Application to constant-velocity case

In the constant-velocity case, the coefficients in finite-difference migration operators will not vary laterally. In other words, the matrix from Table 1 will contain a constant value along each of its five diagonals. The Givens rotation scheme described above is particularly well-behaved in such a case.

Figure  shows a plot of such a matrix after several passes of a Givens rotation scheme. Here I have used just the constant values of 1 and -4 from Table 1. In the migration case these coefficients will be complex, but the real values used are more suitable for a demonstration.

In Figure , one pass of Givens rotations (upper right) has turned the outer diagonals into a series of diagonals, all with amplitudes much smaller than the original. After three passes (lower right), these artifacts have effectively been suppressed. The matrix is now tridiagonal, and I can use traditional methods for solving the system.

 

give
Figure 1 Migration operators before (upper left) and after each of three passes of the Givens rotation scheme. The first pass creates some diagonal artifacts, but these are suppressed by later passes.

Note that the rotation matrices that have been applied need also to be applied to the right-hand side of the system of equations being solved. Because the matrix remains sparse throughout the procedure, this method is rather inexpensive.

Figure  illustrates what is happening during one pass of the Givens rotation technique. The snapshots here are all from the first pass. It can be seen that artifacts are introduced as the diagonal elements are zeroed, and that the zeroed elements are filled in again (with smaller values) as the algorithm proceeds.

 

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Figure 2 givestep

Snapshots from first pass of Givens rotation scheme. As the algorithm steps down along the main diagonal, the elements of the outermost diagonal are zeroed. The artifacts can be eliminated by additional passes.