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The diffraction term of the 45 equation Claerbout (1985) can
be rewritten as the following matrix equation, by inserting the rational
part of the implicit extrapolator () into
equation ():
| |
(38) |

| (39) |

where the complex coefficient, can be
calculated, and is a finite-difference representation of the
Laplacian, .
The right-hand-side of equation () is known. The
challenge is to find the vector by inverting
the matrix, .Given the wavefield on the surface, this equation provides a way to
downward-continue in depth.

The matrices in equation () represent convolution with
a scaled finite-difference Laplacian, with its main diagonal
stabilized.
In the two-dimensional problem, the operator acts only in
the *x*-direction, and can be represented by the three-point
convolutional filter, *d*=(1,-2,1). The matrix, ,therefore, has a tridiagonal structure, which can be inverted
efficiently with a recursive solver.

In three-dimensional wavefield extrapolation, the operator
acts in both the *x* and *y*-directions.
therefore represents a 2-D convolution, and
*d* can be represented by the a simple 5-point filter,

| |
(40) |

or a more isotropic 9-point filter Iserles (1996),
| |
(41) |

The vectors and contain the
wavefield at every point in the (*x*,*y*)-plane.
Therefore, the convolution matrices that operate on
them are square with dimensions .
As an illustration, for a spatial plane, the structure of
matrix with the five-point approximation and transient
boundary conditions, will be the blocked-tridiagonal matrix
| |
(42) |

This blocked system cannot be easily
inverted, even for the case of constant velocity, since the missing
coefficients on the second diagonals break the Toeplitz structure.

** Next:** Helical boundary conditions
** Up:** Implicit extrapolation theory
** Previous:** Implicit extrapolation theory
Stanford Exploration Project

5/27/2001