Propagator Matrices Research Articles

Waves in space-dependent and time-dependent materials: A systematic comparison

Waves in space-dependent and in time-dependent materials obey similar wave equations, with interchanged time- and space-coordinates. However, since the causality conditions are the same in both types of material (i.e., without interchangement of time- and space-coordinates), the solutions are dissimilar.We present a systematic treatment of wave propagation and scattering in 1D space-dependent and in 1D time-dependent materials. After formulating unified equations, we discuss Green’s functions and simple wave field representations for both types of material. Next we discuss propagation invariants, i.e., quantities that are independent of the space coordinate in a space-dependent material (such as the net power-flux density) or of the time coordinate in a time-dependent material (such as the net field-momentum density). A discussion of general reciprocity theorems leads to the well-known source–receiver reciprocity relation for the Green’s function of a space-dependent material and a new source–receiver reciprocity relation for the Green’s function of a time-dependent material. A discussion of general wave field representations leads to the well-known expression for Green’s function retrieval from the correlation of passive measurements in a space-dependent material and a new expression for Green’s function retrieval in a time-dependent material.After an introduction of a matrix–vector wave equation, we discuss propagator matrices for both types of material. Since the initial condition for a propagator matrix in a time-dependent material follows from the boundary condition for a propagator matrix in a space-dependent material by interchanging the time- and space-coordinates, the propagator matrices for both types of material are interrelated in the same way. This also applies to representations and reciprocity theorems involving propagator matrices, and to Marchenko-type focusing functions.

Plug-and-play analytical paradigm for the scattering of plane waves by ‘layer-cake’ periodic systems

We investigate the scattering of scalar plane waves in two dimensions by a heterogeneous layer that is periodic in the direction parallel to its boundary. On describing the layer as a union of periodic laminae, we develop a solution of the scattering problem by merging the concept of propagator matrices and that of Bloch eigenstates featured by the unit cell of each lamina. The featured Bloch eigenstates are obtained by solving the quadratic eigenvalue problem (QEVP) that seeks a complex-valued wavenumber normal to the layer boundary given (i) the excitation frequency, and (ii) real-valued wavenumber parallel to the boundary—that is preserved throughout the system. Spectral analysis of the QEVP reveals sufficient conditions for discreteness of the eigenvalue spectrum and the fact that all eigenvalues come in complex-conjugate pairs. By deploying the factorization afforded by the propagator matrix approach, we demonstrate that the contribution of individual eigenvalues (and so eigenmodes) to the solution diminishes exponentially with absolute value of their imaginary part, which then forms a rational basis for truncation of the factorized Bloch (FB)-wave solution. The proposed methodology caters for the optimal design of rainbow traps, energy harvesters and metasurfaces, whose potency to manipulate waves is decided not only by the individual dispersion characteristics of the component laminae, but also by ordering and generally fitting of the latter into a composite layer. Using the FB approach, evaluation of trial configurations—as generated by the permutation and window translation/stretching of the component laminae—can be accelerated by decades.

Electromechanical coupling characteristics of multilayered piezoelectric quasicrystal plates in an elastic medium

AbstractQuasicrystals (QCs) have attracted tremendous attention of researchers for their unusual properties. In this paper, an exact electric‐elastic solution of the simply supported and multilayered three‐dimensional (3D) cubic piezoelectric quasicrystal (PQC) nanoplate with the nonlocal effect is derived. Based on the basic elasticity equation of 3D QCs, we construct the linear eigenvalue system in terms of the pseudo‐Stroh formalism, from which the general solutions of the extended displacements and stresses in any homogeneous layer can be obtained. The two‐parameter foundation model is utilized to simulate the interaction between the nanoplate and elastic medium. The propagator matrices are employed to connect the field variables at the upper interface to those at the lower interface of each layer. Based on the boundary conditions of the upper and lower surfaces of the laminate and foundation model, the solutions are employed to derive from the global propagator matrix. Compared with the conventional propagator matrix method, a new propagator method is reestablished to deal with numerical instabilities of the case of large aspect ratio and high‐order frequencies for QC laminates. Finally, typical numerical examples are presented to illustrate the influence of nonlocal parameters and elastic medium coefficients on phonon, phason, and electric variables of 3D PQC nanoplates.

