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Seismic wave imaging in complex media requires an accurate wavefield simulation method that can accurately describethe wave propagation in realistic media. Reverse time‐depth migration is the preferred method for seismic wave imaging in complex media. Although it is relatively expensive, its imaging accuracy is usually better than migrations based on the ray method. Migration of primary reflection data requires a wave propagation simulation method that can accurately describe primary reflected/scattered wave energy and incorporate anisotropy. Accordingly, we propose the simulation of wave propagation in tilted transversely isotropic media using a 15° one‐way wave equation in a ray‐centred coordinate system, combining the flexibility of ray theory and the accuracy of wave theory. We use this equation to describe the propagation of body waves in a single ray tube, a ‘beam’. The wavefield along the beam, guided by its central raypath, has an angle limit defined only by the ray angle; therefore, wave propagation in complex and steeply dipping media can be simulated with a 15° one‐way wave equation. Numerical experiments show that the simulation results for beam propagation using the 15° equation in the ray‐centred coordinate system have good accuracy. For prestack depth migration in tilted transversely isotropic media, we built a beam imaging method using this propagator, and this migration method yielded accurate images with greater efficiency than reverse time‐depth migration.

Subsurface rocks (e.g. shale) may induce seismic anisotropy, such as transverse isotropy. Traveltime computation is an essential component of depth imaging and tomography in transversely isotropic media. It is natural to compute the traveltime using the wavefront marching method. However, tracking the 3D wavefront is expensive, especially in anisotropic media. Besides, the wavefront marching method usually computes the traveltime using the eikonal equation. However, the anisotropic eikonal equation is highly non-linear and it is challenging to solve. To address these issues, we present a layer-by-layer wavefront marching method to compute the P-wave traveltime in 3D transversely isotropic media. To simplify the wavefront tracking, it uses the traveltime of the previous depth as the boundary condition to compute that of the next depth based on the wavefront marching. A strategy of traveltime computation is designed to guarantee the causality of wave propagation. To avoid solving the non-linear eikonal equation, it updates traveltime along the expanding wavefront by Fermat’s principle. To compute the traveltime using Fermat’s principle, an approximate group velocity with high accuracy in transversely isotropic media is adopted to describe the ray propagation. Numerical examples on 3D vertical transverse isotropy and tilted transverse isotropy models show that the proposed method computes the traveltime with high accuracy. It can find applications in modelling and depth migration.

Least-squares reverse time migration provides better imaging result than conventional reverse time migration by reducing the migration artefacts, improving the resolution of the image and balancing the amplitudes of the reflectors. However, it is computationally intensive. To reduce its computational cost, we propose an efficient amplitude encoding least-squares reverse time migration scheme in the time domain. Although the encoding scheme is effective in increasing the computational efficiency, it also introduces the well-known crosstalk noise in the gradient that degrades the quality of the imaging result. We analyse the cause of the crosstalk noise using an encoding correlation matrix and then develop two numerical schemes to suppress the crosstalk noise during the inversion process. We test the proposed method with synthetic and field data. Numerical examples show that the proposed scheme can provide better imaging result than reverse time migration, and it also generates images comparable with those from common shot least-squares reverse time migration but with less computational cost.

Compared with planar hyperplane, fitting data on the sphere has been an important and an active issue in geoscience, metrology, brain imaging, and so on. In this paper, with the help of the Jackson-type theorem of polynomial approximation on the sphere, we construct spherical feed-forward neural networks to approximate the continuous function defined on the sphere. As a metric, the modulus of smoothness of spherical function is used to measure the error of the approximation, and a Jackson-type theorem on the approximation is established. Copyright © 2011 John Wiley & Sons, Ltd.

Local angular Hessian can be used to improve wave equation least square migration images. By decomposing the original Hessian operator into the local wavenumber domain or the local angle domain, the least square migration image is obtained as the solution of a linearized least-squares inversion in the frequency and local angle domains. The local angular Hessian contains information about the acquisition geometry and the propagation effects based on the given velocity model. The inversion scheme based on the local angular Hessian avoids huge computation on the exact inverse Hessian matrix. To reduce the instability in the inversion, damping factors are introduced into the deconvolution filter in the local wavenumber domain and the local angle domain. The algorithms are tested using the SEG/EAGE salt2D model and the Sigsbee2A model. Results show improved image quality and amplitudes.

