Application of the preconditioned conjugate gradient method to reconstruction of seismic data

Seismic data reconstruction based on the least-squares Fourier method is ultimately transformed into solving a linear equation. The coefficient matrix is Toeplitz matrix. The conjugate gradient method can be used to solve the linear equations. Pathological extent of matrix affects iterations of the conjugate gradient method. The more irregular sampled seismic data is, the more pathological the matrix is, then it is more difficult to get convergence and reasonable results. We study different construction methods of preconditions based on Toeplitz matrix and the effects of convergence of the conjugate gradient method. Through the use of preconditions, we can speed up the iterative speed of the conjugate gradient method, improve the convergence of conjugate gradient method and the efficiency of computation. Numerical examples and real seismic data reconstruction experiment show that the preconditioned conjugate gradient method has greatly improved efficiency of calculation.