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Formulations

A standard formulation for calculating PEFs from known data is to solve a linear least-squares problem like
 (1)
where is a vector containing the PEF coefficients, is a filter coefficient selector matrix, and denotes convolution with the input data. The coefficient selector 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 is a vector that holds the initial value of the residual, .If the unknown filter coefficients are given initial values of zero, then contains a copy of the input data. 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 ).

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).

Next: Testing Up: Convergence versus accuracy Previous: Convergence versus accuracy
Stanford Exploration Project
10/25/1999