Choice of roughening filter, ${\bf R}$

The most important consideration in the choice of roughening filter is that it is easily invertible. A Fourier domain roughener would meet this criterion; however, we apply a time-space operator that is both cheaper, and less prone to Fourier artifacts such as wrap-around and Gibbs' phenomenom. Claerbout (1998b) describes how to construct invertible multi-dimensional time-space operators by applying helical boundary conditions to the problem. Helical operators cost O(N) operations to apply and invert rather than $O(N \log N)$ for an equivalent Fourier operator.

For the results shown in this paper, we choose ${\bf R}$ to be the helical derivative operator that roughens isotropically in the midpoint-time plane. A cascade of two one-dimensional derivative filters first along the time axis and then along the midpoint axis also works well. Anisotropic smoothing can be controlled by tweaking the ``micropatch'' parameters described below.