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Catalunya
INTERFEROMETRIC RADIOMETRY
MEASUREMENT CONCEPT: THE VISIBILITY
EQUATION
I. Corbella, F. Torres, N. Duffo, M.
Martín-Neira
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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.
28th July 2011
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Interferometry: Fringes
distant point source
z
vd
2A2
Δr=d cos α0
A2
α0
x
d
b1
b1  A cos(t  r / c)
b2
Δℓ
b2  A cos(t   / c)
Δr/λ
Quadratic detector
vd  b1  b2   b12  b22  2 b1b2
2
Total power
28th July 2011
Δℓ/λ
vd  A2  A2 cos 2 (r /    /  )
Cross-correlation
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Fringe Visibility
vd
distant small source with
constant intensity I
z
uΔξ
2I
Δξ
0
0.5
Δr=d cos α0
0.75
I
α0
x
d
b1
b2
Δℓ
ξ0=cos α0 u=d/λ
Δr/λ=uξ0
Δr/λ
Δℓ/λ
vd  I  I sinc u cos 2 (r /    /  )
vd
Michelson’s “Fringe Visibility”:
Cross-correlation for Δℓ=0:
28th July 2011
1
fringe maxima  fringe minima
 sinc u
fringe maxima  fringe minima
b1b2  I sinc u cos 2u0
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Complex Visibility
Definition V (u )  I sinc u e  j 2 u0
• 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)]
z
ξ=cos α
I(ξ)
Δξ
V(u)
I0Δξ=I
I0
Δξ
ξ0
α
b1
d
u=d/λ
28th July 2011
b2
x
ξ
   0 

I ( )  I 0




IGARSS 11. Vancouver. Canada
u
V (u )  I 0  sinc u e  j 20u
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Interferometric radiometres
Use Brightness Temperature (TB)
instead of intensity (I):
1-D
y
u

V (u)   TB ( )e j 2u d
x

The spatial resolution is achieved
• by synthesized beam in ξ
• by antenna pattern in η
 1
d
y
2-D
u
V (u, v)  



 
v
d
d
28th July 2011
x
 2  2  1
TB ( , )e j 2 (u v ) dd
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
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Spatial resolution: Synthetic beam
V (u, v)  
Direct equation



 
Fourier inversion
TB ( , )  


TB ( , )e j 2 (u v ) dd

 
V (u, v)e j 2 (u v ) dudv
Only limited values of (u,v) are available: The measured visibility function is
necessarily windowed.
Retrieved brightness
temperature

ˆ
TB ( , )  


  

ˆ
TB ( , )  


  
W (u, v)V (u, v)e j 2 (u v ) dudv
TB ( , ) AF (   ,  )d d 
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)
28th July 2011
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Comparison with real apertures
Rectangular u-v coverage and no window
v
AF  sinc 2uM  sinc 2vM 
vM
A 
B 

AF   sinc 2   sinc 2  
 
 

uM
B

E
A
28th July 2011
x
A  
B 

t ( , )   sinc    sinc  
  
 

(for small angles
around boresight)

B

0.8
2
0.6
Interferometric
Real
t()

H
2
vM 
Comparison between Interferometric and Real apertures
1
Physical aperture with uniform fields
y
A
A=Δxmax, B=Δymax: Maximum
distance between antennas in
each direction
u
-vM
-uM
uM 
0.4
0.60
0.2
0.88
0
-0.5
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0
(A/) 
0.5
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Examples of Synthetic beam
Y-shape instrument (19 antennas per arm)
Rectangular window
Blackmann window
= 1.73 deg
28th July 2011
IGARSS 11. Vancouver. Canada
= 2.46 deg
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Microwave Radiometry formulation
Extended source of
thermal radiation
TB(θ,)
r1
b1
• Power spectral density:
Antenna temperature
b1 ( f )  kTA1
2
b2 ( f )  kTA2
2
r2
b2
• Cross-Power spectral density:
Visibility
b1 ( f )b2* ( f )  kV12
1
TA1, 2 
TB ( ,  )t ( ,  )d
(units: Kelvin)

