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Since the numerical solution is an approximation of the exact solution,
it does not exactly satisfy the continuous partial differential equation at
hand, but it satisfies a *modified * equation.
In this case, I use the Crank-Nicolson method
to solve the diffraction equation
| |
(32) |

| |
(33) |

| |
(34) |

By Taylor series expansion of every term about *P*^{n}_{j}, I obtain the
modified wave equation that the numerical solution actually satisfies:
| |
(35) |

| |
(36) |

The error term shows that the trace spacing interval
plays a larger role than the depth step size of downward continuation in determining the accuracy of the numerical solution.

** Next:** DISCUSSION
** Up:** Mo: Numerical analysis
** Previous:** The Crank-Nicolson method
Stanford Exploration Project

11/17/1997