Wave Propagation In Heterogeneous Media Research Articles

Modeling a pure qP wave in attenuative transverse isotropic media using a complex-valued fractional viscoacoustic wave equation

Seismic anisotropy, characterized by direction-dependent seismic velocity and attenuation, is a fundamental property of the earth, widely observed in field and laboratory measurements. This anisotropy significantly influences seismic wavefield attributes, such as amplitude, phase, and traveltime. Accurate quantification of anisotropy-related seismic wave propagation is essential for global- and regional-scale studies in seismology, such as imaging and inversion. Because the elastic assumption of the earth requires contending with S waves, which are often weak in our seismic recording, we derive a new general complex-valued spatial fractional Laplacian (SFL) viscoacoustic anisotropic pure quasi-P (qP) wave equation (WE) in attenuative transverse isotropic (A-TI) media under the acoustic assumption and use the fixed-order SFL operator to quantify seismic wave propagation in heterogeneous media. Our WE accurately captures anisotropic effects in seismic velocity and attenuation, even in strongly attenuative and anisotropic environments. It also describes the decoupled energy attenuation and velocity dispersion behaviors of the frequency-independent quality factor Q, beneficial for seismic imaging and inversion. Compared with existing viscoacoustic anisotropic pure qP WEs in A-TI media, our WE simplifies computational complexity and is easy to implement, as it decomposes into a complex-valued scalar operator and two differential operators. The evaluated dispersion curves further confirm the accuracy of our approach. Numerical examples in complex geologic settings demonstrate the new equation’s precision, stability, and efficiency in modeling seismic wave propagation.

Efficient modeling of fractional Laplacian viscoacoustic wave equation with fractional finite-difference method

The fractional viscoacoustic/viscoelastic wave equation, which accurately quantifies the frequency-independent anelastic effects, has been the focus of seismic industry in recent years. The pseudo-spectral (PS) method stands as one of the most widely used numerical methods for solving the fractional wave equation. However, the PS method often suffers from low accuracy and efficiency, particularly when modeling wave propagation in heterogeneous media. To address these issues, we propose a novel and efficient fractional finite-difference (FD) method for solving the wave equation with fractional Laplacian operators. This method develops an arbitrary high-order FD operator via the generating function of our fractional FD (F-FD) scheme, enhancing accuracy with L2-optimal FD coefficients. Similar to classic FD methods, our F-FD method is characterized by straightforward programming and excellent 3D extensibility. It surpasses the PS method by eliminating the need for Fast Fourier Transform (FFT) and inverse-FFT (IFFT) operations at each time step, offering significant benefits for 3D applications. Consequently, the F-FD method proves more adept for wave-equation-based seismic data processes like imaging and inversion. Compared with existing F-FD methods, our approach uniquely approximates the entire fractional Laplacian operator and stands as a local numerical algorithm, with an adjustable F-FD operator order based on model parameters for enhanced practicality. Accuracy analyses confirm that our method matches the precision of the PS method with a correctly ordered F-FD operator. Numerical examples show that the proposed method has good applicability for complex models. Finally, we have carried out reverse time migration on the Marmousi-2 model, and the imaging profiles indicate that the proposed method can be effectively applied to seismic imaging, demonstrating good practicability.

A quantum computing concept for 1-D elastic wave simulation with exponential speedup

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.

A time-domain SGFD-FK hybrid method for 2D teleseismic elastic wave modeling and inversion

AbstractFull waveform inversion (FWI) has proved to be a reliable tool for high-resolution imaging of lithospheric structures at various depths down to the upper mantle. However, when the size of the model is large, the computational burden is significative and applications are restricted to low frequencies. To tackle this issue, we developed a new 2D time-domain hybrid method to simulate high-frequency teleseismic body waves propagating through a local heterogeneous elastic Earth model: The frequency-wavenumber (FK) integration method is coupled with the staggered grid finite difference method (SGFD). The FK method is used to compute the wavefield due to obliquely incident plane P and SV waves in a 1D multilayered half space that excites a local heterogeneous region. Inside this region, the velocity-stress staggered grid FD method (SGFD) is used to accurately deal with wave propagation in heterogeneous media. Spurious waves that might be generated at the boundaries of the local region are avoided using convolutional perfectly matched layers (CPML). This new hybrid method inherits the low-memory requirements of the FK method and the accuracy, efficiency and easy implementation of the SGFD. The new hybrid method is benchmarked against the analytical FK method for some canonical models and shows good agreement with analytical solutions. Subsequently, our modeling tool is incorporated into a full waveform inversion algorithm adapted for teleseismic configurations to invert the incident P wave and its coda. The inversion is carried out using a gradient approach that is efficiently implemented via the adjoint-state method. The results suggest that our hybrid method and FWI algorithm represent a valuable tool for 2D forward and inverse regional applications using teleseismic data sets.

