One-way Wave Equation Research Articles

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 timespace 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.

Migration velocity analysis using traveltime wavefield tomography

We introduce an integrated wave-equation technique for migration velocity analysis (MVA) that consists of three steps: (1) forming the extended data, (2) approximating the correct transmitted wavefield, and (3) using wavefield tomography to update the velocity model. In the first step, the crosscorrelation imaging condition is relaxed to produce other nonzero-lag common image gathers (CIG) that, combined, form a common image cube (CIC). Slicing the CIC at different crosscorrelation lags forms a series of CIGs. Flattened events will occur in the CIGs at a lag other than the zero-lag when an incorrect velocity model is used in the migration. In the second step, for each event on the CIG, we pick the focusing depth and crosscorrelation lag at which it is flattest. We then model a Green’s function by seeding a source at the focusing depth using one-way wave equation modeling, then shift the modeled wavefield with the focusing crosscorrelation lag. This process is repeated for the other primary events at different lateral and vertical positions. The result is a set of modeled data whose wavefield approximates the wavefield that would have been generated if the correct velocity model had been used to simulate these gathers. We then apply wavefield tomography on these data-driven modeled data to update the velocity model. Our inversion scheme is based on wave-equation traveltime tomography that can update the velocity model in the presence of large velocity errors and a complex environment. Tests on synthetic and real 2D seismic data confirm the method’s effectiveness in building velocity models in complex structural areas that have large lateral velocity variations.

3D scalar-wave absorbing boundary conditions with optimal coefficients in the frequency-space domain

To improve the computational efficiency for the solution of the 3D Helmholtz equation in the frequency-space domain, high-order compact forms of finite differences are preferred. We applied a pointwise Padé approximation to develop a 3D 27-point fourth-order compact finite-difference (FD) stencil in the grid interior, with a space-differentiated source term, for the scalar-wave equation; this has similar high-accuracy (4–5 grid points per the shortest wavelength) to another 27-point fourth-order FD stencil using a parsimonious mixed-grid and staggered-grid combination, but is much simpler. For absorbing boundary conditions (ABCs), a damping zone is expensive, and a perfectly matched layer can not be straightforwardly introduced into the compact FD form for the second-order wave equation. Thus, we developed 3D one-way wave equation (OWWE) ABCs with adjustable coefficients. They have different angle approximations and FD forms for the six faces, twelve edges, and eight corners in 3D models to fit with the interior compact FD form. By adjusting the coefficients to the optimum, the OWWE ABCs have wider-angle absorbing ability than those without optimal coefficients. Finally, all the interior and boundary FD forms were combined into a sparse complex-valued impedance matrix of the frequency-space modeling equation, and solved for each frequency. Because the storage of the sparse impedance matrix was determined by the 3D discrete grid size, the OWWE ABCs with only one outer layer needed the minimum grid size compared with other ABCs, thus were the most efficient for the solution of the impedance matrix. The modeling algorithm was performed on multicore processors using a MPI parallel direct solver. Numerical tests on homogeneous and heterogeneous models gave satisfactory absorbing effects.

Compensation for transmission losses based on one-way propagators in the mixed domain

Most true-amplitude migration algorithms based on one-way wave equations involve corrections of geometric spreading and seismic [Formula: see text] attenuation. However, few papers discuss the compensation of transmission losses (CTL) based on one-way wave equations. Here, we present a method to compensate for transmission losses using one-way wave propagators for a 2D case. The scheme is derived from the Lippmann-Schwinger integral equation. The CTL scheme is composed of a transmission term and a phase-shift term. The transmission term compensates amplitudes while the wave propagates through subsurfaces. The transmission term is a function of the vertical wavenumbers of two adjacent heterogeneous screens. The phase-shift term is a Fourier finite-difference (FFD) propagator implemented in a mixed domain via Fourier transform. The transmission term can be flexibly incorporated into the conventional phase-shift migration algorithm, i.e., FFD, at every depth step. We analyze the effects of frequency, lateral velocity contrast, and vertical velocity ratio on the accuracy of the presented formulae. Numerical examples from a flat model and a fault model with lateral velocity variations are presented to demonstrate the ability of the proposed scheme for compensation of transmission losses.

