Locci-Lopez, Daniel (School of Geosciences, University of Louisiana at Lafayette) | Zhang, Rui (School of Geosciences, University of Louisiana at Lafayette) | Oyem, Arnold (Department of Earth and Atmospheric Sciences, University of Houston) | Castagna, John (Department of Earth and Atmospheric Sciences, University of Houston)
Summary Multi-resolution spectral decomposition methods such as the S-transform and the Continuous Wavelet Transform, are known to distort spectral attributes such as peak frequency. We introduce a spectral decomposition approach via a multi-scale Fourier Transform that utilizes a frequency-dependent temporal window to achieve any desired combination of temporal and frequency resolution. We investigate a specific frequency-dependent window that focusses the analysis on the full-width at half-maximum of a frequency-dependent Gaussian function. The resulting time-frequency analysis has significantly improved timeresolution relative to the S-transform. This is demonstrated on real seismic data in the Permian Basin.
Summary Spectral decomposition using regularized inversion calculates time-varying Fourier series coefficients by global optimization with applied constraints. This spectral decomposition method mitigates the tradeoff effect between time and frequency resolution, which the traditional spectral decomposition methods like the shorttime Fourier transform and the continuous wavelet transform cannot avoid. Quantitative resolution analysis via the Heisenberg uncertainty principle on synthetic data and real seismic data shows that regularized spectrum inversion results in a superior combination of time and frequency resolution. Introduction Liu and Fomel (2009) and Puryear et al. (2012) have shown that regularized inversion can be used to solve directly for time-varying Fourier series coefficients. This has advantages over conventional spectral decomposition methods.
Principal component analysis (PCA) can be used to generate frequency-dependent spectral attributes for improved delineation and visualization of frequency-dependent features. A Varimax (Kaiser, 1958) PCA rotation algorithm relates individual principal components (PCs) to characteristic frequency bands. Synthetic and field data results show that Varimax rotated PC spectral attributes effectively respond to geological variations. For a synthetic wedge model, Varimax rotated PC spectral attributes serve well as a thickness indicator. For a real data case, a karst related sink hole is delineated better using Varimax rotated PC attributes than by conventional spectral decomposition or conventional principal component analysis.
Spectral decomposition data are sorted and visualized in a variety of ways. Useful sorting schemes include:
(1) pseudo-spectral volumes or “tuning cubes”, (2) timefrequency gathers, (3) space-frequency gathers, and (4) common frequency volumes. Sorting spectral data into maps and cross sections has useful application in visualizing a combination of time, frequency and offset dimensions
IntroductionSpectral decomposition data are multivalued at a given time and spatial position. Interpretation of subtle stratigraphic features from spectra may, thus, involve multidimensional visualization challenges. Consequently, the spectral data are sorted and visualized in a variety of ways (e.g. Partyka et. al., 1999). Useful sorting schemes include: (1) pseudo spectral volumes, such as “tuning cubes”, which are essentially frequency dependent amplitude maps that can be loaded into common interpretation platforms for visualization (2) timefrequency gathers, which show frequency spectra as function of time for a single seismic trace, (3) space-frequency gathers, which show lateral variation of spectra for a given time window or along a horizon, and (4) common frequency volumes.
Theory Summary We compare the spectra of Short Time Window Fourier Transform (STFT) and Constrained Least Squares Spectral Analysis (CLSSA) for spectral minima periodicity. Using time thickness equals 1/df, where df is frequency period, we show that spectral minima approach using CLSSA gives more accurate time thicknesses than STFT for the same analysis window. We cross plot apparent time thicknesses derived from CLSSA and STFT line spectra using the approach above, against true time thicknesses of the wedge model. The result shows apparent CLSSA thicknesses that are strongly correlated with true time thicknesses We extend this study to real seismic data from Hitts Lake Field, onshore Texas and show that the results are consistent with the results from model studies. The time separation T, between the reflection pair can be determined by 1/df, where df is the frequency period or frequency separation between two spectral minima.