Single and multiple scattering quantitative ultrasound methods in muscle

Increased intramuscular fat—or fatty infiltration (FI)—often develops as a consequence of various musculoskeletal and neuromuscular disorders. FI is an important factor for the assessment and intervention of surgical candidacy. Therefore, there is a need for an accessible, non-invasive quantitative, and objective evaluation of FI in muscle. We can exploit the complexity of ultrasound wave propagation in heterogeneous media to extract quantitative features of the fat distribution in muscle. From finite-difference time-domain simulations conducted with maps created from magnetic resonance imaging scans of healthy and injured shoulders, we acquire radiofrequency (RF) ultrasound data. Single and multiple scattering (SS and MS) components of the RF data are separated by using singular value decomposition and eigenvalue thresholding. Spectral and envelope quantitative ultrasound (QUS) parameters are computed from the SS component. MS-based QUS parameters are obtained by extracting the diffusion constant and tracking the SS intensity decay rate. We investigate the relationship between SS and MS-based QUS parameters and an increasing FI, because more FI will lead to more multiple scattering and modified ultrasonic signatures.

A Hybrid Chebyshev Pseudo-Spectral Finite-Difference Time-Domain Method for Numerical Simulation of 2D Acoustic Wave Propagation

In this study, a hybrid Chebyshev pseudo-spectral finite-difference time-domain (CPS-FDTD) algorithm is proposed for simulating 2D acoustic wave propagation in heterogeneous media, which is different from the other traditional numerical schemes such as finite element and finite difference. This proposed hybrid method integrates the efficiency of the FDTD approach in the time domain and the high accuracy of the CPS technique in the spatial domain. We present the calculation formulas of this novel approach and conduct simulation experiments to test it. The biconjugate gradient is solved by combining the large symmetric sparse systems stabilized algorithm with an incomplete LU factorization. Three numerical experiments are further presented to illustrate the accuracy, efficiency, and flexibility of the hybrid CPS-FDTD algorithm.

A theragnostic HIFU transducer and system for inherently registered imaging and therapy.

One big challenge with high intensity focused ultrasound (HIFU) is the difficulty in accurate prediction of focal location due to the complex wave propagation in heterogeneous medium even with imaging guidance. This study aims to overcome this by combining therapy and imaging guidance with one single HIFU transducer using the vibro-acoustography (VA) strategy. Based on the VA imaging method, a HIFU transducer consisting of 8 transmitting elements was proposed for therapy planning, treatment and evaluation. Inherent registration between the therapy and imaging created unique spatial consistence in HIFU transducer's focal region in the above three procedures. Performance of this imaging modality was first evaluated through in-vitro phantoms. In-vitro and ex-vivo experiments were then designed to demonstrate the proposed dual-mode system's ability in conducting accurate thermal ablation. Point spread function of the HIFU-converted imaging system had a full wave half maximum of about 1.2 mm in both directions at a transmitting frequency of 1.2 MHz, which outperformed the conventional ultrasound imaging (3.15 MHz) in in-vitro situation. Image contrast was also tested on the in-vitro phantom. Various geometric patterns could be accurately 'burned out' on the testing objects by the proposed system both in vitro and ex vivo. Implementation of imaging and therapy with one HIFU transducer in this manner is feasible and it has potential as a novel strategy for addressing the long-standing problem in the HIFU therapy, possibly pushing this non-invasive technique forward towards wider clinical applications.

