用双曲正切和反正切函数拟合面波频散曲线的适应性分析
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摘要
双曲正切函数和反正切函数的形态与面波频散曲线较相似,将其用于频散曲线拟合可减少数据噪音对反演稳定性的影响。这两类函数的曲线形态都是单调变化的,无法完全刻画某些特殊地层结构形成的频散曲线的复杂变化情形,如果用这两类曲线来拟合频散曲线则会存在本质上的模型误差。通过一系列正演模拟,分别运用这两类曲线拟合随层厚、P波速度、S波速度和密度变化模型的频散曲线,定量分析每个参数的变化对拟合误差的影响。结果表明,拟合结果随着P波速度、密度和层厚变化的分布很稳定,误差值保持在0.03~0.11。在S波速度随深度单调增加的情况下,拟合误差变化起伏比较大,误差范围为0.00~0.14,表明这两类三角函数模型的适应性会随着地层间S波速度差异的增大而降低。当模型中含有一定厚度的倒转低速层的时候,拟合误差可高达0.30,表明双曲正切函数和反正切函数频散曲线模型均不适用于拟合含强烈反差的低速倒转地层的频散曲线。实验发现,严重影响频散曲线拟合的误差突跳难题,可以通过多个随机种子的组合得到有效解决。
The shape of hyperbolic tangent and arctangent functions are similar with the dispersion curves of surface waves,hence it is used on dispersion curve fitting to reduce the influence of the noise. As both types of functions are monotonous curves,they are impossible to completely characterize the change of dispersion curves from some special stratigraphic structures. Adaptability of these two types of curves was tested by modeling a series of geological models. The models were designed to characterize the effects on dispersion curves of layer thickness,P-wave velocity,S-wave velocity and density. The results show that the misfits of 0. 03 ～ 0. 11 are stable due to the changes of P-wave velocity,density and layer thickness. On the other hand,shear wave velocity variation plays a very significant role in the fitting quality. When S-wave velocity monotonously increases with depth,fluctuation of fitting errors is very significant,ranging from 0. 00 to 0. 14. It means that the adaptability of these two kinds of trigonometric function models will decrease with the increase of S wave velocity variation. When a reversed layer is embedded in a geological structure,it can affect the fitting quality very significantly,with errors up to 0. 30. This indicates that neither the hyperbolic tangent nor arctangent function model is suitable for fitting the dispersion curve from a geological structure embedded with reversed low velocity layers. The experiment revealed that the error fluctuation problem,which seriously affects the fitting quality,can be effectively resolved by a combination of multi random seeds.
The shape of hyperbolic tangent and arctangent functions are similar with the dispersion curves of surface waves,hence it is used on dispersion curve fitting to reduce the influence of the noise. As both types of functions are monotonous curves,they are impossible to completely characterize the change of dispersion curves from some special stratigraphic structures. Adaptability of these two types of curves was tested by modeling a series of geological models. The models were designed to characterize the effects on dispersion curves of layer thickness,P-wave velocity,S-wave velocity and density. The results show that the misfits of 0. 03 ～ 0. 11 are stable due to the changes of P-wave velocity,density and layer thickness. On the other hand,shear wave velocity variation plays a very significant role in the fitting quality. When S-wave velocity monotonously increases with depth,fluctuation of fitting errors is very significant,ranging from 0. 00 to 0. 14. It means that the adaptability of these two kinds of trigonometric function models will decrease with the increase of S wave velocity variation. When a reversed layer is embedded in a geological structure,it can affect the fitting quality very significantly,with errors up to 0. 30. This indicates that neither the hyperbolic tangent nor arctangent function model is suitable for fitting the dispersion curve from a geological structure embedded with reversed low velocity layers. The experiment revealed that the error fluctuation problem,which seriously affects the fitting quality,can be effectively resolved by a combination of multi random seeds.
