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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.

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... We also apply a related wavefield interpolation as described by Fu (2004) to enable image output at finer depth sampling than the extrapolation step. As pointed out in Zhang et al. (2003) there is an often neglected operator aliasing effect in prestack wave-equation migration. We correct for this effect by generating output at half of the receiver interval, so that the spatial Nyquist wavenumber for the image is twice that of the extrapolated wavefields. ...

Prestack wave-equation migration of isotropic or anisotropic elastic seismic data is described as vector wavefield extrapolation, plus an imaging condition for combinations of shot and receiver wave-modes. For azimuthally anisotropic data, the effect is to combine the (normally separate) steps of shear-wave splitting correction and migration into a single migration step. This enables a more accurate correction of the shear waves based upon the local propagation direction. The algorithm is extended to laterally varying medium with two different forms of generalized phase shift operators. The first, which we call "phase shift plus adaptive windowing" (PSPAW), is appropriate for anisotropic media described by several parameters. The second, based on conventional phase shift plus interpolation (PSPI), has been formulated for isotropic media, but is computationally intractable for general anisotropic media. In both cases, the spatial interpolation methodology is applied both to the phase shift and to the modal decomposition and recomposition steps. The PSPAW algorithm has been applied to modelled data, first for a faulted isotropic model, and then for a model with a faulted layer which is transversely isotropic with a horizontal symmetry axis (HTI). The anisotropic elastic migration unravels the effect of shear-wave splitting as a natural consequence, a task which we show isotropic migration fails to do. The isotropic PSPI algorithm has recently been applied to a new elastic version of the well-known Marmousi model, to test the ability of this algorithm with highly variable media. The preliminary results are encouraging, especially for the shallow imaging of the converted wave data.

... The effect of coherent noise by induced multipath on the back-propagation result was examined in several studies on imaging techniques. 35,36) The spatial aliasing causes the blurring of the imaging result of back-propagation when the spacing between the receivers is not sufficiently narrower than the wavelength. 36) The effect of aliasing in time reversal communication is hardly mentioned. ...

In this study, the performance of passive time reversal (PTR) communication techniques in multipath rich underwater acoustic environments is investigated. It is recognized empirically and qualitatively that a large number of multipath arrivals could generally raise the demodulation result of PTR. However, the relationship between multipath and the demodulation result is hardly evaluated quantitatively. In this study, the efficiency of the PTR acoustic communication techniques for multipath interference cancelation was investigated quantitatively by applying a PTR-DFE (decision feed-back filter) scheme to a synthetic dataset of a horizontal underwater acoustic channel. Mainly, in this study, we focused on the relationship between the signal-to-interference ratio (SIR) of datasets and the output signal-to-noise ratio (OSNR) of demodulation results by a parametric study approach. As a result, a proportional relation between SIR and OSNR is confirmed in low-SNR datasets. It was also found that PTR has a performance limitation, that is OSNR converges to a typical value depending on the number of receivers. In conclusion, results indicate that PTR could utilize the multipath efficiently and also withstand the negative effects of multipath interference at a given limitation.

... Aliasing can arise in wave-equation migration during the application of the imaging condition even when propagating wavefields are alias free. Imaging condition aliasing, as discussed by Zhang et al. (2003), occurs when the image space is inappropriately discretized. We address an aliasing problem that arises when, during the application of the imaging condition, wavenumbers improperly map into the image-space due to sampling changes of the data axes. ...

With the widespread adoption of wavefield continuation methods for prestack migration, the concept of operator aliasing warrants revisiting. While zero-offset migration is unaffected , prestack migrations reintroduce the issue of operator aliasing. Some situations where this problem arises include subsampling the shot-axes to save shot-profile migration costs and limited cross-line shot locations due to acquisition strategies. These problems are overcome in this treatment with the use of an appropriate source function or band-limiting the energy contributing to the image. We detail a synthetic experiment that shows the ramifications of subsampling the shot axis and the efficacy of addressing the problems introduced with our two approaches. Further, we explain how these methods can be tailored in some situations to include useful energy residing outside of the Nyquist limits.

