CN104750917A - Determination method for layered medium rough surface electromagnetic scattering coefficients - Google Patents

Abstract

The invention discloses a determination method for layered medium rough surface electromagnetic scattering coefficients. The problems that an existing technology is large in calculated amount and slow in calculated speed of the layered medium rough surface electromagnetic scattering coefficients. The determination method for the layered medium rough surface electromagnetic scattering coefficients comprises the steps of establishing an arbitrary layered medium rough surface model; establishing a model of a layered plane corresponding to the layered rough surface, calculating a zero-order electric field in the layered plane, and determining the relative dielectric constant difference between the medium layered rough surface and the medium layered plane on the basis of the two above models; determining a first-order disturbance field in the layered medium rough surface; calculating the layered medium rough surface electromagnetic scattering coefficients according to the zero-order electric field, the relative dielectric constant difference and the first-order disturbance field. The determination method for the layered medium rough surface electromagnetic scattering coefficients reduces the complexity degree of the scattering solution derivation process, is suitable for solving the electromagnetic scattering coefficients of an arbitrary layered medium rough surface, improves the analysis accuracy of background rough surface characteristics and can be used for the analysis of target electromagnetic characteristics under the earth sea background.

Description

The defining method of stratified medium Rough Surface EM Scattering coefficient

Technical field

The invention belongs to radar electromagnetic simulation technique field, relate generally to the defining method of Rough Surface EM Scattering coefficient, can be used for Electromagnetic Characters of Target under extra large background analyze in some characteristic of background extraction uneven surface.

Background technology

In natural rugged face background, there are many hierarchies being similar to oil film sea, the electromagnetic scattering research of such stratified medium uneven surface is had great importance in theoretical analysis and practical application.When airborne or spaceborne radar carries out electromagnetic surveying to the target in the backgrounds such as stratified medium uneven surface time, the electromagnetic scattering signal of stratified medium rugged face background is contained in the echoed signal of radar, by analyzing these echoed signals, and then some characteristic of layering uneven surface can be drawn.

In the past few decades, many electromagnetic simulation technique are proposed, in order to the electromagnetic scattering problems on sea, process ground, to be roughly divided into analytic method and numerical method by scholar.Numerical method is widely used because keeping higher simulation accuracy, but requires high to allocation of computer, and memory consumption is comparatively large, during the machine of at substantial simultaneously.Compared to numerical method, the advantage of analytic method consumes that internal memory is low, analysis speed fast, but analytic method is that its precision is often lower based on specific approximate, and the applicability of analytic solution and accuracy still to be tested.Usually perturbation method is adopted to the research of stratified medium Rough Surface EM Scattering, though perturbation method can obtain multi-form analytic solution based on different being similar to, but these calculating of separating mostly need tediously long mathematical derivation, lack rational physical interpretation and are only applicable to the less situation of the uneven surface number of plies, cannot the electromagnetic property of accurate analysis background uneven surface.

Summary of the invention

The object of the invention is to the deficiency overcoming prior art, a kind of method that stratified medium Rough Surface EM Scattering coefficient is determined is provided, to simplify computation process, obtain the electromagnetic scattering coefficient of any multilayer uneven surface, improve the specificity analysis accuracy of background uneven surface.

For achieving the above object, technical scheme of the present invention comprises the steps:

(1) set up any multilayer and plant by (N+1) the media plane geometric model that medium and N number of plane form, if the relative permeability of often kind of medium is 1, s kind medium relative dielectric coefficient is ε from top to bottom _s, s={0,1,2...N}, the degree of depth of m plane is-d _m, m={0,1,2...N-1}, d _mfor non-negative and d ₀=0, two adjacent interplanar distances are Δ _t=d _t-d _t-1, t={1,2...N-1}, wherein N be greater than 1 integer;

(2) the unifrequency plane wave of any polarization mode is incided first plane of described stratified medium plane geometry model from upper half-space, choose rectangular coordinate system and define incident electric fields, utilize GENERALIZED REFLECTION COEFFICIENT and transmission coefficient, calculate the zeroth order electric field in the various medium of described stratified medium plane geometry model;

(3) all planes in described stratified medium plane geometry model are all changed into produced by bivariate stochastic process uneven surface, be stratified medium uneven surface geometric model, utilize Heaviside unit-step function to represent the relative dielectric coefficient of stratified medium plane geometry model and stratified medium uneven surface geometric model, and obtain the difference of the relative dielectric coefficient of abbreviation latter two model;

(4) body perturbation theory be applied in stratified medium uneven surface geometric model, integrating step (2) and step (3), obtain the first-order perturbation field of stratified medium uneven surface:

$\left[\begin{array}{c}{E}_0^{\mathrm{sv}}\\ E_0^{\mathrm{sh}}\end{array}\right]=\mathrm{\π}{k}_0^2\frac{e^{j{k}_0{r}_0}}{r_0}\underset{m=0}{\overset{N-1}{\mathrm{\Σ}}}{\widetilde{\mathrm{\ζ}}}_m(k_{\⊥}^s-k_{\⊥}^i)\×{\widetilde{\mathrm{\α}}}^{m,m+1}(k^s,k^i)\left[\begin{array}{c}{E}_0^{\mathrm{iv}}\\ E_0^{\mathrm{ih}}\end{array}\right]$

Wherein:

${\widetilde{\mathrm{\α}}}^{m,m+1}(k^s,k^i)=\left[\begin{array}{cc}{\widetilde{\mathrm{\α}}}_{\mathrm{vv}}^{m,m+1}(k^s,k^i)& {\widetilde{\mathrm{\α}}}_{\mathrm{vh}}^{m,m+1}(k^s,k^i)\\ {\widetilde{\mathrm{\α}}}_{\mathrm{hv}}^{m,m+1}(k^s,k^i)& {\widetilde{\mathrm{\α}}}_{\mathrm{hh}}^{m,m+1}(k^s,k^i)\end{array}\right]$

$\begin{array}{c}{\widetilde{\mathrm{\α}}}_{\mathrm{vv}}^{m,m+1}(k^s,k^i)=({\mathrm{\ϵ}}_{m+1}-{\mathrm{\ϵ}}_m)\\ \×\frac{1}{{\left(k_0{\mathrm{\ϵ}}_m\right)}^2}[\frac{{\mathrm{\ϵ}}_m}{{\mathrm{\ϵ}}_{m+1}}{k}_{\⊥}^s{\mathrm{\ξ}}_{0\→m}^{+v}(k_{\⊥}^s,z)k_{\⊥}^i{\mathrm{\ξ}}_{0\→m}^{+v}(k_{\⊥}^i,z)\\ -({\hat{k}}_{\⊥}^s\·{\hat{k}}_{\⊥}^i)k_{\mathrm{zm}}^s{\mathrm{\ξ}}_{0\→m}^{-v}(k_{\⊥}^s,z)k_{\mathrm{zm}}^i{\mathrm{\ξ}}_{0\→m}^{-v}(k_{\⊥}^i,z)]\end{array}$

