Multiscale finite-difference method for frequency-domain acoustic wave modeling

Geophysics ◽  
2021 ◽  
pp. 1-64
Author(s):  
Wei Jiang ◽  
Xuehua Chen ◽  
Shuaishuai Jiang ◽  
Jie Zhang
Keyword(s):  
Linear System ◽  
Finite Difference ◽  
Helmholtz Equation ◽  
Frequency Domain ◽  
Multiscale Method ◽  
Fine Scale ◽  
Coarse Mesh ◽  
Computational Costs ◽  
Time Costs ◽  
Frequency Domain Methods

Conventional finite-difference frequency-domain (FDFD) methods can describe wave attenuation and velocity dispersion more easily than time-domain methods. However, there are significant challenges associated with computational costs for solving the linear system when frequency-domain methods are applied in models with large dimensions or fine-scale property variations. Direct-iterative solvers and parallel strategies attempt a tradeoff between memory and time costs. We follow the general framework of heterogeneous multiscale method and develop a multiscale FDFD approach to solve the Helmholtz equation with lower memory and time costs. To achieve this, the discrete linear system approximating the Helmholtz equation is constructed on a coarse mesh, making its dimension much smaller than that of conventional methods. The coefficient matrix in the linear system of dimension-reduction captures fine-scale heterogeneity in the media by coupling fine- and coarse-scale meshes. Several test models are used to verify the accuracy of our multiscale method and investigate potential sources of error. Numerical results demonstrate that our method accurately approximates the wavefields of fine-scale solutions at low frequencies of the source, and could produce solutions with small errors by reducing the size of the coarse mesh cells at high frequencies as well. Comparisons of computational costs with conventional FDFD methods show that the proposed multiscale method significantly reduces computation time and memory consumption.

An accurate and efficient multiscale finite-difference frequency-domain method for the scalar Helmholtz equation

Geophysics ◽  
2021 ◽  
pp. 1-84
Author(s):  
Wei Jiang ◽  
Xuehua Chen ◽  
Bingnan Lv ◽  
shuaishuai jiang
Keyword(s):  
Linear System ◽  
Finite Difference ◽  
Helmholtz Equation ◽  
Frequency Domain ◽  
Seismic Wave ◽  
Multiscale Method ◽  
Basis Functions ◽  
Difference Frequency ◽  
Frequency Domain Method ◽  
Finite Difference Frequency Domain

Frequency-domain numerical modeling of the seismic wave equation can readily describe frequency-dependent seismic wave behaviors, yet is computationally challenging to perform in finely discretized or large-scale geological models. Conventional finite-difference frequency-domain (FDFD) methods for solving the Helmholtz equation usually lead to large linear systems that are difficult to solve with a direct or iterative solver. Parallel strategies and hybrid solvers can partially alleviate the computational burden by improving the performance of the linear system solver. We develop a novel multiscale FDFD method to eventually construct a dimension-reduced linear system from the scalar Helmholtz equation based on the general framework of heterogeneous multiscale method (HMM). The methodology associated with multiscale basis functions in the multiscale finite-element method (MsFEM) is applied to the local microscale problems of this multiscale FDFD method. Solved from frequency- and medium-dependent local Helmholtz problems, these multiscale basis functions capture fine-scale medium heterogeneities and are finally incorporated into the dimension-reduced linear system by a coupling of scalar Helmholtz problem solutions at two scales. We use several highly heterogeneous models to verify the performance in terms of the accuracy, efficiency, and memory cost of our multiscale method. The results show that our new method can solve the scalar Helmholtz equation in complicated models with high accuracy and quite low time and memory costs compared with the conventional FDFD methods.

