Wave Propagation In Random Media Research Articles

Scaling turbulent dissipation and finestructure: The wave-propagation method.

In the 1980s the team of Flatté et al. developed a theory relating the turbulent kinetic energy dissipation rate in the open ocean to finestructure feature observations (1–100 m scale), building on work of Henyey and Pomphrey. Formulas derived from this work are now prevalently used for indirectly estimating cross‐isopycnal ocean mixing, a process of critical interest, using platforms and sensors unable to directly measure viscous‐scale dissipation and associated turbulent transport processes. The method originally involved the internal‐wave energy level. Revisions of the method introduced by follow‐on investigators involve internal‐wave shear and strain. The physics behind the method is that short‐wavelength shear‐rich internal waves propagate in a background of larger‐scale internal waves, with action conserved, and can have their energy concentrated spatially. The concentrated waves are inferred to break, and in doing so provide the energy to support diapycnal mixing. The approach is one of “wave propagation in random media,” a specialty of Flatté.

Experimental validation of percolation‐based models for propagation prediction

AbstractIn this letter, an experimental validation of percolation‐based approaches for the prediction of wave propagation in random media is presented. Measurements are collected in a real controlled environment, where the obstacles are stochastically placed in a two‐dimensional grid according to a known nonuniform density distribution. The obtained results show that, in spite of their simplicity, percolation‐based approaches can be applied in real propagation problems. © 2008 Wiley Periodicals, Inc. Microwave Opt Technol Lett 50: 3190–3192, 2008; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.23937

Statistics of quality factors in three-dimensional disordered magneto-optical systems and its applications to random lasers

We have investigated the distribution of quality factors $P(Q)$ of three-dimensional disordered systems composed of pointlike magneto-optical scatterers under the influence of an external magnetic field, both in the diffusive and localized regimes of light transport. $P(Q)$ is shown to exhibit the same behavior of the distribution of delay times obtained in previous studies within different models to describe wave propagation in random media, such as the Anderson and the kicked rotator models. From the analysis of $P(Q)$ we conclude that exponentially localized (and prelocalized) electromagnetic quasimodes depend on time-reversal symmetry. We suggest that this dependence could be explored in the development of a magneto-optical random laser, in which the threshold could be tuned by the application of a magnetic field. We also argue that such laser could be used to experimentally test if prelocalized modes are at the controversial origin of random lasing with coherent feedback in diffusive systems.

Analysis of acoustic backscattering from the ocean bottom using radiative transfer theory

Radiative Transfer (RT) theory is a heuristic formulation for analysis of wave propagation in random media based on the principle of conservation of energy rather than on the solution of the wave equation. The theory has been applied to electromagnetics and more recently to ultrasound and seismics, and it has the advantage of being computationally tractable for complex random environments. In this paper the RT formulation is proposed for the problem of layered ocean bottom sediment with random scatterers. The volume backscattering level from tenuous media is obtained by solving the RT equation at each individual layer and by applying the corresponding reflection and transmission coefficients at the rough boundaries. Simulations of the backscattered acoustic intensity from a finite layer with elastic spherical scatterers are presented. The results obtained from the RT equation are compared to those based on the wave equation. Single frequency steady state solutions are considered for different cases of sediment attenuation and layer thickness. The flexibility of the RT method is demonstrated by showing the individual effect of propagating longitudinal and shear waves in the elastic sediment. It is proved that the contribution of shear waves in consolidated sediments is considerable.

Random Backscattering in the Parabolic Scaling

In this paper we revisit the parabolic approximation for wave propagation in random media by taking into account backscattering. We obtain a system of transport equations for the moments of the components of reflection and transmission operators. In the regime in which forward scattering is strong and backward scattering is weak, we obtain closed form expressions for physically relevant quantities related to the reflected wave, such as the beam width, the spectral width and the mean spatial power profile. In particular, we analyze the enhanced backscattering phenomenon, that is, we show that the mean power reflected from an incident quasi-plane wave has a maximum in the backscattered direction. This enhancement can be observed in a small cone around the backscattered direction and we compute the enhancement factor as well as the shape of the enhanced backscattering cone.