A Novel Coupled Meshless Model for Simulation of Acoustic Wave Propagation in Infinite Domain Containing Multiple Heterogeneous Media

This study presents a novel coupled meshless model for simulating acoustic wave propagation in heterogeneous media, based on the singular boundary method (SBM) and Kansa’s method (KS). In the proposed approach, the SBM was used to model the homogeneous part of the propagation domain, while KS was employed to model a heterogeneity. The interface compatibility conditions associated with velocities and pressures were imposed to couple the two methods. The proposed SBM–KS coupled approach combines the respective advantages of the SBM and KS. The SBM is especially suitable for solving external sound field problems, while KS is attractive for nonlinear problems in bounded non-homogeneous media. Moreover, the new methodology completely avoids grid generation and numerical integration compared with the finite element method and boundary element method. Numerical experiments verified the accuracy and effectiveness of the proposed scheme.

Numerical simulation of anti-plane wave propagation in heterogeneous media

This paper formulates a semi-analytical boundary meshless method, the singular boundary method (SBM), to simulate the anti-plane wave motion in the heterogeneous media. Two different continuous inhomogeneity variations are studied, and the corresponding fundamental solutions are derived by means of a simple variable transformation. The derivation reveals that the fundamental solutions are determined by the wavenumbers of the resulted governing equations after the variable transformation. The source singularities of these fundamental solutions are overcome by the novel origin intensity factors (OIFs). The fundamental solutions and OIFs make the SBM applicable for the simulation of the anti-plane wave scattering and diffraction in nonhomogeneous media. Numerical experiments, including the finite and semi-infinite cases with different variations and boundary conditions, are presented to illustrate the convergence, accuracy and effectiveness of the proposed SBM.

Features of the Propagation of Long Waves in Phononic Crystals, the Influence of the Concentration and Polydispersity of the Components

The relevance of the research is determined by the need to simulate wave propagation in heterogeneous media based on phononic crystals. These studies are necessary for the tasks of flaw detection, the development of non-destructive testing methods, geophysics, engineering geophysics and other methods of acoustic research. During the study, it was determined that the fragmentation of the components of two-component phononic crystals into 2, 3, … N equal parts in the unit cell leads to a catastrophic rearrangement of the dispersion relations and the appearance of new elementary wave packets for acoustic waves. Using numerical-analytical methods, it is shown that in two-component phononic crystals, the polydispersity of the components can significantly affect the average characteristics of the propagation of long acoustic waves when the polydisperse crystal is close to monodisperse with a smaller unit cell size. For polydisperse crystals close to monodisperse with the same period (unit cell size), their averaged characteristics for long waves coincide.

Corrections to "mSOUND: An Open Source Toolbox for Modeling Acoustic Wave Propagation in Heterogeneous Media".

In the above article [1]>, the authors regret that there were some mistakes pertaining to (1)-(3) and (5).

Hybrid High-Order Methods for the Acoustic Wave Equation in the Time Domain

We devise hybrid high-order (HHO) methods for the acoustic wave equation in the time domain. We first consider the second-order formulation in time. Using the Newmark scheme for the temporal discretization, we show that the resulting HHO-Newmark scheme is energy-conservative, and this scheme is also amenable to static condensation at each time step. We then consider the formulation of the acoustic wave equation as a first-order system together with singly-diagonally implicit and explicit Runge-Kutta (SDIRK and ERK) schemes. HHO-SDIRK schemes are amenable to static condensation at each time step. For HHO-ERK schemes, the use of the mixed-order formulation, where the polynomial degree of the cell unknowns is one order higher than that of the face unknowns, is key to benefit from the explicit structure of the scheme. Numerical results on test cases with analytical solutions show that the methods can deliver optimal convergence rates for smooth solutions of order $$\mathcal{O}(h^{k+1})$$ in the $$H^1$$ -norm and of order $$\mathcal{O}(h^{k+2})$$ in the $$L^2$$ -norm. Moreover, test cases on wave propagation in heterogeneous media indicate the benefits of using high-order methods.

