Feasibility of solving least squares Gazdag migration using method of multigrid.

Abdolnaser Yousefzadeh and John C. Bancroft

ABSTRACT

Methods of multigrid have been widely used in solving partial differential equations in physics and mathematics. With the ability of faster recovery of the low frequency components of the solution, they have been used in solving some problems in the exploration geophysics. Our previous studies showed that the standard method of multigrid is not viable to solve the Kirchhoff least squares prestack migration equation for two reasons. First, the kernel of the main problem is not a diagonally dominant matrix and solvable by the typical multigrid solvers, Jacobi and Guess Seidel. Second, kernel matrices are extremely large to work with. This study investigates the feasibility of using multigrid methods in solving a system of Gazdag least squares migration. It is shown by doing least squares migration for each temporal frequency at the time the kernel matrix become smaller, but it is a diagonally non-dominant matrix. By implementing least squares for each temporal and spatial frequency separately, the kernel matrix remains non-diagonal dominant. Best scenario is performing least squares migration for each temporal frequency and depth separately at a time. The kernel matrix reduces to a diagonal and easily invertible matrix. 1.

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