Equivalent accuracy at a fraction of the cost: Overcoming temporal dispersion

Numerical dispersion in finite difference modeling produces coherent artifacts, severely constraining the resolution of advanced imaging and inversion schemes. Conventionally, we deal with this by increasing the order of accuracy of the finite difference operators and resign ourselves to paying the high computational cost that incurs. But is there a way to reduce such dispersion without increasing cost or, conversely, decrease the cost without increasing numerical dispersion? To tackle this, we separate the finite difference numerical dispersion into pure time and pure space dispersion and address them independently. In this article, we focus on time dispersion. We show that finite difference time dispersion is virtually independent of the medium velocity and the spatial grid for propagation, and only depends on the time stepping scheme and the propagation time. Based on this, we devise post-propagation filtering to collapse the time dispersion effect of the finite difference modeling. Our dispersion correction filters are designed by comparing the input waveform with dispersive waveforms obtained by 1D propagation of that waveform forward in time. These filters are then applied on multi-dimensional shot records to eliminate the time dispersion by two schemes: (1) stationary filtering plus interpolation and (2) non-stationary filtering. We show with both 1D and 2D examples that the time dispersion is effectively removed by our post-propagation filtering at nearly no additional cost.