随机介质中波散射与传播的随机泛函分析
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摘要
作为极具实用性和理论挑战性的课题,随机介质中波的辐射、传播和散射问题一直以来都是电磁理论研究中最为活跃、发展最快的领域之一,并且已成为现代电波传播、通信、遥感、集成光学、集成电子线路、目标识别分类、环境系统监测、生物医学诊断、工程材料测试等众多不同学科共同感兴趣的课题。作为此领域内的博士论文,本文以电磁波理论、随机泛函理论以及概率统计理论为基础,致力于建立一套分析波在随机介质中散射和传播、及其相互作用的崭新的定量理论和方法。
本文对三种随机介质分别展开讨论,将随机泛函理论和概率统计理论应用到随机介质中波散射与传播问题的研究当中,形成一整套完整的研究体系,旨在发展当前的电磁理论基础研究。全文分为连续随机介质中波的传播及其局域性、随机粗糙面及其上方物体的面体复合散射、以及作为扩展研究的随机降雨层中的矢量辐射传输分析三部分。
在连续随机介质中波的传播及其局域性研究中,首先讨论了二维各向同性均匀随机介质中平面波的传播及其局域性。在标量波动方程中引入随机介电起伏,建立随机波动方程。基于介质的均匀特性,由平移算子给出平面波的一般表达式。假设随机介质具有中心位于两倍波数处的窄带高斯谱,将随机介电起伏展开成Wiener积分式,并将积分区域沿波矢量方向条带分割。经过上述处理,将平面波的一般表达式和随机介电起伏的Wiener积分条带分割式代入随机波动方程,并用随机泛函理论求解。得到了二维随机平面波的解析近似解及其幅度和相位的定量统计特性。结果表明随机介质中的平面波以驻波形式存在,其幅度指数衰减,波数随机振荡,证实了随机介质中波的局域化现象。最后数值模拟验证了所得解析解的正确性,并模拟了二维随机介质及其中的平面驻波。
在上述平面波的研究基础上,进一步分析二维随机介质中柱面波的传播特性。同上述研究类似,假设介质具有窄带高斯谱,将随机介电起伏展开成Wiener积分式,并将积分区域条带分割。不同之处在于此时的高斯谱中心不再局限于二倍波数处,而是任意位置。同时由内外行柱面波的线性和给出波场一般表达式。将波场一般表达式和随机介电起伏的Wiener积分条带分割式代入随机波动方程,并在柱坐标系下求解。得到了随机介电起伏对柱面波幅度与相位调制作用的解析表达式,并给出局域化长度。结果表明柱面波在随机介质中传播时受到随机介电起伏的调制作用,其幅度指数衰减,波数随机振荡。且此调制作用主要发生在随机介质的高斯谱密度中心位于二倍波数和三倍波数区间内时。数值模拟给出了柱面波能量的空间分布,以及相位被调制后表现出的不同角模式柱面波。
综合上述讨论,给出二维随机介质中的平面波和柱面波的随机波转换方程,并对二者的传播特性进行对比。结果表明二维随机介质对于其中的平面波和柱面波具有同样的幅度和波数调制作用。不同之处在于,由于柱面波的波阵面为圆形,其能流密度除了和平面波一样被指数调制外,还与1／r成正比,会随传播距离更快的衰减。
在讨论了连续随机介质之后,接下来考虑随机粗糙面及其上方物体之间的复合散射问题。首先介绍用于分析随机粗糙面散射问题的随机泛函方法。以平面波入射到二维各向同性均匀随机粗糙面为例,由随机Floquet定理给出粗糙面散射场的一般表达式,包括入射波、镜面反射波、粗糙面漫散射波三部分。将其进行Wiener-Hermite展开,分别代入Dirichlet和Neumann两种边界条件表达式并求解。并由此进一步讨论了随机粗糙面散射的统计特性：相干散射幅度、等效阻抗、非相干散射能量的空间角分布等,推导出粗糙面散射的能量守恒定律。数值模拟对上述结果进行验证和进一步讨论,给出Neumann边界条件下在大角度入射时的异常散射现象,以及相关长度、入射角和起伏方差对相干散射强度的影响。