1, 2 4
Antenna power pattern
1
 jk ( r1  r2 )
*
V12 
T
(

,

)
F
(

,

)
F
(

,

)
e
d
B
n1
n2

1 2 4
Antenna field patterns
28th July 2011
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(complex valued)
phase difference
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The anechoic chamber paradox
anechoic chamber at
constant temperature
• Power spectral density: Antenna temperature
2
b1, 2  kTA1, 2
T
b1
1
 kT
1, 2
 t
1, 2
4
T
( ,  )d  kT
TA=T (OK!)
• Cross Power spectral density: Visibility
b2
T
b1b2*  kV12 
kT
1 2
*
jkr
F
(

,

)
F
(

,

)
e
d
n2
 n1
4
V12 is apparently non-zero and antenna dependent
r  r2  r1
But V12 should be zero (Bosma Theorem)
Experiments confirm that V12=0
28th July 2011
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The “–Tr” term
The solution is found when all noise
contributors are taken into account.
a1
T
b1
a2
b2
k
bb 
1 2
*
1 2
Tr
Tr
1
1 2
*
jkr
*
*
F
(

,

)
F
(

,

)
e
d



(
S
S

S
S
)
n
1
n
2
11
21
12
22

4
Cross power spectral density for
total output waves:
 jk ( r  r )
*


T
(

,

)

T
F
(

,

)
F
(

,

)
e
d
r
n1
n2
 B
1
2
4
*
Consistent with Bosma theorem: if TB ( ,  )  Tr  b1b2  0
• 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.
28th July 2011
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Empty chamber visibility
Result from IVT at ESA’s Maxwell Chamber
10
10
0
-1
Tch
1 2
K
10
Visibility of an empty chamber at 293K
1
10
10
10
 F
n1
( ,  ) Fn*2 ( ,  )e jkr d
4
-2
-3
-4
0
Tch  Tr
1 2
No -Tr term
Theory
Measurement
*
jkr
F
(

,

)
F
(

,

)
e
d
n
1
n
2

4
5
10
15
20
Antenna separation normalized to wavelength
28th July 2011
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Cold Sky Visibility
Sky
Sky
Arm A
Chamber
Blue:
SMOS at ESA’s
Maxwell Chamber
Chamber
Sky
Arm C
Chamber
28th July 2011
Arm B
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Red:
SMOS on flight
during external
calibration
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Limited bandwidth and time correlation
Bandwidth: B1
Gain: G1
bs1
b1
Receiver 1
 dt
Receiver 2
b1 (t )
b2 (t )
 dt
Centre frequency: f0
bs2
Average power
b2
Bandwidth: B2
Gain: G2
 dt
b1,2(t): Analytic signals
~
r12 (t ) 
28th July 2011
e
B1B2 G1G2


0
 2kG1B1 (TA1  TR1 )
2
 2kG2 B2 (TA2  TR 2 )
TA: Antenna temperature (K)
TR: Receiver noise temperature (K)
Complex correlation
b1 (t )b2* (t )  2k G1G2 B1 B2 G12V12
1
V12: Visibility (K) V12 
1 2
Fringe washing function
 j 2 f 0t
2
H1 ( f ) H 2* ( f )e j 2ft df
jk r
* ~


T

T
F
F
r
(

r
/
c
)
e
d
 B r n1 n 2 12
0
4
~
r12 (t )  ~
r12 (t ) / ~
r12 (0)
k0 
2f 0
c
G12  ~
r12 (0)
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Director cosines and antenna spacing
At large distances (R>>d1)
distant source point
z
x
d1
θ R

r1
d12  x
y
z 
r1  R 
  x1  y1  z1 
2R  R
R
R 
Antenna location at
coordinates (x1,y1,z1)
y
Director cosines
x
y
   sin  cos     sin  sin 
R
R
For two close antennas in the x-y plane: r2  r1   ( x2  x1 )  ( y2  y1 )
 Phase difference: kr  k (r2  r1 )  2 ( u  v)
Antenna normalized spacing u 
28th July 2011
x2  x1

v
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y2  y1

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The visibility equation
Vkj (ukj , vkj ) 
1
k  j

