Common-shot Migration Research Articles

Surface-offset gathers from elastic reverse time migration and velocity analysis

Angle-domain common-image gathers (ADCIGs) from elastic reverse time migration (ERTM) are valuable tools for seismic elastic velocity estimation. Traditional ADCIGs are based on the concept of common-offset domains, but common-shot domain implementations are often favored for computational cost considerations. Surface-offset gathers (SOGs) built from common-offset migration may serve as an alternative to the common-shot ADCIGs. We have developed a theoretical kinematic framework between these two domains, and we determined that the common SOG gives an alternative measurement of kinematic correctness in the presence of incorrect velocity. Specifically, we exploit analytical expressions for the image misposition between these two domains, with respect to the traveltime perturbation caused by velocity errors. Four formulations of the PP and PS residual moveout functions are derived and provide insightful information of the velocity error, angle, and PS velocity ratio contained in ERTM gathers. The analytical solutions are validated with homogeneous examples with a series of varied parameters. We found that the SOGs may perform in the way of simplicity and linearity as an alternative to the common-shot migration. To make a full comparison with ADCIGs, we have developed a cost-effective workflow of ERTM SOGs. A fast vector P- and S-wave decomposition can be obtained via spatial gradients at selected time steps. A selected ERTM imaging condition is then modified in which the migration is done by offset groups between each source and receiver pair for each P- and S-wave decomposition. Two synthetic (marine and land) examples are used to demonstrate the feasibility of our methods.

Plane-wave migration for steep reflectors imaging

The application of plane-wave migration for the steep reflectors imaging is investigated. Due to the limited aperture, a single common shot migration based on the one-way wave equation is not effective for the dipping structures imaging. Plane-wave migration, which can produce equal or higher quality images with much less computation cost than conventional shot migration, doesn?t suffer from the aperture problems of common-shot records since their recording aperture is the length of the seismic survey. We have found that the corresponding relation between the angle of dipping reflectors and the surface ray parameters P can be estimated from the initial velocity information. On one hand, we improve the wavefield extrapolation accuracy only for common P sections corresponding the selected steep targets, in which way, the imaging accuracy of the selected targets can be improved; on the other hand, the plane-wave integrals can be pruned to concentrate the image on the selected targets, in which way, the computation time can be further reduced. Synthetic left salt body dataset from the BP 2004 velocity benchmark and VTI synthetic dataset from Amerada Hess are used to demonstrate the methods respectively for the isotropic and anisotropic case. Then we extend it to real data of Luzong area in China for steep structures imaging. Synthetic and real data examples show that plane-wave migration generates high-quality images of steeply dipping reflectors with very low computation cost.

Extraction of amplitude-preserving angle gathers based on vector wavefield reverse-time migration

Angle-domain common-image gathers (ADCIGs) transformed from the shotdomain common-offset gathers are input to migration velocity analysis (MVA) and prestack inversion. ADCIGs are non-illusion prestack inversion gathers, and thus, accurate. We studied the extraction of elastic-wave ADCIGs based on amplitude-preserving elastic-wave reversetime migration for calculating the incidence angle of P-and S-waves at each image point and for different source locations. The P-and S-waves share the same incident angle, namely the incident angle of the source P-waves. The angle of incidence of the source P-wavefield was the difference between the source P-wave propagation angle and the reflector dips. The propagation angle of the source P-waves was obtained from the polarization vector of the decomposed P-waves. The reflectors’ normal direction angle was obtained using the complex wavenumber of the stacked reverse-time migration (RTM) images. The ADCIGs of P-and S-waves were obtained by rearranging the common-shot migration gathers based on the incident angle. We used a horizontally layered model, the graben medium model, and part of the Marmousi-II elastic model and field data to test the proposed algorithm. The results suggested that the proposed method can efficiently extract the P-and S-wave ADCIGs of the elastic-wave reverse-time migration, the P-and S-wave incident angle, and the angle-gather amplitude fidelity, and improve the MVA and prestack inversion.