引文
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6李传金,刘建欢,杜亚楠,等.扩展空间自相关法用于线性阵列时加权拟合问题研究.科学技术与工程,2017;17(8):111-114Li Chuanjin,Liu Jianhuan,Du Yanan,et al.Study on weighted fitting problem when expanded spatial autocorrelation method applied to linear arrays.Science Technology and Engineering,2017;17(8):111-114
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9 Park C B,Miller R D,Xia J.Imaging dispersion curves of surface waves on multi-channel record.SEG Technical Program Expanded Abstracts 1998.Tulsa:Society of Exploration Geophysicists,1998;1377-1380
10 Xia J,Xu Y,Miller R D.Generating an image of dispersive energy by frequency decomposition and slant stacking.Pure and Applied Geophysics,2007;164(5):941-956
11潘冬明,胡明顺,崔若飞,等.基于拉东变换的瑞雷面波频散分析与应用.地球物理学报,2010;53(11):2760-2766Pan Dongming,Hu Mingshun,Cui Ruofei,et al.Dispersion analysis of Rayleigh surface waves and application based on Radon transform.Chinese Journal of Geophysics,2010;53(11):2760-2766
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13何华.频散曲线拟合瑞雷波勘探方法及其实现.武汉:中国地质大学(武汉),2001He Hua.Dispersion curve fitting Rayleigh wave exploration method and its realization.Wuhan:China University of Geosciences(Wuhan),2001
14陈祥.瑞雷波频散曲线拟合分析及其应用.北京:中国地质大学(北京),2004Chen Xiang.Approach to dispersion curve fitting of rayleigh wave and its application.Beijing:China University of Geosciences(Beijing),2004
15 Durdaˇg D.Using Levenberg-Marquardt method in dispersion curve fitting.9th Congress of the Balkan Geophysical Society.Houten:EAGA,2017:10.3997/2214-4609.201702594
16 Tang X M,Li C,Patterson D J.A curve-fitting technique for determining dispersion characteristics of guided elastic waves.Geophysics,2010;75(3):E153-E160
17唐大雄.工程岩土学.北京:地质出版社,1999Tang Daxiong.Engineering geotechnics.Beijing:Geological Publishing House,1999
18马中高,解吉高.岩石的纵、横波速度与密度的规律研究.地球物理学进展,2005;20(4):905-910Ma Zhonggao,Jie Jigao.Relationship among compressional wave,shear wave velocities and density of rocks.Progress in Geophysics,2005;20(4):905-910
19余嘉顺,贺振华.一种模拟随机数字地层模型的方法.地震研究,2004;27(4):344-349Yu Jiashun,He Zhenhua.A method for modelling randomly layered earth models.Journal of Seismological Research,2004;27(4):344-349
20 Xia J,Miller R D,Park C B.Estimation of near-surface shear-wave velocity by inversion of Rayleigh waves.Geophysics,1999;64(3):691-700
21 Wu B,Chen X.Stable,accurate and efficient computation of normal modes for horizontal stratified models.Geophysical Journal International,2016;206(2):1281-1300
22 Nelder J A,Mead R.A simplex method for function minimization.The Computer Journal,1965;7(4):308-313
23 Press W H,Teukolsky S A,Vetterling W T,et al.Numerical recipes in C:The art of scientific computing.Cambridge:Cambridge University Press,1992
24韩凌辉.模拟退火算法与非线性单纯形算法的混合算法.北京:北京工业大学,2009Han Linghui.Hybrid algorithm combination of simulated annealing algorithm and the non-linear Simplex algorithm.Beijing:Beijing University of Technology,2009
25 Aarts E H L,Van Laarhoven P J M.Statistical cooling:A general approach to combinatorial optimization problems.Philips Journal of Research,1985;40(4):193-2260
26 Cardoso M F,Salcedo R L,Azevedo S F D.The simplex-simulated annealing approach to continuous non-linear optimization.Computers&Chemical Engineering,1996;20(9):1065-1080
27 Aarst E,Korst J.Simulated annealing and boltzmann machinesA stochastic approach to combinatorial optimization and neural computers.New York:Wiley,1989
2刘庆华,鲁来玉,王凯明.主动源和被动源面波浅勘方法综述.地球物理学进展,2015;30(6):2906-2922Liu Qinghua,Lu Laiyu,Wang Kaiming.Review on the active and passive surface wave exploration method for the near-surface structure.Progress in Geophysics,2015;30(6):2906-2922
3张维,何正勤,胡刚,等.用面波联合勘探技术探测浅部速度结构.地球物理学进展,2013;28(4):2199-2206Zhang Wei,He Zhengqin,Hu Gang,et al.Detection of the shallow velocity structure with surface wave prospection method.Progress in Geophysics,2013;28(4):2199-2206
4丰赟,沙椿.面波联合勘探在深厚覆盖层地区应用实例分析.物探与化探,2018;42(2):392-397Feng Yun,Sha Chun.Combined use of active and passive surface waves in the deep overburden area.Geophysical and Geochemical Exploration,2018;42(2):392-397
