Identification of the Velocity Field of 2D and 3D Tunnel Models with Frequency Domain Full Waveform Inversion

Full Waveform Inversion (FWI see [1]) has capability to identify the velocity field of a domain with a good precision. Its power comes from the fact that this approach does not only try to fit travel times of waves, but it tries to fit the whole seismogram. This work is about the application of acoustic full waveform inversion to 2D and 3D tunnel models. The necessary boundary conditions are applied to the models and the acoustic equation is solved by higher-order Finite Elements Method. The Conjugate Gradient (CG) method is utilized to minimize the misfit function. The results were verified with synthetic models. The synthetic tunnel models contain few bodies with different shapes and locations. Starting from a homogeneous velocity field, the synthetic model is sought over iterations. To avoid the ill-posedness, the locations and numbers of the source and receiver points have to be successfully chosen. Apart from this, the frequency set has also to be carefully constructed. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

基于二维随机震源的地震波全波形反演研究

全波形反演利用全部的波场信息做反演求解,兼顾了地震波的运动学特征和动力学特征,是一种直接基于波动方程描述地震波在地下介质中的传播过程,能够获得地质结构和岩性资料的方法.但是作为一种非线性反演算法,如何提高全波形反演的计算速度和成像精度是目前优化反演的难点和重点.针对全波形反演的效率问题,采用分层和模块化的matlab工...     

Mathematical simulation of acoustic wave scattering in fractured media

The simulation of acoustic waves in fractured media is considered. A self-consistent field model is proposed that describes the formation of a scattered field and the attenuation of the incident field. For the total field, a wave equation with a complex velocity is derived and the corresponding dispersion equation is studied. A frequency-dependent field damping law and an energy variation law are established. An initial and a boundary value problem for waves in a fractured medium is addressed. A finite-difference scheme for the initial value problem is constructed, and a condition for its stability is established. Numerical results are presented.     

Linearization Ill-Posedness for 2.5-D Ware Equation Inversion Model

Abstract For the weakly inhomogeneous acoustic medium in Ω={(x,y,z):z>0},we consider the inverse problemof determining the density function ρ(x,y).The inversion input for our inverse problem is the wave field givenon a line.We get an integral equation for the 2-D density perturbation from the linearization.By virtue of theintegral transform,we prove the uniqueness and the instability of the solution to the integral equation.Thedegree of ill-posedness for this problem is also given.     

GL method for solving equations in math-physics and engineering

In this paper, we propose a GL method for solving the ordinary and the partial differential equation in mathematical physics and chemics and engineering. These equations govern the acustic, heat, electromagnetic, elastic, plastic, flow, and quantum etc. macro and micro wave field in time domain and frequency domain. The space domain of the differential equation is infinite domain which includes a finite inhomogeneous domain. The inhomogeneous domain is divided into finite sub domains. We present the solution of the differential equation as an explicit recursive sum of the integrals in the inhomogeneous sub domains. Actualy, we propose an explicit representation of the inhomogeneous parameter nonlinear inversion. The analytical solution of the equation in the infinite homogeneous domain is called as an initial global field. The global field is updated by local scattering field successively subdomaln by subdomain. Once all subdomains are scattered and the updating process is finished in all the sub domains, the solution of the equation is obtained. We call our method as Global and Local field method, in short , GL method. It is different from FEM method, the GL method directly assemble inverse matrix and gets solution. There is no big matrix equation needs to solve in the GL method. There is no needed artificial boundary and no absorption boundary condition for infinite domain in the GL method. We proved several theorems on relationships between the field solution and Green＇s function that is the theoretical base of our GL method. The numerical discretization of the GL method is presented. We proved that the numerical solution of the GL method convergence to the exact solution when the size of the sub domain is going to zero. The error estimation of the GL method for solving wave equation is presented. The simulations show that the GL method is accurate, fast, and stable for solving elliptic, parabolic, and hyperbolic equations. The GL method has advantages and wide applications in the 3D electromagnetic （EM）     