${\widetilde{\mathrm{\α}}}_{\mathrm{vh}}^{m,m+1}(k^s,k^i)=({\mathrm{\ϵ}}_{m+1}-{\mathrm{\ϵ}}_m)\×\hat{z}\·({\hat{k}}_{\⊥}^i\×{\hat{k}}_{\⊥}^s)\frac{k_{\mathrm{zm}}^s}{k_0{\mathrm{\ϵ}}_m}{\mathrm{\ξ}}_{0\→m}^{-v}(k_{\⊥}^s,z){\mathrm{\ξ}}_{0\→m}^{+h}(k_{\⊥}^i,z)$

${\widetilde{\mathrm{\α}}}_{\mathrm{hv}}^{m,m+1}(k^s,k^i)=({\mathrm{\ϵ}}_{m+1}-{\mathrm{\ϵ}}_m)\×\hat{z}\·({\hat{k}}_{\⊥}^i\×{\hat{k}}_{\⊥}^s){\mathrm{\ξ}}_{0\→m}^{+h}(k_{\⊥}^s,z)\frac{k_{\mathrm{zm}}^i}{k_0{\mathrm{\ϵ}}_m}{\mathrm{\ξ}}_{0\→m}^{-v}(k_{\⊥}^i,z)$

${\widetilde{\mathrm{\α}}}_{\mathrm{hh}}^{m,m+1}(k^s,k^i)=({\mathrm{\ϵ}}_{m+1}-{\mathrm{\ϵ}}_m)\×({\hat{k}}_{\⊥}^s\·{\hat{k}}_{\⊥}^i){\mathrm{\ξ}}_{0\→m}^{+h}(k_{\⊥}^s,z){\mathrm{\ξ}}_{0\→m}^{+h}(k_{\⊥}^i,z)$

Wherein: with it is first-order perturbation field in unit vector and unit vector on projection, height relief function ζ on planar layered structure _m(r _⊥) two-dimensional Fourier transform, that incident wave is in unit vector with and unit vector the component in direction, with be perpendicular to k ⁱplane on any two mutually perpendicular unit vectors, r ₀for the distance that initial point is shown up a little, the electromagnetic scattering coefficient of m layer uneven surface, by with composition, represents vertical-vertical polarization respectively, vertical-horizontal polarizes, horizontal vertical polarizes and the electromagnetic scattering coefficient of level-horizontal polarization, ε _m+1be the relative dielectric constant of m+1 layer medium, ε _mbe the relative dielectric constant of m layer medium, k ⁱand k ^sbe respectively incident wave vector and scattering wave vector, k ₀the wave number in vacuum, with for the mould that incident wave resultant scattering wave vector projects on xoy face, with for horizontal unit vector, for vertical unit vector, with be the mould that in m layer medium, incident wave resultant scattering wave vector projects in z-axis, k _zmbe the mould that the space wave number in m layer medium projects in z-axis, z is the mould projected in z-axis of position, arbitrfary point r in model, for intermediate variable, with be GENERALIZED REFLECTION COEFFICIENT and the transmission coefficient of m layer plane, { h, v} represent polarization mode to p ∈, and { i, s} represent incident and scattering mark to q ∈, and j represents imaginary unit;

(5) according to the zeroth order electric field obtained with first-order perturbation field composition total scattering field obtain the electromagnetic scattering coefficient σ (θ of stratified medium uneven surface _s):

$\mathrm{\σ}\left({\mathrm{\θ}}_s\right)=10{\lg }_{10}\left(2\mathrm{\πr}{\left|\frac{E_s}{E_i}\right|}^2\right),$

Wherein, θ _sbe the scattering angle of radar, r is the distance that true origin is put in field, E _iit is incident electric fields.

Tool of the present invention has the following advantages:

First, body perturbation theory in nonhomogeneous media is applied in the electromagnetic scattering of stratified medium uneven surface, layering scattering from rough surface field is decomposed into zeroth order electric field and first-order perturbation field, reduce the complexity of scattering solution derivation, and these analytic solution are applicable to the electromagnetic scattering coefficient solving any multilayered medium uneven surface.

The second, physical hypothesis of the present invention is reasonable, and gained analytic solution physical meaning is clear and definite, and numerical result and the method for moment of electromagnetic scattering coefficient fit like a glove, and its applicability and accuracy all can be verified.

Accompanying drawing explanation

Fig. 1 is realization flow figure of the present invention;

Fig. 2 is the plane layered model schematic diagram in the present invention;

Fig. 3 is the uneven surface hierarchical model schematic diagram in the present invention;

Fig. 4 is the comparison diagram calculating two layer medium Rough Surface EM Scattering coefficient by the present invention and existing method of moment;

Fig. 5 is the comparison diagram calculating three layers of medium rough surface electromagnetic scattering coefficient by the present invention and existing method of moment.

Embodiment

With reference to Fig. 1, specific implementation step of the present invention is as follows:

Step 1: set up stratified medium model of rough surface.

Carry out defining and suppose to any multilayer model of rough surface, this model is made up of N kind medium, (N-1) individual uneven surface as shown in Figure 3, and often kind of medium is all uniform and its relative permeability is 1, and m kind medium relative dielectric constant is ε from top to bottom _m, m={0,1,2...N}, the degree of depth of m uneven surface is-d _m, m={0,1,2...N-1}, d _mfor non-negative and d ₀mean distance between=0, two uneven surfaces is Δ _m, m={1,2...N-1}, be considered as planar medium hierarchy and add small body disturbance by model.

Step 2: determine zeroth order electric field in stratified medium plane.

2a) by shown in Fig. 2, the unifrequency plane wave polarized arbitrarily is incided plane layered medium from upper half-space, by the unifrequency plane wave that polarizes arbitrarily with incident angle incide plane layered medium, in local rectangular coordinate system from upper half-space under, incident electric fields is expressed as follows:

$E_0^i\left(r\right)=[E_0^{\mathrm{ih}}{\hat{h}}_0^i+E_0^{\mathrm{iv}}{\hat{v}}_0^i]e^{j{k}_{\⊥}^i\·r_{\⊥}}{e}^{-j{k}_{z0}^{i}z}$

Wherein r=(r _⊥, z) be locus coordinate, that incident wave is in unit vector with and unit vector the component in direction, and with be perpendicular to plane on any two mutually perpendicular unit vectors, incident wave line of propagation is by incident wave vector determine: the two-dimensional projection of incident wave vector in z=0 plane, the unit vector of incident wave vector on z=0 planar projecting direction, the projection of incident wave vector in z-axis, for with the angle of z-axis forward, for with the angle of x-axis forward, under representing overall rectangular coordinate system respectively, the unit vector on xyz tri-directions.