Self-Averaging from Lateral Diversity in the Itô–Schrödinger Equation

We consider the random Schrödinger equation as it arises in the paraxial regime for wave propagation in random media. In the white noise limit it becomes the Itô–Schrödinger stochastic partial differential equation which we analyze here in the high frequency regime. We also consider the large lateral diversity limit where the typical width of the propagating beam is large compared to the correlation length of the random medium. We use the Wigner transform of the wave field and show that it becomes deterministic in the large diversity limit when integrated against test functions. This is the self-averaging property of the Wigner transform. It follows easily when the support of the test functions is of the order of the beam width. We also show with a more detailed analysis that the limit is deterministic when the support of the test functions tends to zero but is large compared to the correlation length.

The wave propagation in a beam on a random elastic foundation

This paper presents a study of wave propagation in an infinite beam on a random Winkler foundation. The spatial variation of the foundation spring constant is modelled as a random field and the influence of the correlation length is studied. As it is impossible to determine the general stochastic Green’s function, the configurational average of the Green’s function and its correlation function are considered. These functions are found through the transformation of the stochastic equation of motion into the Dyson equation for the mean or coherent field and the Bethe–Salpeter equation for the field correlation, as used in the study of wave propagation in random media. The approximate solutions of the Dyson and the Bethe–Salpeter equations are validated by means of a Monte Carlo simulation and compared with the results of a classical Neumann expansion method. Although both methods only involve the second order statistics of the random field, the approximation of the Dyson and the Bethe–Salpeter equations gives better results than the Neumann expansion, at the expense of a larger computational effort. Furthermore, the results show that a small spatial variation of the spring constant has an influence on the response if the correlation length and the wavelength have a similar order of magnitude, while the waves in the beam cannot resolve the spatial variation in the case where the correlation length is much smaller than the wavelength.

Correlation functions of temperature and velocity fluctuations in a turbulent atmosphere

von Kármán spectra of temperature and velocity fluctuations are widely used in theories of turbulence and wave propagation in random media, including sound propagation through a turbulent atmosphere. They are probably the simplest generalization of the Kolmogorov spectrum of homogeneous and isotropic turbulence which accounts for the outer scale of turbulence. However, the correlation functions of temperature and velocity fluctuations corresponding to the von Kármán spectra are proportional to the modified Bessel functions and are rather involved. In this paper, we propose to use much simpler correlation functions of temperature and longitudinal velocity fluctuations whose dependence on the distance R between observation points is given by exp(−R2/3/L2/3), where L is the outer length scale. It is shown that the spectra of the proposed correlation functions coincide with the Kolmogorov spectrum in the inertial subrange and are bounded in the energy subrange. For some problems, the proposed correlation functions can simplify analysis of sound propagation through a turbulent atmosphere. Examples of the use of these correlation functions in acoustic tomography of the atmosphere and calculations of the coherence function of a sound field are presented. [Work supported by ARO, Grant DAAD19-03-1-0104.]

Validity of the Markov approximation in ocean acoustics

Moment equations and path integrals for wave propagation in random media have been applied to many ocean acoustics problems. Both these techniques make use of the Markov approximation. The expansion parameter, which must be less than one for the Markov approximation to be valid, is the subject of this paper. There is a standard parameter (the Kubo number) which various authors have shown to be sufficient. Fourth moment equations have been successfully used to predict the experimentally measured frequency spectrum of intensity in the mid-ocean acoustic transmission experiment (MATE). Yet, in spite of this success, the Kubo number is greater than 1 for the measured index of refraction variability for MATE, arriving at a contradiction. Here, that contradiction is resolved by showing that the Kubo parameter is far too pessimistic for the ocean case. Using the methodology of van Kampen, another parameter is found which appears to be both necessary and sufficient, and is much smaller than the Kubo number when phase fluctuations are dominated by large scales in the medium. This parameter is shown to be small for the experimental regime of MATE, justifying the applications of the moment equations to that experiment.

Measurement of phase fluctuations for transmitted waves in random media

Transmission fluctuations of log‐amplitude and phase can be used to construct coherence functions to infer the statistical characteristics of crustal and mantle small‐scale heterogeneities. Recent numerical simulations of wave propagation in random media have revealed an apparent discrepancy between simulations and theory for phase coherence functions. In this paper, we show that the reported discrepancy is a result of the inappropriate phase estimates adopted in some literature. Using unwrapped phase fluctuations, instead of estimates of the fluctuations obtained by picking first arrivals by the threshold method, we find that the theory and the results derived from finite difference simulations are consistent. The method of first arrival picking does not yield adequate estimates of phase fluctuation due to the scattering dispersion of waves in the random medium. Phase unwrapping is viable and allows correct inference of medium properties.

