Weir-Jones, Application of Kalman filtering techniques for microseismic event detection, Pure and Applied Geophysics, 2002,159:449-471
[44] 李月,杨宝俊,赵雪平等,检测地震勘探微弱同相轴的混沌振子算法,地球物理学报,2005,48(6):1428～1433
[45] 胡天跃, 地震资料叠前去噪技术的现状与未来.地球物理学进展,2002,17(2):218～223
[46] 崔若飞, 王辉, 多项式拟合技术在煤田地震勘探中的应用, 地球物理学进展,2000,15(2):47～54
[47] 董恩清,刘贵忠,张宗平, 自适应 Kalman 滤波反褶积的快速实现方法,地球物理学报,2001,44(2):255～262
[48] 王振国,周熙襄,李晶, 基于小波变换的最小光滑滤波去噪, 石油地球物理勘探,2002,37(6):594～600
[49] 张中杰, 地震各向异性研究进展, 地球物理学进展,2002,17(2):281～293
[50] 杨选辉,沈萍,刘希强,郑治真, 地震与核爆识别的小波包分量比方法, 地球物理学报,2005,48(1):148～156
[51] Brian C. Lovell,Robert C.Williamson,B.Boashash, The relationship between Instantaneous frequency and time-frequency representations, IEEE Trans. Signal Processing,1993,41(3):1458～1461 .
[52] Taner M T and Koehler F, Velocity spectra-digital computer derivation and applications of velocity function, Geophysics, 1969, 34(6):859～881
[53] 刘洋,双曲时距方程对地震资料处理的影响,石油地球物理勘探,2004,39(2):133～138
[54] 俞寿朋,宽带Ricker子波,石油地球物理勘探,1996,31(5):605～616
[55] 杨宝俊,牛滨华,闫贫,勘探地震学导论,吉林科学技术出版社,1981, 56～98
[56] Kallweit R S, Wood L C, The limits of resolution of zero-phase wavelets, Geophysics,1982, 47(7):1035~1046
[57] Quan Y L, Harris J M, Seismic attenuation tomography using the frequency shift method, Geophysics,1997,62(3):895~905
[58] Neidell N S and Poggiogliolmi E, Stratigraphic modeling and inter-pretation-Geophysical principles and techniques, AAPG Memoir, 1977, 26:389~416
[59] 云美厚,丁伟,地震子波频率浅析,石油物探,2005,44(6):578～ 581
[60] Dobrin, M.B, Introduction to Geophysical Prospecting, McGraw-Hill Book Co., New York, NY., 1976.
[61] Taner M T, Koehler F and Sheriff R E, Complex seismic trace analysis,Geophysics,1979, 44(6):1041~1063
[2] 王宏禹,非平稳随机信号分析与处理,国防工业出版社,1999
[3] 张贤达,保铮,非平稳信号分析与处理.国防工业出版社,1998
[4] Koening R,Dunn H K, Lacy L Y, The sound spectrograph. J.Acoust. Soc.Amcr.,1946,18:19~49
[5] Wigner E P, On the quantum correction for thermodynamic equilibrium, Phys.Rev,1932,40
[6] Zhenyu Guo, Louis-Gilles, Durand, Howard C. Lee, The Time-Frequency distributions of. Nonstationary Signals Based on a Bessel Kernel, IEEE Trans. on Sig. Proc., 1993, 41(4):1589~1602
[7] Choi H I,Williams W J, Improved time-frequency representation of multicomponent signals using exponential kernels, IEEE Trans. Acoust., Speech, Signal Processing, 1989.37:862~871
[8] Cohen L, Generalized phase-space distribution functions,Math.Phys,1966
[9] Baraniuk T G, Jones D L, A signal-dependent time-frequency analysis using a radially Gaussian kernel, Signal Processing, 1993,32(3):263~284
[10] Daubechies O, The wavelet transform, time frequency localization and signal analysis, IEEE Trans IT, 1990,36(5):961～1005
[11] 保铮,孙晓兵.时频平面分析非平稳信号的研究进展.电子科技导报 1996,4:2~4
[12] Barkat B and Boashash B, Design of higher order polynomial Wigner–Ville distributions IEEE Trans. Signal Processing, 1999,47(9):2608~2611.
