Stochastic Functional Analysis For Properties Of Waves Propagation In Random Media

As a research topic of both practical value and theoretical challenging, waves in random and complex media has always been the most dynamic field with fast development. It has been paid common attention in a lot of disciplines such as Radio Propagation, Communication, Remote Sensing, Integrated Optics, Integrated Circuits, Targets Identification and Classification, Environmental System Inspect, Medical Diagnose, and so on. As a doctoral dissertation in this field, this thesis manages to establish a whole new system of quantitative theory and approach for waves in random and complex media and interactions therein based on the Electromagnetic theory as well as the stochastic functional theory.There are nine chapters in this dissertation and they could be divided into three parts logically. The first part is the fundamental theory from chapter 1 to chapter 3, including the Electromagnetic theory and the stochastic functional theory. As the main part of this thesis, the second part covering chapters 4 to 8 gives a through description of the original research work during my past five years. The third part, chapter 9, is conclusion. It is worth mention that the second part could be further divided into two distinct parts, one is the primary research work including the random continua and the rough surfaces, and the other is the secondary research work about random discrete scatterers. All nine chapters form a whole complete theoretical system for the issue of waves in random media.In the primary research part, we first discuss the propagation and localization of plane waves in two-dimensional homogeneous and isotropic Gaussian random media. A random permittivity fluctuation is introduced into the scalar wave equation, which yield the random wave equation. The random medium is then assumed to have a narrowband Gaussian spectrum centered at twice the wave number, and a general expression for plane waves in derived from the translational operator. Therefore, the random wave equation could now be solved with the stochastic functional approach. The analytical approximate solution for random standing plane waves in the two-dimensional random medium as well as the quantitative description of the amplitude and phase are then given. Both the theoretical and numerical results show the localization of plane waves in random media. Numerical simulations of standing plane waves in two-dimensional media is demonstrated to validate the analytical results, meanwhile, the exponentially decaying amplitude justifies the localization phenomenon.Based on the above research about the plane waves in two-dimensional random media, we solve the random wave equation with random permittivity fluctuation in the cylindrical coordinate system. To be more general, the random medium is assumed to have a narrowband Gaussian spectrum centered at any position. The random permittivity is first expanded into the Wiener integral expression in the cylindrical coordinate system using the stochastic functional approach, then the integral area is divided into narrow bands with fixed length equals the wave number. Meanwhile, the wave field is denoted by linear superposition of outgoing and incoming waves. The random wave equation then could be solved.The analytical expression for the wave field is given to show the modulation effect the random permittivity imposed on the amplitude and the phase, as well as the spatial distribution of energy, which demonstrates the localization phenomenon. The localization length is also given.With the help of discussions of plane waves and cylindrical waves in two-dimensional random media above, we give the wave transfer equation of plane waves and cylindrical waves in two-dimensional random media, which is then compared to that of the deterministic case. As far as for the random wave transfer equation, the random permittivity has a modulation effect on the waves therein, which results in an exponentially decaying amplitude and a randomly fluctuating wave number. Also the numerical simulations are given to show the modulation effects and the localization phenomenon, as well as the cylindrical waves with different angular number after the wave number being modulated. Besides, the numerical simulations illustrate results for Gaussian spectrum with various centers, and it has been found that the most dynamic localization phenomenon and wave number fluctuation occurs with the center of the Gaussian spectrum locates in the interval of [2k,3k]. Comprehensively, the random permittivity in two-dimensional media has the same modulation effect on waves therein, despite different shapes of wavefront. However, the power density for cylindrical waves is proportional to 1/r besides being exponentially modulated.Next, scattering of objects above two-dimensional rough surface is discussed. We first introduce the stochastic functional approach for rough surface scattering with an example of plane wave incident on a two-dimensional homogeneous and isotropic random rough surface. The wave field is derived from the stochastic Floquet theorem and then expanded into the Wiener-Hermite expansion to show the incident waves, spectacular reflected wave, and the scattered wave. The statistical properties of wave field, such as the coherent amplitude, the equivalent impedance, the angular distribution of incoherent scattered power, are then given. The optical theorem is then derived. For concrete calculation, the Wiener-Hermite expansion is substitute into the equivalent boundary condition to obtain the analytical solution for both the Dirichlet and Neumann boundary conditions. Numerical simulations validate the analytical results, and demonstrate the statistical properties. The optical theorem could be used to justify the results and to find the application area of stochastic functional approach for rough surface scattering. Then the Green function for the two-dimensional rough surface is given, as well as its asymptotic form in the far field approximation.With the rough surface Green function obtained, we discuss the scattering of object above the rough surface. Based on the Gauss theorem, a novel four-path model is proposed to include the multiple scattering of object and rough surface. Then we take a sphere scattering for example to show the separate variable iteration method, which could improve the iteration speed efficiently and is further used to calculate the difference scattering of object-surface interactions. Numerical simulations are also given to show the influences the parameters such as incident angle, height of object to the surface, and the size of the object have on the difference scattering. As a totally analytical method, the stochastic functional approach circumvents the multiple scattering problem, and describes the object-surface interactions from another point of view. A good precision as well as a high efficiency could be obtained.Finally, we discuss the vector radiative transfer in random discrete scatterers with an example of stochastic precipitation. Being different from above discussions, this part is carried on with vector radiative transfer theory instead of analytical theory. A stochastic precipitation model composed with two different rain rates is established, then the stochastic vector radiative transfer equation is solved with the help of classical probabilistic theory. The bi-static scattering coefficient and the backscattering coefficient are both obtained.We studied the problem of waves in random and complex media by the stochastic functional approach, which is superior to traditional methods in its successfully circumventing the divergent problem and multiple scattering problem, obtaining a balance of high efficiency and good precision. As a summary of my five years' research work, this dissertation is expected to provide some elicitation and instructions for those who are devoting their lives to the subject of waves in random and complex media. It will be much better if it could promote the relevant independent research work as well as application in practical fields for our country. However, many aspects and issues are still remained for further study, not mentioning the rapidly developing sciences and techniques.