The sampling theorem requires minimum two points per wavelength to ensure the proper reconstruction of a wavefield. Violation of this limit will result in data aliasing. With the increased demand for high-frequency seismic processing, data recorded in a field may not have sufficient spatial sampling. Reverse time migration (RTM) using a conventional scheme will not be able to produce a high-quality image from such an aliased input data set. However, when the wavefield and its gradients are available, the generalized sampling theorem relaxes the requirement to only one point per wavelength. Stereo-modeling methods use a system of wave equations to compute a vector containing the particle displacement and its spatial gradients in wavefield propagation. Therefore, in cases in which a wavefield and its spatial gradients are available, such as a multicomponent data set, RTM using the stereo-modeling methods can be potentially much more accurate and efficient than those commonly used ones, such as Lax-Wendroff correction. We investigated the merits of one such method, called the nearly analytic central difference, in RTM to produce high-quality images. Numerical experiments on different data sets produce consistent results with the generalized sampling theorem, which indicates that the stereo-modeling method can yield high-quality images from insufficiently sampled data compared with conventional finite-difference methods.

Velocity modeling and imaging of salt domes are bottlenecks in subsalt reservoir exploration. Because of large velocity variance between salt domes and surrounding rocks and large thickness changing in spatial direction, many problems are present in subsalt seismic exploration, such as seismic wave filed complexity, and structural distortion in time domain. According to the complex geological characters of Area H, we can learn more about the subsalt reservoirs through technological innovation. As to the difficulties of salt velocity modeling, this paper proposes an idea called "multi-information constrained layer-bound solid modeling technique", which uses sequential Gauss simulation and CVI method to solve velocity anomalies in subsalt structure, and can remarkably improve the precision of velocity modeling. As to the complex subsalt structure imaging, through studying based on the finite difference method, this paper proposes a precise and efficient finite difference scheme for reverse-time wave field extrapolation. Based on GPU/CPU cooperating system, we extrapolate the wave field forwardly and reversely in time domain that needs the maximum amount of calculation through GPU. This paper adopts reverse time migration in depth domain, which can precisely image the reflected waves from steep dip (even exceeding 90 degrees in the adjacent beds of a salt dome).By the theoretical test of salt dome model, we verify the correctness of the algorithms and methods mentioned above. Through the reverse time migration method, we solve the problem of imaging sub-salt structure and the adjacent beds, so we can migrate the salt dome and the adjacent beds to their true positions, and eliminate structure distortion of underlying beds in time domain due to the velocity anomaly of the salt dome. So fnianlly we can get precise migration results of sub-salt structure in depth domain.

This paper describes least-squares reverse-time migration. The method provides the exact adjoint operator pair for solving the linear inverse problem, thereby enhancing the convergence of gradient-based iterative linear inversion methods. In this formulation, modified source wavelets are used to correct the source signature imprint in the predicted data. Moreover, a roughness constraint is applied to stabilise the inversion and reduce high-wavenumber artefacts. It is also shown that least-squares migration implicitly applies a deconvolution imaging condition. Three numerical experiments illustrate that this method is able to produce seismic reflectivity images with higher resolution, more accurate amplitudes, and fewer artefacts than conventional reverse-time migration. The methodology is currently feasible in 2-D and can naturally be extended to 3-D when computational resources become more powerful.

We develop an anti-Aliasing filter for reverse-time migration (RTM). It is similar to the traditional antialiasing filter used for Kirchhoff migration in that it low-pass filters the migration operator so that the dominant wavelength in the operator is greater than two times the trace sampling interval, except it is applied to both primary and multiple reflection events. Instead of applying this filter to the data in the traditional RTM operation, we apply the anti-Aliasing filter to the generalized diffraction-stack migration operator. This gives the same migration image as computed by anti-Aliased RTM.

This paper describes least squares reverse-time migration in a matrix-based formulation providing the exact adjoint operator pair for solving the linear inverse problem and thereby enhancing the convergence of gradient-based iterative linear inversion methods. In this formulation, modified source wavelets are used to correct the source signature imprint in the predicted data. Moreover, a roughness constraint is applied to stabilize the inversion and reduce high-wavenumber artefacts. It is verified by the mathematical proof provided that a deconvolution imaging condition is implicitly applied in least squares migration. Three numerical experiments illustrate that this new formulation is able to produce seismic reflectivity images with higher resolution, more accurate amplitudes, and fewer artefacts than conventional reverse-time migration. The methodology is currently feasible in 2-D, but as computational restraints decrease in the future, it will naturally extend to the 3-D application.

