One of the most accepted geologic models is the relation between reflector curvature and the presence of open and closed fractures. Such fractures, as well as other small discontinuities, are relatively small and below the imaging range of conventional seismic data. Depending on the tectonic regime, structural geologists link open fractures to either Gaussian curvature or to curvature in the dip or strike directions. Reflector curvature is fractal in nature, with different tectonic and lithologic effects being illuminated at the [Formula: see text] and [Formula: see text] scales. Until now, such curvature estimates have been limited to the analysis of picked horizons. We have developed what we feel to be the first volumetric spectral estimates of reflector curvature. We find that the most positive and negative curvatures are the most valuable in the conventional mapping of lineations — including faults, folds, and flexures. Curvature is mathematically independent of, and interpretatively complementary to, the well-established coherence geometric attribute. We find the long spectral wavelength curvature estimates to be of particular value in extracting subtle, broad features in the seismic data such as folds, flexures, collapse features, fault drags, and under- and overmigrated fault terminations. We illustrate the value of these spectral curvature estimates and compare them to other attributes through application to two land data sets — a salt dome from the onshore Louisiana Gulf Coast and a fractured/karsted data volume from Fort Worth basin of North Texas.
The Hilbert transform (HT) has been used in seismic data processing and interpretation for many years. A well-known application of HT is seismic complex-trace analysis using instantaneous phase, frequency, and amplitude (Taner et al. 1979). We present a new generalized Hilbert transform (GHT), which has many advantages over the traditional HT-particularly robustness to noise and variety of applications.A simplified definition of GHT is that it can be considered a windowed traditional HT. A more accurate description of GHT, and the mathematical definition of GHT, will be given later.Recently, the windowed Fourier transform was developed to allow more flexibility in applications and the windowed Fourier transform, indeed, has been used in such fields as timefrequency analysis in geophysics and speech recognition/ speaker identification in speech science. Similarly, GHT can be used to create new seismic attributes and extend existing applications of HT. It is believed that GHT will have more diverse applications in seismic data analysis and other fields engaged in digital data processing. In this article, two seismic data analysis examples will be discussed.In the first example, GHT is used to produce a new type of complex seismic trace that is less sensitive to noise than conventional HT. In the second example, we applied GHT for seismic edge-detection or coherence-cube technology. In this application, GHT showed a very desirable ability to capture subtle geologic features that are hardly detected by existing algorithms.Theory of GHT. The traditional Hilbert transform in the frequency domain can be expressed (Claerbout, 1976):where X(ω) is the Fourier transform of an input trace x(t), hi(ω) is the Hilbert transform of x(t) in the frequency domain, Sign(ω) is the sign function and i = √-1.Applying the inverse Fourier transform to hi(ω) and setting t=0 we obtainwhere Σ ω means summation over positive frequencies, and hr(0) and hi(0) are the real and imaginary parts of the complex trace h(0), respectively. Re and Im refer to the real and imaginary parts of a complex value. To make the above HT formula valid for all time values t, we define GHT as hr(t) = {2* Σ ω {Re[X(t,ω)]} n + Re[X(t, 0)] n } 1/n , hi(t) = {2* Σ ω {Im[X(t,ω)]} n } 1/n , h(t) = hr(t) + i • hi(t).where hi(t) is the L n -order (n > 0) GHT of the input data, h(t) is the complex trace, and X(t,ω) is the Fourier transform of the input data within a window centered at t. X(t,ω) reduces to the well-known Gabor transform if a Gaussian window is used. GHT extends HT in two aspects, by introducing the window and the order of n. The L n -order of GHT is analogous to the definition of the L n norm in Hilbert norm spaces. For any input signal (or a seismic trace), the output of the traditional Hilbert transform is unique. However, the output of GHT depends on the choice of the order n, the window shape, and the window length. When the order n is 1 and the window is a box function with infinite length, GHT produces identical results as the traditional Hilbert tra...
Recently developed seismic attributes such as volumetric curvature and amplitude gradients enhance our ability to detect lineaments. However, because these attributes are based on derivatives of either dip and azimuth or the seismic data themselves, they can also enhance high-frequency noise. Recently published structure-oriented filtering algorithms show that noise in seismic data can be removed along reflectors while preserving major structural and stratigraphic discontinuities. In one implementation, the smoothing process tries to select the most homogenous window from a suite of candidate windows containing the analysis point. A second implementation damps the smoothing operation if a discontinuity is detected. Unfortunately, neither of these algorithms preserves thin or small lineaments that are only one voxel in width. To overcome this defect, we evaluate a suite of nonlinear feature-preserving filters developed in the image-processing and synthetic aperture radar (SAR) world and apply them to both synthetic and real 3D dip-and-azimuth volumes of fractured geology from the Forth Worth Basin, USA. We find that the multistage, median-based, modified trimmed-mean algorithm preserves narrow geologically significant features of interest, while suppressing random noise and acquisition footprint.
The new seismic disorder attribute quantitatively describes the degree of randomness embedded in 3D poststack seismic data. We compute seismic disorder using a filter operation that removes simple structures including constant values, constant slopes, and steps in axial directions. We define the power of the filtered data as the seismic disorder attribute, which approximately represents data randomness. Seismic data irregularities are caused by a variety of reasons, including random reflection, diffraction, near-surface variations, and acquisition noise. Consequently, the spatial distribution of the seismic disorder attribute may help hydrocarbon exploration in several ways, including identifying geologic features such as fracture zones, gas chimneys, and terminated unconformities; indicating the signal-to-noise ratio to assess data quality; and providing a confidence index for reservoir simulation and engineering projects. We present three case studies and a comparison to other noise-estimation methods.
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