地震子波提取中非线性优化算法的应用研究
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摘要
准确的地震子波提取是地震资料反褶积处理、地震波阻抗反演和正演的基础,是地震勘探中的重要一环。由于在保持信号相位信息的同时可以压制任意加性高斯噪声,高阶统计量被广泛用于混合相位的地震子波提取。
在地震子波为非最小相位、噪声为加性高斯噪声的假设下,可通过构建地震记录四阶累积量与地震子波参数模型四阶矩的拟合目标函数来提取地震子波。为验证参数模型解的唯一性,本文从地震子波模型的三谱出发,对基于四阶累积量的地震子波提取方式进行了理论分析。公式推导结果表明,除在幅度和时延上有一定的线性偏移外,基于四阶统计量的模型参数提取方法所提取的子波为地震记录中地震子波的准确参数模型,所得的地震子波波形符合地震反褶积处理的要求。
对上述子波估计模型的求解最终可归结为对一多维多峰值目标函数的非线性优化问题,对该目标函数所用优化算法须兼有对参数向量整体的全局随机搜索能力和对单个参数的深度搜索能力,传统的优化算法很难求得其最优解。由于结合了有向和随机两种搜索方式,遗传算法在对搜索空间进行深度搜索和广度搜索之间维持了较好的平衡性,但是其易“早熟”以及局部搜索能力较差的特点严重制约了其寻优性能。为此,本文并从多样性的测度与维持、基本算子的设置与选择以及小邻域搜索等几个方面对基于实数编码的遗传算法进行了分析和改进,最终提出一种基于多样性双重维持的遗传算法。仿真结果表明:该算法能够有效克服“早熟”收敛,具有强大的全局搜索能力。
为克服算法变异方向的随机性,保证变异向优良的方向进行转化,本文从混沌序列的分布特性、混沌映射的遍历性及其计算效率等角度出发,对Logistic映射、Tent映射以及Arnold映射等进行了仿真上的比较分析,最终将基于Arnold映射的混沌优化算法融入遗传算法,提出一种适于高维多峰函数寻优的新型混沌遗传算法,以进行精确的子波估计。高维测试函数仿真实验表明,该算法能够迅速收敛到全局最优解,在搜索效率、搜索精度均较单一算法有了很大提高,并且具有良好的稳定性。其在基于MA和ARMA模型地震子波描述下地震子波拟合提取实验中的应用结果表明:该算法能够迅速有效地提取精确的地震子波,具有良好的适用性。
Accurate seismic wavelet estimation is the base of deconvolution processing, inverse and forward models of seismic wave impedance, and it is significant in seismic exploration fields. Because the higher order statistic retains the phase information of the signal and is able to eliminate the Gaussian noise, it is studied extensively in the field of seismic wavelet estimation.
In this thesis, on the assumption that seismic wavelet was mixed-phase and the noise was Gaussian, the wavelet estimation objective function was constructed via the fourth-order cumulants of seismic data matching the fourth-order moments of parametric wavelet model. In order to validate the uniqueness of the model solution, the wavelet estimation methods based on fourth-order cumulants were analyzed and studied according to the wavelet three-order spectral. Theoretical analysis demonstrated that this approach could estimate the wavelet model parameters precisely with some linear migration of the amplitude or the phase of the seismic wavelet, and the accuracy of the seismic wavelet estimated through this approach meets the requirement of seismic signal processing.
The solution of the objective function leads to a multidimensional and multimodal non-linear optimization problem. The minimizing process of the objective function should able to search the parameter vectors globally, and the deep-searching ability is also the necessary characteristic of the optimization algorithm, this makes it difficult to get the global optimal via the general algorithm. The genetic algorithm perfectly balances the global searching capacity and the local search capability, but often gets premature in the late searching process. To overcome this flaw and get the optimal solution more effectively and stably, this paper did some reasearch on the measurement and maintenance of the population, the setting of the basic operaters, and the small neighborhood seaeching respectively. Then the improved Genetic Algorithm, which could overcome the premature and search the global optimal effectively, is proposed in this paper, simulation alse demonstrated this characteristic of this algorithm.
