Reference-wave solutions for the high-frequency fields in inhomogeneous-background random media

Ray theory plays an important role in determining the propagation properties of high-frequency fields and their statistical measures in complicated random environments. For computations of the statistical measures it is therefore desirable to have a solution for the high-frequency field propagating along an isolated ray trajectory. A new reference wave is applied to obtain an analytic solution of the parabolic wave equation that describes propagation along the ray trajectory of the deterministic-background medium. The methodology is based on defining a paired-field measure as a product of an unknown field propagating in a disturbed medium and the complex-conjugate component propagating in a medium without random fluctuations. When a solution of the equation for the paired-field measure is obtained, the solution of the deterministic component can be extracted from the paired solution to determine the solution of the unknown field in an explicit form.     

Modeling of high-frequency propagation in inhomogeneous background random media

When a high-frequency electromagnetic wave propagates in a complicated scattering environment, the contribution at the observer is usually composed of a number of field species arriving along different ray trajectories. In order to describe each contribution separately the parabolic extension along an isolated ray trajectory in an inhomogeneous background medium was performed. This leads to the parabolic wave equation along a deterministic ray trajectory in a randomly perturbed medium with the possibility of presenting the solution of the high-frequency field and the higher-order coherence functions in the functional path-integral form. It is shown that uncertainty considerations play an important role in relating the path-integral solutions to the approximate asymptotic solutions. The solutions for the high-frequency propagators derived in this work preserve the random information accumulated along the propagation path and therefore can be applied to the analysis of double-passage effects where the correlation between the forward-backward propagating fields has to be accounted for. This results in double-passage algorithms, which have been applied to analyze the resolution of two point scatterers. Under strong scattering conditions, the backscattering effects cannot be neglected and the ray trajectories cannot be treated separately. The final part is devoted to the generalized parabolic extension method applied to the scalar Helmholtz's equation, and possible approximations for obtaining numerically manageable solutions in the presence of random media.     

Double passage analysis in random media using two-scale random propagators

In location and remote sensing experiments there arise a number of effects related to the double passage of the backscattered field through the same random inhomogeneities as the incident one. To account for the correlation of the forward-backward propagating events, there is a need for a measure in which the random information along the propagation path is preserved. For the generation of even statistical moments, the relevant measure defined in the recently formulated stochastic geometrical theory of diflraction is the two-point random function (TPRF)—a paired field measure which is propagated along the geometrical rays of the deterministic background medium. From this function all even statistical moments can be generated. Here we present an approximate analytical solution for the high-frequency propagator obtained by applying the multiscale expansion asymptotic procedure to the partial differential equation governing the propagation a1 the TPRF. The test of the solution is performed on canonical backscattering problems based on point source-point scatterer and paint source-plane mirror configurations, which justifies its further application for construction of the coherence measures of the rctrareflected field. Coherence properties of the plane and spherical wavefields reflected backward by a plane mirror were investigated. Further, we investigated the intensity enhancement effects observed in the double passage of a Gaussian beam retroreflected from a plane mirror. Asymptotic expressions lor the retroreflected intensity are obtained, and their computations show good agreement with the direct numerical evaluations.     

Double passage analysis in random media using two-scale random propagators

Abstract

In location and remote sensing experiments there arise a number of effects related to the double passage of the backscattered field through the same random inhomogeneities as the incident one. To account for the correlation of the forward–backward propagating events, there is a need for a measure in which the random information along the propagation path is preserved. For the generation of even statistical moments, the relevant measure defined in the recently formulated stochastic geometrical theory of diflraction is the two-point random function (TPRF)—a paired field measure which is propagated along the geometrical rays of the deterministic background medium. From this function all even statistical moments can be generated. Here we present an approximate analytical solution for the high-frequency propagator obtained by applying the multiscale expansion asymptotic procedure to the partial differential equation governing the propagation a1 the TPRF. The test of the solution is performed on canonical backscattering problems based on point source–point scatterer and paint source–plane mirror configurations, which justifies its further application for construction of the coherence measures of the rctrareflected field. Coherence properties of the plane and spherical wavefields reflected backward by a plane mirror were investigated. Further, we investigated the intensity enhancement effects observed in the double passage of a Gaussian beam retroreflected from a plane mirror. Asymptotic expressions lor the retroreflected intensity are obtained, and their computations show good agreement with the direct numerical evaluations.     

