基于奇异值分解的Radon域多次波压制法研究
摘要
地震勘探数据处理中,由于地震剖面中的一次波(有效波)和多次波的性质非常相似,所以很难准确预测出剖面中的多次波,因此要想在地震剖面中有效分离出多次波是地震信号处理中的一个难题。多次波压制的效果不仅影响地震资料的后续处理(如速度分析、偏移成像等),而且也对地震剖面的解释(产生假构造等)造成直接的影响。
国内外地球物理学家们长期以来为解决多次波的压制问题经过了不懈的努力,多种压制多次波的方法被提出。但大体可分为两类:一类是基于一次波和多次波在一些特征和性质上存在差别的滤波法;另一类是基于波动方程的预测相减法。虽然所提出的方法都具有一定的实用性,但同时也存在一些不足。如基于Radon变换法压制多次波要想有效滤除多次波,要求有效波和多次波能量在p/q域中能够准确分离,然而实际数据变换到Radon域时,存在严重的“拖尾”现象,这给多次波的有效去除带来了很大的影响。由于实际地震数据的复杂性,在所采集到的地震记录中,有效波和多次波同相轴有可能重叠,在重叠区域处理效果不佳也是许多多次波压制方法普遍存在的问题。
本文在传统基于Radon变换法压制多次波的基础上进行了改进,相对于传统基于Radon变换法压制多次波,该方法克服了在Radon域中存在“拖尾”现象而造成不能有效压制多次波,有效信号丢失等问题,且具有较强的容噪能力。其处理过程大致分为以下几步:(1)应用奇异值分解技术对经部分动校正处理后的地震记录进行多次波预测,由于奇异值分解技术不可能准确地估计出记录中的多次波,这样预测的多次波中含有有效波成分,同时在地震记录中仍然残留多次波,因此我们将该预测的多次波作为后续滤波处理的初始多次波模型;(2)应用高分辨率抛物Radon变换将原始地震记录及预测的初始多次波模型变换到抛物Radon域;(3)在抛物Radon域中应用滤波函数进行多次波压制。对于一些比较复杂的实际数据,可以考虑引入多重滤波思想,将前一次滤波处理得到的多次波作为后续滤波处理的初始多次波模型,这样得到的处理效果会更好,但是计算量也将会相应的加大,所以是否进行多次滤波处理,取决于当前滤波处理的效果。
通过对合成记录试算和实际数据处理,处理效果表明该方法能够很好的压制多次波,同时对有效波有较好的保幅作用,并且能够压制动校正后记录中残余的随机噪声,在有效波和多次波同相轴交叉部位也具有理想的处理效果。
Multiple attenuation is always a hot and difficult problem in processing seismic signal. As the nature of primary significant wave and Multiple on seismic profile is very similar, it is difficult to accurately predict the Multiple of the profile; therefore, how to separate the Multiple is a difficult problem of seismic signal processing. The quality of Multiple suppression not only affects the subsequent processing of seismic data (such as velocity analysis, migration imaging, etc), but also directly affects the interpretation of seismic profile (producing false structures, etc).
Over the years, geophysicists at home and abroad have made unremitting efforts to solve the problem of Multiple suppression, proposed a variety of methods to suppress Multiple. Generally they can be divided into two types: one is the filtering method based on the differ of the characteristics and nature of the primary significant wave and Multiple,;the other is predicting subtraction based on wave equations. Although these traditional methods are practical, but there are also some disadvantages,such as the Radon transform, in order to effectively suppress Multiple, requiring the energy of significant wave and Multiple in the p / q domain can be accurately separated, however, when the real seismic data transformed to the Radon domain, there is serious "tail" phenomenon, which gives a great influence on Multiple attenuation. Because the real seismic signal is complicated, there are some overlap section of same phase axis of effective signal and Multiple. However many traditional methods have a generalize deficiency of bad removing Multiple effect in same phase axis overlap section of effective signal and Multiple.
