Invariant Imbedding Method Research Articles

Scattering of ultrasonic waves by defective adhesion interfaces in submerged laminated plates

We present an analytical-numerical method to simulate time-harmonic ultrasonic scattering from nonhomogeneous adhesive defects in anisotropic elastic laminates. To that end, we combine the quasistatic approximation (QSA) with a very high-order (tens or hundreds of terms) regular perturbation series to allow modeling of nonuniform interfacial flaws. To evaluate each term in the perturbation series, we use a recursive algorithm based on the invariant imbedding method. It is applicable to solve wave propagation problems in arbitrarily anisotropic layered plates and it is stable for high frequencies. We demonstrate examples of convergence and divergence of the perturbation series, and validate the method against the exact solution of plane wave reflection from a layered plate immersed in water. We present a further example of scattering of a Gaussian beam by an inhomogeneous interfacial flaw in the layered plate. We discuss how results of our simulations can be used to indicate the frequencies and angles of incidence where scattering from potential defects is strongest. These parameters, presumably, offer the best potential for flaw characterization.

Invariant imbedding theory of mode conversion in inhomogeneous plasmas. I. Exact calculation of the mode conversion coefficient in cold, unmagnetized plasmas

This is the first of a series of papers devoted to the development of the invariant imbedding theory of mode conversion in inhomogeneous plasmas. A new version of the invariant imbedding theory of wave propagation in inhomogeneous media allows one to solve a wide variety of coupled wave equations exactly and efficiently, even in the cases where the material parameters change discontinuously at the boundaries and inside the inhomogeneous medium. In this paper, the invariant imbedding method is applied to the mode conversion of the simplest kind, that is, the conversion of p-polarized electromagnetic waves into electrostatic modes in cold, unmagnetized plasmas. The mode conversion coefficient and the field distribution are calculated exactly for linear and parabolic plasma density profiles and compared quantitatively with previous results.

Detecting and classifying adhesive flaws between bonded elastic plates

Nondestructively evaluting the quality of an adhesive bond is challenging. The reason is that the only way to measure bond strength is by measuring the force required to break a bond, after which the bond is broken. Localized adhesion flaws that diminish bond strength, however, also tend to diminish bond stiffness. For example, pockets of elevated porosity, microcracking, or other damage in the adhesive layer will simultaneously lower local bond strength and local effective bond stiffness. So guided, we formulate and solve an inverse scattering problem to reconstruct the effective stiffness distribution of an adhesive layer in a layered elastic plate. Our formulation is based on the method of invariant imbedding, and applies to isotropic and anisotropic elastic layers. We present two solutions of the inverse problem: the Born approximation and the exact solution. Both solutions are unstable, as is the nature of inverse scattering problems, and require some regularization in the presence of noise. We present computed examples and discuss the role of regularization.

Dispersive Wave Attenuation Due to Orographic Forcing

The O'Doherty--Anstey (ODA) approximation was originally formulated in the seismological literature for acoustic pulse propagation through a disordered stratified medium [Geophys. Prospecting, 19 (1971), pp. 430--458]. It explains the mechanism for amplitude attenuation (and pulse shaping) promoted by the variable coefficient, conservative hyperbolic model. This work generalizes the one-dimensional ODA theory for linear weakly dispersive water waves forced by a disordered orography. The analysis is performed through the recently formulated terrain-following Boussinesq system. This is achieved by applying the invariant imbedding method. As a result, dispersion alters the medium's correlation function which controls the apparent attenuation mechanism. On the other hand, orography affects the dispersive mechanism for the Airy function--like formation. A nonlinear Boussinesq solver was implemented, and theoretical results were validated for different values of the parameters of interest. The theoretical resul...

Fixed-point smoothing with non-independent uncertainty using covariance information

Recursive filtering and fixed-point smoothing algorithms, using covariance information, are designed in systems with uncertain observations, when the variables describing the uncertainty are not necessarily independent. It is assumed that the observations are perturbed by white plus coloured noises, and the autocovariance functions of the signal and coloured noise are given in a semidegenerate kernel form. The estimators are obtained by the orthogonal projection technique and using an invariant imbedding method. The algorithms can be applied for estimating stationary and non-stationary signals.

Influence of the Parameters of a Random Medium on the Characteristics of Transient Reflections

Based on the invariant imbedding method, we study numerically the statistical characteristics of the kernel of the backscattering operator in the case of normal incidence of a plane wave on a one-dimensional random medium with strong fluctuation intensities and various correlation radii of the irregularities. The local reflection coefficient of the medium is modelled by a centered Gaussian process with an exponential correlation function. The first eight one-point cumulants and the correlation functions of delta-pulse reflection are considered and the fluctuation phenomena are analyzed. The transition to the diffusion scattering regime is studded, and the numerical results are compared with the known analytical solutions.

