An alternative to using a direct method of solution of the
pentadiagonal system is to use an iterative scheme. Cole (1989)
investigated the use of Jacobi and Gauss-Seidel methods to solve the
system. He found stability problems with these methods and that rates
of convergence were very low for the low frequencies. The
pentadiagonal system is less diagonally dominant at low frequencies. I
decided to use the Conjugate Gradient method as it is guaranteed to
converge for positive-definite operators. In order to use the CG
method I must be able to apply the operator *A* and its conjugate. The
operator and its conjugate are given by

Application of these operators involves multiplication, addition, and
use of the Laplacian operator. On a mesh-connected parallel computer
all these operations can be implemented efficiently, making full use
of all the processors. The algorithm is unchanged if I choose to use a
more accurate Laplacian operator. The only difference is that a higher
order Laplacian will produce a less diagonally dominant operator so
the method will converge more slowly. The cost of applying the
Laplacian operator is *O*(*n ^{2}*). If the CG method is run to completion
it will take

I implemented a simple CG method on the Connection Machine. I use the *L _{2}*
norm of the residuals as the indicator of convergence of the
algorithm. If the ratio of the norm of the residual to the
norm of the RHS of the equation was less than 10

Figure shows a time slice of the migration impulse response from the CG method. The impulse response is circular and does not suffer from the numerical anisotropy introduced by the splitting method.

filtdata.slice
Impulse response of implicit migration.
Figure 1 |

12/18/1997