Universitat Politècnica de Catalunya INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY...
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Transcript of Universitat Politècnica de Catalunya INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY...
![Page 1: Universitat Politècnica de Catalunya INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY EQUATION I. Corbella, F. Torres, N. Duffo, M. Martín-Neira.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649de55503460f94addd8b/html5/thumbnails/1.jpg)
Universitat Politècnica de Catalunya
•Remote•Sensing•Laboratory
INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY
EQUATION
I. Corbella, F. Torres, N. Duffo, M. Martín-Neira
![Page 2: Universitat Politècnica de Catalunya INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY EQUATION I. Corbella, F. Torres, N. Duffo, M. Martín-Neira.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649de55503460f94addd8b/html5/thumbnails/2.jpg)
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.
![Page 3: Universitat Politècnica de Catalunya INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY EQUATION I. Corbella, F. Torres, N. Duffo, M. Martín-Neira.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649de55503460f94addd8b/html5/thumbnails/3.jpg)
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
![Page 4: Universitat Politècnica de Catalunya INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY EQUATION I. Corbella, F. Torres, N. Duffo, M. Martín-Neira.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649de55503460f94addd8b/html5/thumbnails/4.jpg)
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
![Page 5: Universitat Politècnica de Catalunya INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY EQUATION I. Corbella, F. Torres, N. Duffo, M. Martín-Neira.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649de55503460f94addd8b/html5/thumbnails/5.jpg)
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)
![Page 6: Universitat Politècnica de Catalunya INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY EQUATION I. Corbella, F. Torres, N. Duffo, M. Martín-Neira.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649de55503460f94addd8b/html5/thumbnails/6.jpg)
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
![Page 7: Universitat Politècnica de Catalunya INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY EQUATION I. Corbella, F. Torres, N. Duffo, M. Martín-Neira.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649de55503460f94addd8b/html5/thumbnails/7.jpg)
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
![Page 8: Universitat Politècnica de Catalunya INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY EQUATION I. Corbella, F. Torres, N. Duffo, M. Martín-Neira.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649de55503460f94addd8b/html5/thumbnails/8.jpg)
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)
![Page 9: Universitat Politècnica de Catalunya INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY EQUATION I. Corbella, F. Torres, N. Duffo, M. Martín-Neira.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649de55503460f94addd8b/html5/thumbnails/9.jpg)
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
![Page 10: Universitat Politècnica de Catalunya INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY EQUATION I. Corbella, F. Torres, N. Duffo, M. Martín-Neira.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649de55503460f94addd8b/html5/thumbnails/10.jpg)
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
![Page 11: Universitat Politècnica de Catalunya INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY EQUATION I. Corbella, F. Torres, N. Duffo, M. Martín-Neira.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649de55503460f94addd8b/html5/thumbnails/11.jpg)
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
![Page 12: Universitat Politècnica de Catalunya INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY EQUATION I. Corbella, F. Torres, N. Duffo, M. Martín-Neira.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649de55503460f94addd8b/html5/thumbnails/12.jpg)
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
![Page 13: Universitat Politècnica de Catalunya INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY EQUATION I. Corbella, F. Torres, N. Duffo, M. Martín-Neira.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649de55503460f94addd8b/html5/thumbnails/13.jpg)
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
![Page 14: Universitat Politècnica de Catalunya INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY EQUATION I. Corbella, F. Torres, N. Duffo, M. Martín-Neira.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649de55503460f94addd8b/html5/thumbnails/14.jpg)
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
![Page 15: Universitat Politècnica de Catalunya INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY EQUATION I. Corbella, F. Torres, N. Duffo, M. Martín-Neira.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649de55503460f94addd8b/html5/thumbnails/15.jpg)
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
![Page 16: Universitat Politècnica de Catalunya INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY EQUATION I. Corbella, F. Torres, N. Duffo, M. Martín-Neira.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649de55503460f94addd8b/html5/thumbnails/16.jpg)
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
![Page 17: Universitat Politècnica de Catalunya INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY EQUATION I. Corbella, F. Torres, N. Duffo, M. Martín-Neira.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649de55503460f94addd8b/html5/thumbnails/17.jpg)
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:
![Page 18: Universitat Politècnica de Catalunya INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY EQUATION I. Corbella, F. Torres, N. Duffo, M. Martín-Neira.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649de55503460f94addd8b/html5/thumbnails/18.jpg)
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
![Page 19: Universitat Politècnica de Catalunya INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY EQUATION I. Corbella, F. Torres, N. Duffo, M. Martín-Neira.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649de55503460f94addd8b/html5/thumbnails/19.jpg)
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
![Page 20: Universitat Politècnica de Catalunya INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY EQUATION I. Corbella, F. Torres, N. Duffo, M. Martín-Neira.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649de55503460f94addd8b/html5/thumbnails/20.jpg)
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
![Page 21: Universitat Politècnica de Catalunya INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY EQUATION I. Corbella, F. Torres, N. Duffo, M. Martín-Neira.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649de55503460f94addd8b/html5/thumbnails/21.jpg)
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
![Page 22: Universitat Politècnica de Catalunya INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY EQUATION I. Corbella, F. Torres, N. Duffo, M. Martín-Neira.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649de55503460f94addd8b/html5/thumbnails/22.jpg)
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 ),(
![Page 23: Universitat Politècnica de Catalunya INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY EQUATION I. Corbella, F. Torres, N. Duffo, M. Martín-Neira.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649de55503460f94addd8b/html5/thumbnails/23.jpg)
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
![Page 24: Universitat Politècnica de Catalunya INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY EQUATION I. Corbella, F. Torres, N. Duffo, M. Martín-Neira.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649de55503460f94addd8b/html5/thumbnails/24.jpg)
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
![Page 25: Universitat Politècnica de Catalunya INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY EQUATION I. Corbella, F. Torres, N. Duffo, M. Martín-Neira.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649de55503460f94addd8b/html5/thumbnails/25.jpg)
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
![Page 26: Universitat Politècnica de Catalunya INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY EQUATION I. Corbella, F. Torres, N. Duffo, M. Martín-Neira.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649de55503460f94addd8b/html5/thumbnails/26.jpg)
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
![Page 27: Universitat Politècnica de Catalunya INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY EQUATION I. Corbella, F. Torres, N. Duffo, M. Martín-Neira.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649de55503460f94addd8b/html5/thumbnails/27.jpg)
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
![Page 28: Universitat Politècnica de Catalunya INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY EQUATION I. Corbella, F. Torres, N. Duffo, M. Martín-Neira.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649de55503460f94addd8b/html5/thumbnails/28.jpg)
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
![Page 29: Universitat Politècnica de Catalunya INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY EQUATION I. Corbella, F. Torres, N. Duffo, M. Martín-Neira.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649de55503460f94addd8b/html5/thumbnails/29.jpg)
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
![Page 30: Universitat Politècnica de Catalunya INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY EQUATION I. Corbella, F. Torres, N. Duffo, M. Martín-Neira.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649de55503460f94addd8b/html5/thumbnails/30.jpg)
Universitat Politècnica de Catalunya
•Remote•Sensing•Laboratory
28th July 2011 IGARSS 11. Vancouver. Canada 30/31
SMOS sky image
![Page 31: Universitat Politècnica de Catalunya INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY EQUATION I. Corbella, F. Torres, N. Duffo, M. Martín-Neira.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649de55503460f94addd8b/html5/thumbnails/31.jpg)
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.