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The theory of electromagnetic
Remote Sensing to random sea waves
Chang Man
Research Institute No. 207 Ministry of Aero-space industry
P.0.Box 142-207, beijing China
Kuang Kui
Depart of Filming Bejing TV Station
Research Institute No. 207 Ministry of Aero-space industry
P.0.Box 142-207, beijing China
Kuang Kui
Depart of Filming Bejing TV Station
Abstract
This paper describes stochastic Green function method in solving random wave equation of electromagnetic field. The volume scattering coefficients of the sea surface of containing bubbles are given. Also when incident angle is less then 300 is pointed out. The surface scattering model is not sufficient. The surface scattering model is not sufficient. The joint problem of volume scattering and surface scattering coefficient in the neighbor hood of incident angle 30° is discussed.
Introduction
In theory of electromagnetic remote sensing, we are concerned about scattering problem of electromagnetic wave, and the interactive medium is the inhomogeneous medium with random rough surface.
The scattering coefficients of electromagnetic wave contain two parts: the first is the surface scattering coefficient; the second is the volume scattering of inhomogeneous medium. At present, there are many methods to calculate the surface scattering coefficients. For example, two-scale model, when incident angle is 30°<q< 80°, at case of down-wind, used to this model, the theoretical results are basic agreement with experimental data. When incident wave near vertical direction, Rich K. Moor (1979)show out the two-scale model that is not used. Because, in this case, the electromagnetic wave is incident on white caps and bubbles of sea waves. For this case, the two-scale model is not applied, and in the case of upwind, the surface scattering coefficient consists with experimental data to be very poor. This result is the same season. In general, the theoretical result of surface scattering then experimental data is big 8dp.So,if we attempt to improve the theoretical result of the scattering coefficients, we must consider affect of the bubbles in sea waves to dielectric coefficient of sea water, and must use new model-the volume scattering model. On this field, A. K. Fung (1981)and J.A.Kong (1985) had made an important study. Especially, J.A.Kong(1985)used matrix equation to study scattering coefficients to plane surface layer of containing bubbles of different shape and size. For the inhomogeneous medium, in general, Born approximation or Dyson equation method was used. The advantage of these methods is simple in mathematics. In this paper, we used average field method to give random differential equation of scattering field, then, to solve it by stochastically Green method and to give Out expression of the volume scattering coefficients, and to make numerical calculation on the near vertical incident direction. In discussion we have given out weight joint formula and numerical results of the volume scattering and surface shuttering coefficients in the neighbor hood of incident angle 30°
The volume scattering equations
Model
The scattering medium is inhomogeneous medium layer of containing bubbles, and its top surface is random rough surface with two-dimension slope Zx and Zy If measurement is at the incident angle qi direction (i.e. backscattering case), the, the probability distribution of slopes can be written (2.1)
Pq(Zx, Zy) = (1+Zxtgq)P(Zx, Zy)
from Fig.1 we can know, the slope can be written as
| Zx = cosjntgqn | |
| Zy = cosjntgqn |
so expression (2.2) can be re-written as
The dielectric coefficient e(z) of sea water of containing bubbles can composite from average eielectric ea and fluctuating dielectric coefficient ef(z)(Zeng.Q.A.1983) i.e
and let a = < ea (z) > , < ef ³ 0 . The dielectric coefficient of pure sea water is
In above expression, the dielectric coefficient of high frequency is e¥ The static state dielectric coefficient es, the delay time t and conductivity of sea water si can be found from Zeng Q. A. (1983) In some extent of wind speed, the content of bubbles volume in sea waves is expressed by R so the average dielectric coefficient of sea water with the bubbles can be written as
The correlation coefficient of fluctuating dielectric coefficient is
Where Iz is horizental correlation length, and Iv is vertical correlation length,s is fluctuating variance of dielectric coefficient (L,Tsang 1985)
where
The equation of volume scattering field
If correlation length of fluctuating medium is bigger than the incident wave length and neigligical affect of depolarization,the wave equation of inhomogeneous medium can be written as
where
is total electric field in air
where
Field approximation the transmitted field satisfies equation
Let us substitue (2.10) in (2.9), and take note of (2,11), then we can obtain
Let
Can be written equation (2.12) to standard form of random differential equation
Where static state operator
The boundary conditions
In the geographic coordinate system 0-xyz, the incident wave respectively is horizontal polarization wave and vertical polarization wave. The boundary conditions can be given respectively.