[13] Ristic B and Boashash B The relationship between polynomial and the high order Wigner-Ville distributions IEEE Signal Processing letter. 1995, 2(12):227~229
[14] Prabhu K M M and Giridhar J Detection performance of an adaptive MTD withWVD as a Doppler filter bank ELSEVIER Signal Processing, 2001, 81:693~698
[15] 李国发,李观寿,张立勤. 利用时空域预测实现地震资料随机噪声衰减[J]. 石油地球物理勘探,1998,33(1):122～128
[16]Canales L. Random noise reduction. the 54th Annual Meeting of SEG, 1984
[17] Yu S, Cai X, Su Y. Seismic signal enhancement by polynomial fitting[J]. Applied Geophysics, 1989, 1:57～65
[18] 张军华, 吕宁, 田连玉, 陆文志, 钟磊. 地震资料去噪方法、技术综合评述[J]. 地球物理学进展,2005,20(4):1083~1091
[19] Christian B?nnemann,Burkhard Buttkus. Estimation of seismic velocities from deep reflection in the τ -pdomain. Pure and Applied Geophysics, 1999,156:279~301
[20]Alan R.Michell, Panos G.Kelamis. Efficient tau-p hyperbolic velocity filtering. Geophysics, 1990, 55(5):619~625
[21] Theophanis S, Queen J. Color display of the localized spectrum. Geophysics, 2000, 65(4):1330~1340
[22] 夏洪瑞,朱勇,周开明.小波变换及其在去噪中的应用.石油地球物理勘探,1994,29(3):274～285.
[23] 王振国,周熙襄,李晶.基于小波变换的最小光滑滤波去噪[J].石油地球物理勘探,2002,37(6):594～600.
[24] 邹文,陈爱萍,顾汉明. 联合时频分析在地震勘探中的应用[J]. 勘探地球物理进展,2004,27(4),246～250
[25] 吴正国,夏立,尹为民,现代信号处理技术,武汉大学出版社,2003
[26] 邹红星,周小波,李衍达,时频分析:回搠与前瞻,电子学报,2000,28(9):78～84
[27] Haykin S, Neural Networks Expand SP's Horizons, IEEE Signal Processing Magazine,1996,24~46
[28] Auger F, Flandrin P, Improving the readability of time-frequency and time-scale representations by the reassignment method, IEEE Trans. Signal Processing, 1995, 43:1068~1089
[29] Haykin S, Analog and Digital Communications, New York: Wiley, 1989.
[30] Boashash B, Estimating and interpreting the instantaneous frequency of a signal—Part 1: fundamentals;part 2: algorithms and applications, Proc. IEEE, 1992,80:520~538
[31] Amin M G, Time-frequency spectrum analysis and estimation for nonstationary random processes, in Time-Frequency Signal Analysis, B.Boualem Boashash, Ed. London, U.K.: Longman Cheshire, 1992,208~232.
[32] Boashash B and O'Shea P J, Polynomial Wigner–Ville distributions and their relationship to time-varying higher order spectra, IEEE Trans.Signal Processing, 1992, 42:216~220.
[33] Wilbur J and Bono J, Wigner/cycle spectrum analysis of spread spectrum and diversity transmissions, IEEE J. Ocean. Eng.,1991, 16:98~105
[34] Roessgen M, A comparative study of spectral estimation techniques for noisy nonstationary EEG data, in Proc. Workshop Signal Process, 1993,191~198
[35] Arnold M J, Roessgen M, and Boashash B, Signal filtering using frequency encoding and the time-frequency plane, in Proc. Asilomar Conf., Pacific Grove, CA,1993,1459~1463.