Multidimensional dust acoustic solitary waves in inhomogeneous magnetized dusty plasmas with nonthermal ions

The linear dispersion relation and a modified variable coefficients Korteweg–de Vries (MKdV) equation governing the three-dimensional dust acoustic solitary waves are obtained in inhomogeneous dusty plasmas comprised of negatively charged dust grains of equal radii, Boltzmann distributed electrons and nonthermally distributed ions. The numerical results show that the inhomogeneity, the nonthermal ions, the external magnetic field and the collision have strong influence on the frequency and the nonlinear properties of dust acoustic solitary waves and both dust acoustic solitary holes (soliton with a density dip) and positive solitons (soliton with a density hump) are excited.     

数学物理中的总体和局部场GL方法

为了求解物理化学生物材料和金融中的微分方程,提出了一种总体(Global)和局部(Local)场方法.微分方程的求解区域可以是有限域,无限域,或具曲面边界的部分无限域.其无限域包括有限有界不均匀介质区域.其不均匀介质区域被分划为若干子区域之和.在这含非均匀介质的无限区域,将微分方程的解显式地表示为在若干非均匀介质子区域上和局部子曲面的积分的递归和.把正反算的非线性关系递归地显式化.在无限均匀区域,微分方程的解析解被称为初始总体场.微分方程解的总体场相继地被各个非均匀介质子区域的局部散射场所修正.这种修正过程是一个子域接着另个子域逐步相继地进行的.一旦所有非均匀介质子区域被散射扫描和有限步更新过程全部完成后,微分方程的解就获得了.称其为总体和局部场的方法,简称为GL方法.GL方法完全地不同于有限元及有限差方法,GL方法直接地逐子域地组装逆矩阵而获得解.GL方法无需求解大型矩阵方程,它克服了有限元大型矩阵解的困难.用有限元及有限差方法求解无限域上的微分方程时,人为边界及其上的吸收边界条件是必需的和困难的,人为边界上的吸收边界条件的不精确的反射会降低解的精确度和毁坏反算过程.GL方法又克服了有限元和有限差方法的人为边界的困难.GL方法既不需要任何人为边界又不需要任何吸收边界条件就可以子域接子域逐步精确地求解无限域上的微分方程.有限元和有限差方法都仅仅是数值的方法,GL方法将解析解和数值方法相容地结合起来.提出和证明了三角的格林函数积分方程公式.证明了当子域的直经趋于零时,波动方程的GL方法的数值解收敛于精确解.GL方法解波动方程的误差估计也获得了.求解椭圆型,抛物线型,双曲线型方程的GL模拟计算结果显示出我们的GL方法具有准确,快速,稳定的许多优点.GL方法可以是有网,无网和半网算法.GL方法可广泛应用在三维电磁场,三维弹塑性力学场,地震波场,声波场,流场,量子场等方面.上述三维电磁场等应用领域的GL方法的软件已经由作者研制和发展了。     

非均匀介质中声波的远场分布的性质

本文运用泛函分析和积分方程的方法,讨论了非均匀介质中声波的远场分布的性质,并应用Тихонов正则化方法讨论了不适定的逆散射问题.     

On compactness of the velocity field in the incompressible limit of the full Navier–Stokes–Fourier system on large domains

The incompressible limit for the full Navier–Stokes–Fourier system is studied on a family of domains containing balls of the radius growing with a speed that dominates the inverse of the Mach number. It is shown that the velocity field converges strongly to its limit locally in space, in particular, the effect of the sound waves is eliminated by means of the local decay estimates for the acoustic wave equation. Copyright © 2008 John Wiley & Sons, Ltd.     

