ae502 polarimetric synthetic aperture radaree157/lecture_note/polarimetric radar.pdftransmitter...

EE/Ae 157 b

Polarimetric Synthetic Aperture Radar (1)

POLARIMETRIC SYNTHETIC APERTURE RADAR

Jakob van Zyl 2

OUTLINE

• PRINCIPLES OF POLARIMETRY

– Polarization representations

– Scatterer as polarization transformer

– Characterization of scatterers

– Implementation of polarimeters

– Polarization signatures

• APPLICATIONS OF POLARIMETRY

– Observed polarization signatures

– Unsupervised land cover characterization

– Cloude’s decomposition theorem

– Soil moisture and surface roughness estimation


Jakob van Zyl 3

INTRODUCTION

SPATIAL INFORMATION

RADAR IMAGERS

SPECTRAL INFORMATION

RADAR SPECTROMETERS

IMA

GIN

G R

AD

AR

S

PE

CT

RO

ME

TE

RS

POLARIZATION INFORMATION

RADAR POLARIMETERS

MULTI-FREQUENCY

RADAR POLARIMETERS

IMAGING RADAR

POLARIMETERS

MULTI-FREQUENCY IMAGING RADAR POLARIMETERS


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WAVE POLARIZATIONS: EXAMPLES

HORIZONTAL (LINEAR) VERTICAL (LINEAR)

RIGHT-HAND CIRCULAR LEFT-HAND CIRCULAR


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PRINCIPLES OF POLARIMETRY: FIELD DESCRIPTIONS

POLARIZATION VECTOR POLARIZATION ELLIPSE STOKES VECTOR

S

S0

S1

S2

S3

S0 ah2 av

2

S1 ah2

av2

S0 cos 2 cos 2

S2 2 ahav* S0 cos 2 sin 2

S3 2 ahav* S0 sin 2

, , a ah2

av2

p ah

av

v

av

ah

POLARIZATION

ELLIPSE

MAJOR

AXIS

MINOR

AXIS


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WAVE POLARIZATIONS: GEOMETRICAL REPRESENTATIONS

h

v

av

ah

POLARIZATION

ELLIPSE

MAJOR

AXIS

MINOR

AXIS

S2

450

S0

S3

S1

2

2

LINEAR

POLARIZATION

(HORIZONTAL)

00; 0

0

RIGHT-HAND

CIRCULAR

LEFT-HAND

CIRCULAR

450 LINEAR

POLARIZATION

(VERTICAL)

1800; 0

0

POLARIZATION ELLIPSE POINCARE SPHERE


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EXAMPLE POLARIZATIONS

h

v

h

v

h

v

LINEAR HORIZONTAL LINEAR VERTICAL LEFT-HAND CIRCULAR

p 1

0

p

0

1

p

1

2

1

i

00; 0

0 0

0; 45

0 90

0; 0

0

S

1

1

0

0

S

1

1

0

0

S

1

0

0

1


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DEFINITION OF ELLIPSE ORIENTATION ANGLES

Sometimes the polarization ellipse orientation angle is defined with respect to the

vertical direction. In that case, linear horizontal polarization has an ellipse

orientation angle of +90 degrees or -90 degrees, and linear vertical polarization

is characterized by an ellipse orientation angle of 0 degrees. Both conventions

are used in this viewgraph package.

WARNING


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• Transverse electromagnetic waves are characterized mathematically as 2-

dimensional complex vectors. When a scatterer is illuminated by an

electromagnetic wave, electrical currents are generated inside the scatterer.

These currents give rise to the scattered waves that are reradiated.

• Mathematically, the scatterer can be characterized by a 2x2 complex scattering

matrix that describes how the scatterer transforms the incident vector into the

scattered vector.

• The elements of the scattering matrix are functions of frequency and the

scattering and illuminating geometries.

