直角坐标系下立体层析FRECHET导数求取修正问题探讨

摘要

The Frechet derivative of stereotomography in the Cartesian coordinate system is a first-order derivative of disturbance quantity observed in surface to initial model space.If the endpoint of the paraxial ray has to be corrected to the surface,a significant error will be introduced into the Frechet derivative.The inversion accuracy will be reduced and should be modified.Differing from the conventional method to solve this problem by tracing the central ray a little longer,we use the triangular relationship among the coordinates of the emergence point for the central ray at the surface,the endpoint of the paraxial ray at the surface and the lateral difference between the endpoint of the paraxial ray and the expected emergence point at the surface,to convert a linear relationship between the endpoint coordinate difference of the central ray and paraxial ray in the surface and the initial perturbation of the ray start-point.Then the correct Frechet derivatives can be achieved based on this conversion.The method presented is exactly equivalent to the previous method with a lower computational cost.Numerical examples demonstrate the correctness and the necessity of the presented correction for stereotomography in the Cartesian coordinate system.%直角坐标系下立体层析反演所需的FRECHET导数是地表观测到的扰动量对模型空间初始扰动量的一阶导数.当傍轴射线没有到达地表时,FRECHET导数的计算会产生误差,从而影响反演精度,必须对其进行修正.利用中心射线地表出射点坐标、傍轴射线最后一点坐标以及傍轴射线最后一点与其地表预期出射点坐标差之间存在的三角关系,换算出中心射线地表出射点与傍轴射线地表预期出射点的坐标差和初始位置扰动之间的线性关系,从而求取正确的FRECHET导数.上述换算与前人提出的多追踪一小段射线的计算公式等价,但是计算成本更低.数值算例证实了修正方法的正确性以及直角坐标系下立体层析实施这种修正的必要性.