Reverse time migration (RTM) is a depth migration algorithm that can image overturned reflectors. Unfortunately, due to the numerical performance of finite difference operators, oversampling of the timestep and grid spacing make the algorithm slow relative to other migration algorithms. Pseudospectral methods calculate the spatial derivatives in the wavenumber domain but use a finite-difference approximation in the time domain and thus also suffer from numerical dispersion. The phase-shift time stepping (PSTS) equation propagates a solution to the two-way variable-velocity acoustic wave equation by calculating the spatial derivatives of the wavefield in the Fourier domain but does not use a finite-difference approximation for the time derivative. Instead, the PSTS equation adapts the exact solution for constant velocity medium to variable velocity medium by a locally homogeneous approximation. While usually faster than finite differencing, PSTS still imposes a considerable computational burden. A number of numerical approximations of the PSTS equations are derived. Firstly, we propose a method of timestepping in a linear velocity gradient. Secondly, a method that takes a number of time steps in the Fourier domain before rewindowing in the space domain. Thirdly, a method that propagates the derivative of the wavefield and the wavefield forward in time which has no limitation on the size of the timestep because of stability or aliasing.
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