2b) theoretical according to the electric wave of propagated forward matrix and GENERALIZED REFLECTION COEFFICIENT, introduce GENERALIZED REFLECTION COEFFICIENT and transmission coefficient, use respectively with represent,

be at (u-1) individual plane place, (u-1) plant the ratio of upward traveling wave and down going wave in medium, u kind medium internal electric field and the ratio of incident electric fields, with be the Fresnel reflection and transmission coefficients in (u-1) individual plane, { h, v} represent polarization mode to p ∈, k _zube the mould that in u layer medium, space wave number is projecting in z-axis, Δ u is two adjacent interplanar distances;

2c) integrating step (2a) and step (2b) use backstepping method, calculate the zeroth order electric field in the various medium of stratified medium plane geometry model:

$\begin{array}{c}{E}_t^{\left(0\right)}\left(r\right)=e^{j{k}_{\⊥}^i\·r_{\⊥}}[{\hat{h}}_0^i{\mathrm{\ξ}}_{0\→t}^{+h}(k_{\⊥}^i,z)E_0^{\mathrm{ih}}\\ +{\hat{k}}_{\⊥}^i\frac{k_{\mathrm{zt}}^i}{k_0{\mathrm{\ϵ}}_t}{\mathrm{\ξ}}_{0\→t}^{-v}(k_{\⊥}^i,z)E_0^{\mathrm{iv}}+\hat{z}\frac{k_{\⊥}^i}{k_0{\mathrm{\ϵ}}_t}{\mathrm{\ξ}}_{0\→t}^{+v}(k_{\⊥}^i,z)E_0^{\mathrm{iv}}]\end{array},$

Wherein, for intermediate variable, with be GENERALIZED REFLECTION COEFFICIENT and the transmission coefficient of t layer plane, { h, v} represent polarization mode to p ∈, k _ztbe the mould that in t layer medium, space wave number projects on xoy face, d _trepresent the modulus value of t plane depth, t={1,2,3...N-1}.

Step 3: the difference determining dielectric stratifying uneven surface and dielectric stratifying plane relative dielectric constant.

Heaviside unit-step function 3a) is utilized to represent stratified medium plane geometry model relative dielectric coefficient, this coefficient ε ⁽⁰⁾r () is expressed as form:

${\mathrm{\ϵ}}^{\left(0\right)}\left(r\right)={\mathrm{\ϵ}}^{\left(0\right)}\left(z\right)={\mathrm{\ϵ}}_0+\underset{m=0}{\overset{N-1}{\mathrm{\Σ}}}({\mathrm{\ϵ}}_{m+1}-{\mathrm{\ϵ}}_m)u(-z-d_m),$

Wherein u () is Heaviside unit-step function, dielectric coefficient ε ⁽⁰⁾z in (), namely subscript 0 represents that this dielectric coefficient is flat dielectric coefficient, z represents the locus coordinate in z-axis, ε _mrepresent the relative dielectric constant of m layer medium, ε ₀represent the relative dielectric coefficient of the superiors, d _mbe the modulus value of m plane depth, m={0,1,2...N-1};

3b) stratified medium uneven surface is considered as the disturbance on planar layered structure, this disturbance height relief ζ _m(x, y)=ζ _m(r _⊥) represent, produce by a 2D stochastic process, stratified medium uneven surface geometric model relative dielectric coefficient ε (r at this moment _⊥, z) form is as follows:

$\mathrm{\ϵ}(r_{\⊥},z)={\mathrm{\ϵ}}_0+\underset{m=0}{\overset{N-1}{\mathrm{\Σ}}}({\mathrm{\ϵ}}_{m+1}-{\mathrm{\ϵ}}_m)u(-z-d_m+{\mathrm{\ζ}}_m\left(r_{\⊥}\right))$

Wherein (r _⊥, z) be locus coordinate, r _⊥for the position coordinates on xoy face, z represents the locus coordinate in z-axis, ε _mrepresent the relative dielectric constant of m layer medium, ε _m+1represent the relative dielectric constant of m+1 layer medium, ε ₀represent the relative dielectric coefficient of the superiors, d _mbe the modulus value of m plane depth, ζ _m(r _⊥) represent height relief function in plane;

3c) according to the character of Heaviside function, obtain the difference of the relative dielectric coefficient of stratified medium uneven surface and stratified medium plane two kinds of models, this value of delta ε (r _⊥, z) representation formula is as follows:

$\mathrm{\δ\ϵ}(r_{\⊥},z)=\mathrm{\ϵ}(r_{\⊥},z)-{\mathrm{\ϵ}}^{\left(0\right)}\left(z\right)\≅\underset{m=0}{\overset{N-1}{\mathrm{\Σ}}}({\mathrm{\ϵ}}_{m+1}-{\mathrm{\ϵ}}_m){\mathrm{\ζ}}_m\left(r_{\⊥}\right)\mathrm{\δ}(-z-d_m),$

Wherein δ () is Dirac delta function, (r _⊥, z) be locus coordinate, r _⊥for the position coordinates on xoy face, z represents the locus coordinate in z-axis, ε _mrepresent the relative dielectric constant of m layer medium, ε _m+1represent the relative dielectric constant of m+1 layer medium, d _mbe the modulus value of m plane depth, ζ _m(r _⊥) represent height relief function in plane, m={0,1,2...N-1}.

Step 4: determine the first-order perturbation field in stratified medium uneven surface.

4a) by far field currents source the radiated electric field produced is radiated on plane layered medium, and this radiated electric field is the incident field on plane layered medium, and the representation formula of incident field is as follows:

${\stackrel{\‾}{E}}_0^i\left(r\right)=e^{-j{k}_{\⊥}^s\·r_{\⊥}}{e}^{-j{k}_{z0}^{s}z}{\stackrel{\‾}{E}}_0^i\hat{t},$

Wherein: r _⊥for the position coordinates on xoy face, z represents the locus coordinate in z-axis, for the two-dimensional projection of incident wave vector in z=0 plane, for incident wave vector projection in a z-direction, incident field amplitude is: k ₀represent the wave beam in vacuum, η ₀for free space wave impedance, r _sfor the distance that initial point is shown up a little, be the unit vector of any direction, subscript i and s represents incident and scattering mark respectively, and the unification in order to guarantor unit makes current source current intensity J=1Am;

4b) pass through incident field obtain the non-Perturbation in plane layered medium its representation formula is as follows:

$\begin{array}{c}{\stackrel{\‾}{E}}_m^{\left(0\right)}\left(r\right)=e^{j{k}_{\⊥}^s\·r_{\⊥}}[{\hat{h}}_0^s{\mathrm{\ξ}}_{0\→m}^{+h}(k_{\⊥}^s,z){\stackrel{\‾}{E}}_0^{\mathrm{ih}}\\ -{\hat{k}}_{\⊥}^s\frac{k_{\mathrm{zm}}^s}{k_0{\mathrm{\ϵ}}_m}{\mathrm{\ξ}}_{0\→m}^{-v}(k_{\⊥}^s,z){\stackrel{\‾}{E}}_0^{\mathrm{iv}}+\hat{z}\frac{k_{\⊥}^s}{k_0{\mathrm{\ϵ}}_m}{\mathrm{\ξ}}_{0\→m}^{+v}(k_{\⊥}^s,z){\stackrel{\‾}{E}}_0^{\mathrm{iv}}]\end{array},$

Wherein for incident wave vector two-dimensional projection in z=0 plane, represent modulus value, represent direction for intermediate variable, with be GENERALIZED REFLECTION COEFFICIENT and the transmission coefficient of m layer plane respectively, { h, v} represent polarization mode to p ∈, k _zmbe the mould that in m layer medium, space wave number projects on xoy face, d _mand d _m-1represent the modulus value of m and m-1 plane depth, ε _mrepresent the relative dielectric constant of m layer medium, m={1,2,3...N-1}, with that incident wave exists with the component in direction, with be perpendicular to plane on any two mutually perpendicular unit vectors, subscript i and s represents incident and scattering mark respectively;

4c) according to reciprocity principle, by the zeroth order electric field in medium the difference δ ε (r of the relative dielectric coefficient of stratified medium uneven surface and stratified medium plane two kinds of models _⊥, z), the non-Perturbation that far field currents source produces in plane layered medium bring first-order perturbation field into in expression formula, obtain first-order perturbation field matrix form as follows:

$\left[\begin{array}{c}{E}_0^{\mathrm{sv}}\\ E_0^{\mathrm{sh}}\end{array}\right]=\mathrm{\π}{k}_0^2\frac{e^{j{k}_0{r}_0}}{r_0}\underset{m=0}{\overset{N-1}{\mathrm{\Σ}}}{\widetilde{\mathrm{\ζ}}}_m(k_{\⊥}^s-k_{\⊥}^i)\×{\widetilde{\mathrm{\α}}}^{m,m+1}(k^s,k^i)\left[\begin{array}{c}{E}_0^{\mathrm{iv}}\\ E_0^{\mathrm{ih}}\end{array}\right]$

Wherein, ${\widetilde{\mathrm{\α}}}^{m,m+1}(k^s,k^i)=\left[\begin{array}{cc}{\widetilde{\mathrm{\α}}}_{\mathrm{vv}}^{m,m+1}(k^s,k^i)& {\widetilde{\mathrm{\α}}}_{\mathrm{vh}}^{m,m+1}(k^s,k^i)\\ {\widetilde{\mathrm{\α}}}_{\mathrm{hv}}^{m,m+1}(k^s,k^i)& {\widetilde{\mathrm{\α}}}_{\mathrm{hh}}^{m,m+1}(k^s,k^i)\end{array}\right]$

$\begin{array}{c}{\widetilde{\mathrm{\α}}}_{\mathrm{vv}}^{m,m+1}(k^s,k^i)=({\mathrm{\ϵ}}_{m+1}-{\mathrm{\ϵ}}_m)\\ \×\frac{1}{{\left(k_0{\mathrm{\ϵ}}_m\right)}^2}[\frac{{\mathrm{\ϵ}}_m}{{\mathrm{\ϵ}}_{m+1}}{k}_{\⊥}^s{\mathrm{\ξ}}_{0\→m}^{+v}\left(k_{\⊥}^s\right)k_{\⊥}^i{\mathrm{\ξ}}_{0\→m}^{+v}\left(k_{\⊥}^i\right)\\ -({\hat{k}}_{\⊥}^s\·{\hat{k}}_{\⊥}^i)k_{\mathrm{zm}}^s{\mathrm{\ξ}}_{0\→m}^{-v}\left(k_{\⊥}^s\right)k_{\mathrm{zm}}^i{\mathrm{\ξ}}_{0\→m}^{-v}\left(k_{\⊥}^i\right)]\end{array}$

${\widetilde{\mathrm{\α}}}_{\mathrm{vh}}^{m,m+1}(k^s,k^i)=({\mathrm{\ϵ}}_{m+1}-{\mathrm{\ϵ}}_m)\×\hat{z}\·({\hat{k}}_{\⊥}^i\×{\hat{k}}_{\⊥}^s)\frac{k_{\mathrm{zm}}^s}{k_0{\mathrm{\ϵ}}_m}{\mathrm{\ξ}}_{0\→m}^{-v}\left(k_{\⊥}^s\right){\mathrm{\ξ}}_{0\→m}^{+h}\left(k_{\⊥}^i\right)$

${\widetilde{\mathrm{\α}}}_{\mathrm{hv}}^{m,m+1}(k^s,k^i)=({\mathrm{\ϵ}}_{m+1}-{\mathrm{\ϵ}}_m)\×\hat{z}\·({\hat{k}}_{\⊥}^i\×{\hat{k}}_{\⊥}^s){\mathrm{\ξ}}_{0\→m}^{+h}\left(k_{\⊥}^s\right)\frac{k_{\mathrm{zm}}^i}{k_0{\mathrm{\ϵ}}_m}{\mathrm{\ξ}}_{0\→m}^{-v}\left(k_{\⊥}^i\right)$

${\widetilde{\mathrm{\α}}}_{\mathrm{hh}}^{m,m+1}(k^s,k^i)=({\mathrm{\ϵ}}_{m+1}-{\mathrm{\ϵ}}_m)\×({\hat{k}}_{\⊥}^s\×{\hat{k}}_{\⊥}^i){\mathrm{\ξ}}_{0\→m}^{+h}\left(k_{\⊥}^s\right){\mathrm{\ξ}}_{0\→m}^{+h}\left(k_{\⊥}^i\right)$

Wherein: with it is first-order perturbation field in unit vector and unit vector on projection, height relief function ζ on planar layered structure _m(r _⊥) two-dimensional Fourier transform, that incident wave is in unit vector and unit vector the component in direction, with be perpendicular to k ⁱplane on any two mutually perpendicular unit vectors, r ₀for the distance that initial point is shown up a little, the electromagnetic scattering coefficient of m layer uneven surface, by with composition, represents vertical-vertical polarization respectively, vertical-horizontal polarizes, horizontal vertical polarizes and the electromagnetic scattering coefficient of level-horizontal polarization, ε _m+1be the relative dielectric constant of m+1 layer medium, ε _mbe the relative dielectric constant of m layer medium, k ⁱand k ^sbe respectively incident wave vector and scattering wave vector, k ₀the wave number in vacuum, with for the mould that incident wave resultant scattering wave vector projects on xoy face, with for horizontal unit vector, for vertical unit vector, with be the mould that in m layer medium, incident wave resultant scattering wave vector projects in z-axis, k _zmbe the mould that the space wave number in m layer medium projects in z-axis, z is the mould projected in z-axis of position, arbitrfary point r in model, for intermediate variable, with be GENERALIZED REFLECTION COEFFICIENT and the transmission coefficient of m layer plane, { h, v} represent polarization mode to p ∈, and { i, s} represent incident and scattering mark to q ∈.

