One-way Wave Equation Research Articles

One-Way Wave Equation Derived from Impedance Theorem

The wave equations for longitudinal and transverse waves being used in seismic calculations are based on the classical force/moment balance. Mathematically, these equations are 2nd order partial differential equations (PDE) and contain two solutions with a forward and a backward propagating wave, therefore also called “Two-way wave equation”. In order to solve this inherent ambiguity many auxiliary equations were developed being summarized under “One-way wave equation”. In this article the impedance theorem is interpreted as a wave equation with a unique solution. This 1st order PDE is mathematically more convenient than the 2nd order PDE. Furthermore the 1st order wave equation being valid for three-dimensional wave propagation in an inhomogeneous continuum is derived.

Depth migration with the near surface: A case study for mineral exploration in China

Seismic reflection data used in mineral exploration are usually of poor quality because of rough topography and small-scale heterogeneities. However, there is an increasing demand for accurate imaging processes that can obtain such data. Here, we have developed a practical rugged depth migration method based on the finite-difference one-way wave equation. It can be used to improve imaging quality when faced with rough topography and complex geologic features. First, synthetic data are used to illustrate the errors and problems that cause unsatisfactory imaging results. This is usually because no consideration is given to rough topography during conventional processing procedures. To address this, a depth migration method that begins with the rugged topography is introduced. It can be used to produce better images. Real data from mineral exploration activities in Fujian, China, are then used to demonstrate the practical use of this method. We also compare two computational processes: The first corrects the data back to the original surface and then builds a velocity model before conducting rugged depth migration; the second process involves performing rugged depth migration on processed data featuring a smoothed surface following short-wavelength static correction and residual correction. The imaging results indicate that the section obtained from the processed data has better quality. They also indicate that the imaging quality of the original surface is affected by the built velocity model, which cannot include near-surface and small-scale structures.

3D Prestack Fourier Mixed-Domain (FMD) depth migration for VTI media with large lateral contrasts

Although many 3D One-Way Wave-equation Migration (OWEM) methods exist for VTI media, most of them struggle either with the stability, the anisotropic noise or the computational cost. In this paper we present a new method based on a mixed space- and wavenumber-propagator that overcome these issues very effectively as demonstrated by the examples. The pioneering methods of phase-shift (PS) and Stolt migration in the frequency-wavenumber domain designed for laterally homogeneous media have been followed by several extensions for laterally inhomogeneous media. Referred many times to as phase-screen or generalized phase-screen methods, such extensions include as main examples of the Split-step Fourier (SSF) and the phase-shift plus interpolation (PSPI). To further refine such phase-screen techniques, we introduce a higher-order extension to SSF valid for a 3D VTI medium with large lateral contrasts in vertical velocity and anisotropy parameters. The method is denoted Fourier Mixed-Domain (FMD) prestack depth migration and can be regarded as a stable explicit algorithm. The FMD technique was tested using the 3D SEG/EAGE salt model and the 2D anisotropic Hess model with good results. Finally, FMD was applied with success to a 3D field data set from the Barents Sea including anisotropy.

One-way propagators based on matrix multiplication in arbitrarily lateral varying media with GPU implementation

Conventional one-way migration methods have some shortcomings, such as their limitations in imaging large angles. A novel strategy is presented here to solve these issues. The Helmholtz operator of the two-way wave equation is a power of the square root operator of the one-way wave equation. Based on this connectivity, the computation of the square root of the Helmholtz operator based on matrix multiplication and the calculation of a one-way propagator with a Taylor series expansion, is studied. The one-way propagators used in this paper involve complete matrix multiplication, which is a distinct feature of this method and is suitable for parallel implementation. Analysing the values of matrix square roots generated by the classical eigenvalues decomposition method and an algorithm of computing matrix square roots with matrix multiplication, we prove that our algorithm can calculate a stable result, which lays a solid foundation for further study. The impulse response test proves the advantage of our proposed method over conventional one-way migration in describing the wavefield propagation of large angles in a medium with a strong lateral velocity variation. Moreover, we implement GPU and CPU versions of the algorithm and compared their efficiencies. The post-stack migration result of the Salt model and the pre-stack migration result of the Marmousi model further illustrate the superiority of our proposed migration method in imaging the complex and fine-scale structures compared to the conventional one-way migration method. This provides a promising practical application in seismic exploration.