Scattering of an electromagnetic wave from a slightly random cylindrical surface: horizontal polarization

The scattering of an electromagnetic wave from a random cylindrical surface ir studied for a plane-wave incidence with S-(TE) polarization, by means ofthe stochastic scattering theory developed by Nakayama, Ogura. Sakati et al. The theory is based on the Wiener-Ito stochastic functional calculus combined with the group-theoretic consideration concerning the homogeneity of the random surface. The random surface is assumed to be a homogeneous Gaussian random field on the cylinder C, homogeneous with respect to the group of motiolrs on C: translations along the axis and rotations around the axis. An operator D operating on a random field on C is introduced in such a way that D keeps the homogeneous random surface invariant This gives a reprerentation of the cylbdrical group and commutes with the boundary condition and the Maxwell equation. Thus, for an injection of the mth cylindrical TE or TM wave, which is a vector eigenfunction of the D operator, the scattered random wave field is an eigenfunctiou with the same eigenvalue: it satisfies the Maxwell equation and is a stoch-tic Iunctional of the Gaussian random surface, BO that it can be expressed in a vector form of the Wiener-Ito expansion in t e m of TE and TM waves and orthogonal functional. of the Gaussian random measures associated with the random cylindrical surface. In the analysis the random surface is modelled by an approximate boundaiy condition representing a perfectly conducting cylindrical surface with a slight roughness. The boundary condition on the random cylinder is transformed into a hierarchy of equations for the Wiener kernels which can be solved approximately. The random wave field for a plane-wave injection is obtained by summing these fields over m. From the stochastic representation of the electromagnetic field so obtained, various statistical characteristics can be calculated the coherent scattering amplitude. total coherent power flow, incoherent power flow, differential sections for coherent rcatlerhig and incoherent scattering, etc. The power conservation law is cast into a stochastic electromagnetic version of the optical theorem stating that the total scatteiing cross section is given by the imaginary part of the forward coherent scattering amplitude. Numerical calculations are made for a planewave injection with S-(TE) polarization. The case of p-(TM) polarization can be treated in a similar manner.     

Backscattering response from periodic spatial patterns in random media

Abstract

In wave-based remote sensing or radio-location of distant objects in a random medium, a high-frequency electromagnetic wave is scattered by object discontinuities, and portions of the scattered radiation can traverse the same random inhomogeneities as the initial incident field. The statistical dependence of the forward–backward travelling events results in an anomaly in the backscattered intensity pattern that carries information about the scattering object. The quality of this information depends on the ability to resolve the fine-structure elements. In this work we investigate the resolving properties of periodic spatial objects by using the random propagators of the stochastic geometrical theory of diffraction.     

Ray theory for a locally layered random medium

We consider acoustic pulse propagation in inhomogeneous media over relatively long propagation distances. Our main objective is to characterize the spreading of the travelling pulse due to microscale variations in the medium parameters. The pulse is generated by a point source and the medium is modelled by a smooth three-dimensional background that is modulated by stratified random fluctuations. We refer to such media as locally layered .

We show that, when the pulse is observed relative to its random arrival time, it stabilizes to a shape determined by the slowly varying background convolved with a Gaussian. The width of the Gaussian and the random travel time are determined by the medium parameters along the ray connecting the source and the point of observation. The ray is determined by high-frequency asymptotics (geometrical optics). If we observe the pulse in a deterministic frame moving with the effective slowness , it does not stabilize and its mean is broader because of the random component of the travel time. The analysis of this phenomenon involves the asymptotic solution of partial differential equations with randomly varying coefficients and is based on a new representation of the field in terms of generalized plane waves that travel in opposite directions relative to the layering.     

