Projected Gabor deconvolution

Abstract

Gabor deconvolution consists of approximating a nonstationary problem as several stationary subproblems via the Gabor frame, solving each subproblem independently, and then recombining/projecting the subsolutions into an approximate solution to the original nonstationary problem. The approximations, however, cause inherent instability due to systematic errors that prevent the algorithm from converging to the true solution even for noise-free cases. Furthermore, the method will be time consuming when a nonlinear optimization is used for shaping the reflectivity structure. An alternative projected Gabor deconvolution is considered, which is based on the Gabor expansion. However, in contrast to the conventional form, in the new method, first the problem is projected into the time domain, and then an inversion is performed to obtain the final reflectivity. Compared with the Gabor deconvolution, the projected alternative (1) exhibits an improved convergence property, (2) is more efficient for sparse deconvolution because only a single optimization is required to be solved, and (3) is more flexible for incorporating prior information about noise and reflectivity structure via a least-squares method. Numerical tests using simulated and field data are presented showing that the new method generates more accurate and stable reflectivity models compared with the Gabor deconvolution.