Mark Spivack

A numerical approach to rough‐surface scattering by the parabolic equation method

The parabolic equation method is an effective approach when the acoustic wave field is incident at low grazing angles onto a rough surface. The method consists of an integral equation and an integral, the first of which yields the surface field derivative. The main part of this paper is concerned with an approximation to this equation, valid when wavenumber times surface height is up to order unity. The approximation has several advantages. First, it allows a decomposition of the equation into deterministic and stochastic components. The stochastic part depends only locally upon the surface in certain regimes, and this can give rise to a great reduction in computational expense. Some basic statistical moments of the stochastic component are also considered. These are nonstationary, but for the incident field a simple stationary transformation is found, which can therefore be compared with Monte Carlo simulations using far fewer realizations. These results are demonstrated computationally. The final part o...

Accurate Numerical Solution of the Fourth-moment Equation for Very Large Values of Γ

Abstract The parabolic fourth-moment equation is integrated numerically using a method based on operator splitting. The numerical scheme is unconditionally stable and convergent, allowing accurate results to be obtained for very large values of the strength parameter Γ. Intensity fluctuation spectra are obtained for values of Γ up to 105, three orders of magnitude larger than previously achieved. These spectra are compared with analytical solutions. Laws governing the position and height of the scintillation index peak are deduced.

Failure prediction and diagnosis for satellite monitoring systems using Bayesian networks

Predicting failure in complex systems, such as satellite network systems, is a challenging problem. A satellite earth terminal contains many components, including high-powered amplifiers, signal converters, modems, routers, and generators, any of which may cause system failure. The ability to estimate accurately the probability of failure of any of these components, given the current state of the system, may help reduce the cost of operation. Probabilistic graphical models, in particular Bayesian networks, provide a consistent framework in which to address problems containing uncertainty and complexity. Building a Bayesian network for failure prediction in a complex system such as a satellite earth terminal requires a large quantity of data. Software monitoring systems have the potential to provide vast amounts of data related to the operating state of the satellite earth terminal. Measurable nodes of the Bayesian network correspond to states of measurable parameters in the system and unmeasurable nodes represent failure of various components. Nodes for environmental factors are also included. A description of Bayesian networks will be provided and a demonstration of inference on the Bayesian network, such as the calculation of the marginal probability of failure nodes given measurements and the maximum probability state of the system for failure diagnosis will be given. Using the data to learn local probabilities of the network will be covered. An interface between MaxView monitoring and control software and a Bayesian network API will also be described.

Direct solution of the inverse problem for rough surface scattering at grazing incidence

Considers the inverse scattering problem for a scalar wavefield incident at grazing angles on a one-dimensional rough surface. The problem is formulated first as a pair of coupled integral equations in two unknown functions, knowledge of which immediately yields the surface. A method is described for the direct approximate solution of this system. Preliminary results are presented in groups of complicated rough surfaces which are closely recaptured in all details except for scale.

Solution of the inverse-scattering problem for grazing incidence upon a rough surface

The inverse-scattering problem for a scalar wave field incident at grazing angles upon a one-dimensional moderately rough surface is considered. The problem is solved directly by treating the scattering integral as an integral equation in the unknown field derivative at the surface and coupling this to a simple equation relating the derivative to the surface itself. An algorithm is described for the solution of this system, and results in which complicated rough surfaces are accurately reconstructed are presented.

Moments and angular spectrum for rough surface scattering at grazing incidence

This paper considers the statistics of a plane wave scattered at low grazing angles from a one‐dimensional rough surface, when the interaction with the surface is mainly in the forward direction. The mean intensity and autocorrelation of the scattered field and the corresponding angular spectrum, to second order in surface height is found. The diffuse component of the spectrum vanishes with the square of the grazing angle, while the specular part approaches unity linearly. The higher moments of the field at the mean surface are also obtained. It is shown that to first order in the grazing angle the moments of the scattered field change only by a phase factor with distance into the medium, so that in the small angle limit the moments in the medium can be derived from those at the surface. The results depend explicitly upon the autocorrelation function of the surface and its first derivative.

Coherent field and specular reflection at grazing incidence on a rough surface

The coherent component of the field scattered at grazing angles from a slightly rough pressure release surface is found. This is valid for multiple scattering and is based upon the parabolic integral equation method. Also examined are the scattering of planes waves under the method and in particular, the effect of truncating the boundary integral. It is shown that the coherent field remains invariant when the source and receiver are displaced vertically by equal and opposite distances, as was found numerically in a previous paper. In general, this can be shown to hold because the coherent field due to any plane wave is specular; however, under the parabolic equation method reflection is not specular, and thus the result is of particular interest. Reflection coefficients are given in closed form for several surface statistics, valid asymptotically at large distances.

Sound propagation in an irregular two-dimensional waveguide

A method is presented for the numerical calculation of a scalar wave propagating along a two-dimensional rough-sided or irregular waveguide. In this situation the wave becomes multiply scattered, with simultaneous interaction at the two boundaries. The field can be expressed in terms of a pair of coupled integral equations; these are derived and solved in an approach based on the parabolic integral equation method, which assumes that all energy is carried in a forward direction. An extended formulation encompassing backscatter is also derived, and a method given for its treatment. This paper serves in part to explain the computational results presented in B. J. Uscinski, “High-frequency propagation in shallow water,” J. Acoust. Soc. Am. 98, 2702–2707 (1995).

Wave Propagation in Finite Periodically Ribbed Structures with Fluid Loading

The paper studies the problem of wave transmission along a fluid-loaded plane elastic membrane supported by a finite array of equally spaced ribs. One of the ribs is driven by a time-harmonic line force and the rest have infinite impedance, so that fluid loading provides the only mechanism for the transmission of energy. Existing solutions for the infinite analogue exhibit a stop/pass band frequency structure, in which the energy is, alternately, exponentially localized around the driving force and constant along the array. However, at pass band frequencies this is inconsistent with numerical studies of finite arrays, which reveal marked amplitude fluctuations. In this paper an exact solution is given for a general finite configuration. This is used to explain and further explore the response. In particular it is shown that as the array length increases the pass band response becomes increasingly sensitive to frequency, and the solution cannot approach an asymptotic limit. The results give the forces along the array as an interference pattern, which may be thought of as propagating inwards from each end. This solution is obtained by forming a 2 x 2 matrix which relates the forces at any pair of adjacent ribs to those at the next pair. From the action of this matrix the response can be found everywhere, and the detailed properties of the solution are determined by those of the matrix. Special treatment is needed to deal with the band edges, which conform neither to stop nor pass band behaviour.

Forward and inverse scattering from rough surfaces at low grazing incidence

In acoustic wave scattering from rough surfaces at low grazing angles the usual approximations are known to break down. In the forward‐scattering regime this case is often treated by the parabolic integral equation method. The scattering of plane waves within this formulation is considered, and effective reflection coefficients are derived for slightly rough surfaces. Of the higher‐order statistics, the dependence of the diffuse and specular components upon incident angle is also examined. The problem of surface reconstruction can be expressed in terms of coupled integral equations; it is explained how in the parabolic case these are easily treated. A major problem, however, is backscatter; a nonparabolic approach to this problem for low‐grazing angles is described. [Work supported by Smith Institute Research Fellowship.]