Formulations

A standard formulation for calculating PEFs from known data is to solve a linear least-squares problem like  

$$\bold 0 \approx \bold Y \bold C \bold a + \bold r_0,$$ (1)

where $\bold a$ is a vector containing the PEF coefficients, $\bold C$ is a filter coefficient selector matrix, and $\bold Y$ denotes convolution with the input data. The coefficient selector $\bold C$ is like an identity matrix, with a zero on the diagonal placed to prevent the fixed 1 in the zero lag of the PEF from changing. The $\bold r_0$ is a vector that holds the initial value of the residual, $\bold Y \bold a_0$.If the unknown filter coefficients are given initial values of zero, then $\bold r_0$ contains a copy of the input data. $\bold r_0$ makes up for the fact that the 1 in the zero lag of the filter is not included in the convolution (it is knocked out by $\bold C$).

When there are many coefficients, it makes sense to add damping equations and/or to precondition the problem. Inserting the preconditioned variable $\bold S \bold p$ (where $\bold S$ is a somewhat arbitrary smoother) for $\bold a$ and adding the also somewhat arbitrary roughener $\bold R = \bold S^{-1}$to regularize the model, gives a formulation like

$$\begin{eqnarray} \bold 0 &\approx & \bold Y \bold K \bold S\bold p + \bold r_0 \\ \bold 0 &\approx & \epsilon\ \bold I \bold p\end{eqnarray}$$ (2) (3)

This is like the formulation used in Clapp 1999. Claerbout 1997 and Crawley 1999 set $\epsilon$ to zero and just limit the number of iterations. $\bold S$ is a smoother, but if it is a helical smoother, it can still be quite small (2 or 3 points), so it does not add much to the cost of computation Claerbout (1998). In this case, the smoother works radially Crawley et al. (1998).