基于三维电磁散射参数化模型的SAR目标识别方法

文贡坚 朱国强 殷红成 邢孟道 杨虎 马聪慧 闫华 丁柏圆 钟金荣

文贡坚, 朱国强, 殷红成, 邢孟道, 杨虎, 马聪慧, 闫华, 丁柏圆, 钟金荣. 基于三维电磁散射参数化模型的SAR目标识别方法[J]. 雷达学报, 2017, 6(2): 115-135. doi: 10.12000/JR17034
引用本文: 文贡坚, 朱国强, 殷红成, 邢孟道, 杨虎, 马聪慧, 闫华, 丁柏圆, 钟金荣. 基于三维电磁散射参数化模型的SAR目标识别方法[J]. 雷达学报, 2017, 6(2): 115-135. doi: 10.12000/JR17034
Wen Gongjian, Zhu Guoqiang, Yin Hongcheng, Xing Mengdao, Yang Hu, Ma Conghui, Yan Hua, Ding Baiyuan, Zhong Jinrong. SAR ATR Based on 3D Parametric Electromagnetic Scattering Model[J]. Journal of Radars, 2017, 6(2): 115-135. doi: 10.12000/JR17034
Citation: Wen Gongjian, Zhu Guoqiang, Yin Hongcheng, Xing Mengdao, Yang Hu, Ma Conghui, Yan Hua, Ding Baiyuan, Zhong Jinrong. SAR ATR Based on 3D Parametric Electromagnetic Scattering Model[J]. Journal of Radars, 2017, 6(2): 115-135. doi: 10.12000/JR17034

基于三维电磁散射参数化模型的SAR目标识别方法

doi: 10.12000/JR17034
基金项目: 国家部委基金
详细信息
    作者简介:

    文贡坚(1972–),男,湖南宁乡人,教授,博士生导师,研究方向为遥感图像处理。

    朱国强(1959–),男,湖北武汉人,教授,博士生导师,研究方向为复杂目标电磁散射、电磁场理论与工程应用。

    殷红成(1967–),男,江西余江人,研究员,现为电磁散射重点实验室专业总师,博士生导师,主要研究方向为电磁散射、雷达目标特性、目标识别等。E-mail: yinhc207@126.com

    邢孟道(1975–),男,浙江嵊州人。西安电子科技大学教授,博士生导师,主要研究方向为雷达成像、目标识别和天波超视距雷达信号处理。E-mail: xmd@xidian.edu.cn

    杨虎:杨   虎(1973–),男,安徽安庆人,教授,博士生导师,研究方向为电磁场与微波技术。

    马聪慧(1987–),女,湖北襄阳人,博士研究生,研究方向为SAR自动目标识别。

    闫华:闫   华(1981–),男,黑龙江哈尔滨人,高级工程师,研究方向为目标电磁散射建模与特性分析。

    丁柏圆(1990–),男,安徽池州人,博士研究生,研究方向为SAR自动目标识别。

    钟金荣(1985–),男,广西玉林人,博士,研究方向为SAR自动目标识别。

    通讯作者:

    马聪慧   ma_conghui@yeah.net

  • 中图分类号: TN957

SAR ATR Based on 3D Parametric Electromagnetic Scattering Model

Funds: The National Minstries Foundation
  • 摘要: 合成孔径雷达目标识别是雷达数据解译中一个长期研究的难点问题。近年来,基于模型的SAR目标识别方法由于在扩展条件下的识别性能表现良好而备受关注。在联合国内多家研究单位进行攻关的基础上,该文简要阐述了对该问题的初步研究成果及思考。首先从3个方面出发梳理了散射部件模型发展的技术脉络并对其进行了补充完善;然后从正向推算和逆向反演两条技术途径提出了复杂目标电磁散射参数化建模方法;最后提出了基于复杂目标电磁散射参数化模型的目标识别新框架。论文最后对基于模型的SAR目标识别下一步研究方向进行了展望。

     

  • 图  1  二面角结构及其RCS比较

    Figure  1.  Dihedral and the RCS comparison

    图  2  基础模型的发展脉络

    Figure  2.  The development of basic model

    图  3  正向建模流程图

    Figure  3.  The diagram for forward target modeling

    图  4  简易坦克目标及其正向建模结果

    Figure  4.  Simplified tank target and its modeling result

    图  5  目标电磁散射参数化模型所成图像与仿真数据对比

    Figure  5.  Comparison between the scattering center model generated image and the simulated image

