Modeling acoustic waves in locally enhanced meshes with a staggered-grid finite difference approach

Finite difference is a well-suited technique for modeling acoustic wave propagation in heterogeneous media as well as for imaging and inversion. Typically, the method aims at solving a set of partial differential equations for the unknown pressure field by using a regularly spaced grid. Although finite differences can be fast and cheap to implement, the accuracy of the solution is always restricted by the computational resources. This is a fundamental key point to treat when dealing with large-scale problems. In this work, we present and test a method that uses a non-uniform distribution of grid points to improve on accuracy or to reduce the required computational resources. The applied grid is generated through a coordinate transformation. Differential geometry and generalized coordinates are used to handle and analyze the effect of using a non-uniform grid. Results obtained with the presented method show that the applied transformation as well as the number of points-per-wavelength influences the stability and dispersion in the solution. We exploit this observation to locally improve the accuracy of our simulations. The work presented in this paper allows us to conclude that differential geometry for finite differences can be used to reduce dispersion and hence improve the accuracy when modeling acoustic wave propagation in heterogeneous media. In addition, it can be used to avoid oversampling through the optimization of the number of grid nodes required to have an accurate solution or just honor to the boundaries.