Time‐phase amplitude spectra based on a modified short‐time Fourier transform

The short-time Fourier transform allows calculation of the amplitude and initial phase distribution of the real signal as functions of time and frequency, whereas the wavelet transform allows calculation of the amplitude and instantaneous phase distribution of the real signal as functions of time and scale. However, for a complete description of the non-stationary signal, we should obtain not only the amplitude, initial phase, and instantaneous phase distribution as functions of time and frequency simultaneously with high precision but also the amplitude distribution as a function of time and phase referred to as the time–phase amplitude spectrum. In this paper, the time–phase amplitude spectrum is presented based on the high-precision time–frequency amplitude spectrum and initial and instantaneous phase spectra that are generated simultaneously by the proposed modified short-time Fourier transform. To minimise the effect of noise on the high-precision time–frequency amplitude spectrum, initial and instantaneous phase spectra, and time–phase amplitude spectrum, the modified short-time Fourier transform is applied to the real signal reconstructed by the peak high-precision time–frequency amplitude spectrum and the high-precision time–frequency instantaneous phase spectrum at that location to obtain the stable high-precision time–frequency amplitude spectrum, initial and instantaneous phase spectra, and stable time–phase amplitude spectrum. Compared with the short-time Fourier transform and wavelet transform, the time–frequency amplitude spectrum and initial and instantaneous phase spectra obtained by the modified short-time Fourier transform have higher precision than those obtained by the short-time Fourier transform and wavelet transform. Analysis of synthetic data shows that the modified short-time Fourier transform can be used not only for the calculation of the high-precision time–frequency amplitude spectrum, initial and instantaneous phase spectra, and time–phase amplitude spectrum but also for signal reconstruction, stable high-precision time–frequency amplitude spectrum, initial and instantaneous phase spectra, and stable time–phase amplitude spectrum. Analysis of real seismic data applications demonstrates that the stable time–phase amplitude spectrum reveals seismic events with high sensitivity and is well-matched for seismic data processing and interpretation.