1

Wave Polarization,Polarimetric SAR, and

Polarimetric Scattering Models

Yisok Oh

Dept. of Radio Engineering, Hong-Ik University

Seoul National University, February 16-19, 2000

PACRIM Training Course (Workshop)

2

NASA/JPL AirSAR Experiment, PACRIM II

실험지역: ‘논산-익산’(9월 2일: 현장답사)(9월 22-23일: 1차취득)(9/29-10/1: 취득)(10/3, 10/7-9: 3차취득)

Ground Truth Data 취득-. 휴경지-. 논/ 벼벤논-. 무,고구마밭-. 산

Scatterometer (5.3 GHz)운용 (9/30-10/1)

3

Contents

1. Wave Polarization1.1 Wave Properties1.2 Polarization Synthesis

2. Polarimetric Radar System2.1 A Scatterometer System2.2 NASA/JPL POLSAR System

3. Polarimetric Scattering Models3.1 Surface Scattering3.2 Volume Scattering

4

1.1 Wave Properties

-. What is the “Field”?-. Waves : Electromagnetic Waves by Maxwell-. Planewave Propagation in free space-. Polarization : Basic concepts-. Microwave Generation : DC to AC-. Microwave Guidance by Waveguides/ Trans. lines-. Microwave Radiation by Antennas-. EM Wave Reflection from infinite planes-. Microwave Scattering from

-. Point Targets-. Distributed Targets

5

Electromagnetic FieldsFields: Spatial distribution of a physical quantity.

Static Fields

Electromagnetic Fields

No time-variationSeparation of Electric Field andMagnetic Field

Time-varying Fields

Dynamic Fields

Co-existence ofElectric and Magnetic fields

: DC : AC

6

Electrostatic FieldsE : Electric Field(showing flux lines)

Assuming infinite plates, ( )mVdVE /=

V

Conducting Plates

E d+

-

Direction of E : From + charges to - charges

7

Magnetostatic Fields

IEEE Emblem

I : Current

H : Magnetic Field(showing flux line)

voltage

current

Electric Fields

Magnetic Fields

Assuming an infinite current line, ( )mAIH /2πρ

=

I

H

ρ

Direction of H : Right-hand rule, ( )RIH ˆˆˆ ×=

8

Dynamic (Time-Varying) Fields

⎥⎦

⎤⎢⎣

⎡⋅

∂∂

+⋅=⋅∂∂

+=×∇

⎥⎦

⎤⎢⎣

⎡⋅

∂∂

−=⋅∂∂

−=×∇

∫ ∫∫

∫ ∫

c SS

c S

sdtDsdJldH

tDJH

sdtBldE

tBE

ElectromotiveForce(Voltage Source)

ConductionCurrent

DisplacementCurrent

Maxwell’sEquations

Time-varyingElectric Field

E(r,t)

Time-varyingMagnetic Field

H(r,t)

9

WavesConsider Water wave in a pond.

Cut water surface at once(t=t0) with Kwan-Woo’s Sword (청룡언월도) and look

X(Spatial Displacement)

Wave Height

Log the height ofFishing Buoy (x=x0)as a time function

t(Time)

Wave Height

Even though the wave comes toward me, the water doesn’t !

10

Electromagnetic (EM) Waves

( )xkztEtrE ˆcos),( 00 φ+−ω=

Magnitude(source,distance,etc.)

SinusoidalWave

TimeVariation

Z-directedpropagation

Vector(Polarization)

An Example of an EM wave:

Tπ

=ω2

λπ

=2k

Time

TxE

z, distanceλxE

11

Phase Velocity

0φ+−ω kzt = constantSame Phase

Assume these circles are surfing boards.

Phase velocity = velocity of the equi-phase point

ktk

constt

tzvp

ω=

∂

⎟⎠⎞

⎜⎝⎛ −φ+ω∂

=∂∂

=

0

Poynting Vector: ( ) ( ) ( )trHtrEtrS ,,, ×=

: Real Power Flow (Magnitude and Direction)

12

Time-Harmonic FieldsTime-harmonic Assumption: tje ω Time variation

xeErE jkz ˆ)( 0−=

yeErH jkz ˆ)( 0 −

η=

HjE ωμ−=×∇ (Maxwell Equation)

( )vacuuminΩ=εμ

=η 377

zyx ˆˆˆ =× z : wave prop. direction

(Phasor form)

13

Planewave Propagation

x

y

z

Hy

Ex

zHE ˆ⊥⊥

xeErE jkz ˆ)( 0−=

Planewave: wavefront is plane

Spherical wave near an antenna

Approximate Planewave

in the Far-zone ⎟⎟⎠

⎞⎜⎜⎝

⎛λ

>22DR

Z-directed propagatingLinear polarized (x-direction)Wave

14

Polarization: shape of the locus of the E vector tip

at a given point in space as a function of time.

( ) ( ) jkzjyx eeyaxazE −δ+= ˆˆ

π=δ ,0

Polarization

Linear

Circular

Elliptical2

, π±=δ= yx aa

Other Cases

Conditions Examples

xeEzE jkz ˆ)( 0−=

( ) ( ) jkzeyjxzE −−= ˆˆ

( ) ( ) jkzeyjxzE −−= ˆ2ˆ

15

Exercise (determination of polarization)

( ) ( ) jkzeyxzE −+= ˆˆ2Find polarization of the wave,

Find instantaneous electric field:

Plot electric field:

Determinepolarization

( ) ( ) [ ]{ }( ) ( )kztyx

tjzEtzE−ω+=ω=

cosˆˆ2expRe,

( ) ( ) ( )tyxtE ω+= cosˆˆ2,0

2

1

Ex

Ey

α0=ωt

π=ωt

Linear pol. with ( )21tan 1−=α

16

Polarization Ellipse

( )( ) δα=χ

δα=ψsin2sin2tancos2tan2tan

0Lin. Pol.

