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The most popularly used numerical method of solving a stiff system of ODEs
such as (11) is the Crank-Nicolson method, chosen because of its
unconditional stability and good accuracy.
First let us look at the Crank-Nicolson (trapezoidal) method
for a single first-order ODE.
Applying the Crank-Nicolson method to the model equation
| |
(21) |

yields
| |
(22) |

or
| |
(23) |

Thus the solution of (21) can be written as
| |
(24) |

where *y*_{0} is the initial condition and
| |
(25) |

For the migration wavefield extrapolation equation (19), the eigenvalue
() is purely
imaginary. Thus

| |
(26) |

Because , the Crank-Nicolson method as applied to
migration wavefield
extrapolation does not generate any amplitude error.
This result agrees with the
one-way migration wavefield extrapolation that is a dampless sinusoidal
wave propagation. However, there is phase error. The numerical solution is
| |
(27) |

| |
(28) |

| |
(29) |

But the exact solution of the ODE (21) is
| |
(30) |

The phase error is
| |
(31) |

The numerical solution lags behind the exact solution.
Thus applying the Crank-Nicolson method to the diffraction
equation causes overmigration - the experimental error.
And the eigenvalues with larger moduli have larger degrees of overmigration.

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Stanford Exploration Project

11/17/1997