RBF-FD analysis of 2D time-domain acoustic wave propagation in heterogeneous media

Radial Basis Function-generated Finite Differences (RBF-FD) is a popular variant of local strong-form meshless methods that do not require a predefined connection between the nodes, making it easier to adapt node-distribution to the problem under consideration. This paper investigates an RBF-FD solution of time-domain acoustic wave propagation in the context of seismic modeling in the Earth's subsurface. Through a number of numerical tests, ranging from homogeneous to highly-heterogeneous velocity models including non-smooth irregular topography, we demonstrate that the present approach can be further generalized to solve large-scale seismic modeling and full waveform inversion problems in arbitrarily complex models enabling more robust interpretations of geophysical observations.

MSOUND: An Open Source Toolbox for Modeling Acoustic Wave Propagation in Heterogeneous Media.

mSOUND is an open-source toolbox written in MATLAB. This toolbox is intended for modeling linear/ nonlinear acoustic wave propagation in media (primarily biological tissues) with arbitrary heterogeneities, in which, the speed of sound, density, attenuation coefficient, power-law exponent, and nonlinear coefficient are all spatially varying functions. The computational model is an iterative one-way model based on a mixed domain method. In this article, a general guideline is given along with three representative examples to illustrate how to set up simulations using mSOUND. The first example uses the transient mixed-domain method (TMDM) forward projection to compute the transient acoustic field for a given source defined on a plane. The second example uses the frequency-specific mixed-domain method (FSMDM) forward projection to rapidly obtain the pressure distribution directly at the frequencies of interest, assuming linear or weakly nonlinear wave propagation. The third example demonstrates how to use TMDM backward projection to reconstruct the initial acoustic pressure field to facilitate photoacoustic tomography (PAT). mSOUND (https://m-sound.github.io/mSOUND/home) is designed to be complementary to existing ultrasound modeling toolboxes and is expected to be useful for a wide range of applications in medical ultrasound including treatment planning, PAT, transducer design, and characterization.

Explicit and Energy-Conserving Constraint Energy Minimizing Generalized Multiscale Discontinuous Galerkin Method for Wave Propagation in Heterogeneous Media

In this work, we propose a local multiscale model reduction approach for the time domain scalar wave equation in a heterogenous media. A fine mesh is used to capture the heterogeneities of the coef...

Domain decomposition preconditioners for high-order discretizations of the heterogeneous Helmholtz equation

Abstract We consider one-level additive Schwarz domain decomposition preconditioners for the Helmholtz equation with variable coefficients (modelling wave propagation in heterogeneous media), subject to boundary conditions that include wave scattering problems. Absorption is included as a parameter in the problem. This problem is discretized using $H^1$-conforming nodal finite elements of fixed local degree $p$ on meshes with diameter $h = h(k)$, chosen so that the error remains bounded with increasing $k$. The action of the one-level preconditioner consists of the parallel solution of problems on subdomains (which can be of general geometry), each equipped with an impedance boundary condition. We prove rigorous estimates on the norm and field of values of the left- or right-preconditioned matrix that show explicitly how the absorption, the heterogeneity in the coefficients and the dependence on the degree enter the estimates. These estimates prove rigorously that, with enough absorption and for $k$ large enough, GMRES is guaranteed to converge in a number of iterations that is independent of $k,p$ and the coefficients. The theoretical threshold for $k$ to be large enough depends on $p$ and on the local variation of coefficients in subdomains (and not globally). Extensive numerical experiments are given for both the absorptive and the propagative cases; in the latter case, we investigate examples both when the coefficients are nontrapping and when they are trapping. These experiments support (i) our theory in terms of dependence on polynomial degree and the coefficients; and (ii) the sharpness of our field of values estimates in terms of the level of absorption required.

