﻿ 海洋内波对海面电磁散射特性的影响分析
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 雷达学报  2015, Vol. 4 Issue (3): 326–333  DOI: 10.12000/JR15060 0

### 引用本文 [复制中英文]

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Wei Yi-wen, Guo Li-xin, and Yin Hong-cheng. Analysis of the scattering characteristics of sea surface with the influence from internal wave[J]. Journal of Radars, 2015, 4(2): 326–333. DOI: 10.12000/JR15060.
[复制英文]

### 文章历史

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Analysis of the Scattering Characteristics of Sea Surface with the Influence from Internal Wave
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State Key Laboratory of Integrated Services Networks, Xidian University, Xi’an 710071, China
School of Physics and Optoelectronic Engineering, Xidian University, Xi’an 710071, China
National Electromagnetic Scattering Laboratory, Beijing 100854, China
Abstract: The internal wave travels beneath the sea surface and modulate the roughness of the sea surface through the wave-current interaction. This makes some dark and bright bands can be observed in the Synthetic Aperture Radar (SAR) images. In this paper, we first establish the profile of the internal wave based on the KdV equations; then, the action balance equation and the wave-current interaction source function are used to modify the sea spectrum; finally, the two-scale theory based facet model is combined with the modified sea spectrum to calculate the scattering characteristics of the sea. We have simulated the scattering coefficient distribution of the sea with an internal wave traveling through. The influence on the scattering coefficients and the Doppler spectra under different internal wave parameters and sea state parameters are analyzed.
Key words: Internal wave    Electromagnetic scattering    Doppler spectrum
1 引言

2 理论模型 2.1 KdV方程

 $\frac{{\partial \eta }}{{\partial t}} + ({C_0} + \alpha \eta + {\alpha _1}{\eta ^2})\frac{{\partial \eta }}{{\partial x}} + \beta \frac{{{\partial ^3}\eta }}{{\partial {x^3}}} = 0$ (1)

 $\alpha = \frac{{3{C_0}({h_1} - {h_2})}}{{2{h_1}{h_2}}} \hspace{115pt}$ (2)
 ${\alpha _1} = \frac{{3{C_0}}}{{h_1^2h_2^2}}\left[{\frac{7}{8}{{\left( {{h_2} - {h_1}} \right)}^2} - (h_2^2 - {h_1}{h_2} + h_1^2)} \right]$ (3)
 $\beta = \frac{{{C_0}{h_1}{h_2}}}{6} \hspace{140pt}$ (4)

 ${C_0} = \sqrt {\frac{{g\Delta \rho }}{\rho }\frac{{{h_1}{h_2}}}{{{h_1} + {h_2}}}}$ (5)

 $\eta (x,t) = \pm {\eta _0}{\rm{sec}}{{\rm{h}}^2}\left[{\frac{{x - C \,_0^\prime t}}{l}} \right]$ (6)

 $C \,_0^\prime = {C_0} + \alpha {\eta _0}/3 = {C_0}\left[{1 + \frac{{{\eta _0}({h_2} - {h_1})}}{{2{h_2}{h_1}}}} \right]$ (7)
 $l = \frac{{2{h_1}{h_2}}}{{\sqrt {3{\eta _0}\left| {{h_2} - {h_1}} \right|} }} \hspace{95pt}$ (8)

 ${U_{\rm{x}}} = \frac{{{C_0}{\eta _0}}}{{{h_1}}}\sec \! {{\rm{h}}^{\rm{2}}}\left[{\frac{{{\rm{x - C}}_{\rm{0}}^\prime }}{{\rm{l}}}} \right]$ (9)

 图 1 内波示意图 Fig.1 Schematic plot of internal wave
2.2 内波对表面波高频谱的调制

 $\frac{{\partial \psi ({\bf{k}})}}{{\partial t}} + ({c_{\rm{g}}} + U)\nabla \psi ({\bf{k}}) \\ \qquad \; = {S_{{\rm{in}}}}({\bf{k}}) + {S_{{\rm{nl}}}}({\bf{k}}) + {S_{{\rm{ds}}}}({\bf{k}}) + {S_{{\rm{cu}}}}({\bf{k}})$ (10)

