A 2.5D finite-difference algorithm for elastic wave modeling using near-optimal quadratures

Constraint energy minimizing generalized multiscale discontinuous Galerkin method

A modified mixed-domain method is presented in this paper to model wave propagation in heterogeneous media. The previous mixed domain method is only accurate for weakly heterogeneous media. Both the phase and amplitude corrections are introduced to the mixed-domain method in order to model wave propagation in more strongly heterogeneous media. Two cases are studied to validate the modified method. A two-layered medium with density and speed of sound variation is first studied. A human skull with density and speed of sound variation is then used to further evaluate the accuracy of this method. Result from the angular spectrum approach is used as the benchmark for the layered medium and result from the MATLAB toolbox k-Wave is used as the benchmark for the case of skull. Simulation results show that the modified mixed-domain method is more accurate for strongly heterogeneous media.

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.

Numerical simulation of elastic and acoustic wave propagation utilizes increasingly large and complex models, providing more realistic and useful results. However, significant challenges remain in applications such as propagation in fractured media, as complex distributions of fracture systems can be difficult to represent on typical, uniform grids with spacing on the order of 10-20 m. While in some cases, effective medium theories may be useful, in other situations the distribution of fracturing or other heterogeneities may have more complex effects on waves. We describe initial results of a new multiscale finite element algorithm for simulating acoustic wave propagation in heterogeneous media that addresses these problems by combining fineand coarse-scale grids. The wave equation is solved on a coarse grid using multiscale basis functions, using a global coupling mechanism to to related information between scales. Time stepping is applied on the coarse grid, leading to additional savings. Numerical results demonstrate the utility of the method. Long term developments have strong potential to enhance inversion algorithms, since the basis functions need not be regenerated, allowing faster simulations for repeated calculations needed for inversion.

We have extended a new highly accurate numerical scheme for unstructured 2D and 3D meshes based on the discontinuous Galerkin approach to simulate seismic wave propagation in heterogeneous media containing fluid-solid interfaces. Because of the formulation of the wave equations as a unified first-order hyperbolic system in velocity stress, the fluid can be in movement along the interface. The governing equations within the moving fluid are derived from well-known first principles in fluid mechanics. The discontinuous Galerkin approach allows for jumps of the material parameters and the solution across element interfaces, which are handled by Riemann solvers or numerical fluxes. The use of Riemann solvers at the element interfaces makesthe treatment of the fluid particularly simple bysetting the shearmodulus in the fluid region to zero. No additional compatibility relations, such as vanishing shear stress or continuity of normal stresses, are necessary to couple the solid and fluid along an interface. The Riemann solver automatically recognizes the jump of the material coefficients at the interface and provides the correct numerical fluxes for fluid-solid contacts. Therefore, wave propagation in the entire computational domain containing heterogeneous media, namely moving fluids and elastic solids, can be described by a uniform set of acoustic and elastic wave equations. The accuracy of the proposed scheme is confirmed by comparing numerical results against analytic solutions. The potential of the new method was demonstrated in a 3D model problem typical for marine seismic exploration with a fluid-solid interface determined by a complicated bathymetry.

Numerical simulation of anti-plane wave propagation in heterogeneous media

Summary Hamilton’s wave theory is used to study the effect on seismic surface wave propagation of lateral variations in the composition of the crust and mantle. For cases with small lateral variations the distortion of the propagation paths from the great circle paths is determined by a perturbation method. Moreover, surface wave paths from the Mid-Atlantic Ridge toward the NORSAR and LASA arrays are computed by numerical integration of the ray equations. The computed wave paths agree well with existing observations. The model of the surface wave velocity distribution used in this study cannot, however, explain all the observed multipathing effects. Studies by several authors have shown that the lateral variations in the structure of the Earth’s crust and upper mantle may lead to large distortions of the propagation paths for seismic surface waves. Notably among these works are the works by Evernden (1953, 1954) and the works by Capon (1970, 1971). These works show that surface waves are refracted or reflected at the continental margins and at major geological structures. The oceanic ridge systems are an example to the latter. Estimates of the propagation path for surface waves are presented in the work cited above. Apart from this and a study by Julian (1972) little has been done in order to compute the propagation paths for surface waves. In various branches of Physics, Hamilton’s wave theory (or the eikonal approximation) has successfully been used to study wave propagation in heterogeneous media. Within the theory of seismic surface waves the method has received less attention although the applicability of it has been demonstrated in an elegant study by Backus (1962). The application of this theory to surface wave propagation in cases with lateral variation in the structure of the crust and upper mantle deserves more exploration. A fundamental assumption in Hamilton’s wave theory is that the physical parameters which are important for the wave propagation vary only by a small fraction of their values over a distance comparable to a wavelength. The theory has therefore very misleadingly been referred to as a short wavelength approximation. The propagation of seismic surface waves are for example fundamentally effected by the value of the physical parameters in a surface layer with thickness comparable to the wavelength. Therefore the lateral variations of the parameters describing the wave motion may be smaller for long waves than for short waves. This is confirmed by the observations

