Sergey Fomel

Angle‐domain common‐image gathers by wavefield continuation methods

Migration in the angle domain creates seismic images for different reflection angles. We present a method for computing angle-domain common-image gathers from seismic images obtained by depth migration using wavefield continuation. Our method operates on prestack migrated images and produces the output as a function of the reflection angle, not as a function of offset ray parameter as in other alternative approaches. The method amounts to a radial-trace transform in the Fourier domain and is equivalent to a slant stack in the space domain. We obtain the angle gathers using a stretch technique that enables us to impose smoothness through regularization. Several examples show that our method is accurate, fast, robust, and easy to implement. The main anticipated applications of our method are in the areas of migration-velocity analysis and amplitude-versus-angle analysis.

Applications of plane-wave destruction filters

Plane‐wave destruction filters originate from a local plane‐wave model for characterizing seismic data. These filters can be thought of as a time–distance (T‐X) analog of frequency‐distance (F‐X) prediction‐error filters and as an alternative to T‐X prediction‐error filters. The filters are constructed with the help of an implicit finite‐difference scheme for the local plane‐wave equation. Several synthetic and real data examples show that finite‐difference plane‐wave destruction filters perform well in applications such as fault detection, data interpolation, and noise attenuation.

Time-shift imaging condition in seismic migration

Seismic imaging based on single-scattering approximationisintheanalysisofthematchbetweenthesourceandreceiver wavefields at every image location. Wavefields at depth are functions of space and time and are reconstructed from surface data either by integral methods Kirchhoff migration or by differential methods reverse-time or wavefieldextrapolationmigration.Differentmethodscanbeused toanalyzewavefieldmatching,ofwhichcrosscorrelationisa popular option. Implementation of a simple imaging condition requires time crosscorrelation of source and receiver wavefields, followed by extraction of the zero time lag. A generalized imaging condition operates by crosscorrelation in both space and time, followed by image extraction at zero time lag. Images at different spatial crosscorrelation lags are indicators of imaging accuracy and are also used for imageangle decomposition. In this paper, we introduce an alternative prestack imaging condition in which we preserve multiple lags of the time crosscorrelation. Prestack images are described as functions of time shifts as opposed to space shifts betweensourceandreceiverwavefields.Thisimagingcondition is applicable to migration by Kirchhoff, wavefield extrapolation, or reverse-time techniques. The transformation allows construction of common-image gathers presented as functionsofeithertimeshiftorreflectionangleateverylocationinspace.Inaccuratemigrationvelocityisrevealedbyangle-domain common-image gathers with nonflat events. Computational experiments using a synthetic data set from a complex salt model demonstrate the main features of the method.

Shaping regularization in geophysical-estimation problems

Regularization is a required component of geophysical-estimation problems that operate with insufficient data. The goal of regularization is to impose additional constraints on the estimated model. I introduce shaping regularization, a general method for imposing constraints by explicit mapping of the estimated model to the space of admissible models. Shaping regularization is integrated in a conjugate-gradient algorithm for iterative least-squares estimation. It provides the advantage of better control on the estimated model in comparison with traditional regularization methods and, in some cases, leads to a faster iterative convergence. Simple data interpolation and seismic-velocity estimation examples illustrate the concept.

Poststack velocity analysis by separation and imaging of seismic diffractions

Small geologic features manifest themselves in seismic data in the form of diffracted waves, which are fundamentally different from seismic reflections. Using two field-data examples and one synthetic example, we demonstrate the possibility of separating seismic diffractions in the data and imaging them with optimally chosen migration velocities. Our criteria for separating reflection and diffraction events are the smoothness and continuity of local event slopes that correspond to reflection events. For optimal focusing, we develop the local varimax measure. The objectives of this work are velocity analysis implemented in the poststack domain and high-resolution imaging of small-scale heterogeneities. Our examples demonstrate the effectiveness of the proposed method for high-resolution imaging of such geologic features as faults, channels, and salt boundaries.