Step 5: the electromagnetic scattering coefficient calculating stratified medium uneven surface.

According to the zeroth order electric field obtained with first-order perturbation field composition total scattering field obtain the electromagnetic scattering coefficient σ (θ of stratified medium uneven surface _s):

$\mathrm{\σ}\left({\mathrm{\θ}}_s\right)=10{\lg }_{10}\left(2\mathrm{\πr}{\left|\frac{E_s}{E_i}\right|}^2\right),$

Wherein, θ _sbe the scattering angle of radar, r is the distance that true origin is put in field, E _iit is incident electric fields.

Effect of the present invention can be further illustrated by following test

In natural rugged face background, there are many hierarchies being similar to oil film sea, significant in theoretical analysis and practical application to the electromagnetic scattering research of this layering uneven surface.The present invention is a kind of method that stratified medium Rough Surface EM Scattering coefficient is determined, can solve the electromagnetic scattering problems of any multilayered medium uneven surface by the method.Experimental facilities required for the present invention is a computing machine with Fortran language compilation environment.

Experiment one:

The centre frequency selecting radar incident wave is 1GHz, incidence angle θ _i=30 °, uneven surface number of plies N=2, the relative dielectric constant of three kinds of media is from top to bottom ε successively ₀=(1.0,0.0), ε ₁=(3.0,0.0), ε ₂=(9.5,0.00055), uneven surface is the gaussian random uneven surface that Monte Carlo method produces, and sampling interval is λ/10, and sampling number is 512, supposes that the root-mean-square height of ground floor and second layer uneven surface is equal with persistence length: δ ₀=δ ₁=δ=0.03 λ, l ₀=l ₁=l=0.5 λ, between two-layer uneven surface, mean distance is Δ ₁=1.15 λ, λ are incident wavelength.

Electromagnetic scattering coefficient after averaging with 100 samples that the present invention and existing method of moment calculate two layer medium uneven surface respectively under the above parameters, two kinds of method result of calculations are to such as Fig. 4, wherein Fig. 4 (a) represents the electromagnetic scattering coefficient under level-horizontal polarization, and Fig. 4 (a) represents the electromagnetic scattering coefficient under vertical-vertical polarization.

As can be seen from Figure 4, the present invention and existing method of moment are coincide better; In computing time, the computing time of the inventive method is 1.4 points, and the computing time of method of moment is 120 points, demonstrates in the electromagnetic scattering calculating two layer medium uneven surface, the present invention not only ensure that its applicability and accuracy, and greatly shortens computing time.

Experiment two:

The centre frequency selecting radar incident wave is 1GHz, incidence angle θ _i=30 °, uneven surface number of plies N=3, four kinds of medium relative dielectric constants are from top to bottom ε successively ₀=(1.0,0.0), ε ₁=(2.5,0.0), ε ₂=(3.5,0.0), ε ₃=(5.5,0.0), three uneven surfaces are all the random Gaussian uneven surfaces produced by Monte Carlo method, and sampling interval is λ/10, sampling number is 512, supposes that the root-mean-square height of ground floor, the second layer and third layer uneven surface is equal with persistence length: δ ₀=δ ₁=δ ₂=δ=0.03 λ, l ₀=l ₁=l ₂=l=0.5 λ.Mean distance Δ between ground floor uneven surface and second layer uneven surface ₁=1.15 λ, mean distance Δ between second layer uneven surface and third layer uneven surface ₂=7 λ, λ are incident wavelength.

Electromagnetic scattering coefficient after averaging with 100 samples that the present invention and existing method of moment calculate three layers of medium rough surface respectively under the above parameters, two kinds of method result of calculations are to such as Fig. 5, wherein Fig. 5 (a) represents the electromagnetic scattering coefficient under level-horizontal polarization, and Fig. 5 (a) represents the electromagnetic scattering coefficient under vertical-vertical polarization.

As can be seen from Figure 5, the present invention and existing method of moment are coincide better, and demonstrate in the electromagnetic scattering of calculating three layers of medium rough surface, the inventive method has applicability and accuracy.

More than describing is only two instantiations of the present invention; obviously for the professional in this area; after having understood content of the present invention and principle; can carry out the various correction in form and in details and change, but these corrections based on inventive concept and change are still within claims of the present invention.

Claims (6)

1. a defining method for stratified medium Rough Surface EM Scattering coefficient, comprises the steps:

(1) set up any multilayer and plant by (N+1) the media plane geometric model that medium and N number of plane form, if the relative permeability of often kind of medium is 1, s kind medium relative dielectric coefficient is ε from top to bottom _s, s={0,1,2...N}, the degree of depth of m plane is-d _m, m={0,1,2...N-1}, d _mfor non-negative and d ₀=0, two adjacent interplanar distances are Δ _t=d _t-d _t-1, t={1,2...N-1}, wherein N be greater than 1 integer;

(2) the unifrequency plane wave of any polarization mode is incided first plane of described stratified medium plane geometry model from upper half-space, choose rectangular coordinate system and define incident electric fields, utilize GENERALIZED REFLECTION COEFFICIENT and transmission coefficient, calculate the zeroth order electric field in the various medium of described stratified medium plane geometry model;

(3) all planes in described stratified medium plane geometry model are all changed into produced by bivariate stochastic process uneven surface, be stratified medium uneven surface geometric model, utilize Heaviside unit-step function to represent the relative dielectric coefficient of stratified medium plane geometry model and stratified medium uneven surface geometric model, and obtain the difference of the relative dielectric coefficient of abbreviation latter two model;

(4) body perturbation theory be applied in stratified medium uneven surface geometric model, utilize reciprocity principle, integrating step (2) and step (3), obtain the first-order perturbation field of stratified medium uneven surface:

$\left[\begin{array}{c}{E}_0^{\mathrm{sv}}\\ E_0^{\mathrm{sh}}\end{array}\right]=\mathrm{\π}{k}_0^2\frac{e^{j{k}_0{r}_0}}{r_0}\underset{m=0}{\overset{N-1}{\mathrm{\Σ}}}{\widetilde{\mathrm{\ζ}}}_m(k_{\⊥}^s-k_{\⊥}^i)\×{\widetilde{\mathrm{\α}}}^{m,m+1}(k^s,k^i)\left[\begin{array}{c}{E}_0^{\mathrm{iv}}\\ E_0^{\mathrm{ih}}\end{array}\right]$

Wherein:

