Seismic Waveform Simulation Using Finite Difference Method In A Polar Coordinate System With Irregular Topography

Finite difference method(FDM)is one of the most popular numerical methods used in seismological research and the applying of traction free surface boundary condition is a key issue for FDM.When irregular topography exists,we need to introduce boundary-conforming grid to avoid artificial scattering wave.For Cartesian coordinate system FDM,previous researches show that Traction Image method,assuming the traction are anti-symmetric by the free surface,satisfy the traction free condition with high accuracy and high efficiency.For seismic wave modeling in regional or global scales,the effect of Earth's curvature must be considered.The cylindrical/polar coordinate system FDM is more convenience to calculate and easier to apply the cyclic boundary condition.Since there is no such high accuracy and high efficiency boundary condition method as Traction Image,we are going to introduce the Traction Image method into cylindrical/polar coordinate system FDM.To save on the computational resource and reduce the complex,here we argue for usage of Traction Image on polar coordinate system FDM.After introducing the boundary-conforming grid system(?,?)to conform the irregular topography,we first rewrite the Hamiltonian operator in(?,?),and also rewrite the expressions of stress,strain,the Hook's law,elastodynamic equations,the unit vector perpendicular to the free surface and the traction free boundary condition.The singularity in velocity derivation for grid on the free surface can be eliminated by the time domain derivation of the traction free condition.For the grid just under the free surface,this singularity is avoided by using compact scheme for velocity derivation.After setting the traction to be anti-symmetric by free surface,the singularity in stress derivation for grid on and near the free surface can be expressed in term of stress,traction,the spatial derivation of traction and its image.Our method is testified by several examples:regular polar area,regular area with local smooth irregular topography(hill,valley,hill and valley);area with planar surface(the planar surface is global curved in polar coordinate system),area with planar surface and local smooth irregular topography(hill,valley,hill and valley);regular polar area with a step corner,area with planar surface and step corner.Our result in most of the examples are great consistency with reference method result.We also presented an optimized version.