Seislet transform and seislet frame

We introduce a digital waveletlike transform, which is tailored specifically for representing seismic data. The transform provides a multiscale orthogonal basis with basis functions aligned along seismic events in the input data. It is defined with the help of the wavelet-lifting scheme combined with local plane-wave destruction. In the 1D case, the seislet transform is designed to follow locally sinusoidal components. In the 2D case, it is designed to follow local plane-wave components with smoothly variable slopes. If more than one component is present, the transform turns into an overcomplete representation or a tight frame. In these terms, the classic digital wavelet transform is simply a seislet transform for a zero frequency (in one dimension) or zero slope (in two dimensions). The main objective of the new transform is an effective seismic-data compression for designing efficient data-analysis algorithms. Traditional signal-processing tasks such as noise attenuation and trace interpolation become simply defined in the seislet domain. When applied in the offset direction on common-midpoint or common-image-point gathers, the seislet transform finds an additional application in optimal stacking of seismic records.

Azimuth moveout for 3-D prestack imaging

We introduce a new partial prestack‐migration operator called “azimuth moveout” (AMO) that rotates the azimuth and modifies the offset of 3-D prestack data. Followed by partial stacking, AMO can reduce the computational cost of 3-D prestack imaging. We have successfully applied AMO to the partial stacking of a 3-D marine data set over a range of offsets and azimuths. When AMO is included in the partial‐stacking procedure, high‐frequency steeply dipping energy is better preserved than when conventional partial‐stacking methodologies are used. Because the test data set requires 3-D prestack depth migration to handle strong lateral variations in velocity, the results of our tests support the applicability of AMO to prestack depth‐imaging problems. AMO is a partial prestack‐migration operator defined by chaining a 3-D prestack imaging operator with a 3-D prestack modeling operator. The analytical expression for the AMO impulse response is derived by chaining constant‐velocity DMO with its inverse. Equivalentl...

Predictive painting of 3D seismic volumes

Predictive painting is a numerical algorithm that spreads information in 3D volumes according to the local structure of seismic events. The algorithm consists of two steps. First, local spatially variable inline and crossline slopes of seismic events are estimated by the plane-wave-destruction method. Next, a seed trace is inserted in the volume, and the information contained in that trace is spread inside the volume, thus automatically “painting” the data space. Immediate applications of this technique include automatic horizon picking and flattening in applications to both prestack and poststack seismic data analysis. Synthetic and field data tests demonstrate the effectiveness of predictive painting.

Adaptive multiple subtraction using regularized nonstationary regression

Stationary regression is the backbone of seismic data-processing algorithms including match filtering, which is commonly applied for adaptive multiple subtraction. However, the assumption of stationarity is not always adequate for describing seismic signals. I have developed a general method of nonstationary regression and that applies to nonstationary match filtering. The key idea is the use of shaping regularization to constrain the variability of nonstationary regression coefficients. Simple computational experiments demonstrate advantages of shaping regularization over classic Tikhonovs regularization, including a more intuitive selection of parameters and a faster iterative convergence. Using benchmark synthetic data examples, I have successfully applied this method to the problem of adaptive subtraction of multiple reflections.

Guest Editors' Introduction: Reproducible Research

The articles in this special issue provide independent solutions for practical reproducible research systems. The use of Matlab-based tools such as the famous Wavelab and Sparselab packages in promoting reproducible research in computational harmonic analysis has been presented. In particular, the authors point to the success of the reproducible research discipline in increasing the reliability of computational research and reflect on the effort necessary for implementing this discipline in a research group and overcoming possible objections to it. An article also describes a Python interface to the well-known Clawpack package for solving hyperbolic partial differential equations that appear in wave propagation problems. The author argues strongly in favor of reproducible computations and shows an example using a simplified Python interface to Fortran code. An article also represents the field of bioinformatics, which has been a stronghold of reproducible research. It describes the cacher package, which is built on top of the R computing environment. Cacher enables a modular approach to reproducible computations by storing results of intermediate computations in a database. The special issue ends with an article on the legal aspects of reproducible research, including copyright and licensing issues.