The operator equations of Lippmann–Schwinger type for acoustic and electromagnetic scattering problems in L 2

This article is concerned with the scattering of acoustic and electromagnetic time harmonic plane waves by an inhomogeneous medium. These problems can be translated into volume integral equations of the second kind – the most prominent example is the Lippmann–Schwinger integral equation. In this work, we study a particular class of scattering problems where the integral operator in the corresponding operator equation of Lippmann–Schwinger type fails to be compact. Such integral equations typically arise if the modelling of the inhomogeneous medium necessitates space-dependent coefficients in the highest order terms of the underlying partial differential equation. The two examples treated here are acoustic scattering from a medium with a space-dependent material density and electromagnetic medium scattering where both the electric permittivity and the magnetic permeability vary. In these cases, Riesz theory is not applicable for the solution of the arising integral equations of Lippmann–Schwinger type. Therefore, we show that positivity assumptions on the relative material parameters allow to prove positivity of the arising volume potentials in tailor-made weighted spaces of square integrable functions. This result merely holds for imaginary wavenumber and we exploit a compactness argument to conclude that the arising integral equations are of Fredholm type, even if the integral operators themselves are not compact. Finally, we explain how the solution of the integral equations in L ² affects the notion of a solution of the scattering problem and illustrate why the order of convergence of a Galerkin scheme set up in L ² does not suffer from our L ² setting, compared to schemes in higher order Sobolev spaces.     

Comparison of additional second-order terms in finite-difference Euler equations and regularized fluid dynamics equations

In recent years, an area of research in computational mathematics has emerged that is associated with the numerical solution of fluid flow problems based on regularized fluid dynamics equations involving additional terms with velocity, pressure, and body force. The inclusion of these functions in the additional terms has been physically substantiated only for pressure and body force. In this paper, the continuity equation obtained geometrically by Euler is shown to involve second-order terms in time that contain Jacobians of the velocity field and are consistent with some of the additional terms in the regularized fluid dynamics equations. The same Jacobians are contained in the inhomogeneous right-hand side of the wave equation and generate waves of pressure, density, and sound. Physical interpretations of the additional terms used in the regularized fluid dynamics equations are given.     

A viscous approximation for a 2-D steady semiconductor or transonic gas dynamic flow: Existence theorem for potential flow

In this paper we solve a boundary value problem in a two-dimensional domain O for a system of equations of Fluid-Poisson type, that is, a viscous approximation to a potential equation for the velocity coupled with an ordinary differential equation along the streamlines for the density and a Poisson equation for the electric field. A particular case of this system is a viscous approximation of transonic flow models. The general case is a model for semiconductors. We show existence of a density ρ, velocity potential φ, and electric potential Φ in the bounded domain O that are C^1,α(O¯), C^2,α(O¯), and W^2,α(O¯) functions, respectively, such that ρ, φ, Φ, the speed |Δφ|, and the electric field E = ΔΦ are uniformly bounded in the viscous parameter. This is a necessary step in the existing programs in order to show existence of a solution for the transonic flow problem. © 1996 John Wiley & Sons, Inc.     

Asymptotic Analysis of Reacting Materials with Saturated Explosion. II. High-Frequency Waves

The interaction of small-scale material inhomogeneities with high-frequency acoustic waves is known to have a prominent role in accelerating the heat-release rate in liquid and solid explosive materials. In the present paper, simplified asymptotic equations are studied which incorporate the above interaction, and which include reactant depletion at leading order. Because fuel may be completely exhausted, singularities do not always form in the model equations; it is conjectured that when a singularity does form, the material has initiated. The detailed mechanisms by which shock formation and resonant wave interaction can either enhance or retard reaction are explored. In a realistic model for inhomogeneous condensed-phase reaction, with pressure-dependent reaction rate and nonconstant initial fuel concentration, initiation of the material depends on correct placement of the fuel relative to the acoustic waves.     

Excitation of a wave field in a triangular domain with impedance boundary conditions

The Helmholtz equation in a closed domain that is an equilateral triangle with inhomogeneous impedance boundary conditions is considered. A functional equation in which the unknown function is the Fourier-image of a wave field on the boundary of the domain is constructed. This functional equation is solved for the case of homogeneous boundary conditions (the problem on eigenvalues), as well as for the case of inhomogeneous boundary conditions in the absence of the resonance. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 300–318. Translated by A. V. Shanin.     