SCATTERER AS POLARIZATION TRANSFORMER

INCIDENT WAVE

SCATTERER

SCATTERED WAVES


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SCATTERING MATRIX

• Far-field response from scatterer is fully characterized by four complex numbers

• Scattering matrix is also known as Sinclair matrix or Jones matrix

• Must measure a scattering matrix for every frequency and all incidence angles

Eh

Ev

sc

Shh Shv

Svh Svv

Eh

Ev

inc


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COORDINATE SYSTEMS

• All matrices and vectors shown in this package are measured using the

backscatter alignment coordinate system. This system is preferred when

calculating radar-cross sections, and is used when measuring them:

s

i

si

ˆ x

ˆ z

ˆ y

ˆ h t

ˆ v t

ˆ k tˆ k r

ˆ h r

ˆ v rTransmitting

AntennaReceiving

Antenna

Scatterer


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POLARIMETER IMPLEMENTATION

• To fully characterize the scatterer, one must measure the full scattering matrix:

• Setting one of the elements of the transmit vector equal to zero allows one to

measure two components of the scattering matrix at a time:

• This technique is commonly used to implement airborne and spaceborne SAR

polarimeters, such as AIRSAR and SIR-C.

Eh

Ev

rec

Shh Shv

Svh Svv

Eh

Ev

tr

Shh

Svh

Shh Shv

Svh Svv

1

0

inc

;Shv

Svv

Shh Shv

Svh Svv

0

1

inc


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TIMING

Transmission:

Horizontal

Vertical

Reception:

Horizontal

Vertical

HH HH HHHV HV

VH VV VH VV VH

Transmitter

Receiver

Receiver

BLOCK DIAGRAM

Horizontal

Vertical


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• In the implementation shown, the scattering matrix elements are measured in

pairs. The two pairs are measured at different times, i.e. from (slightly) different

viewing directions.

• Due to the speckle effect, this means that the two pairs of signals may not be

fully correlated.

• In the AIRSAR case, this decorrelation is of little consequence, since the

distance traveled by the aircraft between successive pulses (~ 50 cm) is small

compared to the system resolution (~ 2-5 m).

• In spaceborne cases, like SIR-C, the decorrelation distance is much closer in

length to the distance traveled by the antenna during the interpulse period.

Therefore, significant decorrelation could occur. This could be corrected

through resampling (interpolation) of one channel with respect to the other.


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MATHEMATICAL CHARACTERIZATION OF SCATTERERS:

SCATTERING MATRIX

• The radiated and scattered electric fields are related through the complex 2x2

scattering matrix:

• The (complex) voltage measured at the antenna terminals is given by the scalar

product of the receiving antenna polarization vector and the received wave

electric field:

• The measured power is the magnitude of the (complex) voltage squared:

V prec S p rad

Esc S prad

P VV* p

rec S prad

2

NOTE: Radar cross-section is proportional to power


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COVARIANCE MATRIX

• We can rewrite the expression for the voltage as follows:

• The first vector contains only antenna parameters, while the second contains

only scattering matrix elements. Using this expression in the power expression,

one finds

• The matrix is known as the covariance matrix of the scatterer

*

* * * * *;P VV AT TA ATT A A C A C TT

C

V prec S prad

phrec ph

radShh phrecpv

radShv pvrec ph

radSvh pvrecpv

radSvv

phrec ph

rad phrec pv

rad pvrec ph

rad pvrec pv

rad

Shh

Shv

Svh

Svv

˜ A T


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STOKES SCATTERING OPERATOR

• The power expression can also be written in terms of the antenna Stokes

vectors. First consider the following form of the power equation:

• The vector in the expression above is a function of the transmit antenna

parameters as well as the scattering matrix elements.

P prec E sc prec E sc *

phrecEh

sc pvrecEv

sc phrecEhsc pvrecEvsc *

phrecphrec* Eh

scEhsc* pv

recpvrec* Ev

scEvsc* ph

recpvrec* Eh

scEvsc* pv

recphrec* Ev

scEhsc*

phrecphrec*

pvrecpvrec*

phrecpvrec*

pvrecphrec*

EhscEhsc*

EvscEvsc*

EhscEvsc*

EvscEhsc*

grec X

X


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• Using the fact that , it can be shown that can also be written as

where

• This means that the measured power can also be expressed as:

XEsc S prad

X W grad

W

ShhShh* ShvShv

* ShhShv* ShvShh

*

SvhSvh* SvvSvv

* SvhSvv* SvvSvh

*

ShhSvh* ShvSvv

* ShhSvv* ShvSvh

*

SvhShh* SvvShv

* SvhShv* SvvShh

*

P grec W g red


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• From the earlier definition of the Stokes vector, we note that the Stokes vector

can be written as:

• This means that we can express the measured power as:

• The matrix is known as the Stokes scattering operator. It is also called

Stokes matrix.

