** Next:** Wave extrapolation
** Up:** Rickett & Claerbout: Factoring
** Previous:** Poisson's equation

Starting from the full wave equation in three-dimensions:
| |
(5) |

we can Fourier transform the time axis, and look for )solutions of the form:
| |
(6) |

For a single frequency, the wave equation therefore reduces to the
Helmholtz (time-independent diffusion) equation
| |
(7) |

where .
We aim to factor this equation on a helix, as with the Poisson
equation above. However, before we can, we need to ensure
that it is a `level-phase' function Claerbout (1998), that
is to say the spectrum of the operator does not touch the negative
real axis on the complex plane.
The spectrum of the Helmholtz operator can be obtained by taking the
Fourier transform of equation (7).

| |
(8) |

clearly becomes negative real for small
values of ; so as it stands, this equation is not
factorable. Fortunately, however, replacing by , where is a small positive number,
successfully stabilizes the spectrum, by pushing the function off the
negative real axis. The physical effect of is to provide
damping as the wave propagates, differentiating between the forward
and backward extrapolation directions.

Before factorization, equation (7) should therefore
be rewritten to include the stabilization term

| |
(9) |

Following the helix solution to Poisson's equation above, a simple
finite-difference approximation to the Laplacian, , produces the matrix equation:

| |
(10) |

Alternatively, and more accurately, we can form a rational
approximation to the Laplacian operator,
| |
(11) |

where is Claerbout's 1985 adjustable
`one-sixth' parameter, and
again represents convolution with a simple finite-difference
filter, *d*.
Inserting equation (11) into
equation (9) yields a matrix equation of
similar form, but with increased accuracy at high spatial wavenumbers:

| |
(12) |

| (13) |

The operator on the left-hand-side of equation (13)
represents a three-dimensional convolution matrix, that can be mapped
to an equivalent one-dimensional convolution by applying helical
boundary conditions.
Although the complex coefficients on the main diagonal cause
the matrix not to be Hermitian, the spectrum of the matrix
is of level-phase. Therefore, for constant , it can be
factored into causal and anti-causal (triangular) components with any
spectral factorization algorithm that has been adapted for
cross-spectra Claerbout (1998).

| |
(14) |

The challenge of extrapolation is to find that
satisfies both the above equation and our initial conditions,
. Starting from , we can invert recursively to obtain a function that satisfies both the
initial conditions, and

| |
(15) |

Hence will also satisfy equation (14).

** Next:** Wave extrapolation
** Up:** Rickett & Claerbout: Factoring
** Previous:** Poisson's equation
Stanford Exploration Project

7/5/1998