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

(中国科学院地质与地球物理研究所，北京100029)

Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029, China

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 equation is similar to the inverse problem of 1D wave equation. The 1D wave equation inverse scattering problem includes many procedures such as time-depth conversion, Z transform, 1D spectral factorization, reflection coefficient and transmission coefficient,et al. In multi-dimension or in the lateral velocity varying situation, the counterpart conception in 1D case should be changed to image ray coordinate, one-way wave operator, multi-dimension spectral factorization based on Witt multiplication, and the plane wave response of reflection and transmission operator. The important contents in studying multi-dimension inverse scattering problem are effective expression of one-way operator integral, differential form of the wave equation in ray coordinate, wide application of Witt multiplication and the development of multi-dimension spectral factorization. Theoretical and actual examples show that the spectrum decomposition results of scattering data has better focus effect, which is beneficial for further analyzing the velocity and for finding the reflection coefficient.