Universitat Politècnica de Catalunya INTERFEROMETRIC RADIOMETRY MEASUREMENT CONCEPT: THE VISIBILITY EQUATION I. Corbella, F. Torres, N. Duffo, M. Martín-Neira

Universitat Politècnica de Catalunya 28th July 2011IGARSS 11. Vancouver. Canada2/31 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 28th July 2011IGARSS 11. Vancouver. Canada3/31 Interferometry: Fringes distant point source d α0α0 ΔℓΔℓ x z Δr=d cos α 0 b1b1 b2b2 vdvd Δℓ/λΔℓ/λ Δr/λΔr/λ A2A2 2A 2 Quadratic detector Cross-correlationTotal power

Universitat Politècnica de Catalunya 28th July 2011IGARSS 11. Vancouver. Canada4/31 Michelson’s “Fringe Visibility”: Fringe Visibility distant small source with constant intensity I Cross-correlation for Δℓ=0 : d α0α0 Δξ ΔℓΔℓ ξ 0 =cos α 0 x z vdvd u=d/λ Δr=d cos α 0 b1b1 b2b2 vdvd Δℓ/λΔℓ/λ Δr/λΔr/λ I 2I uΔξ Δr/λ=uξ 0

Universitat Politècnica de Catalunya 28th July 2011IGARSS 11. Vancouver. Canada5/31 Definition 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 =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 Δξ ξ=cos α x z b1b1 b2b2 u=d/λ ΔξΔξ ξ0ξ0 I0I0 ξu I0Δξ=II0Δξ=I I(ξ)I(ξ)V(u)V(u)

Universitat Politècnica de Catalunya 28th July 2011IGARSS 11. Vancouver. Canada6/31 x y d The spatial resolution is achieved by synthesized beam in ξ by antenna pattern in η x y d v 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 ( T B ) instead of intensity ( I ): 1-D 2-D Interferometric radiometres

Universitat Politècnica de Catalunya 28th July 2011IGARSS 11. Vancouver. Canada7/31 Only limited values of (u,v) are available: The measured visibility function is necessarily windowed. Direct equation Fourier inversion Retrieved brightness temperature Convolution integral 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 28th July 2011IGARSS 11. Vancouver. Canada8/31 Comparison with real apertures Rectangular u-v coverage and no window u v uMuM -u M -v M vMvM A=Δx max, B=Δy max : Maximum distance between antennas in each direction Physical aperture with uniform fields x y A B (for small angles around boresight)

Universitat Politècnica de Catalunya 28th July 2011IGARSS 11. Vancouver. Canada9/31 Y-shape instrument (19 antennas per arm)  = 1.73 deg  = 2.46 deg Rectangular windowBlackmann window Examples of Synthetic beam

Universitat Politècnica de Catalunya 28th July 2011IGARSS 11. Vancouver. Canada10/31 r1r1 r2r2 b1b1 b2b2 Power spectral density: Antenna temperature Cross-Power spectral density: Visibility (units: Kelvin) phase difference (complex valued) Microwave Radiometry formulation Antenna field patterns TB(θ,)TB(θ,) Extended source of thermal radiation Antenna power pattern

Universitat Politècnica de Catalunya 28th July 2011IGARSS 11. Vancouver. Canada11/31 The anechoic chamber paradox 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 Power spectral density: Antenna temperature Cross Power spectral density: Visibility T A =T (OK!) b1b1 b2b2 T

Universitat Politècnica de Catalunya 28th July 2011IGARSS 11. Vancouver. Canada12/31 The “–Tr” term T The solution is found when all noise contributors are taken into account. Cross power spectral density for total output waves: TrTr TrTr Consistent with Bosma theorem: T r : 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, T r is their physical temperature. b1b1 b2b2 a1a1 a2a2

Universitat Politècnica de Catalunya 28th July 2011IGARSS 11. Vancouver. Canada13/31 Empty chamber visibility Result from IVT at ESA’s Maxwell Chamber

Universitat Politècnica de Catalunya 28th July 2011IGARSS 11. Vancouver. Canada14/31 Cold Sky Visibility Arm A Chamber Sky Arm B Arm C Blue: SMOS at ESA’s Maxwell Chamber Red: SMOS on flight during external calibration

Universitat Politècnica de Catalunya 28th July 2011IGARSS 11. Vancouver. Canada15/31 Limited bandwidth and time correlation Receiver 1 Receiver 2 b1b1 b2b2 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

Universitat Politècnica de Catalunya 28th July 2011IGARSS 11. Vancouver. Canada16/31 Director cosines and antenna spacing distant source point R x y z Antenna location at coordinates (x 1,y 1,z 1 ) θ  r1r1 Director cosines At large distances (R>>d 1 ) d1d1 For two close antennas in the x-y plane:  Phase difference: Antenna normalized spacing

Universitat Politècnica de Catalunya 28th July 2011IGARSS 11. Vancouver. Canada17/31 Notes: * u kj and v kj are defined in terms of the wavelength at the centre frequency. * The visibility has hermiticity property The visibility equation Physical temperature of receivers T r =(T rk +T rj )/2 Antenna relative spacing: Decorrelation time:

Universitat Politècnica de Catalunya 28th July 2011IGARSS 11. Vancouver. Canada18/31 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 V(0,0)=T A -T r

Universitat Politècnica de Catalunya 28th July 2011IGARSS 11. Vancouver. Canada19/31 Polarimetric brightness temperatures 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

Universitat Politècnica de Catalunya 28th July 2011IGARSS 11. Vancouver. Canada20/31 Polarimetric interferometric radiometer Visibility at pp polarization Visibility at qq polarization Visibility at pq polarization Visibility at qp polarization

Universitat Politècnica de Catalunya 28th July 2011IGARSS 11. Vancouver. Canada21/31 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) Image Reconstruction

Universitat Politècnica de Catalunya 28th July 2011IGARSS 11. Vancouver. Canada22/31 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

Universitat Politècnica de Catalunya 28th July 2011IGARSS 11. Vancouver. Canada23/31 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 )

Universitat Politècnica de Catalunya 28th July 2011IGARSS 11. Vancouver. Canada24/31 Antenna Positions and numbering u 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 u=(x j -x k )/λ 0 v=(y j -y k )/λ 0 pair ( k, j ): Number of antenna pairs: N a (N a -1)/2 Number of unique ( u-v ) points: 3[N EL (N EL +1)] 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 Number of points in the “star”: 6[N EL (N EL +1)] total points

Universitat Politècnica de Catalunya 28th July 2011IGARSS 11. Vancouver. Canada25/31 Unit circle Alias-free Field Of View (FOV): Zone of non-overlapping unit circle aliases Discrete sampling produces spatial periodicity: Aliases Visibility: ( u-v ) domainBrightness temperature: ( ξ - η ) domain Aliasing

Universitat Politècnica de Catalunya 28th July 2011IGARSS 11. Vancouver. Canada26/31 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

Universitat Politècnica de Catalunya 28th July 2011IGARSS 11. Vancouver. Canada27/31 Projection to ground coordinates Swath: 525 km Nadir Boresight

Universitat Politècnica de Catalunya 28th July 2011IGARSS 11. Vancouver. Canada28/31 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 28th July 2011IGARSS 11. Vancouver. Canada29/31 Full polarimetric SMOS snapshot North-west of Australia

Universitat Politècnica de Catalunya 28th July 2011IGARSS 11. Vancouver. Canada30/31 SMOS sky image

Universitat Politècnica de Catalunya 28th July 2011IGARSS 11. Vancouver. Canada31/31 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.