2.5维介质Born近似波速反演唯一性

考虑脉冲源引起的２．５维弱不均匀介质波速反演问题，利用线性化方法得到了波速的二维小扰动满足的积分方程，这是一个积分几何的问题，进而由Ｆｏｕｒｉｅｒ变换和脉冲函数的性质将此二维积分方程化为单变量的积分方程，最后用压缩映象理论证明了积分方程解的唯一性。本文给出了二给波速反演的一种新算法。同时，唯一性结果证明了已有的迭代算法的合理性。     

Well-posedness of the poincar�� problem in a cylindrical domain for the higher-dimensional wave equation

It is known that waves (acoustic waves, radio waves, elastic waves, and electric waves) in cylindrical tubes are described by the wave equation. In the theory of hyperbolic-type partial differential equations, boundary-value problems with data on the whole boundary serve as examples of ill-posedness of the posed problems. In this work, it is shown that the Poincar´e problem in a cylindrical domain for the higher-dimensional wave equation is uniquely solvable. A uniqueness criterion for a regular solution is also obtained.     

The propagation of sound in particle models of compressible fluids

Summary. The propagation of sound in compressible fluids is described by the acoustic equations that result from the linearization of the Euler equations around a state of constant mass density and velocity zero. In this article, it is shown that a stable and convergent discretization of the acoustic wave equation for the velocity field can be recovered from the particle model of compressible fluids recently developed by the author in [Numer. Math. (1997) 76: 111–142] by linearizing the equations of motion for the particles. For particles of proper shape, this discretization is second order accurate, and with an obvious modification of the basic particle model, one can even reach an arbitrarily high order of convergence. Received January 24, 2000 / Published online November 8, 2000     

Low Mach Number Limit of Viscous Polytropic Flows: Formal Asymptotics in the Periodic Case

This article is devoted to the low Mach number limit of weak solutions to the compressible Navier–Stokes equations for polytropic fluids with periodic boundary conditions and ill‐prepared data. We derive formally the equation satisfied by the mean value of the velocity and the equations governing the dynamics of the nonlinear acoustic waves in dimension d= 2 or 3.     

On wellposedness in nonlinear acoustics

Motivated by a medical application from lithotripsy, we study the initial–boundary value problem given by Westervelt equation (1) in a bounded domain Ω. This models the nonlinear evolution of the acoustic pressure u excited at a part Γ₀ of the boundary. Along with the excitation given by Neumann boundary condition as in (1) , we also consider the Dirichlet type of excitation. Whereas shock waves are known to emerge after a sufficiently large time interval for appropriate initial and boundary conditions, we here prove existence and uniqueness as well as stability of a solution u for small data g, u₀ and u₁ or short time T, using a fixed point argument. Moreover we extend the result to the more general model given by the Kuznetsov equation (2) for the acoustic velocity potential ψ. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence and qualitative theory for stratified solitary water waves

This paper considers two-dimensional gravity solitary waves moving through a body of density stratified water lying below vacuum. The fluid domain is assumed to lie above an impenetrable flat ocean bed, while the interface between the water and vacuum is a free boundary where the pressure is constant. We prove that, for any smooth choice of upstream velocity field and density function, there exists a continuous curve of such solutions that includes large-amplitude surface waves. Furthermore, following this solution curve, one encounters waves that come arbitrarily close to possessing points of horizontal stagnation.We also provide a number of results characterizing the qualitative features of solitary stratified waves. In part, these include bounds on the wave speed from above and below, some of which are new even for constant density flow; an a priori bound on the velocity field and lower bound on the pressure; a proof of the nonexistence of monotone bores in this physical regime; and a theorem ensuring that all supercritical solitary waves of elevation have an axis of even symmetry.