Random matrix theory for description of sound scattering on background internal waves in a shallow sea

The problem of propagation of low-frequency sound in a shallow waveguide with random hydrological inhomogeneity caused by background internal waves is considered. A new approach to statistical modeling of acoustic fields, based on the application of the random matrix theory and previously successfully used for deep-water acoustic waveguides, is used to the case of shallow-water waveguides. In this approach, sound scattering on random inhomogeneity is described using an ensemble of random propagator matrices which describe the transformation of the acoustic field in the space of normal waveguide modes. A study of the effect of sound “escaping” from a waveguide was carried out. The term “escaping” here means energy transfer to modes with stronger attenuation due to scattering on internal waves. A model of an underwater sound channel with an axis at a depth of about 45 meters is considered. It is shown that the first few modes propagating inside the water column are very little subject to losses due to the “escaping”. The strongest impact of the leakage scattering is experienced by the middle group of modes capable of reaching the sea surface. It is revealed as significant increasing of losses as compared to a horizontally homogeneous waveguide. On the other hand, the existence of linear mode combinations for which loss enhancement is practically absent has been revealed. These linear combinations correspond to the eigenfunctions of an inhomogeneous waveguide. Statistical analysis of propagator eigenfunctions indicates on qualitative differences of mechanisms of scattering for frequencies of 100 and 500 Hz.

Random Matrix Theory for Sound Propagation in a Shallow-Water Acoustic Waveguide with Sea Bottom Roughness

The problem of sound propagation in a shallow sea with a rough sea bottom is considered. A random matrix approach for studying sound scattering by the water–bottom interface inhomogeneities is developed. This approach is based on the construction of a statistical ensemble of the propagator matrices that describe the evolution of the wavefield in the basis of normal modes. A formula for the coupling term corresponding to inter-mode transitions due to scattering by the sea bottom is derived. The Weisskopf–Wigner approximation is utilized for the coupling between waterborne and sediment modes. A model of a waveguide with the bottom roughness described by the stochastic Ornstein–Uhlenbeck process is considered as an example. Range dependencies of mode energies, modal cross coherences and scintillation indices are computed using Monte Carlo simulations. It is shown that decreasing the roughness correlation length enhances mode coupling and facilitates sound scattering.

Computing Theoretical Seismograms from a Point Source in a Spherical Multilayered Medium

Abstract Computing theoretical seismograms from a point source in a given Earth model is essential for modeling and inversion of observed seismic waveforms for Earth’s structure and earthquake source parameters. Here, we derived the propagator matrices and source terms for a spherical multilayered Earth model using the exact earth flattening transformation. We found that their differences from their counterparts in horizontal layered media are inversely proportional to the nondimensional horizontal wavenumber and its higher order. In addition, all the source terms in a spherical layered model have a source-depth dependent scaling factor that differs from in a horizontal layered model by up to 6% for deep earthquakes. The surface displacement produced by a point source can be obtained in a similar form as in horizontal layered media. Computation of theoretical seismograms was implemented using the generalized reflection and transmission coefficients method. Numerical tests show that our formulae and implementation are correct and efficient for computing full-wave seismograms, including the permanent displacements, at teleseismic distances up to 100°. Individual seismic phases can be isolated and analyzed semianalytically because the generalized reflection and transmission method is used. Furthermore, our analytic expression of displacement in terms of the propagator matrices and source terms can be used to derive analytic derivatives of seismograms for full-wave waveform inversion.

Propagator and transfer matrices, Marchenko focusing functions and their mutual relations

SUMMARY Many seismic imaging methods use wavefield extrapolation operators to redatum sources and receivers from the surface into the subsurface. We discuss wavefield extrapolation operators that account for internal multiple reflections, in particular propagator matrices, transfer matrices and Marchenko focusing functions. A propagator matrix is a square matrix that ‘propagates’ a wavefield vector from one depth level to another. It accounts for primaries and multiples and holds for propagating and evanescent waves. A Marchenko focusing function is a wavefield that focuses at a designated point in space at zero time. Marchenko focusing functions are useful for retrieving the wavefield inside a heterogeneous medium from the reflection response at its surface. By expressing these focusing functions in terms of the propagator matrix, the usual approximations (such as ignoring evanescent waves) are avoided. While a propagator matrix acts on the full wavefield vector, a transfer matrix (according to the definition used in this paper) ‘transfers’ a decomposed wavefield vector (containing downgoing and upgoing waves) from one depth level to another. It can be expressed in terms of decomposed Marchenko focusing functions. We present propagator matrices, transfer matrices and Marchenko focusing functions in a consistent way and discuss their mutual relations. In the main text we consider the acoustic situation and in the appendices we discuss other wave phenomena. Understanding these mutual connections may lead to new developments of Marchenko theory and its applications in wavefield focusing, Green’s function retrieval and imaging.

Wave-field representations with Green's functions, propagator matrices, and Marchenko-type focusing functions.