    图  6  逆向建模基本思路

    Figure  6.  The diagram for inverse target modeling

    图  7  重构的简单坦克宽角度3维属性散射中心模型

    Figure  7.  The rebuilt model for simplified tank target based on 3D attributed scattering center model

    图  8  目标的SAR图像以及基于属性散射中心模型重构的SAR图像

    Figure  8.  Original SAR image and rebuilt SAR image

    图  9  基于3维部件级电磁散射参数化模型的识别框架

    Figure  9.  Framework of 3D parametric electromagnetic part model based SAR ATR

    图  10  简易坦克目标成像

    Figure  10.  Image of simplified tank target

    图  11  部分扩展操作条件下暗室测量得到的测试数据

    Figure  11.  Test data in EOCs

    表  1  几种不同散射中心的 ${α}_{i}$取值

    Table  1.   The ${α}_{i}$ for different scattering centers

    ai 取值 散射中心
    –1 角绕射,尖顶绕射
    –1/2 边缘绕射
    0 点散射,双曲面反射,直边镜面反射
    1/2 单曲面反射
    1 平板法向发射,二面角反射,三面角反射
    下载: 导出CSV

    表  2  我们研究的耦合散射中心模型的频率依赖因子

    Table  2.   The ${α}$ in the proposed model by our team

    电磁散射中心类型 电磁散射中心名称 a取值
    镜面-镜面耦合 平面-平面耦合 1
    单弯曲-平面耦合 0.5
    双弯曲-平面耦合 0
    单弯曲-单弯曲耦合 0.5
    双弯曲-单弯曲耦合 0
    双弯曲-双弯曲耦合 0
    镜面-边缘耦合 平面-直边耦合 0
    单弯曲-直边耦合 0
    双弯曲-直边耦合 –0.5
    曲边-平面耦合 –0.5
    曲边-单弯曲耦合 –0.5
    曲边-双弯曲耦合 –0.5
    边缘-边缘耦合 直边-直边耦合 –0.5
    直边-曲边耦合 –1
    曲边-曲边耦合 –1
    下载: 导出CSV

    表  3  简易坦克的部件级3维电磁散射模型

    Table  3.   The 3D parametric electromagnetic part model for the simplified tank target

    编号 形成原因 可见 范围(°) 参数化 模型形式 RCS峰值 (dB)
    1 车身 俯仰: [19, 39] 方位: [75, 105] 平板 1
    2 车身平板 与立方体 俯仰: [9, 30] 方位: [75, 105] 方形顶帽 32
    28 右履带与车身 前斜平板 俯仰: [2, 30] 方位: [75, 105] 二面角 33
    29 车身平板与炮筒 俯仰: [11, 30] 方位: [75, 105] 平板上 倒圆柱 10
    下载: 导出CSV

    表  4  基于部件级3维电磁散射模型的识别结果

    Table  4.   The recognition results

    操作条件 数据描述 基于目标3维参数化电磁散射模型 方法的识别率(%) 基于模板方法的识别率(%)
    标准 500幅电磁计算数据和75幅暗室测量数据 99.13 91.2
    扩展 62幅电磁计算数据和106幅暗室测量数据 90.48 58.2
    下载: 导出CSV

