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The helix transform Claerbout (1997) provides boundary
conditions that map multi-dimensional convolution into
one-dimension. In this case, the 2-D convolution
operator, , can be recast as
an equivalent 1-D filter.
Helical boundary conditions allow the two-dimensional convolution
matrix, , to be expressed as a one-dimensional
convolution with a filter of length 2 *N*_{x} +1 that has the form

The structure of the finite-difference Laplacian operator, ,
is simplified when compared to equation (9).
| |
(10) |

The 1-D filter can be factored into a causal and anti-causal
parts, and the matrix inverse can be computed by recursive polynomial
division (1-D deconvolution).

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** Previous:** The 45 wave equation
Stanford Exploration Project

7/5/1998