Phil D. Anno

Spectral decomposition of seismic data with continuous-wavelet transform

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 timefrequency 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.

Recent applications of turning-ray tomography

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.

Instantaneous spectral attributes using scales in continuous-wavelet transform

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. F...

Wavelet-Radon domain dealiasing and interpolation of seismic data

Spatial aliasing is unavoidable in some seismic data and has serious effects on the performance of multichannel data processing and migration. Antialias filtering produces distortion of the signal through the removal of high-frequency information. In contrast, dealiasing produces an unaliased estimate of the signal at all frequencies present in the original time series. A new dealiasing algorithm is developed by exploiting the properties of seismic wavefields in the wavelet-Radon transform domain, specifically the overlap of information between wavelet scales at the same frequency. The effectiveness of the wavelet-Radon dealiasing algorithm is demonstrated through the processing of both synthetic and field seismic data.

Analytic correction for wavelet stretch due to imaging

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 2or 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.

Imaging oil-sand reservoir heterogeneities using wide-angle prestack seismic inversion

Mass density, due to its linear relationship with porosity, has long been recognized as a potential seismic indicator of fluid saturation. Given its dependence on mineral composition, density can also be diagnostic for lithology. In this paper we discuss some key aspects of a wide-angle processing and density inversion workflow, and apply it to a bitumen reservoir in Canada for imaging reservoir heterogeneities, such as shales, that can potentially act as permeability baffles. In this field, intrareservoir shales typically have higher densities than surrounding reservoir sands. This wide-angle workflow yields stable density estimates, from reflected P-waves alone, at a resolution suitable for mapping the intrareservoir shales.

Wavelet‐transform‐based prestack multiscale Kirchhoff migration

We propose a Kirchhoff-style algorithm that migrates coefficients obtained by wavelet decomposition of seismic traces over time. Wavelet-based prestack multiscale Kirchhoff migration involves four steps: wavelet decomposition of the seismic data, thresholding of the resulting wavelet coefficients, multiscale Kirchhoff migration, and image reconstruction from the multiscale images. The migration procedure applied to each wavelet scale is the same as conventional Kirchhoff migration but operates on wavelet coefficients. Since only the wavelet coefficients are migrated, the cost of wavelet-based migration is reduced compared to that of conventional Kirchhoff migration. Kirchhoff migration of wavelet-decomposed data, followed by wavelet reconstruction, is kinematically equivalent to and yields similar migrated signal shapes and amplitudes as conventional Kirchhoff migration when data at all wavelet scales are included. The decimation in the conventional discrete pyramid wavelet decomposition introduces a translation-variant phase distortion in the wavelet domain. This phase distortion is overcome by using a stationary wavelet transform rather than the conventional discrete wavelet transform of the data to be migrated. A wavelet reconstruction operator produces a single composite broadband migrated space-domain image from multiscale images. Multiscale images correspond to responses in different frequency windows, and migrating the data at each scale has a different cost. Migrating some, or only one, of the individual scale data sets considerably reduces the computational cost of the migration. Successful 2D tests are shown for migrations of synthetic data for a point-diffractor model, a multilayer model, and the Marmousi model.

Time‐frequency attribute of seismic data using continuous wavelet transform

A time-frequency decomposition that can provide higher frequency resolution at lower frequencies and higher time resolution at higher frequencies is desirable for analyzing seismic data. This is because the hydrocarbons in the reservoir are diagnostic at lower frequencies and thin beds can be resolved with enhanced time resolution at higher frequencies. In this paper we present a new method to compute the time-frequency spectrum using wavelet as a window that achieves this objective. Time-frequency spectrum is commonly used to compute various frequency attributes of seismic signal like single frequency, dominant frequency, center frequency and so forth. The conventional approach is to use short time Fourier transform (STFT) to obtain a time-frequency spectrum. Time-frequency resolution in the STFT is limited by the choice of a window length. The proposed time-frequency spectrum using CWT (TFCWT) in this work has the ability to adapt with the frequency content of the signal. The flexibility of not having to choose a window is an advantage of our method. We present two applications of TFCWT to real data sets in this paper. In the first example, we use TFCWT to enhance low frequency shadows caused by hydrocarbon reservoirs. In the second example, we apply the time frequency spectrum in interpreting time slices from a 3D seismic volume in frequency space to identify thin beds below tuning thickness.

An Accurate And Stable Wave Equation For Pure Acoustic TTI Modeling

One fundamental shortcoming of the conventional pseudoacoustic approximation is that it only prevents shear wave propagation along the symmetry axis of anisotropy and not in other directions. This problem leads to the presence of unwanted shear waves in P-wave simulation results and brings artifacts into P-wave RTM images. More significantly, the pseudo-acoustic wave equations become unstable for anisotropy parameters e < δ and for heterogeneous models with highly varying dip and azimuth angles in tilted transversely isotropic (TTI) media. Pure acoustic anisotropic wave equations completely de-couple the P-wave response from the elastic wavefield and naturally solve all the above-mentioned problems. In this work, we propose a new pure acoustic TTI wave equation and compare it with the conventional coupled pseudo-acoustic wave equation. We derive finite difference solutions to this new equation and use numerical examples to demonstrate that it produces highly accurate P-wave results, very close to results produced by coupled pseudo-acoustic wave equations, but completely free from shear wave artifacts.

Wavelet Reconstruction of Nonuniformly Sampled Signals

For the reconstruction of a nonuniformly sampled signal based on its noisy observations, we propose a level dependent l1 penalized wavelet reconstruction method. The LARS/Lasso algorithm is applied to solve the Lasso problem. The data adaptive choice of the regularization parameters is based on the AIC and the degrees of freedom is estimated by the number of nonzero elements in the Lasso solution. Simulation results conducted on some commonly used 1_D test signals illustrate that the proposed method possesses good empirical properties.