Flux-normalized wavefield decomposition and migration of seismic data

Separation of wavefields into directional components can be accomplished by an eigenvalue decomposition of the accompanying system matrix. In conventional pressure-normalized wavefield decomposition, the resulting one-way wave equations contain an interaction term which depends on the reflectivity function. Applying directional wavefield decomposition using flux-normalized eigenvalue decomposition, and disregarding interaction between up- and downgoing wavefields, these interaction terms were absent. By also applying a correction term for transmission loss, the result was an improved estimate of the up- and downgoing wavefields. In the wave equation angle transform, a crosscorrelation function in local offset coordinates was Fourier-transformed to produce an estimate of reflectivity as a function of slowness or angle. We normalized this wave equation angle transform with an estimate of the plane-wave reflection coefficient. The flux-normalized one-way wave-propagation scheme was applied to imaging and to the normalized wave equation angle-transform on synthetic and field data; this proved the effectiveness of the new methods.

Common-angle image gathers for shot-profile migration: an efficient and stable strategy

Shot-profile-based prestack depth migration is quite attractive for sparse-shot wide-azimuth geometries. Meanwhile, angle-domain common image gathers (ADCIGs) have become an essential tool in seismic processing (e.g. velocity analysis). In this paper, we first revisit the popular schemes of generating ADCIGs by wavefield continuation migration. Then we present an alternative approach to produce common-angle image gathers after shot-profile migration based on a one-way wave equation. The angle information is obtained from a division of two images: the conventional image and the angle-weighted image. The effects of sparse shot sampling on extracting ADCIGs have been investigated. Numerical results demonstrate that the described scheme is efficient and stable and can handle the problem of shot-aliasing well.

Seismic imaging based on spectral differentiation matrix and GPU implementation

article i nfo Article history: Finite-difference depth migration based on one-way wave equation uses second-order, fourth-order, or other finite-order approximations for spatial derivatives. These finite-order approximations often lead to spatial dispersion errors and low accuracy. To avoid these errors, smaller mesh spacings are used, which results in huge increase in computation cost. In this paper, we develop a new spectral differentiation matrix method for approximating spatial derivatives. The approximation of spectral differentiation matrix is of infinite- order, and therefore avoids dispersion errors and increases accuracy by using a smaller number of grid points. Recently-developed GPU computational technology is well suited to the seismic migration based on frequency-domain wavefield-continuation methods because such approaches can be sufficiently decoupled. We present a frequency-domain wavefield-continuation algorithm based on spectral differentiation matrix, and realize GPU implementation of this new algorithm on Marmousi model.

Forward Modeling of the One-Way Acoustic Wave Equation by the Hartley Method

Abstract The conventional approach for prestack modeling has been to use a two-way wave equation which unavoidably produces multiple reflections. In this paper, a modeling method based on the one-way wave equation is discussed. In the method, the receivers are assumed buried under the surface only recording primary reflected waves, and the recorded data are extrapolated upward in depth up to the surface. The scheme economically produces synthetic sections containing primary energy only. We perform the depth extrapolations using the split-step operator in the frequency domain, so to provide the synthetic data for evaluating some migration methods or velocity estimation schemes, we can compute a certain frequency band of the data, which would accelerate the algorithm greatly especially when dealing with 3-D data. Furthermore, the split-step operator is developed in terms of the Hartley transforms in place of the Fourier transforms, this leads to a more efficient and economical program. The feasibility of the methods is demonstrated on numerical examples.

Preliminary Investigation of Wavefield Depth Extrapolation by Two-Way Wave Equations

Most of the wavefield downward continuation migration approaches are relying on one-way wave equations, which move the seismic energy always in one direction along depth. The one-way downward continuation migrations only use the primaries for imaging and do not treat secondary reflections recorded on the surface correctly. In this paper, we investigate wavefield depth extrapolators based on the full acoustic wave equations, which can propagate wave components to opposite directions. Several two-way wavefield downward continuation propagators are numerically tested in this study. Recursively implementing of the depth extrapolator makes it necessary and important to eliminate the unstable wave modes, that is, evanescent waves. For the laterally varying velocity media, distinction between the propagating and evanescent wave mode is less clear. We demonstrate that the spatially localized two-way beamlet propagator is an effective way to remove the evanescent waves while maintain the propagating mode in laterally inhomogeneous media.