=δ

0=χ

Circular Pol.090±=δ

045±=χ

= Rotation Angle

= Ellipticity Angle

ψχ

17

Various Polarization States

LeftCircular pol.

Wave direction ThumbElectric Field Other fingers

of left hand

18

Microwave GenerationOscillators

Tubes Solid State

KlystronTWTMagnetron

Gunn Diode MESFETHEMT, etc.

High Power

Light, Cheap

D.C. Power

Instability

Resonator

Amplifying

Microwave(A.C.)

A MESFETOscillator:

19

Microwave Guidance

Two Conductors

Single Conductor

No Conductor

Waveguides

(Transverse ElectroMagnetic)

Coaxial CableTwo-wireMicrostrip

Rectangular,CircularWaveguides

DielectricWaveguide

TEM wave TE, TM waves (Optical fibers)

EH

directionA Coaxial Cable :

20

Microwave RadiationDipole Antenna :

Transmission Line

(Wave guider)

Radiator(Discontinuity)

* Current : temporalVariation of charges

21

Antennas

WireAntenna

ApertureAntenna

ReflectorAntenna

PrintedAntenna

Microstrip AntennaCoaxial Cable

ElectricField Lines

22

EM Wave ReflectioniE

iH x

z

Perpendicular PolarizationElectric field is perpendicular

to the incidence plane

iE

iH

x

z

Parallel PolarizationElectric field is parallel to the incidence plane

⊗•

Horizontal PolarizationElectric field is horizontal

to Earth surface

Vertical PolarizationMagnetic field is horizontal

to Earth surface

Infiniteplane

23

Microwave Scattering

Radar System

Radar System

PointTarget

DistributedTarget

( )σ

π

λ= 43

2

4 RGGPP rt

tr

( )0

3

200

4σ

π

λ= ill

rttr AGGPP

( )( )

dsR

ggA

areaillum

rtill ∫ φθ

φθ=

.4 ,

,where

σ : Radar Cross Section ( )2m

0σ : Scattering Coefficient

24

1.2 Polarization Synthesis

Coordinate System

kh

v

khv ˆˆˆ ×=

yxkzkzh ˆcosˆsinˆˆ

ˆˆˆ φ+φ−=×

×=

khv ˆˆˆ ⊥⊥

x

z

yθ

φ

zyxk ˆcosˆsinsinˆcossinˆ θ+φθ+φθ=

zyxv ˆsinˆsincosˆcoscosˆ θ−φθ+φθ=

25

Scattering Matrix

sshs

sv

si

ihi

iv

i

hEvEE

hEvEEˆˆ

ˆˆ

+=

+=i

jkrs ES

reE−

=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟

⎟⎠

⎞⎜⎜⎝

⎛ −

ih

iv

hhhv

vhvvjkr

sh

sv

EE

SSSS

re

EE

ScatteringMatrix

26

Stokes Vector : F

Stokes Parameters

VUQI ,,,0

( )( ) ⎥

⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

χχψχψ

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

−

+

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

2sin2cos2sin2cos2cos

Im2Re2

0

0

0

0

*

*

22

220

III

I

EEEE

EE

EE

VUQI

F

hv

hv

hv

hv

= Rotation Angle

= Ellipticity Angle

ψχ

27

A : Normalized Stokes Vector

0IFA

rr =

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

χχψχψ

==

t

tt

ttt

t

IFA

2sin2cos2sin2cos2cos

1

0

Polarization Synthesis Equation(point targets)

( ) trttrrrt AMA ⋅π=χψχψσ 4,;,

Where

M : Stokes Scattering Operator

28

Stokes Scattering Operator

11 −−= RWRM T

where

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

****

****

****

****

vvhhvhhvvhhhvvhv

hvvhhhvvhhvhhvvv

hvhhhhhvhhhhhvhv

vvvhvhvvvhvhvvvv

SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS

W

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−

−=

ii

R

00110000110011

1−TR : Inverse of the transpose of R

29

Polarization Synthesis Equation(distributed targets)

( ) trttrrrt AMA

A⋅

π=χψχψσ

4,;,0

StokesScatteringOperator

Area

EnsembleAverage

NormalizedStokesVector

ScatteringMatrix

30

An example of polarization synthesis(a large conducting sphere)

224 aSvvvv π=π=σ

⎟⎟⎠

⎞⎜⎜⎝

⎛=

1001

2aS

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−

=

1000010000100001

8

2aM

VariousValues of ( )ttrrrt χψχψσ ,;,

tt χψ ,

Stokes Vectors

rt AA ,

31

Result of polarization synthesis(a large conducting sphere)

Co-pol tr AA =

trtr χ−=χπ+ψ=ψ ,2

Co-pol Response Cross-pol Response

Cross-pol.

32

Measured Polarization Response(a grass surface at L-band)

Co-pol.

500

300

Cross-pol.

