Sparse Time–Frequency Analysis of Seismic Data: Sparse Representation to Unrolled Optimization

Time–frequency analysis (TFA) is widely used to describe local time–frequency (TF) features of seismic data. Among the commonly used TFA tools, sparse TFA (STFA) is an excellent one, which can obtain a TF spectrum with good readability. However, many STFA algorithms suffer from expensive calculation time and unavoidable prior knowledge, such as the iterative shrinkage-thresholding algorithm (ISTA) and the sparse reconstruction by separable approximation (SpaRSA). Inspired by the unrolled algorithm and its successful applications in signal processing, we propose a deep learning (DL)-based ISTA unrolled algorithm, which is named the sparse time–frequency analysis network (STFANet). The STFANet contains two parts, i.e., the sparse TF spectrum generator and the reconstruction module. The former learns how to transform a 1-D seismic signal from a large amount of unlabeled data into a 2-D sparse TF spectrum, which is implemented based on the proposed unrolled iterative dynamic shrinkage-thresholding (UIDST) algorithm. Note that the UIDST algorithm is carried out by using a simplified DL network. The latter serves as a physical constraint of model training to ensure that our generator obtains an accurate TF spectrum, which is actually an inverse TF transform. In this study, the traditional inverse short-time Fourier transform (STFT) is utilized in the reconstruction module. To test the effectiveness of the proposed model, we apply it to 3-D poststack field data. The results show that, compared with the traditional TFA tools, the STFANet can availably compute the TF spectrum with better readability, which benefits seismic attenuation delineation.

I. Introduction

Time–frequency analysis (TFA) plays a vital role in describing time–frequency (TF) features of nonstationary signals, such as seismic signals, physiological signals, and radar signals [1], [2], [3], [4]. There are kinds of TFA tools, which can be classified into three categories, i.e., linear, bilinear, and nonlinear. The traditional linear-based methods mainly include short-time Fourier transform (STFT) [5], [6], continuous wavelet transform (CWT) [7], [8], and S-transform and its improvements [9], [10]. These transforms are subject to the Heisenberg uncertainty principle, which affects the readability of the TF representation [11]. The bilinear TFA methods, such as Wigner–Ville distribution (WVD) and its generalized versions [12], [13], can achieve TF representation with high TF resolution. However, these bilinear TF tools inevitably suffer from cross-term interference [14], which makes their TF representation unsuitable for seismic signal [15].