 2  2 1
 ukj  vkj   j 2 (ukj  vkj )
e
Fnk ( , ) Fnj* ( , ) ~
rkj  
dd
2
2
f0
1   


TB ( , )  Tr
Physical temperature of receivers Tr=(Trk+Trj)/2
Antenna relative spacing:
ukj 
x j  xk
vkj 
0
ukj  vkj
r rj  rk
Decorrelation time:


c
c
f0
y j  yk
0
Notes:
* ukj and vkj are defined in terms of the wavelength at the centre frequency.
* The visibility has hermiticity property V jk  Vkj*
28th July 2011
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The zero baseline
putting u=v=0
Vk (0,0) 
1
k

 2  2 1
TB ( , )  Tr
1   2  2
Fnk ( , ) dd  TAk  Tr
2
V(0,0)=TA-Tr
• 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)
28th July 2011
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Polarimetric brightness temperatures
Observation point
ΔΩ
Thermal radiation
E
Spectral power density:

if E  E p pˆ  Eq qˆ
2
0
2

E p  Eq
(p,q): orthogonal polarization basis (linear, circular, …)
2kTB
2
2
0

Brightness temperature at p polarisation: TBpp  E p E *p  E p
2
Brightness temperature at q polarisation: TBqq  Eq Eq*  Eq
2
Complex Brightness temperature at p-q
polarisations:
Relation with Stokes parameters:
I  TBpp  TBqq
28th July 2011


k TBpp  TBqq
2
 
TBpq  E p Eq*
TBqp  Eq E*p  TBpq
Q  TBpp  TBqq U  2e[TBpq ]
IGARSS 11. Vancouver. Canada
*
V  2m[TBpq ]
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Polarimetric interferometric radiometer
p output
Antenna 1
OMT
Fp1 ( ,  )
bp1
q output
Fq1 ( ,  ) bq1
p output
Antenna 2
OMT
Fp 2 ( ,  )
bp2
Visibility at pp polarization
1
 jk ( r1  r2 )
*
pp
V12pp 
F
(

,

)
F
(

,

)(
T

T
)
e
d
p1
p2
B
r

 p1 p 2 4
*
 
V21pp  V12pp
Visibility at qq polarization
1
 jk ( r1  r2 )
*
qq
V12qq 
F
(

,

)
F
(

,

)(
T

T
)
e
d
q1
q2
B
r

 q1 q 2 4
Visibility at pq polarization
V12pq 
q output
Fq 2 ( ,  ) bq2
1
 q1 p 2
*
*
pq  jk ( r1  r2 )
F
(

,

)
F
(

,

)
T
d
q2
B e
 p1
4
Visibility at qp polarization
V12qp 
28th July 2011
1
 p1 q 2
 
V21qq  V12qq
 
V21pq  V12qp
*
*
qp  jk ( r1  r2 )
F
(

,

)
F
(

,

)
T
d
p2
B e
 q1
4
IGARSS 11. Vancouver. Canada
 
V21qp  V12pq
*
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Image Reconstruction
Visibility: For any pair of antennas k,j (k≠j)
1
Vkj (ukj , vkj ) 
k  j
V jk  Vkj*

 2  2 1
TB ( , )  Trkj
1   2  2
 ukj  vkj   j 2 (ukj  vkj )
e
Fnk ( , ) Fnj* ( , ) ~
rkj  
dd
f0


(hermiticity)
Physical temperature of receivers: Trkj=(Trk+Trj)/2
Antenna relative spacing:
ukj 
x j  xk
0
vkj 
y j  yk
0
Antenna Temperature: For any single antenna k
TA k 
1
k

 2  2 1
28th July 2011
TB ( , )
1   2  2
Fnk ( , ) dd
2
k  1 N a
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The Flat-Target response
Definition
FTR(k , j ) 

