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As a simple illustration of how helical boundary conditions can lead
to recursive solutions to partial differential equations, consider
Poisson's equation, which in the constant coefficient case relates
potential, *u*, to source density, *f*, through the Laplacian
operator:
| |
(13) |

Poisson's equation crops up in many different branches of physics: for
example, in electrostatics, gravity, fluid dynamics (where the fluids
are incompressible and irrotational), and steady-state temperature
studies. It also serves as a simple analogue to the wave-propagation
equations which provide the main interest of this thesis.
To solve Poisson's equation with constant coefficients on a regular
grid Claerbout (1997), we can approximate the
differential operator, , by a convolution with a small
finite-difference filter, *d*.
Taking the operator, , to represent convolution with filter,
*d*, Poisson's equation becomes

| |
(14) |

Although itself is a multi-dimensional convolution operator
that is not easily invertible, helical boundary
conditions Claerbout (1997) allow us to reduce the
dimensionality of the convolution to an equivalent one-dimensional
filter, which we can factor into the product of a lower-triangular
matrix, , and its transpose, . These triangular
matrices represent causal and anti-causal convolution with a
minimum-phase filter, in the form
| |
(15) |

The operator, , is the *helical derivative*, discussed in
more detail by Claerbout (1998a).
We can calculate *u* directly from equation ()
since and its transpose are easily invertible by recursive
polynomial division:
| |
(16) |

The right panel of Figure shows the solution of the
Poisson's equation with the single source and sink shown in the left
panel. The center panel shows the intermediate result,
.
**lapfac
**

Figure 1
Deconvolution by a filter whose autocorrelation
is the two-dimensional Laplacian operator.
This amounts to solving the Poisson equation.
After Claerbout (1997).

** Next:** The Helmholtz equation
** Up:** Helical factorization of the
** Previous:** Introduction
Stanford Exploration Project

5/27/2001