A NOTE ON SCATTERING OPERATOR SYMBOLS FOR ELLIPTIC WAVE PROPAGATION

The ill-posed elliptic wave propagation problems can be transformed into well-posed initial value problems of the reflection and transmission operators characterizing the material structure of the given model by the combination of wave field splitting and invariant imbedding methods. In general, the derived scattering operator equations are of first-order in range, nonlinear, nonlocal, and stiff and oscillatory with a subtle fixed and movable singularity structure. The phase space and path integral analysis reveals that construction and reconstruction algorithms depend crucially on a detailed symbol analysis of the scattering operators. Some information about the singularity structure of the scattering operator symbols is presented and analyzed in the transversely homogeneous limit.

The Influence of the Initial-Signal Duration on Nonstationary Reflections in a Random Medium

Using the invariant imbedding method, we study numerically the statistical characteristics of backscattering of plane-wave pulses incident normally on a random one-dimensional medium. The local reflection coefficient of the medium is simulated by a centered Gaussian process with an exponential correlation function. We consider the time dependences of the average intensity and the kurtosis coefficient of reflections for various durations and carrier frequencies of the initial signal. It is shown that the attenuation and fluctuations of reflections increase with the initial-signal duration normalized to the time scale of the medium irregularities and depend on the signal carrier. Numerical results for the average intensity are compared with previously obtained analytical solutions. We discuss the appearance conditions of a wave stochastic resonance in nonstationary reflections.

State Estimation of Nonlinear Dynamical Systems on the Basis of Nonquadratic Cost Functions

In the present paper, the problem of improving the reliability of state estimation of nonlinear dynamical systems under the conditions of parametric and statistical uncertainty is solved. The solution of this problem is based on the combination of the possibilities of adaptive robust and guaranteeing techniques for signal processing, The proposed technique relies on the theory of multilevel optimization of stochastic systems on the basis of nonclassical cost functions. This technique includes the following: the key parameter is formed for the first level of optimization, which characterizes the accuracy of estimation; the generalized parameter is formed, which characterizes the reliability of estimation; tolerances on the generalized parameter are specified; the likelihood function is formed, which includes the parameters of the first and second levels of optimization; using the maximum principle technique and the method of invariant imbedding, the two-point boundary-value problem is solved. The algorithms synthesized for adaptive robust data processing with guaranteeing tuning from the generalized parameter is shown.

Photon diffusion and invariant imbedding

This paper examines multiple scattering processes from the points of view of the source function of radiative transfer and the emergence probability of photon diffusion. It extends previous results to the case of layered slabs, i.e., inhomogeneous plane-parallel media. It derives equations for basic functions of multiple scattering and photon diffusion and obtains relationships among them. The invariant imbedding method is used to physically and analytically derive initial-value problems for functions which also satisfy integral equations. The initial-value problems are useful for numerically solving direct problems, and for tackling inverse problems of remote sensing.

Chandrasekhar-type filter using covariance information for white Gaussian plus colored observation noise

This paper proposes recursive least-squares (RLS) Chandrasekhar-type filtering equations using the covariance information in the case of white Gaussian plus colored observation noise in linear continuous-time wide-sense stationary stochastic systems. Here, it is assumed that the system matrices in the state-space models for the signal and the colored noise, the cross-variance functions of the state variables for the signal and the colored noise with the observed value, the observation vectors for the signal and the colored noise, the variance of white Gaussian noise and the observed value are known. The number of differential equations included in the current Chandrasekhar-type filter is less than the previous filter using the covariance information. The Chandrasekhar-type equations are derived based on the invariant imbedding method. The Chandrasekhar-type filter brings numerically efficient algorithm in computation time.

Reflection of light from a random amplifying medium with disorder in the complex refractive index: Statistics of fluctuations

The probability distribution of the reflection coefficient for light reflected from a one-dimensional random amplifying medium with cross-correlated spatial disorder in the real and the imaginary parts of the refractive index is derived using the method of invariant imbedding. The statistics of fluctuations have been obtained for both the correlated telegraph noise and the Gaussian white-noise models for the disorder. In both cases, an enhanced backscattering (with a reflection coefficient greater than unity) results because of coherent feedback due to Anderson localization and coherent amplification in the medium. The results show that the effect of randomness in the imaginary part of the refractive index on localization and reflection is qualitatively different.