Sixth-order accurate pseudo-spectral method for solving one-way wave equation

In this paper, we present a pseudo-spectral method to solve the one-way wave equation. The approach is a generalization of the phase-shift plus interpolation technique which is used in geophysical applications. We construct a solution at each depth layer as a linear combination of the solutions corresponding to the models with uniform reference velocities. We suggest using three-term relations to interpolate the solution with the sixth order of accuracy to the deviation from the vertical direction. Standard phase-shift plus interpolation technique uses two-terms relation interpolating the solution with the fourth order. As a result, the numerical error of the suggested approach is one half of that of the PSPI methods for a fixed set of reference velocities for a wide range of spatial discretizations and directions of wave propagation. Consequently, to compute a solution with prescribed accuracy, the presented approach allows using 20% fewer reference velocities than the PSPI. Additionally provided experiments illustrate the efficiency of the suggested approach for simulation of down-going wave propagation in complex geological media, making the algorithm a promising one for the seismic imaging procedures.

Plane-wave migration for steep reflectors imaging

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

Extra-wide-angle parabolic equations in motionless and moving media.

Wide-angle parabolic equations (WAPEs) play an important role in physics. They are derived by an expansion of a square-root pseudo-differential operator in one-way wave equations, and then solved by finite-difference techniques. In the present paper, a different approach is suggested. The starting point is an extra-wide-angle parabolic equation (EWAPE) valid for small variations of the refractive index of a medium. This equation is written in an integral form, solved by a perturbation technique, and transformed to the spectral domain. The resulting split-step spectral algorithm for the EWAPE accounts for the propagation angles up to 90° with respect to the nominal direction. This EWAPE is also generalized to large variations in the refractive index. It is shown that WAPEs known in the literature are particular cases of the two EWAPEs. This provides an alternative derivation of the WAPEs, enables a better understanding of the underlying physics and ranges of their applicability, and opens an opportunity for innovative algorithms. Sound propagation in both motionless and moving media is considered. The split-step spectral algorithm is particularly useful in the latter case since complicated partial derivatives of the sound pressure and medium velocity reduce to wave vectors (essentially, propagation angles) in the spectral domain.

A Comparison of the Classical OWWE Migration Methods: Analysis of Feasibility in Subsalt Plays Exploration

Subsalt seismic imaging is particularly difficult due to the complex overburdens, caused by the movement of salt that usually results in steeply dipping structures, which along with strong lateral velocity contrast at the sedimentssalt interface distort the structural position and the stratigraphic resolution of the subsalt reservoirs. Despite it was proven that two-way wave equation migration provides the best results illuminating these reservoirs, it has huge computational cost, especially in three dimensions. One-way wave equation (OWWE) migration techniques are good alternative in this case as providing the acceptable quality of seismic sections with an appropriate computational cost. To know the advantages and limitations of the OWWE techniques in subsalt imaging, three classical OWWE algorithms were evaluated for depth migration of prestack data (PSDM): phase-shift-plus-interpolation (PSPI), splitstep Fourier (SSF), and Fourier finite-difference (FFD). These algorithms were tested with three different fully elastic synthetic models which simulates the structural complexity showed in subsalt plays. It was concluded that FFD gives very accurate results when the lateral velocity variation was strong with acceptable computational cost. The PSPI provided the best quality results but required about twice the computer time needed for FFD, and SSF was the fastest but clearly the least accurate.

Broadband Least-Squares Wave-Equation Migration

We introduce an effective iterative Least-Squares Wave-Equation Migration (LS-WEM) solution for broadband imaging. Least-Squares Migration (LSM) solutions are designed to produce images of the subsurface corrected for wavefield distortions caused by acquisition and propagation effects. They implicitly solve for the earth reflectivity by means of data residual reduction in an iterative fashion, which usually demands intensive computation. The LS-WEM is implemented using a visco-acoustic anisotropic one-way wave-equation wavefield propagator that is able to fully utilize both the broader seismic bandwidth and the high-resolution velocity information from Full Waveform Inversion (FWI). Our implementation combines the one-way extrapolator with fast linear inversion solvers into an efficient migration inversion system. Application to the 2D Sigsbee2b synthetic model improves the sub-salt illumination by balancing the image amplitudes and reducing the effects of the shadow zones, enhances temporal resolution by broadening the frequency spectrum, balances the wavenumber content and improves images of faults and dipping salt flanks. In addition, LS-WEM converges rapidly to the true solution, reducing the data residuals by 90% in only four iterations. Application to real 3D datasets from the Gulf of Mexico and the North Sea demonstrates high-resolution imaging with reduced acquisition footprint effects, improved spatial frequency content, and better structural imaging at all depths.

Local discontinuous Galerkin method for the nonlocal one-way water wave equation

In this paper, we develop a local discontinuous Galerkin (LDG) method for numerically solving the nonlocal one-way water wave equation. Based on the features of fractional derivative, the considered model is first coupled into a classical first derivative and a nonlocal fractional integral. Then LDG algorithm is used in space discretization by properly choosing the numerical fluxes. Numerical examples are provided to show the accuracy and effectiveness.

