Migration with arbitrarily wide-angle wave equations

Geophysics ◽  
2005 ◽  
Vol 70 (3) ◽  
pp. S61-S70 ◽  
Author(s):  
Murthy N. Guddati ◽  
A. Homayoun Heidari
Keyword(s):  
Impulse Response ◽  
Wave Equations ◽  
Heterogeneous Media ◽  
Computational Effort ◽  
Downward Continuation ◽  
Wide Angle ◽  
Auxiliary Variables ◽  
Space Domain ◽  
Marching Scheme ◽  
Migration Technique

We develop a new scalar migration technique that is highly accurate for imaging steep dips in heterogeneous media. This method is based on arbitrarily wide-angle wave equations (AWWEs) that are highly accurate space-domain one-way wave equations and have a form similar to the 15° equation. The accuracy of the proposed method is increased by introducing auxiliary variables, as well as adjusting the parameters of the approximation. Poststack migration is carried out by downward continuation using the AWWE, for which we have developed a stable, explicit, double-marching scheme. Up to 80° accuracy is achieved by second-order AWWE migration with only 2.3 times the computational effort of the 15° equation and requiring almost the same storage. We illustrate the performance of AWWE migration using impulse-response graphs, a single-dipping reflector, and a slice of the SEG/EAGE salt model.

Time-space-domain implicit finite-difference methods for modeling acoustic wave equations

Geophysics ◽  
2018 ◽  
Vol 83 (4) ◽  
pp. T175-T193 ◽  
Author(s):  
Enjiang Wang ◽  
Jing Ba ◽  
Yang Liu
Keyword(s):  
Finite Difference ◽  
Acoustic Wave ◽  
Wave Equations ◽  
Finite Difference Methods ◽  
Second Order ◽  
Computational Time ◽  
Time Step ◽  
Space Domain ◽  
Time Space ◽  
Temporal Dispersion

It has been proved that the implicit spatial finite-difference (FD) method can obtain higher accuracy than explicit FD by using an even smaller operator length. However, when only second-order FD in time is used, the combined FD scheme is prone to temporal dispersion and easily becomes unstable when a relatively large time step is used. The time-space domain FD can suppress the temporal dispersion. However, because the spatial derivatives are solved explicitly, the method suffers from spatial dispersion and a large spatial operator length has to be adopted. We have developed two effective time-space-domain implicit FD methods for modeling 2D and 3D acoustic wave equations. First, the high-order FD is incorporated into the discretization for the second-order temporal derivative, and it is combined with the implicit spatial FD. The plane-wave analysis method is used to derive the time-space-domain dispersion relations, and two novel methods are proposed to determine the spatial and temporal FD coefficients in the joint time-space domain. First, we fix the implicit spatial FD coefficients and derive the quadratic convex objective function with respect to temporal FD coefficients. The optimal temporal FD coefficients are obtained by using the linear least-squares method. After obtaining the temporal FD coefficients, the SolvOpt nonlinear algorithm is applied to solve the nonquadratic optimization problem and obtain the optimized temporal and spatial FD coefficients simultaneously. The dispersion analysis, stability analysis, and modeling examples validate that the proposed schemes effectively increase the modeling accuracy and improve the stability conditions of the traditional implicit schemes. The computational efficiency is increased because the schemes can adopt larger time steps with little loss of spatial accuracy. To reduce the memory requirement and computational time for storing and calculating the FD coefficients, we have developed the representative velocity strategy, which only computes and stores the FD coefficients at several selected velocities. The modeling result of the 2D complicated model proves that the representative velocity strategy effectively reduces the memory requirements and computational time without decreasing the accuracy significantly when a proper velocity interval is used.

scholarly journals Perfectly matched absorbing layer for modelling transient wave propagation in heterogeneous poroelastic media

2019 ◽  
Author(s):  
Yanbin He ◽  
Tianning Chen ◽  
Jinghuai Gao
Keyword(s):  
Wave Equations ◽  
Heterogeneous Media ◽  
Near Field ◽  
Weak Form ◽  
Transient Wave ◽  
Absorbing Boundary ◽  
Poroelastic Media ◽  
Elastic Wave Equations ◽  
Absorbing Layer ◽  
Biot’S Equations

