Operator Aliasing Research Articles

The value of constrained conjugate-gradient least-squares migration in seismic inversion: Application to a fractured-basement play, Texas Panhandle

Seismic inversion has become almost routine in quantitative 3D seismic interpretation. To ensure the quality of the seismic inversion, the input seismic data need to have a high signal-to-noise ratio. With the current low oil price environment, seismic reprocessing is often preferred over reacquisition to improve data quality. Common filter pairs include forward and inverse [Formula: see text]-[Formula: see text] and Radon transforms. Forward and inverse migrations (i.e., migration and demigration) are a more recently introduced transform pair that, when used together in an iterative workflow, results in a least-squares migration algorithm. Least-squares migration compensates for surface variation in data density and, when combined with a filter applied to prestack migrated images, suppresses the operator and data aliasing. We apply a least-squares migration workflow to a fractured-basement data set from the Texas Panhandle to demonstrate the enhancement in signal-to-noise ratio, the reduction in acquisition footprint and migration artifacts, and the improvement in the P-impedance inversion result.

Aliasing in wavefield extrapolation prestack migration

Aliasing during migration (“operator aliasing”) is widely recognized as a problem for Kirchhoff migration. It occurs when high-frequency reflection data are swept out at steep angles, with the problem being worst for very coarse input-trace spacing. The problem can be solved either by data interpolation to a finer-spaced grid of input traces or, more commonly, by anti-aliasing the migration operator (Gray, 1992; Lumley et al., 1994; Abma et al., 1999; Biondi, 2001, Zhang et al., 2001a). Although there is no general agreement on the best way to perform Kirchhoff migration anti-aliasing for all problems, especially those involving irregular spatial sampling, there is agreement on the underlying principle. That is, the diffraction surface used by the migration to accumulate data into the image at a single point should sample the input traces adequately (Figure 1). For two-dimensional (2D) poststack migration, “adequately” means that the diffraction curve must sample adjacent traces with a time delay no greater than one-half period at any frequency present in the data.

Kirchhoff imaging beyond aliasing

Crucial image resolution may be lost when spatially aliased data are imaged with Kirchhoff algorithms that employ standard antialiasing methods. To maximize resolution, I introduce a method that enables the proper imaging of some aliased components in the data, while avoiding aliasing artifacts. The proposed method is based on a detailed analysis of the different types of aliasing that affect Kirchhoff imaging. In particular, it is based on the observation that operator aliasing depends on the dip spectrum of the data. A priori knowledge on the characteristics of the dip spectrum of the data, in particular on its asymmetry, can thus be exploited to enable “imaging beyond aliasing.” The method is not of general applicability, but it successfully improves the image resolution when a priori assumptions on the data dips are realistic. The imaging of salt‐dome flanks in the Gulf of Mexico has been enhanced by the application of the proposed method.

Anti-aliased Kirchhoff-Helmholtz transformations

Non-aliased integral (Kirchhoff-type) transformations for forward and downward wavefield extrapolations in inhomogeneous media with interfaces are described. Special weights are computed to compensate for operator aliasing and finite-aperture effects, even when the data are spatially aliased and irregularly sampled. Basic components of the algorithm, such as Green's function computation, can be replaced by alternative solutions in conjunction with ray-tracing methods. Applications of this algorithm to model and real data in both two and three dimensions are discussed in terms of its impact on seismic modelling, multiple prediction and prestack imaging.

FREQUENCY‐SELECTIVE DESIGN OF THE KIRCHHOFF MIGRATION OPERATOR1

AbstractIntegral migration techniques perform a sum over an aperture of input traces to obtain output at a single point. The length of the aperture is limited by a spatial Nyquist criterion, which typically prohibits imaging very steep dips at very high frequencies without generating severe migration artifacts (migration operator aliasing). For time‐domain Kirchhoff migration, this can be a fatal shortcoming. The standard way to address this problem is to interpolate traces spatially before migration. This reduces the trace spacing, thereby increasing the frequency content which can be migrated without aliasing at steep dips.An alternative remedy to the operator aliasing problem is to modify the phase response of the Kirchhoff migration operator. This operator is frequency‐selective across the migration aperture: it passes all temporal frequencies of the input traces in the innermost portion of the aperture (referring to the shallow dips), and gradually cuts out the higher frequencies as it approaches the outer portion of the aperture. Thus, while all frequencies of the input data contribute to the shallow‐dip portion of the migrated image, only the permissible low frequencies of the input data contribute to imaging the steepest dips.Using a simple realization of a frequency‐selective Kirchhoff migration operator, this technique is illustrated on a synthetic data set involving greater than vertical dips.

A FILTER, DELAY AND SPREAD TECHNIQUE FOR 3D DMO1

The calculation of dip moveout involves spreading the amplitudes of each input trace along the source-receiver axis followed by stacking the results into a 3D zero-offset data cube. The offset-traveltime (x–t) domain integral implementation of the DMO operator is very efficient in terms of computation time but suffers from operator aliasing. The log-stretch approach, using a logarithmic transformation of the time axis to force the DMO operator to be time invariant, can avoid operator aliasing by direct implementation in the frequency-wavenumber (f–k) domain. An alternative technique for log-stretch DMO corrections using the anti-aliasing filters of the f–k approach in the x-log t domain will be presented. Conventionally, the 2D filter representing the DMO operator is designed and applied in the f–k domain. The new technique uses a 2D convolution filter acting in single input/multiple output trace mode. Each single input trace is passed through several 1D filters to create the overall DMO response of that trace. The resulting traces can be stacked directly in the 3D data cube. The single trace filters are the result of a filter design technique reducing the 2D problem to several ID problems. These filters can be decomposed into a pure time-delay and a low-pass filter, representing the kinematic and dynamic behaviour of the DMO operator. The low-pass filters avoid any incidental operator aliasing. Different types of low-pass filters can be used to achieve different amplitude-versus-offset characteristics of the DMO operator.