An implicit preconditioned conjugate gradient scheme is derived to implement nonstationary phase shift for irregularly sampled seismic data using least squares. This implicit scheme gives fast convergence at all frequencies at the cost of an approximation to the evanescent filter. This results in some error in the resulting phase shifted and regularized data. Our implicit scheme suggests an explicit scheme which unfortunately does not perform any better than the standard unconditioned scheme. The fast implicit scheme suggests that an appropriate preconditioner can be found that will reduce the runtime of the algorithm without sacrificing accuracy, and this will result in a robust trace regularization and statics algorithm for use in heterogeneous media.
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