Abstract The perfectly matched layer (PML) has been demonstrated to be an efficient absorbing boundary for near-field wave simulation. For heterogeneous media, the property of the PML needs to be carefully specified to avoid numerical instability and artificial reflection because part of it lies at the discontinuous interface. Coupled acoustic-poroelastic (A-P) media or coupled elastic-poroelastic (E-P) media often arise in the field of geophysics. However, PMLs that appropriately terminate these heterogeneous poroelastic media are still lacking. The main purpose of this paper is to explore the application of unsplit PMLs for transient wave modeling in infinite, heterogeneous, coupled A-P media or coupled E-P media. To this end, a consistent derivation of memory-efficient PML formulations for the second-order Biot's equations, elastic wave equations and acoustic wave equations is performed based on complex coordinate transformation using auxiliary differential equations. Furthermore, the interface boundary conditions inside the absorbing layer are rigorously derived for the considered A-P and E-P cases. Finally, the weak form of PML formulations for coupled poroelastic problems is presented. The finite element method is used to validate the proposed PML based on several two-dimensional benchmarks. The accuracy and stability of weak PML formulations are investigated. In particular, for coupled acoustic-poroelastic PML, two extreme (open-pore and sealed-pore) interface conditions are considered and PML results are compared with known analytical solutions. This study demonstrates the ability of the PML to effectively eliminate outgoing bulk waves and surface waves in coupled poroelastic media.

scholarly journals Effective models for the multidimensional wave equation in heterogeneous media over long time and numerical homogenization

2016 ◽  
Vol 26 (14) ◽  
pp. 2651-2684 ◽  
Author(s):  
Assyr Abdulle ◽  
Timothée Pouchon
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Heterogeneous Media ◽  
Multiscale Model ◽  
Bloch Wave ◽  
Time Interval ◽  
Numerical Homogenization ◽  
Effective Equation ◽  
Long Time ◽  
Effective Models

A family of effective equations that capture the long time dispersive effects of wave propagation in heterogeneous media in an arbitrary large periodic spatial domain [Formula: see text] is proposed and analyzed. For a wave equation with highly oscillatory periodic spatial tensors of characteristic length [Formula: see text], we prove that the solution of any member of our family of effective equations is [Formula: see text]-close to the true oscillatory wave over a time interval of length [Formula: see text] in a norm equivalent to the [Formula: see text] norm. We show that the previously derived effective equation in [T. Dohnal, A. Lamacz and B. Schweizer, Bloch-wave homogenization on large time scales and dispersive effective wave equations, Multiscale Model. Simulat. 12 (2014) 488–513] belongs to our family of effective equations. Moreover, while Bloch wave techniques were previously used, we show that asymptotic expansion techniques give an alternative way to derive such effective equations. An algorithm to compute the tensors involved in the dispersive equation and allowing for efficient numerical homogenization methods over long time is proposed.

Time‐domain behavior of wide‐angle one‐way wave equations

Geophysics ◽  
1991 ◽  
Vol 56 (3) ◽  
pp. 382-384
Author(s):  
A. H. Kamel
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Source Function ◽  
Plane Waves ◽  
Wide Angle ◽  
Wave Processes ◽  
Inhomogeneous Wave ◽  
Dirac Delta ◽  
Inhomogeneous Wave Equation ◽  
Standard Tool

The constant‐coefficient inhomogeneous wave equation reads [Formula: see text], Eq. (1) where t is the time; x, z are Cartesian coordinates; c is the sound speed; and δ(.) is the Dirac delta source function located at the origin. The solution to the wave equation could be synthesized in terms of plane waves traveling in all directions. In several applications it is desirable to replace equation (1) by a one‐way wave equation, an equation that allows wave processes in a 180‐degree range of angles only. This idea has become a standard tool in geophysics (Berkhout, 1981; Claerbout, 1985). A “wide‐angle” one‐way wave equation is designed to be accurate over nearly the whole 180‐degree range of permitted angles. Such formulas can be systematically constructed by drawing upon the connection with the mathematical field of approximation theory (Halpern and Trefethen, 1988).

Optimum split-step Fourier 3D depth migration: Developments and practical aspects

Geophysics ◽  
2007 ◽  
Vol 72 (3) ◽  
pp. S167-S175 ◽  
Author(s):  
Jianfeng Zhang ◽  
Linong Liu
Keyword(s):  
Finite Difference ◽  
Heterogeneous Media ◽  
Fourier Transforms ◽  
Computational Cost ◽  
Fourier Method ◽  
Approximate Algorithm ◽  
Wide Angle ◽  
Data Set ◽  
Depth Migration ◽  
Varying Coefficients

We present an efficient scheme for depth extrapolation of wide-angle 3D wavefields in laterally heterogeneous media. The scheme improves the so-called optimum split-step Fourier method by introducing a frequency-independent cascaded operator with spatially varying coefficients. The developments improve the approximation of the optimum split-step Fourier cascaded operator to the exact phase-shift operator of a varying velocity in the presence of strong lateral velocity variations, and they naturally lead to frequency-dependent varying-step depth extrapolations that reduce computational cost significantly. The resulting scheme can be implemented alternatively in spatial and wavenumber domains using fast Fourier transforms (FFTs). The accuracy of the first-order approximate algorithm is similar to that of the second-order optimum split-step Fourier method in modeling wide-angle propagation through strong, laterally varying media. Similar to the optimum split-step Fourier method, the scheme is superior to methods such as the generalized screen and Fourier finite difference. We demonstrate the scheme’s accuracy by comparing it with 3D two-way finite-difference modeling. Comparisons with the 3D prestack Kirchhoff depth migration of a real 3D data set demonstrate the practical application of the proposed method.