Finite Element Analysis with Paraxial & Viscous Boundary Conditions for Elastic Wave Propagation

In this study, two studies are performed. One is to apply paraxial boundary conditions which are local boundary conditions based on paraxial approximations of the one-way wave equations to finite element analysis. To do this, a penalty functional is proposed and the existence and uniqueness of the extremum of the proposed functional is demonstrated. The other is to improve the capacity of viscous boundary conditions using dashpots. To do this, customary viscous boundary conditions are modified to maximize the efficiency according to angles of incidence and materials. For the numerical analysis of elasticity with paraxial boundary conditions and the modified viscous boundary conditions, the coding of the finite element models is implemented, and the efficiency of those boundary conditions is investigated.

Implementation of Absorbing Boundary Conditions Based on the Second-Order One-Way Wave Equation in the LOD- and the ADI-FDTD Methods

This letter describes the implementation of second-order one-way wave-equation absorbing boundary conditions (ABCs) in two unconditionally stable finite-difference time-domain (FDTD) methods-namely the locally one-dimensional (LOD)- and the alternating-direction implicit (ADI)-FDTD methods. The Higdon second-order absorbing operator is discretized in the same way as when it is used with the conventional FDTD method. The resulting discrete expression is directly applied to the electric field in each time substep of the LOD- and the ADI-FDTD methods. Numerical examples are given to illustrate the validity of the proposed approach.

A Mur Type Analytical Absorbing Boundary Condition for Multidimensional Wave Analysis with the Directional Splitting Technique

A Mur type analytical absorbing boundary condition (A-ABC), which is based on the one-dimensional one-way wave equation, is proposed for multidimensional wave analysis by introducing the directional splitting technique. This new absorbing boundary condition is expansion of the first-order Mur. The absorbing ability, required memory, and calculation speed of the Mur type A-ABC are evaluated by comparison with those of conventional ABCs. The result indicated that absorbing ability of the proposed ABC is higher than the first-order Mur and lower than the second-order Mur at large incident angle. While, our proposed ABC has advantage in both required memory and calculation speed by comparison with the second-order Mur. Thus, effectivity of the proposed Mur type A-ABC is shown.

A Fourier integral algorithm and its GPU/CPU collaborative implementation for one-way wave equation migration

Pre-stack one-way wave equation (OWE) is a useful tool for seismic imaging and modeling. Since the idea of OWE appeared in the 1970s, geophysicists have made great effort to improve the accuracy of the one-way wave equation extrapolators. In this paper, we present the idea of solving OWE using the Fourier integral method, which represents OWE as a Fourier integral equation and solves it in dual spaces (both space and wave-number domains). By doing this, we can propagate wave-fields up to nearly 90° angle from the vertical direction in the presence of lateral velocity variations. The proposed method is stable, does not suffer from the numerical dispersion, and overcomes the azimuthal anisotropy problem when extended to three dimensions. The computation cost of the Fourier integral method is too high and was considered impractical for a conventional computer. In this paper, we take advantages of the Graphic Processing Unit (GPU) and use the matrix multiplication technique to accelerate the algorithm. The speedup ratio we obtained is tens of hundreds times so that the method can be applied to a real project for pre-stack depth imaging.

Wide-azimuth TTI imaging at Tahiti: Reducing structural uncertainty of a major deepwater subsalt field

