Adaptive Time-frequency Localization of SLTFT

The key different between S-transform and STFT is that S-transform employs a frequency-varying Gaussian taper, which provides $t\textrm{-}f$ spectrum with variable $t\textrm{-}f$ localization. Considering that the localization scalar $\varepsilon$ controls the $t\textrm{-}f$ localization of SLTFT, we design the new localization scalar varying over frequency

$$\varepsilon = \varepsilon(f),$$ (21)

where $f$ is the frequency. Increment of the localization scalar enhances the time localization of the spectrum, albeit at the cost of the reduced frequency localization, and vice versa. This inverse relationship allows us to adjust the balance between time and frequency localization according to specific requirements.

We use two synthetic nonstationary models to to illustrate the time-frequency characterization of the proposed method. The first model (refered to as signal 1) comprises two hyperbolic signals $s_1(t)$, $s_2(t)$ and Gaussian noise, where

The signal 1 with a time interval of 0.5 ms contains 2000 samples between each sample. We windowed the periodic signals by a length-fixed cosine taper. The first signal $s_1(t)$ has a time duration ranging from 0.1 to 0.7 seconds, while the second signal $s_2(t)$ spans from 0.25 to 0.75 seconds. The S-transform provides a variable $t$-$f$ localization (see Fig.4a) that allows the spectrum to exhibit enhanced frequency resolution at lower frequencies and superior time localization at higher frequencies. Fig.4c shows the spectrum calculated from the STFT with fixed window length. It is noted that the spectrum exhibits improved frequency localization at lower frequencies. However, this enhancement comes at the expense of reduced time localization around higher frequencies. Therefore, we can apply the frequency-varying localization scalers (see Fig.4g) to the SLTFT, and the result (see Fig.4e) provides a reasonable time-frequency spectrum similar to that of the S-transform.

We selected two different hyperbolic signals $s_3(t)$, $s_4(t)$ in replace those in the second model (refered to as signal 2) to further test the SLTFT, where

In this case, the S-transform encounters difficulties in producing an accurate spectrum shown in Fig.4b, which displays significant aliasing since its high time resolution leads to diminished frequency localization at higher frequencies. This observation shows that the variable time-frequency localization inherent in the S-transform introduces a trade-off between the representation of low and high frequency components. For comparison, the STFT with constant window length is hard to achieve a balance between time and frequency localization. As illustrated in Fig.4d, the spectrum obtained by the SLTFT with constant localization scaler exhibits better frequency localization (around higher frequencies), but comes at the expense of diminished time localization (at lower frequencies). Fig.4g shows the frequency-varying localization scalers applied in this case, which has a trend opposite to that of signal 1. The flexible frequency-varying parameters lead to a resonable spectrum obtained by the proposed SLTFT (see Fig.4f).

st,st1,stft,stft1,sltft,sltft1,epss
Figure 4. The time-frequency map of synthetic signal 1 obtained by (a) S-transform, (c) STFT with fixed window length and (e) SLTFT with frequency-varying localization scalers. The time-frequency map of synthetic signal 2 obtained by (b) S-transform, (d) STFT with fixed window length and (f) SLTFT with frequency-varying localization scalers. (g) The frequency-varying localization scalers of SLTFT used in Fig.4e (solid line) and Fig.4f (dash line).

The above examples show the adaptive and flexible frequency sampling of SLTFT in signal analysis, which provides an adjustable time-frequency representation. Next, we use several field datasets to show its performance in seismic data processing tasks.