New velocity data in addition to literature data derived from sonic log, seismic, and laboratory measurements are analyzed for elastic silicate rocks. These data demonstrate simple systematic relationships between compressional and shear wave velocities. For water-saturated elastic silicate rocks, shear wave velocity is approximately linearly related to compressional wave velocity and the compressional-to-shear velocity ratio decreases with increasing compressional velocity. Laboratory data for dry sandstones indicate a nearly constant compressional-to-shear velocity ratio with rigidity approximately equal to bulk modulus. Ideal models for regular packings of spheres and cracked solids exhibit behavior similar to the observed watersaturated and dry trends. For dry rigidity equal to dry bulk modulus, Gassmann' s equations predict velocities in close agreement with data from the water-saturated rock.

Instantaneous spectral analysis (ISA) is a continuous timefrequency analysis technique that provides a frequency spectrum for each time sample of a seismic trace. ISA achieves both excellent time and frequency localization utilizing wavelet transforms to avoid windowing problems that complicate conventional Fourier analysis. Applications of the method include enhanced resolution, improved visualization of stratigraphic features, thickness estimation for thin beds, noise suppression, improved spectral balancing, and direct hydrocarbon indication. We have seen four distinct ways in which ISA can help in the detection of hydrocarbons: (1) anomalously high attenuation in thick or very unconsolidated gas reservoirs, (2) low-frequency shadows in reservoirs where the thickness is not sufficient to result in significant attenuation, (3) preferential illumination at the "tuning" frequency which can be different for gas or brine-saturated rocks, and (4) frequency-dependent AVO. In this paper, we describe the ISA technique, compare it to other spectral decomposition methods, and show some examples of the use of ISA to detect lowfrequency shadows beneath gas reservoirs.The ISA method involves the following steps: (1) decompose the seismogram into constituent wavelets using wavelet transform methods such as Mallat's matching pursuit decomposition, (2) sum the Fourier spectra of the individual wavelets in the time-frequency domain to produce "frequency gathers," and (3) sort the frequency gathers to produce common (constant) frequency cubes, sections, time slices, and horizon slices. The results can be viewed using animation techniques available in commercial interpretation and visualization packages. Figure 1 shows a synthetic seismic trace and the corresponding ISAtime-frequency analysis. The time-frequency plot shows amplitude spectra for each time sample. We refer to this kind of plot as a "frequency gather." The first arrival on the synthetic seismogram results from an isolated reflector. The frequency spectrum is the spectrum of the wavelet. Note that the duration of the spectrum is identical to the duration of the arrival in the time domain as opposed to Fourier-based methods in which the time duration is equal to the window length. The second event is a composite of two events of differing center frequency arriving precisely at the same time. The frequency spectrum indicates a low-frequency arrival spread over time and a higher-frequency arrival that is more localized in time. The third event is caused by two interfering arrivals of the same frequency. Although the presence of two arrivals is not immediately apparent on the seismogram, the time-frequency decomposition clearly shows two distinct arrivals. The fourth event is a composite of four waveforms arriving at two distinct times evident on the time-frequency analysis. The final event consists of three arrivals of the same frequency that are very closely spaced in time. The three distinct arrivals cannot be resolved at low frequencies, but the separation is clearly...