33

Phase Information

Co-pol. Difference Cross-pol. Difference

hhvv φ−φ hhhv φ−φ

vhhv φ−φ : delta function ?( )

34

2 Polarimetric Radar System

• System Set-up • Calibration Techniques

• System Characteristics• Data Compression• SAR Calibration

2.2 NASA/JPL POLSAR System

2.1 A Scatterometer System

35

Polarimetric Scatterometers

Tx

Rx

Circulator

Transmitter

Receiver

SW-1

SW-2

Circulator

V

V SW-1

H

HSW-2

V - V H - V V - H H - H

OMT

HornAntenna

- A Single Antenna System -

E

36

Tx

Rx

Transmitter

SW

Receiver

V

HOMT

HornAntennas

- A Two-Antenna System -

OMT

V H

E

E

37

Polarimetric Scatterometer Set-up ( University of Michigan )

38

-Hong-Ik Scatterometer System-( NWA-Based Ku-band Polarimetric Scatterometer )

12

NetworkAnalyzer

Source Unit(0.45 MHz-20 GHz)S-Parameter UnitIF UnitDisplay Unit

V

H

Transmit Receive Pol.1 1 V - V

1 2 H - V2 1 V - H2 2 H - H

E

39

Antenna Support Motor Control Cable Polarimetric Antennas RF Cable 2m Network Analyzer

S-parameter set/ Source

Rotator Computer/

Controller/ 2m Power Source

Polarimetric ScatterometerSet-up

(Hong-Ik University)

40

Calibration Techniques of the Scatterometer

0EvvT

hvT

vvR

vhR

hhR

hvR

0ETE vvtv =

0ETE hvth =

rvE

rhE

0EvhT

hhT0ETE vh

tv =

0ETE hhth =

V-polarized transmit

h-polarized transmit

Dashed line: Polarization Coupling.“No coupling for isolated antennas”

41

( )σ

π

λ= 43

2

4 RGGPP rt

tr

Calibration Technique -continued-

0

2

2η= rr EP 2

4 pqpq Sπ=σ

: For “V-polarized transmit” case

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛π⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

π

λη=

−

hv

vv

hhhv

vhvv

hhhv

vhvvrttkrj

rTT

SSSS

RRRRGGP

reE 4

42 2

1

3

20

2

2

ScatteringMatrixamplitude and phase

errors in receiver

42

Calibration Technique -continued-

amplitude and phase

errors in transmitter

( )

21

2

20

42

⎟⎟⎠

⎞⎜⎜⎝

⎛

π

λη= rtt GGPK

tkrj

r pTSRKr

eE 2

2−= , where

⎟⎟⎠

⎞⎜⎜⎝

⎛=

hhhv

vhvv

TTTT

T

In general

⎟⎟⎠

⎞⎜⎜⎝

⎛==

01

vpt : for v-pol. transmit

⎟⎟⎠

⎞⎜⎜⎝

⎛==

10

hpt : for h-pol. transmit

General Calibration Technique(GCT)

43

Isolated Antenna Calibration Technique (IACT)

If the scatterometer has good cross-polarization isolation

Distortion Matrices are Diagonal.

0==== hvvhhvvh RRTT

rttr

krj

rt STRKr

eE 2

2−= A simple form

44

V-polarizedTransmitter vT

H-polarizedTransmitter hT

V-polarizedReceiver vR

H-polarizedReceiver hR

Simplified block diagram of a dual-polarized radar system

Target

rtG

rG

Isolated Antenna Calibration Technique –continued-

MetalSphere

Tilted Cylinder

Calibration Targets

ro

rc

45

Isolated Antenna Calibration Technique –continued-

Metal Sphere Theoretical Scattering Amplitude: 0,0 ==≡= hvvhhhvv SSSSS

Measurements of Received Fields:

02

20

0

02

20

0

0

0

STRerKE

STRerKE

hhkrj

hh

vvkrj

vv

−

−

=

=

Measurements of Received Fields:

cvhhv

krj

c

cvh

chvvh

krj

c

cvh

STRerKE

STRerKE

c

c

22

22

−

−

=

=

Tilted Cylinder

46

Isolated Antenna Calibration Technique –continued-

Measurements of Received Fields:

uvvvv

krjuvv STRe

rKE 2

2−=

Unknown Target

Scattering Amplitude:

( )

( )0

22

00

02

2

00

0

0

Serr

EES

Serr

EES

rrkj

hh

uhhu

hh

rrkj

vv

uvvu

vv

−−

−−

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛= ( )

( )0

22

02

1

02

2

021

0

0

SerrE

KKS

Serr

KKES

rrkjuhv

uvh

rrkjuhvu

hv

−−

−−

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

where⎟⎟⎠

⎞⎜⎜⎝

⎛=≡⎟⎟

⎠

⎞⎜⎜⎝

⎛=≡ − 2

04

40

200

210, STRTRe

rKEEK

TRTR

EEK hhvv

krjhhvv

hv

vhcvh

chv

L=uvh

uhv

uhh EEE ,,Similarly,

47

Other Polarimetric Calibration TechniquesGeneralized Calibration

Technique (GCT)Sphere,

45o cylinder0o cylinder

DifficultWhit et al.,

IEEE Trans. Ant. Prop. Jan. 1991

Isolated Antenna Calibration Technique

(IACT)

Sphere,45o cylinder easy

easy

Correct phase

Phase, cross-

talk, etc.

