Tán xạ của sóng điện từ P4

204 5 SCATTERING AND EMISSION BY LAYERED MEDIA.. We can obtain the reflection coefficient for the stratified medium Rh = Ao/ Bo by using the following procedure.. 208 5 SCATTERING AND EM

Chapter 5

Scattering of Electromagnetic Waves: Theories and Applications

Leung Tsang, Jin Au Kong, Kung-Hau Ding Copyright  2000 John Wiley & Sons, Inc ISBNs: 0-471-38799-1 (Hardback); 0-471-22428-6 (Electronic)

200 5 SCATTERING AND EMISSION BY LAYE.RED MEDIA

dI -

(5.1.1) gives for the tth layer

Ided

- = A e-KalSeCelZ + Tl

(5.1.3)

51 Incoherent Approach of Radiative Transfer 201

+ F+l

(5.1.4) (5.1.5)

2

&.J(~l) = rlIdl(h) + (1 - r&I,(l+l)(B~+l)

nl+l

2 Id(l+l) @+l) = rziT,(l+~)(h+~) + (1 - rl)~Idl(@l)

(5.1.6)

(5.1.7) where 01 and 81+1 are related by Snell’s law

202 5 SCATTERING AND EMISSION BY LAYERED MEDIA

polarization

nz+1 cos 81 - nz cos ez+1 2

7-l =

nz cos 81 - nz+1 COSQZ+l 2

rl =

from below & and from above dl are the same The extra factors of nF/nF+l

Ale tc,l set &dl + 3 = q(Ble K,~ set 01 dl + E)

Bz+le ~~(l+l) set &+d + F+l = rz(Az+le G(~+I) set &+A + X+1)

+ (1 - rl> [Bze -TV set Ol dl + q (5.1.16b)

(5.1.17)

52 Wave Approach 203

2.1 Reflection and Transmission

204 5 SCATTERING AND EMISSION BY LAYERED MEDIA

with Hz = Ho e-ilc, Zeix-L l FL incident on the stratified medium, the total field

regions are equal We note that in region 0 where I = 0, we have

A 1+1e -ik(,+l,zdr + Rl(l+l) Bl+leik(l+lJzdl 1 (5.2.10a)

4(1+1)Al+le ilc(z+l)A + Bl+leik(z+l)zdz 1 (5.2.10b)

&!.l Reflection and Transmission 205

where

Rl(l+l) =

Pl+lklz - Plk(l+l)z Pl+lklz + Pl~(l+l)z

(5.2.11)

Bl Al+1 e-i2k(l+l)zdz+l

(5.2.12) a+1

Rl(l+l) e i2k(l+l,z (4+1-d) + 1

e i2k(r+l)sdl+l

e-i2k(l+z)rdl+z ’ , and so on, until which can in turn the transmitted be expressed region t is reached where in terms of (Al+2/Bl+$

AtI& = 0 We can obtain the reflection coefficient for the stratified medium

Rh = Ao/ Bo by using the following procedure Since At /Bt = 0 and t = n+ 1,

and Rh = Ao/Bo is calculated

~1, ~1 -+ 61, El -+ Hl, and Hl + -El

A Reflection from a One-Layer (Half-Space) Medium

5 SCATTERING AND EMISSION BY LAYERED MEDIA

Observation angle (deg)

~1 = 3.2(1 + iO.l)~,, dl = 50 cm, et = HOE,, and frequency at 1 GHz

52.2 Dyadic Green’s Function for Stratified Medium 207

Frequency (GHz)

3.2(1 + iO.l)c*, dl = 50 cm, and et = 80~~

B Reflection from a Two-Layer Medium

ROl + & e i2kl,dl

R - -

208 5 SCATTERING AND EMISSION BY LAYERED MEDIA

due to the response of the layered medium

52.2 Dyadic Green’s Amction for Stratified Medium 209

+ RTM~(lc,)ei~.“~~-~~)~-~~g~’

find that

Al e -ikl,dl + Bl e’klxdl = Al+1 e ik(l+l)zdl + Bl+l eik(l+l),dl (5.2.18)

210 5 SCATTERING AND EMLSSION BY LAYERED MEDIA

Rz(z+l) =

klz - $+l)z klz + k(l+l)z

%(l+l) =

~l+l~~z - ~lk(l+l)z Q+lklz + d+l+l)z

(5.2.22a)

(5.2.223)

Rz(z+l) and Q+l) are the reflection coefficients for TE and TM waves, re-

Al+1 e ik(z+l,.&+l

B eik(Z+l)zdl+l I+1

(5.2.23a) and

CZ+l e -+l+l,.dz+l

D 1+1 eik(l+l)zdl+l 1 = =TM

-

(5.2.25)