Optical properties of an echelette grating in the short-wave approximation

The problem of plane-wave scattering by an echelette grating with a right angle is considered in the case of high-frequency approximation (the wavelength is assumed to be small compared with the period of grating). We present the results of short-wave asymptotic analysis for a ray optical solution to the problem that was derived on the basis of the method of summation of multiple diffracted fields that is well known in the geometric theory of diffraction. A new effect of perfect blazing for an echelette grating in the high-frequency approximation is also discussed.     

Coherent effects of multiple scattering for scalar and electromagnetic fields: Monte-Carlo simulation and Milne-like solutions

Based on an expansion of the Bethe-Salpeter equation in scattering orders a semi-analytic approach for simulation of coherent phenomena of multiple scattering in random media has been developed. We found that for scalar field the manifestation of these phenomena, observed as temporal field correlation function and coherent backscattering, are universal and well agreed with the results predicted by diffusion approximation. For the electromagnetic field the temporal correlation function and coherent backscattering are noticeably differ from those found for the scalar field, depending strongly on the scattering anisotropy. The obtained numerical results, for the first time to our knowledge, are compared directly with the known generalizations of the Milne solution.     

Two families of non-local scattering models and the weighted curvature approximation

There are several nonlocal scattering models available in the literature. Most of them are given with little or no mention of their expected accuracy. Moreover, high- and low-frequency limits are rarely tested. The most important limits are the low-frequency or the small perturbation method (SPM) and the high-frequency Kirchhoff approximation (KA) or the geometric optics (GO). We are interested in providing some insight into two families of non-local scattering models. The first family of models is based on the Meecham-Lysanov ansatz (MLA). This ansatz includes the non-local small slope approximation (NLSSA) by Voronovich and the operator expansion method by Milder (OEM). A quick review of this first family of models is given along with a novel derivation of a series of kernels which extend the existing models to include some more fundamental properties and limits. The second family is derived from formal iterations of geometric optics which we call the ray tracing ansatz (RTA). For this family we consider two possible kernels. The first is obtained from iteration of the high-frequency Kirchhoff approximation, while the second is an iteration of the weighted curvature approximation (WCA). In the latter case we find that most of the required limits and fundamental conditions are fulfilled, including tilt invariance and reciprocity. A study of scattering from Dirichlet sinusoidal gratings is then provided to further illustrate the performance of the models considered.     

Scattering of an electromagnetic wave from a slightly random cylindrical surface: horizontal polarization

Abstract

The scattering of an electromagnetic wave from a random cylindrical surface ir studied for a plane-wave incidence with S-(TE) polarization, by means ofthe stochastic scattering theory developed by Nakayama, Ogura. Sakati et al. The theory is based on the Wiener-Ito stochastic functional calculus combined with the group-theoretic consideration concerning the homogeneity of the random surface. The random surface is assumed to be a homogeneous Gaussian random field on the cylinder C, homogeneous with respect to the group of motiolrs on C: translations along the axis and rotations around the axis. An operator D operating on a random field on C is introduced in such a way that D keeps the homogeneous random surface invariant This gives a reprerentation of the cylbdrical group and commutes with the boundary condition and the Maxwell equation. Thus, for an injection of the mth cylindrical TE or TM wave, which is a vector eigenfunction of the D operator, the scattered random wave field is an eigenfunctiou with the same eigenvalue: it satisfies the Maxwell equation and is a stoch-tic Iunctional of the Gaussian random surface, BO that it can be expressed in a vector form of the Wiener-Ito expansion in t e m of TE and TM waves and orthogonal functional. of the Gaussian random measures associated with the random cylindrical surface. In the analysis the random surface is modelled by an approximate boundaiy condition representing a perfectly conducting cylindrical surface with a slight roughness. The boundary condition on the random cylinder is transformed into a hierarchy of equations for the Wiener kernels which can be solved approximately. The random wave field for a plane-wave injection is obtained by summing these fields over m. From the stochastic representation of the electromagnetic field so obtained, various statistical characteristics can be calculated the coherent scattering amplitude. total coherent power flow, incoherent power flow, differential sections for coherent rcatlerhig and incoherent scattering, etc. The power conservation law is cast into a stochastic electromagnetic version of the optical theorem stating that the total scatteiing cross section is given by the imaginary part of the forward coherent scattering amplitude. Numerical calculations are made for a planewave injection with S-(TE) polarization. The case of p-(TM) polarization can be treated in a similar manner.     

