Angle-domain Common-image Gathers Research Articles

Stacking angle-domain common-image gathers for normalization of illumination

Unequal illumination of the subsurface highly impacts the quality of seismic imaging. Different image points receive different folds of reflection-angle illumination, which can be caused by irregular acquisition or by wave propagation in complex media. Illumination problems can deteriorate amplitudes in migrated images. To address this problem, we present a method of stacking angle-domain common-image gathers, in which we use local similarity with soft thresholding to determine the folds of local illumination. Normalization by local similarity regularizes local illumination of reflection angles for each image point of the subsurface model. This approach compensates for irregular illumination by selective stacking in the image space, regardless of the cause of acquisition or propagation irregularities. Additional migration is not required because the methodology is implemented in the reflection angle domain after migration. We use two synthetic examples to demonstrate that our method can normalize migration amplitudes and effectively suppress migration artefacts.

Residual curvature migration velocity analysis for angle domain common imaging gathers

Pre-stack depth migration velocity analysis is one of the keys to influencing the imaging quality of pre-stack migration. In this paper we cover a residual curvature velocity analysis method on angle-domain common image gathers (ADCIGs) which can depict the relationship between incident angle and migration depth at imaging points and update the migration velocity. Differing from offset-domain common image gathers (ODCIGs), ADCIGs are not disturbed by the multi-path problem which contributes to imaging artifacts, thus influencing the velocity analysis. On the basis of horizontal layers, we derive the residual depth equation and also propose a velocity analysis workflow for velocity scanning. The tests to synthetic and field data prove the velocity analysis methods adopted in this paper are robust and valid.

Angle-domain common-image gathers in generalized coordinates

The theory of angle-domain common-image gathers (ADCIGs) is extended to migrations performed in generalized 2D coordinate systems. I have developed an expression linking the definition of reflection opening angle to differential traveltime operators and spatially varying weights derived from the non-Cartesian geometry. Generalized-coordinate ADCIGs can be calculated directly using Radon-based offset-to-angle approaches for coordinate systems satisfying the Cauchy-Riemann differentiability criteria. The canonical examples of tilted-Cartesian, polar, and elliptical coordinates can be used to illustrate the ADCIG theory. I have compared analytically and numerically generated image volumes for a set of elliptically shaped reflectors. Experiments with a synthetic data set showed that elliptical-coordinate ADCIGs better resolve the reflection opening angles of steeply dipping structure, relative to conventional Cartesian image volumes, because of improved large-angle propagation and enhanced sensitivity to steep structural dips afforded by coordinate system transformations.

Residual depth moveout and velocity determination for parametric media

For the case when prestack depth migration has been performed with inaccurate velocities, it can be shown that second-order approximations can be used to predict the residual depth moveout of common-image and angle-domain common-image gathers. Corrections may be made in terms of offset and common-reflection angles that can be estimated from the migrated data using moveout analysis schemes. The residual moveout determined can be used to improve the signal-to-noise ratio of the migrated data and to determine the variation of parameters of well-known formation-dependent velocity laws away from borehole positions. Although the method is developed for small offsets, reflection angles, and velocity differences, realistic examples may be used to demonstrate that satisfactory approximations to the residual moveout curve can be obtained over a relatively wide range of offsets or reflection angles and for moderate velocity differences. A zero-offset inversion was applied to the estimated residual radii of curvature. This inversion was performed layerwise. For each layer, the velocity was assumed to vary according to a known depth-varying and a laterally varying component to be determined. Iterative applications of the suggested approach have rendered satisfactory results for several representative velocity models. For each application, appropriate initial models were provided. Applications to initial models with large velocity differences and to seismic data with large offsets have to be investigated.

