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.

The frequency-dependent width of the Gaussian window function used in the S-transform may not be ideal for all applications. In particular, in seismic reflection prospecting, the temporal resolution of the resulting S-transform time-frequency spectrum at low frequencies may not be sufficient for certain seismic interpretation purposes. A simple parameterization of the generalized S-transform overcomes the drawback of poor temporal resolution at low frequencies inherent in the S-transform, at the necessary expense of reduced frequency resolution. This is accomplished by replacing the frequency variable in the Gaussian window with a linear function containing two coefficients that control resolution variation with frequency. The linear coefficients can be directly calculated by selecting desired temporal resolution at two frequencies. The resulting transform conserves energy and is readily invertible by an inverse Fourier transform. This modification of the S-transform, when applied to synthetic and real seismic data, exhibits improved temporal resolution relative to the S-transform and improved resolution control as compared with other generalized S-transform window functions.

In unconventional reservoirs with significant organic content, the Greenberg-Castagna (GC) S-wave velocity prediction method does not yield accurate S-wave velocity predictions, with observed mean errors varying from 6% to 16% in a variety of unconventional reservoirs rich in organic content. This is because kerogen content is not explicitly taken into account in the GC S-wave velocity prediction method. Two alternative approaches for bedding-normal S-wave velocity prediction from P-wave velocity and other well logs in relatively straight holes drilled in unconventional reservoirs are investigated. The first method is purely empirical, requiring minimal information such as P-wave velocity and total organic carbon content. This approach implicitly accounts for compositional and stress effects on mudrock elasticity. The second method can be classified as a hybrid technique, comprising the following three steps: (1) computing a nonkerogen phase P-wave velocity using the Vernik-Kachanov (VK) model, (2) determination of the nonkerogen S-wave velocity from the GC approach, and (3) using a simplified VK model to mix the nonkerogen matrix with nanoporous kerogen to predict the bedding-normal S-wave velocity of the organic mudrock. The second method explicitly takes into account all the variables that control elastic properties of organic mudrock reservoirs. Tests in nine wells from seven different oil and gas shale reservoirs indicate that both methods have prediction accuracy better than 3% error when input data are accurate.

Various postprocessing methods can be applied to seismic data to extend the spectral bandwidth and potentially increase the seismic resolution. Frequency invention techniques, including phase acceleration and loop reconvolution, produce spectrally broadened seismic sections but arbitrarily create high frequencies without a physical basis. Tests in extending the bandwidth of low-frequency synthetics using these methods indicate that the invented frequencies do not tie high-frequency synthetics generated from the same reflectivity series. Furthermore, synthetic wedge models indicate that the invented high-frequency seismic traces do not improve thin-layer resolution. Frequency invention outputs may serve as useful attributes, but they should not be used for quantitative work and do not improve actual resolution. On the other hand, under appropriate circumstances, layer frequency responses can be extrapolated to frequencies outside the band of the original data using spectral periodicities determined from within the original seismic bandwidth. This can be accomplished by harmonic extrapolation. For blocky earth structures, synthetic tests show that such spectral extrapolation can readily double the bandwidth, even in the presence of noise. Wedge models illustrate the resulting resolution improvement. Synthetic tests suggest that the more complicated the earth structure, the less valid the bandwidth extension that harmonic extrapolation can achieve. Tests of the frequency invention methods and harmonic extrapolation on field seismic data demonstrate that (1) the frequency invention methods modify the original seismic band such that the original data cannot be recovered by simple band-pass filtering, whereas harmonic extrapolation can be filtered back to the original band with good fidelity and (2) harmonic extrapolation exhibits acceptable ties between real and synthetic seismic data outside the original seismic band, whereas frequency invention methods have unfavorable well ties in the cases studied.

Any seismic trace can be decomposed into a 2D function of amplitude versus time and phase. We call this process phase decomposition, and the amplitude variation with time for a specific seismic phase is referred to as a phase component. For seismically thin layers, phase components are particularly useful in simplifying seismic interpretation. Subtle lateral impedance variations occurring within thin layers can be greatly amplified in their seismic expression when specific phase components are isolated. For example, the phase component corresponding to the phase of the seismic wavelet could indicate isolated interfaces or any other time symmetrical variation of reflection coefficients. Assuming a zero-phase wavelet, flat spots and unresolved water contacts may show directly on the zero-phase component. Similarly, thin beds and impedance ramps will show up on components that are 90° out of phase with the wavelet. In the case of bright spots caused by hydrocarbons in thin reservoirs because these occur when the reservoir is of an anomalously low impedance, it is safe to assume that the brightening caused by hydrocarbons occurs on the component $-90°$ out of phase with the wavelet. Amplitudes of other phase components associated with bright reflection events, resulting perhaps from differing impedances above and below the reservoir, thus obscure the hydrocarbon signal. Assuming a zero-phase wavelet, bright-spot interpretation is thus greatly simplified on the $-90°$ phase component. Amplitude maps for the Teal South Field reveal that the lateral distribution of amplitudes is greatly different for the original seismic data and the $-90°$ phase component, exhibiting very different prospectivity and apparent areal distribution of reservoirs. As the impedance changes laterally, the interference pattern for composite seismic events also changes. Thus, waveform peaks, troughs, and zero crossings, may not be reliable indicators of formation top locations. As the waveform phase changes laterally due to lateral rock properties variations, the position of a formation top on the waveform also changes. By picking horizons on distinct phase components, this ambiguity is reduced, and more consistent horizon picking is enabled.

Borehole measurements of salt-body P-wave interval velocities from 55 wells in the vicinity of the Keathley Canyon and Walker Ridge areas of the Gulf of Mexico vary from $\mathrm{13,966}\mathrm{ft}/\mathrm{s}$ ($4256\mathrm{m}/\mathrm{s}$) to $\mathrm{18,535}\mathrm{ft}/\mathrm{s}$ ($5649\mathrm{m}/\mathrm{s}$) with mean velocity of $\mathrm{14,920}\mathrm{ft}/\mathrm{s}$ ($4547\mathrm{m}/\mathrm{s}$) and standard deviation of $726\mathrm{ft}/\mathrm{s}$ ($221\mathrm{m}/\mathrm{s}$). The velocities vary significantly with latitude. Five different interval velocity zones have been identified with each having specific associated mineralogies within a latitude range. In the midlatitude zones, sylvite, various clastics, and small traces of anhydrite and gypsum are found within the salt, yielding a salt interval velocity variation from 14,388 to $\mathrm{14,965}\mathrm{ft}/\mathrm{s}$ ($4386$–$4544\mathrm{m}/\mathrm{s}$). The salt interval velocity in the southern limits of the study area is higher than $\mathrm{15,000}\mathrm{ft}/\mathrm{s}$ ($4572\mathrm{m}/\mathrm{s}$) and is associated with more gypsum. The northernmost wells are anhydrite rich and exhibit the highest velocities. The interval velocities are relatively uncorrelated and are insensitive to factors such as wellbore temperature, depth, and overburden pressure. Composite medium modeling of mixed salt-body mineralogies and multiple regression analysis indicate that compositional variation within the salt body can explain the observed velocity variations.