Classical acoustic wave-field representations consist of volume and boundary integrals, of which the integrands contain specific combinations of Green's functions, source distributions, and wave fields. Using a unified matrix-vector wave equation for different wave phenomena, these representations can be reformulated in terms of Green's matrices, source vectors, and wave-field vectors. The matrix-vector formalism also allows the formulation of representations in which propagator matrices replace the Green's matrices. These propagator matrices, in turn, can be expressed in terms of Marchenko-type focusing functions. An advantage of the representations with propagator matrices and focusing functions is that the boundary integrals in these representations are limited to a single open boundary. This makes these representations a suitable basis for developing advanced inverse scattering, imaging and monitoring methods for wave fields acquired on a single boundary.

Static response and free vibration analysis for cubic quasicrystal laminates with imperfect interfaces

Quasicrystals have attracted the tremendous attention of researchers for their unusual properties. In this paper, an analytical treatment is presented for the static response and free vibration of a multilayered three-dimensional, cubic quasicrystal rectangular simply supported plate with bonding imperfections. Based on the basic elasticity equation of three-dimensional quasicrystals, we construct the linear eigenvalue system in terms of the pseudo-Stroh formalism, from which the general solutions of the extended displacements and stresses in any homogeneous layer can be obtained. Furthermore, these solutions along the thickness direction can be utilized to solve any physical variables under given boundary conditions. For multilayered plates, the propagator matrices are employed to connect the field variables at the upper interface to those at the lower interface of each layer. The special spring model, which describes the discontinuity of the physical quantities across the interface, is introduced into the overall propagator relationship of the structure. Compared with the conventional propagator matrix method, a new propagator relation is established to resolve numerical instabilities of the case of large aspect ratio and high-order frequencies for QC laminates with imperfect interface. In addition, the traction-free boundary condition on the top and bottom surfaces of the layered plate is considered to investigate the free vibration characteristics of the laminates. Finally, typical numerical examples are presented to illustrate the influence of imperfect interfaces on static response and free vibration of cubic quasicrystal plates.

Small Matrix Path Integral for System-Bath Dynamics.

A small matrix decomposition of the path integral expression (SMatPI) that yields the reduced density matrix of a system interacting with a dissipative harmonic bath is obtained by recursively spreading the entangled influence functional terms over longer time intervals while simultaneously decreasing their magnitude until these terms become negligible. This allows summation over the path integral variables one by one through multiplication of small matrices with dimension equal to that of the bare system. The theoretical framework of the decomposition is described using a diagrammatic approach. Analytical and numerical calculations show that the necessary time length for the temporal entanglement to become negligible is practically the same as the bath-induced memory. The properties and structure of the propagator matrices are discussed, and applications to multistate systems are presented.

Evaluation of complex spectrum of the symmetrical Lamb waves for three-layered plates at low frequency. Part I: Foundations

A new method combining the asymptotic and iteration approaches to study the total wavenumber spectrum for a plate made of isotropic layers is suggested. In this part we use the propagator matrices to deduce the dispersion equation and its static limit in a closed form with the use of symbolic calculations. The roots in statics which are analyzed at the first step play the key role. Then, the low frequency approximations for the dispersion curves are derived. The method is aimed to evaluate a dispersion curve of any order n and to describe it in a closed form by an asymptotic formula. Especially it concerns the case n→+∞ which causes difficulty for the pure numerical approaches.

Dynamic analysis of multilayered magnetoelectroelastic plates based on a pseudo-Stroh formalism and Lagrange polynomials

In this article, a semi-analytical three-dimensional modeling of dynamic behavior of the multilayered magnetoelectroelastic plates under simply supported edges boundary conditions is derived. A combination of pseudo-Stroh formalism and the Lagrange polynomials is elaborated for the space and time response. The time domain is subdivided into small intervals that are discretised using the associated Tchybechev points. The layer-time solution is elaborated in time-dependent matrix form. The propagator matrices are used for the laminated multifunctional plates with an arbitrary number of layers. Extended-traction vectors are obtained for mechanical, electrical, and magnetic excitations. To validate the elaborated numerical procedure, the dynamic behavior of the three layered plates made of piezoelectric material [Formula: see text] and piezomagnetic material [Formula: see text] is investigated. The lower surface of the plate is assumed to be traction free, whereas the upper surface is subjected to a dynamic sinusoidal loading. The obtained results are in good agreement with the available ones based on the Layer wise and the state-space approaches. These results demonstrated that a magnetoelectric coupling coefficient is time-independent but depends strongly on the kind of imperfect interfaces and the taking sequences of the multilayered plates. Furthermore, it is established that the imperfect interfaces have a strong influence on the dynamic behavior of the laminated structures.