Regularization in slope tomography

Seismic imaging in depth is limited by the accuracy of velocity model estimation. Slope tomography uses the slowness components and traveltimes of picked reflection or diffraction events for velocity model building. The unavoidable data incompleteness requires additional information to assure stability to inversion. One natural constraint for ray-based tomography is a smooth velocity model. We propose a new, reflection-angle-based kind of smoothness constraint as regularization in slope tomography and have compared its effects to three other, more conventional constraints. The effects of these constraints were evaluated through angle-domain common-image gathers, computed with wave-equation migration using the estimated velocity model. We found that the smoothness constraints have a distinct effect on the velocity model but a weaker effect on the migrated data. In numerical tests on synthetic data, the new constraint leads to geologically more consistent models.

Wave-equation angle-domain common-image gathers for converted waves

Wavefield-extrapolation methods can produce angle-domain common-image gathers (ADCIGs). To obtain ADCIGs for converted-wave seismic data, information about the image dip and the P-to-S velocity ratio must be included in the computation of angle gathers. These ADCIGs are a function of the half-aperture angle, i.e., the average between the incidence angle and the reflection angle. We have developed a method that exploits the robustness of computing 2D isotropic single-mode ADCIGs and incorporates both the converted-wave velocity ratio [Formula: see text] and the local image dip field. It also maps the final converted-wave ADCIGs into two ADCIGs, one a function of the P-incidence angle and the other a function of the S-reflection angle. Results with both synthetic and real data show the practical application for converted-wave ADCIGs. The proposed approach is valid in any situation as long as the migration algorithm is based on wavefield downward continuation and the final prestack image is a function of the horizontal subsurface offset.

Attenuation of specular and diffracted 2D multiples in image space

In complex areas, the attenuation of specular and diffracted multiples in image space is an attractive alternative to surface-related multiple elimination (SRME) and to data space Radon filtering. We present the equations that map, via wave-equation migration, 2D diffracted and specular water-bottom multiples from data space to image space. We show the equations for both subsurface-offset-domain common-image-gathers (SODCIGs) and angle-domain common-image-gathers (ADCIGs). We demonstrate that when migrated with sediment velocities, the over-migrated multiples map to predictable regions in both SODCIGs and ADCIGs. Specular multiples focus similarly to primaries, whereas diffracted multiples do not. In particular, the apex of the residual moveout curve of diffracted multiples in ADCIGs is not located at the zero aperture angle. We use our equation of the residual moveout of the multiples in ADCIGs to design an apex-shifted Radon transform that maps the 2D ADCIGs into a 3D model space cube whose dimensions are depth, curvature, and apex-shift distance. Well-corrected primaries map to or near the zero-curvature plane and specularly reflected multiples map to or near the zero apex-shift plane. Diffracted multiples map elsewhere in the cube according to their curvature and apex-shift distance. Thus, specularly reflected as well as diffracted multiples can be attenuated simultaneously. We show the application of our apex-shifted Radon transform to a 2D seismic line from the Gulf of Mexico. Diffracted multiples originate at the edges of the salt body and we show that we can successfully attenuate them, along with the specular multiples, in the image Radon domain.

Residual moveout in anisotropic angle-domain common-image gathers

I develop the fundamental concepts for quantitatively relating perturbations in anisotropic parameters to the corresponding reflector movements in angle-domain common-image gathers (ADCIGs) after anisotropic wavefield-continuation migration. The proposed theory potentially enables the application of residual moveout (RMO) analysis of ADCIGs to velocity estimation in realistic anisotropic conditions. I demonstrate that linearization of the relationship between anisotropic velocity parameters and reflector movements can be derived by assuming stationary raypaths. This assumption leads to a fairly simple analytical derivation. I then apply the general method to the particular case of RMO analysis of reflections from flat reflectors in a vertical transverse isotropic (VTI) medium. This analysis yields expressions to predict RMO curves in migrated ADCIGs. These RMO expressions are functions of both the phase aperture angle and the group aperture angle. Several numerical examples demonstrate the accuracy of the RMO curves predicted by my kinematic analysis. The synthetic examples also show that approximating the group angles with the phase angles in the application of the RMO expressions may lead to substantial errors for events reflected at wide aperture angles. The results obtained by migrating a 2D line extracted from a Gulf of Mexico 3D data set confirm the accuracy of the proposed method. The RMO curves predicted by the theory match the RMO function observed in the ADCIGs computed from the real data.