基于上述讨论,给出二维随机粗糙面的半空间格林函数解析表达式。该表达式清晰的反映出入射波、镜面反射波、以及粗糙面满散射波三部分。运用鞍点法分别得到格林函数相干部分和非相干部分在远场近似下的渐近表达式,给出辐射能量空间角分布特性。由于粗糙面格林函数同时满足波动方程和粗糙面的边界条件,因此已经包括了粗糙面对上方物体的作用。为下文分析粗糙面及其上方物体之间面体复合散射提供了理论基础。
在得到了随机粗糙面的格林函数之后,继续讨论粗糙面及其上方物体之间的面体复合散射。为了去除粗糙面截断及照明波束宽度对结果的影响,用射线法描述了面体相互作用的过程,并由此引入了差场散射的概念。以二维良导体粗糙面及其上方导体目标为例,由高斯定理结合粗糙面格林函数,提出一种新的四路径模型,将面体之间的多次散射作用包括在内,精确刻画了差场散射。接着以自由空间的单球散射为例,给出后续计算中用到的分离变量迭代法。根据新四路径模型,并采用分离变量迭代法,计算了面体之间的差场复合散射。数值结果给出物体大小、入射角、物体距离粗糙面高度等参数对差场散射能量空间角分布特性的影响。由于采用了纯粹的解析方法——随机泛函理论,并且避开了面体复合散射中最复杂的多次散射部分,在保留计算精度的同时,大大提高了计算效率。
扩展研究部分,本文以随机降雨层为例讨论离散粒子随机介质中的波传播问题。与之前的讨论不同,这一部分在处理随机介质中的多次散射问题时,不再采用随机泛函理论,而是采用概率统计理论分析了随机降雨层矢量辐射传输。首先建立由两种雨强交替构成的随机降雨层模型,由此模型在矢量辐射传输方程中引入随机量,建立随机矢量辐射传输方程,再由概率统计理论求解,得到用于大气遥感的双站和后站散射系数。最后数值模拟讨论了随机降雨层的参数对结果的影响。
本文将随机泛函理论用到了电磁波随机问题的研究当中,建立起清晰明确的物理模型,有效地解决了发散问题、多次散射等传统方法束手无策的难题,兼具解析方法的高效率和数值方法的高精度,为解决此类问题提供了一个崭新的思路。作为作者五年来博士研究工作的总结,本文被期望于为广大致力于研究随机介质中波传播问题的科研工作者提供一些启发和提示,促进我国在相关领域深层次的、自主的基础研究,并加快、加大在相关领域中的应用。当然,科学技术的发展日新月异,本文工作中还有许多有待完善和深入研究的地方,希望广大读者不吝指教。
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.
本文对三种随机介质分别展开讨论,将随机泛函理论和概率统计理论应用到随机介质中波散射与传播问题的研究当中,形成一整套完整的研究体系,旨在发展当前的电磁理论基础研究。全文分为连续随机介质中波的传播及其局域性、随机粗糙面及其上方物体的面体复合散射、以及作为扩展研究的随机降雨层中的矢量辐射传输分析三部分。
在连续随机介质中波的传播及其局域性研究中,首先讨论了二维各向同性均匀随机介质中平面波的传播及其局域性。在标量波动方程中引入随机介电起伏,建立随机波动方程。基于介质的均匀特性,由平移算子给出平面波的一般表达式。假设随机介质具有中心位于两倍波数处的窄带高斯谱,将随机介电起伏展开成Wiener积分式,并将积分区域沿波矢量方向条带分割。经过上述处理,将平面波的一般表达式和随机介电起伏的Wiener积分条带分割式代入随机波动方程,并用随机泛函理论求解。得到了二维随机平面波的解析近似解及其幅度和相位的定量统计特性。结果表明随机介质中的平面波以驻波形式存在,其幅度指数衰减,波数随机振荡,证实了随机介质中波的局域化现象。最后数值模拟验证了所得解析解的正确性,并模拟了二维随机介质及其中的平面驻波。
在上述平面波的研究基础上,进一步分析二维随机介质中柱面波的传播特性。同上述研究类似,假设介质具有窄带高斯谱,将随机介电起伏展开成Wiener积分式,并将积分区域条带分割。不同之处在于此时的高斯谱中心不再局限于二倍波数处,而是任意位置。同时由内外行柱面波的线性和给出波场一般表达式。将波场一般表达式和随机介电起伏的Wiener积分条带分割式代入随机波动方程,并在柱坐标系下求解。得到了随机介电起伏对柱面波幅度与相位调制作用的解析表达式,并给出局域化长度。结果表明柱面波在随机介质中传播时受到随机介电起伏的调制作用,其幅度指数衰减,波数随机振荡。且此调制作用主要发生在随机介质的高斯谱密度中心位于二倍波数和三倍波数区间内时。数值模拟给出了柱面波能量的空间分布,以及相位被调制后表现出的不同角模式柱面波。
