interferometric radiometry measurement concept: the visibility equation

Universitat Politècnica de Catalunya

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INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY

EQUATION

I. Corbella, F. Torres, N. Duffo, M. Martín-Neira

Universitat Politècnica de Catalunya

•Remote•Sensing•Laboratory

28th July 2011 IGARSS 11. Vancouver. Canada 2/31

Interferometric Radiometry• Technique to enhance spatial resolution without

large bulk antennas.• Based on cross-correlating signals collected by pairs

of ”small” antennas (baselines).• Image obtained by a Fourier technique from

correlation measurements. No scanning needed.• Examples:

– Precedent: Michelson (end of 19th century). Astronomical observations at optical wavelengths.

– Radioastronomy: Very Large Array (1980). 27 dish antennas, 21 km arm length Y-shape. Various frequencies.

– Earth Observation: SMOS (2009). 69 antennas, 4m arm length Y-shape. L-band.

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2122

21

221 2 bbbbbbvd

Interferometry: Fringes

distant point source

d

α0

Δℓ

x

z

Δr=d cos α0

b1 b2

)/(cos1 crtAb )/(cos2 ctAb

vd

Δℓ/λΔr/λ

A2

2A2

)//(2cos22 rAAvd

Quadratic detector

Cross-correlationTotal power

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Michelson’s “Fringe Visibility”:

usincminimafringemaximafringe

minimafringemaximafringe

Fringe Visibility

)//(2cossinc ruIIvd

distant small source with constant intensity

I

Cross-correlation for Δℓ=0: 021 2cossinc uuIbb

d

α0

Δξ

Δℓ ξ0=cos α0

x

z

vd

u=d/λ

Δr=d cos α0

b1 b2

vd

Δℓ/λΔr/λ

I

2I0

1

0.5

0.75

uΔξ

Δr/λ=uξ0

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Definition 02sinc)( ujeuIuV

Complex Visibility

• Michelson’s “fringe visibility” is the amplitude of the complex visibility |V(u)|=I·|sinc uΔξ| normalized to the total intensity of the source.

• The cross correlation between both signals for Δℓ=0 is the real part of the complex visibility <b1 b2>=Re[V(u)]. The imaginary part is obtained by adding a 90º phase shift (quarter wavelength) to one of the signals.

• The complex visibility is the Fourier Transform of the Intensity distribution expressed as a function of the director cosine ξ: V(u)=F[I(ξ)]

d

α

Δξξ=cos α

x

z

b1 b2

u=d/λ

Δξ

ξ0

I0

ξ u

0

0)( II ujeuIuV 020 sinc)(

I0Δξ=II(ξ) V(u)

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x

y

d

deTuV ujB

2)()(

ddeTvuV vujB

)(2),(),(

The spatial resolution is achieved• by synthesized beam in ξ • by antenna pattern in η

x

y

dv

u

The spatial resolution is achieved by synthesized beam in both dimensions (ξ and η).Different options for geometry:

• Y-shape, Rectangular, T-shape, Circle, Others

d

u

Use Brightness Temperature (TB) instead of intensity (I):1-

D

2-D

Interferometric radiometres

1

122

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Only limited values of (u,v) are available: The measured visibility function is necessarily windowed.

ddeTvuV vujB

)(2),(),(

dudvevuVT vujB

)(2),(),(

dudvevuVvuWT vujB

)(2),(),(),(ˆ

ddAFTT BB ),(),(),(ˆ

Direct equation

Fourier inversion

Retrieved brightness temperature

Convolution integral

• Array Factor: Inverse Fourier transform of the window• It is the “synthetic beam”. It sets the spatial resolution• Its width depends on the maximum (u,v) values (antenna maximum

spacing)

Spatial resolution: Synthetic beam

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-0.5 0 0.50

0.2

0.4

0.6

0.8

1

(A/)

t( )