[36] Arnold M J, Roessgen M, and Boashash B, Filtering real signals through frequency modulation and peak detection in the time-frequency plane, in Proc. ICASSP, Adelaide, Australia, 1994, 345~348.
[37] Claasen T A C M. and Mecklenbrauker , W F G. , The Wigner distribution—A tool for time-frequency signal analysis, Philips J. Res., pt. 1, 1981, 35: 217~250 (Part 1), 276~300 (Part 2), 372~389 (Part 3)
[38] Boashash B, Time-frequency signal analysis, in Advances in Spectral Estimation and Array Processing, S. Haykin, Ed. Englewood Cliffs, NJ: Prentice-Hall, 1990, 1:418~517.
[39] Papoulis A, Random modulation: a review, IEEE Trans. Acoust., Speech, Signal Processing, 1983, vol. ASSP-31:96~105
[40] Stankovic′ L J and Katkovnik V, The Wigner distribution of noisy signals with adaptive time-frequency varying window, IEEE Trans. Signal Processing ,1999, 47(4):1099~1108
[41] Boashash B, Signal enhancement by time-frequency peak Filtering, IEEE Trans. Signal Processing,2004, 52(4):929~937
[42] Ristau J P, Wooil M. Moon, Adaptive filtering of random noise in 2-D geophysical data, Geophysics, 2001, 66:342～349.
[43] Erick Baziw, Iain Weir-Jones, Application of Kalman filtering techniques for microseismic event detection, Pure and Applied Geophysics, 2002,159:449-471
[44] 李月,杨宝俊,赵雪平等,检测地震勘探微弱同相轴的混沌振子算法,地球物理学报,2005,48(6):1428～1433
[45] 胡天跃, 地震资料叠前去噪技术的现状与未来.地球物理学进展,2002,17(2):218～223
[46] 崔若飞, 王辉, 多项式拟合技术在煤田地震勘探中的应用, 地球物理学进展,2000,15(2):47～54
[47] 董恩清,刘贵忠,张宗平, 自适应 Kalman 滤波反褶积的快速实现方法,地球物理学报,2001,44(2):255～262
[48] 王振国,周熙襄,李晶, 基于小波变换的最小光滑滤波去噪, 石油地球物理勘探,2002,37(6):594～600
[49] 张中杰, 地震各向异性研究进展, 地球物理学进展,2002,17(2):281～293
[50] 杨选辉,沈萍,刘希强,郑治真, 地震与核爆识别的小波包分量比方法, 地球物理学报,2005,48(1):148～156
[51] Brian C. Lovell,Robert C.Williamson,B.Boashash, The relationship between Instantaneous frequency and time-frequency representations, IEEE Trans. Signal Processing,1993,41(3):1458～1461 .
[52] Taner M T and Koehler F, Velocity spectra-digital computer derivation and applications of velocity function, Geophysics, 1969, 34(6):859～881
[53] 刘洋,双曲时距方程对地震资料处理的影响,石油地球物理勘探,2004,39(2):133～138
[54] 俞寿朋,宽带Ricker子波,石油地球物理勘探,1996,31(5):605～616
[55] 杨宝俊,牛滨华,闫贫,勘探地震学导论,吉林科学技术出版社,1981, 56～98
[56] Kallweit R S, Wood L C, The limits of resolution of zero-phase wavelets, Geophysics,1982, 47(7):1035~1046
[57] Quan Y L, Harris J M, Seismic attenuation tomography using the frequency shift method, Geophysics,1997,62(3):895~905
[58] Neidell N S and Poggiogliolmi E, Stratigraphic modeling and inter-pretation-Geophysical principles and techniques, AAPG Memoir, 1977, 26:389~416
[59] 云美厚,丁伟,地震子波频率浅析,石油物探,2005,44(6):578～ 581
[60] Dobrin, M.B, Introduction to Geophysical Prospecting, McGraw-Hill Book Co., New York, NY., 1976.
[61] Taner M T, Koehler F and Sheriff R E, Complex seismic trace analysis,Geophysics,1979, 44(6):1041~1063