The Finite Difference Numerical Method For Eikonal Equation

The eikonal equation is the basis of the finite difference traveltime forward method.The fast marching method and the fast sweeping method based on the eikonal equation are two common finite difference traveltime calculation methods.In this paper,the numerical solution of eikonal equation is studied deeply.In view of the shortcomings of conventional method,the improved opinions and methods are put forward in three directions.(1)A fast sweeping method for polar/sphere coordinates is proposed for solve the source singularity problem in the finite difference traveltime method.In the Cartesian coordinate system,the finite difference traveltime calculation method defaults the wavefronts as a plane which perpendicular to the wavefields propagation direction.But under the point source condition,the actual wavefronts near the source is preceded by a surface with large radians,which is difficult to fit into a plane.Due to the mismatch between the actual wavefronts and the assumed wavefronts,the error is generated.The errors generated by the source point region will gradually accumulate with the operation,which will affect the overall calculation accuracy.The grid morphology of the polar/sphere coordinates is similar to the wavefields in the point source condition.The default wavefronts of the polar/sphere coordinates method go also matches the actual wavefronts,which can greatly suppress the source singularity problem.In this paper,the formula for solving eikonal equation is derived in polar/sphere coordinate system,and the detailed numerical method is also given.The fast marching method and the fast sweeping method of the polar/sphere coordinates can be realized by the relevant numerical solution formula.(2)The order of traveltime field is discussed,and a fast marching method based on insertion sorting method is proposed.The fast marching method uses the narrow band technology,which requires frequent searching for the minimum traveltime points in narrow-band point array.The conventional method defaults the narrow band point array to the disordered array,but actually according to the causality condition of the eikonal equation,the traveltime field value of ideal state has the law of distribution from small to large.In the regular Cartesian coordinate algorithm,the order of the traveltime field is disturbed because of the singularity problem of the source point.For the traveltime calculation method which has suppressed the source singularity problem,the order of the traveltime field is enhanced.Replacing the conventional heap sorting method with the insertion sorting method can effectively improve the accuracy and efficiency of the algorithm.(3)A high-order WENO-Z++scheme for fast sweeping method is proposed.WENO-Z++format is introduced based on the conventional high-order WENO-JS format fast sweeping method.The improved high-order format can effectively improve the accuracy and iterative convergence speed of the algorithm.Finally,through numerical simulation,the improved research methods of the above three aspects are tested and verified.