Random focusing of sound into spatially coherent regions

Abstract

Oceanographic variability creates a weak random propagation medium for acoustic waves. The impact on acoustic transmission is becoming increasingly appreciated as the deterministic modelling of sound propagation in the ocean has become tractable and better understood. Beyond the near field (where phase fluctuations are weak) and the far field (where the scintillation index becomes saturated) multiple-scattering theory predicts that random focusing will greatly enhance the acoustic energy density over small volumetric regions, which we call ‘ribbons’. In 1986 an experiment was carried out in the eastern Mediterranean to test this prediction using acoustic propagation along distinct, resolvable ray paths. This experiment is one of the few to map the spatial structure of acoustic intensity with such a large vertical aperture, and as far as the authors are aware it remains the only experiment to attempt to detect the two-dimensional structure of the predicted focused ribbons for individual energy paths. Renewed impetus to publish the results has been provided by the recent focus on moderate- to high-frequency acoustics in near-shore and shallow-water environments. The experiment is described and high-intensity regions consistent with the theoretical predictions are reported. A 3.5 kHz pulsed signal was transmitted over ranges of 11–23 km and sampled over a vertical aperture of 250–350 m and horizontal apertures of 4–4.5 km. The acoustic signals travelling along individual ray paths developed randomly focused regions of 6–18 dB over regions with a vertical dimension of about 20 m and whose horizontal length could possibly be up to 1 km. The understanding of these features allows system limitations to be estimated quantitatively and opens up the way to their constructive tactical use. The results are applicable to many systems from towed array sonars to high-frequency bathymetric sidescan and minehunting.     

Sound propagation and scattering in media with random inhomogeneities of sound speed, density and medium velocity

This review presents both classical and new results of the theory of sound propagation in media with random inhomogeneities of sound speed, density and medium velocity (mainly in the atmosphere and ocean). An equation for a sound wave in a moving inhomogeneous medium is presented, which has a wider range of applicability than those used before. Starting from this equation, the statistical characteristics of the sound field in a moving random medium are calculated using Born-approximation, ray, Rytov and parabolic-equation methods, and the theory of multiple scattering. The results obtained show, in particular, that certain equations previously widely used in the theory of sound propagation in moving random media must now be revised. The theory presented can be used not only to calculate the statistical characteristics of sound waves in the turbulent atmosphere or ocean but also to solve inverse problems and develop new remote-sensing methods. A number of practical problems of sound propagation in moving random media are listed and the further development of this field of acoustics is considered.     

Two families of non-local scattering models and the weighted curvature approximation

Abstract

There are several nonlocal scattering models available in the literature. Most of them are given with little or no mention of their expected accuracy. Moreover, high- and low-frequency limits are rarely tested. The most important limits are the low-frequency or the small perturbation method (SPM) and the high-frequency Kirchhoff approximation (KA) or the geometric optics (GO). We are interested in providing some insight into two families of non-local scattering models. The first family of models is based on the Meecham–Lysanov ansatz (MLA). This ansatz includes the non-local small slope approximation (NLSSA) by Voronovich and the operator expansion method by Milder (OEM). A quick review of this first family of models is given along with a novel derivation of a series of kernels which extend the existing models to include some more fundamental properties and limits. The second family is derived from formal iterations of geometric optics which we call the ray tracing ansatz (RTA). For this family we consider two possible kernels. The first is obtained from iteration of the high-frequency Kirchhoff approximation, while the second is an iteration of the weighted curvature approximation (WCA). In the latter case we find that most of the required limits and fundamental conditions are fulfilled, including tilt invariance and reciprocity. A study of scattering from Dirichlet sinusoidal gratings is then provided to further illustrate the performance of the models considered.     