IEEE Trans. Geoscience

Rem. Sens. 70-75, 1990

Single Target Calibration Technique

(STCT)

Sphere (or trihedral)

IEEE Trans. GeoscienceRem. Sens. 1022- , 1990

Differential Mueller Matrix Calibration

Technique

PolarimetricAntenna Pattern

IEEE Trans. Antenna Prop.

1524-1532, 1992

Imaging Radar Calibration Techniques

Trihedral or PARC

IEEE Trans. Geoscience

Rem. Sens. 942-, 1991

Distributed

Targets

PointTargets

48

2.2 NASA/JPL POLSAR SystemConceptual view of a SAR

RemoteSensingRadars

Altimeter

Scatterometer

Imaging Radar

Real Aperture Radar (RAR)

Synthetic Aperture Radar (SAR)

: measure surface heights

: measure scattering coefficients

49

Conceptual view of a SAR -continue-

Real Aperture Radar

rX

aX

aX : Azimuth Resolutionby antenna beam width

rX : Range Resolutionby pulsing

λ≈θ

λ≈θ

L

L

N

hp

2

1

Antenna length=L

vAntenna

Antenna beamfoot-print

h

(SLAR)

: half-powerbeam width

: null-to-nullbeam width

50

Conceptual view of a SAR -continue-

Synthetic Aperture

x1 x2 x3 xi xN

D

( ) DRARX a =

D : record length of SAR

Xi : recording points

an array antenna with an aperture of D

( )

Lhh

DRARX

N

aλ

=θ≈

=2

Maximum array length=DAzimuth resolution of the array:

( )

!!!2

,L

Dh

hSARX arrayhpa

≈λ

≈

θ≈

51

POLSAR P-, L- and C-band polarimetry(HH, HV, VH and VV polarization combinations).

XTI1 C-band single-baseline cross track interferometryin VV polarization only. L- and P-band polarimetry.

XTI1P C-band double-baseline cross track interferometryin VV polarization only. L- and P-band polarimetry.

ATI* L- and C-band double-baseline along track interferometryin VV polarization only.

XTI2* L- and C-band single-baseline cross track interferometryin VV polarization only. P-band polarimetry.

XTI2P* L- and C-band double-baseline cross track interferometryin VV polarization. P-band polarimetry.

POLTOP* C-band double-baseline XTI in quad polarization;L- and P-band polarimetry.

* : operating as experimental modes.

Definition of SAR Modes

52

P-bandPolSAR

1.87 x 0.91m

L-band PolSAR(1.61 x 0.45 m)

C-bandATI

C-bandPolSAR

1.35 x 0.17m

C-bandTopSAR

(XTI)

L-bandTopSAR

(XTI)

L-band ATI (not shown)

AIRSAR Antenna Configuration

Nose of DC-8

53

BandsP (0.45 Ghz, 67 cm),L (1.26 Ghz, 23 cm),C (5.31 Ghz, 5.7 cm)

Polarization Full polBandwidth 20, 40, 80 MHz

Resolution, range

7.5, 3.75, 1.875 m

Resolution, azimuth 1 m single lookPulse width 5 or 10 micro-seconds

PRF 17/25 or 34/25 * Gnd speed(272 or 544 pps @ 400 kts)

AirSAR Instrument Characteristic Table

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛=

×××

=

==τ

=τ

mMHzX

resolutionrangeSlantXcB

r

r

5.710202

10320

2,1

6

8

54

Recording, Quantization10 MB/s per frequency (30

MB/s for all three frequencies),8 bits

Peak power (out of transmitter) 1 kW (P), 6 kW (L), 2 kW (C)

Number range cells variable (~1200 or ~2500)

Swath width 10 km (nominal); 17 km (max)

Incidence angles 0-75 deg (usually 20 - 60 degrees)

ATIL & C-band,

antenna separation: 19.8 m at L-band,1.93 m at C-band

Spotlight No

Look direction left

AirSAR Instrument Characteristic Table -continue-

55

Ant. el. beamwidthP: 55 deg., L: 66 deg., C: 64 deg.

Ant. az. beamwidthP: 24 deg., L: 10 deg., C: 2.5 deg.

Ant. stabilization Body mounted

Aircraft DC-8

Range/Endurance 5000 mi

Nominal Speed 420 knots

Altitude 8 km typical

AirSAR Instrument Characteristic Table -continue-

56

Data CompressionFrom polarization synthesis,

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟

⎟⎠

⎞⎜⎜⎝

⎛ −

ih

iv

hhhv

vhvvjkr

sh

sv

EE

SSSS

re

EE

( )2

,, trrt ESEKP ⋅φθλ=

ScatteringMatrix S

where ( ) ( )2

0

,2

,,rE

gK φθηπ

=φθλ

( ) trrt AMAKP ⋅φθλ= ,,

or

StokesScatteringOperator

57

Averaging to Reduce Statistical Variation

Averaging over N measurements

Reduce Statistical Variationand reduce data volume

( )∑∑==

⋅φθλ==N

n

tn

rN

nnrt ESEKPP

1

2

1,,

( ) t

N

nnr

N

nnrt AMAKPP ⎟

⎟⎠

⎞⎜⎜⎝

⎛⋅φθλ== ∑∑

== 11,,

M : single matrix (Stokes Matrix)

Need N ScatteringMatrices !