Chaos optimization algorithm is also absorbed for further improvement of the mutation ability of this algorithm. The characteristics of Logistic mapping, Tent mapping and Arnold mapping are discussed in the view of the egodicity and the computational efficiency of the chaotic mapping or the distribution of the chaotic sequence. Then the Arnold mapping is absorbed in the Genetic Algorithm to extract the wavelet. The high dimensional function simulation demonstrated that this algorithm could improve the searching ability and get the global optimal effectively and stably. The application of seismic data processing results demonstrated that the novel chaos genetic algorithm is able to estimate the accuracy seismic wavelet fastly and effectively.
在地震子波为非最小相位、噪声为加性高斯噪声的假设下,可通过构建地震记录四阶累积量与地震子波参数模型四阶矩的拟合目标函数来提取地震子波。为验证参数模型解的唯一性,本文从地震子波模型的三谱出发,对基于四阶累积量的地震子波提取方式进行了理论分析。公式推导结果表明,除在幅度和时延上有一定的线性偏移外,基于四阶统计量的模型参数提取方法所提取的子波为地震记录中地震子波的准确参数模型,所得的地震子波波形符合地震反褶积处理的要求。
对上述子波估计模型的求解最终可归结为对一多维多峰值目标函数的非线性优化问题,对该目标函数所用优化算法须兼有对参数向量整体的全局随机搜索能力和对单个参数的深度搜索能力,传统的优化算法很难求得其最优解。由于结合了有向和随机两种搜索方式,遗传算法在对搜索空间进行深度搜索和广度搜索之间维持了较好的平衡性,但是其易“早熟”以及局部搜索能力较差的特点严重制约了其寻优性能。为此,本文并从多样性的测度与维持、基本算子的设置与选择以及小邻域搜索等几个方面对基于实数编码的遗传算法进行了分析和改进,最终提出一种基于多样性双重维持的遗传算法。仿真结果表明:该算法能够有效克服“早熟”收敛,具有强大的全局搜索能力。
为克服算法变异方向的随机性,保证变异向优良的方向进行转化,本文从混沌序列的分布特性、混沌映射的遍历性及其计算效率等角度出发,对Logistic映射、Tent映射以及Arnold映射等进行了仿真上的比较分析,最终将基于Arnold映射的混沌优化算法融入遗传算法,提出一种适于高维多峰函数寻优的新型混沌遗传算法,以进行精确的子波估计。高维测试函数仿真实验表明,该算法能够迅速收敛到全局最优解,在搜索效率、搜索精度均较单一算法有了很大提高,并且具有良好的稳定性。其在基于MA和ARMA模型地震子波描述下地震子波拟合提取实验中的应用结果表明:该算法能够迅速有效地提取精确的地震子波,具有良好的适用性。
Accurate seismic wavelet estimation is the base of deconvolution processing, inverse and forward models of seismic wave impedance, and it is significant in seismic exploration fields. Because the higher order statistic retains the phase information of the signal and is able to eliminate the Gaussian noise, it is studied extensively in the field of seismic wavelet estimation.
In this thesis, on the assumption that seismic wavelet was mixed-phase and the noise was Gaussian, the wavelet estimation objective function was constructed via the fourth-order cumulants of seismic data matching the fourth-order moments of parametric wavelet model. In order to validate the uniqueness of the model solution, the wavelet estimation methods based on fourth-order cumulants were analyzed and studied according to the wavelet three-order spectral. Theoretical analysis demonstrated that this approach could estimate the wavelet model parameters precisely with some linear migration of the amplitude or the phase of the seismic wavelet, and the accuracy of the seismic wavelet estimated through this approach meets the requirement of seismic signal processing.
The solution of the objective function leads to a multidimensional and multimodal non-linear optimization problem. The minimizing process of the objective function should able to search the parameter vectors globally, and the deep-searching ability is also the necessary characteristic of the optimization algorithm, this makes it difficult to get the global optimal via the general algorithm. The genetic algorithm perfectly balances the global searching capacity and the local search capability, but often gets premature in the late searching process. To overcome this flaw and get the optimal solution more effectively and stably, this paper did some reasearch on the measurement and maintenance of the population, the setting of the basic operaters, and the small neighborhood seaeching respectively. Then the improved Genetic Algorithm, which could overcome the premature and search the global optimal effectively, is proposed in this paper, simulation alse demonstrated this characteristic of this algorithm.