Angle-domain common-image gathers from anisotropic migration

I present a general methodology for computing angle-domain common-image gathers (ADCIGs) in conjunction with anisotropic wavefield-continuation migration. The method is based on transforming the prestack image from the subsurface-offset domain to the angle domain using slant stacks. The processing sequence is the same as that for computing ADCIGs for the isotropic case, though the interpretation of the relationship between the slopes measured in the prestack image and the aperture angles is more complex. I demonstrate that the slopes measured by performing slant stacks along the subsurface-offset axis of the prestack image provide a good approximation of the phase aperture angles, and they are exactly equal to the phase aperture angles for flat reflectors in vertical transversly isotropic (VTI) media. In the general case of dipping reflectors, the angles computed using slant stacks can be easily corrected by applying the relationships that I present in this paper, and the accurate aperture angles can be determined as a function of the reflector dip and anisotropic slowness at the reflector. I derive these relationships from both plane-wave and ray viewpoints. This theoretical development links the kinematics in ADCIGs with migration-velocity errors. I apply the proposed method to compute ADCIGs from the prestack image obtained by anisotropic migration of a 2D line recorded in the Gulf of Mexico. I analyze the error introduced by neglecting the difference between the true phase aperture angle and the angle computed through slant stacks, showing that, at least for this data set, these errors are negligible and can be safely ignored. In contrast, group aperture angles can be quite different from phase aperture angles; thus, ignoring the distinction between these two angles can be detrimental to practical applications.

True-amplitude, angle-domain, common-image gathers from one-way wave-equation migrations

True-amplitude wave-equation migration provides a quality migrated image of the earth’s interior. In addition, the amplitude of the output provides an estimate of the angular-dependent reflection coefficient, similar to the output of Kirchhoff inversion. Recently, true-amplitude wave-equation migration for common-shot data has been proposed to generate amplitude-reliable, shot-domain, common-image gathers in heterogeneous media. We present a method to directly produce angle-domain common-image gathers from both common-shot and shot-receiver wave-equation migration. Generating true-amplitude, shot-domain, common-image gathers requires a deconvolution-type imaging condition using the ratio of the upgoing and downgoing wavefield, each downward-projected to the image point. Producing true-amplitude, angle-domain, common-image gathers requires, instead, the product of the upgoing wavefield and the complexconjugate of the downgoing wavefield in the imaging condition. Since multiplication is a more stable computational process than division, the new methods proposed provide more stable ways of inverting seismic data. Furthermore, the resulting common-image gathers can be directly used for migrated amplitude-variation-with angle analysis and tomography-based velocity analysis. Shot-receiver wave-equation migration requires new true-amplitude, one-way wave equations with one depth variable and transverse variables for the coordinates corresponding to sources and receivers, hence, two transverse coordinates in 2D and four transverse coordinates in 3D. We propose a modified double-square-root one-way wave equation to produce true amplitude common-image angle gathers. We also demonstrate the new methods with some synthetic examples. Some numerical examples show that the new methods we propose give better amplitude performance on the migrated angle gathers.

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.

Multiple attenuation in the image space

Multiples can be suppressed in the angle-domain image space after migration. For a given velocity model, primaries and multiples have different angle-domain moveout and, therefore, can be separated using techniques similar to the ones employed in the data space prior to migration. We use Radon transforms in the image space to discriminate between primaries and multiples and employ accurate functions describing angle-domain moveouts. Since every individual angle-domain common-image gather incorporates complex 3D propagation effects, our method has the advantage of working with 3D data and complicated geology. Therefore, our method offers an alternative to the more expensive surface-related multiple-elimination (SRME) approach operating in the data space. Radon transforms are cheap but not necessarily ideal for separating primaries and multiples, particularly at small angles where the moveout discrepancy between the two kinds of events are not large. Better techniques involving signal/noise separation using prediction-error filters can be employed as well, although such approaches fall outside the scope of this paper. We demonstrate, using synthetic and real data examples, the power of our method in discriminating between primaries and multiples after migration by wavefield extrapolation, followed by transformation to the angle domain.