A method using a filter function in High Precision Parabolic Radon domain to suppress Multiple is presented in this paper. it improved some traditional way of multiple attenuation based on Radon Transform, Compared with these traditional ways of multiple attenuation, This method overcomes the phenomenon of Radon domain with bad tail, and it obtained a good effect with a high precision. The process of this method in this paper is roughly divided into the following steps: firstly, forecasting Multiple model from partial NMO seismic record by using SVD(singular value decomposition) technique, we make the Multiple obtained by SVD as the initial Multiple model, In fact using such prediction, the Multiple contains effective signal, while effective signal also contains multiple. secondly, transforming original seismic records and prediction model of the initial Multiple to the parabolic Radon domain by using high resolution parabolic Radon transform. lastly, suppressing Multiple by using a filter function in the high resolution parabolic Radon domain. For some real seismic signal is more complicated, we can consider introducing the repetitive filtering thoughts. making Multiple obtained by the last filtering processing as the next time filtering initial Multiple model, the processing effect will be better, but the computation will increase, whether the repeating filtering should be made, it is decided by the preceding treatment effect.
Through the process of synthetic signal and real seismic signal, we find that this method can suppress Multiple effectively, meanwhile, it has a good performance on keeping the amplitude of effective signal, lastly, it can also remove the random noise in the profile after NMO and has a good performance in the area that the overlap of primary significant wave and Multiple.
国内外地球物理学家们长期以来为解决多次波的压制问题经过了不懈的努力,多种压制多次波的方法被提出。但大体可分为两类:一类是基于一次波和多次波在一些特征和性质上存在差别的滤波法;另一类是基于波动方程的预测相减法。虽然所提出的方法都具有一定的实用性,但同时也存在一些不足。如基于Radon变换法压制多次波要想有效滤除多次波,要求有效波和多次波能量在p/q域中能够准确分离,然而实际数据变换到Radon域时,存在严重的“拖尾”现象,这给多次波的有效去除带来了很大的影响。由于实际地震数据的复杂性,在所采集到的地震记录中,有效波和多次波同相轴有可能重叠,在重叠区域处理效果不佳也是许多多次波压制方法普遍存在的问题。
本文在传统基于Radon变换法压制多次波的基础上进行了改进,相对于传统基于Radon变换法压制多次波,该方法克服了在Radon域中存在“拖尾”现象而造成不能有效压制多次波,有效信号丢失等问题,且具有较强的容噪能力。其处理过程大致分为以下几步:(1)应用奇异值分解技术对经部分动校正处理后的地震记录进行多次波预测,由于奇异值分解技术不可能准确地估计出记录中的多次波,这样预测的多次波中含有有效波成分,同时在地震记录中仍然残留多次波,因此我们将该预测的多次波作为后续滤波处理的初始多次波模型;(2)应用高分辨率抛物Radon变换将原始地震记录及预测的初始多次波模型变换到抛物Radon域;(3)在抛物Radon域中应用滤波函数进行多次波压制。对于一些比较复杂的实际数据,可以考虑引入多重滤波思想,将前一次滤波处理得到的多次波作为后续滤波处理的初始多次波模型,这样得到的处理效果会更好,但是计算量也将会相应的加大,所以是否进行多次滤波处理,取决于当前滤波处理的效果。
通过对合成记录试算和实际数据处理,处理效果表明该方法能够很好的压制多次波,同时对有效波有较好的保幅作用,并且能够压制动校正后记录中残余的随机噪声,在有效波和多次波同相轴交叉部位也具有理想的处理效果。
Multiple attenuation is always a hot and difficult problem in processing seismic signal. As the nature of primary significant wave and Multiple on seismic profile is very similar, it is difficult to accurately predict the Multiple of the profile; therefore, how to separate the Multiple is a difficult problem of seismic signal processing. The quality of Multiple suppression not only affects the subsequent processing of seismic data (such as velocity analysis, migration imaging, etc), but also directly affects the interpretation of seismic profile (producing false structures, etc).
Over the years, geophysicists at home and abroad have made unremitting efforts to solve the problem of Multiple suppression, proposed a variety of methods to suppress Multiple. Generally they can be divided into two types: one is the filtering method based on the differ of the characteristics and nature of the primary significant wave and Multiple,;the other is predicting subtraction based on wave equations. Although these traditional methods are practical, but there are also some disadvantages,such as the Radon transform, in order to effectively suppress Multiple, requiring the energy of significant wave and Multiple in the p / q domain can be accurately separated, however, when the real seismic data transformed to the Radon domain, there is serious "tail" phenomenon, which gives a great influence on Multiple attenuation. Because the real seismic signal is complicated, there are some overlap section of same phase axis of effective signal and Multiple. However many traditional methods have a generalize deficiency of bad removing Multiple effect in same phase axis overlap section of effective signal and Multiple.
