Lowrank Finite-difference Method for Elastic Wave Propagation

Elastic wave extrapolation in the time domain is significant for an elastic wave equation-based processing. To improve the simulation reliability and accuracy of decoupled elastic P- and S- waves, we propose the staggered-grid lowrank finite-difference method based on the elastic wave decomposition. For elastic wave propagation, a lowrank finite-difference method based on the staggered grid is derived to improve the accuracy. Regarding the application of the decoupled elastic wave equation, we derive the finite-difference scheme coefficients which are dependent on velocity. Based on the elastic wave decomposition and plane wave theories, we formulate the elastic wave-extrapolation operators, which contain trigonometric adjustment factors. Accordingly, by applying the lowrank method to approximating the operators, the finite- difference scheme is designed to discretize the decoupled wave equation. The derivation processing implies the combination of elastic wave-mode decomposition and extrapolation. The proposed method enables elastic P- and S-waves to extrapolate in the time-space domain separately and produces accurate P-and S-wave components simultaneously. Dispersion analysis suggests that our proposed method is reliable and accurate in a wide range of wavenumber. Numerical simulation tests on a simple model and the Marmousi2 model validate the accuracy and effectiveness of the method, showing its ability in handling complex structures. Although the operators are accurate only when the medium is homogeneous, they are of high accuracy when the velocity gradient is quite small and are applicable when the velocity gradient is large. The subsequent results of reverse time migration for the Marmousi2 model also suggest that the proposed method is enough to serve as an extrapolator in elastic reverse time migration.

Implicit staggered‐grid finite‐difference methods are attractive for elastic wave modelling due to significantly enhanced spatial accuracy compared to explicit ones. However, the central‐grid finite‐difference operators used to approximate the temporal derivatives result in a limited accuracy in time. Temporal high‐order finite‐difference methods have the ability to weaken the temporal dispersion and improve the modelling stability. It is noted that the previous temporal high‐order and spatial implicit finite‐difference methods are all designed in the space domain for performing acoustic wave propagation. To implement 2‐D elastic wave simulation with high‐order accuracy both in space and time, we propose two time–space domain implicit staggered‐grid finite‐difference schemes, in which the spatial derivatives are approximated by the weighted average of a few extra off‐axial nodes and axial nodes of the conventional cross‐stencil. We derive the P‐ and S‐wave dispersion relations of the whole elastic wave equation and estimate the finite‐difference coefficients via a variable substitution‐based Taylor‐series expansion. Our Taylor‐series expansion‐based new scheme yields high‐order temporal and spatial accuracy. Besides, the spatial accuracy can be further enhanced by our newly proposed linear optimization strategy, which benefits from easy implementation since we only optimize the axial spatial coefficients via a least‐squares strategy and set the off‐axial temporal coefficients the same as the solution of the Taylor‐series expansion method. Besides, the P‐ and S‐wave separation approach is adopted to propagate the P‐ and S‐wavefields with the P‐ and S‐wave dispersion relation‐based finite‐difference operators, respectively. Our two new schemes are more capable of suppressing the numerical dispersion and exhibit better stability performance compared to conventional one, as we will illustrate via a detailed analysis of dispersion, stability and numerical experiments. In addition, a comparison of computation times demonstrates the efficiency advantage of two new schemes since small operator lengths and large time steps are allowed.

A unified variational framework for the space discontinuous Galerkin method for elastic wave propagation in anisotropic and piecewise homogeneous media

A particle-based continuum model is presented for the simulation of seismic wave propagation. Accurate simulation of seismic ground motion for the model with arbitrary free surface of the ground is a key issue for not only scientific interest but also in practical fields to estimate seismic signals traveled through subsurface structure. In the present study, a Hamiltonian particle method (HPM) is proposed to seismological numerical modeling. HPM has some advantages comparing to traditional continuous and discontinuous methods in terms of; (i) the introduction of the traction-free boundary conditions is easier than that in finite difference methods (FDM), (ii) data structure in HPM is simpler than that in finite element methods (FEM) because of no need to consider the connectivity between nodes and elements when discretizing object of analysis, and (iii) no need to calibrate material parameters that is a cumbersome process in distinct element method (DEM) with a hexagonal arrangement. In this paper, we first calculate the seismic wave propagation in a homogeneous media and compare the results with that from FDM to demonstrate the performance of the method. Next, we simulate surface wave propagation in a model with arbitrary surface topology. These results indicate that HPM can calculate elastic and surface wave propagation with sufficient accuracy for models with a free surface of arbitrary shape.