Recursive Kirchhoff wavefield extrapolation in the space-frequency domain can be thought of as a simple convolutional filter that calculates a single output point at depth z+dz using a weighted summation of all input points within the extrapolator aperture at depth z. The desired velocity values for the extrapolator are the ones that provide the best approximation of the true phase (propagation time) of the seismic wavefield between the input points and the output point. Recursive Kirchhoff extrapolators can be designed to handle lateral variations in velocity in a number of ways: a PSPI-type extrapolator uses only the velocity at the output point, a NSPS-type extrapolator uses the velocities at the input points; a SNPS-type extrapolator incorporates two extrapolation steps of dz/2 where the first step uses the velocities at the input points (NSPS-type) and the second step uses the velocity at the output point (PSPI-type); while the Weyl-type extrapolator uses an average of the velocities between each input point and the output point. Here, we introduce the PAVG-type extrapolator, which uses velocity values calculated by an average of slowness along straight raypaths between each input point and the output point. A simple synthetic with a lateral step in velocity shows that the PAVG Kirchhoff extrapolator is very close to the exact desired response. Tests using the Marmousi synthetic data set suggest that the extrapolator behaviour is only one of many considerations that must be addressed for accurate depth imaging. Other important considerations include preprocessing, aperture size, taper width, and imaging condition.

We propose a wave equation based migration scheme for imaging the steep dipping structures with 3-D sparsely sampled dataset, especially sparsely sampled in the crossline direction. The scheme is developed based on the hybrid domain (wavenumber and space) wave equation based migration methods. The problem for imaging the steep dipping structures can be solved by applying anti-alias one-way propagator together with filling zero traces into the original sparsely sampled data (to generate dataset with a sufficient sampling). The proposed scheme overcomes a main difficulty when applying wave equation based migration methods to the field data. As a result, good imaging can be obtained for complex structures by using wave equation based migration methods. The proposed scheme is demonstrated by 2-D Marmousi dataset and 3-D field data. The proposed scheme will be much helpful to the current 3-D dataset.

We propose a frequency-dependent varying-step depth extrapolation scheme and a table-driven, one point wavefield interpolation technique for the wave equation based migration methods. The former reduces the computational cost of wavefield depth extrapolation, and the latter reconstructs the extrapolated wavefield with an equal, desired vertical step with high computational efficiency. The proposed varying-step depth extrapolation plus one-point interpolation scheme results in 2/3 reduction in omputational cost when compared to the conventional equal-step depth extrapolation of wavefield, but gives the almost same imaging. We present the scheme using the optimum split-step Fourier method on the 2-D Marmousi dataset and 3-D field dataset. The results demonstrate the high computational efficiency of the scheme in the absence of loss of accuracy. The proposed scheme can also be used by other wave equation based migration methods of the frequency domain.

We reformulate the equation of reverse-time migration so that it can be interpreted as summing data along a series of hyperbola-like curves, each one representing a different type of event such as a reflection or multiple. This is a generalization of the familiar diffraction-stack migration algorithm where the migration image at a point is computed by the sum of trace amplitudes along an appropriate hyperbola-like curve. Instead of summing along the curve associated with the primary reflection, the sum is over all scattering events and so this method is named generalized diffraction-stack migration. This formulation leads to filters that can be applied to the generalized diffraction-stack migration operator to mitigate coherent migration artefacts due to, e.g., crosstalk and aliasing. Results with both synthetic and field data show that generalized diffraction-stack migration images have fewer artefacts than those computed by the standard reverse-time migration algorithm. The main drawback is that generalized diffraction-stack migration is much more memory intensive and I/O limited than the standard reverse-time migration method.

To avoid spatial aliasing problems in broad band high resolution seismic sections, I present a high density migration processing solution. I first analyze the spatial aliasing definition for stack and migration seismic sections and point out the differences between the two. We recognize that migration sections more often show spatial aliasing than stacked sections. Second, from wave propagation theory, I know that migration output is a new spatial sampling process and seismic prestack time migration can provide the high density sampling to prevent spatial aliasing on high resolution migration sections. Using a 2D seismic forward modeling analysis, I have found that seismic spatial aliasing noise can be eliminated by high density spatial sampling in prestack migration. In a 3D seismic data study for Daqing Oilfield in the Songliao Basin, I have also found that seismic sections obtained by high-density spatial sampling (10 × 10 m) in prestack migration have less spatial aliasing noise than those obtained by conventional low density spatial sampling (20 × 40 m) in prestack migration.

