Numerical Simulation Of Two-dimensional Seismic Wave Propagation In Isotropic And Anisotropic Media By SBP-SAT Method

The velocity-stress differential equations used to describe the propagation of seismic waves are often solved by the finite difference method.This method has the characteristics of fast calculation,simple principle and strong adaptability,and is widely used in seismic wave field simulation.At present,the finite difference method has derived a variety of high-order precision methods,which not only greatly reduces the error,but also significantly reduces the degree of freedom for calculation,saves memory occupation,and improves the calculation efficiency.However,when dealing with complex geometric models,the finite difference method is prone to generate spurious diffraction,corner scattering,and even simulation instability.When dealing with boundary conditions,it is also necessary to modify the difference scheme.The accuracy is generally lower than that of the interior of the medium and there is also a problem of stability.These difficulties are particularly evident in the higher-order method,variable-step method and non-traditional mesh generation method.Many researchers often use SBP-SAT method.It is based on the nodal method and keeps the form of the differential equation.The derivative is approximately expressed in the linear combination form of the nodal solution.The combination of space discrete SBP operator and SAT boundary method has many advantages.The operator in matrix product form makes the programming easier and the calculation efficiency higher.At the same time,the boundary conditions can also adapt to various boundary conditions.These greatly expand the application scope of SBP-SAT method.In addition,SBP operator can also be used in other numerical methods,and has a wide application potential.In this paper,the multi-block SBP-SAT method is applied to obtain the wave field information of isotropic media and anisotropic media,the symmetric matrix form(SMF)of elastic wave equation is derived,and the rectangular grid-spacing ratios is obtained and extended to the curve domain.Based on the SMF framework,we apply non-reflective boundary conditions,and carry out the energy analysis of the system with the help of the energy method.The main contents of the paper include:(1)The elastic wave equations expressed in velocity and stress is integrated into the SMF system by matrix symmetry.On this basis,the discrete form is established using the multi-block SBP-SAT method and the stability is proved using the energy method.We transform the wave equation in a rectangular domain into a curved domain,divide the solution domain into an appropriate number of sub computational domains using SAT,and apply appropriate boundary conditions to achieve an elastic wave SMF discrete framework.(2)Numerical simulation of wave field in isotropic media: We program the derived equations to simulate the propagation of seismic waves.The results were compared with the simulation results of COMSOL software to verify the correctness and timeliness of the results.Through the wave field snapshots and single shot records simulated by isotropic media model,layered media model,and Corner-edge model,we discussed the characteristics and propagation laws of the wave field.We also discuss the effects of spatial step size and operator order on simulation results in order to reduce numerical dispersion.(3)Numerical simulation of wave field in anisotropic media: We describe the properties of transversely isotropic media and obtain an elastic wave equation in matrix symmetric form.Based on this,we numerically simulate the VTI media model,TTI media model,layered VTI media model,curved layered complex media model,and VTI media model with cracks,summarizing the wave field characteristics and propagation laws.The influence of Thomsen parameters on wave field is discussed with the help of VTI media model,providing reference for geological exploration and engineering applications.