Post-stack reverse-time migration using a finite difference method based on triangular grids

Compared with other migration methods, reverse-time migration is based on a precise wave equation, not an approximation, and performs extrapolation in the depth domain rather than the time domain. It is highly accurate and not affected by strong subsurface structure complexity and horizontal velocity variations. The difference method based on triangular grids maintains the simplicity of the difference method and the precision of the finite element method. It can be used directly for forward modeling on models with complex top surfaces and migration without statics preprocessing. We apply a finite difference method based on triangular grids for post-stack reverse-time migration for the first time. Tests on model data verify that the combination of the two methods can achieve near-perfect results in application.

Poststack reverse-time migration using a non-reflecting recursive algorithm on surface relief

Presently the research based on the accurate seismic imaging methods for surface relief, complex structure, and complicated velocity distributions is of great significance. Reverse-time migration is considered to be one of highly accurate methods. In this paper, we propose a new non-reflecting recursive algorithm for reverse-time migration by introducing the wave impedance function into the acoustic wave equation and the algorithm for the surface relief case is derived from the coordinate transformation principle. Using the exploding reflector principle and the zero-time imaging condition of poststack reverse- time migration, poststack numerical simulation and reverse-time migration with complex conditions can be realized. The results of synthetic and real data calculations show that the method effectively suppresses unwanted internal reflections and also deals with the seismic imaging problems resulting from surface relief. So, we prove that this method has strong adaptability and practicality.

A study of perfectly matched layers for joint multicomponent reverse-time migration

Reverse-time migration in finite space requires effective boundary processing technology to eliminate the artificial truncation boundary effect in the migration result.On the basis of the elastic velocity-stress equations in vertical transversely isotropic media and the idea of the conventional split perfectly matched layer（PML）,the PML wave equations in reverse-time migration are derived in this paper and then the high order staggered grid discrete schemes are subsequently given.Aiming at the＂reflections＂from the boundary to the computational domain,as well as the effect of seismic event＇s abrupt changes at the two ends of the seismic array,the PML arrangement in reverse-time migration is given.The synthetic and real elastic,prestack,multi-component,reverse-time depth migration results demonstrate that this method has much better absorbing effects than other methods and the joint migration produces good imaging results.

Elastic reverse-time migration based on amplitude-preserving P- and S-wave separation

Imaging the PP- and PS-wave for the elastic vector wave reverse-time migration requires separating the P- and S-waves during the wave field extrapolation. The amplitude and phase of the P- and S-waves are distorted when divergence and curl operators are used to separate the P- and S-waves. We present a P- and S-wave amplitude-preserving separation algorithm for the elastic wavefield extrapolation. First, we add the P-wave pressure and P-wave vibration velocity equation to the conventional elastic wave equation to decompose the P- and S-wave vectors. Then, we synthesize the scalar P- and S-wave from the vector P- and S-wave to obtain the scalar P- and S-wave. The amplitude-preserved separated P- and S-waves are imaged based on the vector wave reverse-time migration （RTM）. This method ensures that the amplitude and phase of the separated P- and S-wave remain unchanged compared with the divergence and curl operators. In addition, after decomposition, the P-wave pressure and vibration velocity can be used to suppress the interlayer reflection noise and to correct the S-wave polarity. This improves the image quality of P- and S-wave in multicomponent seismic data and the true-amplitude elastic reverse time migration used in prestack inversion.

Least squares reverse-time migration in the pseudodepth domain and reservoir exploration

Least squares reverse-time migration （LSRTM） is an inversion method that removes artificial images and preserves the amplitude of reflectivity sections. LSRTM has been used in reservoir exploration and processing of 4D seismic data. LSRTM is, however, a computationally costly and memory-intensive method. In this study, LSRTM in the pseudodepth domain was combined with the conjugate gradient method to reduce the computational cost while maintaining precision. The velocity field in the depth domain was transformed to the velocity field in the pseudodepth domain; thus, the total number of vertical sampling points was reduced and oversampling was avoided. Synthetic and field data were used to validate the proposed method. LSRTM in the pseudodepth domain in conjunction with the conjugate gradient method shows potential in treating field data.