SUMMARY Quantum computing has attracted considerable attention in recent years because it promises speedups that conventional supercomputers cannot offer, at least for some applications. Though existing quantum computers (QCs) are, in most cases, still too small to solve significant problems, their future impact on domain sciences is already being explored now. Within this context, we present a quantum computing concept for 1-D elastic wave propagation in heterogeneous media with two components: a theoretical formulation and an implementation on a real QC. The method rests on a finite-difference approximation, followed by a sparsity-preserving transformation of the discrete elastic wave equation to a Schrödinger equation, which can be simulated directly on a gate-based QC. An implementation on an error-free quantum simulator verifies our approach and forms the basis of numerical experiments with small problems on the real QC IBM Brisbane. The latter produce simulation results that qualitatively agree with the error-free version but are contaminated by quantum decoherence and noise effects. Complementing the discrete transformation to the Schrödinger equation by a continuous version allows the replacement of finite differences by other spatial discretization schemes, such as the spectral-element method. Anticipating the emergence of error-corrected quantum chips, we analyse the computational complexity of the best quantum simulation algorithms for future QCs. This analysis suggests that our quantum computing approach may lead to wavefield simulations that run exponentially faster than simulations on classical computers.

In this paper, a new stable finite-difference (FD) method for solving elastodynamic equations is presented and applied on the Biot and Biot/squirt (BISQ) models. This method is based on the operator splitting theory and makes use of the characteristic boundary conditions to confirm the overall stability which is demonstrated with the energy method. Through the stability analysis, it is showed that the stability conditions are more generous than that of the traditional algorithms. It allows us to use the larger time step τ in the procedures for the elastic wave field solutions. This context also provides and compares the computational results from the stable Biot and unstable BISQ models. The comparisons show that this FD method can apply a new numerical technique to detect the stability of the seismic wave propagation theories. The rigorous theoretical stability analysis with the energy method is presented and the stable/unstable performance with the numerical solutions is also revealed. The truncation errors and the detailed stability conditions of the FD methods with different characteristic boundary conditions have also been evaluated. Several applications of the constructed FD methods are presented. When the stable FD methods to the elastic wave models are applied, an initial stability test can be established. Further work is still necessary to improve the accuracy of the method.

A time-space domain stereo finite difference method for 3D scalar wave propagation

The propagation of elastic transient waves in a multi–layered medium subjected to in–plane loadings is investigated in this study. One of the objectives of this study is to develop an effective analytical method for determining transient full–field solutions in the layered medium. A matrix method is developed by expanding the matrix solution obtained directly from the boundary–value problem in the integral transform domain into a power series of the phase–related reflection and transmission matrix which characterizes the multiple reflections and transmissions of all waves in every layer. The transient response of the multi–layered medium is decomposed into infinite wave groups in which the waves are either reflected by, or transmitted through, the interfaces. The connection between the proposed matrix method and the generalized ray method is established for the layered medium in the transform domain. The matrix representation of the solution enables us to calculate the transient response of the layered medium without tracing the ray path manually. The obtained analytical solution can easily be applied to numerical calculations. The double inverse transform is performed based on Cagniard9s method and the theoretical transient solution for a layered half–space subjected to an in–plane dynamic force will be presented in part II. An experimental set–up that simulates the plane stress condition for a layered half–space was established to obtain the dynamic displacement response. The experimental result agrees well with the theoretical solution. The proposed methodology in this study can be extended to solve more complicated problems such as waves propagating in three–dimensional space.

Large-eddy simulation of turbulent channel flows with conservative IDO scheme

In this paper, we apply the CIP (Cubic Interpolated Profile) method to the simulation of elastic waves as a highly accurate and stable algorithm to solve characteristic equations. The key idea of the CIP is that not only the physical value itself but also its first spatial derivative obeys the same characteristic equations. Using this property the solution is interpolated by cubic polynomials and interpolation coefficients can be evaluated arithmetically. We implemented this idea to the elastic wave simulation by derivation of the characteristic equations from the basic equations of motion. The characteristic equations can be interpreted as the combined one-way wave equations for each mode of wave. Then, we define boundary conditions by using the merit of one-side propagation; free surface, solid-fluid boundary, irregular topography. From simulation study and the stability evaluation, we recognize the method of characteristics with the CIP is a very powerful simulation technique for the elastic wave propagation. Numerical dispersion is negligible, requiring about half the number of grid cells per wavelength than other solvers. This allows accurate, high-frequency, full-wavefield simulation in models with highly variable, random elastic contrasts with fluid-solid mixed media and complex topographic media.