58

Stokes Scattering Operator(or Stokes Matrix)

Reciprocity

vhhv SS =

Symmetric Stokes Matrix

In Backscattering Mode,

44332211 MMMM ++=

9 independentStokes matrix elements

( 9 Real numbers )

9 x 4 = 36 bytes

10 bytes

QuantizationTechnique

59

Quantization / Data Compression

8 bits/byte: 25628 = Assume 12711

128 22 ≤≤− M

Byte(1) for the exponent Byte(2) for mantissa11M

⎟⎟⎠

⎞⎜⎜⎝

⎛=

2loglog)1( 11MIntbyte ⎟⎟

⎠

⎞⎜⎜⎝

⎛

⎭⎬⎫

⎩⎨⎧ −= 5.1

2254)2( )1(

11byteMIntbyte

Other elements are normalized by )1(25.1254

)2( bytebytex ⋅⎟⎠⎞

⎜⎝⎛ +=

byte(3), …., byte(10)

4434332423141312 ,,,,,,, MMMMMMMM

60

Radiometric CalibrationInternal Calibration:

External Calibration:

Losses and gains of the systemBy power meter measurement, precise antenna pattern, and preflight

Use calibration targets of Trihedral, PARC (polarimetric active radar calibrator)

1. A. Freeman, “SAR Calibration: An Overview”, IEEE Trans. GeoscienceRemote Sensing, vol. 30, pp.1107-1121, 1992.

2. H.A. Zebker, et al., “Calibrated Imaging Radar Polarimetry”, IEEE Trans. Geoscience Remote Sensing, vol. 29, pp.942-961, 1991.

3. F.T. Ulaby and C. Elachi, Radar Polarimetry for Geoscience Applications, Artech House, 1990.

References:

61

Trihedral(A Typical Passive SAR Calibrator)

62

Scattering Pattern of the Trihedral

63

PARC (Polarimetric Active Radar Calibrator)( Single Antenna Type )

64

3.1 Surface Scattering

3. Polarimetric Scattering Models

3.2 Volume Scattering

-. Scattering Mechanism-. Theoretical Models-. Numerical Analysis-. Experimental Models

-. Radiative Transfer Model-. Numerical Analysis

65

Scattering Mechanism

( ) ( )

( ) ( ) ( )trEt

trEJJtrH

trHt

trE

dc ,,,

,,

∂∂

ε+σ=+=×∇

∂∂

μ−=×∇

Conduction Current

Displacement Current

Maxwell’s Equations:

Constitutive Parameters:

εμσ

: Permittivity: Permeability: Conductivity

Intrinsic Impedance: ε

μ=η

} Material Characteristics

:& HE Exist togetherfor EM Wave

66

E

Induced Currents

iH

iE

ik

kHE ˆ⊥⊥in the far-zone

A Target

sH

sE

sk

Secondary Sources

Primary Source

Fields &Currents

J

J

sk

sk

ik

Scattering Mechanism

67

( ) ( ) ( )∇ ×∇× − = −E r k E r j J r2 ω μ

Wave Equation for sinusoidal waves:

μεω=kwhere J : primary or secondary currents

μ μ μ μ μ= = =0 0 1r r;

( ) ( ) ( ) ( ) ( )rEjrEjjrErEjrH εω≡⎟⎟⎠

⎞⎜⎜⎝

⎛ε′ωσ

−ε′ω=σ+ε′ω=×∇ 1

( ) tan11 0 ε ′′−ε′≡δ−εε≡⎟⎟⎠

⎞⎜⎜⎝

⎛ε′ωσ

−ε′=ε jjj r

ε′ωσ

=δtan rεε=ε′ 0: Loss Tangent rε : Dielectric Constant

Earth Surface:

Scattering Mechanism

68

( ) ( ) ( )∇ + = −2 2A r k A r J rμ

( ) ( ) [ ]A r J rr r

jk r r dvV

=−

− −∫ ∫ ∫μπ

''

exp ' '14

εω×∇

=μ×∇

=j

HEAHs

ss ,

Simpler form of Wave equation:

Unknown Current

Simple Targets: Compute J and evaluate A exactlyComplex Targets: Approximate evaluation of A

AThen,

Scattering Mechanism

69

Ei ks

Hi

y Es

ki Hs

W z x Conducting Strip

Scattering from a conducting strip

( ) ( )J x n H xi≈ ×2(1) Theoretical Computation

= Approximate current(physical optics approx.)

(2) Experimental Measurement

Measure RCSusing a scatterometer

(3) Numerical Computation

( ) ( )E kZ J x H k dxzs z

W

=−

−∫4 002' 'ρ ρ '

( )E E x W yzi zs+ = ≤ ≤ =0 0 0,

(IntegralEquation)

(Boundary Condition)

Current Jx

(precise)

70

Radar Clutter

Radar Scattering fromDistributed Targets

From World War II.For Military Applications.

High ResolutionImaging Radar

(SAR)

Radar Remote SensingFrom 1960s.For civilian, military,and environmental applications.

