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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 Pnj, 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