A method using a filter function in High Precision Parabolic Radon domain to suppress Multiple is presented in this paper. it improved some traditional way of multiple attenuation based on Radon Transform, Compared with these traditional ways of multiple attenuation, This method overcomes the phenomenon of Radon domain with bad tail, and it obtained a good effect with a high precision. The process of this method in this paper is roughly divided into the following steps: firstly, forecasting Multiple model from partial NMO seismic record by using SVD(singular value decomposition) technique, we make the Multiple obtained by SVD as the initial Multiple model, In fact using such prediction, the Multiple contains effective signal, while effective signal also contains multiple. secondly, transforming original seismic records and prediction model of the initial Multiple to the parabolic Radon domain by using high resolution parabolic Radon transform. lastly, suppressing Multiple by using a filter function in the high resolution parabolic Radon domain. For some real seismic signal is more complicated, we can consider introducing the repetitive filtering thoughts. making Multiple obtained by the last filtering processing as the next time filtering initial Multiple model, the processing effect will be better, but the computation will increase, whether the repeating filtering should be made, it is decided by the preceding treatment effect.
Through the process of synthetic signal and real seismic signal, we find that this method can suppress Multiple effectively, meanwhile, it has a good performance on keeping the amplitude of effective signal, lastly, it can also remove the random noise in the profile after NMO and has a good performance in the area that the overlap of primary significant wave and Multiple.
引文
[1]A. B. WEGLEIN. Multiple-attenuation: An overview of recent advan- ces and the road ahead [J]. The Leading Edge. 1999, 18(1):40-44.
[2]YILMAZ O Z. Seismic data analysis [M]. USA: Society of Exploration Geophysicists, 1993.
[3]SHERIFF R E,GELDART L P,初英等译.勘探地震学[M].北京:石油工业出版社,1999.
[4]陆基孟.地震勘探原理[M].东营:石油大学出版社,1993.164-172.
[5]A. B. WEGLEIN. Multiple attenuation: Recent advances and the road ahead [C]. 65th Ann Internet. Mtg.Soc.Expl. Geophysics. Expanded Abstracts. 1995,1492-1495.
[6]LANDA E, BELFER I and KEYDAR S. Multiple attenuation in the parabolic domain using wavefront characteristics of multiple generating primaries [J]. Geophysics,1999,64:1806-1815..
[7]FOSTER D J, MASHER C C. Suppression of multiple reflection using the radon transform [J]. Geophysics,1992,57:386-395. .
[8]P W CARY. The simplest discrete Radon transform [C].68th Annual SEG Meeting,1998:1999-2002.
[9]MILO M. BACKUS. Water reverberation.their nature and elimation [J].Geophysics,1959,24(2):233-261.
[10]ENDERS A. ROBINSON1, SVEN TREITEL. Principle of digital Wiener filtering [J]. Geophysical Prospecting,1967,15(3):311-333.
[11]AJ HARDING. Slowness-time mapping of near offset seismic reflection data [J]. Geophysical Journal of the Royal Astronomical Society,1985,80:463-492.
[12]D HAMPSON. Inverse velocity stacking of Multiple elimination [J]. Journal of the Canadian society of Exploration Geophysic, 1986,22:44-55.
[13]YILMAZ O Z. Velocity stack processing [J]. Geophysical Prospect- ing,1989, 37(4):357-382.
[14]IF JONES,S lEVY. Signal to noise ratio enhancement in multi-channel seismic data via the Karhunen-Loeve transform [J].Geophy- sical Prospecting,1987,35:12-32.
[15]AL.YAHYA,K.M.Application of the partial Karhunen-Loeve trans- form to suppress random noise in seismic sections [J]. Geophysical Prospeeting,1993,39:77-99.
[16]杨文采.非线性K-L滤波及其在反射地震资料处理中的应用[J].石油物探,1996,35(2):17-26.
[17]谢风兰,赵改善.利用SVD法重建地震剖面[J].石油物探,1990,29(4):33-40.
[18]SERGIO L. M. FREIRE and TAD J.ULRYCH. Application of singular value decomposition to vertical seismic profiling [J]. Geophysics. 1988,53(6): 778-785.
