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

(1) |

When there are many coefficients, it makes sense to add damping equations and/or to precondition the problem. Inserting the preconditioned variable (where is a somewhat arbitrary smoother) for and adding the also somewhat arbitrary roughener to regularize the model, gives a formulation like

(2) | ||

(3) |

This is like the formulation used in Clapp 1999. Claerbout 1997 and Crawley 1999 set to zero and just limit the number of iterations. 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).

10/25/1999