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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 Nx +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