[19]T HU,RE White. Robust Multiple suppression using adaptive beamforming [J]. Geophysical Prospecting,1998,46:227-248.
[20]TIANYUE HU,FEI HONG. Multiple attenuation using beamforming onshore and offshore China [J]. The Leading Edge,2002,21(9):906-910.
[21]胡大跃,王润秋,REWHIET.地震资料处理中的聚束滤波方法[J].地球物理学报,2000,43:105-115.
[22]L MORLEY. Predictive deconvolution in shot receiver space. Geophysics, 1983,48(5):515-531.
[23]JW WIGGINS. Attenuation of complex water bottom multiples by wave equation based prediction and subtraction [J]. Geophysics, 1988,53: 1527-1539.
[24]VERSCHUUR,D.J,BERKHOUT,A.Jand.WAPENAAR,C.P.A.Adaptive surface related multiple elimination [J]. Geophysics, 1992, 57: 1166-1177.
[25]D J VERSCHUUR, A J BERKHOUT and P G KELAMIS. Estimation of mul- tiple scattering by iterative inversion [C]. PartⅡ:Examples on marine and land data:65th Ann.Intermat.Mtg.Soc.Expl.Geophysics, Expanded Abstracts,1995, 1470-1473.
[26]A J BERKHOUT. Multiple removal based on the feedback model [J].The Leading Edge,1999,18(1):127-131.
[27]A B WEGLEIN,F A GASPAROTTO, P M CARVALHO and R H STOLT. An inverse scattering series method for attenuating multiples in seismic reflection data [J]. Geophysics,1997,62:1975-1989.
[28]CARVALHO P.M.,WEGLEIN A.B.,and STOLT,R.H. Examples of a nonlinear inversion method based on the T matrix of scattering theory: Application to multiple suppression [C]. 61st Ann Internet. Mtg.Soc.Expl.Geophysics, Expanded Abstracts,1991,1319-1322.
[29] GUOCHUN LU,BJRN URSIN,and JAN LUTRO. Model based removal of water layer multiple reflections[J]. Geophysics, 1999, 64:1816-1827.
[30]LASSE AMUNDSEN. Elimination of free surface related multiples without need of the source wavelet [J]. Geophysics, 2001, 66:327-341.
[31]董敏煜.地震勘探[M].东营:石油大学出版社,2000.48-64.
[32]杨有发,张建祥.海洋地震勘探[M].长春:吉林科学技术出版社,1997. 29-31.
[33]ROBERT E, SHERIFF. Eneyelopedie Dictionary of Exploration Geophysics[J]. Society of Exploration Geophysics,1991.199-200.
[34]罗小明,牛滨华,于延庆,黄新武.海上某区二维地震多次波的识别和压制[J].现代地质,2003,17:474-478.
[35]王汝珍.多次波识别与衰减[J].勘探地球物理进展,2003,26(5.6):423-432.
[36]刘长辉,王建鹏,曹军.海洋多次波特征和压制方法初探[J].石油天然气学报,2005,27(4):608-609.
[37]陈瑜,魏赟,葛勇.东海盆地L凹陷多次波分析与压制[J].中国海上油气, 2004,16(6):373-376.
[38]BELTRAMI E. Sulle funzioni bilineari,Giomale di Mathematiche ad Uso Student Delle Uninersita.1873,11:98-106. An English translation by D Boley is available as University of the Minnesota,Department of Computer Science,Technical Report, 1990, 90-37
[39]JORDAN C. Memoire sur les formes lineaires [J]. J Math Pures Appl,Deuxieme Serie,1874,19:35-54.
[40]AUTONNE L. Sur les groupes lineaires,reelles et orthogonaus [J].Bull Soc Math,France,1902,30:121-133.
[41]C. ECKART and G. YOUNG. A principal axis transformation for non hermitian matrices [J]. NullAmer.Math.Soc.,1939,45:118-121..
[42]张贤达.矩阵分析与应用[M].北京:清华大学出版社,2004.344-354.
[43]徐树方.矩阵计算的理论与方法[M].北京:北京大学出版社,1995.
[44]陈良瑜,朱振福,刘忠领,李军伟,车国锋.图像奇异值特征矢量缩放不变性分析及应用[J].红外与激光工程, 2003,32(5):498-501.
[45]宋伟,侯建军,李赵红.基于SVD和伪随机循环链的图像认证水印算法[J].北京交通大学学报, 2008,32(02):71-75..