S

phph* pv pv

*

phph* pv pv

*

phpv* ph

*pv

i( phpv* ph

*pv)

1 1 0 0

1 1 0 0

0 0 1 1

0 0 i i

phph*

pv pv*

phpv*

ph*pv

R g g R 1

S

M

P Srec R

1W R

1Srad

Srec M Srad~


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POLARIZATION SYNTHESIS

• Once the scattering matrix, covariance matrix, or the Stokes matrix is known,

one can synthesize the received power for any transmit and receive antenna

polarizations using the polarization synthesis equations:

Scattering matrix:

Covariance Matrix:

Stokes scattering operator:

• Keep in mind that all matrices in the polarization synthesis equations must be

expressed in the backscatter alignment coordinate system.

P Srec M Srad

P A C A*

P prec S prad

2


Jakob van Zyl 21

POLARIZATION SIGNATURE

• The polarization signature (also known as the polarization response) is a

convenient graphical way to display the received power as a function of

polarization.

• Usually displayed assuming identical transmit and receive polarizations (co-

polarized) or orthogonal transmit and receive polarizations (cross-polarized).


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Polarization Response

• Another way to look at the signature is that the scatter changes the co-polarized

return from a sphere (using the Poincare representation) to some other figure:


Jakob van Zyl 23

POLARIZATION SIGNATURE OF TRIHEDRAL CORNER REFLECTOR

S c1 0

0 1

C c

2

1 0 0 1

0 0 0 0

0 0 0 0

1 0 0 1

M 1

2c2

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

ˆ v t

ˆ h t

l

c k0l

2

12


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Jakob van Zyl 25


Using the Stokes matrix approach, we can write the received power of the trihedral corner reflector as

P k02l4

24 21 cos2 2 cos2 2 sin2 2 cos2 2 sin2 2

k02l4

24 21 cos

22 sin2 2

k02l4

2421 cos 4

where the top sign is for the co-polarized case, and the bottom sign is for the cross-polarized case. It isimmediately apparent that the received power is independent of the ellipse orientation angle, as shownby the signatures on the previous page.

For the co-polarized case, the maximum is found when 0 , i.e. for linear polarizations, and the

minimum occurs when 450

, i.e. for circular polarizations. The positions of the maxima and minimaare reversed in the case of the cross-polarized signatures.

This behavior is explained by the fact that the reflected waves have the opposite handedness than thetransmitted ones. Therefore, if either circular polarization is transmitted, the reflected wave is polarizedorthogonally to the transmitted wave, leading to maximum reception in the cross-polarized, andminimum reception in the co-polarized case.


Jakob van Zyl 26

POLARIZATION SIGNATURE OF A DIHEDRAL CORNER REFLECTOR

ˆ v t

ˆ h t

S c1 0

0 1

C c

2

1 0 0 1

0 0 0 0

0 0 0 0

1 0 0 1

M 1

2c2

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

2b

a

c k0ab


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Three-Dimensional Response


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POLARIZATION SIGNATURE OF A SHORT THIN CYLINDER

S c0 0

0 1

C c

2

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 1

M 1

4c2

1 1 0 0

1 1 0 0

0 0 0 0

0 0 0 0

c k02l3

3 ln 4l /a 1 ; a radius of cylinder

ˆtv

ˆth

l


Jakob van Zyl 29



Jakob van Zyl 30


S c1 1

1 1

C c

2

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

M c2

1 0 1 0

0 0 0 0

1 0 1 0

0 0 0 0

c k02l3

6 ln 4l / a 1 ; a radius of cylinder

ˆtv

ˆth

l


Jakob van Zyl 31

POLARIZATION SIGNATURE OF A 45o SHORT THIN CYLINDER


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POLARIZATION SIGNATURE OF RIGHT-HANDED HELIX

S 1

2

1 i

i 1

C

1

4

1 i i 1

i 1 1 i

i 1 1 i

1 i i 1

M 1

4

1 0 0 1

0 0 0 0

0 0 0 0

1 0 0 1

ˆ v t

ˆ h t


Jakob van Zyl 33

RIGHT-HANDED HELIX


Jakob van Zyl 34

POLARIZATION SIGNATURE OF LEFT-HANDED HELIX

S 1

2

1 i

i 1

C

1

4

1 i i 1

i 1 1 i

i 1 1 i

1 i i 1

M 1

4

1 0 0 1

0 0 0 0

0 0 0 0

1 0 0 1

ˆ v t

ˆ h t


Jakob van Zyl 35

LEFT-HANDED HELIX


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SCATTERING MODELS AND

OBSERVATIONS


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Signatures


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Rough Surface Scattering - Roughness


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Rough Surface Scattering – Dielectric Constant


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Surface Characteristics – Geometrical Properties


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Surface Characteristics – Dielectric Properties

0

5

10

15

20

25

30

35

40

45

0 0.1 0.2 0.3 0.4 0.5

Volumetric Soil Moisture

Rea

l Pa

rt of Ep

silon

Hallikainen

Wang and Schmugge

Brisco et al

Dobson et al.


