For the case where velocity varies laterally, this transformation is prohibitively expensive because the matrix loses its sparseness as the algorithm proceeds. However, using a mean rotation angle can significantly improve the diagonal dominance of the matrix, while keeping the matrix sparse. This improved operator can then be used in solution methods such as the Jacobi iterative scheme and conjugate gradients, and should improve the performance of these methods considerably.
More work needs to be done to determine whether Givens rotations can really aid in solving the 3-D migration problem. I need to estimate the cost of reducing the matrix to tridiagonal form and solving the resulting tridiagonal system. Then this cost should be compared with the costs of solving the original system by a good direct method, solving with iterative methods (with of without Givens rotations as a preconditioner), and solving via splitting. Only when these different methods have been implemented and the costs compared will I be able to decide what place, if any, Givens rotations have in 3-D migration.