[46]LIU RUI.ZHEN,TAN TIE.NIU. An SVD-based watermarking scheme for protecting rightful ownership [J] . IEEE Transactions on Multimedia, 2002, 4(1):121-128.
[47]李亚峻,李月,杨宝俊、高颖.SVD与小波变换相结合抑制面波和随机噪声[J].计算机工程与应用,2007,43(31):182-185.
[48]罗仁泽,冉瑞生,王汝言.基于奇异值分解的基图像的人脸识别[J].电讯技术, 2008,48(02):25-28.
[49]倪传坤,周建中.基于SVD的水电机组轴心轨迹自动识别[J].水电自动化与大坝监测, 2003,27(06):22-24.
[50]陈遵德.地震记录的奇异值分布特征[J].石油物探快讯,1990,5: 44-42.
[51]高静怀,朱光明,王玉贵.奇异值分解在VSP中的应用[J].西安地质学院学报,1992,14(3):71-78.
[52]吴律编著.τ? p变换及应用[M].北京:石油工业出版社,1990.1-20.
[53]C.H.CHAPMAN and P.W.CARY. The circular harmonic Radon transform [J]. Inverse Problems,1986,2:23-49.
[54]T S DURRANI,D BISSET. The Radon transform and its properties [J]. Geophysics,1984,49:1180-1187.
[55]牛滨华,孙春岩,张中杰,沈操.多项式Radon变换[J].地球物理学报,2001,44:263-271.
[56]GREG TURNER. Aliasing in theτ-P transform and the removal ofspatially aliased coherent noise [J]. GeoPhysies,1990,55:496- 503.
[57]SACCHI M D. Aperture Compensated Radon and Fourier Transforms [D]. Doetor Thesis of the University of British Columbia,1996.
[58]王维红,崔宝文.同分辨率抛物Radon变换法速度分析[J],大庆石油地质与开发,2007,26(5):129-132.
[59]刘喜武,刘洪,李幼铭.高分辨率Radon变换方法及其在地震信号处理中的应用[J].地球物理学进展,2004,19(1):8-15.
[60]BINZHONG ZHOU and STEWART A.GREENHALGH.Wave-equation extrapolation based multiple attenuation:2-D filtering in the f-k domain [J]. Geophysics, 1994,55(9):1377-1391.
[61]SHOUDONG HUO and YANGHUA Wang. Improving adaptive subtraction in seismic multiple attenuation. Geophysics,2009,74(4):59-67.
[62]黄新武,吴律.抛物线Radon变换中的参数采样与假频[J].石油人学学报(白然科学版),2003,27:27-31.
[63]王维红,崔宝文,刘洪.表面多次波衰减的研究现状与进展[J].地球物理学进展,2007,22(1):156-164.
[64]张军华,吕宁,雷凌,田连玉,郭见乐.抛物线拉冬变换消除多次波的应用要素分析[J].石油地球物理勘探,2004,39(4):398-405.
[2]YILMAZ O Z. Seismic data analysis [M]. USA: Society of Exploration Geophysicists, 1993.
[3]SHERIFF R E,GELDART L P,初英等译.勘探地震学[M].北京:石油工业出版社,1999.
[4]陆基孟.地震勘探原理[M].东营:石油大学出版社,1993.164-172.
[5]A. B. WEGLEIN. Multiple attenuation: Recent advances and the road ahead [C]. 65th Ann Internet. Mtg.Soc.Expl. Geophysics. Expanded Abstracts. 1995,1492-1495.
[6]LANDA E, BELFER I and KEYDAR S. Multiple attenuation in the parabolic domain using wavefront characteristics of multiple generating primaries [J]. Geophysics,1999,64:1806-1815..
[7]FOSTER D J, MASHER C C. Suppression of multiple reflection using the radon transform [J]. Geophysics,1992,57:386-395. .
[8]P W CARY. The simplest discrete Radon transform [C].68th Annual SEG Meeting,1998:1999-2002.
[9]MILO M. BACKUS. Water reverberation.their nature and elimation [J].Geophysics,1959,24(2):233-261.
[10]ENDERS A. ROBINSON1, SVEN TREITEL. Principle of digital Wiener filtering [J]. Geophysical Prospecting,1967,15(3):311-333.
