Approximation Of Wave Equation Research Articles

An ambiguity in attenuation scattering imaging

SUMMARY Migration constructs a subsurface image by mapping band-limited seismic data to reflectors in the Earth, given a background velocity model that describes the kinematics of the seismic waves. Classically, the reflectors correspond to impedance perturbations on length scales of the order of the seismic wavelength. The Born approximation of the visco-acoustic wave equation enables the computation of synthetic data for such a model. Migration then amounts to solving the linear inverse problem for perturbations in density, velocity, and attenuation. Here, the problem is simplified by assuming the density to be constant, leaving only velocity and attenuation perturbations. In the frequency domain, a single complex-valued model parameter that depends on subsurface position describes both. Its real part is related to the classic reflectivity, its imaginary part also involves attenuation variations. Attenuation scattering is usually ignored but, when included in the migration, might provide information about, for instance, the presence of fluids. We found, however, that it is very difficult to solve simultaneously for both velocity and attenuation perturbations. The problem already occurs when computing synthetic data in the Born approximation for a given scattering model: after applying a weighted Hilbert transform in the depth coordinate to a given scattering model, we obtained almost the same synthetic data if the scatterers had small dip and were located at not-too-shallow depths. This implies that it will be nearly impossible to simultaneously determine the real and imaginary part of the scattering parameters by linearized inversion without imposing additional constraints. The oil and gas industry acquires seismic data to obtain an image of the subsurface that may reveal hydrocarbon bearing formations. Because a 3-D full visco-elastic inversion is computationally still out of reach, various approximations of the wave equation are employed, often based on ray tracing or one-way wave equations. These are often sufficiently accurate to obtain a structural image. A precise characterization of the subsurface that allows for volumetric estimates of the amounts of hydrocarbons in place requires a more accurate description. Visco-acoustic or visco-elastic full waveform inversion is computationally tractable in a 2-D approximation. However, the presence of local minima in the least-squares misfit functional makes the solution of the inverse problem difficult. One cause for this problem is the absence of low frequencies, below 8–10 Hz, in the seismic data. The inverse problem becomes considerably simpler when linearized. Classic methods for velocity analysis provide a background velocity model. An operation called migration maps the band-limited seismic data to reflectors in the subsurface. Mathematically, this method amounts to a single iteration of a gradient based minimization of the least-squares misfit functional between observed and modelled data, using the Born approximation of the wave equation. Because this approximation assumes single scattering, the direct arrival and multiple reflections should be removed from the data before migration. Examples (Ostmo et al. 2002; Mulder & Plessix 2004) show that with proper weighting or preconditioning (Plessix & Mulder 2004), one or a few iterations with the conjugate-gradient method suffice to obtain a solution to the inverse problem when using the Born approximation of the constant-density visco-acoustic wave equation. The method works in the frequency-domain and reconstructs scatterers that represent perturbations of the background velocity model. In the frequency-domain, these are represented by complex numbers. The real part is almost entirely related to impedance perturbations, usually caused by abrupt changes in rock properties. If we include the imaginary part, we can formally obtain both velocity and attenuation perturbations. The latter, however, appeared to have no relation to physically realistic values in our numerical experiments. To better understand why it may be difficult to reconstruct the imaginary part of the scattering perturbations, we consider the simple case of a 1-D model consisting of horizontally layered scatterers in a homogeneous background for the constant-density

The physical mechanism (viscosity related) of low‐frequency acoustic wave attenuation in sandy/silty sediments.

Considered saturated sediments contain diverse rock pebbles (characteristic size of 0.1 mm). The weight of higher pebbles holds lower pebbles in contact sufficiently that acoustically induced solid displacements vary slowly over several adjacent pebbles. Apart from contact areas, each is nearly surrounded by water at a nearly uniform pressure. An appropriate first approximation predicts that the elastic stress tensor in the pebbles is diagonal, with components equal to the negative of the acoustic pressure in the neighboring fluid. The assumptions of Mallock and Wood apply: the mass weighted local average velocity is proportional to the negative gradient of the pressure in the water. The no‐slip condition at the interfaces tends to force the water to move with the pebbles, but the finite viscosity allows the fluid at small distances from the interfaces to move at a different velocity than the pebbles. The apparent driving force for the oscillations of the interstitial water relative to the pebbles is associated with the inertia of the water and is proportional to the difference in densities. The derived approximate wave equation predicts attenuation proportional to frequency squared, proportional to the square of the difference of the densities, and inversely proportional to viscosity. The derived dimensionless proportionality constant is consistent with recent experiments.