Boundary effects on the resonant coupling of the excitation modes of semiconductor quantum wells

An alternative approach based on the invariant imbedding method and the random phase approximation is used to study the excitation spectra of both neutral and parabolic quantum wells. Compared to other methods, this approach is efficient and leads to more stable solutions. It also provides a convenient way for studying boundary effects. The calculated excitation spectra for both types of quantum well agree well with experimental observations and previous calculations. The effects of the boundary condition on the excitation spectra were extensively studied. It was found that the positions and relative strengths of resonant peaks are sensitive to the boundary condition. The evolution of some of the resonant peaks can be attributed to the coupling between collective plasma modes and the intersubband transitions.

Investigation of statistical characteristics of waves in layered media with regular and random inhomogeneities based on the invariant imbedding method

We consider a model of a stationary problem of wave propagation in a layered half-space with regular and random inhomogeneities. The choice of regular perturbation corresponds to a linear profile of the wave velocity. Random inhomogeneities are simulated in the framework of the white noise model. We analyse the influence of inhomogeneities on the probability distribution of the reflection-coefficient phase and the imaginary value of the average wavefield at the boundary of the half-space.

Coherently amplifying random medium: Statistics of super-reflection

Light-wave propagation in a randomly scattering but uniformly and coherently amplifying optical medium is analysed for the statistics of the coefficient of reflectionr(l) that may now exceed unity, hence super-reflection. Uniform coherent amplification is introduced phenomenologically through a constant negative imaginary part added to the otherwise real dielectric constant of the medium, assumed random. The probability densityp(r,l) for the reflection coefficient, calculated in a random phase approximation, tends to an asymptotic formp(r, ∞) having a long tail with divergent mean 〈r〉 in the limit of weak disorder but with the sample lengthl(measured in units of the localization length in the absence of gain) ⪢1. This super-reflection is attributed to a synergetic effect of localization and coherent amplification, as distinct from the classically diffusive path-length prolongation. Our treatment is based on the invariant-imbedding method for the one-channel (single-mode) case, and is in the spirit of the scattering approach to wave transport as pioneered by Landauer. Generalization to the multichannel case is pointed out. Relevance to random lasers, and some recent results for a sub-meanfree path sample are also briefly discussed.

COMPARISON OF INVARIANT IMBEDDING AND LAYER-PEELING METHODS IN INVERSE PROBLEM

This paper examines an implementation of a layer-peeling method to inverse scattering problems in the time domain, which is applied here to the case of one-dimensional lossless discontinuous medium. The method is based on the fast Schur recursion applied directly to the discrete problem. The method is tested on several numerical examples in order to compare its performance to the standard invariant imbedding algorithm. The reconstructions of media from both synthetic data and measured data are presented. The layer-peeling method is shown to be considerably faster than the invariant imbedding method without the loss of precision.

Integrodifferential equations for the two-dimensional transition kernels of invariant imbedding

Invariant imbedding methods for two spatial dimensions are applied to derive a set of integrodifferential equations and related boundary conditions, in the kernels of transmission, left-turn, and right-turn transition operators for a two-dimensional particle transport theory problem. The operators describe the transition from incident to exiting physical states over rectangular regions. The set of equations is coupled only through the reflection kernel. This work is a continuation of and complements the earlier effort of the authors wherein an integrodifferential boundary-value problem for the reflection kernel is derived.

Inverse problem in nonstationary multidimensional medium

The problem of a scalar wave propagation from the point impulsive source in the layer of a nonstationary multidimensional medium is considered. The boundary problem for the wave equation is reformulated in the problem with the initial condition using the invariant imbedding method. The integral-differential inverse procedures of the various orders were obtained from the imbedding equations using the singularities method. The order of inverse procedure is defined by the degree of a polynomial in the analytical representation of the medium characteristic near the layer boundary. It was shown that the coefficients of the polynomial are calculated with the help of the differential characteristics of the point impulsive source in the inhomogeneous medium. The cause and character of the multidimensional inverse problem overdefiniteness are considered. The application of the proposed procedure for a statistical problem is discussed.

Multipole edge plasma modes of parabolic quantum wells

We study the excitation modes in parabolic quantum wells via an approach based on the invariant-imbedding method in the random-phase approximation. For large values of wave vectors, several higher-multipole edge plasmon modes have been observed for the first time below the intrasubband plasmon in parabolic quantum wells in addition to the normal surface plasmon modes. It is shown that the edge electron density profile has an important effect on the higher-multipole edge plasmon modes.

Wave propagators for transient waves in one-dimensional media

Wave propagators are introduced for transient wave propagation in a general one-dimensional medium. The definition of wave propagators is based upon a wave splitting technique. Several properties for the propagators are given and the relations between the method of propagators and the invariant imbedding method and the Green function approach are discussed.