5刘若飞.被动源面波方法在城市中的应用研究.武汉:中国地质大学(武汉),2015Liu Ruofei.Research on applications of passive surface-wave methods in urban areas.Wuhan:China University of Geosciences(Wuhan),2015
6李传金,刘建欢,杜亚楠,等.扩展空间自相关法用于线性阵列时加权拟合问题研究.科学技术与工程,2017;17(8):111-114Li Chuanjin,Liu Jianhuan,Du Yanan,et al.Study on weighted fitting problem when expanded spatial autocorrelation method applied to linear arrays.Science Technology and Engineering,2017;17(8):111-114
7 Mcmechan G A,Yedlin M J.Analysis of dispersive waves by wave field transformation.Geophysics,1981;46(6):869-874
8 Gabriels P,Snieder R,Nolet G.In situ measurements of shear-wave velocity in sediments with higher-mode Rayleigh waves.Geophysical Prospecting,1987;35(2):187-196
9 Park C B,Miller R D,Xia J.Imaging dispersion curves of surface waves on multi-channel record.SEG Technical Program Expanded Abstracts 1998.Tulsa:Society of Exploration Geophysicists,1998;1377-1380
10 Xia J,Xu Y,Miller R D.Generating an image of dispersive energy by frequency decomposition and slant stacking.Pure and Applied Geophysics,2007;164(5):941-956
11潘冬明,胡明顺,崔若飞,等.基于拉东变换的瑞雷面波频散分析与应用.地球物理学报,2010;53(11):2760-2766Pan Dongming,Hu Mingshun,Cui Ruofei,et al.Dispersion analysis of Rayleigh surface waves and application based on Radon transform.Chinese Journal of Geophysics,2010;53(11):2760-2766
12 Louie J N.Faster,better:Shear-wave velocity to 100 meters depth from refraction microtremor arrays.Bulletin of the Seismological Society of America,2001;91(2):347-364
13何华.频散曲线拟合瑞雷波勘探方法及其实现.武汉:中国地质大学(武汉),2001He Hua.Dispersion curve fitting Rayleigh wave exploration method and its realization.Wuhan:China University of Geosciences(Wuhan),2001
14陈祥.瑞雷波频散曲线拟合分析及其应用.北京:中国地质大学(北京),2004Chen Xiang.Approach to dispersion curve fitting of rayleigh wave and its application.Beijing:China University of Geosciences(Beijing),2004
15 Durdaˇg D.Using Levenberg-Marquardt method in dispersion curve fitting.9th Congress of the Balkan Geophysical Society.Houten:EAGA,2017:10.3997/2214-4609.201702594
16 Tang X M,Li C,Patterson D J.A curve-fitting technique for determining dispersion characteristics of guided elastic waves.Geophysics,2010;75(3):E153-E160
17唐大雄.工程岩土学.北京:地质出版社,1999Tang Daxiong.Engineering geotechnics.Beijing:Geological Publishing House,1999
18马中高,解吉高.岩石的纵、横波速度与密度的规律研究.地球物理学进展,2005;20(4):905-910Ma Zhonggao,Jie Jigao.Relationship among compressional wave,shear wave velocities and density of rocks.Progress in Geophysics,2005;20(4):905-910
19余嘉顺,贺振华.一种模拟随机数字地层模型的方法.地震研究,2004;27(4):344-349Yu Jiashun,He Zhenhua.A method for modelling randomly layered earth models.Journal of Seismological Research,2004;27(4):344-349
20 Xia J,Miller R D,Park C B.Estimation of near-surface shear-wave velocity by inversion of Rayleigh waves.Geophysics,1999;64(3):691-700
21 Wu B,Chen X.Stable,accurate and efficient computation of normal modes for horizontal stratified models.Geophysical Journal International,2016;206(2):1281-1300
22 Nelder J A,Mead R.A simplex method for function minimization.The Computer Journal,1965;7(4):308-313
23 Press W H,Teukolsky S A,Vetterling W T,et al.Numerical recipes in C:The art of scientific computing.Cambridge:Cambridge University Press,1992
24韩凌辉.模拟退火算法与非线性单纯形算法的混合算法.北京:北京工业大学,2009Han Linghui.Hybrid algorithm combination of simulated annealing algorithm and the non-linear Simplex algorithm.Beijing:Beijing University of Technology,2009
25 Aarts E H L,Van Laarhoven P J M.Statistical cooling:A general approach to combinatorial optimization problems.Philips Journal of Research,1985;40(4):193-2260
26 Cardoso M F,Salcedo R L,Azevedo S F D.The simplex-simulated annealing approach to continuous non-linear optimization.Computers&Chemical Engineering,1996;20(9):1065-1080
27 Aarst E,Korst J.Simulated annealing and boltzmann machinesA stochastic approach to combinatorial optimization and neural computers.New York:Wiley,1989