${\widetilde{\mathrm{\α}}}^{m,m+1}(k^s,k^i)=\left[\begin{array}{cc}{\widetilde{\mathrm{\α}}}_{\mathrm{vv}}^{m,m+1}(k^s,k^i)& {\widetilde{\mathrm{\α}}}_{\mathrm{vh}}^{m,m+1}(k^s,k^i)\\ {\widetilde{\mathrm{\α}}}_{\mathrm{hv}}^{m,m+1}(k^s,k^i)& {\widetilde{\mathrm{\α}}}_{\mathrm{hh}}^{m,m+1}(k^s,k^i)\end{array}\right]$

$\begin{array}{c}{\widetilde{\mathrm{\α}}}_{\mathrm{vv}}^{m,m+1}(k^s,k^i)=({\mathrm{\ϵ}}_{m+1}-{\mathrm{\ϵ}}_m)\\ \×\frac{1}{{\left(k_0{\mathrm{\ϵ}}_m\right)}^2}[\frac{{\mathrm{\ϵ}}_m}{{\mathrm{\ϵ}}_{m+1}}{k}_{\⊥}^s{\mathrm{\ξ}}_{0\→m}^{+v}(k_{\⊥}^s,z)k_{\⊥}^i{\mathrm{\ξ}}_{0\→m}^{+v}(k_{\⊥}^i,z)\\ -({\hat{k}}_{\⊥}^s\·{\hat{k}}_{\⊥}^i)k_{\mathrm{zm}}^s{\mathrm{\ξ}}_{0\→m}^{-v}(k_{\⊥}^s,z)k_{\mathrm{zm}}^i{\mathrm{\ξ}}_{0\→m}^{-v}(k_{\⊥}^i,z)]\end{array}$

${\widetilde{\mathrm{\α}}}_{\mathrm{vh}}^{m,m+1}(k^s,k^i)=({\mathrm{\ϵ}}_{m+1}-{\mathrm{\ϵ}}_m)\×\hat{z}\·({\hat{k}}_{\⊥}^i\×{\hat{k}}_{\⊥}^s)\frac{k_{\mathrm{zm}}^s}{k_0{\mathrm{\ϵ}}_m}{\mathrm{\ξ}}_{0\→m}^{-v}(k_{\⊥},z){\mathrm{\ξ}}_{0\→m}^{+h}(k_{\⊥}^i,z)$

${\widetilde{\mathrm{\α}}}_{\mathrm{hv}}^{m,m+1}(k^s,k^i)=({\mathrm{\ϵ}}_{m+1}-{\mathrm{\ϵ}}_m)\×\hat{z}\·({\hat{k}}_{\⊥}^i\×{\hat{k}}_{\⊥}^s){\mathrm{\ξ}}_{0\→m}^{+h}(k_{\⊥}^s,z)\frac{k_{\mathrm{zm}}^i}{k_0{\mathrm{\ϵ}}_m}{\mathrm{\ξ}}_{0\→m}^{-v}(k_{\⊥}^i,z)$

${\widetilde{\mathrm{\α}}}_{\mathrm{hh}}^{m,m+1}(k^s,k^i)=({\mathrm{\ϵ}}_{m+1}-{\mathrm{\ϵ}}_m)\×({\hat{k}}_{\⊥}^s\×{\hat{k}}_{\⊥}^i){\mathrm{\ξ}}_{0\→m}^{+h}(k_{\⊥}^s,z){\mathrm{\ξ}}_{0\→m}^{+h}(k_{\⊥}^i,z)$

Wherein: with it is single order scattered field (r ₀) in unit vector and unit vector on projection, height relief function ζ on planar layered structure _m(r _⊥) two-dimensional Fourier transform, that incident wave is in unit vector with and unit vector the component in direction, with be perpendicular to k ⁱplane on any two mutually perpendicular unit vectors, r ₀for the distance that initial point is shown up a little, the electromagnetic scattering coefficient of m layer uneven surface, by with composition, represents vertical-vertical polarization respectively, vertical-horizontal polarizes, horizontal vertical polarizes and the electromagnetic scattering coefficient of level-horizontal polarization, ε _m+1be the relative dielectric constant of m+1 layer medium, ε _mbe the relative dielectric constant of m layer medium, k ⁱand k ^sbe respectively incident wave vector and scattering wave vector, k ₀the wave number in vacuum, with for the mould that incident wave resultant scattering wave vector projects on xoy face, with for horizontal unit vector, for vertical unit vector, with be the mould that in m layer medium, incident wave resultant scattering wave vector projects in z-axis, k _zmbe the mould that the space wave number in m layer medium projects in z-axis, z is the mould projected in z-axis of position, arbitrfary point r in model, for intermediate variable, with be GENERALIZED REFLECTION COEFFICIENT and the transmission coefficient of m layer plane, { h, v} represent polarization mode to p ∈, and { i, s} represent incident and scattering mark to q ∈, and j represents imaginary unit;

(5) according to the zeroth order electric field obtained with first-order perturbation field composition total scattering field obtain the electromagnetic scattering coefficient σ (θ of stratified medium uneven surface _s):

$\mathrm{\σ}\left({\mathrm{\θ}}_s\right)={101g}_{10}\left(2\mathrm{\πr}{\left|\frac{E_s}{E_i}\right|}^2\right),$

Wherein, θ _sbe the scattering angle of radar, r is the distance that true origin is put in field, E _iit is incident electric fields.

2. the defining method of stratified medium Rough Surface EM Scattering coefficient according to claim 1, it is characterized in that: described in step (2), utilize GENERALIZED REFLECTION COEFFICIENT and transmission coefficient, calculate the zeroth order electric field in the various medium of described stratified medium plane geometry model, carry out as follows:

(2a) by the unifrequency plane wave that polarizes arbitrarily with incident angle incide plane layered medium, in local rectangular coordinate system from upper half-space under, incident electric fields is expressed as follows:

$E_0^i\left(r\right)=[E_0^{\mathrm{ih}}{\hat{h}}_0^i+E_0^{\mathrm{iv}}{\hat{v}}_0^i]e^{j{k}_{\⊥}^i\·r_{\⊥}}{e}^{-j{k}_{z{0}^z}^i}$

Wherein r=(r _⊥, z) be locus coordinate, that incident wave is in unit vector with and unit vector the component in direction, and with be perpendicular to plane on any two mutually perpendicular unit vectors, incident wave line of propagation is by incident wave vector determine: the two-dimensional projection of incident wave vector in z=0 plane, the unit vector of incident wave vector on z=0 planar projecting direction, the projection of incident wave vector in z-axis, for with the angle of z-axis forward, for with the angle of x-axis forward, under representing overall rectangular coordinate system respectively, the unit vector on xyz tri-directions.

