# A High-Resolution Spectral Analysis Method Based on Fast Iterative Least Squares Constraints
## 4 Field Data Example  
The main area of operation is located in southern China. The sedimentary characteristics in the study area are complex. To better identify sedimentary layers, this study applies time-frequency analysis to seismic data, using different objective functions to assist in mapping stratigraphic features. Four spectral decomposition methods--STFT, CWT, CLSSA, \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & **CWT** & **STFT** & **ST** & **Our Method** \\ \hline Elapsed time (s) & 0.086 & 0.077 & 0.175 & 0.0021 \\ \hline Speed-up ratio & 40.9 & 36.6 & 83.3 & 1.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Elapsed time for different time–frequency spectra. Figure 2: (**a**) A single waveform with four wavelets with noise of 10 dB using (**b**) CWT, (**c**) STFT, (**d**) ST, and (**e**) our new time spectrum method. and CLSSA based on the Greedy FISTA--are applied to actual seismic data to analyze their time-frequency spectrum characteristics. This study compares their computational efficiency, effectiveness, and stability. Figure 3 shows the actual seismic section profile. Figure 4 presents the time-frequency analysis frequency attribute maps using the STFT method. Specifically, Figure 4a shows the 10 Hz profile, Figure 4b displays the 20 Hz profile, and Figure 4c,d depict the 40 Hz and 60 Hz time-frequency spectrum profiles, respectively. In STFT, spectral distortion is more pronounced within short windows. In this study, a 40 ms time window is used for time-frequency analysis to ensure sufficient time resolution at low frequencies. As shown in Figure 4, the STFT method exhibits good lateral continuity of the time-frequency spectrum. However, due to the spectral tailing effect, the frequency energy in the STFT method appears blurry and shifts toward higher frequencies across all characteristic frequency attribute maps. This causes the frequency details to be less clearly highlighted. Figure 5 shows the time-frequency analysis frequency attribute maps using the CWT method. Specifically, Figure 5a presents the 10 Hz profile, Figure 5b shows the 20 Hz profile, Figure 5c displays the 40 Hz profile, and Figure 5d shows the 60 Hz profile. By comparing Figure 4a,b and Figure 5a,b, it is evident that the CWT method exhibits very low time resolution at low frequencies, failing to effectively display spectral characteristics at high resolution. The primary reason for this is that the CWT method does not use a time window to isolate seismic events, which prevents local analysis, resulting in very low time resolution at low frequencies. Figure 6 illustrates the time-frequency analysis frequency attribute maps using the CLSSA method. Similar to the STFT and CWT methods, Figure 6a,b represent the 10 Hz and 20 Hz time-frequency spectrum profiles, while Figure 6c,d correspond to the 40 Hz and 60 Hz profiles. A 30 ms time window is used for time-frequency analysis. Through comparison with Figure 4, it can be seen that the least squares constrained spectral analysis  
Figure 4: STFT time–frequency analysis characteristic frequency profile. (**a**) 10 Hz profile, (**b**) 20 Hz profile, (**c**) 40 Hz profile, and (**d**) 60 Hz profile. STFT uses 40 ms time window. Figure 5: CWT time–frequency analysis characteristic frequency profile. (**a**) 10 Hz profile, (**b**) 20 Hz profile, (**c**) 40 Hz profile, and (**d**) 60 Hz profile. method can achieve high-resolution spectral analysis at both low and high frequencies, with higher resolution than the STFT method. Figure 7 shows the time-frequency analysis frequency attribute maps using the CLSSA method based on the Greedy FISTA. Figure 7a-d correspond to the 10 Hz, 20 Hz, 40 Hz, and 60 Hz profiles, respectively. By comparing this method with the traditional CLSSA method, it is evident that the proposed method achieves higher time-frequency resolution at both high and low frequencies, with a resolution significantly better than that of the STFT and CWT methods. Figure 6: CLSSA time–frequency analysis characteristic frequency profile.