Wave Equation Method Research Articles

Full wave-equation methods for complex imaging challenges

Despite being at the end of a long and full Annual Meeting week, the workshop “Full wave-equation methods for complex imaging challenges,” held 30 October 2009, drew an audience which filled the large meeting room. They were rewarded with a full day of presentations which mostly focused on emerging techniques for velocity model estimation rather than imaging itself: the consensus seemed to be that it is indeed there that the true imaging challenges lie. Nonetheless, significant contributions on the imaging side also demonstrated continuing progress in improving the quality and efficiency of imaging, and in interpreting the results.

On an inverse problem for tree‐like networks of elastic strings

AbstractWe consider the in‐plane motion of elastic strings on tree‐like network, observed from the ‘leaves’. We investigate the inverse problem of recovering not only the physical properties, i.e. the ‘optical lengths’ of each string, but also the topology of the tree which is represented by the edge degrees and the angles between branching edges. To this end we use the Boundary Control method for wave equations on graphs established in [4, 7]. It is shown that under generic assumptions the inverse problem can be solved by applying measurements at all leaves, the root of the tree being fixed.

Scaling Behavior of Electronic Excitations in Assemblies of Molecules with Degenerate Ground States

The behavior of long space-time excitations in many-electron systems with ground state degeneracy is explored via multiscale analysis. The analysis starts with an ansatz for the wave function's dual dependence on the N-electron configuration (i.e., both by direct means and by indirect means via a set of order parameters). It is shown that a Dirac-like equation form of the wave equation emerges in the limit where the ratio epsilon (of the average nearest-neighbor distance to the characteristic length of the long-scale phenomenon of interest) is small. Examples of the long scale are the size of a quantum dot, nanotube, or wavelength of a density disturbance. The velocities in the Dirac-like equation are the transition moments of the single-particle momentum operator connecting degenerate ground states. While detailed band structure and the independent quasi-particle picture could underlie the behavior of some systems (as commonly suggested for graphene), the present scaling law results show it is not necessarily the only explanation. Rather, it can follow from the scaling properties of low-lying, long spatial scale excitations and ground state degeneracy, even in strongly interacting systems. The generality of our findings suggests graphene may be just one of many examples of Dirac-like equation behavior. A preliminary validation of our quantum scaling law for molecular arrays is presented. As our scaling law constitutes a coarse-grained wave equation, path integral or other methods derived from it hold great promise for calibration-free, long-time simulation of many-particle quantum systems.

Deep water prestack multiple suppression on reflection data from ionian sea (offshore western greece)

Multiple attenuation comprises a very important phase of seismic reflection processing because it removes undesir- able effects occurring during propagation of seismic signal in the Earth. The present work aims to present the effec- tiveness of modern multiple suppression methods applied on streamer seismic reflection data. The seismic dataset represents an area with moderately dipping seafloor. The specific dataset was chosen to present the elimination of the water-column reverberations and internal multiples in dipping layers. Strong seafloor and internal multiples, lat- eral reflections and coherent noise impose difficulties in discriminating primary reflections in the seismic image of the present work. The prestack demultiple scheme combines wave equation method and deconvolution to predict and attenuate water-column reverberations and internal multiple reflections respectively. In particular, the region of mod- erately dipping seafloor chosen for the application of the proposed scheme images the Frontal Bulge in the outer Hel- lenic arc. The prestack demultiple method improved the quality of the stacked and migrated sections by attenuating the majority of multiple reflections and extracting information on the structural units of the study area.

Symplectic wavelet collocation method for Hamiltonian wave equations

This paper introduces a novel symplectic wavelet collocation method for solving nonlinear Hamiltonian wave equations. Based on the autocorrelation functions of Daubechies compactly supported scaling functions, collocation method is conducted for the spatial discretization, which leads to a finite-dimensional Hamiltonian system. Then, appropriate symplectic scheme is employed for the integration of the Hamiltonian system. Under the hypothesis of periodicity, the properties of the resulted space differentiation matrix are analyzed in detail. Conservation of energy and momentum is also investigated. Various numerical experiments show the effectiveness of the proposed method.

