Most finite-difference simulation algorithms use fixed-length spatial operators to compute spatial derivatives. The choice of length is dictated by computing cost, stability, and dispersion criteria that are satisfied globally. We propose finite-difference schemes with adaptive variable-length spatial operators to decrease computing costs significantly without reducing accuracy. These schemes adopt long operators in regions of low velocity and short operators in regions of high velocity. Two methods automatically determine variable operator lengths. Dispersion analysis, along with 1D and 2D modeling, demonstrates the validity and efficiency of our schemes. In addition, a hybrid absorbing boundary condition helps reduce unwanted reflections from model boundaries. Our scheme is more efficient than those based on variable-grid methods for modeling, migration, and inversion of models with complex velocity structures because the latter require local grid refinement, which usually increases memory requirements and computing costs.

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