Multidimensional recursive filters via a helix

Abstract

Wind a wire onto a cylinder to create a helix. I show that a filter on the 1-D space of the wire mimics a 2-D filter on the cylindrical surface. Thus 2-D convolution can be done with a 1-D convolution program. I show some examples of 2-D recursive filtering (also called 2-D deconvolution or 2-D polynomial division). In 2-D as in 1-D, the computational advantage of recursive filters is the speed with which they propagate information over long distances. We can estimate 2-D prediction‐error filters (PEFs) that are assured of being stable for 2-D recursion. Such 2-D and 3-D recursions are general‐purpose preconditioners that vastly speed the solution of a wide class of geophysical estimation problems. The helix transformation also enables use of the partial‐differential equation of wave extrapolation as though it were an ordinary‐differential equation.