Statistics of a coherent lidar signal in a turbulent atmosphere

Absolute and conditional statistical properties of a pulse coherent Doppler lidar signal in a turbulent atmosphere are studied. Upon coherent receiving of optical fields scattered by a large number of particles, the lidar signal is shown to be a nonstationary non-Gaussian random process with Gaussian conditional statistical characteristics. The appearance of non-Gaussian properties of the signal is caused by correlation of turbulent fluctuations of the wind velocity field within the scattering volume. For the considered signal model, which corresponds to the single scattering approximation and is a sum of a large number of random variables, the central limit theorem is found to be untrue due to the statistical dependence of particles’ positions in a turbulent atmosphere. The results of numerical calculations show that, for a homogeneous and isotropic turbulence, the behavior of the signal statistics significantly depends on the size of the scattering volume and on the state of atmospheric turbulence. A Gaussian statistics is observed at small heights; with an increase in height, the non-Gaussian component becomes considerable in fluctuations of the lidar signal.     

Sound propagation and scattering in media with random inhomogeneities of sound speed,density and medium velocity

Abstract

This review presents both classical and new results of the theory of sound propagation in media with random inhomogeneities of sound speed, density and medium velocity (mainly in the atmosphere and ocean). An equation for a sound wave in a moving inhomogeneous medium is presented, which has a wider range of applicability than those used before. Starting from this equation, the statistical characteristics of the sound field in a moving random medium are calculated using Born-approximation, ray, Rytov and parabolic-equation methods, and the theory of multiple scattering. The results obtained show, in particular, that certain equations previously widely used in the theory of sound propagation in moving random media must now be revised. The theory presented can be used not only to calculate the statistical characteristics of sound waves in the turbulent atmosphere or ocean but also to solve inverse problems and develop new remote-sensing methods. A number of practical problems of sound propagation in moving random media are listed and the further development of this field of acoustics is considered.     

Scattering of a TM plane wave from periodic random surfaces

This paper deals with the scattering of a TM plane wave from conductive periodic random surfaces. By means of the stochastic functional approach, the scattered field is expressed in terms of a harmonic series representation, in which the coefficients are homogeneous random functions and are given by Wiener-Hermite expansions. An approximate solution for the Wiener kernels is obtained up to the second order. Several anomalies appear in the angular distribution of the incoherent scattering because of combinations of scattering due to surface randomness and diffraction due to surface periodicity. These are incoherent Wood's anomalies associated with guided surface waves propagating along the surface, enhanced backscattering and diffracted backscattering enhancement. The physical reasons for these anomalies and numerical results are discussed.     

A two-dimensional tomography model for the oceanic inhomogeneity reconstruction with wave and ray representations of acoustic field

A model reconstruction of two-dimensional combined oceanic inhomogeneities (of refractive and kinetic types) in tomographic experiments with ray and wave representations of acoustic field is considered. The possibility of a complete reconstruction of two-dimensional flows from the scattering data alone is illustrated. For the realization of the tomographic scheme, a nonorthogonal redundant basis consisting of a number of intersecting stripes is used. The results of reconstruction are presented for model inhomogeneities of kinetic and combined (refractive-kinetic) types. The iterative reconstruction of the flow velocity vector distribution is considered. The tomographic problem in the ray representation is solved by taking into account both the time delays in the signal propagation along the rays and the ray trajectory distortions due to the inhomogeneity of the medium.