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INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY EQUATION I. Corbella, F. Torres, N. Duffo, M. Martín-Neira
Interferometric Radiometry <ul><li>Technique to enhance spatial resolution without large bulk antennas. </li></ul><ul><li>...
Interferometry: Fringes 28th July 2011 IGARSS 11. Vancouver. Canada /31 distant point source d α 0 Δ ℓ x z Δ r=d  cos  α 0...
Fringe Visibility 28th July 2011 IGARSS 11. Vancouver. Canada /31 Michelson’s “Fringe Visibility”: distant  small  source ...
Complex Visibility 28th July 2011 IGARSS 11. Vancouver. Canada /31 Definition <ul><li>Michelson’s “fringe visibility” is t...
Interferometric radiometres 28th July 2011 IGARSS 11. Vancouver. Canada /31 x y d <ul><li>The spatial resolution is achiev...
28th July 2011 IGARSS 11. Vancouver. Canada /31 Only limited values of (u,v) are available: The measured visibility functi...
28th July 2011 IGARSS 11. Vancouver. Canada /31 Comparison with real apertures Rectangular u-v coverage and no window u v ...
28th July 2011 IGARSS 11. Vancouver. Canada /31 Y-shape instrument (19 antennas per arm)  = 1.73 deg  = 2.46 deg Recta...
Microwave Radiometry formulation 28th July 2011 IGARSS 11. Vancouver. Canada /31 r 1 r 2 b 1 b 2 <ul><li>Power spectral de...
The anechoic chamber paradox 28th July 2011 IGARSS 11. Vancouver. Canada /31 T V 12  is apparently non-zero and antenna de...
The “–Tr” term 28th July 2011 IGARSS 11. Vancouver. Canada /31 T The solution is found when all noise contributors are tak...
Empty chamber visibility 28th July 2011 IGARSS 11. Vancouver. Canada /31 Result from IVT at ESA’s Maxwell Chamber
Cold Sky Visibility 28th July 2011 IGARSS 11. Vancouver. Canada /31 Arm A Chamber Chamber Chamber Sky Sky Sky Arm B Arm C ...
Limited bandwidth and time correlation 28th July 2011 IGARSS 11. Vancouver. Canada /31 Receiver 1 Receiver 2 b 1 b 2 Compl...
Director cosines and antenna spacing 28th July 2011 IGARSS 11. Vancouver. Canada /31 distant source point R x y z Antenna ...
The visibility equation 28th July 2011 IGARSS 11. Vancouver. Canada /31 Notes: *  u kj  and  v kj  are defined in terms of...
The zero baseline <ul><li>V (0,0)  is equal to the difference between the antenna temperature and the receivers’ physical ...
Polarimetric brightness temperatures 28th July 2011 IGARSS 11. Vancouver. Canada /31 ΔΩ Observation point Brightness tempe...
Polarimetric interferometric radiometer 28th July 2011 IGARSS 11. Vancouver. Canada /31 Visibility at pp polarization Visi...
Image Reconstruction 28th July 2011 IGARSS 11. Vancouver. Canada /31 Visibility:  For any pair of antennas  k , j  ( k ≠ j...
28th July 2011 IGARSS 11. Vancouver. Canada /31 The Flat-Target response Definition The visibility of a completely unpolar...
28th July 2011 IGARSS 11. Vancouver. Canada /31 Image reconstruction consists of solving for  T ( ξ , η ) in the following...
28th July 2011 IGARSS 11. Vancouver. Canada /31 Antenna Positions and numbering u 1 7 13 19 8 14 v Example:  N EL =6;  d =...
28th July 2011 IGARSS 11. Vancouver. Canada /31 Unit circle Alias-free Field Of View (FOV): Zone of non-overlapping unit c...
28th July 2011 IGARSS 11. Vancouver. Canada /31 Strict and extended alias-free field of view Zone of non-overlapping Earth...
28th July 2011 IGARSS 11. Vancouver. Canada /31 Projection to ground coordinates Swath:  525 km Nadir Boresight
Geo-location 28th July 2011 IGARSS 11. Vancouver. Canada /31 Regular grid in director cosines Irregular grid in lat-lon <u...
Full polarimetric SMOS snapshot 28th July 2011 IGARSS 11. Vancouver. Canada /31 North-west of Australia
SMOS sky image 28th July 2011 IGARSS 11. Vancouver. Canada /31
Conclusions <ul><li>Interferometric radiometry has a long heritage that goes back to the 19th century. SMOS has demonstrat...
