** Next:** Cross-spectral factorization
** Up:** Implicit extrapolation theory
** Previous:** Matrix representation of implicit

As discussed previously, the
helix transform Claerbout (1998b) 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.
The 5-point 2-D filter becomes the sparse 1-D 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 ().

| |
(43) |

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).

** Next:** Cross-spectral factorization
** Up:** Implicit extrapolation theory
** Previous:** Matrix representation of implicit
Stanford Exploration Project

5/27/2001