This paper presents a new depth migration method, adaptive spatial-division split-step Fourier （ASDSSF） migration. In this method we introduce the idea of a stratified phase shift migration into the split-step Fourier （SSF） migration to make an accurate and efficient wave field image when sharp discontinuities appear in the velocity field. In principle, the ASDSSF migration is a multi-reference slowness （reciprocal of velocity） （MRS） migration. Compared to previous MRS migration methods, this method uses fewer reference slowness values without accuracy loss. The reference slowness is determined in this paper according to an error-control parameter of the perturbation term in the SSF operator and the variation of the complet velocity field. The velocity corresponded to reference slowness can define a spatial division. Each division can also be divided into several discontinuous spatial subdivisions to effectively reduce the number of reference slowness values needed. The choice of reference slowness, including the number of reference slowness values needed and how to construct the spatial divisions, is adaptive and reasonable at each extrapolation step （depth step）. A simple and economical smoothing filter in the wave number-frequency domain is designed to avoid artifacts in the wave field extrapolation due to the presence of sharp discontinuities in the velocity field. For comparable conditions the present approach to migration is expected to be computationally more efficient and accurate than other MRS migration methods. The performance of the method is demonstrated on a simple 2D prestack model and the prestack SEG/EAEG salt dataset.
