Seismic migration by Chebychev transform : a novel approach

Chebychev semi-discretizations for both ordinary and partial differential equations are explored. The Helmholtz, heat, Schrӧdinger and 15° migration equations are investigated. The Galerkin, pseudospectral and tau projection operators are employed, while the Crank-Nicolson scheme is used for the integration of the time (depth) dependence. The performance of the Chebychev scheme is contrasted with the performance of the finite difference scheme for Dirichlet and Neumann boundary conditions. Comparisons between all finite difference, Fourier and Chebychev migration algorithms are drawn as well. Chebychev expansions suffer from neither the artificial dispersion dispersion of finite difference approximations nor the demand for a periodic boundary structure of Fourier expansions. Thus, it is shown that finite difference schemes require at least one order of magnitude more points in order to match the accuracy level of the Chebychev schemes. In addition, the Chebychev migration algorithm is shown to be free of the wraparound problem, inherent in migration procedures based on Fourier transform.Science, Faculty ofEarth, Ocean and Atmospheric Sciences, Department ofGraduat