Feasibility of solving least squares prestack Kirchhoff migration using multigrid methods

Abdolnaser Yousefzadeh, John C. Bancroft

The multigrid technique is a powerful method for solving a linear matrix equation; for finding the low frequency components of the solution as rapidly as the high frequency components.

We investigate the feasibility of different approaches for using the multigrid method in solving the linear system of Kirchhoff Least-Squares Prestack Time Migration equation.

To date, this study shows that the standard multigrid method is not able to solve the Kirchhoff Least-Squares Prestack Time Migration equation for at least two reasons. First, the kernel of the main problem, G'G, is not a diagonally dominant matrix, therefore, Jacobi or Gauss-Seidel iteration, the standard iterative algorithms used in multigrid methods, are not effective. Second, matrices are too large and dense to be loaded in the memory of today's computers.

However, the performance of the Conjugate Gradient and Kaczmarz methods as multigrid solvers is examined for some synthetic data sets, including Marmousi, and results are shown. Convergence rate of the Conjugate Gradient is independent of the frequency content of the solution. Therefore, it does not converge more quickly with high frequency contents of the solution. Since Conjugate Gradient does not smoothe, it should not be considered for the multigrid iterative solver.

Using the Conjugate Gradient as an iterative solver for the multigrid method may slightly reduce the number of iterations for the same rate of convergence in the Conjugate Gradient itself. However, it does not reduce the total computational cost.