The Tahiti field is a recent major development in the deepwater Gulf of Mexico. The field’s prolific Miocene reservoir section lies below a thick salt canopy with structural dips as high as 80 degrees, adjacent to a near-vertical salt root. Successful appraisal and initial development was enabled by interpretation of proprietary depth imaging products generated from narrow-azimuth seismic data. However, reservoir-scale mapping and fault definition remained problematic due to seismic imaging and illumination challenges. In 2009–2010, the Tahiti partnership initiated a reimaging project using multiclient wide-azimuth seismic data. The project employed current technologies for multiple attenuation, tilted transverse isotropy velocity modeling, and migration. Increased azimuthal coverage and inherent multiple suppression provided by wide azimuth acquisition delivered significant imaging enhancements. Advanced noise and multiple attenuation techniques provided cleaner data with improved signal-to-noise. Earth models representing multiazimuth subsurface velocities and anisotropy parameters calibrated to well control with detailed salt interpretation resulted in higher confidence structural imaging. Comparison of Gaussian beam, one-way wave equation, and reverse time migration algorithms shows that reverse time migration generally provides superior subsalt and salt-body data quality, with improved event positioning, higher resolution, and enhanced steep dip imaging. The resulting seismic volumes enable accurate mapping of reservoir horizons and faulting. This will improve resource determination and future well placement in the next phase of field development.

Migration velocity analysis using the common image cube (CIC)

Migration velocity analysis (MVA) is commonly performed in the image domain in conjunction with ray-based tomography to update the velocity model. This approach can be challenging in the presence of large velocity errors as it may require many MVA iterations before converging to a model that can focus the events in the image domain. We introduced a downward continuation-based domain for carrying out MVA that is more flexible than conventional domains. This approach consists of two steps: (1) forming the common image cube (CIC) and (2) modeling the Green’s functions. In the first step, the cross-correlation imaging condition is relaxed to produce more than the zero lag common image gather (CIG). Slicing these data at different lags forms a series of CIGs, whereas a conventional CIG can be obtained by slicing the cube at the zero lag. When the velocity model used for the migration differs from the true velocity model, properly flattened events may occur in CIGs other than the zero lag. In the second step, for each event on the CIG, we picked the cross-correlation lag and depth at which it flattens best. For each event, we modeled a Green’s function by seeding a source at the focusing depth using one-way wave-equation modeling. This process is then repeated for other events at different lateral positions. The result is a set of Green’s functions whose wavefield approximates the ones that would have been generated if the correct velocity model was used to simulate these gathers. The updated Green functions are easier to work with than the raw data as they have less noise. Wavefield tomography can then be applied on these data-driven, modeled Green’s functions to build the final velocity model. Tests on synthetic and real 2D data confirm the method’s effectiveness in building velocity models in complex structural areas with large lateral velocity variations.

3D angle gathers from reverse time migration

Common-image gathers are an important output of prestack depth migration. They provide information needed for velocity model building and amplitude and phase information for subsurface attribute interpretation. Conventionally, common-image gathers are computed using Kirchhoff migration on common-offset/azimuth data volumes. When geologic structures are complex and strong contrasts exist in the velocity model, the complicated wave behaviors will create migration artifacts in the image gathers. As long as the gather output traces are indexed by any surface attribute, such as source location, receiver location, or surface plane-wave direction, they suffer from the migration artifacts caused by multiple raypaths. These problems have been addressed in a significant amount of work, resulting in common-image gathers computed in the reflection angle domain, whose traces are indexed by the subsurface reflection angle and/or the subsurface azimuth angle. Most of these efforts have concentrated on Kirchhoff and one-way wave-equation migration methods. For reverse time migration, subsurface angle gathers can be produced using the same approach as that used for one-way wave-equation migration. However, these approaches need to be revisited when producing high-quality subsurface angle gathers in three dimensions (reflection angle/azimuth angle), especially for wide-azimuth data. We have developed a method for obtaining 3D subsurface reflection angle/azimuth angle common-image gathers specifically for the amplitude-preserved reverse time migration. The method builds image gathers with a high-dimensional convolution of wavefields in the wavenumber domain. We have found a windowed antileakage Fourier transform method that leads to an efficient and practical implementation. This approach has generated high-resolution angle-domain gathers on synthetic 2.5D data and 3D wide-azimuth real data.