Telecommunication in a disordered environment with iterative time reversal

Recent researches in the area of wave propagation in random media applied to telecommunications showed that, contrary to intuition, reverberation or scattering of waves in a disordered medium can actually help to increase the information transfer rate. The key element therein is the ability of a communication system to exploit independent channels of propagation. We present a method to transmit digital information through a highly scattering medium. It is based on iterations of a time-reversal process, and permits to focus short pulses, both spatially and temporally, from a base antenna to different users. This iterative technique is shown to be more efficient (lower inter-symbol interference and lower error rate) than classical time-reversal communication, while being computationally light and stable. Experiments are presented: digital information is conveyed from 15 transmitters to 15 receivers by ultrasonic waves propagating through a highly scattering slab. From a theoretical point of view, the iterative technique achieves the inverse filter of propagation in the subspace of non null singular values of the time reversal operator. We also investigate the influence of external additive noise, and show that the number of iterations can be optimized to give the lowest error rate.

Scintillation of short duration records

We have investigated propagation of sound in the ocean using the theory of wave propagation in random media. The theory classifies specific source/receiver geometries as either unsaturated, partially saturated, or saturated. In the unsaturated regime the variance of the intensity fluctuations will generally be quite small such that the scintillation index is less than unity. The partially saturated regime can give rise to scintillation indices larger than unity. In the fully saturated regime the scintillation index asymptotically approaches unity. We have conducted an experiment in which the two-dimensional sound-speed field was measured directly using a towed CTD chain, and relatively short acoustic records were captured using a number of source/receiver pairs. The length scales and index of refraction variation place the experiment in the saturated regime. However, measured scintillation indices are much less than 1. We provide an explanation for this difference and utilize local current measurements to bring the theory into agreement with the data. Our results are applicable to operational sonar systems, where short duration records are ordinarily utilized to transmit information or detect targets. [Work sponsored by ONR Code 321US.]

Dynamic correlation in wave propagation in random media.

Field spectra are analyzed to yield the time-resolved statistics of pulsed transmission through quasi-one-dimensional dielectric media with static disorder. The normalized intensity correlation function with displacement and polarization rotation for an incident pulse of linewidth sigma at delay time t is a function only of the field correlation function, which is identical to that found for steady-state excitation, and of kappa(sigma)(t), the residual degree of intensity correlation at points at which the field correlation function vanishes. The dynamic probability distribution of normalized intensity depends only upon kappa(sigma)(t). Steady-state statistics are recovered in the limit sigma-->0, in which kappa(sigma=0) is the steady-state degree of correlation.

A survey of ionospheric effects on space-based radar

In this survey, we fully review almost all potential ionospheric effects on the performance of space-based radar systems (SBRs), which operate in the ambient ionosphere environment; in particular, we review the use of space-based synthetic aperture radar systems (SARs) for imaging. There are two families of effects involved. One is the effects of the background ionosphere (non-turbulent ionosphere), such as dispersion, group delay, refraction, Faraday rotation, and phase shift. The other is the effects due to ionospheric irregularities, such as refractive index fluctuation, phase perturbation, angle-of-arrival fluctuation, pulse broadening, clutter, and amplitude scintillation. These effects adversely affect SAR imaging in several respects, such as by causing image shift in the range, and degradations of the range resolution, azimuthal resolution, and/or the resolution in height (elevation). We also review ionospheric irregularity characteristics and descriptions, propagation channel statistics, received signal statistics, and detection performance of SBRs in ionospheric scintillation situations. First, a brief outline of SBR systems and principles, and theories of wave propagation in random media, especially some effects caused by two-way propagation, is given. Finally, several of the most probable directions of future research are pointed out.

Laser characterization of ultrasonic wave propagation in random media.

Lasers can be used to excite and detect ultrasonic waves in a wide variety of materials. This allows the measurement of absolute particle motion without the mechanical disturbances of contacting transducers. In an ultrasound transmission experiment, the wave field is usually accessible only on the boundaries of a sample. Using optical methods, one can measure the surface wave field, in effect, within the scattering region. Here, we describe noncontacting (laser source and detector) measurements of ultrasonic wave propagation in randomly heterogeneous rock samples. By scanning the surface of the sample, we can directly visualize the complex dynamics of diffraction, multiple scattering, mode conversion, and whispering gallery modes. We will show measurements on rock samples that have similar elastic moduli and intrinsic attenuation, but different grain sizes, and hence, different scattering strengths. The intensity data are well fit by a radiative transfer model, and we use this fact to infer the scattering mean free path.