Application of the [formula omitted]-expansion to travelling wave solutions of the Broer–Kaup and the approximate long water wave equations

By using the G ′ G -expansion proposed recently the travelling wave solutions involving parameters of the Broer–Kaup equations and the approximate long water wave equations are found out. The travelling wave solutions are expressed by three types of functions which are the hyperbolic functions, the trigonometric functions and the rational functions. When the parameters are taken as special values the solitary wave solutions are obtained.

Wide-angle FD and FFD migration using complex Padé approximations

Seismic migration by downward continuation using the one-way wave-equation approximations has two shortcomings: imaging steep-dip reflectors and handling evanescent waves. Complex Padé approximations allow a better treatment of evanescent modes, stabilizing finite-difference migration without requiring special treatment for the migration domain boundaries. Imaging of steep-dip reflectors can be improved using several terms in the Padé expansion. We discuss the implementation and evaluation of wide-angle complex Padé approximations for finite-difference and Fourier finite-difference migration methods. The dispersion relation and the impulsive response of the migration operator provide criteria to select the number of terms and coefficients in the Padé expansion. This ensures stability for a prescribed maximum propagation direction. The implementations are validated on the Marmousi model data set and SEG/EAGE salt model data.

A NOVEL MARCHING ALGORITHM FOR RADIO WAVE PROPAGATION MODELING OVER ROUGH SURFACES

Abstract—In this paper, the parabolic approximation of wave equation will be solved by the method of least squares. At first, the radio wave propagation in homogeneous media will be considered. The electromagnetic field will be expanded by proper expansion functions, which satisfy the parabolic equation in homogeneous media. The expansion coefficients will be derived by the least square method through enforcing initial and boundary conditions. The least square functionals satisfy the initial and boundary conditions. Similar to the split step method, the field in the inhomogeneous media with known profile of refractive index can be obtained by proper phase shifting of the field in homogeneous media. The proposed method is more reliable than the split step method and can be applied over rough boundary without any excess computations. In comparison with the finite difference method, the proposed method is very fast.

Virtual boundary method for the analysis of elliptically cross-sectional photonic-crystal fibers with elliptical pores

This paper introduces a new semianalytical method for the analysis of propagation characteristics of elliptically cross-sectional photonic-crystal fibers (PCFs) with elliptical pores. This method known as a virtual boundary method (VBM) is based on the equivalency between an actual PCF and a three-layered, transversely inhomogeneous waveguide. The complicated refractive-index profile of the PCF is written as a double Fourier series, and an approximate separable wave equation is found in an elliptical coordinate system for the longitudinal field components. The exact solution to the derived equation is expressed in terms of higher order transcendental functions, such as regular and irregular Coulomb-wave functions and Mathieu functions. After having expressed all the field components, boundary conditions are imposed on the boundaries, and then, a transcendental equation for the propagation constant is derived, which is solved numerically. The validity of the method is ensured by comparing various quantities, such as effective indexes, modal birefringences, and electromagnetic field distributions, with those from an accurate full-vector finite-element method (FEM) simulator, showing relatively good agreement between the results. The method correctly confirms some of the unique PCFs' properties, such as strong localization of light within the fiber and enhancement of modal birefringence as a function of the topology of hole arrangement.

On wave propagation in an inhomogeneous non–stationary medium with an inhomogeneous non-stationary flow

An approximate wave equation that models scalar wave propagation in a moving fluid whose ambient properties and flow are inhomogeneous both in space and time is considered. Asymptotic solutions for both non–caustic and caustic regions and some Hamiltonian properties of the equation in both non–caustic and caustic regions are developed.