    1  10种典型体参数化模型表达式列表

    1.   The parametric models for ten scatterers

    典型散射部件名称 图形 参数化模型 模型适用角度范围 各参数排列顺序和含义的解释
    长方体 $\begin{array}{l}S_{\rm vv,hh}^{\rm cuboid} = {S_1} + {S_2} + {S_3} + {S_{\rm 4vv,hh}} + {S_{\rm 5vv,hh}} + {S_{\rm 6vv,hh}}\\{S_1} = - {\rm j}\frac{{kbc}}{{\sqrt {π} }}{l_x}{\rm sinc}\left( {kb{l_y}} \right){\rm sinc}\left( {kc{l_z}} \right){{\rm e}^{{\rm j}k\left( {a{l_x} + c{l_z}} \right)}}\\{S_2} = - {\rm j}\frac{{kca}}{{\sqrt {π} }}{l_y}{\rm sinc}\left( {kc{l_z}} \right){\rm sinc}\left( {ka{l_x}} \right){{\rm e}^{{\rm j}k\left( {b{l_y} + c{l_z}} \right)}}\\{S_3} = - {\rm j}\frac{{kab}}{{\sqrt {π} }}{l_z}{\rm sinc}\left( {ka{l_x}} \right){\rm sinc}\left( {kb{l_y}} \right){{\rm e}^{{ \rm j}2kc{l_z}}}\\{S_{4{\rm vv,hh}}} = \frac{{\sqrt 3 }}{9}\frac{c}{{\sqrt {π} }}\left\{ \begin{array}{l}\left[ { \mp 1 - \frac{{\sqrt 3 }}{2}\frac{{\sin \displaystyle\frac{1}{3}\left( {{π} /2 - \phi } \right)}}{{\cos \displaystyle\frac{\phi }{3}\sin \displaystyle\frac{2}{3}\left( {{π} - \phi } \right)}}} \right]\\ \cdot {\rm sinc}\left( {kc{l_z}} \right){{\rm e}^{{\rm j}k\left( {a{l_x} - b{l_y} + c{l_z}} \right)}}\\ + \left[ { \mp 1 + \frac{{\sqrt 3 }}{4}\frac{1}{{\cos \displaystyle\frac{1}{3}\left( {{π} /2 - \phi } \right)\cos \displaystyle\frac{\phi }{3}}}} \right]\\\cdot {\rm sinc}\left( {kc{l_z}} \right){{\rm e}^{{\rm j}k\left( {a{l_x} + b{l_y} + c{l_z}} \right)}}\\ + \left[ { \mp 1 - \frac{{\sqrt 3 }}{2}\frac{{\sin \displaystyle\frac{\phi }{3}}}{{\cos \displaystyle\frac{1}{3}\left( {{π} /2 - \phi } \right)\sin \displaystyle\frac{2}{3}\left( {{π} /2 + \phi } \right)}}} \right]\\\cdot {\rm sinc}\left( {kc{l_z}} \right){{\rm e}^{{\rm j}k\left( {a{l_x} - b{l_y} + c{l_z}} \right)}}\end{array} \right\}\end{array}$ $\begin{array}{l}\theta \in \left[ 0^{°},{{90}^{°}} \right]\\\phi \in \left[ 0^{°},{{90}^{°}} \right]\end{array}$ a, b, c:长方体边长
    ${S_{5{\rm vv,hh}}} = \frac{{\sqrt 3 }}{9}\frac{a}{{\sqrt {π} }}\left\{ \begin{array}{l}\left[ { \pm 1 - \frac{{\sqrt 3 }}{2}\frac{{\sin \displaystyle\frac{\theta }{3}}}{{\cos \displaystyle\frac{1}{3}\left( {{π} /2 - \theta } \right)\sin \displaystyle\frac{2}{3}\left( {{π} /2 + \theta } \right)}}} \right]\\\cdot {\rm sinc}\left( {ka{l_x}} \right){{\rm e}^{{\rm j}kb{l_y}}}\\ + \left[ { \pm 1 + \frac{{\sqrt 3 }}{4}\frac{1}{{\cos \displaystyle\frac{\theta }{3}\cos\displaystyle \frac{1}{3}\left( {{π} /2 - \theta } \right)}}} \right]\\\cdot {\rm sinc}\left( {ka{l_x}} \right){{\rm e}^{{\rm j}k\left( {b{l_y} + 