Wave‐equation migration velocity analysis. I. Theory

ABSTRACTWe present a migration velocity analysis (MVA) method based on wavefield extrapolation. Similarly to conventional MVA, our method aims at iteratively improving the quality of the migrated image, as measured by the flatness of angle‐domain common‐image gathers (ADCIGs) over the aperture‐angle axis. However, instead of inverting the depth errors measured in ADCIGs using ray‐based tomography, we invert ‘image perturbations’ using a linearized wave‐equation operator. This operator relates perturbations of the migrated image to perturbations of the migration velocity. We use prestack Stolt residual migration to define the image perturbations that maximize the focusing and flatness of ADCIGs.Our linearized operator relates slowness perturbations to image perturbations, based on a truncation of the Born scattering series to the first‐order term. To avoid divergence of the inversion procedure when the velocity perturbations are too large for Born linearization of the wave equation, we do not invert directly the image perturbations obtained by residual migration, but a linearized version of the image perturbations. The linearized image perturbations are computed by a linearized prestack residual migration operator applied to the background image. We use numerical examples to illustrate how the backprojection of the linearized image perturbations, i.e. the gradient of our objective function, is well behaved, even in cases when backprojection of the original image perturbations would mislead the inversion and take it in the wrong direction.We demonstrate with simple synthetic examples that our method converges even when the initial velocity model is far from correct. In a companion paper, we illustrate the full potential of our method for estimating velocity anomalies under complex salt bodies.

Angle‐domain common‐image gathers for migration velocity analysis by wavefield‐continuation imaging

We analyze the kinematic properties of offset‐domain common image gathers (CIGs) and angle‐domain CIGs (ADCIGs) computed by wavefield‐continuation migration. Our results are valid regardless of whether the CIGs were obtained by using the correct migration velocity. They thus can be used as a theoretical basis for developing migration velocity analysis (MVA) methods that exploit the velocity information contained in ADCIGs.We demonstrate that in an ADCIG cube, the image point lies on the normal to the apparent reflector dip that passes through the point where the source ray intersects the receiver ray. The image‐point position on the normal depends on the velocity error; when the velocity is correct, the image point coincides with the point where the source ray intersects the receiver ray. Starting from this geometric result, we derive an analytical expression for the expected movements of the image points in ADCIGs as functions of the traveltime perturbation caused by velocity errors. By applying this analytical result and assuming stationary raypaths (i.e., small velocity errors), we then derive two expressions for the residual moveout (RMO) function in ADCIGs. We verify our theoretical results and test the accuracy of the proposed RMO functions by analyzing the migration results of a synthetic data set with a wide range of reflector dips.Our kinematic analysis leads also to the development of a new method for computing ADCIGs when significant geological dips cause strong artifacts in the ADCIGs computed by conventional methods. The proposed method is based on the computation of offset‐domain CIGs along the vertical‐offset axis and on the “optimal” combination of these new CIGs with conventional CIGs. We demonstrate the need for and the advantages of the proposed method on a real data set acquired in the North Sea.