[11]AJ HARDING. Slowness-time mapping of near offset seismic reflection data [J]. Geophysical Journal of the Royal Astronomical Society,1985,80:463-492.
[12]D HAMPSON. Inverse velocity stacking of Multiple elimination [J]. Journal of the Canadian society of Exploration Geophysic, 1986,22:44-55.
[13]YILMAZ O Z. Velocity stack processing [J]. Geophysical Prospect- ing,1989, 37(4):357-382.
[14]IF JONES,S lEVY. Signal to noise ratio enhancement in multi-channel seismic data via the Karhunen-Loeve transform [J].Geophy- sical Prospecting,1987,35:12-32.
[15]AL.YAHYA,K.M.Application of the partial Karhunen-Loeve trans- form to suppress random noise in seismic sections [J]. Geophysical Prospeeting,1993,39:77-99.
[16]杨文采.非线性K-L滤波及其在反射地震资料处理中的应用[J].石油物探,1996,35(2):17-26.
[17]谢风兰,赵改善.利用SVD法重建地震剖面[J].石油物探,1990,29(4):33-40.
[18]SERGIO L. M. FREIRE and TAD J.ULRYCH. Application of singular value decomposition to vertical seismic profiling [J]. Geophysics. 1988,53(6): 778-785.
[19]T HU,RE White. Robust Multiple suppression using adaptive beamforming [J]. Geophysical Prospecting,1998,46:227-248.
[20]TIANYUE HU,FEI HONG. Multiple attenuation using beamforming onshore and offshore China [J]. The Leading Edge,2002,21(9):906-910.
[21]胡大跃,王润秋,REWHIET.地震资料处理中的聚束滤波方法[J].地球物理学报,2000,43:105-115.
[22]L MORLEY. Predictive deconvolution in shot receiver space. Geophysics, 1983,48(5):515-531.
[23]JW WIGGINS. Attenuation of complex water bottom multiples by wave equation based prediction and subtraction [J]. Geophysics, 1988,53: 1527-1539.
[24]VERSCHUUR,D.J,BERKHOUT,A.Jand.WAPENAAR,C.P.A.Adaptive surface related multiple elimination [J]. Geophysics, 1992, 57: 1166-1177.
[25]D J VERSCHUUR, A J BERKHOUT and P G KELAMIS. Estimation of mul- tiple scattering by iterative inversion [C]. PartⅡ:Examples on marine and land data:65th Ann.Intermat.Mtg.Soc.Expl.Geophysics, Expanded Abstracts,1995, 1470-1473.
[26]A J BERKHOUT. Multiple removal based on the feedback model [J].The Leading Edge,1999,18(1):127-131.
[27]A B WEGLEIN,F A GASPAROTTO, P M CARVALHO and R H STOLT. An inverse scattering series method for attenuating multiples in seismic reflection data [J]. Geophysics,1997,62:1975-1989.
[28]CARVALHO P.M.,WEGLEIN A.B.,and STOLT,R.H. Examples of a nonlinear inversion method based on the T matrix of scattering theory: Application to multiple suppression [C]. 61st Ann Internet. Mtg.Soc.Expl.Geophysics, Expanded Abstracts,1991,1319-1322.
[29] GUOCHUN LU,BJRN URSIN,and JAN LUTRO. Model based removal of water layer multiple reflections[J]. Geophysics, 1999, 64:1816-1827.
[30]LASSE AMUNDSEN. Elimination of free surface related multiples without need of the source wavelet [J]. Geophysics, 2001, 66:327-341.
[31]董敏煜.地震勘探[M].东营:石油大学出版社,2000.48-64.
[32]杨有发,张建祥.海洋地震勘探[M].长春:吉林科学技术出版社,1997. 29-31.
[33]ROBERT E, SHERIFF. Eneyelopedie Dictionary of Exploration Geophysics[J]. Society of Exploration Geophysics,1991.199-200.
[34]罗小明,牛滨华,于延庆,黄新武.海上某区二维地震多次波的识别和压制[J].现代地质,2003,17:474-478.
[35]王汝珍.多次波识别与衰减[J].勘探地球物理进展,2003,26(5.6):423-432.
[36]刘长辉,王建鹏,曹军.海洋多次波特征和压制方法初探[J].石油天然气学报,2005,27(4):608-609.
