Background

For practical purposes, seismic data consists locally of the superposition of n plane waves. Disregarding aliasing effects and the data's wavelet, the slopes of the n plane waves fully and uniquely parameterize the data locally. Claerbout (1992) casts the problem of single-slope estimation as a linear, univariate optimization problem. Fomel extends the problem to the estimation of two slopes, and utilizes the estimated slopes for the interpolation of missing data and signal/noise separation. He iteratively solves a linearization of a nonlinear problem, and applies a model regularization term to enforce smoothness of the estimated slopes.

In this paper, I present another method for solving the two-slope estimation problem. It is a nonlinear extension of Claerbout's methodology, and differs from Fomel's in the sense that it is a strictly local method. Because mine is a local method, it runs much faster than Fomel's. Theoretically, my method is sensitive to aliased data, unlike Fomel's. Like Fomel's, the estimated slope depends on the starting guess. The existence of local minima appears to be an inherent weakness of the two-slope estimation problem in general.

Fomel successfully applies ``plane-wave destructor'' filters, derived from estimated slopes, to the signal/noise separation problem. Analogously, I use the estimated slopes to construct ``steering filters'' of a form derived by Clapp et al. (1997). Like Fomel, I find that when the signal and noise slopes are too similar, my method converges to (incorrect) local minima, unless the slope estimation is ``guided'' with a prior model of the signal or the noise. Using this constrained approach, I obtain excellent separation results on three different real data examples. Most encouragingly, in all cases, very simple, easily-obtained prior models sufficed.