A Low-Order Series Approximation of Thin-Bed PP-Wave Reflections

The study of thin-bed seismic phenomena is important in crustal, exploration and engineering seismology. Presently, seismic reflectivity theories based on single-interface assumption are widely used though they are only suitable for thick deposits. Thin-bed reflectivity theories are established on complex propagator matrices and are difficult to be applied to reveal thin-bed properties directly. Therefore, an approximation of thin-bed PP-wave reflection coefficients (RPP) is derived in this paper. First, the relationship between thin-bed RPP and incidence angles is analyzed through series expansion method. For PP-wave, its reflection coefficients are even power series functions of sine incidence angles. Then, for small incidence, RPP of the thin bed is further simplified into a second-order series approximation with respect to the sine incidence angles. Simulations and accuracy analyses of the approximate formula show that approximation errors are smaller than 5% as the incidence angles smaller than 20 degrees. Based on this approximate formula, an approach is given for estimating thin-bed properties including P-wave impedance ratios and thickness. The estimation approach is applied in properties estimation of a thin bed model. Perfect performances of the model example show the future potentiality of the approximate formula in thin-bed Amplitude-Versus-Offset (AVO) analysis and inversion.

Breit-Wigner approximation for propagators of mixed unstable states

For systems of unstable particles that mix with each other, an approximation of the fully momentum- dependent propagator matrix is presented in terms of a sum of simple Breit-Wigner propagators that are multiplied with finite on-shell wave function normalisation factors. The latter are evaluated at the complex poles of the propagators. The pole structure of general propagator matrices is carefully analysed, and it is demonstrated that in the proposed approximation imaginary parts arising from absorptive parts of loop integrals are properly taken into account. Applying the formalism to the neutral MSSM Higgs sector with complex parameters, very good numerical agreement is found between cross sections based on the full propagators and the corresponding cross sections based on the described approximation. The proposed approach does not only technically simplify the treatment of propagators with non-vanishing off-diagonal contributions, it is shown that it can also facilitate an improved theoretical prediction of the considered observables via a more precise implementation of the total widths of the involved particles. It is also well-suited for the incorporation of interference effects arising from overlapping resonances.

Mode coherence in random matrix theory simulations

Random matrix theory (RMT) can be used to simulate the effect of internal waves on broadband acoustic mode time series in deep water, as described by Hegewisch and Tomsovic in several papers [Europhys. Lett. 2012; J. Acoust. Soc. Am. 2013]. Using RMT, narrowband mode propagation consists of the multiplication of a series of propagator matrices designed to model the mode coupling due to internal waves at a single frequency. In a recent paper, we varied the correlation of the mode coupling matrices with frequency and examined the effect on the time series generated via Fourier synthesis [Van Uffelen & Wage, Inst. of Acoustics Conf., 2016]. This talk focuses on another key aspect of the RMT model, i.e., mode-to-mode coherence at a single frequency. Modes decorrelate as they propagate through internal waves. Building on Creamer's work, Colosi and Morozov use transport theory to analyze mode-to-mode coherence and mode energy in deep water environments [J. Acoust. Soc. Am., 2009]. This talk compares the energy and coherence results obtained with RMT methods to those from transport theory. [Work sponsored by ONR.]

Random matrix theory simulation of propagation through internal waves

Hegewisch and Tomsovic proposed using random matrix theory (RMT) to simulate the effect of internal wave scattering on long-range propagation in deep water [JASA, 2013]. They model mode propagation as the multiplication of a series of unitary propagator matrices whose statistics are derived using perturbation theory. A simplifying assumption facilitates the generation of broadband time series. One potential advantage of the RMT approach is that it requires fewer computations than a full wave simulation to produce a large set of statistical realizations of the received time series [Hegewisch & Tomosovic, Europhys. Lett. 2012]. Efficient simulation techniques are especially important for applications that require predictions of the timefront and/or its statistics at a large number of possible receiver ranges. For example, simulating data sets for acoustic gliders operating within a tomographic array is computationally intensive. While Hegewisch and Tomsovic illustrated the performance of their approach for one deep water simulation environment, they also highlighted the need for additional research. This talk investigates the feasibility of using RMT to efficiently simulate broadband time series for use in analysis of glider data. The underlying assumptions of the RMT approach will be assessed based on results of prior experimental work. [Work sponsored by ONR.]