By transforming the classical double-square-root (DSR) one-way wave equation from depth domain to two-way vertical traveltime (τ) domain, we derive a one-way DSR wave propagator which can be applied to implement the imaging concept of “survey sinking”. Its algorithm to recursively continue the source and receiver wavefields, which includes a wavenumber domain phase shift in a constant background medium followed by a phase correction in the space domain that accommodates lateral velocity variations, can tackle the effects of lateral velocity variations under complex overburdens. Applying the zero-offset, zero-time imaging condition, we develop a DSR equation prestack migration method in which the wavefield continuation and imaging are operated in the τ space. To address the problems that full volume 3-D DSR equation migration could meet with in practical application, we present a feasible common-azimuth prestack τ migration approach based on the theory of cross-line common-offset migration. Numerical tests show that our prestack τ migration provides a significant improvement over the traditional prestack time migration in laterally varying media.

Common reflection surface (CRS) stack is a macro-model independent zero-offset (ZO) imaging method. The traditional CRS stack method is implemented by determination of a set of wavefield attributes by which the CRS operator is constructed, wherein the attributes are searched and optimized automatically by a data-driven strategy such that a high quality ZO section will be simulated by stacking along the optimized CRS operator. However, the data-driven strategy also causes an inevitable “dip-angle discrimination phenomenon” which leads to loss of weak reflections and distortion of kinematical characteristic of the ZO section. In this paper, dip decomposition CRS stack is proposed to handle above problems which makes the CRS stack method more practical.

After reviewing the basic principle of the migration methods based on the double square root (DSR) equation, we introduce a fast algorithm of angle domain imaging to facilitate migration velocity analysis (MVA). Then we discuss two approaches to migrate the narrow azimuth seismic data. One has restriction to the input data set, the other has limitation of the wave propagation direction. The numerical tests show that the DSR equation prestack depth migration methods, such as common azimuth migration, have a promising prospect of application in areas with strong lateral variations of velocity where bsalt, reef or burred hills exist.

Travel time calculation in 3-D complicated media is the key for Kirchhoff integral migration and tomography. A 2-D main energy travel time calculation approach put forward by Nichols is extended to 3-D. With the help of the factor decomposition finite-difference scheme, the Helmholtz equation in the spherical coordinate system in frequency band can be solved efficiently. Some numerical examples and pre-stack depth migration results on SEG/EAGE salt-dome model demonstrate that this travel time calculation method is very suitable for imaging of complex geological structures.

The conventional shot-record prestack depth migration costs a very huge amount of computations required for wavefield extrapolations. Based on the fast areal shot migration technology, the controlled illumination prestack depth migration aims to efficiently achieve the high-quality image of the target structure by controlling the source wavefield at the target. The synthesis operators needed for areal short records are obtained by inversely extrapolating the predefined source wavefield at the target upward to the surface. We propose a more efficient scheme of controlled illumination, the surface rotation controlled illumination, which performs target-oriented illumination control over the source wavefield directly at the surface and can produce high-quality image of the target zones. A surface synthesis operator function is simply constructed by geometrically rotating a certain kernel synthesis operator, instead of by complicated wave extrapolations. This new controlled illumination method is as efficient as the general areal shot migration technology, but has much greater ability to image the complicated target zone. The application of the proposed technique to the Marmousi model gives a very ideal image of the deep complex structure.

Common reflection surface (CRS) stack is a macro-model independent seismic imaging method. Performing CRS stack depends on determination of three attributes which are the angle of emergence (α), the radii of curvature of the NIP-wave (RNIP), and the normal-wave (RN). Based on the practical theoretical foundation of CRS stack, this paper presents a detailed description of implementing CRS stack on real data. Firstly, initial parameter triplet (α,RN,Rnip) is determined by a concise one-dimensional coherency analysis on CMP stack section. Then a three-parametric optimization method is performed on the corresponding prestack data space, the optimized attributes will be much better suited to collect energy from underground reflectors than initial attributes. Finally, optimized attributes are used to realize CRS stack.

