S-transform and Fourier transform frequency spectra of broadband seismic signals

Abstract

The S-transform is one way to transform a 1D seismogram into a 2D time-frequency analysis. We have investigated its use to compute seismic interpretive attributes, such as peak frequency and bandwidth. The S-transform normalizes a frequency-dependent Gaussian window by a factor proportional to the absolute value of frequency. This normalization biases spectral amplitudes toward higher frequency. At a given time, the S-transform spectrum has similar characteristics to the Fourier spectrum of the derivative of the waveform. For narrowband signals, this has little impact on the peak frequency of the time-frequency analysis. However, for broadband seismic signals, such as a Ricker wavelet, the S-transform peak frequency is significantly higher than the Fourier peak frequency and can thus be misleading. Numerical comparisons of spectra from a variety of waveforms support the general rule that S-transform peak frequencies are equal to or greater than Fourier-transform peak frequencies. Comparisons on real seismic data suggest that this effect should be considered when interpreting S-transform spectral decompositions. One solution is to define the unscaled S-transform by removing the normalization factor. Tests comparing the unscaled S-transform with the S-transform and the short-windowed Fourier transform indicate that removing the scale factor improves the time-frequency analysis on reflection seismic data. This improvement is most relevant for quantitative applications.