Approximating the signal PEF

In this section, we describe a method that computes the signal PEF needed in equations (2) and (4). Spitz (1999) showed that for uncorrelated signal and noise, the signal PEF can be expressed in terms of 2 PEF's: a PEF ${\bf D}$, estimated from the data ${\bf d}$, and a PEF ${\bf N}$, estimated from the noise model such that  

$$\bold S = \bold D \bold N^{-1}.$$ (9)

Equation (9) states that the signal PEF equals the data PEF deconvolved by the noise PEF. Spitz formulated the problem in the f-x domain, but the helix transform Claerbout (1998) permits stable inverse filtering with multidimensional t-x domain filters.