Quantitative Resolution Analysis for Seismic Spectral Decomposition Methods



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The S-transform is one way to transform a one-dimensional seismogram into a two-dimensional time-frequency analysis. This study investigates 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 towards higher frequency. At a given time, the S-transform spectrum has similar characteristics to the Fourier spectrum of the derivative of the waveform. For narrow-band signals, this has little impact on the peak frequency of the time-frequency analysis. However, for broad-band seismic signals, such as a Ricker wavelet, the S-transform peak frequency is significantly higher than 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. Spectral decomposition using regularized inversion calculates time-varying Fourier series coefficients by global optimization with applied constraints. This spectral analysis method mitigates to some extent the trade-off effect between time and frequency resolution due to the Heisenberg Uncertainty Principle. The Heisenberg uncertainty product can be used as an effective evaluation criterion in spectral resolution analysis. Synthetic examples show that regularized spectrum inversion has smaller uncertainty product values than do the short-time Fourier Transform and the Continuous Wavelet Transform. Real seismic data examples illustrate that regularized spectrum inversion results in a superior combination of time and frequency resolution.



Spectral decomposition, Resolution