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The first-order approximations of the one-way wave equation by both
continued-fraction expansion and Taylor-series expansion result
in the same equation: the conventional 15-degree equation.
The 15-degree migration equation in the domain (Claerbout, 1985) is given by
| |
(6) |

After a second difference approximation to the second partial derivative
and subsequent simplification, the equation (6) becomes
| |
(7) |

where the matrix M takes the form
Rearranging,
with
and
The solution to equation (7) is

| |
(8) |

The calculation of the exponentiated matrix requires matrix
diagonalization, which is numerically expensive.
However, we can approximate the exponential of the matrix by
writing (Richardson, et al., 1991)
| |
(9) |

where the matrix M is split into two matrices *M*=*M*_{e}+*M*_{o}, with
and
The approximate depth-stepping operator,

| |
(10) |

forms the basis for our unconditionally stable, explicit algorithm for
migration. To compute the matrix exponentials, notice that both *M*_{e}
and *M*_{o} are block diagonal and we need only consider the exponential
of the 2 by 2 matrix
By using the eigenvalue decomposition of the matrix, we see that
where *Q* represents the matrix whose columns are eigenvectors of the
matrix *E*, and is the diagonal eigenvalue matrix.
Since both *M*_{e} and *M*_{o} are block diagonal, exponentiating them amounts to
exponentiating *E*. Both and are also block diagonal. Since the eigenvalues of are and with *a* and *b* imaginary, it follows that
and .To prove the unconditional stability of the algorithm we need only show that
. This follows immediately since , each of
which 1 according to the preceding discussion.

In Figure , the impulse responses of both implicit and
explicit operators of 15-degree migration are compared for the same
extrapolation step .
In the case of the implicit scheme, we used a Crank-Nicholson
implementation which has an accuracy on the order of .For the explicit scheme, we performed a second-order approximation
using a split *M* = *M*_{e} + *M*_{o} whose accuracy is on the order of .We see that the impulse response
of the explicit method has the shape of an ellipse, which is
characteristic of the 15-degree extrapolation operator.
The impulse response of the explicit method shows some dispersion
than that of the implicit scheme due to a poor approximation of
the matrix exponent.
The way to get a more accurate approximation is discussed
in the following section.

**fig1
**

Figure 1 The impulse responses (a) of explicit 15-degree migration
with split *M* = *M*_{e} + *M*_{o} and (b) of implicit 15-degree migration with
Crank-Nicholson approximation. Both extrapolations are performed with the
same extrapolation step .

** Next:** Accuracy
** Up:** Ji and Biondi: Explicit
** Previous:** ONE-WAY WAVE EQUATION AND
Stanford Exploration Project

12/18/1997