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Transcript of "igarss11_1126_corbella.ppt"

  1. 1. INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY EQUATION I. Corbella, F. Torres, N. Duffo, M. Martín-Neira
  2. 2. Interferometric Radiometry <ul><li>Technique to enhance spatial resolution without large bulk antennas. </li></ul><ul><li>Based on cross-correlating signals collected by pairs of ”small” antennas (baselines). </li></ul><ul><li>Image obtained by a Fourier technique from correlation measurements. No scanning needed. </li></ul><ul><li>Examples: </li></ul><ul><ul><li>Precedent: Michelson (end of 19th century). Astronomical observations at optical wavelengths. </li></ul></ul><ul><ul><li>Radioastronomy : Very Large Array (1980). 27 dish antennas, 21 km arm length Y-shape. Various frequencies. </li></ul></ul><ul><ul><li>Earth Observation : SMOS (2009). 69 antennas, 4m arm length Y-shape. L-band. </li></ul></ul>28th July 2011 IGARSS 11. Vancouver. Canada /31
  3. 3. Interferometry: Fringes 28th July 2011 IGARSS 11. Vancouver. Canada /31 distant point source d α 0 Δ ℓ x z Δ r=d cos α 0 b 1 b 2 v d Δ ℓ / λ Δ r / λ A 2 2A 2 Quadratic detector Cross-correlation Total power
  4. 4. Fringe Visibility 28th July 2011 IGARSS 11. Vancouver. Canada /31 Michelson’s “Fringe Visibility”: distant small source with constant intensity I Cross-correlation for Δ ℓ =0 : d α 0 Δξ Δ ℓ ξ 0 = cos α 0 x z v d u=d/λ Δ r=d cos α 0 b 1 b 2 Δ r/ λ =u ξ 0 v d Δ ℓ / λ Δ r / λ I 2I 0 1 0.5 0.75 u Δξ
  5. 5. Complex Visibility 28th July 2011 IGARSS 11. Vancouver. Canada /31 Definition <ul><li>Michelson’s “fringe visibility” is the amplitude of the complex visibility | V ( u )|=I·|sinc u Δξ| normalized to the total intensity of the source. </li></ul><ul><li>The cross correlation between both signals for Δℓ=0 is the real part of the complex visibility < b 1 b 2 >=Re[ V ( u )] . The imaginary part is obtained by adding a 90º phase shift (quarter wavelength) to one of the signals. </li></ul><ul><li>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 )] </li></ul>d α Δξ ξ= cos α x z b 1 b 2 u=d/λ Δ ξ ξ 0 I 0 ξ u I 0 Δ ξ = I I ( ξ ) V ( u )
  6. 6. Interferometric radiometres 28th July 2011 IGARSS 11. Vancouver. Canada /31 x y d <ul><li>The spatial resolution is achieved </li></ul><ul><li>by synthesized beam in ξ </li></ul><ul><li>by antenna pattern in η </li></ul>x y d v u <ul><li>The spatial resolution is achieved by synthesized beam in both dimensions ( ξ and η ). </li></ul><ul><li>Different options for geometry: </li></ul><ul><ul><li>Y-shape, Rectangular, T-shape, Circle, Others </li></ul></ul>d u Use Brightness Temperature ( T B ) instead of intensity ( I ): 1-D 2-D
  7. 7. 28th July 2011 IGARSS 11. Vancouver. Canada /31 Only limited values of (u,v) are available: The measured visibility function is necessarily windowed. Direct equation Fourier inversion Retrieved brightness temperature Convolution integral <ul><li>Array Factor: Inverse Fourier transform of the window </li></ul><ul><li>It is the “synthetic beam”. It sets the spatial resolution </li></ul><ul><li>Its width depends on the maximum (u,v) values (antenna maximum spacing) </li></ul>Spatial resolution: Synthetic beam
  8. 8. 28th July 2011 IGARSS 11. Vancouver. Canada /31 Comparison with real apertures Rectangular u-v coverage and no window u v u M -u M -v M v M A = Δ x max , B = Δ y max : Maximum distance between antennas in each direction Physical aperture with uniform fields x y A B 0.60 0.88 (for small angles around boresight)
  9. 9. 28th July 2011 IGARSS 11. Vancouver. Canada /31 Y-shape instrument (19 antennas per arm)  = 1.73 deg  = 2.46 deg Rectangular window Blackmann window Examples of Synthetic beam