2c{l_z}} \right)}}\\ + \left[ { \pm 1 - \frac{{\sqrt 3 }}{2}\frac{{\sin \displaystyle\frac{1}{3}\left( {{π} /2 - \theta } \right)}}{{\cos \displaystyle\frac{\theta }{3}\sin \displaystyle\frac{2}{3}\left( {{π} - \theta } \right)}}} \right]\\\cdot {\rm sinc}\left( {ka{l_x}} \right){{\rm e}^{{\rm j}k\left( { - b{l_y} + 2c{l_z}} \right)}}\end{array} \right\}\\{S_{6{\rm vv,hh}}} = \frac{{\sqrt 3 }}{9}\frac{b}{{\sqrt {π} }}\left\{ \begin{array}{l}\left[ { \pm 1 - \displaystyle\frac{{\sqrt 3 }}{2}\frac{{\sin \displaystyle\frac{1}{3}\left( {{π} /2 - \theta } \right)}}{{\cos \displaystyle\frac{\theta }{3}\sin \frac{2}{3}\left( {{π} - \theta } \right)}}} \right]\\\cdot {\rm sinc}\left( {kb{l_y}} \right){{\rm e}^{{\rm j}k\left( { - a{l_x} + 2c{l_z}} \right)}}\\ + \left[ { \pm 1 + \displaystyle\frac{{\sqrt 3 }}{4}\frac{1}{{\cos \displaystyle\frac{1}{3}\left( {{π} /2 - \theta } \right)\cos \displaystyle\frac{\theta }{3}}}} \right]\\\cdot {\rm sinc}\left( {kb{l_y}} \right){{\rm e}^{{\rm j}k\left( {a{l_x} + 2c{l_z}} \right)}}\\ + \left[ { \pm 1 - \frac{{\sqrt 3 }}{2}\frac{{\sin \displaystyle\frac{\theta }{3}}}{{\cos \displaystyle\frac{1}{3}\left( {{π} /2 - \theta } \right)\sin \displaystyle\frac{2}{3}\left( {{π} /2 + \theta } \right)}}} \right]\\\cdot {\rm sinc}\left( {kb{l_y}} \right){{\rm e}^{{\rm j}ka{l_x}}}\end{array} \right\}$ $\begin{array}{l}\theta \in \left[ 0^{°},{{90}^{°}} \right]\\\phi \in \left[ 0^{°},{{90}^{°}} \right]\end{array}$ a, b, c:长方体边长
    圆柱体 $S_{\rm vv,hh}^{\rm cylinder} = {S_{1{\rm vv,hh}}} + {S_{2{\rm vv,hh}}} + {S_{3{\rm vv,hh}}}$ 当 $2kr\sin \theta > 2.44$且 $kh\cos \theta > 2.25$ $\begin{array}{l}{S_{1{\rm vv,hh}}} = \sqrt {\frac{{{\rm j}r}}{{3k\sin \theta }}} \left[ { \mp \frac{2}{3} - {{\left( {\frac{1}{2} + \cos \frac{{4\theta }}{3}} \right)}^{ - 1}}} \right]{{\rm e}^{{\rm j}2kr\sin \theta }}\\{S_{2{\rm vv,hh}}} = \sqrt {\frac{{{\rm j}r}}{{3k\sin \theta }}} \left[ { \mp \frac{2}{3} - {{\left( {\frac{1}{2} + \cos \frac{{4\left( {{π} - \theta } \right)}}{3}} \right)}^{ - 1}}} \right]{{\rm e}^{{\rm j}2k\left( {r\sin \theta + h\cos \theta } \right)}}\\{S_{3{\rm vv,hh}}} = {\rm j}\sqrt {\frac{{{\rm j}r}}{{3k\sin \theta }}} \left[ { \mp \frac{2}{3} - {{\left( {\frac{1}{2} + \cos \frac{{4\left( {{π} /2 - \theta } \right)}}{3}} \right)}^{ - 1}}} \right]{{\rm e}^{{\rm j}2k\left( { - r\sin \theta + h\cos \theta } \right)}}\end{array}$ 当 $kh\cos \theta \le 2.25$ ${\left( {{S_1} + {S_2} + {S_3}} \right)_{\theta \to \displaystyle\textstyle\frac{{π} }{2}}} = - \sqrt {{\rm j}kr{h^2}\sin \theta } {\rm sinc}\left( {kh\cos \theta } \right){{\rm e}^{{\rm j}k\left( {h\cos \theta + 2r\sin \theta } \right)}}$ 当 $2kr\sin \theta \le 2.44$ ${\left( {{S_1} + {S_2} + {S_3}} \right)_{\theta \to 0}} = - {\rm j}2\sqrt {π} k{r^2}\left| {\cos \theta } \right|\frac{{{J_1}\left( {2kr\sin \theta } \right)}}{{2kr\sin \theta }}{{\rm e}^{{\rm j}kh\left( {\cos \theta + \left| {\cos \theta } \right|} \right)}}$ $\begin{array}{l}\theta \in \left[ {{0^{°}},{{90}^{°}}} \right]\\\phi \in \left[ {{0^{°}},{{90}^{°}}} \right]\end{array}$ r:圆柱半径
    h:圆柱的长
    方形顶帽 $\begin{array}{l}S_{{\rm vv,hh}}^{{\rm cuboid - hat}} = {S_1} + {S_2} + {S_3} + {S_{4{\rm vv,hh}}} + {S_{5{\rm vv,hh}}}\\{S_1} = - {\rm j}\frac{{kbc}}{{\sqrt {π} }}{l_x}{\rm sinc}\left( {kb{l_y}} \right){\rm sinc}\left( {kc{l_z}} \right){{\rm e}^{{\rm j}k\left( {a{l_x} + c{l_z}} \right)}}\\{S_2} = - {\rm j}\frac{{kca}}{{\sqrt {π} }}{l_y}{\rm sinc}\left( {kc{l_z}} \right){\rm sinc}\left( {ka{l_x}} \right){{\rm e}^{{\rm j}k\left( {b{l_y} + c{l_z}} \right)}}\\{S_3} = - {\rm j}\frac{{kab}}{{\sqrt {π} }}{l_z}{\rm sinc}\left( {ka{l_x}} \right){\rm sinc}\left( {kb{l_y}} \right){{\rm e}^{{\rm j}2kc{l_z}}}\\{S_{4{\rm vv,hh}}} =\!\! \mp \frac{1}{{\sqrt {π} }}\!\! \frac{{\min \left\{ {c{l_x},{d_1}{l_z}} \right\}}}{{{l_y}}}\left\{ \begin{array}{l}{\!\!\! {\rm e}^{{\rm j}k\left( {b - {\delta _1}} \right){l_y}}} {\rm sinc}\left( {k{\delta _1}{l_y}} \right)\! - \!{{\rm e}^{ - {\rm j}kb{l_y}}},{\delta _1} \! \le \! b\\\frac{b}{{{\delta _1}}}\left[ {{\rm sinc}\left( {kb{l_y}} \right) - {{\rm e}^{ - {\rm j}kb{l_y}}}} \right],\quad{\delta _1} > b\end{array} \right\}{{\rm e}^{{\rm j}ka{l_x}}}\\{S_{5{\rm vv,hh}}} = \mp \frac{1}{{\sqrt {π} }}\frac{{\min \left\{ {c{l_y},{d_2}{l_z}} \right\}}}{{{l_x}}}\left\{ \begin{array}{l}{{\rm e}^{{\rm j}k\left( {a - {\delta _2}} \right){l_x}}}{\rm sinc}\left( {k{\delta _2}{l_x}} \right) - {{\rm e}^{ - {\rm j}ka{l_x}}},{\delta _2} \le a\\\frac{a}{{{\delta _2}}}\left[ {{\rm sinc}\left( {ka{l_x}} \right) - {{\rm e}^{ - {\rm j}ka{l_x}}}} \right],\quad\ \ {\delta _2} > a\end{array} \right\}{{\rm e}^{{\rm j}kb{l_y}}}\end{array}$ $\begin{array}{c}\theta \in \left[ {{{20}^{°}},{{90}^{°}}} \right]\\\!\!\!\!\phi \in \left[ {{0^{°}},{{15}^{°}}} \right]\\\quad \cup \left[ {{{75}^{°}},{{90}^{°}}} \right]\end{array}$ a, b, c:长方体边长 d1x方向“帽沿” 宽度 d2y方向“帽沿” 宽度
