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.

Pseudoacoustic anisotropic wave equations are simplified elastic wave equations obtained by setting the S-wave velocity to zero along the anisotropy axis of symmetry. These pseudoacoustic wave equations greatly reduce the computational cost of modeling and imaging compared to the full elastic wave equation while preserving P-wave kinematics very well. For this reason, they are widely used in reverse time migration (RTM) to account for anisotropic effects. One fundamental shortcoming of this pseudoacoustic approximation is that it only prevents S-wave propagation along the symmetry axis and not in other directions. This problem leads to the presence of unwanted S-waves in P-wave simulation results and brings artifacts into P-wave RTM images. More significantly, the pseudoacoustic wave equations become unstable for anisotropy parameters [Formula: see text] and for heterogeneous models with highly varying dip and azimuth angles in tilted transversely isotropic (TTI) media. Pure acoustic anisotropic wave equations completely decouple the P-wave response from the elastic wavefield and naturally solve all the above-mentioned problems of the pseudoacoustic wave equations without significantly increasing the computational cost. In this work, we propose new pure acoustic TTI wave equations and compare them with the conventional coupled pseudoacoustic wave equations. Our equations can be directly solved using either the finite-difference method or the pseudospectral method. We give two approaches to derive these equations. One employs Taylor series expansion to approximate the pseudodifferential operator in the decoupled P-wave equation, and the other uses isotropic and elliptically anisotropic dispersion relations to reduce the temporal frequency order of the P-SV dispersion equation. We use several numerical examples to demonstrate that the newly derived pure acoustic wave equations produce highly accurate P-wave results, very close to results produced by coupled pseudoacoustic wave equations, but completely free from S-wave artifacts and instabilities.

The presence of wavelet stretch due to imaging presents serious difficulty in AVO or inversion analysis, especially for 3-term wide-angle analysis. Wavelet stretch significantly alters the gradient and wide-angle coefficient and reduces resolution of stacks. In this paper we present a method for correcting wavelet stretch that is exact for any v(z) (layered) medium. It does not depend on an underlying AVO/AVA approximation and is therefore applicable for 2-or 3-term AVA analysis. The required input is an extracted wavelet from any known reflection angle. The resulting correction operator is stationary over the time coordinate of the angle domain and is robustly implemented by a Weiner-Levinson method. This filter corrects angle gathers for wavelet stretch, producing improved resolution in subsequent angle stacks or gradient computations. Wavelet stretch correction is essential for linear inversion for density.

Instantaneous spectral properties of seismic data — center frequency, root-mean-square frequency, bandwidth — often are extracted from time-frequency spectra to describe frequency-dependent rock properties. These attributes are derived using definitions from probability theory. A time-frequency spectrum can be obtained from approaches such as short-time Fourier transform (STFT) or time-frequency continuous-wavelet transform (TFCWT). TFCWT does not require preselecting a time window, which is essential in STFT. The TFCWT method converts a scalogram (i.e., time-scale map) obtained from the continuous-wavelet transform (CWT) into a time-frequency map. However, our method includes mathematical formulas that compute the instantaneous spectral attributes from the scalogram (similar to those computed from the TFCWT), avoiding conversion into a time-frequency spectrum. Computation does not require a predefined window length because it is based on the CWT. This technique optimally decomposes a multiscale signal. For nonstationary signal analysis, spectral decomposition from [Formula: see text] has better time-frequency resolution than STFT, so the instantaneous spectral attributes from CWT are expected to be better than those from STFT.

Recent applications of 2D and 3D turning-ray tomography show that near-surface velocities are important for structural imaging and reservoir characterization. For structural imaging, we used turning-ray tomography to estimate the near-surface velocities for static corrections followed by prestack time migration and the near-surface velocities for prestack depth migration. Two-dimensional acoustic finite-difference modeling illustrates that wave-equation prestack depth migration is very sensitive to the near-surface velocities. Field data demonstrate that turning-ray tomography followed by prestack time migration helps to produce superior images in complex geologic settings. When the near-surface velocity model is integrated into a background velocity model for prestack depth migration, we find that wave propagation is very sensitive to the velocities immediately below the topography. For shallow-reservoir characterization, we developed and applied azimuthal turning-ray tomography to investigate observed apparent azimuthal-traveltime variations, using a wide-azimuth land seismic survey from a heavy-oil field at Surmont, Canada. We found that the apparent azimuthal velocity variations are not necessarily related to azimuthal anisotropy, or horizontal transverse isotropy (HTI), induced by the stress field or fractures. Near-surface heterogeneity and the acquisition footprint also could result in apparent azimuthal variations.