VolumeScattering

SurfaceScattering

Radar Remote Sensing

71

Radar Scattering Model

Radar Inversion Model

Surface RoughnessSoil Moisture

Radar ScatteringCoefficients

Exact estimation ofsurface roughnessand soil Moisture

Exact models forradar scattering

Scattering Model vs. Inversion Model

72

Surface Scattering

00 3 6 9 12

k l

ks

0.5

1.0

1.5

2.0

2.5

SPMPO

GO

지표면 거칠기 구역( )마이크로파에서

Theoretical Models

0.3,3.0,3.0

≤≤≤

klksm

25.0,76.2

,0.62

≤λ≥

≥

msl

kl

10)cos2(,76.2

,0.6

2

2

≥θ

λ≥

≥

kssl

kl

Horizontal Roughness

Vertic

al Roughness

Rougher

Rougher

73

Theoretical Models

σπ

θ α θpp ppk W k004 4 24 2 0= cos ( sin , ),

SPM (small perturbation model)

22

22

2

2

sincos

)sin1(sin)1(,sincos

sincos

⎟⎠⎞⎜

⎝⎛ θ−ε−θε

θ+ε−θ−ε=α

θ−ε+θ

θ−ε−θ=α

rr

rrvv

r

rhh

Fresnel reflection Coefficient (hh-pol.)

[ ] )(exp)sin(21)0,sin2(

)()0,sin2(5.1222

)sin(22 2

onentialkllskW

GaussianelskW

e

klG

−

θ−

θ+π=θ

π=θWhere roughness spectrum for backscattering is

For Co-pol. (1st-order approximation)

74

Theoretical Models –continue-

SPM (small perturbation model)

( )

( ) ( )∫ ∫ θ+θ−ε+

⋅

−−εθππ

=σ=σ

∞

∞−yxyxyx

zrz

yx

hvrhvvh

dkdkkkkWkkkWkk

kk

RRk

,sin,sin

))(1(cos22

1

21

22

2242

00

For Cross-pol. (2nd-order approximation)

{ }[ ] 5.122222

44)sin(

22

22201

2220

2/)sin(1),sin(

,),sin(

,,2222

−

−θ±

−

+θ±+π=θ±

π=θ±

−−ε=−−=

lkkklskkkW

eelskkkW

kkkkkkkk

yxyxe

lklkk

yxG

yxrzyxz

yx

where

75

Theoretical Models –continue-

Kirchhoff Approximation (KA)

!

)cos2(cos1

2)cos2(22

200

,2

0I∑

∞

=

θ− ⋅θ

θπ

=σn

nks

ainaa nkseRk

th

[ ] 5.122

2)sin(2

)sin(2,

2

θ+

π=

π=

θ−

kln

nlenl

en

kl

G II

( )( )

σ θθ

θaa

sRm m

02

2 4

2

2

02 2

= −⎡

⎣⎢

⎤

⎦⎥cos

exptan

PO (physical optics) Model

GO (geometrical optics) Model

m sl= 2m is the rms slope, for Gaussian surface

76

A Numerical ModelRough Surface Generation

( ) ( ) ( )∑−

+⋅=M

MkjXjWkZ

=j

( ) ( ) ( )[ ]2j2-exp 2 LLsjW π=

( )kjX +

GaussianRandom Vector

77

C ),(21-=

n)(E

),(G-

),(G)(

11y

2d2

2d21

′∈ρρ⎭⎬⎫

′∂

ρ′∂ρ′ρ

⎩⎨⎧

′∂ρ′ρ∂

ρ′∫

y

C y

Edi

nE

E e P

EF f P

y n n

yy n n

n

N

1

11

1

( )= ( ) ,

( )n

( )

n=1

N

′ ′

′′

≡ = ′

∑

∑=

ρ ρ

∂ ρ∂

ρ

[ ] . [ ] [ ][ ] . [ ] [ ]

[ ][ ]

[ ][ ]

Z I ZZ I Z

EF

Vmn mn

mn mnn

n

m11 12

21 22

0 50 5 0

−+

⎡

⎣⎢

⎤

⎦⎥ ⋅⎡

⎣⎢

⎤

⎦⎥ =

⎡

⎣⎢

⎤

⎦⎥

dl )(

),(G- ),(G)( 12d1

2d111

⎭⎬⎫

′∂

ρ′∂ρ′ρ

⎩⎨⎧

′∂ρ′ρ∂

ρ′= ∫ nE

nEE y

c ysy

( )σ θπρ

θρpp pp

spps

DE E0 22= lim

→∞−⎧⎨⎩

⎫⎬⎭

Moment Method

Numerical Algorithm

78

Numerical Results

Can usefor any surface roughnessconditions !

Other NumericalModels:FDTD

79

An Experimental Model

00 vvhvq σσ≡ ( )[ ] ( )[ ]{ }ks0Γ−−−θ+Γ= 6.14.1exp1sin1.025.0 9.00

( ) ( )[ ]p kshh vv≡ = − ⋅ − σ σ θ π0 0 0 314 21 2 0. expΓ

kWkseklkshpkseo

vv 6.0)cos2( 05.025.3)(cos 2)( 1

2.0)(4,1 5.13 θ−−θΓ−=σ

[ ]W kl

klk l

k lk =

+−

−

+

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

( )( . s in )

. ( . s in )

( . s in )

2

2

2

2 21 2 61 0 7 1 1 3 2 6

1 2 6θθ

θ

σ σhh0 = p vv

0 σ σhv vvq0 0=

where

Measurements by a Scatterometer from Soil Surfaces

Very Good

May need correction for SAR

80

Comparison with the old SEM(1992 version)[Oh, et al., IEEE TGRS, 1992]

