An Amplitude- and Frequency- Preserving S Transform

The time–frequency analysis is very useful for attenuation compensation, quality factor Q estimation, anomaly detection, and so on. Among plenty of time–frequency analysis methods, the S transform (ST) and its extensions are widely used because of their self-adjustable flexibility, compared with the short-time Fourier transform and Gabor transform. However, the traditional ST has a poor amplitude-preserving property near the boundary while being implemented in the time domain, because the partition of unity cannot be guaranteed. Besides, the frequency distribution biases the actual Fourier spectrum because of the linear-frequency-dependent term in the analytical window, which can decrease the accuracy of attenuation estimation. To preserve the amplitude and frequency, a new analytical window is designed, and the corresponding comprehensive window is derived in the time domain. The frequency-domain formulae are derived in detail for an efficient implementation, in which the time-domain convolution is achieved through multiplication. Numerical examples on the synthetic layered model and pseudorandom time series demonstrate the validity of the proposed method in amplitude- and frequency-preserving quantitatively. Examples at a well location of field data further demonstrate its frequency-preserving property qualitatively. Furthermore, the proposed method can have wide applications in exploration geophysics, seismology, or signal analysis fields, combining with the synchrosqueezing transform.