Fast ℓ1-regularized Radon transforms for seismic data processing

Abstract

The efficiency of the recently developed algorithms for sparse Radon transforms depends heavily on the efficiency of the inverse transformation and its adjoint. In this paper, we propose a fast algorithm for the implementation of these canonical transforms, which runs in complexity O(NfNlog⁡N+NxNtlog⁡Nt) for a signal of size Nt×Nx, as opposed to O(NfNxNp+NxNtlog⁡Nt) of the direct computation, where N depends on the maximum frequency and offset in the data set, the maximum curvature in the Radon space, and Np. These transforms are utilized within the split Bregman iteration to solve the sparse Radon transform as an ℓ2/ℓ1 optimization problem. The computations involved in each iteration of the proposed algorithm, are carried out by a fast Fourier transform (FFT) algorithm. Furthermore, in the new algorithm, the amount of regularization is controlled by iteration number. We obtain a good estimate of predictive error via the generalized cross validation (GCV) analysis and automatically determine the optimum number of iterations at a negligible cost, thus leading to an automatic fast algorithm. This allows a time-domain sparse Radon transformation of large-scale data that are otherwise intractable. Numerical tests using simulated data confirm high efficiency of the proposed algorithm for processing large-scale seismic data.