Ray Path Calculation Method Based On Orthogonal Polynomial Approximated Traveltime

Seismic wave travel time and ray path calculation are important components of seismic wave forward modeling.It is one of the important methods to solve seismic wave propagation,imaging and inversion problems in any inhomogeneous medium.The seismic wave travel time calculation method using the Eikonal equation has the characteristics of fast calculation speed and no calculation of blind spots,and is an ideal method for solving the seismic wave travel time and fast calculation of the ray path.However,only the seismic wave travel time can only be calculated by using the equation of travel equation,and the ray path cannot be calculated.In order to solve the calculation problem of the ray path,a gradient is generally calculated by a finite difference or interpolation function method.However,the accuracy and stability of the travel time gradient calculation are limited by the travel time accuracy of the grid points.To solve the above problem,after obtaining the discrete travel time calculated by the fast-pushing method,multiple continuous derivative function operators are introduced into the discrete travel time field,and the traveltime approximation function based on orthogonal polynomials is obtained,and the approximation function is obtained based on various approximation functions.The travel time gradient direction of the operator starts from the detection point and tracks the ray path to the source or to the artificial interface in the negative direction of the gradient.Combined with multi-level multi-reflection calculations,a ray based on orthogonal polynomial approximation travel time is proposed.Path calculation method,obtained the following research results:1.Use the orthogonal polynomial to approximate the travel time field,use the derivative stability of the continuous derivative function to avoid the time-varying numerical value of the uncertainty problem,and extend the function to three-dimension;2.Using the error of the approximation function to achieve the minimum characteristic in the least-squares sense,and the low-order polynomial approximation of the Chebyshev polynomial with high-order accuracy to improve the gradient calculation accuracy,and to ensure that the calculation method is satisfactory in the velocity mutation region.The calculation effect.3.Combine the multi-level zoning method to apply this ray calculation method to the partitioned multi-level travel time field to calculate any sub-transmission and reflected ray paths.The test results based on a number of simple or complex velocity models show that the ray path calculation method based on orthogonal polynomial approximation travel time has higher computational speed,and eases the influence of grid point travel time accuracy on accuracy and stability of gradient calculation.The most accurate conclusion is that the radiation calculated by the first-order Chebyshev(second-order)polynomial approximation travel time is obtained.Because the method ray in this paper is unilaterally based on the characteristics of travel time calculation,it means that the method can combine arbitrary travel time fields,and retain all the characteristics of the travel time calculation method.