(2b) introduce GENERALIZED REFLECTION COEFFICIENT and transmission coefficient, use respectively with represent,

be at (u-1) individual plane place, (u-1) plant the ratio of upward traveling wave and down going wave in medium, u kind medium internal electric field and the ratio of incident electric fields, with be the Fresnel reflection and transmission coefficients in (u-1) individual plane, { h, v} represent polarization mode to p ∈, k _zube the mould that in u layer medium, space wave number is projecting in z-axis, Δ u is two adjacent interplanar distances;

(2c) integrating step (2a) and step (2b) use backstepping method, and the zeroth order electric field obtained in t kind medium is:

$\begin{array}{c}{E}_t^{\left(0\right)}\left(r\right)=e^{j{k}_{\⊥}^i\·r_{\⊥}}[{\hat{h}}_0^i{\mathrm{\ξ}}_{0\→t}^{+h}(k_{\⊥}^i,z)E_0^{\mathrm{ih}}\\ +{\hat{k}}_{\⊥}^i\frac{k_{\mathrm{zt}}^i}{k_0{\mathrm{\ϵ}}_t}{\mathrm{\ξ}}_{0\→t}^{-v}(k_{\⊥}^i,z)E_0^{\mathrm{iv}}+\hat{z}\frac{k_{\⊥}^i}{k_0{\mathrm{\ϵ}}_t}{\mathrm{\ξ}}_{0\→t}^{+v}(k_{\⊥}^i,z)E_0^{\mathrm{iv}}]\end{array}$

Wherein, for intermediate variable, with be GENERALIZED REFLECTION COEFFICIENT and the transmission coefficient of t layer plane, { h, v} represent polarization mode to p ∈, k _ztbe the mould that in t layer medium, space wave number projects on xoy face, d _trepresent the modulus value of t plane depth, t={1,2,3...N-1}.

3. the defining method of stratified medium Rough Surface EM Scattering coefficient according to claim 1, it is characterized in that: utilize Heaviside unit-step function to represent stratified medium plane geometry model relative dielectric coefficient in described step (3), its representation formula is as follows:

${\mathrm{\ϵ}}^{\left(0\right)}\left(z\right)={\mathrm{\ϵ}}_0+\underset{m=0}{\overset{N-1}{\mathrm{\Σ}}}({\mathrm{\ϵ}}_{m+1}-{\mathrm{\ϵ}}_m)u(-z-d_m),$

Wherein u () is Heaviside unit-step function, dielectric coefficient ε ⁽⁰⁾z in (), namely subscript 0 represents that this dielectric coefficient is flat dielectric coefficient, z represents the locus coordinate in z-axis, ε _mrepresent the relative dielectric constant of m layer medium, ε ₀represent the relative dielectric coefficient of the superiors, d _mbe the modulus value of m plane depth, m={0,1,2...N-1}.

4. the defining method of stratified medium Rough Surface EM Scattering coefficient according to claim 1, it is characterized in that: utilize Heaviside unit-step function to represent stratified medium uneven surface geometric model relative dielectric coefficient in described step (3), its representation formula is as follows:

$\mathrm{\ϵ}(r_{\⊥},z)={\mathrm{\ϵ}}_0+\underset{m=0}{\overset{N-1}{\mathrm{\Σ}}}({\mathrm{\ϵ}}_{m+1}-{\mathrm{\ϵ}}_m)u(-z-d_m+{\mathrm{\ζ}}_m\left(r_{\⊥}\right))$

Wherein (r _⊥, z) be locus coordinate, r _⊥for the position coordinates on xoy face, z represents the locus coordinate in z-axis, ε _mrepresent the relative dielectric constant of m layer medium, ε _m+1represent the relative dielectric constant of m+1 layer medium, ε ₀represent the relative dielectric coefficient of the superiors, d _mbe the modulus value of m plane depth, ζ _m(r _⊥) represent height relief function in plane.

5. the defining method of stratified medium Rough Surface EM Scattering coefficient according to claim 1, it is characterized in that: according to the character of Heaviside function in described step (3), obtain the difference δ ε (r of the relative dielectric coefficient of stratified medium uneven surface and stratified medium plane two kinds of models after abbreviation _⊥, z), its representation formula is as follows:

$\mathrm{\δ\ϵ}(r_{\⊥},z)=\mathrm{\ϵ}(r_{\⊥},z)-{\mathrm{\ϵ}}^{\left(0\right)}\left(z\right)\widetilde{=}\underset{m=0}{\overset{N-1}{\mathrm{\Σ}}}({\mathrm{\ϵ}}_{m+1}-{\mathrm{\ϵ}}_m){\mathrm{\ζ}}_m\left(r_{\⊥}\right)\mathrm{\δ}(-z-d_m),$

δ () is Dirac delta function, wherein (r _⊥, z) be locus coordinate, r _⊥for the position coordinates on xoy face, z represents the locus coordinate in z-axis, ε _mrepresent the relative dielectric constant of m layer medium, ε _m+1represent the relative dielectric constant of m+1 layer medium, d _mbe the modulus value of m plane depth, ζ _m(r _⊥) represent height relief function in plane, wherein m={0,1,2...N-1}.

6. the defining method of stratified medium Rough Surface EM Scattering coefficient according to claim 1, is characterized in that: described step (4) is fallen into a trap the first-order perturbation field of point counting layer medium rough surface, carries out as follows:

(6a) by far field currents source the radiated electric field produced is radiated on plane layered medium, and this radiated electric field is the incident field on plane layered medium, and incident field representation formula is as follows:

${\stackrel{\‾}{E}}_0^i\left(r\right)=e^{-j{k}_{\⊥}^s\·r_{\⊥}}{e}^{-j{k}_{z{0}^Z}^s}{\stackrel{\‾}{E}}_0^i\hat{t},$

Wherein: r _⊥for the position coordinates on xoy face, z represents the locus coordinate in z-axis, for incident wave vector is in the projection of z=0 planar, for incident wave vector projection in a z-direction, incident field amplitude is: k ₀represent the wave beam in vacuum, η ₀for free space wave impedance, r _sfor the distance that initial point is shown up a little, be the unit vector of any direction, the unification in order to guarantor unit makes current source current intensity J=1Am.