The horizontal polarization incident wave [0,Ey,O]
The Ey's boundary condition can be given from targetial components continuity of boundary.
The Vertical Polarization incident wave[Ex,Ey,0]
The boundary conditions of component of electrical field are
The boundary conditions in local coordinate system.
For the convenience of calcuatian, let us introduce local coordinate system 0-xyz, and the normal of surface n is line with z axis. We easily obtain the boundary conditions in local coordinate system.
The horizontal polarization
The vertical polarization:
Where Ex,Ey and Ez are components of electrical field in local coordinate system
The stochastic green function solution of scattering field equation
One of methods to solve random differential equation can be that static state solution of equation is first given and then stochastic solution of equation is obtained.
In terms of this idea, George Adomian intrduced method of stochastic Green fundtion in 1983. This method gives out series solution of random differential equation . This solution is very clear on the physical picture.
Because the random quantity is satisfied
<a> = R02< Sf > = 0
So, The average value of 2n+1 order correlation coefficients in stochastic Green function equivalent to zero. For this, 2n order solution of seriesis only retained. So, random defferential equation (2.12), the average solution only needs to retain the following stochestic Green functions.
The volume scattering coefficients
The transmitted Green Functions
The static state Green function g in (4.1), for the volume scattering question, the g is used as transmitted green function of wave field from inhomogeneous volume scattering medium into air. As we know, Fourier transform of static state Green function of horizontal polarization scattering field in air is.
The fourier transform of static state Green function in sea waves with bubbles is
in the above two expressions,
k's = (k02 - kx2 - ky2)½
using boundary conditions (3.5) and (3.6), we can obtain boundary conditions of Green functions and at Z=0
gy2 = gy1----------------------------------(5.3)
For the same reason, for vertical polarization wave, the fourier tramsform of Green function in air is
The fourier transform of Green function in sea waves with bubbles is
using (3.8) and (3.9), we can obtain boundary conditions of green function in local coordinate system at z=0
substitutline g1 and g2 in (5.7) and (5.8) into (5.9) and (5.10) respectively, we can obtain
The correlation between incident angle and transmitted angle in local coordinate system can be determined by Snell law
due to the vector of incident wave
the normal of tangential plane
So, the correlation between incident angles in local system and in geographic system is
The volume scattering coefficients
The horizontal polarization wave.
if incident field is horizonal polarization wave, its form as
The transmitted field in sea water of containing bubbles
where
The transmitted field from sea water of containing bubbles transmit into air to be
so, the fourier transform of scattering field of horizonal ploarization wave is
where
Based on expressions (2.2) and (2.6), we can obtain
Where symbol
Based on the scattering coefficients, we can give out
The substituting Kx , Kyspectrum of
The vertical ploarization wave
The incident field
The transmitted field in sea water of containing bubbles is
where
We can us the same deductive method with horizontal polarization, the back-scattering coefficent of vertical polarization can be obtained
The expressions of (5.20) and (5.21) are functions of x and y, So the scattering coeffcient average of probability of slope is
The symbol pp can be hh or vv . Because of the value m of mean squre root is small generally, since the scattering coefficient 0pp is sensitive function of k2 ?n so, we can apply method of steepest descent to above integral.
From this, we can obtain final numerical result.
