GD Equation-Based Transient-Extracting Transform for Seismic Time-Frequency Analysis

The time-reassignment method (RM) and time-synchrosqueezing transform (SST) show a good ability in impulse-like signal analysis. This kind of method calculates the group delay (GD) estimators at the spread time-frequency (TF) locations first and then relocates the spread TF energy into the estimated GD trajectories to yield a high-concentration TF representation. However, computing the GD estimators for every TF energy point can lead to inaccurate location and energy diffusion. To address this issue, a new feature extractor called the second-order GD equation is proposed, which focuses only on the TF points on the GD to characterize the frequency-varying models with N-order amplitude and second-order phase. The theoretical analysis of the second-order GD equation is highlighted. By combining a fixed-point iterative algorithm with the extracting transform, we introduce a novel weighted transient-extracting transform (TET) based on the solutions of the second-order GD equation. This transform enhances TF distribution concentration while retaining reconstruction capability. Numerical simulations demonstrate that our proposed method improves the average performance for TF concentration by 3% across various noise levels and enhances the accuracy of GD location by over 0.2 within an SNR range from -1 to 20 dB, compared with current state-of-the-art TF analysis methods. Finally, the proposed TF transform is applied to analyze seismic data for low-frequency shadow (LFS) attributes and thin-layer characterization. The results clearly illustrate its effectiveness in seismic processing and interpretation.