Chaos optimization algorithm is also absorbed for further improvement of the mutation ability of this algorithm. The characteristics of Logistic mapping, Tent mapping and Arnold mapping are discussed in the view of the egodicity and the computational efficiency of the chaotic mapping or the distribution of the chaotic sequence. Then the Arnold mapping is absorbed in the Genetic Algorithm to extract the wavelet. The high dimensional function simulation demonstrated that this algorithm could improve the searching ability and get the global optimal effectively and stably. The application of seismic data processing results demonstrated that the novel chaos genetic algorithm is able to estimate the accuracy seismic wavelet fastly and effectively.
引文
[1]陆基孟.地震勘探原理[M].山东:石油大学出版社.1993
[2]牟永光.地震勘探资料数字处理方法[M].北京:石油工业出版社,1981
[3]地震资料处理技术交流会论文集[P].大庆高分辨率地震资料处理技术与应用.石油工业出版社.2003
[4]俞寿朋.高分辨率地震勘探[M].北京:石油工业出版社.1993
[5]梁光河.测井约束地震子波外推方法研究[J].石油地球物理勘探,1998,33(3):296-304
[6]粱光河.地震子波提取方法研究[J].石油物探,1998,37(1):31-39
[7]尹成,唐斌,谢桂生.地震子波估计-高阶累积量矩阵方程法[J].信号处理,2000,16(增刊):83-87
[8] Leinbach, J., Wiener spiking de-convolution and minimum-phase wavelets[J]. A tutorial: The Leading Edge, 1995, 14(3):189-192
[9] Porsani, M. J., and Ursin, B., Mixed-phase de-convolution[J]. Geophysics, 1998,63(2): 637-647
[10]李翠萍,谢红卫.基于高阶累积量方法的非高斯非最小相位ARMA模型辨识[J].上海大学学报(自然科学版),2001,7(5):438-453
[11] Gregory D.Lazear. Mixed-phase wavelet estimation using fourth-order cumulants[J]. Geophysics, 1993, 58(7): 1042-1051
[12]唐斌,尹成.基于高阶统计的非最小相位地震子波恢复[J].地球物理学报, 2001, 44(3):404-410
[13]谢桂生,石玉梅,尹成.高阶累积量法地震子波估计与处理[J].石油物探,2000,39(4):19-25
[14]石殿祥,李岩,殷福亮等.基于高阶累积量的非最小相位地震子波提取[J].石油地球物理勘探,1999,34(5):491-499
[15] Robinson, E. A. Predictive decomposition of seismic traces[J]. Geophysics, 1957,22(4): 767-778
[16] Gregory D. Lazear. An examination of the exponential decay method of mixed- phase wavelet estimation[J]. Geophysics,1984,49(12):2094-2099
[17] Arild Buland and Henning More. Bayesian wavelet estimation from seismic and well data[J]. Geophysics, 2003,68(6): 2000-2009
[18]杨文采.非线性地球物理反演方法:回顾与展望[J].地球物理学进展,2002,17(2): 255-261
[19] Enders A. Robinson and Sven Treitel. Geophysical signal analysis[M]. Englewood Cliffs : Prentice-Hall, 1980
[20] D. W. Oldenburg, S. Levy& and K. P. Whittall. Wavelet estimation and de- convolution [J]. Geophysics, 1981, 46(11):1528~1542
[21] Tadeusz J. Ulrych, Danilo R. Velis, and Mauricio D. Saucchi. Wavelet estimation revisited [J]. The Leading Edge. 1995,14(11):1139-1143
[22] Velis D R,Ulrych T J.Simulated annealing wavelet estimation via fourth-order cumulant matching[J]. Geophysics, 1996, 61(6): 1939-1948
[23]尹成,周翼,谢桂生等.基于综合的混沌优化算法估计地震子波[J].物探化探计算技术,2001,23(2):97-100
[24]尹成,谢桂生,吕中育等.地震反演与非线性随机优化方法[J].物探化探计算技术,2001,23(1):6-10