Spatial aliasing is an important issue in seismic migration. In this paper, we discuss anti-aliasing schemes in the wavefield continuation which is a wave-equation-based migration method. We describe the difference in interpolation applied pre and post the imaging condition, and compare the effectiveness of the linear interpolation and the sinc function interpolation. Our conclusion is that a five-sample sinc function interpolation applied before the imaging condition is an optimal anti-aliasing scheme according to theoretical and application analysis. We then apply the 3D Born approximation pre-stack depth migration method with optimal anti-aliasing interpolation to a 80 km2 seismic dataset. The migration result shows that the method we propose is better than a conventional wave equation method (not applied anti-spatial aliasing) in the definition of reflection events.

A significant degradation in the quality of Kirchhoff 3-D migration images often arises because the migration operator summation trajectory is too steep for the input seismic trace spacing and frequency content. We present an operator anti-aliasing method that suppresses this problem, based on local triangle filtering. The N-point anti-alias triangles are efficiently applied as 3-point filters after causal and anticausal integration of the seis- mic trace data. We implement our method on a massively parallel CM-5 in a memory and floating-point efficient algorithm, and compare our anti-aliasing method to a standard Kirchhoff migration using a 3-D salt intrusion dataset from the Gulf of Mexico. Our re- sults indicate that our anti-aliasing method greatly enhances the 3-D resolution of steep salt-sediment interfaces and faults, and suppresses false reflections caused by conventional Kirchhoff-migration aliasing artifacts.

Wavefield sampling theory imposes anti-aliasing restrictions on prestack Kirchhoffmigration; less obviously, aliasing problems also affect prestack migrations based on wavefield extrapolation. We investigate these problems, which we call "imaging-condition aliasing," in both common-shot migration and shot-receiver migration. To remedy them, we propose spatial interpolation of the downward-continued wavefields before applying the imaging condition.

Schemes for seismic mapping of reflectors in the presence of an arbitrary velocity model, dipping and curved reflectors, diffractions, ghosts, surface elevation variations, and multiple reflections are reviewed and reduced to a single formula involving up and downgoing waves. The mapping formula may be implemented without undue complexity by means of difference approximations to the relativistic Schroedinger equation.

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.

Migration anti-aliasing is accomplished trivially for frequency-wavenumber (f-k) migration, but not so for Kirchhoff migration. Although Kirchhoff migration anti-aliasing can be applied reasonably efficiently, some doubt persists as to the correct criterion for applying it in 3-D. Here, we derive anti-aliasing from first principles, showing that the Kirchhoff integral (of seismic data that are an appropriate continuously sampled version of the actual discretely sampled input traces) is naturally anti-aliased, with a filter that can be applied to the Kirchhoff sum of the actual data. This leads to an unambiguous criterion for 3-D anti-aliasing, and it provides a unified framework for discussing several other anti-aliasing issues, such as anti-aliasing in the image space and in beam migration. We also discuss briefly the effect of anti-aliasing on migrated amplitudes, as a function of both incidence angle and azimuth angle.

This paper addresses some practical aspects of an-tialiasing in Kirchhoff migration. We show that the effec-tive trace spacing related to the aliasing-frequency limits for directions between the in-line and cross-line direc-tions is smaller than values previously used and that the frequency limits for antialiasing may be increased. We present a corrected expression for the migration opera-tor spatial derivative for 3-D time migration that avoids the overfiltering at nonzero offsets produced by a previ-ous approximation. Simple operator truncation is shown to reduce aliasing noise; it is inexpensive but may lead to errors in interpretation. Triangle filtering and Gray's method both produce similar results, but the relative costs of these two methods will vary with the size of the migration and the computational environment. Gray's method takes more setup time for each input trace than does triangle filtering, but Gray's method might be more cost effective when each input trace contributes to a large number of output traces.

Anti-aliasing methods in Kirchhoff migration: Geophysics

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Theory of migration anti-aliasing: 71st Ann

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