2D full-waveform modeling of seismic waves in layered karstic media

We have developed a new propagator-matrix scheme to simulate seismic-wave propagation and scattering in a multilayered medium containing karstic voids. The propagator matrices can be found using the boundary element method. The model can have irregular boundaries, including arbitrary free-surface topography. Any number of karsts can be included in the model, and each karst can be of arbitrary geometric shape. We have used the Burton-Miller formulation to tackle the numerical instability caused by the fictitious resonance due to the finite size of a karstic void. Our method was implemented in the frequency-space domain, so frequency-dependent [Formula: see text] can be readily incorporated. We have validated our calculation by comparing it with the analytical solution for a cylindrical void and to the spectral element method for a more complex model. This new modeling capability is useful in many important applications in seismic inverse theory, such as imaging karsts, caves, sinkholes, and clandestine tunnels.

Spate: AnRPackage for Spatio-Temporal Modeling with a Stochastic Advection-Diffusion Process

The R package spate implements methodology for modeling of large space-time data sets. A spatio-temporal Gaussian process is defined through a stochastic partial differential equation (SPDE) which is solved using spectral methods. In contrast to the traditional geostatistical way of relying on the covariance function, the spectral SPDE approach is computationally tractable and provides a realistic space-time parametrization. This package aims at providing tools for simulating and modeling of spatio-temporal processes using an SPDE based approach. The package contains functions for obtaining parametrizations, such as propagator or innovation covariance matrices, of the spatio-temporal model. This allows for building customized hierarchical Bayesian models using the SPDE based model at the process stage. The functions of the package then provide computationally efficient algorithms needed for doing inference with the hierarchical model. Furthermore, an adaptive Markov chain Monte Carlo (MCMC) algorithm implemented in the package can be used as an algorithm for doing inference without any additional modeling. This function is flexible and allows for application specific customizing. The MCMC algorithm supports data that follow a Gaussian or a censored distribution with point mass at zero. Spatio-temporal covariates can be included in the model through a regression term.

Numerical reproducibility of time-dependent motions of spatial waves in air-columns of wind instruments

Since Schumacher introduced a time-domain model of single-reed instruments and McIntyre et al. gave the general concept of time-domain models of wind instruments, the time-domain models, namely, delayed feedback models, have become an important numerical tool for study of wind instruments due to their simplicity, easiness to handle, and reliability. However, those models only reproduce wave oscillations observed in mouthpieces of wind instruments. In this talk, we will propose a numerical technique, which is able to reproduce time dependent motions of spatial waves in an air-column. It is composed of inversed wave propagator matrices combined with the forward and backward Fourier transformations. The resultant spatial waves in the air-column exhibit very similar time-dependent behavior to those observed by an experiment for the clarinet. Actually, backward and forward rounded-off step waves are observed. We will also discuss difference in wave shapes and their time-dependent behavior depending on the shapes of air-columns, cylindrical one like the clarinet, conical one like the saxophone and horn-shaped one like brass instruments. For the conical and horn-shaped air-columns, Helmholtz-like waves are observed rather than the step waves observed for the cylindrical air-column.

Spatio‐temporal smoothing and EM estimation for massive remote‐sensing data sets

The use of satellite measurements in climate studies promises many new scientific insights if those data can be efficiently exploited. Due to sparseness of daily data sets, there is a need to fill spatial gaps and to borrow strength from adjacent days. Nonetheless, these satellites are typically capable of conducting on the order of 100,000 retrievals per day, which makes it impossible to apply traditional spatio‐temporal statistical methods, even in supercomputing environments. To overcome these challenges, we make use of a spatio‐temporal mixed‐effects model. For each massive daily data set, dimension reduction is achieved by essentially modelling the underlying process as a linear combination of spatial basis functions on the globe. The application of a dynamical autoregressive model in time, over the reduced space, allows rapid sequential computation of optimal smoothing predictions via the Kalman smoother; this is known as Fixed Rank Smoothing (FRS). The dimension‐reduced mixed‐effects model contains a number of unknown parameters, including covariance and propagator matrices, which describe the spatial and temporal dependence structure in the reduced‐dimensional process. We take an empirical‐Bayes approach to inference, which involves estimating the parameters and substituting them into the optimal predictors. Method‐of‐moments (MM) parameter estimation (currently used in FRS) is typically inefficient compared to maximum likelihood (ML) estimation and can result in large sampling variability. Here, we develop ML estimation via an expectation‐maximization (EM) algorithm, which offers stable computation of valid estimators and makes efficient use of spatial and temporal dependence in the data. The two parameter‐estimation approaches, MM and ML, are compared in a simulation study. We also apply our methodology to global satellite CO2 measurements: We optimally smooth the sparse daily CO2 maps obtained by the Atmospheric InfraRed Sounder (AIRS) instrument on the Aqua satellite; then, using FRS with EM‐estimated parameters, a complete sequence of the daily global CO2 fields can be obtained, together with their associated prediction uncertainties.