二阶微扰法随机粗糙面透射特性计算方法

摘要

本发明提供了一种二阶微扰法随机粗糙面透射特性计算方法，考虑了二阶项解对粗糙面透射特性的影响，利用微扰法给出了二阶透射场、二阶透射率和二阶双向透射系数的计算方法，扩大了适用范围，提高了精度，满足微波辐射遥感定量信息反演的精度要求。

主权项

1.二阶微扰法随机粗糙面透射特性计算方法，用于地物粗糙表面的透射特性计算，以满足微波辐射遥感定量信息反演的精度要求，包含透射场、透射率和双向透射系数的计算步骤，具体为：令一平面波由介质0入射到随机粗糙面上进入介质1中，介质0的介电常数为ε₀，介质1的介电常数ε₁，其中入射波矢量入射波矢量的水平分量为对随机粗糙面建立三维直角坐标系，由z＝f(x,y)随机函数描述此粗糙面，对函数取集平均得＝0；(1)计算地物粗糙表面透射场其表示为：$\begin{array}{c}{\bar{E}}_t^{\left(2\right)}\left(\bar{r}\right)=\int d{\bar{k}}_{\perp }{e}^{i{\bar{k}}_{i\perp }·{\bar{r}}_{\perp }-{\mathrm{ik}}_{1z}z}\left\{{\hat{e}}_1\left({-k}_{1z}\right)\right[f_{\mathrm{ee}}^{t\left(2\right)}({\bar{k}}_{\perp },{\bar{k}}_{i\perp })({\hat{e}}_1\left({-k}_{1\mathrm{zi}}\right)·{\hat{e}}_i)+f_{\mathrm{eh}}^{t\left(2\right)}({\bar{k}}_{\perp },{\bar{k}}_{i\perp })({\hat{h}}_1\left({-k}_{1\mathrm{zi}}\right)·{\hat{e}}_i)\\ +{\hat{h}}_1\left({-k}_{1z}\right)[f_{\mathrm{he}}^{t\left(2\right)}({\bar{k}}_{\perp },{\bar{k}}_{i\perp })({\hat{e}}_1\left({-k}_{1\mathrm{zi}}\right)·{\hat{e}}_i)+f_{\mathrm{hh}}^{t\left(2\right)}({\bar{k}}_{\perp },{\bar{k}}_{i\perp })({\hat{h}}_1\left({-k}_{1\mathrm{zi}}\right)·{\hat{e}}_i)]\}\end{array}$其中，$\begin{array}{c}{f}_{\mathrm{ee}}^{t\left(2\right)}({\bar{k}}_{\perp },{\bar{k}}_{i\perp })=\frac{k_{\mathrm{iz}}}{k_{\mathrm{iz}}+k_{1\mathrm{iz}}}\frac{k_1^2-k^2}{k_{1z}+k_z}\{\cos ({\phi }_k-{\phi }_i)·(k_{1\mathrm{zi}}-k_z)F^{\left(2\right)}({\bar{k}}_{\perp }-{\bar{k}}_{i\perp })\\ -2(k_1^2-k^2)\int d{\bar{k}}_{\perp }^{\prime }F({\bar{k}}_{\perp }^{\prime }-{\bar{k}}_{i\perp })F({\bar{k}}_{\perp }-{\bar{k}}_{\perp }^{\prime })\\ ·[-\sin ({\phi }_k-{{\phi }_k}^{\prime })\sin ({{\phi }_k}^{\prime }-{\phi }_i)\frac{k{{}_z}_{}\prime k_{1z}\prime }{k_1^2{k}_z^{\prime }+k^2{k}_{1z}^{\prime }}+\cos ({\phi }_k-{{\phi }_k}^{\prime })\cos ({{\phi }_k}^{\prime }-{\phi }_i)\frac{1}{k_{1z}^{\prime }+k_z^{\prime }}]\}\end{array}$$\begin{array}{c}{f}_{\mathrm{eh}}^{t\left(2\right)}({\bar{k}}_{\perp },{\bar{k}}_{i\perp })=\frac{k_1^2-k^2}{k_{1z}+k_z}\frac{{\mathrm{kk}}_{\mathrm{iz}}}{{k_1}^2{k}_{\mathrm{iz}}+k^2{k}_{1\mathrm{zi}}}\{({k_1}^2-k_{1\mathrm{zi}}{k}_z)F^{\left(2\right)}({\bar{k}}_{\perp }-{\bar{k}}_{i\perp })\sin ({\phi }_k-{\phi }_i)\\ -2\int d{\bar{k}}_{\perp }^{\prime }F({\bar{k}}_{\perp }^{\prime }-{\bar{k}}_{i\perp })F({\bar{k}}_{\perp }-{\bar{k}}_{\perp }^{\prime })[\sin ({\phi }_k-{\phi }_k^{\prime })\frac{{k_1}^2{k}_{\rho }^{\prime }{k}_{\mathrm{\rho i}}}{k_{\rho }^{\prime 2}k+k_z^{\prime }{k}_{1z}^{\prime }}+\frac{(k_1^2-k^2)k_{1\mathrm{zi}}}{k_1^2{k}_z^{\prime }+k^2{k}_{1z}^{\prime }}\\ ·(\sin ({\phi }_k-{{\phi }_k}^{\prime })\cos ({{\phi }_k}^{\prime }-{\phi }_i)k_z^{\prime }{k}_{1z}^{\prime }+\cos ({\phi }_k-{{\phi }_k}^{\prime })\sin ({{\phi }_k}^{\prime }-{\phi }_i)(k_{\rho }^{\prime 2}+k_z^{\prime }{k}_{1z}^{\prime })\left]\right\}\end{array}$$\begin{array}{c}{f}_{\mathrm{he}}^{t\left(2\right)}({\bar{k}}_{\perp },{\bar{k}}_{i\perp })=\frac{k_1^2-k^2}{k_{1\mathrm{zi}}+k_{\mathrm{iz}}}\frac{k_1{k}_{\mathrm{iz}}}{k_1^2{k}_z+k^2{k}_{1z}}\{(k^2-k_{1\mathrm{zi}}{k}_z)F^{\left(2\right)}({\bar{k}}_{\perp }-{\bar{k}}_{i\perp })\sin ({\phi }_k-{\phi }_i)\\ +2\int d{\bar{k}}_{\perp }^{\prime }F({\bar{k}}_{\perp }^{\prime }-{\bar{k}}_{i\perp })F({\bar{k}}_{\perp }-{\bar{k}}_{\perp }^{\prime })[\sin ({\phi }_k^{\prime }-{\phi }_i)\frac{k_1^2{k}_{\rho }^{\prime }{k}_{\rho }}{k_{\rho }^{\prime 2}+k_z^{\prime }{k}_{1z}^{\prime }}+\frac{(k_1^2-k^2)k_z}{k_1^2{k}_z^{\prime }+k^2{k}_{1z}^{\prime }}\\ ·(k_z^{\prime }{k}_{1z}^{\prime }\sin ({\phi }_k-{{\phi }_k}^{\prime })\cos ({{\phi }_k}^{\prime }-{\phi }_i)+(k_{\rho }^{\prime 2}+k_z^{\prime }{k}_{1z}^{\prime })\cos ({\phi }_k-{{\phi }_k}^{\prime })\sin ({{\phi }_k}^{\prime }-{\phi }_i))\left]\right\}\end{array}$$\begin{array}{c}{f}_{\mathrm{hh}}^{t\left(2\right)}({\bar{k}}_{\perp },{\bar{k}}_{i\perp })=\frac{({k_1}^2-k^2)}{({k_1}^2{k}_z+k^2{k}_{1z})}\frac{k_1{k}_{1z}}{k_1^2{k}_{1z}+k^2{k}_{1\mathrm{zi}}}\{k(k_1^2{k}_z-k^2{k}_{1\mathrm{zi}})F^{\left(2\right)}({\bar{k}}_{\perp }-{\bar{k}}_{i\perp })\cos ({\phi }_k-{\phi }_i)\\ +2\int d{\bar{k}}_{\perp }^{\prime }F({\bar{k}}_{\perp }^{\prime }-{\bar{k}}_{i\perp })F({\bar{k}}_{\perp }-{\bar{k}}_{\perp }^{\prime })[{\mathrm{kk}}_z{k}_{1\mathrm{zi}}\frac{(k_1^2-k^2)}{k_z^{\prime }+k_{1z}^{\prime }}\sin ({\phi }_k-{{\phi }_k}^{\prime })\sin ({{\phi }_k}^{\prime }-{\phi }_i)\\ +\frac{1}{k_1^2{k}_z^{\prime }+k^2{k}_{1z}^{\prime }}[-k({k_1}^2-k^2)k_{\rho }{k}_{\rho }^{\prime 2}{k}_{\mathrm{\rho i}}+k_{\rho }{k}_{\prime \rho }^{\prime }{k}^3(k_z^{\prime }+k_{1z}^{\prime })k_{1\mathrm{zi}}\cos ({{\phi }_k}^{\prime }-{\phi }_i)-k_{\rho }^{\prime }{k}_{\mathrm{\rho i}}{{\mathrm{kk}}_1}^2\\ ·(k_z^{\prime }+k_{1z}^{\prime })k_z\cos ({\phi }_k-{{\phi }_k}^{\prime })-{\mathrm{kk}}_z^{\prime }{k}_{1z}^{\prime }({k_1}^2-k^2)k_z{k}_{1\mathrm{zi}}\cos ({\phi }_k-{{\phi }_k}^{\prime })\cos ({{\phi }_k}^{\prime }-{\phi }_i)\left]\right\}\end{array};$(2)计算地物粗糙表面透射率t(π-θ_i,φ_i)，其表示为：其中是随机起伏高度的相关函数的Fourier变换，$W({\bar{k}}_{\perp }-{\bar{k}}_{i\perp })=\frac{h^2{l}^2}{4\pi }\exp (-\frac{(k_{\rho }^2+k_{\mathrm{\rho i}}^2)l^2}{4}+\frac{k_{\rho }{k}_{\mathrm{\rho i}}{l}^2}{2}\cos ({\phi }_k-{\phi }_i))$(3)计算地物粗糙表面双向透射系数，其表示为：在入射波h极化，透射波h极化情况下的双向透射系数为：$\begin{array}{c}\frac{1}{4\pi }{\gamma }_{\mathrm{hh}}^t(\pi -{\theta }_t,{\phi }_t;\pi -{\theta }_i,{\phi }_i)=[\frac{k_{1\mathrm{zi}}}{k_{\mathrm{zi}}}{|1+R_{\mathrm{ho}}|}^2+2\mathrm{Re}\left(\frac{k_{1\mathrm{zi}}}{k_{\mathrm{zi}}}(1+R_{\mathrm{ho}})f_{\mathrm{ee}}^{t\left(2\right)}\left({\bar{k}}_{i\perp }\right)\right)]\delta (\cos {\theta }_t-\frac{k_{1\mathrm{zi}}}{k_1})\delta ({\phi }_t-{\phi }_i)\\ +\frac{k_1{k}_{1z}^2}{k_{1z}}W({\bar{k}}_{\perp }-{\bar{k}}_{i\perp }){\left|f_{\mathrm{ee}}^{t\left(1\right)}({\bar{k}}_{\perp },{\bar{k}}_{i\perp })\right|}^2\end{array}$在入射波v极化，透射波h极化情况下，$\frac{1}{4\pi }{\gamma }_{\mathrm{hv}}^t(\pi -{\theta }_t,{\phi }_t;\pi -{\theta }_i,{\phi }_i)=\frac{k_1{k}_{1z}^2}{k_{1z}}W({\bar{k}}_{\perp }-{\bar{k}}_{i\perp }){\left|f_{\mathrm{eh}}^{r\left(1\right)}({\bar{k}}_{\perp },{\bar{k}}_{i\perp })\right|}^2$在入射波h极化，透射波v极化情况下，$\frac{1}{4\pi }{\gamma }_{\mathrm{vh}}^t(\pi -{\theta }_t,{\phi }_t;\pi -{\theta }_i,{\phi }_i)=\frac{k_1{k}_{1z}^2}{k_{1z}}W({\bar{k}}_{\perp }-{\bar{k}}_{i\perp }){\left|f_{\mathrm{he}}^{t\left(1\right)}({\bar{k}}_{\perp },{\bar{k}}_{i\perp })\right|}^2$在入射波v极化，透射波v极化情况下，$\begin{array}{c}\frac{1}{4\pi }{\gamma }_{\mathrm{vv}}^t(\pi -{\theta }_t,{\phi }_t;\pi -{\theta }_i,{\phi }_i)=[\frac{k^2{k}_{1\mathrm{zi}}}{k_1^2{k}_{\mathrm{iz}}}{|1+R_{\mathrm{vo}}|}^2+2\mathrm{Re}\left(\frac{{\mathrm{kk}}_{1\mathrm{zi}}}{k_1{k}_{\mathrm{iz}}}(1+R_{\mathrm{vo}})f_{\mathrm{hh}}^{t\left(2\right)}\left({\bar{k}}_{i\perp }\right)\right)]\\ ·\delta (\cos {\theta }_t-\frac{k_{1\mathrm{zi}}}{k_1})\delta ({\phi }_t-{\phi }_i)+\frac{k_1{k}_{1z}^2}{k_{\mathrm{iz}}}W({\bar{k}}_{\perp }-{\bar{k}}_{i\perp }){\left|f_{\mathrm{hh}}^{t\left(1\right)}({\bar{k}}_{\perp },{\bar{k}}_{i\perp })\right|}^2\end{array}$其中a，b分别为v极化和h极化的两种情况，且表达镜向透射方向应以θ_t,φ_t的形式：$\delta (\cos {\theta }_t-\frac{k_{1\mathrm{zi}}}{k_1})\delta ({\phi }_t-{\phi }_i),$其中$\cos {\theta }_t=\frac{\sqrt{k_1^2-k^2{\sin }^2{\theta }_i}}{k_1}=\frac{k_{1\mathrm{zi}}}{k_1}$时，θ_t才指向透射方向，Re()表示对括号中的数取实部；$f_{\mathrm{ab}}^{t\left(2\right)}\left({\bar{k}}_{i\perp }\right)=<f_{\mathrm{ab}}^{t\left(2\right)}({\bar{k}}_{\perp },{\bar{k}}_{i\perp })>a,b=v,h$$\begin{array}{c}{f}_{\mathrm{ee}}^{t\left(2\right)}\left({\bar{k}}_{i\perp }\right)=\frac{k_{\mathrm{iz}}(k_1^2-k^2)}{{(k_{\mathrm{iz}}+k_{1\mathrm{zi}})}^2}\{(k_{1\mathrm{zi}}-k_{\mathrm{iz}}){\int }_{-\infty }^{\infty }d{\bar{k}}_{\perp }({\bar{k}}_{\perp }-{\bar{k}}_{i\perp })-2(k_1^2-k^2){\int }_{-\infty }^{\infty }d{\bar{k}}_{\perp }W({\bar{k}}_{\perp }-{\bar{k}}_{i\perp })\\ ·[{\sin }^2({\phi }_k-{\phi }_i)\frac{k_z{k}_{1z}}{k_1^2{k}_z+k^2{k}_{1z}}+{\cos }^2({\phi }_k-{\phi }_i)\frac{1}{k_{1z}+k_z}]\}\end{array}$$\begin{array}{c}{f}_{\mathrm{hh}}^{t\left(2\right)}\left({\bar{k}}_{i\perp }\right)=\frac{k_1({k_1}^2-k^2)k_{\mathrm{iz}}}{{({k_1}^2{k}_{\mathrm{iz}}+k^2{k}_{\mathrm{izi}})}^2}\{{\int }_{-\infty }^{\infty }d{\bar{k}}_{\perp }W({\bar{k}}_{\perp }-{\bar{k}}_{i\perp })d(k_1^2{k}_{\mathrm{iz}}-k^2{k}_{1\mathrm{zi}})+\\ 2{\int }_{-\infty }^{\infty }d{\bar{k}}_{\perp }W({\bar{k}}_{\perp }-{\bar{k}}_{i\perp })[-{\mathrm{kk}}_{\mathrm{iz}}{k}_{1\mathrm{zi}}\left(\frac{k_1^2-k^2}{k_z+k_{1z}}\right){\sin }^2({\phi }_k-{\phi }_i)+\\ \frac{1}{k_1^2{k}_z+k^2{k}_{1z}}(-k({k_1}^2-k^2)k_{\mathrm{\rho i}}^2{k}_{\rho }^2+k_{\rho }{k}_{\mathrm{\rho i}}k(k_z+k_{1z})(k^2{k}_{1\mathrm{zi}}-k_1^2{k}_{\mathrm{iz}})\\ ·\cos ({\phi }_k-{\phi }_i)-{\mathrm{kk}}_z{k}_{1z}{k}_{\mathrm{iz}}{k}_{1\mathrm{zi}}({k_1}^2-k^2){\cos }^2({\phi }_k-{\phi }_i)\left)\right]\}\end{array}$$f_{\mathrm{ee}}^{t\left(1\right)}({\bar{k}}_{\perp },{\bar{k}}_{i\perp })=\frac{({k_1}^2-k^2)}{k_z+k_{1z}}\left(\frac{{2k}_{\mathrm{iz}}}{k_{\mathrm{iz}}+k_{\mathrm{izi}}}\right)\cos ({\phi }_k-{\phi }_i),$$f_{\mathrm{he}}^{t\left(1\right)}({\bar{k}}_{\perp },{\bar{k}}_{i\perp })=\frac{k_z{k}_1(k_1^2-k^2)}{k_1^2{k}_z+k^2{k}_{1z}}\left(\frac{{2k}_{\mathrm{iz}}}{k_{\mathrm{iz}}+k_{1\mathrm{zi}}}\right)\sin ({\phi }_k-{\phi }_i),$$f_{\mathrm{eh}}^{t\left(1\right)}({\bar{k}}_{\perp },{\bar{k}}_{i\perp })=\frac{k_{\mathrm{iz}}({k_1}^2-k^2)}{k_z+k_{1z}}\left(\frac{{2\mathrm{kk}}_{1\mathrm{zi}}}{{k_1}^2{k}_{\mathrm{iz}}+k^2{k}_{1\mathrm{zi}}}\right)\sin ({\phi }_k-{\phi }_i),$$f_{\mathrm{hh}}^{t\left(1\right)}({\bar{k}}_{\perp },{\bar{k}}_{i\perp })=\frac{(k_1^2-k^2)}{k_1^2{k}_z+k^2{k}_{1z}}\left(\frac{{2\mathrm{kk}}_1{k}_{\mathrm{iz}}}{k_1^2{k}_{\mathrm{iz}}+k^2{k}_{1\mathrm{zi}}}\right)\{k_{\rho }{k}_{\mathrm{\rho i}}+k_{1\mathrm{zi}}{k}_z\cos ({\phi }_k-{\phi }_i)\},$入射波矢量${\bar{k}}_i={\bar{k}}_{\perp i}+\hat{z}{k}_{\mathrm{zi}}=\hat{x}{k}_{\mathrm{xi}}+\hat{y}{k}_{\mathrm{yi}}+\hat{z}{k}_{\mathrm{zi}},$入射波矢量的水平分量为k_ix＝ksinθ_icosφ_i，k_iy＝ksinθ_isinφ_i，k_iz＝kcosθ_i，k_ρi＝ksinθ_i，入射波水平分量的模散射波矢量$\bar{k}={\bar{k}}_{\perp }+\hat{z}{k}_z=\hat{x}{k}_x+\hat{y}{k}_y+\hat{z}{k}_z,k=\omega \sqrt{{\mu }_0{ϵ}_0},$散射波矢量的水平分量为散射波水平分量的模散射波垂直分量的模透射波矢量${\bar{k}}_1={\bar{k}}_{1\perp }+\hat{z}{k}_{1z}=\hat{x}{k}_{1x}+\hat{y}{k}_{1y}+\hat{z}{k}_{1z},k_1=\omega \sqrt{{\mu }_0{ϵ}_1},$透射波矢量的水平分量为透射波水平分量的模透射波垂直分量的模$k_{1\mathrm{zi}}=\sqrt{k_1^2+k_{i\perp }^2},$入射波的入射角θ_i和方位角透射波的透射角θ_t和方位角和分别代表原点和场点的位置矢量，当入射场为TE极化时入射场为TM极化时μ_o和μ₁分别代表介质0和介质1的磁导率，ε_o和ε₁分别代表介质0和介质1的电导率，ω为角频率，η₁为介质1的波阻抗，粗糙面的均方根高度h和相关长度l，$\hat{e}\left(k_z\right)=\hat{k}\times \hat{z}/|\hat{k}\times \hat{z}|=(1/k_{\rho })(\hat{x}{k}_y-\hat{y}{k}_x),$$\hat{e}(-k_z)=(1/k_{\rho })(\hat{x}{k}_y-\hat{y}{k}_x),$$\hat{e}(-k_{1\mathrm{zi}})={\hat{k}}_1\times \hat{z}/|{\hat{k}}_1\times \hat{z}|=(1/k_{1\mathrm{\rho i}})(\hat{x}{k}_{1\mathrm{yi}}-\hat{y}{k}_{1\mathrm{xi}}),$${\hat{e}}_1\left(k_{1z}\right)={\hat{k}}_1\times \hat{z}/|{\hat{k}}_1\times \hat{z}|=(1/k_{\rho })(\hat{x}{k}_y-\hat{y}{k}_x),$$\hat{h}\left(k_z\right)=\frac{1}{k}\hat{e}\times \hat{k}=\frac{{-k}_z}{{\mathrm{kk}}_{\rho }}(\hat{x}{k}_x+\hat{y}{k}_y)+\frac{k_{\rho }}{k}\hat{z},$$\hat{h}\left({-k}_z\right)=\frac{k_z}{{\mathrm{kk}}_{\rho }}(\hat{x}{k}_x+\hat{y}{k}_y)+\frac{k_{\rho }}{k}\hat{z},$${\hat{h}}_1\left(k_{1z}\right)=\frac{1}{k_1}{\hat{e}}_1\times {\hat{k}}_1=\frac{-k_{1z}}{k_1{k}_{\rho }}(\hat{x}{k}_x+\hat{y}{k}_y)+\frac{k_{\rho }}{k_1}\hat{z},$${\hat{h}}_1\left({-k}_{1z}\right)=\frac{k_{1z}}{k_1{k}_{\rho }}(\hat{x}{k}_{1x}+\hat{y}{k}_{1y})+\frac{k_{\rho }}{k_1}\hat{z},$${\hat{h}}_1\left({-k}_{1\mathrm{zi}}\right)=\frac{k_{1\mathrm{zi}}}{k_1{k}_{\mathrm{\rho i}}}(\hat{x}{k}_{\mathrm{xi}}+\hat{y}{k}_{\mathrm{yi}})+\frac{k_{\mathrm{\rho i}}}{k_1}\hat{z},$傅里叶变换$\frac{1}{4{\pi }^2}\int d{\bar{r}}_{\perp }{e}^{-i{\bar{k}}_{\perp }·{\bar{r}}_{\perp }}f\left({\bar{r}}_{\perp }\right)=F\left({\bar{k}}_{\perp }\right),$$\frac{1}{4{\pi }^2}\int d{\bar{r}}_{\perp }{e}^{-i{\bar{k}}_{\perp }·{\bar{r}}_{\perp }}{f}^2\left({\bar{r}}_{\perp }\right)=F^2\left({\bar{k}}_{\perp }\right),$$F^2\left({\bar{k}}_{\perp }\right)={\int }_{-\infty }^{\infty }d{\bar{k}}_{\perp }^{\prime }F\left({\bar{k}}_{\perp }^{\prime }\right)F({\bar{k}}_{\perp }-{\bar{k}}_{\perp }^{\prime }),$是TE波的菲涅尔透射系数，是TM波的菲涅尔透射系数，φ'_k，k'_ρ，k'_z和k′_1z是积分所需要的中间变量；φ_k等值为透射波的方位角φ_i等值为入射波的方位角