Element-free method and its efficiency improvement in seismic modelling and reverse time migration

Abstract

We present the theory of the element-free method (EFM) and its numerical applications in seismic modelling and reverse time migration. The absence of elements makes the method cheaper and more flexible than the finite element method (FEM). The shape function in the EFM only needs to satisfy the moving-least-squares (MLS) criterion, so it can be generated easily. Besides, it is convenient to deal with the local problems using weight functions and influence domains. Due to the MLS fitting method, the dependent variable and its derivative are both continuous and precise in the EFM. However, as a result of its heavy cost burden, this method seems difficult to be developed in seismic modelling and reverse time migration. The cost is mainly caused by improper storage of some large sparse matrices such as the mass matrix and the stiffness matrix, and improper operations (multiplication and inversion) on them. In this paper we compress the sparse matrices by the compressed sparse column (CSC) format and solve the linear equations instead of inverting sparse matrices, with the help of Intel linear sparse solver ‘PARDISO’. By these strategies, we have saved computer resources significantly. In the end, we show some applications of the improved method in seismic modelling and reverse time migration.