Common Reflection Surface (CRS) stack is the best way known to simulate a zero-offset section. It can adapt to weak laterally inhomogeneous media. Common Reflection Surface (CRS) means a segment around an underground reflector. Its traveltime response in the time-space domain can be regarded as a combination of associated Common Reflection Point (CRP) trajectories on the segment. After a general CRP trajectory expression is derived, with two kinds of eigen wave—Normal wave and Normal incidence point wave being introduced, the traveltime response of CRS can be given by extending CRP trajectories of one reflector to all reflectors in the segment around the reflector according to paraxial approximation. Inspecting this mathematical “transition” from CRP to CRS, this process provides the theoretical foundation for implementation of CRS stack.

Starting from the double square root (DSR) based wave equation, and based on the survey-sinking concept and the perturbation theory, we introduce a prestack depth full migration operator and an approach to construct common imaging gathers (CIGs). Considering the azimuthal characteristics of 3-D prestack data, three DSR migration operators for common azimuth sections, cross-line common offset sections and common offset vector sections are developed respectively by the stationary phase method. Among them the cross-line common offset migration is a theoretically new DSR-based method. The theoretic analysis and examples on synthetic datasets prove that, the full migration operator and common azimuth migration operator have good performance in the case of strong velocity variation, while the other two operators work only when the velocity varies gentlely.

The imaging result of prestack depth migration (Pre-SDM) is quite sensitive to migration velocity field. Reasonable migration velocity modeling becomes the key to obtain the Pre–SDM imaging with high quality. Firstly, common imaging point (CIP) gathers are extracted by using wave equation Pre-SDM with high imaging precision. Then, based on the perturbation method, the quantitative relationship between the errors in migration velocity and those in imaged depths is established through a parameterized velocity function and the modified residual curvature analysis (RCA). Finally, migration velocity modeling is realized by using single/multi parameter joint-iteration inversion. The test on the Marmousi model shows that the method proposed in this paper has a stronger adaptability and better modeling and imaging results for a complex geological body, and it needs only to analyze and control the main reflectors through 3 to 4 iterations to meet the required precision.

The essential shortcoming of the finite-difference migration is that steeply dipping events will be attenuated. To solve this problem and improve the quality of imaging in complicated media, we propose a prestack depth migration operator with coefficient optimized paraxial wave equation on the common shot gathers. Under the imaging condition based on reflection coefficients estimating, the prestack depth migration can be fulfiled. This operator has a form like that of the conventional second-order wave equation. It is accurate to about 70 degrees because of optimizing the coefficients of the equation. And compared with those methods in time-space domain, it has higher computation efficiency, because the finite-difference equation of wave field extrapolation has simpler form in the frequency-space domain. Moreover, it can also accommodate arbitrary velocity variations owing to finite-difference algorithm of depth migration. The impulse responses and the imaged result of the Marmousi model show that this migration operator has good imaging effect in strong laterally varying velocity media.

In this paper, source-receiver migration based on the double-square-root one-way wave equation is modified to operate in the two-way vertical traveltime (τ) domain. This tau migration method includes reasonable treatment for media with lateral inhomogeneity. It is implemented by recursive wavefield extrapolation with a frequency-wavenumber domain phase shift in a constant background medium, followed by a phase correction in the frequency-space domain, which accommodates moderate lateral velocity variations. More advanced τ-domain double-square-root wave propagators have been conceptually discussed in this paper for migration in media with stronger lateral velocity variations. To address the problems that the full 3D double-square-root equation prestack tau migration could meet in practical applications, we present a method for downward continuing common-azimuth data, which is based on a stationary-phase approximation of the full 3D migration operator in the theoretical frame of prestack tau migration of cross-line constant offset data. Migrations of synthetic data sets show that our tau migration approach has good performance in strong contrast media. The real data example demonstrates that common-azimuth prestack tau migration has improved the delineation of the geological structures and stratigraphic configurations in a complex fault area. Prestack tau migration has some inherent robust characteristics usually associated with prestack time migration. It follows a velocity-independent anti-aliasing criterion that generally leads to reduction of the computation cost for typical vertical velocity variations. Moreover, this τ-domain source-receiver migration method has features that could be of help to speed up the convergence of the velocity estimation.