  10. 10. Microwave Radiometry formulation 28th July 2011 IGARSS 11. Vancouver. Canada /31 r 1 r 2 b 1 b 2 <ul><li>Power spectral density: </li></ul><ul><li>Antenna temperature </li></ul><ul><li>Cross-Power spectral density: </li></ul><ul><li>Visibility </li></ul>(units: Kelvin) phase difference (complex valued) Antenna field patterns T B ( θ ,  ) Extended source of thermal radiation Antenna power pattern
  11. 11. The anechoic chamber paradox 28th July 2011 IGARSS 11. Vancouver. Canada /31 T V 12 is apparently non-zero and antenna dependent But V 12 should be zero (Bosma Theorem) anechoic chamber at constant temperature Experiments confirm that V 12 =0 T <ul><li>Power spectral density: Antenna temperature </li></ul><ul><li>Cross Power spectral density: Visibility </li></ul>T A = T (OK!) b 1 b 2 T
  12. 12. The “–Tr” term 28th July 2011 IGARSS 11. Vancouver. Canada /31 T The solution is found when all noise contributors are taken into account. Cross power spectral density for total output waves: T r T r Consistent with Bosma theorem: <ul><li>T r : equivalent temperature of noise produced by the receivers and entering the antennas. This noise is coupled from one antenna to the other. </li></ul><ul><li>If the receivers have input isolators, T r is their physical temperature. </li></ul>b 1 b 2 a 1 a 2
  13. 13. Empty chamber visibility 28th July 2011 IGARSS 11. Vancouver. Canada /31 Result from IVT at ESA’s Maxwell Chamber
  14. 14. Cold Sky Visibility 28th July 2011 IGARSS 11. Vancouver. Canada /31 Arm A Chamber Chamber Chamber Sky Sky Sky Arm B Arm C Blue: SMOS at ESA’s Maxwell Chamber Red: SMOS on flight during external calibration
  15. 15. Limited bandwidth and time correlation 28th July 2011 IGARSS 11. Vancouver. Canada /31 Receiver 1 Receiver 2 b 1 b 2 Complex correlation b s1 b s2 Average power Bandwidth: B 1 Gain: G 1 Bandwidth: B 2 Gain: G 2 T A : Antenna temperature (K) T R : Receiver noise temperature (K) V 12 : Visibility (K) b 1,2 ( t ) : Analytic signals Centre frequency: f 0 Fringe washing function
  16. 16. Director cosines and antenna spacing 28th July 2011 IGARSS 11. Vancouver. Canada /31 distant source point R x y z Antenna location at coordinates (x 1 ,y 1 ,z 1 ) θ  r 1 Director cosines At large distances (R>>d 1 ) d 1 For two close antennas in the x-y plane:  Phase difference: Antenna normalized spacing
  17. 17. The visibility equation 28th July 2011 IGARSS 11. Vancouver. Canada /31 Notes: * u kj and v kj are defined in terms of the wavelength at the centre frequency. * The visibility has hermiticity property Physical temperature of receivers T r = ( T rk +T rj )/2 Antenna relative spacing: Decorrelation time:
  18. 18. The zero baseline <ul><li>V (0,0) is equal to the difference between the antenna temperature and the receivers’ physical temperature. </li></ul><ul><li>It is redundant of order equal to number of receivers. </li></ul><ul><li>At least one antenna temperature must be measured. </li></ul><ul><li>In SMOS, two methods have been considered: </li></ul><ul><ul><li>Three dedicated noise-injection radiometers (NIR) </li></ul></ul><ul><ul><li>All receivers operating as total power radiometers. </li></ul></ul><ul><li>The selected baseline method is the first one (NIR) </li></ul>28th July 2011 IGARSS 11. Vancouver. Canada /31 putting u=v=0 V(0,0)=T A -T r
  19. 19. Polarimetric brightness temperatures 28th July 2011 IGARSS 11. Vancouver. Canada /31 ΔΩ Observation point Brightness temperature at p polarisation: Complex Brightness temperature at p-q polarisations: Relation with Stokes parameters: (p,q): orthogonal polarization basis (linear, circular, …) Spectral power density: if Brightness temperature at q polarisation: Thermal radiation
  20. 20. Polarimetric interferometric radiometer 28th July 2011 IGARSS 11. Vancouver. Canada /31 Visibility at pp polarization Visibility at qq polarization Visibility at pq polarization Visibility at qp polarization