    圆形顶帽 $\begin{array}{l}S_{{\rm vv,hh}}^{{\rm top - hat}} = {S_1} + {S_2} + {S_{3{\rm vv,hh}}}\\{S_1} = - \sqrt {{\rm j}kr{h^2}\sin \theta } {\rm sinc}\left( {kh\cos \theta } \right){{\rm e}^{{\rm j}k\left( {h\cos \theta + 2r\sin \theta } \right)}}\\{S_2} = - {\rm j}2\sqrt {π} k{r^2}\left| {\cos \theta } \right|\frac{{{J_1}\left( {2kr\sin \theta } \right)}}{{2kr\sin \theta }}{{\rm e}^{{\rm j}kh\left( {\cos \theta + \left| {\cos \theta } \right|} \right)}}\\{S_{3{\rm vv,hh}}} = \mp 2\sqrt {{\rm j}kr\sin \theta } \min \left\{ {h,\frac{d}{{\tan \theta }}} \right\}{{\rm e}^{{\rm j}2kr\sin \theta }}\end{array}$ $\begin{array}{l}\theta \in \left[ {{{0.1}^{°}},{{90}^{°}}} \right]\\\phi \in \left[ {{0^{°}},{{90}^{°}}} \right]\end{array}$ r:圆柱半径 h:圆柱的长d:“帽沿”宽度
    平板上倒圆柱 $\begin{array}{c}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! S_{{\rm vv,hh}}^{{\rm cyl - plane}} = {S_1} + {S_{2{{\rm vv,hh}}}} + {S_{3{\rm vv,hh}}}\\\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!{S_1} = - \sqrt {{\rm j}kr\sqrt {1 - l_y^2} } L{\rm sinc}\left( {kL{l_y}} \right){{\rm e}^{{\rm j}2k\left( {r + h} \right){l_z}}}\\{S_{2{\rm vv,hh}}} = \pm \sqrt { - {\rm j}kr{L^2}\sin \theta } \sin \! {\rm c}\left( {kL{l_y}} \right)U\left( {\arctan \left( {\frac{d}{{r + h}}} \right) - \theta } \right){{\rm e}^{{\rm j}2kr\sin \theta }}\\\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!{S_{3{\rm vv,hh}}} = \frac{{2{b_0}}}{{\sqrt {3{π} } }}\left[ { \pm \frac{1}{3} + \frac{1}{{1 + 2\cos \displaystyle\frac{{4\theta }}{3}}}} \right]{\rm sinc}\left( {k{b_0}{l_y}} \right){{\rm e}^{{\rm j}k{a_0}{l_x}}}\\ \quad\quad+ \frac{{2{b_0}}}{{\sqrt {3{π} } }}\left[ { \pm \frac{1}{3} + \frac{1}{{1 + 2\cos \displaystyle\frac{{4\left( {{π} - \theta } \right)}}{3}}}} \right]{\rm sinc}\left( {k{b_0}{l_y}} \right){{\rm e}^{{\rm j}k\left( {{a_0}{l_x} + 2{d_0}{l_z}} \right)}}\\ \quad\quad \quad\ + \frac{{2{b_0}}}{{\sqrt {3{π} } }}\left[ { \pm \frac{1}{3} + \frac{1}{{1 + 2\cos \displaystyle\frac{{4\left( {{π} /2 - \theta } \right)}}{3}}}} \right]{\rm sinc}\left( {k{b_0}{l_y}} \right){{\rm e}^{{\rm j}k\left( { - {a_0}{l_x} + 2{d_0}{l_z}} \right)}}\end{array}$ $\begin{array}{l}\theta \in \left[ {{0^{°}},{{90}^{°}}} \right]\\\phi \in \left[ {{0^{°}},{{10}^{°}}} \right]\end{array}$ r:圆柱半径 L:圆柱的长h:圆柱面到平面距离 a0:平板边长 b0:平板边长 d0:平板厚度
