Study On Numerical Method Of Forward And Inversion Of Variable Coefficient Wave Equation

When geological exploration is conducted,it is particularly significant to grasp the propagation law of seismic waves in the underground medium.After exploration,people found that the information of geological parameters is mostly included in the coefficients of wave equations.Through performing forward and inverse analysis of these wave equations,a large amount of geological information can be obtained,which can provide more accurate data for geological exploration problems.However,as far as the inverse problem of wave equation is concerned,it faces not only the essential difficulties of randomness,noisyness,non-linearity and ill-posedness of the actual seismic data,but also problems of the huge amount of calculation and huge storage memory in actual output.Therefore,the research on forward and inversion algorithms of wave equations has both important theoretical significance and practical application value.First,in view of the character of seismic waves propagate in the form of a spherical surface,the paper uses a two-dimensional wave equation in a hemispherical region as a specific mathematical model.As the research area of the model is hemispherical,and difficult to solve with difference discrete.The mathematical model of the two-dimensional wave equation in the polar coordinate system is obtained by polar coordinate transformation.The finite difference method is used to discretize the wave equation and boundary conditions,arriving at a numerical difference scheme for solving the two-dimensional wave equation forward modeling in a hemispherical region,and numerical algorithm for the forward modeling problem is formed.The algorithm process is programmed and verified with MATLAB.Secondly,based on the above-mentioned research of the two-dimensional wave equation forward modeling in the hemispherical region,a inversion model of the two-dimensional wave equation in the hemispherical region is built.Because the inversion of the wave equation is mostly ill-posed,the Tikhonov regularization method is combined to overcome ill-posedness of solution,making the solution unique.The final solution problem is transformed into an optimization problem,and combined with the basic iterative method theory,an iterative inversion algorithm based on the conjugate gradient method and damped Newton method are built.Finally,in the paper,the two algorithms constructed above with a suitable source function to perform numerical simulation of the two-dimensional wave equation inversion in a hemispherical region.Among them,a three-layer horizontal layered medium model and a single anomalous body medium model are selected,and their velocity parameters are numerically inverted.The above two algorithms are analyzed and summarized from the actual calculated effects.The results show the stability,and efficiency of the construction inversion algorithm.In summary,the paper constructs a forward numerical algorithm for a two-dimensional wave equation in a hemispherical region and an iterative inversion algorithm combining damped Newton's method and conjugate gradient.The algorithm is programmed for numerical simulation,The algorithm is feasible from practical results.And the practical application is flexible,which has certain theoretical significance and research value.