VV-pol. p and q

81

Ground Truth of Soil Surface

Moisture Contents

mVV

VV

WW

WW

cm cmvw

t

w

d

w

w

b

d

w b

d

= = = ⋅ = −

ρρ ρ 3 3

m WW

mg

w

d

v

b

= = × = ×100 100ρ

(%)

2210

210

210

)()()(

v

v

mCcSccmCbSbb

CaSaa

+++

+++++=εSoil Moisture

Dielectric Constant

TraditionalMethod:Oven-Dry

DielectricProbe Method:

Empirical formulaS=sand, C=clay

[El-Rayes, et al., IEEE TGRS, 1985]

82

Ground Truth of Soil Surface -continue-

SurfaceHeightProfile

SurfaceHeightDensity

SurfaceHeightCorrelation

CorrelationLength, l

RMS Height, s

Surface Roughness A typical example for a surface ofs=1.1 cm, l=8.4 cm

83

0 10 20 30 40 50 60 70 80-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

Incidence Angle (degrees)

Bac

ksca

tt. C

oeff.

(dB

)

SPM, VV-pol.f=1.5 GHzs=0.01 m, l=0.1 m

: Exponential Correlation: Gaussian Correlation: Measured Correlation

Scattering Coefficientsby SPM

Two Different Correlation Functions

Role of the Surface Correlation Function

20 25 30 35 40-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Distance (unit)

Hei

ght (

unit)

Exponential Correlation

Gaussian Correlation

s=0.032 unitl=0.46 unit

Exponential

Gaussian

84

3.7 dB ( Max. error )

0 5 10 15 20 25

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

Normalized Spatial Frequency (dkx )

Rou

ghne

ss

Spe

ctru

m

(dB

)

Corresponding Correlation

: Exponential Function: L = 2000 units: L = 50 units

0 50 100 150 200 250 3000

1

2

3

4

5

6

7

8

9

10

Profile Length (correl. length)

Max

. Erro

r in

Rou

ghne

ss S

pect

rum

(dB

) s = 0.2 unit

l = 1.0 unit

Surface profiles should be long enough !

(Fourier Transform of Correlation Function)

RoughnessSpectrum Maximum error

from Expon. function’s

85

An Inversion Algorithm

Two nonlinear equations, p and q

Find Γ and ks

Compute ε

Soil moisture mvSurface roughness s

2210210210 )()()( vv mCcSccmCbSbbCS ++++++α+α+α=ε

2

0 11

r

r

ε+

ε−=Γ

00 vvhvq σσ≡( ) ( )[ ]2314.000 exp21 0 ksp vvhh −⋅πθ−=σσ≡ Γ

( )[ ] ( )[ ]{ }ks0Γ−−−θ+Γ= 6.14.1exp1sin1.025.0 9.00

86

Inversion Results

Surface Roughness Soil Moisture

Estim

ate

d ks

Estim

ate

d M

v

Measured MvMeasured ks

87

Smooth Surface Moderately Rough Rough Surface

JPL AirSAR Measurements (an example)(Pellston, Michigan, 1990)

L-band Data

88

Volume Scattering

Analysis for Volume Scattering(Multiple Scattering)

Wave Approach(Field)

Intensity Approach(Power)

-. Accurate (Maxwell Equation)-. Coherent-. Complicate equations

(Impossible to solve)-. Approximated Computation

-. Approximated Formulations-. Incoherent-. Computable-. Phase function includes

field scattering

“Analytical Theory” “Radiative Transfer Theory”

89

Radiative Transfer TheorysdJsdJsdIsdIId ssaasa κ+κ+κ−κ−=

I I+dI

Absorption Loss

Scattering Loss

Scattering sourceAbsorption Source(ignore in Radar)

( ) ( ) ( ) ( ) ,ˆ,ˆˆ,ˆˆ,ˆˆ,ˆ4∫ π

Ω′+κ−= dsrIssPsrIds

srIde

Phase Matrix

s′ˆ

s′ˆs

sae κ+κ=κ

90

Polarimetric Backscattering Coefficients

( ) ( ) ( )0000000 ,,, φθ−πφθΤ=φθ II ts

t

( ) ( ) ( )000000 ,,, φθΤ+φθΤ=φθΤ gct

( )[ ]( )[ ]( )[ ]( )[ ]12000

021000

022000

011000

0

,cos4

,cos4

,cos4

,cos4

φθΤθπ=σ

φθΤθπ=σ

φθΤθπ=σ

φθΤθπ=σ

tvh

thv

thh

tvv

Canopy Scattering + Ground Scattering

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ),,,1+

,,1+

,,1+

,,1,

001

3000

001

2000/

0

/000

1200

0

/000

11000

/

000

0

0

00

φθ−πΑπ+φθμ

φθΑπ+φθ−πθμ

θφθΑπ+φθμ

θφθΑπ+φθ−πθμ

=φθΤ

−

−μ−

μ−−

μ−−μ−

+

−

−+

EE

EERe

eREE

eREERe

bdk

dKa

dkdkc

e

e

ee

( ) ( ) 00 /04

/00 , μκ−μκ− −+

θ=φθ ddg

ee eGeT

Transfer Matrices

(1)

(2a)

(2b)

(3)

(4)

91

Radar Backscattering MechanismsIncidence Backscatterdirection direction

(1) (2a) (2b) (3) (4) : 4 different mechanisms

z=0 Diffuse Boundary

Vegetation Layer

z= -d Ground Surface

92

Vector Radiative Transfer AlgorithmScattering Matrix (for a leaf)

Mueller MatrixAverage over distributions of Leaf width, length, Elevation, azimuth angles, etc.