[25]格莱克,张淑誉译.混沌:开创新科学[M].上海:上海译文出版社,1990
[26]郝伯林.分叉、混沌、奇怪吸引子、湍流及其他[J].物理学进展,1983,3(3):329-416
[27]王兴元.复杂非线性系统中的混沌[M].北京:电子工业出版社,2003
[28]李庆忠.走向精确勘探的道路—高分辨率地震勘探系统工程剖析[M].北京:石油工业出版社,1995
[29]张贤达,现代信号处理[M].北京:清华大学出版社,2002
[30]邱天爽,张旭秀,李小兵等.统计信号处理-非高斯信号处理及其应用[M].北京:电子工业出版社,2004
[31]张玲华,郑宝玉.随机信号处理[M].北京:清华大学出版社,2003
[32]赵达纲,朱善迎编.应用随机过程[M].北京:机械工业出版社,1993
[33] Tugnait J K. Identification of linear stochastic system via second- and fourth-order cumulant matching [J]. IEEE Trans. On Info. Theory, 1987,33(5):393-407
[34]胡广书,王俊峰.平稳随机信号准确模型的研究[M].清华大学学报,1997,4(37) :68-71
[35] John. H. Holland. Adaptation in Nature and Artificial Systems [M]. Michigan, USA :The University of Michigan Press,1975
[36] Schwefel , H. , Evolution and optimum seeking[M]. New York :Wiley Press, 1995
[37] Koza, J.R., Genetic programming [M]. Cambrige: MIT Press , 1992
[38] Ho, L. L., Simulation of condensed-phase physical, chemical, and biological system on massively parallel computers [D]. New Haven: Yale University, 1997
[39] Larson, C. B., Vibration testing by design: excitation and sensor placement using genetic algorithms [D]. Gainesville: University of Florida, 1996
[40] Chunduru, R. K., Global and hybrid optimization in geophysical inversion[D].Austin: University of Texas-Austin, 1998
[41] G Rudolph. Convergence analysis of canonical genetical algorithms [J]. IEEE Trans on Neural Networks, 1994, 5(1): 96~101
[42]陈国良,王东生等.遗传算法及应用[M].北京:人民邮电出版社,1996
[43]玄光男,程润伟.遗传算法与工程优化[M].北京:清华大学出版社,2004
[44]马春慨,王亚英等.混沌在实数编码遗传算法中的应用[J],上海交通大学学报,2000,34(12):1669-1670
[45]王凌等,智能优化算法及其应用[M].北京:清华大学出版社,2001
[46]武晓今,朱仲英.遗传算法多样性测度问题研究.信息与控制[J],2005,34(4):416- 421
[47] Jang Sung Chun, Jeong. Pil. Lim, Hyun. Kyo. Jung. Optimal Design of Synchronous with Parameter Correction Using Immune Algorithm [J]. IEEE Trans on Energy Conversion, 1999, 14(3): 610-615
[48]王小平,曹立明等.遗传算法—理论、应用与软件实现[M].西安:西安交通大学出版社,2003
[49] Bazaraa, M., J. Javis, and H.Sherail, Linear Programming and Network Flows 2nd edition [M]. New York: Wiley Press,1990
[50] Cheng, R, M. Gen, and T. Tozawa. Genetic algorithms for multi-row machine layout problem[J]. Engineering Design and Automation ,1996
[51]袁晓辉,袁艳斌等.一种新型的自适应混沌遗传算法.电子学报[J],2006,34(4):708-712
[52] Michalewicz , Z., Genetic algorithm + Data Structure=Evolution programs,3rd edition[M]. Springer-Verlag, New Rork,1996
[53]唐焕文,秦学志等.实用最优化方法[M].大连:大连理工大学出版社.2001
[54]马西庚,李媛媛等.一种用于优化计算的自适应免疫遗传算法[J].计算机工程与应用[J],2006,12:41-43
[55]蒋世忠,杨天奇.给予免遗传算法的水机调节参数优化与仿真[J].计算机仿真,2004,21(11):200~202
[56]张楠,李志属,张建华.基于混沌理论的否定选择算法[J].四川大学学报(工程科学版),2006,38(1):124-127
[57]匡胤,刘益和,黄迪明.混沌理论在遗传算法中的应用研究[J].四川理工学院院报,2007,20(4):1-3
[58]杨迪雄,李刚等.非线性函数混沌优化算法比较研究[J].计算力学学报,2004,21(3):257-261
[59]单梁,强浩,李军.基于Tent映射的混沌优化算法[J].控制与决策,2005,20(2):179-182