Comparison between Interferometric and Real apertures

InterferometricReal

Comparison with real apertures

Rectangular u-v coverage and no window

u

v

uM-uM

-vM

vM MM vuAF 2sinc2sinc

A=Δxmax, B=Δymax: Maximum distance between antennas in each direction

A

uM B

vM

Physical aperture with uniform fields

x

y

A

BEH

BA

AF 2sinc2sinc

22

sincsinc),(

BA

t

0.60

0.88

(for small angles around boresight)

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Y-shape instrument (19 antennas per arm)

= 1.73 deg = 2.46 deg

Rectangular window Blackmann window

Examples of Synthetic beam

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42,1

),(),(1

2,1dtTT BA

r1

r2

b1

b2

1

2

1 )( AkTfb

2

2

2 )( AkTfb

• Power spectral density:Antenna temperature

• Cross-Power spectral density:Visibility

12*21 )()( kVfbfb

4

)(*21

21

1221),(),(),(

1deFFTV rrjk

nnB

(units: Kelvin)

phase difference

(complex valued)

Microwave Radiometry formulation

Antenna field patterns

TB(θ,)

Extended source of thermal radiation

Antenna power pattern

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The anechoic chamber paradox

T kTdtkTkTb A

4

2,12,1

2,1

2

2,1 ),(1

V12 is apparently non-zero and antenna dependent

4

*21

21

12*21 ),(),( deFF

kTkVbb rjk

nn

But V12 should be zero (Bosma Theorem)

anechoic chamber at constant temperature

Experiments confirm that V12=0

T

• Power spectral density: Antenna temperature

• Cross Power spectral density: Visibility

TA=T (OK!)

12 rrr

b1

b2

T

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The “–Tr” term

TThe solution is found when all noise contributors are taken into account.

)(),(),(1 *

2212*2111

4

*21

21

SSSSdeFF rjknn

Cross power spectral density for total output waves:

Tr

Tr

Consistent with Bosma theorem:• Tr: equivalent temperature of noise produced by the receivers and

entering the antennas. This noise is coupled from one antenna to the other.

• If the receivers have input isolators, Tr is their physical temperature.

4

)(*21

21

*21

21),(),(),( deFFTTk

bb rrjknnrB

0),(if *21 bbTT rB

b1

b2

a1

a2

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0 5 10 15 2010

-4

10-3

10-2

10-1

100

101

Antenna separation normalized to wavelength

K

Visibility of an empty chamber at 293K

No -Tr termTheoryMeasurement

4

*21

21

ch ),(),( deFFT rjk

nn

4

*21

21

ch ),(),( deFFTT rjk

nnr

Empty chamber visibilityResult from IVT at ESA’s Maxwell Chamber

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Cold Sky Visibility

Arm A

Chamber Chamber

Chamber

SkySky

Sky

Arm B

Arm CBlue:SMOS at ESA’s Maxwell Chamber

Red:SMOS on flight during external calibration

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Limited bandwidth and time correlation

Receiver 1

Receiver 2

b1

b2 Complex correlation

bs1

bs2

Average powerBandwidth: B1

Gain: G1

Bandwidth: B2Gain: G2

)(2)( 1111

2

1 RA TTBkGtb

)(2)( 2222

2

2 RA TTBkGtb

TA: Antenna temperature (K)

12122121*21 2)()( VGBBGGktbtb

TR: Receiver noise temperature (K)

dt

dt

dt

V12: Visibility (K)

b1,2(t): Analytic signals

412

*21

21

120)/(~1

decrrFFTTV rjknnrB

dfefHfHGGBB

etr ftj

tfj

2

0

*21

2121

2

12 )()()(~0

c

fk 0

0

2

Centre frequency: f0

Fringe washing function)0(~/)(~)(~

121212 rtrtr

)0(~1212 rG

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Director cosines and antenna spacing

distant source point

R

x y

z

Antenna location at coordinates (x1,y1,z1)

θ

r1

111

21

1 2z

R

zy

R

yx

R

x

R

dRr

Director cosines

cossinR

x sinsinR

y

At large distances (R>>d1)

d1

For two close antennas in the x-y plane: )()( 121212 yyxxrr

Phase difference: )(2)( 12 vurrkrk

12 xx

u

12 yyv

Antenna normalized spacing

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Notes:

* ukj and vkj are defined in terms of the wavelength at the centre

frequency.

