Instead of one PEF per patch, we estimate a PEF for every output data point; changing the problem from overdetermined to very underdetermined. We can estimate all these filter coefficients by the usual formulation, supplemented with some damping equations, say

(1) |

When the roughening operator is a differential operator, the number of iterations can be large. We can speed the calculation immensely and make the equations somewhat neater by ``preconditioning''. When we define a new variable by and insert it into (1) we get

(2) | ||

(3) |

(4) | ||

(5) |

To reduce clutter, we could drop the damping (5) and keep only (2); then to control the null space, we need only to start from a zero solution and limit the number of iterations. As a practical matter, without (5) we must find a good number of iterations and with it we must find a good value for .

For we can use polynomial division by a Laplacian or by filters with a preferred direction. If the data are CMP gathers, it is attractive to use radial filters, which are explained further down.

7/5/1998