(6b) the non-Perturbation in plane layered medium is obtained by the incident field in step (6a) its representation formula is as follows:

$\begin{array}{c}{\stackrel{\‾}{E}}_m^{\left(0\right)}\left(r\right)=e^{-j{k}_{\⊥}^s\·r_{\⊥}}[{\hat{h}}_0^s{\mathrm{\ξ}}_{0\→m}^{+h}(k_{\⊥}^s,z){\stackrel{\‾}{E}}_0^{\mathrm{ih}}\\ -{\hat{k}}_{\⊥}^s\frac{k_{\mathrm{zm}}^s}{k_0{\mathrm{\ϵ}}_m}{\mathrm{\ξ}}_{0\→m}^{-v}(k_{\⊥}^s,z){\stackrel{\‾}{E}}_0^{\mathrm{iv}}+\hat{z}\frac{k_{\⊥}^s}{k_0{\mathrm{\ϵ}}_m}{\mathrm{\ξ}}_{0\→m}^{+v}(k_{\⊥}^s,z){\stackrel{\‾}{E}}_0^{\mathrm{iv}}]\end{array},$

Wherein for incident wave vector two-dimensional projection in z=0 plane, represent modulus value, represent direction for intermediate variable, with be GENERALIZED REFLECTION COEFFICIENT and the transmission coefficient of m layer plane, { h, v} represent polarization mode to p ∈, k _zmbe the mould that in m layer medium, space wave number projects on xoy face, d _mand d _m-1represent the modulus value of m and m-1 plane depth, ε _mrepresent the relative dielectric constant of m layer medium, m={1,2,3...N-1}, with that incident wave exists with the component in direction; with be perpendicular to plane on any two mutually perpendicular unit vectors.

(6c) according to reciprocity principle, by the zeroth order electric field in medium the difference δ ε (r of the relative dielectric coefficient of stratified medium uneven surface and stratified medium plane two kinds of models _⊥, z), the non-Perturbation that far field currents source produces in plane layered medium bring first-order perturbation field into in expression formula, obtain first-order perturbation field matrix form as follows:

$\left[\begin{array}{c}{E}_0^{\mathrm{sv}}\\ E_0^{\mathrm{sh}}\end{array}\right]=\mathrm{\π}{k}_0^2\frac{e^{j{k}_0{r}_0}}{r_0}\underset{m=0}{\overset{N-1}{\mathrm{\Σ}}}{\widetilde{\mathrm{\ζ}}}_m(k_{\⊥}^s-k_{\⊥}^i)\×{\widetilde{\mathrm{\α}}}^{m,m+1}(k^s,k^i)\left[\begin{array}{c}{E}_0^{\mathrm{iv}}\\ E_0^{\mathrm{ih}}\end{array}\right]$

Wherein, ${\widetilde{\mathrm{\α}}}^{m,m+1}(k^s,k^i)=\left[\begin{array}{cc}{\widetilde{\mathrm{\α}}}_{\mathrm{vv}}^{m,m+1}(k^s,k^i)& {\widetilde{\mathrm{\α}}}_{\mathrm{vh}}^{m,m+1}(k^s,k^i)\\ {\widetilde{\mathrm{\α}}}_{\mathrm{hv}}^{m,m+1}(k^s,k^i)& {\widetilde{\mathrm{\α}}}_{\mathrm{hh}}^{m,m+1}(k^s,k^i)\end{array}\right]$

$\begin{array}{c}{\widetilde{\mathrm{\α}}}_{\mathrm{vv}}^{m,m+1}(k^s,k^i)=({\mathrm{\ϵ}}_{m+1}-{\mathrm{\ϵ}}_m)\\ \×\frac{1}{{\left(k_0{\mathrm{\ϵ}}_m\right)}^2}[\frac{{\mathrm{\ϵ}}_m}{{\mathrm{\ϵ}}_{m+1}}{k}_{\⊥}^s{\mathrm{\ξ}}_{0\→m}^s\left(k_{\⊥}^s\right)k_{\⊥}^i{\mathrm{\ξ}}_{0\→m}^{+v}\left(k_{\⊥}^i\right)\\ -({\hat{k}}_{\⊥}^s\·{\hat{k}}_{\⊥}^i)k_{\mathrm{zm}}^s{\mathrm{\ξ}}_{0\→m}^{-v}\left(k_{\⊥}^s\right)k_{\mathrm{zm}}^i{\mathrm{\ξ}}_{0\→m}^{-v}\left(k_{\⊥}^i\right)]\end{array}$

${\widetilde{\mathrm{\α}}}_{\mathrm{vh}}^{m,m+1}(k^s,k^i)=({\mathrm{\ϵ}}_{m+1}-{\mathrm{\ϵ}}_m)\×\hat{z}\·({\hat{k}}_{\⊥}^i\×{\hat{k}}_{\⊥}^s)\frac{k_{\mathrm{zm}}^s}{k_0{\mathrm{\ϵ}}_m}{\mathrm{\ξ}}_{0\→m}^{-v}\left(k_{\⊥}^s\right){\mathrm{\ξ}}_{0\→m}^{+h}\left(k_{\⊥}^i\right)$

${\widetilde{\mathrm{\α}}}_{\mathrm{hv}}^{m,m+1}(k^s,k^i)=({\mathrm{\ϵ}}_{m+1}-{\mathrm{\ϵ}}_m)\×\hat{z}\·({\hat{k}}_{\⊥}^i\×{\hat{k}}_{\⊥}^s){\mathrm{\ξ}}_{0\→m}^{+h}\left(k_{\⊥}^s\right)\frac{k_{\mathrm{zm}}^i}{k_0{\mathrm{\ϵ}}_m}{\mathrm{\ξ}}_{0\→m}^{-v}\left(k_{\⊥}^i\right)$

${\widetilde{\mathrm{\α}}}_{\mathrm{hh}}^{m,m+1}(k^s,k^i)=({\mathrm{\ϵ}}_{m+1}-{\mathrm{\ϵ}}_m)\×({\hat{k}}_{\⊥}^i\·{\hat{k}}_{\⊥}^s){\mathrm{\ξ}}_{0\→m}^{+h}\left(k_{\⊥}^s\right){\mathrm{\ξ}}_{0\→m}^{+h}\left(k_{\⊥}^i\right)$

Wherein: with it is single order scattered field in unit vector and unit vector on projection, height relief function ζ on planar layered structure _m(r _⊥) two-dimensional Fourier transform, that incident wave is in unit vector and unit vector the component in direction, with be perpendicular to k ⁱplane on any two mutually perpendicular unit vectors, r ₀for the distance that initial point is shown up a little, the electromagnetic scattering coefficient of m layer uneven surface, by with composition, represents vertical-vertical polarization respectively, vertical-horizontal polarizes, horizontal vertical polarizes and the electromagnetic scattering coefficient of level-horizontal polarization, ε _m+1be the relative dielectric constant of m+1 layer medium, ε _mbe the relative dielectric constant of m layer medium, k ⁱand k ^sbe respectively incident wave vector and scattering wave vector, k ₀the wave number in vacuum, with for the mould that incident wave resultant scattering wave vector projects on xoy face, with for horizontal unit vector, for vertical unit vector, with be the mould that in m layer medium, incident wave resultant scattering wave vector projects in z-axis, k _zmbe the mould that the space wave number in m layer medium projects in z-axis, z is the mould projected in z-axis of position, arbitrfary point r in model, for intermediate variable, with be GENERALIZED REFLECTION COEFFICIENT and the transmission coefficient of m layer plane, { h, v} represent polarization mode to p ∈, and { i, s} represent incident and scattering mark to q ∈.