The helical coordinate

Next: Causality in three dimensions Up: KOLMOGOROFF SPECTRAL FACTORIZATION Previous: Constant Q medium

## Causality in two dimensions

Our foundations, the basic convolution-deconvolution pair (1) and (2) are applicable only to filters with all coefficients after zero lag. Filters of physical interest generally concentrate coefficients near zero lag. Requiring causality in 1-D and concentration in 2-D leads to shapes such as these:

 (9)

where are coefficients we find by least squares.

The complete story is rich in mathematics and in concepts; but to sum up, filters fall into two categories according to the numerical values of their coefficients. There are filters for which equations (1) and (2) work as desired and expected. These filters are called minimum phase.'' There are also filters for which equation (2) is a disaster numerically, the feedback process diverging to infinity.

Divergent cases correspond to physical processes that are not simply described by initial conditions but require also reflective boundary conditions, so information flows backward, i.e., anticausally. Equation (2) only allows for initial conditions.

I oversimplify by trying to collapse an entire book FGDP (Fundamentals of Geophysical Data Processing) into a few sentences by saying here that for any fixed 1-D spectrum there exist many filters. Of these, only one has stable polynomial division. That filter has its energy compacted as soon as possible after the 1.0'' at zero lag.

 The helical coordinate

Next: Causality in three dimensions Up: KOLMOGOROFF SPECTRAL FACTORIZATION Previous: Constant Q medium

2015-03-25