We have addressed the problem of seismic data frequencytime filtration based on the S-transform. The S-transform provides frequency-dependent resolution while maintaining a direct relationship with the Fourier spectrum and has been widely used in different seismic data-processing applications. The standard S-transform filtration method is based on its invertibility. In a sense of time localization, this method is not optimal because the calculation of the inverse S-transform includes time averaging. We propose an alternative filtration method based on the reconstruction of a signal by S-transform ridges. We approximate the phase of the Fourier spectrum of the reconstructed signal using the phase of the S-transform ridge. We derive the integral equation, which expresses the S-transform ridge amplitudes in terms of the amplitudes of the Fourier spectrum of the reconstructed signal. The introduced equation is numerically solved to find the signal amplitude using the truncated singular value decomposition. From the results, we obtain both the phase and amplitude of the signal. Thus, the signal can be extracted from the S-transform spectrum only by the ridges. This finding is promising in terms of maximum usage of the time-localization properties of the S-transform during frequencytime filtration. The presented results of ground-roll attenuation from reflection seismic data demonstrate that the proposed S-transform ridge filtration method can be effective in practice.