    直二面角 $\begin{array}{l}S_{{\rm vv,hh}}^{{\rm dihedral}} = {S_1} + {S_2} + {S_{3{\rm vv,hh}}}\\{S_1} = - {\rm j}\frac{{kah}}{{\sqrt {π} }}{l_y}{\rm sinc}\left( {kh{l_z}} \right){\rm sinc}\left( {ka{l_x}} \right){{\rm e}^{{\rm j}k\left( {a{l_x} + h{l_z}} \right)}}\\{S_2} = - {\rm j}\frac{{kbh}}{{\sqrt {π} }}{l_x}{\rm sinc}\left( {kh{l_z}} \right){\rm sinc}\left( {kb{l_y}} \right){{\rm e}^{{\rm j}k\left( {b{l_y} + h{l_z}} \right)}}\\{S_{\rm 3vv,hh}} = \left\{ {\begin{array}{*{20}{c}}{ \pm \frac{1}{{\sqrt {π} }}\frac{{\min \{ b{l_x},a{l_y}\} }}{{{l_z}}}\left\{ {{{\rm e}^{{\rm j}k\left( {h - \delta } \right){l_z}}}{\rm sinc}\left( {k\delta {l_z}} \right) - {{\rm e}^{ - {\rm j}kh{l_z}}}} \right\}{{\rm e}^{{\rm j}kh{l_z}}}},&\!\!{\delta \le h}\\\!\!\!\!\!\!\!\!\!\! { \pm \frac{1}{{\sqrt {π} }}\frac{{\min \{ b{l_x},a{l_y}\} }}{{{l_z}}}\left\{ {\frac{h}{\delta }{\rm sinc}\left( {kh{l_z}} \right) - {{\rm e}^{ - {\rm j}kh{l_z}}}} \right\}{{\rm e}^{{\rm j}kh{l_z}}}},&\!\!{\delta > h}\end{array}} \right.\\\quad\quad\ \delta = \min \left\{ {b{l_z}/{l_y},a{l_z}/{l_x}} \right\}\end{array}$ $\begin{array}{l}\theta \in \left[ {{{40}^{°}},{{90}^{°}}} \right]\\\phi \in \left[ {{0^{°}},{{90}^{°}}} \right]\end{array}$ a, b:二面角两面宽度h:二面角的长度
    直三面角 $\begin{array}{l}S_{{\rm vv,hh}}^{\rm trihedral} = {S_1} + {S_2} + {S_3} + {S_{12{\rm vv,hh}}} + {S_{\rm 23vv,hh}} + {S_{\rm 31vv,hh}} + {S_{123}}\\{S_1} = - {\rm j}\frac{{k{a^2}}}{{\sqrt {π} }}{l_x}{\rm sinc}\left( {ka{l_y}} \right){\rm sinc}\left( {ka{l_z}} \right){{\rm e}^{{\rm j}ka\left( {{l_y} + {l_z}} \right)}}\\{S_2} = - {\rm j}\frac{{k{a^2}}}{{\sqrt {π} }}{l_y}{\rm sinc}\left( {ka{l_z}} \right){\rm sinc}\left( {ka{l_x}} \right){{\rm e}^{{\rm j}ka\left( {{l_z} + {l_x}} \right)}}\\{S_3} = - {\rm j}\frac{{k{a^2}}}{{\sqrt {π} }}{l_z}{\rm sinc}\left( {ka{l_x}} \right){\rm sinc}\left( {ka{l_y}} \right){{\rm e}^{{\rm j}ka\left( {{l_x} + {l_y}} \right)}}\\{S_{\rm 12vv,hh}} = \mp {\rm j}2\frac{{k{a^2}}}{{\sqrt {π} }}\min \left\{ {{l_x},{l_y}} \right\}{\rm sinc}\left( {ka{l_z}} \right){{\rm e}^{{\rm j}ka{l_z}}}\\{S_{\rm 23vv,hh}} = \pm {\rm j}2\frac{{k{a^2}}}{{\sqrt {π} }}\min \left\{ {{l_y},{l_z}} \right\}{\rm sinc}\left( {ka{l_x}} \right){{\rm e}^{{\rm j}ka{l_x}}}\\{S_{\rm 31vv,hh}} = \pm {\rm j}2\frac{{k{a^2}}}{{\sqrt {π} }}\min \left\{ {{l_z},{l_x}} \right\}{\rm sinc}\left( {ka{l_y}} \right){{\rm e}^{{\rm j}ka{l_y}}}\\\begin{aligned}{S_{123}} = & - {\rm j}\frac{{k\sqrt 3 {a^2}}}{{\sqrt {π} }}\min \left\{ {\sin \left( {\theta + {π} /4 - \alpha } \right),\cos \left( {\theta + {π} /4 - \alpha } \right)} \right\} \\ &\cdot \min \left\{ {\sin \phi ,\cos \phi } \right\}\end{aligned}\end{array}$ $\begin{array}{l}\theta \in \left[ {{{10}^{°}},{{80}^{°}}} \right]\\\phi \in \left[ {{{10}^{°}},{{80}^{°}}} \right]\end{array}$ a:三面角边长