Momentum–time flux conservation method for one-dimensional wave equations

We present a conservation element and solution element method in time and momentum space. Several paradigmatic wave problems including simple wave equation, convection–diffusion equation, driven harmonic oscillating charge and nonlinear Korteweg–de Vries (KdV) equation are solved with this method and calibrated with known solutions to demonstrate its use. With this method, time marching scheme is explicit, and the nonreflecting boundary condition is automatically fulfilled. Compared to other solution methods in coordinate space, this method preserves the complete information of the wave during time evolution which is an useful feature especially for scattering problems.

Contraction-Galerkin method for a semi-linear wave equation

In this paper we consider the unique solvability of initial-boundary value problems of semi-linear wave equations with space-time dependent coefficients and special mixed non-homogeneous boundary values which form the so-called boundary-like antiperiodic condition. The procedure in this paper is the combination of the Galerkin method and a contraction.

A new time–space domain high-order finite-difference method for the acoustic wave equation

A new unified methodology was proposed in Finkelstein and Kastner (2007) [39] to derive spatial finite-difference (FD) coefficients in the joint time–space domain to reduce numerical dispersion. The key idea of this method is that the dispersion relation is completely satisfied at several designated frequencies. We develop this new time–space domain FD method further for 1D, 2D and 3D acoustic wave modeling using a plane wave theory and the Taylor series expansion. New spatial FD coefficients are frequency independent though they lead to a frequency dependent numerical solution. We prove that the modeling accuracy is 2nd-order when the conventional ( 2 M ) th-order space domain FD and the 2nd-order time domain FD stencils are directly used to solve the acoustic wave equation. However, under the same discretization, the new 1D method can reach ( 2 M ) th-order accuracy and is always stable. The 2D method can reach ( 2 M ) th-order accuracy along eight directions and has better stability. Similarly, the 3D method can reach ( 2 M ) th-order accuracy along 48 directions and also has better stability than the conventional FD method. The advantages of the new method are also demonstrated by the results of dispersion analysis and numerical modeling of acoustic wave equation for homogeneous and inhomogeneous acoustic models. In addition, we study the influence of the FD stencil length on numerical modeling for 1D inhomogeneous media, and derive an optimal FD stencil length required to balance the accuracy and efficiency of modeling. A new time–space domain high-order staggered-grid FD method for the 1D acoustic wave equation with variable densities is also developed, which has similar advantages demonstrated by dispersion analysis, stability analysis and modeling experiments. The methodology presented in this paper can be easily extended to solve similar partial difference equations arising in other fields of science and engineering.

Conformal multi-symplectic integration methods for forced-damped semi-linear wave equations

Conformal symplecticity is generalized to forced-damped multi-symplectic PDEs in 1 + 1 dimensions. Since a conformal multi-symplectic property has a concise form for these equations, numerical algorithms that preserve this property, from a modified equations point of view, are available. In effect, the modified equations for standard multi-symplectic methods and for space-time splitting methods satisfy a conformal multi-symplectic property, and the splitting schemes exactly preserve global symplecticity in a special case. It is also shown that the splitting schemes yield incorrect rates of energy/momentum dissipation, but this is not the case for standard multi-symplectic schemes. These methods work best for problems where the dissipation coefficients are small, and a forced-damped semi-linear wave equation is considered as an example.

Developing Earth models with full waveform inversion

Exploration in more geologically complex areas requires new methodologies. In its quest to answer these new challenges, the oil and gas industry has moved from ray-based imaging to finite-difference, wave-equation migration to achieve better subsurface descriptions of target zone and reservoirs. Notable in this progression is the movement from ray-traced Kirchhoff algorithms through one-way wave-equation methods to use the acoustic two-way wave equation.