Including lateral velocity variations into true-amplitude common-shot wave-equation migration

In heterogeneous media, standard one-way wave equations describe only the kinematic part of one-way wave propagation correctly. For a correct description of amplitudes, the one-way wave equations must be modified. In media with vertical velocity variations only, the resulting true-amplitude one-way wave equations can be solved analytically. In media with lateral velocity variations, these equations are much harder to solve and require sophisticated numerical techniques. We present an approach to circumvent these problems by implementing approximate solutions based on the one-dimensional analytic amplitude modifications. We use these approximations to show how to modify conventional migration methods such as split-step and Fourier finite-difference migrations in such a way that they more accurately handle migration amplitudes. Simple synthetic data examples in media with a constant vertical gradient demonstrate that the correction achieves the recovery of true migration amplitudes. Applications to the SEG/EAGE salt model and the Marmousi data show that the technique improves amplitude recovery in the migrated images in more realistic situations.

Reply to the discussion

We thank Dr. Stork and Mr. Kapoor for their interest and comments regarding our paper, and while we think that Stork and Kapoor raise interesting points on data decimation before prestack wave-equation migration, we disagree with their assertion that our key equation “is based on the wrong sampling criterion.” We address some of their issues below. First, although we demonstrate that delayed-shot migration can be significantly more efficient than common-shot migration, we do not “claim that delayed-shot WEM is mathematically more efficient than standard-shot WEM migration” in general. In fact, we can easily think of cases where common-shot migration is more efficient than delayed-shot migration.

The Theory of True Amplitude One‐Way Wave Equation Migrations

AbstractIn this technical report, we review the recent development of true amplitude one‐way wave equation migration and summarize the migrated amplitude performance from different one‐way wave equations. To produce correct amplitude from postack phase‐shift migration, we have to apply accurate geometrical spreading compensation in preprocessing. Although Dubrulle's common‐offset phase‐shift migration gives correct imaging position, it is not a true amplitude migration except for zero offset or flat reflectors. We have shown that the conventional common‐shot one‐way wave equation migration does not produce correct migration amplitude. Based on the true amplitude one‐way wave equations and the surface boundary condition corrections, we have developed true amplitude common‐shot migration and have proved that it produces an inversion output that agrees asymptotically to Kirchhoff inversion. In the next step, we propose a way to obtain true amplitude common‐angle image gathers from both single‐square‐root and double‐square‐root wave equation migrations. Our analysis indicates that the imaging condition is a good candidate to produce true amplitude angle domain common imaging gathers. Compared to the pU/pD imaging condition which is required for true amplitude common‐shot migration, the new approach suggests a more stable way of doing inversion and is more attractive to the real seismic data processing.

AVO inversion based on common shot migration

Conventional AVO inversion utilizes the trace amplitudes of CMP gathers. There are three main factors affecting the accuracy of the inversion. First, CMP gathers are based on the hypothesis of horizontal layers but most real layers are not horizontal. Greater layer dip results in a greater difference between the observed CMP gathers and their real location. Second, conventional processing flows such as NMO, DMO, and deconvolution will distort amplitudes. Third, the formulation of reflection coefficient is related to incidence angles and it is difficult to get the relationship between amplitude and incidence angle. Wave equation prestack depth migration has the ability of imaging complex media and steeply dipping layers. It can reduce the errors of conventional processing and move amplitudes back to their real location. With true amplitude migration, common angle gathers abstraction, and AVO inversion, we suggest a method of AVO inversion from common shot gathers in order to reduce the effect of the above factors and improve the accuracy of AVO inversion.

Delayed-shot 3D depth migration

For 3D seismic imaging in structurally complex areas, the use of migration by wavefield extrapolation has become widespread. By its very nature, this family of migration methods operates on data sets that satisfy a wave equation in the context of a single, physically realizable field experiment, such as a common-shot record. However, common-shot migration of data recorded over dipping structures requires a migration aperture much larger than the recording aperture, resulting in extra computations. A different type of wave-equation record, the response to a linear or planar source, can be synthesized from all the common-shot records. Synthesizing these records from common-shot records involves slant-stack processing, or applying delays to the various shots; we call these records delayed-shot records. Delayed-shot records don't suffer from the aperture problems of common-shot records since their recording aperture is the length of the seismic survey. Consequently, delayed-shot records hold potential for efficient, accurate imaging by wavefield extrapolation. We present a formulation of delayed-shot migration in 2D and 3D (linear sources) and its application to 3D marine streamer data. This formulation includes a discussion of sampling theory issues associated with the formation of delayed-shot records. For typical marine data, 2D and 3D delayed-shot migration can be significantly more efficient than common-shot migration. Synthetic and real data examples show that delayed-shot migration produces images comparable to those from common-shot migration.