 $\left[{\frac{\partial }{{\partial t}} + ({c_ \rm g} + U)\frac{\partial }{{\partial x}}} \right]\Delta \psi (k) = {S_\rm{cu}}(k)$ (11)

 ${S_{{\rm{cu}}}} = - \left( {{S_{{a b}}}\frac{{\partial {U_{b}}}}{{\partial {x_{a}}}}} \right)\psi (k)$ (12)

 \begin{aligned} {S_{\alpha \beta }}\frac{{\partial {U_\beta }}}{{\partial {\chi _\alpha }}} = \left[{\frac{{\partial u}}{{\partial x}}{{\cos }^2}\varphi + \frac{{\partial v}}{{\partial y}}{{\sin }^2}\varphi } \right. \qquad \qquad \\ \left. { + \left( {\frac{{\partial u}}{{\partial y}} + \frac{{\partial v}}{{\partial x}}} \right)\cos \varphi \sin \varphi } \right]{\Big/}2 \end{aligned} (13)

 ${S_{\alpha \beta }}{{\partial {U_\beta }} \over {\partial {\chi _\alpha }}} = - \left( {{{Co{{\cos }^2}\varphi } \over {h1l}}} \right){\eta _0} \\ \qquad \qquad \,\, \cdot \sec {h^2}\left( {{{\chi - {C^\prime }{o^t}} \over l}} \right)th\left( {{{\chi - {C^\prime }{o^t}} \over l}} \right)$ (14)

 $\Delta \psi \left( k \right) = - m3{\omega ^{ - 1}}{k^{ - 4}}{\eta _0}\left( {{{Co{{\cos }^2}\varphi } \over {h1l}}} \right) \\ \qquad \qquad \,\, \cdot \sec {h^2}\left( {{{\chi - {C^\prime }{o^t}} \over l}} \right)th\left( {{{\chi - {C^\prime }{o^t}} \over l}} \right)$ (15)
2.3 电磁散射模型 2.3.1 电磁散射场计算

 ${\bf{E}}_{{\rm{pq}}}^{{\rm{scatt}}}{\rm{(}}{{\bf{\hat k}}_{\rm{i}}}{\rm{,}}{{\bf{\hat k}}_{\rm{s}}}{\rm{)}} = 2\pi \frac{{{{\rm{e}} {ik{R_0}}}}}{{i{R_0}}}{S_{{\rm{pq}}}}{\rm{(}}{{\bf{\hat k}}_{\rm{i}}}{\rm{,}}{{\bf{\hat k}}_{\rm{s}}}{\rm{)}}$ (16)

 ${S_{{\rm{pq}}}}({{\bf{k}}_{\rm{i}}},{{\bf{k}}_{\rm{s}}}) = \frac{{{k^2}(1 - e )}}{{8{\pi ^2}}}{F_{{\rm{pq}}}}\iint\limits_{\Delta s} {\zeta ({\bf{r}}) {\rm e^{ - i({{\bf{k}}_{\rm{s}}} - {{\bf{k}}_{\rm{i}}}) \cdot {\bf{r}}}}{\rm{d}}{\bf{r}}}$ (17)

 ${F_{{\rm{vv}}}} = \frac{1}{\varepsilon }[1 + {R_{\rm{v}}}({\theta _{\rm{i}}})][1 + {R_{\rm{v}}}({\theta _{\rm{s}}})]\sin {\theta _{\rm{i}}}\sin {\theta _{\rm{s}}}\\ \quad \quad \quad - [1 - {R_{\rm{v}}}({\theta _{\rm{i}}})][1 - {R_{\rm{v}}}({\theta _{\rm{s}}})]\cos {\theta _{\rm{i}}}\cos {\theta _{\rm{s}}}\cos {\phi _{\rm{s}}}$ (18)
 ${F_{{\rm{vh}}}} = [1 - {R_{\rm{v}}}(\theta _{\rm{i}}^{})][1 + {R_{\rm{h}}}(\theta _{\rm{s}}^{})]\cos \theta _{\rm{i}}^{}\sin \phi _{\rm{s}} \hspace{35pt}$ (19)
 ${F_{{\rm{hv}}}} = [1 + {R_{\rm{h}}}(\theta _{\rm{i}}^{})][1 - {R_{\rm{v}}}(\theta _{\rm{s}}^{})]\cos \theta _{\rm{s}}^{}\sin \phi _{\rm{s}} \hspace{35pt}$ (20)
 ${F_{{\rm{hh}}}} = [1 + {R_{\rm{h}}}(\theta _{\rm{i}}^{})][1 + {R_{\rm{h}}}(\theta _{\rm{s}}^{})]\cos \phi _{\rm{s}} \hspace{60pt}$ (21)