Common Angle Image Gathers Obtained from Gabordaubechies Beamlet Prestack Depth Migration

AbstractDue to the multi‐pathing of wave propagating through complex media, the problem of reflector location ambiguity is often encountered in the conventional offset‐domain common image gathers (CIGs). Angle‐domain CIGs have been proposed as a solution for such a problem and become a natural choice for velocity analysis, AVA and true amplitude migration/imaging. The development of wave‐theory based seismic migration and imaging methods provides reliable ways for the construction of angle‐domain CIGs of high quality. In particular, the beamlet‐domain wavefield extrapolation and migration method, in which both the wavefield decomposition functions and the extrapolation propagators bear localization properties in both space and direction, has been proved to be an effective tool for angle‐related studies. In this paper, beamlet prestack depth migration based on Gabor‐Daubechies frame decomposition is used for angle‐domain imaging. Through different stack procedures, both common reflection‐angle image gathers (CRAIG) and common dip‐angle image gathers (CDAIG) are constructed from the local angle image matrices. As a numerical example, the CRAIGs and CDAIGs for the SEG‐EAGE 2‐D salt model are calculated and analyzed. Based on their distinct features, the applications of these two CIGs to seismic imaging are discussed in detail.

Complex structural velocity model building for depth imaging

The quality of a migration image depends strongly upon the accuracy of the velocity model. For tomographic velocity analysis, which is most popularly used in the seismic industry, pick errors, null space, velocity ambiguity and intrinsic nonlinearity all have a strong effect on the accuracy of the velocity model. The discussion in this paper is based on a post-migration tomography algorithm which can work on either offset domain common-image gathers (CIGs) or angle domain common-image gathers (CAGs). A large number of depth residuals (hyperbolic and non-hyperbolic) and the structural dip field are automatically picked and high-graded based on the event semblance on the CIGs or CAGs and the coherency along the horizon on post-migrated stack volume. In order to further high-grade the picks, a gamma cube is created which is used for QCing and further high-grading the picks. In the solution of the tomographic equations, a few constraints are used, including a preconditioning operator in data space to reduce the influence of extreme picks and a preconditioning operator in model space with a variable-length controlled direction smoother in order to reduce the null space, speed up the convergence and stabilize the solution, and hard constraints that limit the velocity updates within a certain range. The discussion focuses mainly on how to better represent the velocity model, how to automatically pick a large number of events and structural dip field, how to high-grade the picks based on the event semblance, horizon coherency and filtered gamma cube, how to add proper constraints to the solver, and how to further improve the tomographic inverted velocity model by gamma-based inversion. Usually, the tomographic problem is a highly underdetermined problem; even if we pick as many events as possible and add constraints to the tomographic equations, a perfect result is not guaranteed. In some areas, events are still not flat. In order to further improve the velocity model after tomography, a gamma-based inversion is used. Both numerical and real data tests show that our software package for complex structural velocity model building works well and produces encouraging results. Despite many successful applications of tomographic velocity analysis, challenges still remain for this ill-posed problem. The presence of a null space and pick errors guarantees some level of velocity ambiguity. After discussing some of the challenges facing tomographic velocity analysis, we introduce ways to cope with them.

Equivalence of source‐receiver migration and shot‐profile migration

At first glance, shot-profile migration (Jacobs, 1982) and source-receiver (survey-sinking) migration (Claerbout, 1985) seem to be substantially different algorithms. The basic principles used by the two schemes are different. Shot-profile migration is performed by independently propagating the source wavefield and the receiver wavefield. The image is obtained by crosscorrelating the two wavefields (possibly normalized by the amplitude of the source wavefield). Source-receiver migration is based on the concept of survey sinking, by which we recursively synthesize equivalent data sets at increasing depth. At each depth step, imaging is performed by extracting the wavefield at zero time. Recently, the issue of the relation between shot-profile migration and source-receiver migration has become more relevant because of two important developments: (1) the increase in the practical applications of 3D prestack migration by wavefield continuation for imaging complex structures, and (2) the introduction of methods for computing angle-domain common-image gathers (ADCIG), both for velocity estimation (Biondi and Sava, 1999; Prucha et al., 1999; Clapp and Biondi, 2000) and for amplitude variation with angle (AVA) analysis (de Bruin et al., 1990; Wapenaar et al., 1999; Sava et al., 2001).