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

The time-reassignment method and time-synchrosqueezing transform show a good ability in impulse-like signal analysis. This kind of method calculates the group delay (GD) estimators at the spread 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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i>-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 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 a SNR range of -1 dB to 20 dB, compared to current state-of-the-art TF analysis methods. Finally, the proposed TF transform is applied to analyze seismic data for low-frequency shadow attributes and thin layer characterization. The results clearly illustrate its effectiveness in seismic processing and interpretation.