多维波动方程逆散射的基础理论研究  

Study into fundamental theory of inverse scattering of multi-dimensional wave equation.

作　　者：刘洪[1] 刘国峰[1] 武威[1] 袁江华[1] 李幼铭[1] 

机构地区：[1]中国科学院地质与地球物理研究所,北京100029

出　　处：《石油物探》2007年第6期569-581,共13页Geophysical Prospecting For Petroleum

基　　金：国家重点基础研究发展计划(973)(2007CB209603);国家自然科学基金项目(40674072;40474047);中国科学院知识创新工程重要方向(KZCX3-SW-147);国家高技术研究发展计划(863)(2006AA09A102-08)共同资助

摘　　要：探讨了多维(包括二维和三维)波动方程逆散射基础理论。通过对前人研究成果的综合分析,指出多维波动方程逆散射解法的整体框架类似于一维波动方程反问题。三维波动方程逆散射的关键环节可类比于一维波动方程反问题,一维波动方程逆散射中的时深转换、Z变换、一维谱分解和反射与透射系数等环节,在多维波动方程逆散射或速度横向变化介质逆散射的研究中,被替换为射线坐标系、单程波算子、基于Witt积的多维谱分解和反射与透射算子的平面波响应。单程波算子积分表示的有效化、射线坐标系上波动方程的微分形式化、Witt积的深入应用和多维谱分解的现代发展,是多维波动方程逆散射关键基础问题研究的重要组成部分。理论和数值实例表明,散射数据的谱分解结果有更好的聚焦效果,这对于进一步速度分析和反射系数的求取十分有益。The basic theory of multi-dimensional(including 2D and 3D) wave equation inverse scattering was probed.Through analyzing the former research results,we have pointed out that the entire frame of inverse scattering solution on multi-dimensional wave e- quation is similar to the inverse problem of 1D wave equation.The 1D wave equation inverse scattering problem includes many proce- dures such as time-depth conversion,Z transform,1D spectral fac- torization,reflection coefficient and transmission coefficient,et al. In multi-dimension or in the lateral velocity varying situation,the counterpart conception in 1D case should he changed to image ray coordinate,one-way wave operator,multi-dimension spectral fac- torization based on Witt multiplication,and the plane wave re- sponse of reflection and transmission operator.The important con- tents in studying multi-dimension inverse scattering problem are ef- fective expression of one-way operator integral,differential form of the wave equation in ray coordinate,wide application of Witt multi- plication and the development of multi-dimension spectral factoriza- tion.Theoretical and actual examples show that the spectrum de- composition results of scattering data has better focus effect,which is beneficial for further analyzing the velocity and for finding the re- flection coefficient.

关 键 词：保结构算法 拟微分算子理论 射线坐标系 谱因式分解 大步长偏移 指数映射 Lie代数积分方法 

分 类 号：P631.4[天文地球—地质矿产勘探]