    圆边浅凹体 $\begin{array}{l}\quad\quad\quad\quad\quad\quad\quad\quad\ S_{\rm vv,hh}^{\rm pan} = \pm {10^{{p_{\rm vv,hh}}\left( \theta \right)/20}}{{\rm e}^{ - {\rm j}2ka\sin \theta }}\\\quad\quad\quad\quad\quad\quad\quad\quad\ {p_{\rm vv}}\left( \theta \right) = \sum\limits_{i = 1}^{{N_{\rm vv}}} {c_i^{\rm vv}{{\left( {90 - \theta } \right)}^i}} \\\quad\quad\quad\quad\quad\quad\quad\quad\ {p_{\rm hh}}\left( \theta \right) = \sum\limits_{i = 1}^{{N_{\rm hh}}} {c_i^{\rm hh}{{\left( {90 - \theta } \right)}^i}} \end{array}$ $\begin{array}{l}\theta \in \left[ {20}^{°},{{90}^{°}} \right]\\\phi \in \left[ 0^{°},{{90}^{°}} \right]\end{array}$ ${N_{\rm vv,hh}}$:拟合函数阶数 $c_j^{\rm vv,hh}$:拟合系数 a:圆盘半径 kc:中心波数
    矮顶帽 $\begin{array}{l}S_{\rm vv,hh}^{\rm short - tophat} = \pm {10^{{p_{\rm vv,hh}}\left( \theta \right)/20}}{{\rm e}^{{\rm j}2ka\sin \theta }}\\{p_{\rm vv}}\left( \theta \right) = \sum\limits_{i = 1}^{{N_{\rm vv}}} {c_i^{\rm vv}{{\left( {90 - \theta } \right)}^i}} \\{p_{\rm hh}}\left( \theta \right) = \sum\limits_{i = 1}^{{N_{\rm hh}}} {c_i^{\rm hh}{{\left( {90 - \theta } \right)}^i}}\end{array}$ $\begin{array}{l}\theta \in \left[ {{{20}^{°}},{{90}^{°}}} \right]\\\phi \in \left[ {{0^{°}},{{90}^{°}}} \right]\end{array}$ ${N_{\rm vv,hh}}$:拟合函数阶数 $c_j^{\rm vv,hh}$:拟合系数a:顶部半径kc:中心波数
    狭长二面角 $\begin{array}{l}\quad\quad\quad S_{\rm vv,hh}^{\rm dihedral\_smallsurf}\\\quad\quad\quad\quad = \pm {{\rm e}^{ - \frac{2}{3}{{\left( {{π} fL\sin \theta /{\rm c}} \right)}^2}{{\left( {\phi - {π} } \right)}^2}}}{10^{{p_{\rm vv,hh}}\left( \theta \right)/20}}{{\rm e}^{{\rm j}2ka\sin \theta }}\\\quad\quad\quad{p_{\rm vv}}\left( \theta \right) = \sum\limits_{i = 1}^{{N_{\rm vv}}} {c_i^{\rm vv}{{\left( {90 - \theta } \right)}^i}} \\\quad\quad\quad{p_{\rm hh}}\left( \theta \right) = \sum\limits_{i = 1}^{{N_{\rm hh}}} {c_i^{\rm hh}{{\left( {90 - \theta } \right)}^i}} \end{array}$ $\begin{array}{l}\theta \in \left[ {{{20}^{°}},{{90}^{°}}} \right]\\\phi \in \left[ {{0^{°}},{{90}^{°}}} \right]\end{array}$ ${N_{\rm vv,hh}}$:拟合函数阶数 $c_j^{\rm vv,hh}$:拟合系数 d:狭长立板偏置距离 L:狭长立板长度kc:中心波数
    说明: ${l_x} = \sin \theta \cos \phi, \;{l_y} = \sin \theta \sin \phi, \;{l_z} = \cos \theta $ ${J_1}\left( \cdot \right)$为一阶贝塞尔函数;min{a,b}指取a,b中的较小值;U(x)为单位阶跃函数,即x大于等于0时取1,小于0时取0; $k = \frac{{2{π} f}}{\rm c}$为波数。
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出版历程
  • 收稿日期:  2017-03-30
  • 修回日期:  2017-04-18
  • 网络出版日期:  2017-05-04

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