Imaging salt bodies using explicit migration operators offshore Norway

We have tested the performance of 3D shot-profile depth migration using explicit migration operators on a real 3D marine data set. The data were acquired offshore Norway in an area with a complex subsurface containing large salt bodies. We compared shot-profile migration using explicit migration operators with conventional Kirchhoff migration, split-step Fourier migration, and common-azimuth by generalized screen propagator (GSP) migration in terms of quality and computational cost. Image quality produced by the explicit migration operator approach is slightly better than with split-step Fourier migration and clearly better than in common-azimuth by GSP and Kirchhoff migrations. The main differences are fewer artifacts and better-suppressed noise within the salt bodies. Kirchhoff migration shows considerable artifacts (migration smiles) within and close to the salt bodies, which are not present in images produced by the other three wave-equation methods. Expressions for computational cost were developed for all four migration algorithms in terms of frequency content and acquisition parameters. For comparable frequency content, migration cost using explicit operators is four times the cost of the split-step Fourier method, up to 260 times the cost of common-azimuth by GSP migration, and 25 times the cost of Kirchhoff migration. Our results show that in terms of image quality, shot-profile migration using explicit migration operators is well suited for imaging in areas with complex geology and significant velocity changes. However, computational cost of the method is high and makes it less attractive in terms of efficiency.

Digital image processing based mass flow rate measurement of gas/solid two-phase flow

With the rapid growth of the process industry, pneumatic conveying as a tool for the transportation of a wide variety of pulverized and granular materials has become widespread. In order to improve plant control and operational efficiency, it is essential to know the parameters of the particle flow. This paper presents a digital imaging based method which is capable of measuring multiple flow parameters, including volumetric concentration, velocity and mass flow rate of particles in the gas/solid two phase flow. The measurement system consists of a solid state laser for illumination, a low-cost CCD camera for particle image acquisition and a microcomputer with bespoke software for particle image processing. The measurements of particle velocity and volumetric concentration share the same sensing hardware but use different exposure time and different image processing methods. By controlling the exposure time of the camera a clear image and a motion blurred image are obtained respectively. The clear image is thresholded by OTSU method to identify the particles from the dark background so that the volumetric concentration is determined by calculating the ratio between the particle area and the total area. Particle velocity is derived from the motion blur length, which is estimated from the motion blurred images by using the travelling wave equation method. The mass flow rate of particles is calculated by combining the particle velocity and volumetric concentration. Simulation and experiment results indicate that the proposed method is promising for the measurement of multiple parameters of gas/solid two-phase flow.

A generalized expansion method for nonlinear wave equations

A generalized Jacobian/exponential expansion method for finding the exact traveling wave solutions of a nonlinear partial differential equation is discussed. We use this method to construct many new, previously undiscovered exact solutions for the Boussinesq and modified KdV equations. We also apply it to the shallow long wave approximate equations. New solutions are deduced for this system of partial differential equations.

Observability properties of a semi-discrete 1d wave equation derived from a mixed finite element method on nonuniform meshes

The goal of this article is to analyze the observability properties for a space semi-discrete approximation scheme derived from a mixed finite element method of the 1d wave equation on nonuni- form meshes. More precisely, we prove that observability properties hold uniformly with respect to the mesh-size under some assumptions, which, roughly, measures the lack of uniformity of the meshes, thus extending the work (Castro and Micu, Numer. Math. 102 (2006) 413-462) to nonuniform meshes. Our results are based on a precise description of the spectrum of the discrete approximation schemes on nonuniform meshes, and the use of Ingham's inequality. We also mention applications to the boundary null controllability of the 1d wave equation, and to stabilization properties for the 1d wave equation. We finally present some applications for the corresponding fully discrete schemes, based on recent articles by the author.