Fast plane-wave reverse time migration

Plane-wave reverse time migration (RTM) could potentially provide quick subsurface images by migrating fewer plane-wave gathers than shot gathers. However, the time delay between the first and the last excitation sources in the plane-wave source largely increases the computation cost and decreases the practical value of this method. Although the time delay problem is easily overcome by periodical phase shifting in the frequency domain for one-way wave-equation migration, it remains a challenge for time-domain RTM. We have developed a novel method, referred as to fast plane-wave RTM (FP-RTM), to eliminate unnecessary computation burden and significantly reduce the computational cost. In the proposed FP-RTM, we assume that the Green’s function has finite-length support; thus, the plane-wave source function and its responding data can be wrapped periodically in the time domain. The wrapping length is the assumed total duration length of Green’s function. We also determine that only two period plane-wave source and data after the wrapping process are required for generating the outcome with adequate accuracy. Although the computation time for one plane-wave gather is twice as long as a normal shot gather migration, a large amount of computation cost is saved because the total number of plane-wave gathers to be migrated is usually much less than the total number of shot gathers. Our FP-RTM can be used to rapidly generate RTM images and plane-wave domain common-image gathers for velocity model building. The synthetic and field data examples are evaluated to validate the efficiency and accuracy of our method.

Dispersive Behavior of an Energy-Conserving Discontinuous Galerkin Method for the One-Way Wave Equation

The dispersive behavior of the recently proposed energy-conserving discontinuous Galerkin (DG) method by Fu and Shu [10] is analyzed and compared with the classical centered and upwinding DG schemes. It is shown that the new scheme gives a significant improvement over the classical centered and upwinding DG schemes in terms of dispersion error. Numerical results are presented to support the theoretical findings.

An improved hybrid absorbing boundary condition for wave equation modeling

A recently developed hybrid absorbing boundary condition (ABC) can significantly suppress artificial boundary reflections by inserting a transition area between the boundary and the inner area. The wavefield in the transition area is computed by linearly weighting the wavefield from one-way wave equation (OWWE) and that from two-way wave equation. This leads to a smooth variation from the computational area to the boundary via the transition area and thus greatly reduces boundary reflections. In this paper, we propose two techniques to further enhance the absorbing effect. First, we widen the boundary from one point to several points. Second, we adopt nonlinear weighting coefficients instead of linear ones. Numerical simulations with 2D acoustic finite-difference modeling, combined with Clayton–Engquist OWWE, demonstrate that the proposed techniques can significantly improve the absorption effect without increasing computational cost. Tests using Higdon OWWE-based ABC suggest that the second technique can greatly enhance the absorption effect. Experiments with frequency-domain finite-difference modeling demonstrate similar improvement in absorption effect.

Parallel GPU-based Implementation of One-Way Wave Equation Migration

We present an original algorithm for seismic imaging, based on the depth wavefield extrapolation by the one-way wave equation. Parallel implementation of the algorithm is based on the several levels of parallelism. The input data parallelism allows processing full coverage for some area (up to one square km); thus, data are divided into several subsets and each subset is processed by a single MPI process. The mathematical approach allows dealing with each frequency independently and treating solution layer-by-layer; thus, a set of 2D cross-sections instead of the initial 3D common-offset vector gathers are processed simultaneously. This part of the algorithm is implemented suing GPU. Next, each common-offset vector image can be stacked, processed and stored independently. As a result, we designed and implemented the parallel algorithm based on the use of CPU-GPU architecture which allows computing common-offset vector images using one-way wave equation-based amplitude preserving migration. The algorithm was used to compute seismic images from real seismic land data.

One-way true-amplitude migration using matrix decomposition

To meet the requirement of true-amplitude migration and address the shortcomings of the classic one-way wave equations on the dynamic imaging, one-way true-amplitude wave equations were developed. Migration methods, based on the Taylor or other series approximation theory, are introduced to solve the one-way true-amplitude wave equations. This leads to the main weakness of one-way true-amplitude migration for imaging the complex or strong velocity — contrast media — the limited imaging angles. To deal with this issue, we apply a matrix decomposition method to accurately calculate the square-root operator and impose the boundary conditions of the one-way true-amplitude wave equations. Our migration method and the conventional one-way true-amplitude Fourier finite-difference (FFD) migration method are used by us to test and compare the imaging performance. The impulse responses in a strong velocity-contrast model prove that our migration method works for larger imaging angles than the one-way true-amplitude FFD method. The amplitude calculations in a strong-lateral velocity variation media with one reflector and in the Marmousi model demonstrate that our migration method provides better amplitude-preserving performance and offers higher structural imaging quality than the one-way true-amplitude FFD method. We also use field data to indicate the imaging enhancement and the feasibility of our method compared with the one-way true-amplitude FFD method. Our one-way true-amplitude migration method using matrix decomposition fully exploits the features of one-way true-amplitude wave equations with less approximation, and it is capable of producing more accurate amplitude estimations and potentially wider imaging angles.