* The visibility has hermiticity property

The visibility equation

1

)(2

0

*

2222

~),(),(1

),(1),(

dde

f

vurFF

TTvuV kjkj vujkjkj

kjnjnkrB

jk

kjkjkj

0f

vu

c

rr

c

r kjkjkj

Physical temperature of receivers Tr=(Trk+Trj)/2

*kjjk VV

0kj

kj

xxu

0kj

kj

yyv

Antenna relative

spacing:

Decorrelation time:

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The zero baseline

• V(0,0) is equal to the difference between the antenna temperature and the receivers’ physical temperature.

• It is redundant of order equal to number of receivers.• At least one antenna temperature must be measured.• In SMOS, two methods have been considered:

– Three dedicated noise-injection radiometers (NIR)– All receivers operating as total power radiometers.

• The selected baseline method is the first one (NIR)

putting u=v=0 rAknkrB

kk TTddF

TTV

1

2

2222

),(1

),(1)0,0(

V(0,0)=TA-Tr

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Polarimetric brightness temperatures

2*ppp

ppB EEET

ΔΩObservation point

Brightness temperature at p polarisation:

*qp

pqB EET

** pqBpq

qpB TEET

Complex Brightness temperature at p-q polarisations:

qqB

ppB TTI qq

Bpp

B TTQ ][2 pqBTeU ][2 pq

BTmV

Relation with Stokes parameters:

(p,q): orthogonal polarization basis (linear, circular, …)

qEpEE qp ˆˆ

2

0

22

BkTE

Spectral power density:

if

2

0

22

qqB

ppBqp TTkEE

2*qqq

qqB EEET Brightness temperature at q polarisation:

Thermal radiation

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Polarimetric interferometric radiometer

4

)(*21

21

1221),(),(

1deTFFV rrjkpq

Bqp

qp

pq

4

)(*21

21

1221))(,(),(

1deTTFFV rrjk

rpp

Bpp

pp

pp

bp1

bq1

OMT

),(1 pFp output

q output

),(1 qF

Antenna 1

bp2

bq2

OMT

),(2 pFp output

q output

),(2 qF

Antenna 2

Visibility at pp polarization

4

)(*21

21

1221))(,(),(

1deTTFFV rrjk

rqq

Bqq

qq

qq

Visibility at qq polarization

Visibility at pq polarization

4

)(*21

21

1221),(),(

1deTFFV rrjkqp

Bpq

pq

qp

Visibility at qp polarization

*1221pqqp VV

*1221qppq VV

*1221pppp VV

*1221qqqq VV

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1

)(2

0

*

2222

~),(),(1

),(1),(

dde

f

vurFF

TTvuV kjkjkj vujkjkj

kjnjnk

rB

jk

kjkjkj

Visibility: For any pair of antennas k,j (k≠j)

Physical temperature of receivers: Trkj=(Trk+Trj)/2

*kjjk VV

0kj

kj

xxu

0kj

kj

yyv

Antenna relative

spacing:

1

2

2222

),(1

),(1

ddFT

T nkB

kkA aNk 1

Antenna Temperature: For any single antenna k

(hermiticity)

Image Reconstruction

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The Flat-Target response

1

)(2

0

*

2222

~),(),(

1

1),(

dder

FFjkFTR kjkj vuj

f

kjvkju

kj

j

nj

k

nk

Definition

The visibility of a completely unpolarised target having equal brightness temperature in any direction (“flat target”) is:

),()(),( jkFTRTTvuVkjrBkjkj

FTkj

MeasurementIt can be measured by pointing the instrument to a known flat target as the cold sky (galactic pole).