[37]陈瑜,魏赟,葛勇.东海盆地L凹陷多次波分析与压制[J].中国海上油气, 2004,16(6):373-376.
[38]BELTRAMI E. Sulle funzioni bilineari,Giomale di Mathematiche ad Uso Student Delle Uninersita.1873,11:98-106. An English translation by D Boley is available as University of the Minnesota,Department of Computer Science,Technical Report, 1990, 90-37
[39]JORDAN C. Memoire sur les formes lineaires [J]. J Math Pures Appl,Deuxieme Serie,1874,19:35-54.
[40]AUTONNE L. Sur les groupes lineaires,reelles et orthogonaus [J].Bull Soc Math,France,1902,30:121-133.
[41]C. ECKART and G. YOUNG. A principal axis transformation for non hermitian matrices [J]. NullAmer.Math.Soc.,1939,45:118-121..
[42]张贤达.矩阵分析与应用[M].北京:清华大学出版社,2004.344-354.
[43]徐树方.矩阵计算的理论与方法[M].北京:北京大学出版社,1995.
[44]陈良瑜,朱振福,刘忠领,李军伟,车国锋.图像奇异值特征矢量缩放不变性分析及应用[J].红外与激光工程, 2003,32(5):498-501.
[45]宋伟,侯建军,李赵红.基于SVD和伪随机循环链的图像认证水印算法[J].北京交通大学学报, 2008,32(02):71-75..
[46]LIU RUI.ZHEN,TAN TIE.NIU. An SVD-based watermarking scheme for protecting rightful ownership [J] . IEEE Transactions on Multimedia, 2002, 4(1):121-128.
[47]李亚峻,李月,杨宝俊、高颖.SVD与小波变换相结合抑制面波和随机噪声[J].计算机工程与应用,2007,43(31):182-185.
[48]罗仁泽,冉瑞生,王汝言.基于奇异值分解的基图像的人脸识别[J].电讯技术, 2008,48(02):25-28.
[49]倪传坤,周建中.基于SVD的水电机组轴心轨迹自动识别[J].水电自动化与大坝监测, 2003,27(06):22-24.
[50]陈遵德.地震记录的奇异值分布特征[J].石油物探快讯,1990,5: 44-42.
[51]高静怀,朱光明,王玉贵.奇异值分解在VSP中的应用[J].西安地质学院学报,1992,14(3):71-78.
[52]吴律编著.τ? p变换及应用[M].北京:石油工业出版社,1990.1-20.
[53]C.H.CHAPMAN and P.W.CARY. The circular harmonic Radon transform [J]. Inverse Problems,1986,2:23-49.
[54]T S DURRANI,D BISSET. The Radon transform and its properties [J]. Geophysics,1984,49:1180-1187.
[55]牛滨华,孙春岩,张中杰,沈操.多项式Radon变换[J].地球物理学报,2001,44:263-271.
[56]GREG TURNER. Aliasing in theτ-P transform and the removal ofspatially aliased coherent noise [J]. GeoPhysies,1990,55:496- 503.
[57]SACCHI M D. Aperture Compensated Radon and Fourier Transforms [D]. Doetor Thesis of the University of British Columbia,1996.
[58]王维红,崔宝文.同分辨率抛物Radon变换法速度分析[J],大庆石油地质与开发,2007,26(5):129-132.
[59]刘喜武,刘洪,李幼铭.高分辨率Radon变换方法及其在地震信号处理中的应用[J].地球物理学进展,2004,19(1):8-15.
[60]BINZHONG ZHOU and STEWART A.GREENHALGH.Wave-equation extrapolation based multiple attenuation:2-D filtering in the f-k domain [J]. Geophysics, 1994,55(9):1377-1391.
[61]SHOUDONG HUO and YANGHUA Wang. Improving adaptive subtraction in seismic multiple attenuation. Geophysics,2009,74(4):59-67.
[62]黄新武,吴律.抛物线Radon变换中的参数采样与假频[J].石油人学学报(白然科学版),2003,27:27-31.
[63]王维红,崔宝文,刘洪.表面多次波衰减的研究现状与进展[J].地球物理学进展,2007,22(1):156-164.
[64]张军华,吕宁,雷凌,田连玉,郭见乐.抛物线拉冬变换消除多次波的应用要素分析[J].石油地球物理勘探,2004,39(4):398-405.