Phase Matrix Extinction Matrix (Eigen Matrix)

Canopy Scattering Matrix, A

Vegetation Transformation Matrix

Reflection Matrix

93

Mueller Matrix

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

−+−−+

−

−

=

∗∗∗∗∗∗

∗∗∗∗∗∗

∗∗

∗∗

)Re()Im()Im(2)Im(2)Im()Re()Re(2)Re(2

)Im()Re(

)Im()Re(22

22

hvvhhhvvhvvhhhvvhhvhhvvv

hvvhhhvvhvvhhhvvhhvhhvvv

hhhvhhhvhhhv

vhvvvhvvvhvv

SSSSSSSSSSSSSSSSSSSSSSSS

SSSSSS

SSSSSS

L

Phase Matrix

( ) ( ) ( ) ppppiissk

K

kppkkiiss dddbdaLbapN φθφθφθφθφθ=φθφθ ∑ ∫∫∫∫

=,;,;,,;,,;,

1P

kp : Joint probability density function for kth particlesNk : Total number of Kth particles in m3 , k: stem, leaf, grain, etc.

94

Canopy Scattering Matrices

( )[ ]( ) ( )[ ]

( ) ( )[ ]( ) ( ) ( )[ ]

( )[ ]( )

A e

E P E

A e e

ij

d

i j

ij

a ij

d

i j

j

1 0 0 0 00 0 0 0

10 0 0 0 0 0 0 0

2 0 0 0 0

1 0 0 0 0

0 0

θ φ π θ φβ π θ φ π β θ φ θ

π θ φ π π θ φ π θ φ θ φ

θ φ π θ φ

β π θ φ π β θ φ θ

β θ φ θ

, ; ,, , sec

, , ; , ,

, ; ,

, , sec

, sec

+ = −− + +

⋅ − + − +

+ = −

− − + +

−

− ( )

( ) ( )[ ]( ) ( ) ( )[ ]

( )[ ]( ) ( )

( )

− +

−

− − + − −

+ −

⋅ + +

+ = −− − + + −

β θ φ π θ

β π θ φ π θ β π θ φ θ

β θ φ π β θ φ θ

θ φ π θ φ π θ φ θ φ

θ φ π θ φβ π θ φ π β π θ

i

i j

d

i j

ij

b ij

d d

i j

E P E

A e e

0 0

0 0 0 0

0 0 0 0

10 0 0 0 0 0 0 0

2 0 0 0 00 0 0

, sec

, sec , sec

, , sec

, , ; , ,

, ; ,, ( )[ ]

( ) ( ) ( )[ ]( )[ ]

( ) ( )[ ]( ) ( )[ ]( )

, sec

, , ; , ,

, ; ,, , sec

, , ;

, , sec

φ θ

π θ φ π π θ φ π π θ φ π θ φ

θ φ π θ φβ θ φ π β π θ φ θ

θ φ π θ φ π π θ

β θ φ π β π θ φ θ

0

10 0 0 0 0 0 0 0

3 0 0 0 00 0 0 0

10 0 0 0

1 0 0 0 0

⋅ − + − + − −

+ = −+ + −

⋅ + + −

−

− + + −

−

E P E

A e

E P

ij

ij

d

i j

i j

( ) ( )[ ]0 0 0 0, ,φ π θ φEij

−

95

Resistive Sheet Approximation[Senior, Sarabandi, Ulaby, Radio Sci., vol.22, pp.1109-1116, 1987]

,cos211

00

−

⎟⎟⎠

⎞⎜⎜⎝

⎛θ+=Γ

ZR

h

1

00

sec21−

⎟⎟⎠

⎞⎜⎜⎝

⎛θ+=Γ

ZR

v

( )10

0−ετ

=k

iZR0Z

0kτε

Leaf: approximated by an electric current sheet with Resistivity R.

with Physical Optics Approximation

Current (leaf) = Reflection coeff. * Current (Perfect conductor)

: Intrinsic impedance (free space): Wave number (free space): Thickness of a leaf: Complex relative permittivity

Scattering Matrix for a leaf

96

An Example (Lawn)Grass Canopy Parameters:

a = 4 mm, b =10 cm with Gaussian distribution

0< theta <90,0< phi <360 withUniform distribution

d =10 cmN =50,000 개/m3

mgv = 0.5mvs = 0.2

rms height =0.5 cmcorr. length =5 cmat f=15 GHz

Back

scat

terin

g C

oeff.

(dB

)

Angle (degree)

HH

VHVV

HV

97

Effect of Direct-Ground Scattering(VV-polarization Case)

Rough surfaceScattering Model:PO (Physical Optics)Model

ks = 1.57kl = 15.7

VV-pol.

Ground

Vegitation

Back

scat

terin

g C

oeff.

(dB

)

Angle (degree)

98

Comparison of Scattering Mechanisms(VV-polarization case)

(1) (2a) (2b) (3) (4)

Direct scattering fromcanopy is dominantin this case.Ba

cksc

atte

ring

Coe

ff. (

dB)

Angle (degree)

Mech-3

Mech-2

Mech-1

VV-pol.Vegetation Canopy