Atherosclerosis is a chronic vascular disease affecting large and medium sized arteries. Several treatment options are already available for treatment of this disease. Targeting atherosclerotic plaques by high intensity focused ultrasound (HIFU) using dual mode ultrasound arrays (DMUA) was recently introduced in literature. We present a finite difference time domain (FDTD) simulation modeling of the wave propagation in heterogeneous medium from the surface of a 3.5 MHz array prototype with 32-elements. After segmentation of the ultrasound image obtained for the treatment region in-vivo, we integrated this anatomical information into our simulation to account for different parameters that may be caused by these multi-region anatomical complexities. The simulation program showed that HIFU was able to induce damage in the prefocal region instead of the target area. The HIFU lesions, as predicted by our simulation, were well correlated with the actual damage detected in histology.

Piezoelectric ultrasonic transducers typically employ composite structures to improve their transmission and reception sensitivities. The geometry of the composite is regular with one dominant length scale and, since these are resonant devices, this dictates the central operating frequency of the device. In order to construct a wide bandwidth device it would seem natural therefore to utilise resonators that span a range of length scales. In this paper we consider such a device and build a theoretical model to predict its performance. A fractal medium is used as this contains a wide range of length scales and yields to a renormalisation approach. The propagation of an ultrasonic wave in this heterogeneous medium is then analysed and used to construct expressions for the electrical impedance, and the transmission and reception sensitivities of this device as a function of the driving frequency.

Seismic data are routinely used to infer in situ properties of earth materials on many scales, ranging from global studies to investigations of surficial geological formations. While inversion and imaging algorithms utilizing these data have improved steadily, there are remaining challenges that make detailed measurements of the properties of some geologic materials very difficult. For example, the determination of the concentration and orientation of fracture systems is prohibitively expensive to simulate on the fine grid and, thus, some type of coarse-grid simulations are needed. In this paper, we describe a new multiscale finite element algorithm for simulating seismic wave propagation in heterogeneous media. This method solves the wave equation on a coarse grid using multiscale basis functions and a global coupling mechanism to relate information between fine and coarse grids. Using a mixed formulation of the wave equation and staggered discontinuous basis functions, the proposed multiscale methods have the following properties. • The total wave energy is conserved. • Mass matrix is diagonal on a coarse grid and explicit energy-preserving time discretization does not require solving a linear system at each time step. • Multiscale basis functions can accurately capture the subgrid variations of the solution and the time stepping is performed on a coarse grid. We discuss various subgrid capturing mechanisms and present some preliminary numerical results.

We present a scattering attenuation model based on the statistical wave propagation theory in random media. It is suitable for the weak wavefield fluctuation regime and has practically no restriction in the frequency domain. The presented formulas allow to quantify scattering attenuation in complex geological regions using simple statistical estimates from well-log data. This knowledge is important for further petrophysical interpretations of reservoir rocks. To test our theory we perform numerical simulations of seismic wave propagation in 3-D elastic random media using a finite-difference solution of the elastodynamic wave equation. From the synthetic seismograms we determine the scattering attenuation (Q −1) with help of spectral decay methods. We find good agreement of the frequency-dependent Q values and the theoretical predictions.

In the present study seismic wave propagation in heterogeneous media is numerically simulated by using the pseudospectral method with the staggered grid RFFT differentiation in order to clarify the cause for the complicated distribution characteristics of strong ground motion in regions with basin structure. The results show that the maximum amplitudes of simulated ground acceleration waveforms are closely related to the basin structure. Interference of seismic waves in the basin strongly affects the distribution of maximum seismic waveforms, which may result in peak disasters during earthquakes. Peak disasters might be away from basin boundaries or earthquake faults. Seismic energy transmitted into the basin from the bedrock can hardly penetrate the bottom of the basin and then travel back into the bedrock region. The seismic energy is absorbed by basin media, and transferred into the kinematical energy of seismic waves with great amplitude in the basin. Seismic waves between basins may result in serious damage to buildings over the basin. This is significant for aseismatic research. Geological surveys in and around urban areas would benefit aseismatic research and mitigation of seismic disasters of a city. Such geological surveys should involve seismic velocity structure in the media above the bedrock besides such subjects as active faults and geological structure.

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.

P-SV wave propagation in heterogeneous media: velocity-stress finite-difference method : Virieux, J Geophysics V51, N4, April 1986, P889–901

This article deals with a fast numerical method for GPR simulation of great scenes. Electromagnetic wave propagation in heterogeneous media is analyzed using a finite‐difference time‐domain (FDTD)‐based method. The use of a moving subdomain allows dealing with large scale and realistic B‐scan or C‐scan in a reasonable computational time. A simple and fast inversion method based on a coherent superposition of signal phases will also be exposed. The performances of this method for the localization and shape recovering of various buried objects will be pointed out. This process is very easy to implement and also has the advantage to be used for monostatic or multisensor radar records. © 2009 Wiley Periodicals, Inc. Microwave Opt Technol Lett 51: 796–802, 2009; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.24153