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$^\circ$ 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$^\circ$. The high dip, evanescent energy in the 45$^\circ$ movie, propagates correctly in the new approach.

 

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

Figure  compares different value of the `one-sixth' parameter, $\beta$. For this application, the optimal value seems to be $\beta=1/12$.

 

sixth
Figure 4 Helmholtz equation factorization with different values for the `one-sixth' parameter, $\beta$. From left, $\beta =$ 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,  

$$(1-\gamma) \, \left(\begin{array} {ccc} & 1 & \\ 1 & -4 & 1 ... ... {ccc} 1/2 & & 1/2 \\ & -2 & \\ 1/2 & & 1/2\end{array}\right),$$ (33)

where $\gamma$ is an adjustable parameter between 0 and 1. Best results were obtained with $\gamma = 2/3$. The impulse response with $\gamma=0$ 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 $3 \times 3$ finite-difference representations of the Laplacian. From left, $\gamma =$ 0, 1/2, 2/3 and 1.