interferometric radiometry measurement concept: the visibility equation
DESCRIPTION
INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY EQUATION. I. Corbella, F. Torres, N. Duffo, M. Martín-Neira. Interferometric Radiometry. Technique to enhance spatial resolution without large bulk antennas. - PowerPoint PPT PresentationTRANSCRIPT
Universitat Politècnica de Catalunya
•Remote•Sensing•Laboratory
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.
Universitat Politècnica de Catalunya
•Remote•Sensing•Laboratory
28th July 2011 IGARSS 11. Vancouver. Canada 3/31
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
Universitat Politècnica de Catalunya
•Remote•Sensing•Laboratory
28th July 2011 IGARSS 11. Vancouver. Canada 4/31
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
Universitat Politècnica de Catalunya
•Remote•Sensing•Laboratory
28th July 2011 IGARSS 11. Vancouver. Canada 5/31
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)
Universitat Politècnica de Catalunya
•Remote•Sensing•Laboratory
28th July 2011 IGARSS 11. Vancouver. Canada 6/31
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
Universitat Politècnica de Catalunya
•Remote•Sensing•Laboratory
28th July 2011 IGARSS 11. Vancouver. Canada 7/31
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
Universitat Politècnica de Catalunya
•Remote•Sensing•Laboratory
28th July 2011 IGARSS 11. Vancouver. Canada 8/31
-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)
Universitat Politècnica de Catalunya
•Remote•Sensing•Laboratory
28th July 2011 IGARSS 11. Vancouver. Canada 9/31
Y-shape instrument (19 antennas per arm)
= 1.73 deg = 2.46 deg
Rectangular window Blackmann window
Examples of Synthetic beam
Universitat Politècnica de Catalunya
•Remote•Sensing•Laboratory
28th July 2011 IGARSS 11. Vancouver. Canada 10/31
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
Universitat Politècnica de Catalunya
•Remote•Sensing•Laboratory
28th July 2011 IGARSS 11. Vancouver. Canada 11/31
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
Universitat Politècnica de Catalunya
•Remote•Sensing•Laboratory
28th July 2011 IGARSS 11. Vancouver. Canada 12/31
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
Universitat Politècnica de Catalunya
•Remote•Sensing•Laboratory
28th July 2011 IGARSS 11. Vancouver. Canada 13/31
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
Universitat Politècnica de Catalunya
•Remote•Sensing•Laboratory
28th July 2011 IGARSS 11. Vancouver. Canada 14/31
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
Universitat Politècnica de Catalunya
•Remote•Sensing•Laboratory
28th July 2011 IGARSS 11. Vancouver. Canada 15/31
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
Universitat Politècnica de Catalunya
•Remote•Sensing•Laboratory
28th July 2011 IGARSS 11. Vancouver. Canada 16/31
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
Universitat Politècnica de Catalunya
•Remote•Sensing•Laboratory
28th July 2011 IGARSS 11. Vancouver. Canada 17/31
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:
Universitat Politècnica de Catalunya
•Remote•Sensing•Laboratory
28th July 2011 IGARSS 11. Vancouver. Canada 18/31
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
Universitat Politècnica de Catalunya
•Remote•Sensing•Laboratory
28th July 2011 IGARSS 11. Vancouver. Canada 19/31
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
Universitat Politècnica de Catalunya
•Remote•Sensing•Laboratory
28th July 2011 IGARSS 11. Vancouver. Canada 20/31
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
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
Universitat Politècnica de Catalunya
•Remote•Sensing•Laboratory
28th July 2011 IGARSS 11. Vancouver. Canada 21/31
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
Universitat Politècnica de Catalunya
•Remote•Sensing•Laboratory
28th July 2011 IGARSS 11. Vancouver. Canada 22/31
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 ),(
Universitat Politècnica de Catalunya
•Remote•Sensing•Laboratory
28th July 2011 IGARSS 11. Vancouver. Canada 23/31
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
Universitat Politècnica de Catalunya
•Remote•Sensing•Laboratory
28th July 2011 IGARSS 11. Vancouver. Canada 24/31
-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
Universitat Politècnica de Catalunya
•Remote•Sensing•Laboratory
28th July 2011 IGARSS 11. Vancouver. Canada 25/31
-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
Universitat Politècnica de Catalunya
•Remote•Sensing•Laboratory
28th July 2011 IGARSS 11. Vancouver. Canada 26/31
-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
Universitat Politècnica de Catalunya
•Remote•Sensing•Laboratory
28th July 2011 IGARSS 11. Vancouver. Canada 27/31
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
Universitat Politècnica de Catalunya
•Remote•Sensing•Laboratory
28th July 2011 IGARSS 11. Vancouver. Canada 28/31
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
Universitat Politècnica de Catalunya
•Remote•Sensing•Laboratory
28th July 2011 IGARSS 11. Vancouver. Canada 29/31
Full polarimetric SMOS snapshot
xxBT
yyBT
]Re[ xyBT ]Im[ xy
BT
North-west of Australia
Universitat Politècnica de Catalunya
•Remote•Sensing•Laboratory
28th July 2011 IGARSS 11. Vancouver. Canada 30/31
SMOS sky image
Universitat Politècnica de Catalunya
•Remote•Sensing•Laboratory
28th July 2011 IGARSS 11. Vancouver. Canada 31/31
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.