海洋内波对海面电磁散射特性的影响分析
«上一篇
 文章快速检索 高级检索

 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]

[复制中文]
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

, ,

Analysis of the Scattering Characteristics of Sea Surface with the Influence from Internal Wave
, ,
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

 [1] Hughes B A and Gower J F. SAR imagery and surface truth comparisons of internal waves in Georgia Strait, British Columbia, Canada[J]. Journal of Geophysical Research Oceans, 1983, 88(C3): 1809-1824.(1) [2] Hughes B A and Dawson T W. Joint Canada-U.S. ocean wave investigation project: an overview of the Georgia strait experiment[J]. Journal of Geophysical Research Oceans, 1988, 93(C10): 12219-12234.(1) [3] Gasparovic R F and Etkin V S. An overview of the Joint US/Russia internal wave remote sensing experiment[J]. International Ceoscience and Remote Sensing Symposium, IGARSS'94, 1994, 2: 741-743.(1) [4] Alpers W. Theory of radar imaging of internal waves[J]. Nature, 1985, 314(6008): 245-247.(1) [5] West J. Correlation of Bragg scattering from the sea surface at different polarizations[J]. Waves in Random and Complex Media, 2005, 15(3): 345-403.(1) [6] Zheng Quanan, Yuan Y, and Klemas V. Theoretical expression for an ocean internal soliton synthetic aperture radar image and determination of the soliton characteristic half width[J]. Journal of Geophysical Research Oceans, 2001, 106(C12): 31415-31423.(3) [7] Brandt P, Romeiser R, and Rubino A. On the dependence of radar signatures of oceanic internal solitary waves on wind conditions and internal wave parameters[J]. 1998 IEEE International Geoscience and Remote Sensing Symposium Proceedings IGARSS'98, 1998, 3: 1662-1664.(1) [8] 李海艳. 利用合成孔径雷达研究海洋内波[D]. [硕士论文], 中 国海洋大学, 2004: 第一章, 11-19. Li Hai-yan. Studying ocean internal waves with SAR[D].[Master dissertation], Ocean University of China, 2004: Chap. 1, 11-19.(2) [9] Ouyang Yue, Chong Jin-song, and Wu Yi-rong. Simulation studies of internal waves in SAR images under different SAR and wind field conditions[J]. IEEE Transactions on Geoscience and Remote Sensing, 2011, 49(5): 1734-1743.(1) [10] Liu A K, Chang Y S, and Hsu M K. [10] Evolution of nonlinear internal waves in the East and South China Seas[J]. Journal of Geophysical Research Oceans, 1998, 103(C4): 7995-8008. (1) [11] Fuks I M. Wave diffraction by a rough boundary of an arbitrary plane-layered medium[J]. IEEE Transactions on Antennas and Propagation, 2001, 49(4): 630-639.(1) [12] 陈珲. 动态海面及其上目标复合电磁散射与多普勒谱研究[D].[博士论文], 西安电子科技大学, 2012: 第四章, 70-76. Chen Hui. A study of electromagnetic composite scattering and doppler spectra of a target at time-evolving sea surface[D]. [Ph.D. dissertation], Xidian University, 2012: Chap. 4, 70-76.(1) [13] Peterson P and Groesen E V. A direct and inverse problem for wave crests modelled by interactions of two solitons[J]. Physica D, 2001, 141(14): 316-332.(1) [14] Peterson P and Groesen E V. Sensitivity of the inverse wave crest problem[J]. Wave Motion, 2001, 34(4): 391-399.(1) [15] Peterson P, Soomere T, and Engelbrecht J. Soliton interaction as a possible model for extreme waves in shallow water[J]. Nonlinear Processes in Geophysics, 2003, 10: 503-510.(1)