起伏地表条件下的声波散射数值模拟的积分方程法
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摘要
从散射理论的角度来看,起伏地表可以看作是一种特殊的扰动介质,因此应用散射积分方程求解起伏地表条件下的散射场在理论上是可行的。从三维频率域声波方程出发,由格林函数定理,得出起伏地表条件下的散射积分方程。散射积分方程为关于起伏地表的面积分和与速度扰动体有关的体积分之和,同时给出了格林函数在奇异点的积分方法。由于数值离散求解积分方程存在着计算时间太长和存储内存不足的问题,采用电磁散射积分方程的拟解析近似的方法。在假设反射函数为缓变函数的基础上,最终得到其近似表达式,因此散射场的数值求解不必再借助于代数方程组,只要进行数值积分即可。这种方法避开了传统数值计算方法存在的问题,为地震散射波场快速正演模拟打下了基础。理论分析表明,这种方法适用于小扰动的问题。当扰动较大时,拟解析近似会产生较大的误差。
According to scattering theory,irregular surface can be considered to be a special kind of perturbation.So it is available to calculate scattered field by means of scattered integral on irregular topography.In this paper,beginning with acoustic formulation in the frequency domain of threedimension,based on the theorem of Green function,we obtain the integral equation for scattered field,which is the sum form of the area integral relating to irregular topography and the volume integral relating to velocity perturbation.Following,we present the integral about the singularity of Green function.Because,numerical methods bring about the problems of calculating time and computer storage,in order to avoiding these problems,we introduce a new method: quasianalytical approximation.Based on the assumption that reflection function varies slowly in studying area,we obtain the expression of its approximation.By means of this method,the scattered field computation needs no longer to be realized by solving large linear equation systems,a numerical integration is enough.This scheme avoids the problems raised by the traditional numerical method,so that it paves way for fast forward simulation.Academic analysis indicates that this method is available in the case of small perturbation;otherwise it will cause great error.
According to scattering theory,irregular surface can be considered to be a special kind of perturbation.So it is available to calculate scattered field by means of scattered integral on irregular topography.In this paper,beginning with acoustic formulation in the frequency domain of threedimension,based on the theorem of Green function,we obtain the integral equation for scattered field,which is the sum form of the area integral relating to irregular topography and the volume integral relating to velocity perturbation.Following,we present the integral about the singularity of Green function.Because,numerical methods bring about the problems of calculating time and computer storage,in order to avoiding these problems,we introduce a new method: quasianalytical approximation.Based on the assumption that reflection function varies slowly in studying area,we obtain the expression of its approximation.By means of this method,the scattered field computation needs no longer to be realized by solving large linear equation systems,a numerical integration is enough.This scheme avoids the problems raised by the traditional numerical method,so that it paves way for fast forward simulation.Academic analysis indicates that this method is available in the case of small perturbation;otherwise it will cause great error.
引文
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[2]Moses H E.Calculation of the scattering potentialfrom reflection coefficients[J].Physical Review,1955,102(2):559-567.
[3]Weglein A B,Gasparotto F A,Carvalh P M,et al.An inversescattering series method for attenuatingmultiples in seismic reflection data[J].Geophysics,1997,62(6):1975-1989.
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[5]李灿苹,刘学伟,王祥春,杨丽.地震波的散射理论和散射特征及其应用[J].勘探地球物理进展,2005,28(2):81-90.
[6]孙建国.声波散射数值模拟的两种新方案[J].吉林大学学报(地球科学版),2006,36(5):863-869.
[7]Fu L Y,Mu Y G,Yang H J,Forward problem ofnonlinear Fredholm integral equation in referencemedium via velocityweighted wave field function[J].Geophysics,1997,62(2):650-656.
[8]姚海英,聂在平.三维矢量散射分析中奇异积分的准确计算方法[J].电子科学学刊,2000,22(3):471-477.
[9]Zhdanov M S,Dmitriev V I,Fang S,Hurs′an G.Quasianalytical approximations and series inelectromagnetic modeling[J].Geophysics,2000,65(6):1746-1757.
[2]Moses H E.Calculation of the scattering potentialfrom reflection coefficients[J].Physical Review,1955,102(2):559-567.
[3]Weglein A B,Gasparotto F A,Carvalh P M,et al.An inversescattering series method for attenuatingmultiples in seismic reflection data[J].Geophysics,1997,62(6):1975-1989.
[4]Widya I.Stable inverse scattering algorithms for 2Dmedia[J].Annual Meeting Abstracts:Society ofExploration Geophysicists,1984.
[5]李灿苹,刘学伟,王祥春,杨丽.地震波的散射理论和散射特征及其应用[J].勘探地球物理进展,2005,28(2):81-90.
[6]孙建国.声波散射数值模拟的两种新方案[J].吉林大学学报(地球科学版),2006,36(5):863-869.
[7]Fu L Y,Mu Y G,Yang H J,Forward problem ofnonlinear Fredholm integral equation in referencemedium via velocityweighted wave field function[J].Geophysics,1997,62(2):650-656.
[8]姚海英,聂在平.三维矢量散射分析中奇异积分的准确计算方法[J].电子科学学刊,2000,22(3):471-477.
[9]Zhdanov M S,Dmitriev V I,Fang S,Hurs′an G.Quasianalytical approximations and series inelectromagnetic modeling[J].Geophysics,2000,65(6):1746-1757.
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