综合上述讨论,给出二维随机介质中的平面波和柱面波的随机波转换方程,并对二者的传播特性进行对比。结果表明二维随机介质对于其中的平面波和柱面波具有同样的幅度和波数调制作用。不同之处在于,由于柱面波的波阵面为圆形,其能流密度除了和平面波一样被指数调制外,还与1／r成正比,会随传播距离更快的衰减。
在讨论了连续随机介质之后,接下来考虑随机粗糙面及其上方物体之间的复合散射问题。首先介绍用于分析随机粗糙面散射问题的随机泛函方法。以平面波入射到二维各向同性均匀随机粗糙面为例,由随机Floquet定理给出粗糙面散射场的一般表达式,包括入射波、镜面反射波、粗糙面漫散射波三部分。将其进行Wiener-Hermite展开,分别代入Dirichlet和Neumann两种边界条件表达式并求解。并由此进一步讨论了随机粗糙面散射的统计特性：相干散射幅度、等效阻抗、非相干散射能量的空间角分布等,推导出粗糙面散射的能量守恒定律。数值模拟对上述结果进行验证和进一步讨论,给出Neumann边界条件下在大角度入射时的异常散射现象,以及相关长度、入射角和起伏方差对相干散射强度的影响。
基于上述讨论,给出二维随机粗糙面的半空间格林函数解析表达式。该表达式清晰的反映出入射波、镜面反射波、以及粗糙面满散射波三部分。运用鞍点法分别得到格林函数相干部分和非相干部分在远场近似下的渐近表达式,给出辐射能量空间角分布特性。由于粗糙面格林函数同时满足波动方程和粗糙面的边界条件,因此已经包括了粗糙面对上方物体的作用。为下文分析粗糙面及其上方物体之间面体复合散射提供了理论基础。
在得到了随机粗糙面的格林函数之后,继续讨论粗糙面及其上方物体之间的面体复合散射。为了去除粗糙面截断及照明波束宽度对结果的影响,用射线法描述了面体相互作用的过程,并由此引入了差场散射的概念。以二维良导体粗糙面及其上方导体目标为例,由高斯定理结合粗糙面格林函数,提出一种新的四路径模型,将面体之间的多次散射作用包括在内,精确刻画了差场散射。接着以自由空间的单球散射为例,给出后续计算中用到的分离变量迭代法。根据新四路径模型,并采用分离变量迭代法,计算了面体之间的差场复合散射。数值结果给出物体大小、入射角、物体距离粗糙面高度等参数对差场散射能量空间角分布特性的影响。由于采用了纯粹的解析方法——随机泛函理论,并且避开了面体复合散射中最复杂的多次散射部分,在保留计算精度的同时,大大提高了计算效率。
扩展研究部分,本文以随机降雨层为例讨论离散粒子随机介质中的波传播问题。与之前的讨论不同,这一部分在处理随机介质中的多次散射问题时,不再采用随机泛函理论,而是采用概率统计理论分析了随机降雨层矢量辐射传输。首先建立由两种雨强交替构成的随机降雨层模型,由此模型在矢量辐射传输方程中引入随机量,建立随机矢量辐射传输方程,再由概率统计理论求解,得到用于大气遥感的双站和后站散射系数。最后数值模拟讨论了随机降雨层的参数对结果的影响。
本文将随机泛函理论用到了电磁波随机问题的研究当中,建立起清晰明确的物理模型,有效地解决了发散问题、多次散射等传统方法束手无策的难题,兼具解析方法的高效率和数值方法的高精度,为解决此类问题提供了一个崭新的思路。作为作者五年来博士研究工作的总结,本文被期望于为广大致力于研究随机介质中波传播问题的科研工作者提供一些启发和提示,促进我国在相关领域深层次的、自主的基础研究,并加快、加大在相关领域中的应用。当然,科学技术的发展日新月异,本文工作中还有许多有待完善和深入研究的地方,希望广大读者不吝指教。
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.
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