EstimationIt can also be estimated (computed) from antenna patterns and fringe washing functions measurements.

For large antenna separation, FTR≈0

kjrB

FTkj TTVjkFTR ),(

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Image reconstruction consists of solving for T(ξ,η) in the following equation

1in 22 (zero outside)

ddeTvuV vuj )(2),(),(

0

*

22

~),(),(

1

),(

f

vur

FFTT kj

j

nj

k

nk

and V and T depend of the approach chosen:

),( kjkjkj vuVkk rA TT

rB TT ),(

),(),( jkFTRTvuVkjrkjkjkj

kAT ),( BT

),()(),( jkFTRTTvuVkjkj rAkjkjkj

AB TT ),( 0

),( vuV )0,0(V ),( T

#1

#2

#3

Approach

where

T(ξ,η) is only function of (ξ,η)

0, vu

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-10 -5 0 5 10

-8

-6

-4

-2

0

2

4

6

8

u

v

-4 -2 0 2

-4

-3

-2

-1

0

1

2

3

4

x/

y/

Antenna Positions and numberingu

17

13

19

8

14

v

Example: NEL=6; d=0.875

Principal values

Hermitic values

Hexagonal sampling (MIRAS)

u,v points

NEL=6

Na=3NEL+1=19

u=(xj-xk)/λ0

v=(yj-yk)/λ0

pair (k,j):

• Number of antenna pairs: Na(Na-1)/2

• Number of unique (u-v) points: 3[NEL(NEL+1)]

Na: Total number of antennas

NEL : Number of antennas in each arm. An antenna in the centre is considered.

3[NEL(NEL+1)]=126

3[NEL(NEL+1)]=126

• Number of points in the “star”: 6[NEL(NEL+1)]+1

253 total points

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-2 -1 0 1 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-20 -10 0 10 20

-20

-15

-10

-5

0

5

10

15

20

u

v

Unit circle

Alias-free Field Of View (FOV):Zone of non-overlapping unit circle aliases

Discrete sampling produces spatial periodicity: AliasesVisibility: (u-v) domain Brightness temperature: (ξ-η) domain

Aliasing

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-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

hsat=755 km, tilt=32.5º, d=0.875

Strict and extended alias-free field of view

Zone of non-overlapping Earth contours

Earth ContourUnit Circle

Earth aliasesUnit Circle aliases

Alias-Free Field of View Extended Alias-Free Field of view

Antenna Boresight

Zone of non-overlapping unit circles

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Projection to ground coordinates

-1000 -500 0 500 1000

-200

0

200

400

600

800

1000

1200

Cross track coordinate (km)

Alo

ng

tra

ck c

oo

rdin

ate

(km

)

hsat=755 km, tilt=32.50º, d=0.875

Swath: 525 km

Nadir

Boresight

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Geo-location

Regular grid in director cosines Irregular grid in lat-lon

• The regular grid in xi-eta is mapped into irregular grid in longitude-latitude

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Full polarimetric SMOS snapshot

xxBT

yyBT

]Re[ xyBT ]Im[ xy

BT

North-west of Australia

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SMOS sky image

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Conclusions• Interferometric radiometry has a long heritage that

goes back to the 19th century. SMOS has demonstrated its feasibility for Earth Observation from space.

• The complete visibility equation for a microwave interferometer must include the effect of antenna cross coupling and receivers finite bandwidth.

• Image reconstruction is based on Fourier inversion. Improved performance is achieved by using the flat target response.

• Aliasing induces a complex field of view. In SMOS two zones with different data quality exist: Alias-free and extended alias-free.

• Spatial resolution, sensitivity, incidence angle and rotation angle have significant variations inside the Field of view.