Jakob van Zyl 42

POLARIZATION SIGNATURE OF SINGLE SHORT THIN CYLINDER

S c0 0

0 1

C c

2

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 1

M 1

4c2

1 1 0 0

1 1 0 0

0 0 0 0

0 0 0 0

c k02l3

3 ln 4l /a 1 ; a radius of cylinder

ˆtv

ˆth

l


Jakob van Zyl 43

SHORT CYLINDERS ORIENTED COSINE SQUARED AROUND

VERTICAL


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THIN CYLINDERS ORIENTED UNIFORMLY RANDONLY


Jakob van Zyl 45

MODEL OF VEGETATION SCATTERING

Vegetation Layer

Ground Surface

, , , ,p p p p pa l p

, , , ,s s s s sa l p

, , ,g g lh l s

b


Jakob van Zyl 46

SCATTERING MECHANISMS

i

i

ii

b

2

z

3

4 1


Jakob van Zyl 47

OBSERVED POLARIZATION SIGNATURES:

L-BAND POLARIZATION SIGNATURES OF THE OCEAN


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SAN FRANCISCO


Jakob van Zyl 49


SAN FRANCISCO

At each frequency the ocean area scatters in a manner consistent with models of slightly roughsurface scattering, the urban area like a dihedral corner reflector, while the park and natural terrainregions scatter much more diffusely, that is, the signatures possess large pedestals. This indicatesthe dominant scattering mechanisms responsible for the backscatter for each of the targets. Theocean scatter is predominantly single bounce, slightly rough surface scattering. The urban regionsare characterized by two-bounce geometry as the incident waves are twice forward reflected fromthe face of a building to the ground and back to the radar, or vice versa. The apparent diffusenature of the backscatter from the park and natural terrain indicates that in vegetated areas thereexists considerable variation from pixel to pixel of the observed scattering properties, leading to thehigh pedestal. This variation may be due to multiple scatter or to a distinct variation in dominantscattering mechanism between 10m resolution elements.


Jakob van Zyl 50


GEOLOGY


Jakob van Zyl 51


GEOLOGY

The previous viewgraph shows polarization signatures extracted from an AIRSAR image of thePisgah lava flow in California. Signatures correspond to the flow itself (a very rough surface), analluvial fan (medium roughness), and from the playa next to the flow (a very smooth surface). Notethat as the roughness of the surface increases, so does the observed pedestal height. This is quiteconsistent with the predictions of the slightly rough surface models, even though the surface r.m.s.heights exceed the strict range of validity of the model. The exception is the playa case, where thepedestal at P-band is higher than that at L-band. Possible explanations for this behavior includesubsurface scattering due to increased penetration at P-band, or signal-to-noise limitations for thevery smooth surface.


Jakob van Zyl 52


SEA ICE


Jakob van Zyl 53


SEA ICE

Note the pedestal height as a function of frequency for the multi-year ice - the P-band pedestal isquite small, while the C-band pedestal is the greatest of the three. For the first year ice, exactly theopposite behavior is seen. The P-band signature shows the highest and the C-band signature thelowest pedestal. The multi-year ice behavior may be explained if we consider the ice to be formedof two layers, where the upper layer consists of randomly oriented oblong inclusions about the sizeof a C-band wavelength, several centimeters. The lower layer forms a solid, but slightly roughsurface. In this situation the C-band signal would interact strongly with the diffuse scattering upperlayer, giving rise to the high pedestal. The longer wavelength L- and P-band signals would passthrough the upper and be scattered by the lower layer, which is smooth enough to exhibit fairlypolarized backscatter. On the other hand, similar characteristics are also observed for simplerough surface scattering also, as previously explained.

At present we have no model to explain the first year ice behavior.

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