Coda-wave decorrelation sensitivity kernels in 2-D elastic media: a numerical approach

SUMMARYProbabilistic sensitivity kernels based on the analytical solution of the diffusion and radiative transfer equations have been used to locate tiny changes detected in late arriving coda waves. These analytical kernels accurately describe the sensitivity of coda waves towards velocity changes located at a large distance from the sensors in the acoustic diffusive regime. They are also valid to describe the acoustic waveform distortions (decorrelations) induced by isotropically scattering perturbations. However, in elastic media, there is no analytical solution that describes the complex propagation of wave energy, including mode conversions, polarizations, etc. Here, we derive sensitivity kernels using numerical simulations of wave propagation in heterogeneous media in the acoustic and elastic regimes. We decompose the wavefield into P- and S-wave components at the perturbation location in order to construct separate P to P, S to S, P to S and S to P scattering sensitivity kernels. This allows us to describe the influence of P- and S-wave scattering perturbations separately. We test our approach using acoustic and elastic numerical simulations where localized scattering perturbations are introduced. We validate the numerical sensitivity kernels by comparing them with analytical kernel predictions and with measurements of coda decorrelations on the synthetic data.

Modeling acoustic waves in locally enhanced meshes with a staggered-grid finite difference approach

Finite difference is a well-suited technique for modeling acoustic wave propagation in heterogeneous media as well as for imaging and inversion. Typically, the method aims at solving a set of partial differential equations for the unknown pressure field by using a regularly spaced grid. Although finite differences can be fast and cheap to implement, the accuracy of the solution is always restricted by the computational resources. This is a fundamental key point to treat when dealing with large-scale problems. In this work, we present and test a method that uses a non-uniform distribution of grid points to improve on accuracy or to reduce the required computational resources. The applied grid is generated through a coordinate transformation. Differential geometry and generalized coordinates are used to handle and analyze the effect of using a non-uniform grid. Results obtained with the presented method show that the applied transformation as well as the number of points-per-wavelength influences the stability and dispersion in the solution. We exploit this observation to locally improve the accuracy of our simulations. The work presented in this paper allows us to conclude that differential geometry for finite differences can be used to reduce dispersion and hence improve the accuracy when modeling acoustic wave propagation in heterogeneous media. In addition, it can be used to avoid oversampling through the optimization of the number of grid nodes required to have an accurate solution or just honor to the boundaries.

Constraint energy minimizing generalized multiscale discontinuous Galerkin method

Numerical simulation of flow problems and wave propagation in heterogeneous media has important applications in many engineering areas. However, numerical solutions on the fine grid are often prohibitively expensive, and multiscale model reduction techniques are introduced to efficiently solve for an accurate approximation on the coarse grid. In this paper, we propose an energy minimization based multiscale model reduction approach in the discontinuous Galerkin discretization setting. The main idea of the method is to extract the non-decaying component in the high conductivity regions by identifying dominant modes with small eigenvalues of local spectral problems, and define multiscale basis functions in coarse oversampled regions by constraint energy minimization problems. The multiscale basis functions are in general discontinuous on the coarse grid and coupled by interior penalty discontinuous Galerkin formulation. The minimal degree of freedom in representing high-contrast features is achieved through the design of local spectral problems, which provides the most compressed local multiscale space. We analyze the method for solving Darcy flow problem and show that the convergence is linear in coarse mesh size and independent of the contrast, provided that the oversampling size is appropriately chosen. Numerical results are presented to show the performance of the method for simulation on flow problem and wave propagation in high-contrast heterogeneous media.

Effective Models and Numerical Homogenization for Wave Propagation in Heterogeneous Media on Arbitrary Timescales

A family of effective equations for wave propagation in periodic media for arbitrary timescales mathcal {O}(varepsilon ^{-alpha }), where varepsilon ll 1 is the period of the tensor describing the medium, is proposed. The well-posedness of the effective equations of the family is ensured without requiring a regularization process as in previous models (Benoit and Gloria in Long-time homogenization and asymptotic ballistic transport of classical waves, 2017, arXiv:1701.08600; Allaire et al. in Crime pays; homogenized wave equations for long times, 2018, arXiv:1803.09455). The effective solutions in the family are proved to be varepsilon close to the original wave in a norm equivalent to the {mathrm {L}^{infty }}(0,varepsilon ^{-alpha }T;{{mathrm {L}^{2}}(varOmega )}) norm. In addition, a numerical procedure for the computation of the effective tensors of arbitrary order is provided. In particular, we present a new relation between the correctors of arbitrary order, which allows to substantially reduce the computational cost of the effective tensors of arbitrary order. This relation is not limited to the effective equations presented in this paper and can be used to compute the effective tensors of alternative effective models.