3D beam prestack depth migration with examples from around the world

In 1999, we specialized in 2D seismic depth processing. It was necessary to progress into 3D depth migration, but it seemed unrealistic to utilize Kirchhoff or wave-equation methods and be able to compete with large companies, their compute resources, and their economies of scale. Consequently, it was decided to attempt the commercial development of an alternative approach that would hopefully provide similar or improved technical quality, but with capital outlay and economics that were more appropriate to a small company. A very straightforward approach referred to as beam prestack depth migration (BPSDM) was adopted; its origins dated to work performed in the 1930s by Frank Rieber. The anticipated merits of BPSDM were overall simplicity, economy, flexibility, and future development possibilities. The unknowns were the migration accuracy and quality achievable. BPSDM was essentially developed independently, but aspects of it obviously are related to much recently published work on other beam migrations a...

Velocity measurement of pneumatically conveyed particles through digital imaging

A digital imaging based method for measuring the velocity of pneumatically conveyed particles is described in this paper. The measurement system consists of a laser sheet for illumination, a CCD camera for particle image acquisition, and bespoke image processing software. By controlling the exposure time of the camera, blurred images of moving particles are captured, which contain velocity information of the particles. The relationship between the motion blur length and the particle velocity is established using a lens model of the camera. To improve the robustness of the system to noise in the acquired particle images, a traveling wave equation (TWE) method is used to estimate the motion blur length and then deduce the particle velocity. In addition to computer simulations experimental work on a free fall particle flow rig and pneumatically conveying test rig is carried out. The results indicate that the proposed method is feasible for the velocity measurement of pneumatically conveyed particles.

On the immersed interface method for solving time-domain Maxwell's equations in materials with curved dielectric interfaces

This paper deals with accurate numerical simulation of two-dimensional time-domain Maxwell's equations in materials with curved dielectric interfaces. The proposed fully second-order scheme is a hybridization between the immersed interface method (IIM), introduced to take into account curved geometries in structured schemes, and the Lax–Wendroff scheme, usually used to improve order of approximations in time for partial differential equations. In particular, the IIM proposed for two-dimensional acoustic wave equations with piecewise constant coefficients [C. Zhang, R.J. LeVeque, The immersed interface method for acoustic wave equations with discontinuous coefficients, Wave Motion 25 (1997) 237–263] is extended through a simple least squares procedure to such Maxwell's equations. Numerical results from the simulation of electromagnetic scattering of a plane incident wave by a dielectric circular cylinder appear to indicate that, compared to the original IIM for the acoustic wave equations, the augmented IIM with the proposed least squares fitting greatly improves the long-time stability of the time-domain solution. Semi-discrete finite difference schemes using the IIM for spatial discretization are also discussed and numerically tested in the paper.

Operator-splitting methods respecting eigenvalue problems for convection-diffusion and wave equations

We discuss iterative operator-splitting methods for convection-diffusion and wave equations motivated from the eigenvalue problem to decide the splitting process. The operator-splitting methods are well-know to solve such complicated multi-dimensional and multi-physical problems. Often the problem, how to decouple the underlying operators, is not understood well enough. We propose a method based on computing the eigenvalues for the simpler problem to decide the splitting operators and the time steps. We present the analysis and the numerical results.

A convergent finite-difference method for a nonlinear variational wave equation

We establish rigorously convergence of a semidiscrete upwind scheme for the nonlinear variational wave equation u tt ― c(u)(c(u)u x ) x = 0 with u| t=0 = u 0 and u t | t=0 = υ 0 . Introducing Riemann invariants R = u t + cu x and S = u t - cu x , the variational wave equation is equivalent to R t ― cR x = c(R 2 ― S 2 ) and S t + cS x = ―c(R 2 ― S 2 ) with c = c'/(4c). An upwind scheme is defined for this system. We assume that the speed c is positive, increasing and both c and its derivative are bounded away from zero and that R| t=0 , S| t=0 ∈ L 1 (ℝ) n L 3 (ℝ) are nonpositive. The numerical scheme is illustrated on several examples.

Application of Homotopy Analysis Method and Variational Iteration Method for Shock Wave Equation

Application of Homotopy Analysis Method and Variational Iteration Method for Shock Wave Equation