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## The Crank-Nicolson method

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 y0 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.

Next: THE MODIFIED WAVE EQUATION Up: OVERMIGRATION CAUSED BY THE Previous: OVERMIGRATION CAUSED BY THE
Stanford Exploration Project
11/17/1997