Theory of true-amplitude one-way wave equations and true-amplitude common-shot migration

One-way wave operators are powerful tools for forward modeling and migration. Here, we describe a recently developed true-amplitude implementation of modified one-way operators and present some numerical examples. By “true-amplitude” one-way forward modeling we mean that the solutions are dynamically correct as well as kinematically correct. That is, ray theory applied to these equations yields the upward- and downward-traveling eikonal equations of the full wave equation, and the amplitude satisfies the transport equation of the full wave equation. The solutions of these equations are used in the standard wave-equation migration imaging condition. The boundary data for the downgoing wave is also modified from the one used in the classic theory because the latter data is not consistent with a point source for the full wave equation. When the full wave-form solutions are replaced by their ray-theoretic approximations, the imaging formula reduces to the common-shot Kirchhoff inversion formula. In this sense, the migration is true amplitude as well. On the other hand, this new method retains all of the fidelity features of wave equation migration. Computer output using numerically generated data confirms the accuracy of this inversion method. However, there are practical limitations. The observed data must be a solution of the wave equation. Therefore, the data must be collected from a single common-shot experiment. Multiexperiment data, such as common-offset data, cannot be used with this method as presently formulated.

Methods for wave equation prestack depth migration and numerical experiments

In this paper the methods of wave theory based prestack depth migration and their implementation are studied. Using the splitting of wave operator, the wavefield extrapolation equations are deduced and the numerical schemes are presented. The numerical tests for SEG/EAEG model with MPI are performed on the PC-cluster. The numerical results show that the methods of single-shot (common-shot) migration and synthesized-shot migration are of practical values and can be applied to field data processing of 3D prestack depth migration.

Finite difference 3D common-shot migration of data from the West Delta area of the Gulf of Mexico

The advent of the network, or “cluster,” of modern personal computers has enabled prestack 3D “wave equation” methods, such as common-shot migration, to become practical alternatives to standard Kirchhoff depth migration. Clusters have overcome hardware limitations (cost-effective memory and speed) that previously limited the widespread use of these wave equation algorithms. Prestack Kirchhoff migration is efficient and easily adapts to various data acquisition geometries, while its steep-dip and high-resolution imaging capabilities are unparalleled. Traveltime calculations, typically by ray tracing, are fundamental to conventional Kirchhoff migration. However, propagating wavefronts become complicated when traveling through complex geology; for example, a simple spherical wave propagating near its source eventually separates into several (or many) distinct “branches” as the wave encounters large lateral velocity variations. Computing 3D traveltime fields that faithfully represent such wave propagation and actually using these in 3D Kirchhoff migration is difficult and computationally expensive. Conventional Kirchhoff migration instead uses simplified, single-valued traveltime fields produced by criteria such as first arrival or maximum energy. Where several raypaths encounter a grid node, for example, the maximum-energy criterion selects the traveltime from the raypath with largest amplitude. The single-valued traveltime approximation may be poor in complex geology, where the traveltime fields really are multibranched or multivalued. Prestack wave equation imaging, on the other hand, such as finite difference common-shot migration, uses numerical modeling of the one-way wave equation instead of ray tracing. These wave equation methods can be more accurate than ray methods for modeling wave propagation in complex media. The result can be a superior image in areas of complex geology, such as beneath salt bodies. The most serious limitation of these one-way wave equation methods is that they are dip-limited relative to Kirchhoff methods, especially for strong lateral velocity variations. Most of today's prestack wave equation methods use accurate “dual-domain” techniques …