[60]谭广兴,朱燕飞,毛宗源.基于Henon映射的自适应克隆选择优化算法[J].计算机工程与应用,2006,9:73-76
[61]赵立强,李艳坡等.基于广义Arnold混沌映射和Henon混沌映射的小波水印算法[J].河北科技师范学院学报,2006,20(4):32-37
[62]王西文.地震资料处理和解释中的小波分析方法[M].北京:石油工业出版社,2004
[63] Dai Yongshou, Wang Junling, Wang Weiwei,et al. Research on Wavelet Estimation via Cumulant Based ARMA Model Approach[J]. Proceedings of 2007 International Conference on Wavelet Analysis and Pattern Recognition [P]. 2007 4(4): 1882-1886
[2]牟永光.地震勘探资料数字处理方法[M].北京:石油工业出版社,1981
[3]地震资料处理技术交流会论文集[P].大庆高分辨率地震资料处理技术与应用.石油工业出版社.2003
[4]俞寿朋.高分辨率地震勘探[M].北京:石油工业出版社.1993
[5]梁光河.测井约束地震子波外推方法研究[J].石油地球物理勘探,1998,33(3):296-304
[6]粱光河.地震子波提取方法研究[J].石油物探,1998,37(1):31-39
[7]尹成,唐斌,谢桂生.地震子波估计-高阶累积量矩阵方程法[J].信号处理,2000,16(增刊):83-87
[8] Leinbach, J., Wiener spiking de-convolution and minimum-phase wavelets[J]. A tutorial: The Leading Edge, 1995, 14(3):189-192
[9] Porsani, M. J., and Ursin, B., Mixed-phase de-convolution[J]. Geophysics, 1998,63(2): 637-647
[10]李翠萍,谢红卫.基于高阶累积量方法的非高斯非最小相位ARMA模型辨识[J].上海大学学报(自然科学版),2001,7(5):438-453
[11] Gregory D.Lazear. Mixed-phase wavelet estimation using fourth-order cumulants[J]. Geophysics, 1993, 58(7): 1042-1051
[12]唐斌,尹成.基于高阶统计的非最小相位地震子波恢复[J].地球物理学报, 2001, 44(3):404-410
[13]谢桂生,石玉梅,尹成.高阶累积量法地震子波估计与处理[J].石油物探,2000,39(4):19-25
[14]石殿祥,李岩,殷福亮等.基于高阶累积量的非最小相位地震子波提取[J].石油地球物理勘探,1999,34(5):491-499
[15] Robinson, E. A. Predictive decomposition of seismic traces[J]. Geophysics, 1957,22(4): 767-778
[16] Gregory D. Lazear. An examination of the exponential decay method of mixed- phase wavelet estimation[J]. Geophysics,1984,49(12):2094-2099
[17] Arild Buland and Henning More. Bayesian wavelet estimation from seismic and well data[J]. Geophysics, 2003,68(6): 2000-2009
[18]杨文采.非线性地球物理反演方法:回顾与展望[J].地球物理学进展,2002,17(2): 255-261
[19] Enders A. Robinson and Sven Treitel. Geophysical signal analysis[M]. Englewood Cliffs : Prentice-Hall, 1980
[20] D. W. Oldenburg, S. Levy& and K. P. Whittall. Wavelet estimation and de- convolution [J]. Geophysics, 1981, 46(11):1528~1542
[21] Tadeusz J. Ulrych, Danilo R. Velis, and Mauricio D. Saucchi. Wavelet estimation revisited [J]. The Leading Edge. 1995,14(11):1139-1143
[22] Velis D R,Ulrych T J.Simulated annealing wavelet estimation via fourth-order cumulant matching[J]. Geophysics, 1996, 61(6): 1939-1948
[23]尹成,周翼,谢桂生等.基于综合的混沌优化算法估计地震子波[J].物探化探计算技术,2001,23(2):97-100
[24]尹成,谢桂生,吕中育等.地震反演与非线性随机优化方法[J].物探化探计算技术,2001,23(1):6-10
[25]格莱克,张淑誉译.混沌:开创新科学[M].上海:上海译文出版社,1990
[26]郝伯林.分叉、混沌、奇怪吸引子、湍流及其他[J].物理学进展,1983,3(3):329-416
[27]王兴元.复杂非线性系统中的混沌[M].北京:电子工业出版社,2003
[28]李庆忠.走向精确勘探的道路—高分辨率地震勘探系统工程剖析[M].北京:石油工业出版社,1995
[29]张贤达,现代信号处理[M].北京:清华大学出版社,2002
[30]邱天爽,张旭秀,李小兵等.统计信号处理-非高斯信号处理及其应用[M].北京:电子工业出版社,2004
[31]张玲华,郑宝玉.随机信号处理[M].北京:清华大学出版社,2003
[32]赵达纲,朱善迎编.应用随机过程[M].北京:机械工业出版社,1993