Reverse-time migration using multidirectional wavefield decomposition method

Reverse-time migration has attracted more and more attention owing to the advantages of high imaging accuracy, no dip restriction, and adaptation to complex velocity models. Cross-correlation imaging method is typically used in conventional reverse-time migration that produces images with strong low-frequency noise. Wavefield decomposition imaging can suppress such noise; however, some residual noise persists in the imaging results. We propose a 2D multidirectional wavefield decomposition method based on the traditional wavefield decomposition method. First, source wavefields and receiver wavefields are separated into eight subwavefields, respectively. Second, cross-correlation imaging is applied to selected subwavefields to produce subimages. Finally, the subimages are stacked to generate the final image. Numerical examples suggest that the proposed method can eliminate the low-frequency noise effectively and produce high-quality imaging profiles.

Huber inversion-based reverse-time migration with de-primary imaging condition and curvelet-domain sparse constraint

Least-squares reverse-time migration(LSRTM) formulates reverse-time migration(RTM) in the leastsquares inversion framework to obtain the optimal reflectivity image. It can generate images with more accurate amplitudes, higher resolution, and fewer artifacts than RTM. However, three problems still exist:(1) inversion can be dominated by strong events in the residual;(2) low-wavenumber artifacts in the gradient affect convergence speed and imaging results;(3) high-wavenumber noise is also amplified as iteration increases. To solve these three problems, we have improved LSRTM: firstly, we use Hubernorm as the objective function to emphasize the weak reflectors during the inversion;secondly, we adapt the de-primary imaging condition to remove the low-wavenumber artifacts above strong reflectors as well as the false high-wavenumber reflectors in the gradient;thirdly, we apply the L1-norm sparse constraint in the curvelet-domain as the regularization term to suppress the high-wavenumber migration noise. As the new inversion objective function contains the non-smooth L1-norm, we use a modified iterative soft thresholding(IST) method to update along the Polak-Ribie re conjugate-gradient direction by using a preconditioned non-linear conjugate-gradient(PNCG) method. The numerical examples,especially the Sigsbee2 A model, demonstrate that the Huber inversion-based RTM can generate highquality images by mitigating migration artifacts and improving the contribution of weak reflection events.

Target-oriented Q-compensated reverse-time migration by using optimized pure-mode wave equation in anisotropic media

Research on seismic anisotropy and attenuation plays a significant role in exploration geophysics. To enhance the imaging quality for complicated structures, we develop several effective improvements for anisotropic attenuation effects in reverse-time migration (Q-RTM) on surface and vertical seismic profiling (VSP) acquisition geometries. First, to suppress pseudo-shear wave artifact and numerical instability of the commonly used anisotropic pseudo-acoustic wave equations, an optimized pure P-wave dispersion relation is derived and the corresponding pure-mode wave equation is solved by combining the finite-difference and Possion methods. Second, a simplified anisotropic pure-mode visco-acoustic wave equation (PVAWE) based on standard linear solid model is established. Third, a time-dispersion correlation strategy is applied to improve the modeling accuracy. Fourth, we extend a target-oriented scheme to anisotropic attenuated modeling and imaging. Instead of the conventional wavefield modeling and RTM, the proposed approach can extract available wavefield information near the target regions and produce high imaging resolution for target structures. Last, both anisotropic surface and VSP Q-RTMs are executed by combining optimized PVAWE, time-dispersion correlation and target-oriented algorithm. Modeling examples demonstrate the advantages of our schemes. Moreover, our modified Q-compensated imaging workflow can be regarded as a supplement to the classical anisotropic RTM.

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

Angle-domain common-image gathers （ADCIGs） transformed from the shot- domain 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 reverse- time 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.

Stretching correction for amplitude-preserving vector wavefield reverse-time migration

The migration of multi-wave seismic data is aimed at obtaining the P- and S-wave imaging results of the amplitude preserving. But the P- and S-wave stretching effect produced by the reverse time migration of the elastic wave equation will not only reduce the vertical resolution of the migration results and the amplitude preserving of the large reflection angle. In this paper, the reverse time migration technique of amplitude preserving vector wave-field separating is used. Based on the analysis of the stretch mechanism and the influencing factors of stretch magnitude, the paper gave the stretch correcting factors. Then, realize the stretch correction method at the time that after the reverse extrapolation and before the imaging by solving the problem which is how to calculate the P-wave and Ps-wave propagation directions of imaging points at different times. The stretch correction method can improve the vertical resolution and amplitude fidelity of the imaging results and provide high fidelity input data for seismic data interpretation and inversion.