An implicit finite-difference operator for the Helmholtz equation

Geophysics ◽  
2012 ◽  
Vol 77 (4) ◽  
pp. T97-T107 ◽  
Author(s):  
Chunlei Chu ◽  
Paul L. Stoffa
Keyword(s):  
Finite Difference ◽  
Helmholtz Equation ◽  
Frequency Domain ◽  
Difference Operator ◽  
Computational Cost ◽  
Finite Difference Discretization ◽  
Implicit Finite Difference ◽  
Explicit Finite Difference ◽  
The One ◽  
Difference Discretization

We have developed an implicit finite-difference operator for the Laplacian and applied it to solving the Helmholtz equation for computing the seismic responses in the frequency domain. This implicit operator can greatly improve the accuracy of the simulation results without adding significant extra computational cost, compared with the corresponding conventional explicit finite-difference scheme. We achieved this by taking advantage of the inherently implicit nature of the Helmholtz equation and merging together the two linear systems: one from the implicit finite-difference discretization of the Laplacian and the other from the discretization of the Helmholtz equation itself. The end result of this simple yet important merging manipulation is a single linear system, similar to the one resulting from the conventional explicit finite-difference discretizations, without involving any differentiation matrix inversions. We analyzed grid dispersions of the discrete Helmholtz equation to show the accuracy of this implicit finite-difference operator and used two numerical examples to demonstrate its efficiency. Our method can be extended to solve other frequency domain wave simulation problems straightforwardly.

3D marine magnetotelluric modeling and inversion with the finite-difference time-domain method

Geophysics ◽  
2014 ◽  
Vol 79 (6) ◽  
pp. E269-E286 ◽  
Author(s):  
Sébastien de la Kethulle de Ryhove ◽  
Rune Mittet
Keyword(s):  
Finite Difference ◽  
Frequency Domain ◽  
Time Domain ◽  
Finite Difference Time Domain ◽  
Linear Equations ◽  
Time Domain Method ◽  
Domain Method ◽  
Frequency Domain Methods ◽  
Starting Point ◽  
Difference Time

Frequency-domain methods, which are typically applied to 3D magnetotelluric (MT) modeling, require solving a system of linear equations for every frequency of interest. This is memory and computationally intensive. We developed a finite-difference time-domain algorithm to perform 3D MT modeling in a marine environment in which Maxwell’s equations are solved in a so-called fictitious-wave domain. Boundary conditions are efficiently treated via convolutional perfectly matched layers, for which we evaluated optimized parameter values obtained by testing over a large number of models. In comparison to the typically applied frequency-domain methods, two advantages of the finite-difference time-domain method are (1) that it is an explicit, low-memory method that entirely avoids the solution of systems of linear equations and (2) that it allows the computation of the electromagnetic field unknowns at all frequencies of interest in a single simulation. We derive a design criterion for vertical node spacing in a nonuniform grid using dispersion analysis as a starting point. Modeling results obtained using our finite-difference time-domain algorithm are compared with results obtained using an integral equation method. The agreement was found to be very good. We also discuss a real data inversion example in which MT modeling was done with our algorithm.

scholarly journals Finite Difference Time Marching in the Frequency Domain: A Parabolic Formulation for the Convective Wave Equation

1996 ◽  
Vol 118 (4) ◽  
pp. 622-629 ◽  
Author(s):  
K. J. Baumeister ◽  
K. L. Kreider
Keyword(s):  
Wave Equation ◽  
Steady State ◽  
Finite Difference ◽  
Frequency Domain ◽  
Sound Propagation ◽  
Time Dependent ◽  
Frequency Domain Method ◽  
Impedance Boundary ◽  
Frequency Domain Methods ◽  
Time Marching

An explicit finite difference iteration scheme is developed to study harmonic sound propagation in ducts. To reduce storage requirements for large 3D problems, the time dependent potential form of the acoustic wave equation is used. To insure that the finite difference scheme is both explicit and stable, time is introduced into the Fourier transformed (steady-state) acoustic potential field as a parameter. Under a suitable transformation, the time dependent governing equation in frequency space is simplified to yield a parabolic partial differential equation, which is then marched through time to attain the steady-state solution. The input to the system is the amplitude of an incident harmonic sound source entering a quiescent duct at the input boundary, with standard impedance boundary conditions on the duct walls and duct exit. The introduction of the time parameter eliminates the large matrix storage requirements normally associated with frequency domain solutions, and time marching attains the steady-state quickly enough to make the method favorable when compared to frequency domain methods. For validation, this transient-frequency domain method is applied to sound propagation in a 2D hard wall duct with plug flow.

