The W transform and its improved methods for time-frequency analysis of seismic data

The conventional linear time-frequency analysis method cannot achieve high resolution and energy focusing in the time and frequency dimensions at the same time, especially in the low frequency region. In order to improve the resolution of the linear time-frequency analysis method in the low-frequency region, we have proposed a W transform method, in which the instantaneous frequency is introduced as a parameter into the linear transformation, and the analysis time window is constructed which matches the instantaneous frequency of the seismic data. In this paper, the W transform method is compared with the Wigner-Ville distribution (WVD), a typical nonlinear time-frequency analysis method. The WVD method that shows the energy distribution in the time-frequency domain clearly indicates the gravitational center of time and the gravitational center of frequency of a wavelet, while the time-frequency spectrum of the W transform also has a clear gravitational center of energy focusing, because the instantaneous frequency corresponding to any time position is introduced as the transformation parameter. Therefore, the W transform can be benchmarked directly by the WVD method. We summarize the development of the W transform and three improved methods in recent years, and elaborate on the evolution of the standard W transform, the chirp-modulated W transform, the fractional-order W transform, and the linear canonical W transform. Through three application examples of W transform in fluvial sand body identification and reservoir prediction, it is verified that W transform can improve the resolution and energy focusing of time-frequency spectra.