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Equation (5) can be transformed into the space-frequency domain,
resulting in the 15-degree time migration equation

| |
(6) |

This partial differential equation can be solved by splitting it
(Claerbout, 1985) into the diffraction term
| |
(7) |

and the lens term
| |
(8) |

The lens equation can be solved accurately by analytic exponentiation.
Here I consider the diffraction equation, which is a
partial differential equation (PDE). A PDE can be readily converted to a
system of ordinary differential equations (ODEs) by using finite-difference
approximations for derivatives in all but one of the dimensions.
To do so, I discretize coordinate *x* with *N*+1 uniformly spaced grid points,
as follows:

| |
(9) |

Using the second-order central-difference scheme for the second derivative in
*x* results in
| |
(10) |

where and .
With a further assumption of the boundary condition ,
I obtain a system of
ODEs that can be written in matrix form as
| |
(11) |

where *P*_{j}'s are the elements of the vector , and *A* is the
(*N*-1)*(*N*-1) tri-diagonal matrix
| |
(12) |

This system of ODEs can be solved using any of the numerical methods, such as
the Runge-Kutta formulas or multi-step methods.
However, when dealing with systems,
we need to be concerned about stiffness. To investigate numerical stiffness,
we must study the eigenvalue structure of the matrix *A*.
The eigenvalues of *A*
obtained from a known formula for the eigenvalues of a tri-diagonal
matrix with constant entries are

| |
(13) |

It is important to keep in mind that all the eigenvalues of *A* are non-zero
and purely imaginary, since is purely imaginary.
The eigenvalue with the smallest magnitude is
| |
(14) |

For large *N*, the series expansion for ,
| |
(15) |

converges rapidly. Retaining the first two terms in the expansion results in
| |
(16) |

Also, for large *N*
| |
(17) |

Therefore, the ratio of the eigenvalue with the largest modulus to the
eigenvalue with the smallest modulus is
| |
(18) |

Thus for seismic migration, typically traces,
this is an extremely stiff system of ODEs.
There is severe contradiction in choosing the
pseudodepth (time ) step size of downward extrapolation,
so that explicit methods, for example, explicit Euler and Runge-Kutta methods
are unworkable.

Standard decoupling using the matrix of eigenvectors *S* yields

| |
(19) |

where is a diagonal matrix with the eigenvalues of *A* on
the diagonal, *S* is the matrix whose columns are the eigenvectors of *A*,
and . Since the equations are uncoupled,
the following solution can be readily obtained:
| |
(20) |

Thus, purely imaginary eigenvalues result in the oscillation of the solution
that is expected for the wave equation.

** Next:** The Crank-Nicolson method
** Up:** Mo: Numerical analysis
** Previous:** UNDERMIGRATION CAUSED BY PARABOLIC
Stanford Exploration Project

11/17/1997