 \begin{aligned} I( \cdot ) = \iint\limits_{\Delta s} {\zeta (r){{\rm{e}}^{ - i({k_{\rm{s}}} - {k_{\rm{i}}}) \cdot r}}} {\rm{d}}{\bf{r}} \qquad \qquad \qquad \quad \; \\ {\rm{ = }}\frac{1}{n_{z}}\int_{ - \Delta {x_{\rm{g}}}/2}^{\Delta {x_{\rm{g}}}/2} {} \int_{ - \Delta {y_{\rm{g}}}/2}^{\Delta {y_{\rm{g}}}/2} {\zeta r{{\rm{e}} ^{ - i({{\bf{R}}_{\rm{s}}} - {{\bf{R}}_{\rm{i}}}) \cdot {r}}} {\rm{d}}x{\rm{d}}y} \end{aligned} (22)

 $\zeta ({\bf{r}}) = B({{\bf{k}}_{\rm{c}}})\sin ({\bf{k}} \cdot {\bf{r}} - {\omega _{\rm{c}}}t)$ (23)

 $I\left( \cdot \right)\! = \!\frac{{B({{\bf{k}}_{\rm{c}}})}}{{2{n_z}}}{{\rm{e}}^{ - i{\bf{q}} \cdot {{\bf{r}}_0}}}\sum\limits_{n = - \infty }^\infty {{{( - i)}^n}} {J_{\rm{n}}}[{q_z}B({{\bf{k}}_c})]\mathop \smallint \nolimits_{ - \Delta {x_{\rm{g}}}/2}^{\Delta {x_{\rm{g}}}/2} \int_{ - \Delta {y_{\rm{g}}}/2}^{\Delta {y_{\rm{g}}}/2} \\ \qquad \quad {\left\{ {{{\rm{e}}^{i\left\{ {\left[ {(1 + n){k_{{\rm{c}}x}} - {q_x} - {q_z}{Z_x}} \right]{x_c} + \left[ {(1 + n){k_{{\rm{c}}y}} - {q_y} - {q_z}{Z_y}} \right]{y_{\rm{c}}}} \right\}}}} \right.} \\ \qquad \quad \cdot {{\rm{e}}^{ - i(1 + n){\omega _{\rm{c}}}t}} + \left. {{{\rm{e}}^{ - i\left\{ {\left[ {(1 - n){k_{{\rm{c}}x}} + {q_x} + {q_z}{Z_x}} \right]{x_{\rm{c}}} + \left[ {(1 - n){k_{{\rm{c}}y}} + {q_y} + {q_z}{Z_y}} \right]{y_{\rm{c}}}} \right\}}}{{\rm{e}}^{i(1 - n){\omega _{\rm{c}}}t}}} \right\}{\rm{d}}{x_{\rm{c}}}{\rm{d}}{y_{\rm{c}}}$ (24)

 $E_{{\rm{pq}}}^{{\rm{total {\tiny\_} scatt}}}{\rm{(}}{{\bf{\hat k}}_{\rm{i}}}{\rm{,}}{{\bf{\hat k}}_{\rm{s}}}{\rm{)}} = \sum\limits_{}^M {\sum\limits_{}^N {{\bf{E}}_{{\rm{pq}}}^{{\rm{scatt}}}} }$ (25)

