CIP-MOC Modeling of Seismic Wave Propagation in Elastic Media

In many fields such as hydrodynamics and MHD, the CIP method, an upwind difference hyperbolic equation solver, has widely been employed for advection calculation. The CIP scheme was constructed considering that an advected property and its spatial derivative follow same advection equation. This effects low numerical dispersion and relaxed CFL condition in the advection calculation. In the present work, we developed a CIP-MOC (CIP with method of characteristics) scheme for seismic wave propagation in 3D elastic heterogeneous media with flat free surface. 3D elastic wave equations in velocity-stress formulation and their spatial derivatives, as well, are converted into sets of 1D advection equations and non-advection equations for each direction (x,y,z in Cartesian coodinate system) with the method of characteristics. Since the Riemann invariant of each advection equation consists of stress and velocity, updatings of velocity and stress are simultaneous and a collocated grid system is employed. A free surface is modeled as a zero-stress surface. A reflection free boundary is installed by considering no incident wave comes from outside of the boundary. A double coupled seismic point source is introduced as external point stresses. Overall scheme is made up of multiphases employing time-splitting and directional-splitting techniques. Each time step is composed of three directional updating phases each for wave propagation in x, y and z direction. Each directional updating phase is made up of advection phase and non-advection phase. In the advection phase, advection equations are solved with the CIP method. In the non-advection phases, non-advetion equations and boundary conditions are evaluated with central finite differences. We conducted CIP-MOC seismic wave propagation simulations in a half-space, layered and fully heterogeneous media for embedded point source. By comparing our products with those produced with discrete wavenumber method and finite difference method, we found that the newly developed scheme successfully calculated heterogeneous seismic wave propagations.