The conceptual model of seismic data as n locally-crossing plane waves lends itself well to parameterization by a few parameters. The multidimensional prediction-error filter (PEF) is a particularly popular option (see, for example, Claerbout (1998)). Estimated by autoregression against the data, the PEF encodes hidden multiplicity in the data with a few filter coefficients. It has the approximate inverse spectrum of the data from which it was estimated.
By using a model of the noise to obtain a nonstationary noise PEF and deconvolving a PEF estimated from the data by the noise PEF to obtain a signal PEF Spitz (1999), many authors have solved equation (17) to successfully separate coherent noise from signal Brown et al. (1999); Brown and Clapp (2000); Clapp and Brown (2000); Guitton et al. (2001); Spitz (1999).
As noted by Fomel (2000), however, the considerable amount of parameter tuning required to create stable nonstationary PEFs (a requirement for the deconvolution step) remains a significant obstacle to their use in industrial-scale processing environments.
If the signal and noise consist of distinct slopes everywhere, then it is in theory possible to implicitly separate signal from noise in the slope domain with a two-slope estimation algorithm. Fomel uses estimated slope to construct plane-wave destructor filters which are used directly as and in equation (17), without any deconvolution. The filters are guaranteed stable and insensitive to spatially aliased data. Fomel obtains an independent estimate the noise slope from a prior noise model, and then fixes the noise slope as the signal slope is estimated.
I take a slightly different tack at the problem. Like Fomel, I use my two-slope estimation
technique to directly obtain signal and noise slope estimates. I also exploit a prior
noise model and also a prior signal model, in cases where the signal is simpler to model
than the noise. Most importantly, I find that very simple, easily-obtainable signal or
noise models suffice. To overcome aliasing, I apply normal moveout (NMO) to the data.
Rather than plane-wave destructor filters, I (again) use 9-point Lagrange steering
filters derived by Clapp et al. (1997).