Approximate single wave equation derived from Biot’s porous media equations

Manipulation of Biot’s equations using low-frequency and long-wavelength approximations yields wave equation with a dissipation term that predicts plane wave attenuation coefficients proportional to the square of the frequency. Examination of Biot’s original derivation verifies that the wave speed in this equation is what results from Wood’s theory. The derivation predicts that the fluid displacement is a predictable fraction of that of the solid matrix, and this is consistent with the behavior in Brillouin’s model of an elastic spring loaded with masses of alternating magnitude. Biot’s assumption that the pores must be connected is argued to be unnecessary. Unconnected pores may be elongated with constrictions, and the sound field will cause some portions of the porelets to be compressed more than other portions, so that pressure differences across constrictions force viscous flow. A heuristic derivation shows that the wave attenuation coefficient is inversely proportional to viscosity and sound speed, and directly proportional to density, the number of porelets per unit volume, the average length of porelets, and the square of the cross-sectional area of the constrictions and frequency. The heuristic theory is supported with a detailed analysis of the energy dissipation within a dumbbell-shaped porelet within an extended isotropic elastic solid.

Fast, nonperturbative, transmission inverse scattering, quantitative imaging with laboratory and simulated acoustic data

Inverse scattering provides an ultimate method for acoustic imaging for medical and other fields of application (because it is quantitative and is self-correcting for refraction, absorption, and diffraction artifacts). E. Wolf showed that, with plane wave incident excitation, spatial distributions of inhomogeneous acoustic speed of sound and absorption within bodies could be reconstructed under the Born approximation by use of a simple back propagation algorithm operating in spatial frequency space (k-space). This approach, as extended to include the Rytov approximation and/or to real-space, is known as ‘‘diffraction tomography’’ (DT). However, DT fails at frequencies above 1 MHz, for objects with the size and contrast of the human female breast. We have extended DT by replacing its approximate wave equations with an exact wave equation. We present for the first time images of laboratory objects (agar cylinders, excised tissue, etc.) made by inverting transmission data. Present algorithms are at the boundary of real time operations or for 3D imaging with PC level computer networks. We discuss problems characteristic of inverse scattering such as multiple minima solutions and the need for receiver and transmitter element beam patterns. Input of these images for computer aided diagnosis is discussed.

Correlated continuum wave functions for three particles with Coulomb interactions

We present an approximate solution of the Schr\"odinger equation for the three-body Coulomb problem. We write the Hamiltonian in parabolic curvilinear coordinates and study the possible separation of the wave equation as a system of coupled partial differential equations. When two of the particles are heavier than the others, we write an approximate wave equation that incorporates some terms of the Hamiltonian that before had been considered as a perturbation. Its solution can be expressed in terms of a confluent hypergeometric function of two variables. We show that the proposed wave function includes a correlation between the motion of the light particle relative to the heavy particles and verifies the correct asymptotic behavior when all particles are far from each other. Finally, we discuss the possible uses of this function in the calculation of transition matrices and differential cross sections in ionizing collisions.

New full-wave approximation for ocean acoustic travel time predictions

A new full-wave parabolic approximation is introduced that is valid for a wide range of grazing angles. By Fourier synthesis it yields travel times of ocean acoustic multipaths that are insensitive to a reference speed of sound. After depths and sound speeds are transformed to new coordinates, the highly efficient ‘‘split-step Fourier’’ algorithm is used to solve the new approximate wave equation for forward propagation. Accuracy of the new approximation has been tested by comparison to a broadband normal mode model in a range-independent environment. At 1000 km range and with a pulse of resolution 20 ms at center frequency 75 Hz, computed travel times of 24 multipaths agreed with maximum difference 3.4 ms, mean difference 0.9 ms, and rms difference 1.5 ms. This approximation may prove to be an efficient method for accurate travel time predictions of multipaths over a wide range of acoustic frequencies and for basin scale distances.