Reverse-Time Migration from Rugged Topography to Image Ground-Penetrating Radar Data in Complex Environments

In ground-penetrating radar （GPR） imaging, it is common for the depth of investigation to be on the same order as the variability in surface topography, In such cases, migration fails when it is carried out from a datum after the application of elevation statics, We introduce a reverse-time migration （RTM） algorithm based on the second-order decoupled form of Maxwell＇s equations, which requires computation of only the electric field, The wavefield extrapolation is computed directly from the acquisition surface without the need for datuming, In a synthetic case study, the algorithm significantly improves image accuracy over a processing sequence in which migration is performed after elevation statics, In addition, we acquired a field dataset at the Coral Pink Sand Dunes （CPSD） in Utah, USA, The data were acquired over rugged topography and have the complex internal stratigraphy of multiply eroded, modern, and ancient eolian deposits, The RTM algorithm significantly improves radar depth images in this challenging environment,

Elastic reverse-time migration in irregular tunnel environment based on polar coordinates

When seismic exploration is conducted in a special geological environment such as a tunnel space,the traditional imaging method in the Cartesian coordinate system cannot accurately discretize the air column in that environment.Thus,obtaining Thus,obtaining highquality imaging results is diffi cult.Therefore,an elastic-wave reverse-time migration method based on the polar coordinate system is proposed.In this method,three boundary conditions exist:outer,inner,and corner boundaries.In the outer boundary,the polar-coordinated absorbing boundary in the radial direction is used to suppress the artifi cial-boundary refl ection.The free-surface boundary condition is adopted in the tunnel space at the inner boundary.In the angular boundaries,we use two diff erent boundary conditions for two cases.The air column in the tunnel space is usually not an irregular circle.Therefore,the irregular tunnelspace geological body in the polar coordinate system is meshed into curvilinear grids and transformed into a regular one in an auxiliary polar coordinate system using the mapping method.Finally,elastic reverse-time migration technology is applied into the auxiliary polar coordinate system.In the numerical examples,two typical models are used to test the proposed method,which verify that the proposed method can obtain accurate images from the datasets in the tunnel space.

Attenuation compensation in multicomponent Gaussian beam prestack depth migration

Gaussian beam prestack depth migration is an accurate imaging method of subsurface media. Prestack depth migration of multicomponent seismic data improves the accuracy of imaging subsurface complex geological structures. Viscoelastic prestack depth migration is of practical significance because it considers the viscosity of the subsurface media. We use Gaussian beam migration to compensate for the attenuation in multicomponent seismic data. First, we use the Gaussian beam method to simulate the wave propagation in a viscoelastic medium and introduce the complex velocity Q-related and exact viscoelastic Zoeppritz equation. Second, we discuss PP- and PS-wave Gaussian beam prestack depth migration algorithms for common-shot gathers to derive expressions for the attenuation and compensation. The algorithms correct the amplitude attenuation and phase distortion caused by Q, and realize multicomponent Gaussian beam prestack depth migration based on the attenuation compensation and account for the effect of inaccurate Q on migration. Numerical modeling suggests that the imaging resolution of viscoelastic Gaussian beam prestack depth migration is high when the viscosity of the subsurface is considered.

High angle prestack depth migration with absorption compensation

The absorption effect of actual subsurface media can weaken wavefield energy, decrease the dominating frequency, and further lead to reduced resolution. In migration, some actions can be taken to compensate for the absorption effect and enhance the resolution. In this paper, we derive a one-way wave equation with an attenuation term based on the time- space domain high angle one-way wave equation. A complicated geological model is then designed and synthetic shot gathers are simulated with acoustic wave equations without and with an absorbing term. The derived one-way wave equation is applied to the migration of the synthetic gathers without and with attenuation compensation for the simulated shot gathers. Three migration profiles are obtained. The first and second profiles are from the shot gathers without and with attenuation using the migration method without compensation, the third one is from the shot gathers with attenuation using the migration method with compensation. The first and third profiles are almost the same, and the second profile is different from the others below the absorptive layers. The amplitudes of the interfaces below the absorptive layers are weak because of their absorption. This method is also applied to field data. It is concluded from the migration examples that the migration method discussed in this paper is feasible.

Anisotropic converted wave amplitude-preserving prestack time migration by the pseudo-offset method

In this paper, we use the method of pseudo-offset migration （POM） to complete converted wave pre-stack time migration with amplitude-preservation in an anisotropic medium. The method maps the original traces into common conversion scatter point （CCSP） gathers directly by POM, which simplifies the conventional processing procedure for converted waves. The POM gather fold and SNR are high, which is favorable for velocity analysis and especially suitable for seismic data with low SNR. We used equivalent anisotropic theory to compute anisotropic parameters. Based on the scattering wave traveltime equation in a VTI medium, the POM pseudo-offset migration in anisotropic media was deduced. By amplitude-preserving POM gather mapping, velocity analysis, stack processing, and so on, the anisotropic migration results were acquired. The forward modeling computation and actual data processing demonstrate the validity of converted wave pre-stack time migration with amplitude-preservation using the anisotropic POM method.