A Model for the Fast Determination of Modal Sound Transmission of Large Rectangular Opening

2020 ◽  
Vol 142 (6) ◽  
Author(s):  
Jiazhu Li ◽  
Rui Zhang ◽  
Shen Chen ◽  
Can Li ◽  
Jian Chen
Keyword(s):  
Wave Equations ◽  
Three Dimensional ◽  
Sound Transmission ◽  
Computational Effort ◽  
Sound Insulation ◽  
Fast Determination ◽  
High Frequencies ◽  
Higher Order Modes ◽  
Insulation Performance

Abstract The existence of openings affects the sound insulation performance of structures significantly. The determination of sound transmission through large rectangular openings is often time-consuming, because of the large number of modes, especially if there is a need to go to high frequencies. A model is proposed and detailed based on three-dimensional wave equations, the transfer matrix method, and modal superposition. The viscous and thermal boundary layer effects have been concerned; hence, the model accuracy for narrow slits was improved. The computational effort is significantly decreased by neglecting the cross-modal sound transmission. The accuracy of this model is validated by comparing it with the existing model, the measurement, and the acoustic finite element method. The study of sound transmission behavior of higher-order modes is performed. The modal sound transmission is predicted and compared for several modes. The phenomenon that is different from that of the plane wave situation is found and discussed.

Scale-dependent dispersion for solute transport in bounded formations

2021 ◽  
Author(s):  
Qinzhuo Liao ◽  
Gang Lei ◽  
Dongxiao Zhang ◽  
Shirish Patil
Keyword(s):  
Solute Transport ◽  
Heterogeneous Media ◽  
Three Dimensional ◽  
Computational Effort ◽  
Spatial Covariance ◽  
Covariance Functions ◽  
Promising Tool ◽  
Longitudinal Dispersivity ◽  
Eigen Decomposition ◽  
Variance Approach

<p>We present a new method to estimate the displacement covariance and macrodispersivity for solute transport in bounded formations. Here we use circulant embedding, which is based on the fast Fourier transform and is much more efficient than eigen-decomposition for the factorization of random spatial fields. We compute the displacement covariances using the analysis of variance approach and introduce an interpolation process to significantly reduce the number of forward simulations. Once the effect of each eigenvector on the displacement covariance is obtained, it is unnecessary to rerun the simulator for different spatial covariance functions or anisotropy ratios, which saves a large amount of computational effort. The proposed method is validated in various tests in two-dimensional and three-dimensional examples and accurately matches the results from the Monte Carlo simulation. It is found that the longitudinal dispersivity is not sensitive to the boundaries, while the transverse and vertical dispersivities are greatly affected. The method is applied to the Borden site and provides a better explanation of the observed data after considering the effect of vertical boundaries. These results show that our method could serve as a promising tool for studying and predicting the characteristics of solute transport in heterogeneous media.</p>

Including lateral velocity variations into true-amplitude common-shot wave-equation migration

Geophysics ◽  
2010 ◽  
Vol 75 (5) ◽  
pp. S175-S186 ◽  
Author(s):  
Daniela Amazonas ◽  
Rafael Aleixo ◽  
Gabriela Melo ◽  
Jörg Schleicher ◽  
Amélia Novais ◽  
...  
Keyword(s):  
Wave Equations ◽  
Heterogeneous Media ◽  
Synthetic Data ◽  
Approximate Solutions ◽  
Vertical Gradient ◽  
Correct Description ◽  
Lateral Velocity ◽  
One Dimensional ◽  
Velocity Variations ◽  
The One

In heterogeneous media, standard one-way wave equations describe only the kinematic part of one-way wave propagation correctly. For a correct description of amplitudes, the one-way wave equations must be modified. In media with vertical velocity variations only, the resulting true-amplitude one-way wave equations can be solved analytically. In media with lateral velocity variations, these equations are much harder to solve and require sophisticated numerical techniques. We present an approach to circumvent these problems by implementing approximate solutions based on the one-dimensional analytic amplitude modifications. We use these approximations to show how to modify conventional migration methods such as split-step and Fourier finite-difference migrations in such a way that they more accurately handle migration amplitudes. Simple synthetic data examples in media with a constant vertical gradient demonstrate that the correction achieves the recovery of true migration amplitudes. Applications to the SEG/EAGE salt model and the Marmousi data show that the technique improves amplitude recovery in the migrated images in more realistic situations.

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