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## Propagating waves with the Wavemovie program

The following pseudo-code provides an algorithm for propagating waves into the Earth with the the new factorization of the wave equation.
Fourier transform input data over time-axis
Loop over frequency {
Initialize wave at z=0
Factor wave equation for this w/v
Recursively divide input data by factor
Fourier Transform back to time-domain
Sum into output
}

Incorporating this code into the Wavemovie program Claerbout (1985) provides a laboratory for testing the new algorithm.

Figure  compares the results of the new extrapolation procedure with the conventional Crank-Nicolson solution to the 45 equation. The new approach has little dispersion since I use a rational approximation (the one-sixth trick') to the Laplacian on the vertical and horizontal axes. In addition, the new factorization retains accuracy up to 90. The high dip, evanescent energy in the 45 movie, propagates correctly in the new approach.

 vs45 Figure 3 Comparison of the 45 wave equation (left) with the helical factorization of the Helmholtz equation (right).

Figure  compares different value of the one-sixth' parameter, . For this application, the optimal value seems to be .

sixth
Figure 4
Helmholtz equation factorization with different values for the `one-sixth' parameter, . From left, 0, 1/12, 1/8 and 1/6.

Figure  compares different finite-difference Laplacian operators. In all cases the finite-difference Laplacian was given by the linear sum of 5-point filters,
 (33)
where is an adjustable parameter between 0 and 1. Best results were obtained with . The impulse response with only contains energy on every second grid point, since the rotated filter only propagates energy diagonally: as in the game of a chess, if a bishop starts on a white square, it always stays on white.

laplac
Figure 5
Helmholtz equation factorization with different finite-difference representations of the Laplacian. From left, 0, 1/2, 2/3 and 1.

Next: Reducing the filter length Up: The Helmholtz equation Previous: Wave extrapolation
Stanford Exploration Project
5/27/2001