An elongated finite-difference scheme for 2D acoustic and elastic wave simulations in the frequency domain

Geophysics ◽  
2019 ◽  
Vol 84 (2) ◽  
pp. T29-T45
Author(s):  
Junichi Takekawa ◽  
Hitoshi Mikada
Keyword(s):  
Finite Difference ◽  
Phase Velocity ◽  
Elastic Wave ◽  
Frequency Domain ◽  
Impedance Matrix ◽  
Memory Usage ◽  
Numerical Accuracy ◽  
Wave Modeling ◽  
Computational Costs ◽  
Grid Points

We have developed a novel scheme to simulate acoustic and elastic wave propagation in the frequency-domain using a rectangular finite-difference (FD) stencil. One of the main problem of the frequency-domain modeling is its huge computational costs, i.e., the calculation time and memory usage. To overcome this problem, researchers have proposed many schemes to reduce the number of grid points in a wavelength. In general, high-accuracy schemes require large-sized stencils that cause increment in the bandwidth of the impedance matrix. It is, therefore, important to improve the accuracy of numerical schemes without increasing the bandwidth. We have applied an elongated stencil with different sampling ratio between horizontal and vertical directions to circumvent extra numerical bandwidth in the impedance matrix. Optimal FD coefficients and the aspect ratio of the grid cell are determined to minimize the error of the phase velocity. We investigate the dispersion property of the proposed scheme using plane-wave analysis. The dispersion analysis indicates that we could reduce the number of grid points in a wavelength by approximately 2.78 for acoustic wave modeling and by approximately 3.15 for elastic wave modeling so that the error of the phase velocity is less than 1%. We also conduct numerical simulations using homogeneous and inhomogeneous models to demonstrate the effectiveness of our scheme. The comparison of numerical accuracy and computational costs between our scheme and the conventional ones indicates that the computational costs (calculation time, memory usage) can be reduced with high numerical accuracy especially in elastic wave modeling. Because our technique is a simple and a powerful cost-efficient frequency-domain method, the elongated stencil can be an alternative scheme to the conventional ones for acoustic and elastic wave modeling.

A finite-difference iterative solver of the Helmholtz equation for frequency-domain seismic wave modeling and full waveform inversion

Geophysics ◽  
2020 ◽  
pp. 1-43
Author(s):  
Xingguo Huang ◽  
Stewart Greenhalgh
Keyword(s):  
Finite Difference ◽  
Helmholtz Equation ◽  
Frequency Domain ◽  
Iteration Method ◽  
Krylov Subspace ◽  
Waveform Inversion ◽  
Full Waveform Inversion ◽  
Iterative Solver ◽  
Subspace Iteration ◽  
Full Waveform

We present a finite difference iterative solver of the Helmholtz equation for seismic modeling and inversion in the frequency-domain. The iterative solver involves the shifted Laplacian operator and two-level pre-conditioners. It is based on the application of the pre-conditioners to the Krylov subspace stabilized biconjugate gradient method. A critical factor for the iterative solver is the introduction of a new pre-conditioner into the Krylov subspace iteration method to solve the linear system resulting from the discretization of the Helmholtz equation. This new pre-conditioner is based upon a reformulation of an integral equation-based convergent Born series for the Lippmann-Schwinger equation to an equivalent differential equation. We demonstrate that the proposed iterative solver combined with the novel pre-conditioner when incorporated with the finite difference method accelerates the convergence of the Krylov subspace iteration method for frequency-domain seismic wave modeling. A comparison of a direct solver, a one-level Krylov subspace iterative solver and the proposed two-level iterative solver verified the accuracy and accelerated convergence of the new scheme. Extensive tests in full waveform inversion demonstrate the solver applicability to full waveform inversion applications.

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