** Next:** Helical boundary conditions
** Up:** Implicit extrapolation
** Previous:** Implicit extrapolation

The diffraction term of the in the 45 equation Claerbout (1985) can
be rewritten as the following matrix equation, by inserting the rational
part of the implicit extrapolator (3) into
equation (1):
| |
(5) |

| (6) |

where the complex coefficients and can be
calculated, and is a finite-difference representation of the
Laplacian, .
The right-hand-side of equation (6) 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 (6) represent convolution with
a scaled finite-difference Laplacian, with its main diagonal stabilized.
Scaling coefficients, and , are complex and
depend on the ratio, .

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.
and therefore represent 2-D convolution, and
*d* can be represented by the a simple 5-point filter,

| |
(7) |

or a more isotropic 9-point filter Iserles (1996),
| |
(8) |

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
| |
(9) |

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
** Previous:** Implicit extrapolation
Stanford Exploration Project

7/5/1998