Analysis of a fast method for solving the high frequency Helmholtz equation in one dimension

We propose and analyze a fast method for computing the solution of the high frequency Helmholtz equation in a bounded one-dimensional domain with a variable wave speed function. The method is based on wave splitting. The Helmholtz equation is split into one-way wave equations with source functions which are solved iteratively for a given tolerance. The source functions depend on the wave speed function and on the solutions of the one-way wave equations from the previous iteration. The solution of the Helmholtz equation is then approximated by the sum of the one-way solutions at every iteration. To improve the computational cost, the source functions are thresholded and in the domain where they are equal to zero, the one-way wave equations are solved with geometrical optics with a computational cost independent of the frequency. Elsewhere, the equations are fully resolved with a Runge–Kutta method. We have been able to show rigorously in one dimension that the algorithm is convergent and that for fixed accuracy, the computational cost is asymptotically just $\mathcal {O}(\omega^{1/ p})$ for a pth order Runge–Kutta method, where ω is the frequency. Numerical experiments indicate that the growth rate of the computational cost is much slower than a direct method and can be close to the asymptotic rate.

One-way wave-equation migration of compressional and converted waves in a VTI medium

In seismic reflection surveying, by recording both pressure and shear-wave reflections, one can increase the amount of information obtained about the subsurface rather than by recording pressure waves alone. Geologic structures that are not visible by using conventional pressure-data may possibly be imaged using shear waves, thus mitigating the risk in oil and gas exploration and production. Horizontally layered sedimentary rocks exhibit anisotropy that can be approximated by an effective transverse isotropic medium with a vertical axis of symmetry. Taking into account a vertically transverse isotropic earth, we derive phase-slowness expressions for quasi-P and quasi-SV waves that are used in a one-way wave-equation migration scheme. We derive simplified slowness-expressions that are useful for processing of conventional pressure data. Numerical examples demonstrate that the slowness approximations are valid for wide-angle propagation, and the resulting one-way propagators are validated on a series of synthetic tests and applied on a field ocean-bottom seismic data set. The results show that the method accurately images both compressional and converted waves in OBS data over a vertically transverse isotropic medium.

Visibility analysis for target-oriented reverse time migration and optimizing acquisition parameters

To obtain the best possible image quality of complex geologic structures, prestack depth migration has evolved over the past several years from using one-way wave-equation migration (WEM) to using reverse time migration (RTM) in production seismic processing (Etgen et al., 2009). RTM extrapolates both source and receiver wavefields in time (Baysal et al., 1983; McMechan, 1983), rather than along a spatial (usually vertical) axis and is thus able to handle waves that propagate long distances horizontally, turning waves that propagate beyond 90°, and some special classes of multiple reflections, such as duplex waves and prism waves. The huge computational burden of RTM for high-frequency wavefields, especially in tilted anisotropic (TTI) media often prevents its use as an iterative seismic imaging tool. While certainly some imaging projects run multiple iterations of RTM, these projects often require long time frames and substantial expense, something justified only by the highest-profile projects. A wide class of seismic imaging problems being faced today are mostly “target-oriented” in their nature; they usually involve imaging particular details of salt features or other velocity complexities, or they involve creating the best possible image of the reservoir itself. In these cases, it is not necessary to migrate full-volume data to better reconstruct the image around the area of interest for each iteration because most of these data have no physical contributions to the target events and may even introduce noise and, thus, degrade the image quality of the target events. Rather, we should migrate those data having physical contributions to the target events. The visibility analysis we describe provides a solution to quantitatively select the data that need to be migrated for reconstructing the target events (Jin and Xu, 2009; Jin et al., 2010).

Wave-equation angle-based illumination weighting for optimized subsalt images

A methodology has been developed for improving seismic depth images generated in areas of complex structure, such as subsalt. The main goal of the wave-equation angle-based illumination workflow is to calculate a set of illumination weight gathers and apply them to migrated field angle gathers. To obtain meaningful illumination weight gathers, it is important to use the field-acquisition geometry information, migration velocity models, and interpreted horizons. The workflow is model based and requires horizons conformable to real structure, preferably picked on the field seismic depth images. The workflow is implemented using one-way and two-way wave-equation methods and is tested on narrow-azimuth towed-streamer (NATS) and merged NATS with wide-azimuth towed-streamer (XWATS) data sets. The method weights down poorly illuminated areas while preserving strongly illuminated areas, which is opposite to the compensation methods. As a result, an overall improvement in seismic depth images can be observed, especially in subsalt areas. The impact of the workflow on improving well-tie estimation at the Thunder Horse subsalt development in the Gulf of Mexico is demonstrated.