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The conventional way of applying implicit finite-difference schemes
reduces to solving a system of linear equations with a sparse matrix.
For example, to apply the scheme of equation (11), we
can put the filter denominator on the other side of the extrapolation
equation, writing it as
| |
(14) |

where is the identity matrix, is the convolution
matrix for filter (10), and is the vector of
temperature distribution at time level *t*. In the case of
two-dimensional extrapolation, the matrix on the left side of equation
(14) takes the tridiagonal form
| |
(15) |

where , and where, for simplicity, we assume
zero-slope boundary conditions. Like any positive-definite tridiagonal
matrix, matrix can be inverted recursively by an *LU*
decomposition into two bidiagonal matrices. The cost of inversion is
directly proportional to the number of vector components. The same
conclusion holds for the case of depth extrapolation [equation
(13)] with the substitution .
In the case of a laterally constant coefficient *a*, we can take a
different point of view on the tridiagonal matrix inversion. In this
case, the matrix represents a convolution with a
symmetric three-point filter . The *LU*
decomposition of such a matrix is precisely equivalent to filter
*factorization* into the product of a causal minimum-phase filter
with its adjoint. This conclusion can be confirmed by the easily
verified equality

| |
(16) |

where . The inverse of the causal minimum-phase
filter is a recursive inverse filter.
Correspondingly, the inverse of its adjoint pair, , is the same inverse filtering, performed in the adjoint mode
(backwards in space). In the next subsection, we show how this
approach can be carried into three dimensions by applying the helix
transform.

** Next:** Helix and multidimensional deconvolution
** Up:** Spectral factorization and three-dimensional
** Previous:** Spectral factorization and three-dimensional
Stanford Exploration Project

10/9/1997