Exact solutions of regular approximate relativistic wave equations for hydrogen-like atoms

Apart from relativistic effects originating from high kinetic energy of an electron in a flat potential, which are treated in first order by the Pauli Hamiltonian, there are relativistic effects even for low-energy electrons if they move in a strong Coulomb potential. The latter effects can be accurately treated already in the zeroth order of an expansion of the Foldy–Wouthuysen transformation, if the expansion is carefully chosen to be nondivergent for r→0 even for Coulomb potentials, as shown by Van Lenthe et al. [J. Chem. Phys. 99, 4597 (1993)] (cf. also Heully et al. [J. Phys. B 19, 2799 (1986)] and Chang et al. [Phys. Scr. 34, 394 (1986)]). In the present paper, it is shown that the solutions of the zeroth order of this two-component regular approximate (ZORA) equation for hydrogen-like atoms are simply scaled solutions of the large component of the Dirac wave function for this problem. The eigenvalues are related in a similar way. As a consequence, it is proven that under some restrictions, the ZORA Hamiltonian is bounded from below for Coulomb-like potentials. Also, an exact result for the first order regular approximate Hamiltonian is given. The method can also be used to obtain exact results for regular approximations of scalar relativistic equations, like the Klein–Gordon equation. The balance between relativistic effects originating from the Coulombic singularity in the potential (typically core penetrating s and p valence electrons in atoms and molecules) and from high kinetic energy (important for high-energy electrons in a flat potential and also for core-avoiding high angular momentum (d, f, and g states in atoms) are discussed.

Coupled wave equations for numerical calculation of acoustical propagation and scattering by atmospheric turbulence

The usual approximate acoustic wave equations [e.g., A. D. Pierce, J. Acoust. Soc. Am. 87, 2292 (1990)] were developed for ambient conditions (turbulence) in which the scale length a of the spatial variation of temperature and flow velocity is much larger than the acoustical wavelength λ; this is not the case in many applications of interest. An alternative set of coupled acoustical wave equations, valid for any a/λ, is presented. These equations involve no approximations except the usual neglect of the viscosity terms in the Navier–Stokes equation for the acoustic flow. For quasistatic or ‘‘frozen’’ turbulence, the integral forms of these equations are presented, suitable for application of the digitized Green’s function method [G. H. Goedecke and Sean G. O’Brien, Appl. Opt. 27, 2431 (1988)].

WAVELET SOLUTIONS FOR TIME HARMONIC ACOUSTIC WAVES IN A FINITE OCEAN

In this paper, we show how wavelet analysis may be used to solve the parabolic approximate wave equation. The paper is naturally subdivided into three parts. The first entails a crash course in wavelet analysis with the aim of using it to solve partial differential equations. The second part develops the wavelet-Galerkin approximation for solving partial differential equations in variational form. Finally, the third section derives a range-depth adaptive wavelet approach, and, moreover, provides an algorithm for solving the propagation problem.

Wave equation for sound in fluids with unsteady inhomogeneous flow

An approximate wave equation is derived for sound propagation in an inhomogeneous fluid with ambient properties and flow that vary both with position and time. The derivation assumes that the characteristic length scale and characteristic time scale for the ambient medium are larger than the corresponding scales for the acoustic disturbance. For such a circumstance, it is argued that accumulative effects of inhomogeneities and the ambient unsteadiness are satisfactorily taken into account by a wave equation that is correct to first order in the derivatives of the ambient quantities. A derivation that consistently neglects second- and higher-order terms leads to a concise wave equation similar to the familiar ordinary wave equation of acoustics. The wide applicability of this equation is established by showing that it reduces to previously known wave equations for special cases and by showing, with the eikonal approximation, that it yields the geometrical acoustics equations for ray propagation in moving inhomogeneous media.

Coupled-mode theory for corrugated optical waveguides

A new derivation of the coupled-mode equations and coupling coefficient for the troublesome but important case of the Bragg reflection of TM guided waves in a corrugated, planar optical waveguide is presented. The technique solves the (approximate) wave equation directly and clarifies those features of the analysis that are essential to treat the problem correctly.