Amplitude.preserving plane-wave prestack time migration for AVO analysis

To support amplitude variation with offset （AVO） analysis in complex structure areas, we introduce an amplitude-preserving plane-wave prestack time migration approach based on the double-square-root wave equation in media with little lateral velocity variation. In its implementation, a data mapping algorithm is used to obtain offset-plane-wave data sets from the common-midpoint gathers followed by a non-recursive phase-shift solution with amplitude correction to generate common-image gathers in offset-ray-parameter domain and a structural image. Theoretical model tests and a real data example show that our prestack time migration approach is helpful for AVO analysis in complex geological environments.

Beamlet prestack depth migration and illumination： A test based on the Marmousi model

Beamlet sources have strong local and directional character and can easily accomplish local illumination and migration. Besides, they provide better migration results than conventional migration methods. We introduce the basic principles of beamlet prestack depth migration that includes a windowed Fourier transform and frame theory. We explain the Gabor-Daubechies （G-D） frame based on a Gaussian function. Beamlet decomposition provides information on the local space and direction of wavefield. We synthesize the beamlet source and beamlet records in the wavelet domain using both rectangle and Gaussian windows and then extrapolate the synthesized data with a Fourier finite-difference operator. We test the method using the standard Marmousi model. By comparing and analyzing the migration results of single directional beamlet and beamlets with different windows and directions, we demonstrate the validity of the prestack depth migration with Gaussian beamlets method.

Application of Amplitude Ratio Profiles Extracted with Prestack Kirchhoff Integral Migration

The Z component and X component profiles of seismic waves extracted with the prestack Kirchhoff integral migration could approximate to the primary wave （P wave） and converted shear wave （PS wave） profiles under certain conditions. The relative change of their reflection amplitude reflects the formation stress anomaly and subsurface media anisotropy. The principle and method for extracting amplitude ratios were studied and the application of amplitude ratio profiles was also examined when processing and interpreting actual seismic data. The amplitude ratio profile is an effective supplementary means of identifying the stratigraphic boundary and lithology.

Fast least-squares prestack time migration via accelerating the explicit calculation of Hessian matrix with dip-angle Fresnel zone

Amplitude versus offset analysis is a fundamental tool for determining the physical properties of reservoirs but generally hampered by the blurred common image gathers(CIGs).The blurring can be optimally corrected using the blockwise least-squares prestack time migration(BLS-PSTM),where common-offset migrated sections are divided into a series of blocks related to the explicit offsetdependent Hessian matrix and the following inverse filtering is iteratively applied to invert the corresponding reflectivity.However,calculating the Hessian matrix is slow.We present a fast BLS-PSTM via accelerating Hessian calculation with dip-angle Fresnel zone(DFZ).DFZ is closely related to optimal migration aperture,which significantly attenuates migration swings and reduces the computational cost of PSTM.Specifically,our fast BLS-PSTM is implemented as a two-stage process.First,we limit the aperture for any imaging point with an approximated the projected Fresnel zone before calculating the Hessian matrix.Then,we determine whether a seismic trace contributes to the imaging point via DFZ during calculating the Hessian matrix.Numerical tests on synthetic and field data validate the distinct speedup with higher-quality CIGs compared to BLS-PSTM.

Seismic 3D Prestack Time Migration on Parallel Computers

Parallel versions of prestack KirchhofT 3D integral migration algorithm, which is suitable forseismic data processing, are described in this paper. Firstly, the inherent parallel characteristics of seismicdata processing are analyzed. Then some principles in algorithm partition are discussed. Based on these analyses and the system architecture, communication mechanism, this algorithm is divided into four subtasksallocated to four nodes of 990 STAR-l. Then we describe in detail a module-partitioning method-theI / O processing and communication are separated from the computation process, the processes includingI / O processing and communication are allocated to transputer T805 and the other is allocated to processori860. These two processes are synchronized by shared memory and memory-lock mechanism, but the communication betWeen different nodes is implemented through links of transputer. Load balance among fourprocessor modules is performed dynamically. Finally, we discussed the speed--up of the parallel versions ofprestack KirchhofT 3D integral migration algorithm running on four nodes. Some further researches are also melltioned in this paper.