GREENBERG, M.L. and CASTAGNA, J.P. 1992. Shear-wave velocity estimation in porous rocks : theoretical formulation, preliminary verification and applications. Geophysical Prospecting 40, Shear-wave velocity logs are useful for various seismic interpretation applications, including bright spot analyses, amplitude-versus-offset analyses and multicomponent seismic interpretations. Measured shear-wave velocity logs are, however, often unavailable.We developed a general method to predict shear-wave velocity in porous rocks. If reliable compressional-wave velocity, lithology, porosity and water saturation data are available, the precision and accuracy of shear-wave velocity prediction are 9% and 3%, respectively. The success of our method depends on: (1) robust relationships between compressional-and shear-wave velocities for water-saturated, pure, porous lithologies; (2) nearly linear mixing laws for solid rock constituents; (3) first-order applicability of the Biot-Gassmann theory to real rocks.We verified these concepts with laboratory measurements and full waveform sonic logs. Shear-wave velocities estimated by our method can improve formation evaluation. Our method has been successfully tested with data from several locations. 195-209. and rock composition, these have necessarily had limited success, because velocity also depends on effective stress, porous rock structure (pore shape distribution) and degree of lithification. An alternative approach to shear-wave velocity prediction exists, because these factors affect compressional-and shear-wave velocity in a similar way and because compressional-wave velocity data are widely available. We have investigated the use of measured compressional-wave velocity, with porosity and lithology data, to predict shear-wave velocity.Relationships between compressional-and shear-wave velocities are well-known for brine-saturated, pure lithologies (Pickett 1963). These relations are readily applied to mixed lithologies because, as predicted from Hashin and Shtrikman (1963) bounds, the solid rock constituents combine almost linearly.A general method for shear-wave velocity prediction must account for rocks which are not brine saturated. Biot's (1956) theoretical work on elastic-wave propagation in porous media, once coupled to measurable elastic parameters by Geertsma and Smit (1961), showed that Gassmann's (1951) equations are generally applicable to statistically isotropic, porous rocks in the limit of zero-frequency wave propagation. When applied to real rocks, Gassmann's equations yield a first-order prediction for the dependence of elastic velocities on the properties of pore-filling fluids (e.g. Castagna, Batzle and Eastwood 1985).We developed a method for predicting shear-wave velocity in porous, sedimentary rocks which couples empirical relations between shear-and compressionalwave velocities with Gassmann's equations. Mixed lithologies and fluids are accounted for. Laboratory measurements are used for method verification. We find that the mean predicted shear-wave velo...

This paper presents a new methodology for computing a time-frequency map for nonstationary signals using the continuous-wavelet transform (CWT). The conventional method of producing a time-frequency map using the short time Fourier transform (STFT) limits time-frequency resolution by a predefined window length. In contrast, the CWT method does not require preselecting a window length and does not have a fixed time-frequency resolution over the time-frequency space. CWT uses dilation and translation of a wavelet to produce a time-scale map. A single scale encompasses a frequency band and is inversely proportional to the time support of the dilated wavelet. Previous workers have converted a time-scale map into a time-frequency map by taking the center frequencies of each scale. We transform the time-scale map by taking the Fourier transform of the inverse CWT to produce a time-frequency map. Thus, a time-scale map is converted into a time-frequency map in which the amplitudes of individual frequencies rather than frequency bands are represented. We refer to such a map as the time-frequency CWT (TFCWT). We validate our approach with a nonstationary synthetic example and compare the results with the STFT and a typical CWT spectrum. Two field examples illustrate that the TFCWT potentially can be used to detect frequency shadows caused by hydrocarbons and to identify subtle stratigraphic features for reservoir characterization.

Amplitude variation with offset (AVO) interpretation may be facilitated by crossplotting the AVO intercept (A) and gradient (B). Under a variety of reasonable petrophysical assumptions, brine‐saturated sandstones and shales follow a well‐defined “background” trend in the A-B plane. Generally, A and B are negatively correlated for “background” rocks, but they may be positively correlated at very high [Formula: see text] ratios, such as may occur in very soft shallow sediments. Thus, even fully brine‐saturated shallow events with large reflection coefficients may exhibit large increases in AVO. Deviations from the background trend may be indicative of hydrocarbons or lithologies with anomalous elastic properties. However, in contrast to the common assumptions that gas‐sand amplitude increases with offset, or that the reflection coefficient becomes more negative with increasing offset, gas sands may exhibit a variety of AVO behaviors. A classification of gas sands based on location in the A-B plane, rather than on normal‐incidence reflection coefficient, is proposed. According to this classification, bright‐spot gas sands fall in quadrant III and have negative AVO intercept and gradient. These sands exhibit the amplitude increase versus offset which has commonly been used as a gas indicator. High‐impedance gas sands fall in quadrant IV and have positive AVO intercept and negative gradient. Consequently, these sands initially exhibit decreasing AVO and may reverse polarity. These behaviors have been previously reported and are addressed adequately by existing classification schemes. However, quadrant II gas sands have negative intercept and positive gradient. Certain “classical” bright spots fall in quadrant II and exhibit decreasing AVO. Examples show that this may occur when the gas‐sand shear‐wave velocity is lower than that of the overlying formation. Common AVO analysis methods such as partial stacks and product (A × B) indicators are complicated by this nonuniform gas‐sand behavior and require prior knowledge of the expected gas‐sand AVO response. However, Smith and Gidlow’s (1987) fluid factor, and related indicators, will theoretically work for gas sands in any quadrant of the A-B plane.