[33] Tugnait J K. Identification of linear stochastic system via second- and fourth-order cumulant matching [J]. IEEE Trans. On Info. Theory, 1987,33(5):393-407
[34]胡广书,王俊峰.平稳随机信号准确模型的研究[M].清华大学学报,1997,4(37) :68-71
[35] John. H. Holland. Adaptation in Nature and Artificial Systems [M]. Michigan, USA :The University of Michigan Press,1975
[36] Schwefel , H. , Evolution and optimum seeking[M]. New York :Wiley Press, 1995
[37] Koza, J.R., Genetic programming [M]. Cambrige: MIT Press , 1992
[38] Ho, L. L., Simulation of condensed-phase physical, chemical, and biological system on massively parallel computers [D]. New Haven: Yale University, 1997
[39] Larson, C. B., Vibration testing by design: excitation and sensor placement using genetic algorithms [D]. Gainesville: University of Florida, 1996
[40] Chunduru, R. K., Global and hybrid optimization in geophysical inversion[D].Austin: University of Texas-Austin, 1998
[41] G Rudolph. Convergence analysis of canonical genetical algorithms [J]. IEEE Trans on Neural Networks, 1994, 5(1): 96~101
[42]陈国良,王东生等.遗传算法及应用[M].北京:人民邮电出版社,1996
[43]玄光男,程润伟.遗传算法与工程优化[M].北京:清华大学出版社,2004
[44]马春慨,王亚英等.混沌在实数编码遗传算法中的应用[J],上海交通大学学报,2000,34(12):1669-1670
[45]王凌等,智能优化算法及其应用[M].北京:清华大学出版社,2001
[46]武晓今,朱仲英.遗传算法多样性测度问题研究.信息与控制[J],2005,34(4):416- 421
[47] Jang Sung Chun, Jeong. Pil. Lim, Hyun. Kyo. Jung. Optimal Design of Synchronous with Parameter Correction Using Immune Algorithm [J]. IEEE Trans on Energy Conversion, 1999, 14(3): 610-615
[48]王小平,曹立明等.遗传算法—理论、应用与软件实现[M].西安:西安交通大学出版社,2003
[49] Bazaraa, M., J. Javis, and H.Sherail, Linear Programming and Network Flows 2nd edition [M]. New York: Wiley Press,1990
[50] Cheng, R, M. Gen, and T. Tozawa. Genetic algorithms for multi-row machine layout problem[J]. Engineering Design and Automation ,1996
[51]袁晓辉,袁艳斌等.一种新型的自适应混沌遗传算法.电子学报[J],2006,34(4):708-712
[52] Michalewicz , Z., Genetic algorithm + Data Structure=Evolution programs,3rd edition[M]. Springer-Verlag, New Rork,1996
[53]唐焕文,秦学志等.实用最优化方法[M].大连:大连理工大学出版社.2001
[54]马西庚,李媛媛等.一种用于优化计算的自适应免疫遗传算法[J].计算机工程与应用[J],2006,12:41-43
[55]蒋世忠,杨天奇.给予免遗传算法的水机调节参数优化与仿真[J].计算机仿真,2004,21(11):200~202
[56]张楠,李志属,张建华.基于混沌理论的否定选择算法[J].四川大学学报(工程科学版),2006,38(1):124-127
[57]匡胤,刘益和,黄迪明.混沌理论在遗传算法中的应用研究[J].四川理工学院院报,2007,20(4):1-3
[58]杨迪雄,李刚等.非线性函数混沌优化算法比较研究[J].计算力学学报,2004,21(3):257-261
[59]单梁,强浩,李军.基于Tent映射的混沌优化算法[J].控制与决策,2005,20(2):179-182
[60]谭广兴,朱燕飞,毛宗源.基于Henon映射的自适应克隆选择优化算法[J].计算机工程与应用,2006,9:73-76
[61]赵立强,李艳坡等.基于广义Arnold混沌映射和Henon混沌映射的小波水印算法[J].河北科技师范学院学报,2006,20(4):32-37
[62]王西文.地震资料处理和解释中的小波分析方法[M].北京:石油工业出版社,2004
[63] Dai Yongshou, Wang Junling, Wang Weiwei,et al. Research on Wavelet Estimation via Cumulant Based ARMA Model Approach[J]. Proceedings of 2007 International Conference on Wavelet Analysis and Pattern Recognition [P]. 2007 4(4): 1882-1886