The numerical results and discussion
The complete calculation of electromagnetic scattering coefficient is a complex question. In this paper, we apply a method of average field to treat in electromagnetic scattering problem of the sea waves of containing bubbles. In general, the electromagnetic scattering and volume scattering. In different extent of incident angle, the contribution of surface scattering and volume scattering are different. When near vertical incident, the incident angles are 0 qi 20°, the electromagnetic wave main incident on peak or valley of waves, so, the volume scattering is important, and when incident angles are 200 ?i <800, the contribution of surface scattering is important, however when incident angles are in extent of 20°< qi<30°, this case is compleser, i.e. the theoretical results of alone surface scattering or alone volume scattering can not agree with experimental data. i.e., on the mathematics, in this extent, there is a problem of weight joint of the volume scattering coefficient and the surface scattering coefficient . For this, according to the experimental data a formula of weight and joint of volume scattering and surface scattering and surface scattering, i.e when incident angle qi is increased from 20° to 30°,the contribution of volume scattering is decreased from s° to zero, and the contribution of surface scattering is increased from zero to s° the joint formula of scattering
The angle jm can be determined by experimental data. Here, we take jm = 5°. The critical angle ? of joint of surface scattering and volume scattering is correlative with wind speed
qc = 25° + 2jm.V--------------------(6.4)
Where R is amount of bubbles in seawater
R = 7.751 X 10-h - V.231
V is wind speed (m/s), from expression (6.5) we can see if wind speed V=0, then R=0. In this work, we calculate scattering coefficients to four frequencies, four dielectric coefficients, two polarizations (V and H) i.e.
X-Band 8.91 GH ex = 48.3-j34.9
E-Band 4.455 GH ec = 57.1=j36.3
L-Band 1.228 GH e1 =74-j85
P-Band 0.482 GH ep = 73-j165
The wind speed 20.56 m/s
Three wind directions,i.e. down-wind, up-wind and cross-wind. The parameter of volume scattering Koih =8
When incident angles are in extent of 20°~30°, the weight joint of surface scattering and volume scattering coefficient must be made. The numerical results are shown in table 1, Table II and Fig. 2, Fig.3.
Upwind v = 20.56m/s vertical polarization Klv=8, Klz=2, F=8.91 GHz
Table I
| incident angle | 0 | 15 | 30 | 15 | 60 | 70 | 80 | 85 |
| Our | 1.42 | -4.18 | -13.49 | -18.70 | -23.01 | -28.32 | -35.92 | -41.81 |
| A.K.Fung | 11.10 | -3.97 | -13.15 | -19.60 | -25.69 | -30.76 | -39.53 | -49.36 |
| Experiment | 3 | 8 | 14 | -18.5 | -21 | -24.5 | -28.5 | -34.5 |
Upwind speed v= 20.56 m/s , horizontal pol.Klv=8, Klz=2, f=8.91 GHz
Table 2.
| incident angle | 0 | 15 | 30 | 15 | 60 | 70 | 80 | 85 |
| Our | 1.40 | -4.23 | -13.75 | 19.21 | -24.08 | -28.43 | -35.39 | -40.86 |
| A.K.Fung | 8.73 | -5.99 | -14.29 | -19.46 | -24.29 | -28.96 | -38.14 | -48.53 |
| Experiment | 2 | -6.5 | -14.5 | -19.5 | -26.5 | -27.5 | -32 | -39 |
The experimental data are taken from Daley's (1970) result.
We used the model of average field in a sea waves of containing bubbles, the numerical result shows the volume scattering coefficient has more obvious improvement than A.K. Fung's result. The method of stochastic Green Function can give out series solution of field equation, and its zero order solution just be Born approximation solution used by A.K. Fung (1981).
References
- Richard K.Moore and abrian K.Fung , (1979) determination of winds at
Sea, Proc.of the IEEE Vol.67,No.11,PI 504.
- Fawwag T. Ulaby, Richard K.Moore and Adrian K. Fung (1981),
Microwave remotes Sensing, active and passive, Vo1.2, Chapter 15.
- Leung Tsang, Jin au Kong and Rotert T. Shin (1985), Theory of
Microwave Remote Sensing Chapter 3
- Zeng Q.A., Kemes (1983), The Effect Oceanic Whitecaps and foams on
plus-Limited Rader Altimeters . Jaurnal of Geophysical Research,Vol.88 ,
No.4,PP 2575578
- George adomian (1983) , Stochastic Systems , New York
- Daley J.C. W.T.Davis and N.R. Miklls (1970) radar Sea Return in High sea States , AD 713589.