-

(5.2.26)

52.2 Dyadic Green’s Amction for Stratified Medium 211

k jjot (El, 2’) esi’l *?L (5.2.313)

212 5 SCATTERING AND EMISSION BY LAYERED MEDIA

Hence, using (5.2.27) and (5.2.28) in (5.2.33) gives

t c;

TBh(b) =&x-I

-4-l dz’Ti(z’)

0 l=1 % -4

t c;

TBv(b) = & 7, - /

-4-l dz’ T&z’)

O 1=1 &I -dr

2

(5.2.35)

$2.3 Brightness Temperatures for Stratified Medium 213

of (5.2.23) and (5.2.24) are used to calculate the upward and downward wave

214 5 SCATTERING AND EMISSION BY LAYERED MEDIA

Illustrations

* = 46 + i0.6) + p(3 + i0.3) eaz

and AK = ~t20 K It is interesting to observe a maximum for vertical

near the surface region where the temperat ures are different

52.3 Brightness Temperatures for Stratified Medium

216 5 SCATTERING AND EMISSION BY LAYERED MEDIA

10” loo

* 10’ lo2 Frequency (GHz)

53 Incoherent and Coherent Approach 217

Approach

218 5 SCATTERING AND EMISSION BY LAYERED MEDIA

coherent and incoherent models for an abrupt change profile

-: Coherent Model 50-

53 Incoherent and Coherent Approach 219

255 -

245l I I

lo-’ 10” 10’ 10’

Frequency @Hz)

a TE wave

220 5 SCATTERING AND EMISSION BY LAYERED MEDIA

space

54 Applications to Remote Sensing of Soil 221

&u)m = 49 (5.4.3)

2mu(T) = 1.1109 x 10-l’ - 3.824 x 10-12T

stant of soil is also temperature-dependent

is changed from 2% to 35%

222 5 SCATTERING AND EMISSION BY LAYERED MEDIA

18~ I I I I I I I I I I

N 16- -\\ : 3O*h moisture

\

\ 14- \ - : 5% moisture

\

\ 12- \

b imaginary -N -9

6- / / / &I -_ -9 -9 -9 -9 VI

0 -9 -9 -'I 4- 1 0 real -9 -a

/

0

2- 0'

/ 01’ imaginary

5 10 15 20 25 30 35

Moisture Content (%)

Figure 5.4.2 Dielectric constant of soil as a function of moisture content

$4 Applications to Remote Sensing of Soil

120; I

5

I I I I 1 I

10 15 20 25 30 35 Moisture Content (%)

224 5 SCATTERING AND EME’ION BY LAYERED MEDIA

0’ 0 10 I 20 ! 1 I I 1 I I

30 40 50 60 Obsemation Angle (degree) 70 60 90

1.42 GHz and 10.69 GHz is small in this case

the region from z = 0 to x = -d and that the region below -d is uniform

may occur during rain Profiles 2-5 can represent various stages of drying of the soil surface for different soil types Profile 6 can happen when a subsurface water table exists

54 Applications to Remote Sensing of Soil 225

226 5 SCATl-‘ERING AND EMLSSION BY LAYERED MEDIA

into 500 fine layers The thickness of each layer is much less than the wave-

2, 3, and 4 are quite close to each other at low frequency There is not much

for low frequency The emission at lower frequency becomes less sensitive to

10 cm for 0.25 GHz and 0 to 1 cm for 1.42 GHz These average depths for

54 Applications to Remote Sensing of Soil 227

profiles shown in Fig 5.47 Soil moisture profile 3 is used

228 5 SCATTERING AND EMISSION BY LAYERED MEDIA

*o-5cm OO-1Ocm

Average Moisture Content (%)

quency of 1.42 GHz Temperature profile 3 is used

REFERENCES 229

Press, New York

Klein, L A and C T Swift (1977), An improved model for the dielectric constant of sea

York

Lane, J and J Saxton (1952), Dielectric dispersion in pure liquids at very high radio fre-

Millimeter- Wave Passive Structures, T Itoh, Ed., Chap 3, 113-213, Wiley

Peterson, A F., R L Scott, and R Mittra (1998), Computational Methods of Electromagnetic

Stogryn, A (1970), The brightness temperature of a vertically structured layer, J Geophys Res., 80, 4484-4496

Tsang, L., C C Huang, and C H Chan (2000)) S ur f ace electric fields and spatial derivatives

M(4), 288-295