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The problem of approximating the Laplacian operator in two dimensions
not only inherits the inaccuracies of the one-dimensional
finite-difference approximations, but also raises the issue of
azimuthal asymmetry. For example, the usual five-point filter

| |
(26) |

exhibits a clear difference between the grid directions and the
directions at a 45-degree angle to the grid. To overcome this
unpleasant anisotropy, we can consider a slightly larger filter of the
form
| |
(27) |

where the constants and are to be defined. The
Fourier-domain representation of filter (27) is
| |
(28) |

and the isotropic filter that we can try to approximate is defined
analogously to its one-dimensional equivalent, as follows:
| |
(29) |

Comparing equations (28) and (29), we notice that
they match exactly, when either of the wavenumbers *k*_{x} or *k*_{y} is
equal to zero, provided that
| |
(30) |

Therefore, we can reduce the problem to estimating a single
coefficient . Another way of expressing this conclusion is to
represent filter *F*_{9} in equation (28) as a linear
combination of filter *F*_{5} from equation (28) and its
rotated version Cole (1994), as follows:
| |
(31) |

With the value of , filter *F*_{9} takes the value
| |
(32) |

and corresponds precisely to the nine-point McClellan filter
Hale (1991a); McClellan (1973). On the other hand, the value of
gives the least error in the vicinity of the zero
wavenumber *k*. In this case, the filter is
| |
(33) |

Errors of different approximations are plotted in Figure
11^{}
**laplace
**

Figure 11 The numerical anisotropy error of different
Laplacian approximations. Both the five-point Laplacian (plot a) and
its rotated version (plot b) are accurate along the axes, but
exhibit significant anisotropy in between at large wavenumbers. The
nine-point McClellan filter (plot c) has a reduced error, while the
filter with (plot d) has the flattest error around the
origin.

Under the helix transform, a filter of the general form
(27) becomes equivalent to a one-dimensional filter with
the *Z* transform

where *N*_{x} is the helix period (the number of grid points in the *x*
dimension). To find the inverse of a convolution with filter
(34), we factorize the filter into the causal minimum-phase
component and its adjoint:

| |
(35) |

To find the coefficients of the filter *P*, any one-dimensional
spectral factorization method can be applied. It is important to point
out that the result of factorization (neglecting the numerical errors)
does not depend on *N*_{x}. Another approach is to define a residual
error vector for the coefficients of Z in equation (35)
and minimize it for some particular norm. For example, minimizing the
norm when *F*_{9} is the McClellan filter
(32), we discover that the filter *P*, after transforming
back to two dimensions, takes the form
| |
(36) |

The results of applying a recursive deconvolution with filter
(36) are shown in Figure 12. An
essentially similar procedure, only with a different set of filters,
works for implicit wavefield extrapolation.
**inv-laplace
**

Figure 12 Inverting the Laplacian
operator by a helix deconvolution. The top left plot shows the
input, which contains a single spike and the causal minimum-phase
filter *P*. The top right plot is the result of inverse filtering.
As expected, the filter is deconvolved into a spike, and the spike
turns into a smooth one-sided impulse. After the second run, in the
backward (adjoint) direction, we obtain a numerical solution of
Laplace's equation! In the two bottom plots, the solution is shown
with grayscale and contours.

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Stanford Exploration Project

10/9/1997