  2 1
2
*
Fnk ( , ) Fnj ( , ) ~  ukj  vkj   j 2 (ukj  vkj )
rkj   f e
dd
2
2
0


k
j
1   
1
The visibility of a completely unpolarised target having equal brightness
temperature in any direction (“flat target”) is:
VkjFT (ukj , vkj )  (TB  Trkj ) FTR(k , j )
Measurement
It can be measured by pointing the instrument to a known flat target as
the cold sky (galactic pole).
Estimation

FTR(k , j )  VkjFT TB  Trkj

It can also be estimated (computed) from antenna patterns and fringe
washing functions measurements.
For large antenna separation, FTR≈0
28th July 2011
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Image reconstruction consists of solving for T(ξ,η) in the
following equation
V (u, v)  



 
T ( , )e j 2 (u v ) dd
*
Fnk ( , ) Fnj ( , ) ~  u  v 

where T  
rkj  
2
2
f0 
k
j
1   

T ( , )
V (u , v)
u, v  0
#1
Vkj (ukj , vkj )
#2
Vkj (ukj , vkj )  Trkj FTR(k , j )
#3
Vkj (ukj , vkj )  (TAkj  Trkj ) FTR(k , j )
28th July 2011
2

 2  1
(zero outside)
T(ξ,η) is only function of (ξ,η)
and V and T depend of the approach chosen:
Approach
in 
V (0,0)
T ( , )
TAk  Trk
TB ( , )  Tr
T Ak
0
IGARSS 11. Vancouver. Canada
TB ( , )
TB ( , )  TA
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Hexagonal sampling (MIRAS)
Na: Total number of antennas
• Number of antenna pairs: Na(Na-1)/2
• Number of unique (u-v) points: 3[NEL(NEL+1)]
• Number of points in the “star”: 6[NEL(NEL+1)]+1
NEL : Number of antennas in
each arm. An antenna in
the centre is considered.
Example: NEL=6; d=0.875
Antenna Positions
and numbering
3
2
1
3[N
6 EL(NEL+1)]=126
4
u
v
pair (k,j):
8
0
1
7
-1
u=(xj-xk)/λ0
v=(yj-yk)/λ0
14
2
0
-2
-2
NEL=6
-4
-3
Na=3NEL+1=19
-6
-4
-4
-2
0
x/
28th July 2011
u,v points
8 Hermitic values
v
4
y/
13
2
19
Principal values
3[NEL(NEL+1)]=126
-8 253 total points
-10
IGARSS 11. Vancouver. Canada
-5
0
u
5
10
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Aliasing
Discrete sampling produces spatial periodicity: Aliases
Visibility: (u-v) domain
Brightness temperature: (ξ-η) domain
2
20
1.5
15
1
5
0.5
0
0

v
10
-5
-0.5
-10
-1
-15
-1.5
-20
-20
-10
28th July 2011
0
u
10
20
-2
-2
-1
0

1
2
Alias-free Field Of View (FOV):
Unit circle
Zone of non-overlapping unit circle aliases
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Strict and extended alias-free field of view
hsat=755 km, tilt=32.5º, d=0.875
1.5
Unit Circle
1
Earth Contour

0.5
Antenna Boresight
0
-0.5
Unit Circle aliases
Earth aliases
-1
-1.5
-1.5
-1
-0.5
Alias-Free Field of View
Zone of non-overlapping unit circles
28th July 2011
0

0.5
1
1.5
Extended Alias-Free Field of view
Zone of non-overlapping Earth contours
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Projection to ground coordinates
hsat=755 km, tilt=32.50º, d=0.875
Along track coordinate(km)
1200
Boresight
1000
800
600
Nadir
400
200
0
-200
-1000
28th July 2011
Swath: 525 km
-500
0
500
Cross track coordinate (km)
1000
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Geo-location
• The regular grid in xi-eta is mapped into irregular grid in longitude-latitude
Regular grid in director cosines
28th July 2011
Irregular grid in lat-lon
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Full polarimetric SMOS snapshot
TBxx
TByy
North-west
of Australia
Re[TBxy ]
28th July 2011
Im[ TBxy ]
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SMOS sky image
28th July 2011
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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.
28th July 2011
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