 $\sigma _{{\rm{pq}}}^0{\rm{(}}{{\bf{\hat k}}_{\rm{i}}}{\rm{,}}{{\bf{\hat k}}_{\rm{s}}}{\rm{)}}\! =\! 4\pi R_0^2\left\langle {{\bf{E}}_{{\rm{pq}}}^{{\rm{scatt}}}{\rm{(}}{{{\bf{\hat k}}}_{\rm{i}}}{\rm{,}}{{{\bf{\hat k}}}_{\rm{s}}}{\rm{)}}{{\left[{{\bf{E}}_{{\rm{pq}}}^{{\rm{scatt}}}{\rm{(}}{{{\bf{\hat k}}}_{\rm{i}}}{\rm{,}}{{{\bf{\hat k}}}_{\rm{s}}}{\rm{)}}} \right]}^*}} \right\rangle \!{\Big/}\!\Delta S$ (26)

 \eqalign{ \sigma _{{\rm{pq}}}^0({{{\bf{\hat k}}}_{\rm{i}}},{{{\bf{\hat k}}}_{\rm{s}}}) = \pi {k^4}{\left| {\varepsilon - 1} \right|^2}{\left| {{F_{{\rm{pq}}}}} \right|^2}{1 \over {{{\left( {2\pi } \right)}^2}}} \qquad \,\,\,\,\, \\ \qquad \qquad \qquad \,\, \cdot \int\!\!\!\int {\left\langle {\zeta ({r^\prime })\zeta (r)} \right\rangle {{\rm{e}}^{ - iq \cdot ({{\bf{r}}^\prime } - {\bf{r}})}}{\rm{d}}{{\bf{r}}^\prime }{\rm{d}}{\bf{r}}} \cr} (27)

 $\psi ({\bf{k}}) = \frac{1}{{{{\left( {2\pi } \right)}^2}}}\iint {\left\langle {\zeta ({{\bf{r}}^\prime})\zeta ({\bf{r}})} \right\rangle } {{\mathop{\rm e}\nolimits} ^{ - i2k({{{\bf{\hat k}}}_{\rm{s}}} - {{{\bf{\hat k}}}_{\rm{i}}}) \cdot ({{\bf{r}}^\prime} - {\bf{r}})}}{\rm{d}}{{\bf{r}}^\prime}{\rm{d}}{\bf{r}}$ (28)

 $\sigma _{{\rm{pq}}}^0{\rm{(}}{{\bf{\hat k}}_{\rm{i}}}{\rm{,}}{{\bf{\hat k}}_{\rm{s}}}{\rm{)}} = \pi {k^4}{\left| {\varepsilon - 1} \right|^2}{\left| {{F_{{\rm{pq}}}}} \right|^2}\psi \left( {{{\bf{q}}_{\rm{1}}}} \right)$ (29)

Ψ(q1)是表面毛细波的海谱，q1是散射矢量 ${\bf{q}} = k({{\bf{\hat k}}_{\rm{s}}} - {{\bf{\hat k}}_{\rm{i}}})$ 在倾斜面元上的投影。在内波存在的情况下Ψ不再是传统计算海面的谱值，需要进行调制 $\psi = {\psi _{{\rm{sea}}}} + \Delta \psi$ ，而 $\Delta \psi$ 即为2.2节中得到的海面高频谱调制值。

2.3.3 多普勒谱计算

 $S(f) = \frac{1}{T}{\left| {\int\nolimits_0^T {{\bf{{\rm E}}}_{{\rm{pq}}}^{{\rm{total {\tiny\_} scatt}}}{\rm{(}}{{{\bf{\hat k}}}_{\rm{i}}}{\rm{,}}{{{\bf{\hat k}}}_{\rm{s}}}{\rm{,}}t{\rm{)exp(}}i2\pi ft{\rm{)}}{\rm{d}}t} } \right|^2}$ (30)

3 数值结果和分析 3.1 内波对散射系数的影响

 图 2 内波存在和不存在情况下海面散射系数分布 Fig.2 Backscattering coefficient distribution of the sea with and without internal wave

 $\Delta \sigma = \sigma - {\sigma ^0}$ (31)

 图 3 不同参数对调制深度的影响 Fig.3 Dependence of the modulation depth on different parameters

3.2 内波对多普勒谱的影响

 图 4 内波对动态海面多普勒谱的影响 Fig.4 The influence of Doppler spectra from internal wave

 图 5 不同参数对多普勒谱的影响 Fig.5 Dependence of the Doppler spectra on different parameters
4 结束语

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