Near-surface velocity estimation using source-domain full traveltime inversion and early arrival waveform inversion

Accurate estimation of near-surface velocity is a key step for imaging deeper targets. We have developed a new workflow to invert complex early arrivals in land seismic data for near-surface velocities. This workflow is composed of two methods: source-domain full traveltime inversion (FTI) and early arrival waveform inversion (EWI). Source-domain FTI automatically generates the background velocity that kinematically matches the reconstructed plane-wave sources from early arrivals with true plane-wave sources. This method does not require picking first arrivals for inversion, which is one of the most challenging and labor-intensive steps in ray-based first-arrival traveltime tomography, especially when the subsurface medium contains low-velocity zones that cause shingled multivalue arrivals. Moreover, unlike the conventional Born-based method, source-domain FTI can determine if the initial velocity is slower or faster than the true one according to the gradient sign. In addition, the computational cost is reduced considerably by using the one-way wave equation to extrapolate the plane-wave Green’s function. The source-domain FTI tomogram is then used as the starting model for EWI to obtain the short-wavelength component associated with the velocity model. We tested the workflow on two synthetic and one onshore filed data sets. The results demonstrate that source-domain FTI generates reasonable background velocities for EWI even though the first arrivals are shingled, and that this workflow can produce a high-resolution near-surface velocity model.

Computing Where Perturbations Affect the Acoustic Impulse Response in the Ocean

We compute accurate maps of oceanic perturbations affecting transient acoustic signals propagating from source to receiver. The technological advance involves coupling the one-way wave equation (OWWE) propagation model with the theory for the Differential Measure of Influence (DMI) yielding the map. The DMI requires two finite-frequency solutions of the acoustic wave equation obeying reciprocity: from source to receiver and vice versa. OWWE satisfies reciprocity at basin-scales with sound speed varying horizontally and vertically. At infinite frequency, maps of the DMI collapse into rays. Mapping the DMI is useful for understanding measurements of acoustic perturbations at finite frequencies.

3D superwide-angle one-way propagator and its application in seismic modeling and imaging

Traditional one-way wave-equation based propagators have been widely used in past decades. Comparing to two-way propagators, one-way methods have higher efficiency and lower memory demands. These two features are especially important in solving large-scale 3D problems. However, regular one-way propagators cannot simulate waves that propagate in large angles within 90° because of their inherent wide angle limitation. Traditional one-way can only propagate along the determined direction (e.g., z-direction), so simulation of turning waves is beyond the ability of one-way methods. We develop 3D superwide-angle one-way propagator to overcome angle limitation and to simulate turning waves with superwide-angle propagation angle (>90°) for modeling and imaging complex geological structures. Wavefields propagating along vertical and horizontal directions are combined using typical stacking scheme. A weight function related to the propagation angle is used for combining and updating wavefields in each propagating step. In the implementation, we use graphics processing units (GPU) to accelerate the process. Typical workflow is designed to exploit the advantages of GPU architecture. Numerical examples show that the method achieves higher accuracy in modeling and imaging steep structures than regular one-way propagators. Actually, superwide-angle one-way propagator can be applied based on any one-way method to improve the effects of seismic modeling and imaging.

The stabilization of high-order multistep schemes for the Laguerre one-way wave equation solver

This paper considers spectral-finite difference methods of a high-order of accuracy for solving the one-way wave equation using the Laguerre integral transform with respect to time as the base. In order to provide a high spatial accuracy and stability, the Richardson method can be employed. However such an approach requires high computer costs, therefore we consider alternative algorithms based on the Adams multistep schemes. To reach the stability for the one-way equation, the stabilizing procedures using the spline interpolation were developed. This made it possible to efficiently implement a predictor–corrector type method thus decreasing computer costs. The stability and accuracy of the procedures proposed have been studied, based on the implementation of the migration algorithm within a problem of seismic prospecting.

On the accuracy of the Complex-Step-Finite-Difference method

The Complex-Step-Finite-Difference method (CSFDM) is a very simple methodology that can be implemented in well known numerical techniques helping to improve, for instance in the wave propagation problem, time and/or space derivative based wavefields. We clarify differences between the CSFDM and previous implementations of the Complex-Step (CS) derivative approximation in well known numerical techniques. We study dispersion properties for the one-way and two-way wave equations using the Finite-Difference method (FDM), the Pseudospectral method (PSM), the Finite-Element method (FEM) and the CSFDM, under the influence of a plane wave and Ricker source time functions. We show the gain in numerical accuracy offered by the methodology of the CSFDM over the FDM, PSM and FEM. We finally discuss consequences of the CSFDM in future scenarios and propose directions of study in this area.