Sparse Radon transform in the mixed frequency-time domain with ℓ1-2 minimization

Abstract

Due to the finite acquisition aperture and sampling of seismic data, the Radon transform (RT) suffers from a smearing problem which reduces the resolution of the estimated model. In addition, inverting the RT is typically an ill-posed problem. To address these challenges, a sparse RT mixing the [Formula: see text] and [Formula: see text] norms of the RT coefficients in the mixed frequency-time domain is developed, and it is denoted as SRTL1-2. In most conventional sparse RTs, the sparse constraint term often is the [Formula: see text] norm of the Radon model. We prove that the sparsity effect of the [Formula: see text] minimization is better than that of the [Formula: see text] norm alone by comparing and analyzing their 2D distribution patterns and threshold functions. The difference of the convex functions algorithm and the alternating direction method of multipliers algorithm are modified by combining the forward and inverse Fourier transforms to solve the corresponding sparse inverse problem in the mixed frequency-time domain. Our method is compared with three RT methods, including a least-squares RT (LSRT), a frequency-domain sparse RT (FSRT), and a time-invariant RT in the mixed frequency-time domain based on an iterative 2D model shrinkage method (SRTIS). Furthermore, we modify the basis function in SRTL1-2 by including an orthogonal polynomial transform to fit the amplitude-variation-with-offset (AVO) signatures found in seismic data, and we denote this as high-order SRTL1-2. Compared to the SRTL1-2, the high-order SRTL1-2 performs better when processing seismic data with AVO signatures. Synthetic and real data examples indicate that our method has better performance than the LSRT, FSRT, and SRTIS in terms of attenuation of multiples, noise mitigation, and computational efficiency.