Uniform formulation of the problem of distortion of sonic booms or bangs by atmospheric turbulence

A now-classic paper relating to wave propagation through turbulence is S. C. Crow's “Distortion of sonic bangs by atmospheric turbulence” [J. Fluid Mech. 37, 529–563 (1969)]. Crow treats the linear acoustics problem of an (initially) step discontinuity in pressure (weak shock} propagating through a nominally uniform atmosphere with weak turbulence superimposed, uses a first-order perturbation theory to determine the turbulence-induced corrections to the waveform, and obtains some simple predictions concerning the variance at times somewhat later than shock onset, which have subsequently been well confirmed by field data. The singular nature of Crow's solution near the nominal time of shock onset, however, has always been somewhat disquieting to the present author, so a formulation is here given that avoids this singularity using a version of the method of strained coordinates. The recently derived approximate wave equation [Pierce, J. Acoust. Soc. Am. (in press)] for wave propagation through a fluid with inhomogeneous unsteady flow is used as a starting point. The solution for the acoustic pressure is taken in the form of a rippled wavefront (Δp)(1 + ψ)H(1 − c−1x + φ), where φ is a function of position, while ψ is a function of position and time. Here, H(t) is the Heaviside unit step function and Δp is the nominal pressure jump at the start of the boom. Linear perturbation equations governing the functions φ and ψ are developed and solved by approximate methods. [Work supported by NASA Langley Research Center and by the William E. Leonhard endowment to Pennsylvania State University.]

Benchmark calculations for higher-order parabolic equations

Benchmark solutions generated with parabolic equation (PE) models are presented for range-dependent underwater acoustic propagation problems involving both penetrable and perfectly reflecting ocean bottoms. The solution of the wide-angle PE of Claerbout [J. F. Claerbout, Fundamentals of Geophysical Data Processing (McGraw-Hill, New York, 1976), pp. 206–207] agrees with the outgoing coupled-mode solution for the problems involving penetrable bottoms. The solution of the higher-order PE of Bamberger et al. [Bamberger et al., ‘‘Higher Order Paraxial Wave Equation Approximations in Heterogeneous Media,’’ SIAM J. Appl. Math. 48, 129–154 (1988)], which is a generalization of Claerbout’s PE, agrees with the outgoing coupled-mode solution for problems involving large variations in sound speed and propagation nearly orthogonal to the preferred direction. The computer code FEPE was used to generate the benchmark solutions and was found to run several times faster than the IFDPE computer code due to a tridiagonal system solver in FEPE that is optimized for range-dependent problems.

Embedding of focus wave modes into a wider class of approximate wave equation solutions

Focus wave modes are embedded into a wider class of solutions of the wave equation in the paraxial approximation. The generalizations of focus wave modes are made in regard to the following attributes: (1) a wave-packet envelope is admitted that moves with the group velocity in the direction opposite that of the focus envelope and (2) the velocity of the focus envelope can be chosen arbitrarily, in contrast to the group velocity. The Fourier spectrum of the generalized approximate solutions of the wave equation is calculated. This Fourier spectrum becomes singular on the dispersion surface with a strong correlation between frequencies and longitudinal components of the wave vectors in case of focus wave modes. The well-known Gaussian beam solutions with resting focus appear as another special case of the considered class of approximate solutions of the wave equation.

Parabolic approximation method for the mode conversion-tunneling equation

The derivation of the wave equation that governs ion cyclotron range of frequencies (ICRF) wave propagation, absorption, and mode conversion within the kinetic layer in tokamaks has been extended to include diffraction and focusing effects associated with the finite transverse dimensions of the incident wave fronts. The kinetic layer considered consists of a uniform density, uniform temperature slab model in which the equilibrium magnetic field is oriented in the ẑ direction and varies linearly in the x̂ direction. An equivalent dielectric tensor, as well as a two-dimensional energy conservation equation, are derived from the linearized Vlasov–Maxwell system of equations. A generalized, but approximate form of the mode conversion-tunneling equation is then extracted from the Maxwell equations, using the parabolic approximation method in which transverse variations of the wave fields are assumed to be weak in comparison to the variations in the primary direction of propagation. Numerical solutions of this approximate wave equation agree well with solutions to the exact wave equation within the kinetic layer.