Fast ray tracing method in 3-D structure and its proof of positive definiteness

Based on Fermat’s principle, two-point ray tracing method was studied in three-dimensional structure. By means of first order Taylor’s incomplete series expansion (i.e. no expansion to the length of the ray), a symmetry block tridiagonal matrix equation set was deduced. Further, the positive definiteness of coefficient matrix was discussed, and the positive definiteness was accurately proved in a mathematical way. It assured that the algorithm was well-posed. Associated with iterative method, the solution to ray tracing can be got through step-by-step linearized iteration of the nonlinear problem. An algorithm of the whole path iterative ray tracing method in three-dimensional velocity structure was obtained. This method shows a clear and simple as well as explicit computation formula, which makes ray tracing computation easily applicable in practice. The correction vector is obtained through finding the solution to the positive definite block tridiagonal equation set, which ensures the method is robust convergence. This study offers a new kind of feasible and efficient ray tracing method for three dimensional seismic migration and tomography. Meanwhile, it also provides the prerequisite guarantee to design a fast algorithm.

浙江省遂昌县左别源梯级水电站水工隧洞优化设计

水电站引水建筑物主要有隧洞、渠道、管道等形式，由于隧洞具有线路短、适应水位变幅大、施工作业涉及面小、运行安全可靠等特点，小水电工程普遍采用水工隧洞的引水方式。遂昌县左别源梯级水电站工程引水发电隧洞共长6700m，隧洞工程直接投资迭700万元，占工程总投资（2350万元）30％以上，且隧洞工程为电站建设工期控制性工程，故隧洞工程的优化设计对于控制工程投资，加快工程进度，提高工程效益具有至关重要的意义。左别源梯级水电站水工隧洞的设计在遵循设计规范条文的前提下，根据各级电站不同工程布置及地形、地貌、地质条件特点，选取了兼顾施工方便、节省工期、经济合理的设计方案，取得了良好的效益。