3D weak-dispersion reverse time migration using a stereo-modeling operator

Reliable 3D imaging is a required tool for developing models of complex geologic structures. Reverse time migration (RTM), as the most powerful depth imaging method, has become the preferred imaging tool because of its ability to handle complex velocity models including steeply dipping interfaces and large velocity contrasts. Finite-difference methods are among the most popular numerical approaches used for RTM. However, these methods often encounter a serious issue of numerical dispersion, which is typically suppressed by reducing the grid interval of the propagation model, resulting in large computation and memory requirements. In addition, even with small grid spacing, numerical anisotropy may degrade images or, worse, provide images that appear to be focused but position events incorrectly. Recently, stereo-operators have been developed to approximate the partial differential operator in space. These operators have been used to develop several weak-dispersion and efficient stereo-modeling methods that have been found to be superior to conventional algorithms in suppressing numerical dispersion and numerical anisotropy. We generalized one stereo-modeling method, fourth-order nearly analytic central difference (NACD), from 2D to 3D and applied it to 3D RTM. The RTM results for the 3D SEG/EAGE phase A classic data set 1 and the SEG Advanced Modeling project model demonstrated that, even when using a large grid size, the NACD method can handle very complex velocity models and produced better images than can be obtained using the fourth-order and eighth-order Lax-Wendroff correction (LWC) schemes. We also applied 3D NACD and fourth-order LWC to a field data set and illustrated significant improvements in terms of structure imaging, horizon/layer continuity and positioning. We also investigated numerical dispersion and found that not only does the NACD method have superior dispersion characteristics but also that the angular variation of dispersion is significantly less than for LWC. Read More: http://library.seg.org/doi/abs/10.1190/geo2013-0472.1National Natural Science Foundation (China) (Grant 41230210)Massachusetts Institute of Technology. Earth Resources Laboratory (Founding Members Consortium

Arbitrary-order symplectic time integrator for the acoustic wave equation using the pseudo-spectral method

학위논문 (박사)-- 서울대학교 대학원 : 협동과정 계산과학전공, 2017. 2. 신창수.A Hamiltonian system is symplectic. To simulate a Hamiltonian system, symplectic time integrators are generally appliedotherwise, the energy or the generalized energy is not conserved in the volume of interest. In this study, the symplectic nature of the acoustic wave system is proven. Then, a symplectic scheme that can be extended arbitrarily in temporal dimensions is suggested. The method is based on the Lax-Wendroff expansion of the time differentiation of acoustic wave variables, such as pressure and velocity, existing on the staggered time axis, i.e., one is on the integer grid, and the other is defined on the half integer of the time step. The series can be reduced to the pseudo-differential operator, which enables the application of other approximation techniques, such as the Jacobi-Anger expansion. By virtue of considering the property of the nature of the acoustic wave phenomena, the scheme is more stable and accurate than methods that do not consider symplecticity. Moreover, the phase error per time step can be kept sufficiently small to conduct simulation over long periods of time. According to the analysis of the scheme, the larger the time strides are, the more efficient the simulation is in terms of computing power when a sufficient number of multiplications of the map are accumulated. The effectiveness and accuracy are verified through simulation results using a homogeneous model in which the computed wavefield is equivalent to the analytic solution. The numerical results of the wavefield in the heterogeneous model also yield equivalent results irrespective of the time step lengths. The scheme can be applied to the source problemshowever, the time step is confined to describing the entire frequency component of the wavelet.1. Introduction 1 1.1. Background 1 1.2. Overview 8 1.3. Outline 10 2. Theory 11 2.1. Acoustic wave equation 11 2.2. Symplecticity and symplectic time integrator 18 2.2.1. Symplecticity of the transformation map 18 2.2.2. Symplectic time integrator 21 2.3. Arbitrary-order symplectic time integrator 26 3. Analysis 31 3.1. Stability analysis 32 3.2. Dispersion analysis 37 3.3. Phase analysis 45 3.4. Spectral accuracy and compromise 56 3.5. Source wavelet issue 67 4. Numerical Examples 70 4.1. Initial value problems 71 4.1.1. Homogeneous model 71 4.1.2. Synthetic heterogeneous model: Marmousi-2 72 4.2. Source problems 90 4.2.1. Homogeneous model 90 4.2.2. Synthetic heterogeneous model: Marmousi-2 91 4.3. Discussion on factors debasing the accuracy 107 5. Conclusions 112 References 115 Appendix A. Additional formulations 121 A1. Absorbing boundary conditions 121 A2. Analytic solution 126 Appendix B. Matlab codes 128 B1. Arbitrary-order symplectic time operator 128 B2. Analytic solution 131 초록 133Docto

Computational Electromagnetism and Acoustics

It is a moot point to stress the significance of accurate and fast numerical methods for the simulation of electromagnetic fields and sound propagation for modern technology. This has triggered a surge of research in mathematical modeling and numerical analysis aimed to devise and improve methods for computational electromagnetism and acoustics. Numerical techniques for solving the initial boundary value problems underlying both computational electromagnetics and acoustics comprise a wide array of different approaches ranging from integral equation methods to finite differences. Their development faces a few typical challenges: highly oscillatory solutions, control of numerical dispersion, infinite computational domains, ill-conditioned discrete operators, lack of strong ellipticity, hysteresis phenomena, to name only a few. Profound mathematical analysis is indispensable for tackling these issues. Many outstanding contributions at this Oberwolfach conference on Computational Electromagnetism and Acoustics strikingly confirmed the immense recent progress made in the field. To name only a few highlights: there have been breakthroughs in the application and understanding of phase modulation and extraction approaches for the discretization of boundary integral equations at high frequencies. Much has been achieved in the development and analysis of discontinuous Galerkin methods. New insight have been gained into the construction and relationships of absorbing boundary conditions also for periodic media. Considerable progress has been made in the design of stable and space-time adaptive discretization techniques for wave propagation. New ideas have emerged for the fast and robust iterative solution for discrete quasi-static electromagnetic boundary value problems

Nonlinear Evolution Equations: Analysis and Numerics

The qualitative theory of nonlinear evolution equations is an important tool for studying the dynamical behavior of systems in science and technology. A thorough understanding of the complex behavior of such systems requires detailed analytical and numerical investigations of the underlying partial differential equations

On leapfrog-Chebyshev schemes for second-order differential equations

In this thesis the efficient time integration of semilinear second-order ordinary differential equations is investigated. Based on the leapfrog (Störmer, Verlet) scheme a new class of explicit two-step schemes is constructed by utilizing Chebyshev polynomials. For deriving rigorous error bounds of these leapfrog-Chebyshev (LFC) schemes a more general class of two-step schemes is introduced. Precise conditions are stated for this general class guaranteeing stability as well as second-order convergence in time. In addition, the influence of the starting value is analyzed in detail. Furthermore, by combining the leapfrog scheme with this general class of schemes a class of multirate two-step methods is constructed. Sufficient conditions for the stability of these schemes are derived as well as error bounds showing the second-order convergence in time. For both the LFC schemes and the multirate schemes if equipped with the LFC schemes it is shown that in specific situations they outperform the leapfrog scheme. Numerical examples are provided to illustrate the theoretical results