Some validity issues in the theory and modeling of WPRM

In many publications in the recent to older litererature on the theory and modeling of wave propagation in random media, WPRM, in ocean and atmospheric environments there are serious validity issues that arise. The issues discussed will include the following. (1) The validity of using the Markov approximation in theoretical treatments must be established for specific environments before theories based on that approximation can be used. Parameters that are required to be small must be shown to be so before the applicabilty to specific environments can be assured. (2) Most publications that purport to model the pdf’s of intensity do not include the standard ‘‘goodness of fit’’ parameters. The Kolmogorov–Smirnov GoF test emphasizes the center of the distribution and often the pdf’s of WPRM are characterized by very high tailed distributions where the fitting of that component can be important to understanding the scattering physics. GoF tests that include the full range of received intensity are needed. (3) Signal processors invariably require that the quadrature components of their receptions have Gaussian pdf’s. It has been shown by simulations of WPRM that this is often not even close to the case. What significance this effect has on signal processing algorithms is a difficult and unresolved issue.

Coherent backscattering and localization in a self-attracting random walk model

Intensity propagation of waves in dilute 2D and 3D disordered systems is well described by a random walk path-model. In strongly scattering media, however, this model is not quite correct because of interference eects like coherent backscattering. In this letter, coherent backscattering is taken into account by a modied, self-attracting random walk. Straightforward simulations of this model essentially reproduce the results of current theories on \non-classical transport behavior, i.e. Anderson localization in 1D and 2D for any amount of disorder and a phase transition from weak to strong localization in 3D. However, in the strongly scattering regime corrections are necessary to account for the nite number of light modes due to their non-vanishing lateral extention. Within our model this correction leads to the observation that strong localization does not take place. PACS. 42.25.Dd Wave propagation in random media { 71.23.An Theories and models; localized states

Validity conditions for the Markov approximation in theories for internal wave effects on propagation

There are two conditions that, if satisfied, guarantee the validity of the Markov approximation as usually imposed [see, for example, B. J. Uscinski The Elements of Wave Propagation in Random Media (McGraw-Hill, New York, 1977)]. One condition is that the scattering angle be small compared to the anisotropy of the medium, and the second is that there be less than one radian of phase shift in a correlation length. For the MATE experiment on acoustic propagation through oceanic internal waves, the second condition is violated, yet theories based on the Markov approximation work well. In this talk, it is shown that this second condition is much too pessimistic, by explicit construction of the next-order correction in an example that neglects sound channel effects over a correlation length. There is a strong cancellation among terms for the correction, each of which is of the order of the second expansion parameter squared. This cancellation occurs because the horizontal wave number is insensitive to the vertical wave number at small angles and that most of the phase shift comes from large-scale internal waves. With this cancellation, Markov approximation theories should correctly predict the MATE results—as they do. [Work supported by ONR.]

Long distance time reversal and imaging in random media: Numerical simulations

We have conducted a careful time reversal and imaging study for long distance wave propagation in random media in the underwater acoustics regime. In this talk we briefly describe the computational technology, which is based on a phase-screen propagation code for the paraxial wave equation. The advantages and limitations of the code are discussed. We show some time reversal results which demonstrate the super-resolution phenomenon and its statistical stability in the time domain. We also show results of imaging in random media using an array of active transducers.

The PDF of intensity for Ocean WPRM—15 years later

Spatial intensity correlations for wave propagation in random media (WPRM) can be predicted in terms of the index of refraction space-time autocorrelation function, with scattering parameters Gamma (strength) and X (scaled range) in the iso-velocity case. In ocean WPRM the temporal intensity correlations are also well predicted from the modeled index of refraction spectrum for the lower path in sub-CZ propagation with a depth-dependent sound speed profile (Eastern Pacific). For both of these cases the intensity pdf’s have been shown [Ewart and Percival, 80, 1745–1753 (1986)] to be very well fit by generalized Gamma distributions. In that work quantile–quantile goodness of fit tests were used to test the generalized Gamma hypothesis. The upper path intensity correlations, in sub-CZ propagation, however, have not been predicted. The pdf’s of upper path data taken during MATE will be discussed. Several authors have proposed functional forms of intensity pdf’s, e.g., one distribution modulated by another. Quantile-quantile tests will be applied to see if such functions model ocean WPRM intensity pdf’s. [Work supported by ONR.]