Traditionally reservoir simulation has been accomplished by solution of reservoir flow models using variations of the methods of finite differences and less frequently by finite elements. The accuracy of these methods depends heavily on the size of grid blocks and time steps employed for the numerical solution. Consequently, computational time and memory requirements may be overwhelming in many cases. In the present study, a recently proposed method called the finite analytic method is shown to alleviate the difficulties associated with the conventional methods. The finite analytic method is operationally and principally different than the conventional finite difference and finite element methods in that a general analytical solution of the simulation model is used over mesh elements to obtain a numerical scheme. Therefore, for comparable accuracy, finite analytic method can lead to higher order approximations than those achieved by the finite difference and finite element techniques. Finite analytic schemes are derived from the reservoir flow models and therefore they are naturally problem dependent. This is the reason for the efficiency of the finite analytic method. By this method problems such as numerical diffusion, numerical instability, slow convergence, as well as requirements of large number of time steps and grid blocks are significantly reduced. In this study, the methodology of formulating and calculating with this method is illustrated by means of a typical two-phase immiscible flow problem described by the Buckley-Leverett equation. The results are compared with the analytic solution of Buckley-Leverett problem and solutions obtained by finite difference and finite element methods. The present study demonstrates that the finite analytic method is capable of producing accurate flow behavior in porous media even when relatively large time steps and grid blocks are used.

Viscoelastic materials are widely used for vibroacoustic solutions due to their ability to dissipate energy and thus mitigate vibrations and structure-borne sound. Wave propagation methods belong to a growing family of material characterization techniques based on the analysis of waveform patterns in one-dimensional structures. The time evolution of patterns results from material elasticity and damping. Conversely, it is possible to identify these properties once the pattern's time evolution is known. The most frequently used propagation methods, namely, Hopkinson bar methods, assume no dispersion, i.e., the complex elasticity modulus is not frequency dependent. However, this is not relevant for resilient materials such as elastomers. More recent approaches have been developed to measure frequency-dependent properties from a pulse propagating in a slender bar. In the previous works, we showed how to adapt these methods for shorter samples of materials. This represented a real advance, as extrusion is a cumbersome process for many materials. The main concept was to reconstruct the time history of the wave propagating in a composite structure composed of a long incident bar made of a known material and extended by a shorter sample bar. The sample material's viscoelastic properties were then determined in the frequency domain by optimization and wave fitting in the time domain. In industry, most insulation solutions that rely on mounts and bushings have to support structural weights. Consequently, it is particularly useful to measure the tangent viscoelastic properties of the material around the static stressed state and not only in stress-free conditions. Measuring strained materials not only requires modifying the experimental apparatus but also extending the scope of the underlying theory as the waves propagate differently in composite structures made of strained media. Here, we show how to overcome this challenging issue. The theoretical framework and its numerical implementation are described, and the method was validated experimentally by characterizing an elastomer sample under prestrained conditions.

To make time-domain computation more efficient in optical structures, the wave equation is rewritten to track only the envelope of the electric field. The resultant equation is complex in nature and it is solved via a mapping concept and a Pade approximation. Compared to the finite-difference time-domain algorithm, this method takes two orders of magnitude less memory. This method is applied to the propagation of a 500 fsec pulse in a waveguide over 2 mm. Transient behavior and the effects of waveguide parameters are demonstrated. >

In the recent past the Discontinuous Galerkin Method has attracted the interest of researchers in numerical seismology due in part to the flexibility to implement boundary conditions and media discontinuities, the ability to model wave field discontinuities and to approximate the wave field with arbitrary high order polynomials, and because the method can be energy conservative and it is suitable for parallel implementations. In this paper we investigate the applicability of the Interior Penalty Discontinuous Galerkin Method (IP-DGM) for elastic wave propagation by analyzing its grid dispersion. We consider different types of basis functions that naturally yield a diagonal mass matrix. This is relevant to seismology because a diagonal mass matrix is tantamount to an explicit and efficient time marching scheme. We find that the Legendre basis functions that are traditionally used in IP-DGM introduce numerical dispersion and anisotropy. Furthermore, we find that using Lagrange basis functions along with the Gauss nodes has attractive advantages for numerical wave propagation.

SUMMARY Lattice methods are a class of numerical scheme which represent a medium as a connection of interacting nodes or particles. In the case of modelling seismic wave propagation, the interaction term is determined from Hooke’s Law including a bond-bending term. This approach has been shown to model isotropic seismic wave propagation in an elastic or viscoelastic medium by selecting the appropriate underlying lattice structure. To predetermine the material constants, this methodology has been restricted to regular grids, hexagonal or square in 2-D or cubic in 3-D. Here, we present a method for isotropic elastic wave propagation where we can remove this lattice restriction. The methodology is outlined and a relationship between the elasticmaterial properties and an irregular latticegeometry arederived. The numerical method is compared with an analytical solution for wave propagation in an infinite homogeneous body along with comparing the method with a numerical solution for a layered elastic medium. The dispersion properties of this method are derived from a plane wave analysis showing the scheme is more dispersive than a regular lattice method. Therefore, the computational costs of using an irregular lattice are higher. However, by removing the regular lattice structure the anisotropic nature of fracture propagation in such methods can be removed.