  21. 21. Image Reconstruction 28th July 2011 IGARSS 11. Vancouver. Canada /31 Visibility: For any pair of antennas k , j ( k ≠ j ) Physical temperature of receivers: T rkj = ( T rk +T rj )/2 Antenna relative spacing: Antenna Temperature: For any single antenna k (hermiticity)
  22. 22. 28th July 2011 IGARSS 11. Vancouver. Canada /31 The Flat-Target response Definition The visibility of a completely unpolarised target having equal brightness temperature in any direction (“ flat target” ) is: Measurement It can be measured by pointing the instrument to a known flat target as the cold sky (galactic pole). Estimation It can also be estimated (computed) from antenna patterns and fringe washing functions measurements. For large antenna separation, FTR≈0
  23. 23. 28th July 2011 IGARSS 11. Vancouver. Canada /31 Image reconstruction consists of solving for T ( ξ , η ) in the following equation (zero outside) and V and T depend of the approach chosen: #1 #2 #3 Approach where T ( ξ , η ) is only function of ( ξ , η )
  24. 24. 28th July 2011 IGARSS 11. Vancouver. Canada /31 Antenna Positions and numbering u 1 7 13 19 8 14 v Example: N EL =6; d =0.875 Principal values Hermitic values Hexagonal sampling (MIRAS) u,v points N EL =6 N a =3 N EL +1=19 <ul><li>Number of antenna pairs: N a ( N a -1 ) /2 </li></ul><ul><li>Number of unique ( u-v ) points: 3[ N EL ( N EL +1)] </li></ul>N a : Total number of antennas N EL : Number of antennas in each arm. An antenna in the centre is considered. 3[N EL (N EL +1)]=126 3[N EL (N EL +1)]=126 <ul><li>Number of points in the “star”: 6[ N EL ( N EL +1)]+1 </li></ul>253 total points u =( x j -x k )/ λ 0 v =( y j -y k )/ λ 0 pair ( k , j ):
  25. 25. 28th July 2011 IGARSS 11. Vancouver. Canada /31 Unit circle Alias-free Field Of View (FOV): Zone of non-overlapping unit circle aliases Discrete sampling produces spatial periodicity: Aliases Visibility: ( u-v ) domain Brightness temperature: ( ξ - η ) domain Aliasing
  26. 26. 28th July 2011 IGARSS 11. Vancouver. Canada /31 Strict and extended alias-free field of view Zone of non-overlapping Earth contours Earth Contour Unit Circle Earth aliases Unit Circle aliases Alias-Free Field of View Extended Alias-Free Field of view Antenna Boresight Zone of non-overlapping unit circles
  27. 27. 28th July 2011 IGARSS 11. Vancouver. Canada /31 Projection to ground coordinates Swath: 525 km Nadir Boresight
  28. 28. Geo-location 28th July 2011 IGARSS 11. Vancouver. Canada /31 Regular grid in director cosines Irregular grid in lat-lon <ul><li>The regular grid in xi-eta is mapped into irregular grid in longitude-latitude </li></ul>
  29. 29. Full polarimetric SMOS snapshot 28th July 2011 IGARSS 11. Vancouver. Canada /31 North-west of Australia
  30. 30. SMOS sky image 28th July 2011 IGARSS 11. Vancouver. Canada /31
  31. 31. Conclusions <ul><li>Interferometric radiometry has a long heritage that goes back to the 19th century. SMOS has demonstrated its feasibility for Earth Observation from space. </li></ul><ul><li>The complete visibility equation for a microwave interferometer must include the effect of antenna cross coupling and receivers finite bandwidth. </li></ul><ul><li>Image reconstruction is based on Fourier inversion. Improved performance is achieved by using the flat target response. </li></ul><ul><li>Aliasing induces a complex field of view. In SMOS two zones with different data quality exist: Alias-free and extended alias-free. </li></ul><ul><li>Spatial resolution, sensitivity, incidence angle and rotation angle have significant variations inside the Field of view. </li></ul>28th July 2011 IGARSS 11. Vancouver. Canada /31
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