1. Fourth IRAM Millimeter Interferometry School 2004: Atmospheric phase correction 1 Atmospheric phase correction Jan Martin Winters IRAM, Grenoble

2. Fourth IRAM Millimeter Interferometry School 2004: Atmospheric phase correction 2 The problem Atmosphere introduces (complex) refractive index => path delay + absorption/emission Water vapor poorly mixes with dry air => „eddies“ Atmosphere is turbulent => fluctuating path delay Time varying deformation of wavefront => Phase fluctuation => Degradation of source amplitude Degradation of spatial resolution

3. Fourth IRAM Millimeter Interferometry School 2004: Atmospheric phase correction 3 Effect of phase noise An interferometer measures amplitude and phase of the incoming wave (complex visibility). Integration of the signal can be concieved as the summation of vectors, characterized by their length (amplitude) and orientation (phase) V1V1 → V3V3 → V2V2 → V=  V i →→ Without phase noise With phase noise V1V1 → V3V3 → V2V2 → V=  V i →→  Degradation of amplitude + smearing out of structure information |V|=  V i | →→ |V|<  V i | →→

4. Fourth IRAM Millimeter Interferometry School 2004: Atmospheric phase correction 4 The idea Determine the amount of water vapor in front of each telescope by measuring its emission Deduce the path delay caused by this water column Apply a corresponding phase correction

5. Fourth IRAM Millimeter Interferometry School 2004: Atmospheric phase correction 5 The method Atmospheric emission T sky = T Atm (1  e  ) With  d  w  d  pwv Excess path L = L d + L V = L d + 6.52 pwv [cm] Phase delay  L  =  L/  T sky )  T sky => Measure T sky (fluctuating) in front of each telescope Use atmospheric model to derive ( , T Atm,) pwv,  L/  T sky Compute phase correction  and apply it to data

6. Fourth IRAM Millimeter Interferometry School 2004: Atmospheric phase correction 6 In practice (I): Total power radiometry, e.g., in the 1mm band (a factor ~6 more sensitive to pwv than 3mm band) using the astronomical receivers This was the standard method used at the PdBI until August 2004 Problems: Clouds: large , low n => large variations in T sky, but only small effect on the path excess  L Measurement at only one frequency(band): effect of clouds cannot be removed Long-term stability of the astronomical receivers (which are designed for sensitivity) (important for absolute phase correction)

7. Fourth IRAM Millimeter Interferometry School 2004: Atmospheric phase correction 7 In practice (II): B) Multi channel radiometry in a water line (here: at 22GHz) using dedicated instruments (Rem.: ALMA will use the 183GHz line) This is the standard method used at the PdBI since August 2004 Advantages: Effect of clouds can be removed : T sky,H 2 O = T vapor  T cloud = T Atm (1  e  v ) + T Cloud (1  e  c ),  C ~ 2 linearize cloud exponential term, measure at two frequencies, build weighted mean:  T double = T sky,1 – T sky,2 (    ) 2 = T vapor,1 – T vapor,2 (    ) 2 Instruments designed for stability => absolute phase correction

8. Fourth IRAM Millimeter Interferometry School 2004: Atmospheric phase correction 8 22GHz monitor Sampling rate: 1s

9. Fourth IRAM Millimeter Interferometry School 2004: Atmospheric phase correction 9 unstable atmospheric conditions 4.4mm pwv phases @ 110 GHz A-configuration: E23-W27-N29-E16-W23-N13 8 min on NRAO150 Results 22GHz correction (I) Temporal phase variation

10. Fourth IRAM Millimeter Interferometry School 2004: Atmospheric phase correction 10

11. Fourth IRAM Millimeter Interferometry School 2004: Atmospheric phase correction 11 Results 22GHz correction (II) Turbulent conditions, 4.4mm pwv, A-configuration Calibrator NRAO150, strong continuum point source => Factor 2.5 gain in amplitude without phase correction with phase correction

12. Fourth IRAM Millimeter Interferometry School 2004: Atmospheric phase correction 12 Kolmogorov turbulence Turbulence is fed by energy input on large scales L (= outer scale of the turbulent field) This energy is cascaded down to smaller scales (in a stationary process) until it is dissipated into heat on the smallest scales 0 (inner scale) by viscosity The velocity fluctuation associated with linear scale is v, the typical time scale of the fluctuation is  =  / v Per unit mass, the rate at which energy is fed into eddies of size is then  ~ v 2 /  = v 3 /  = v L 3 / L  or v   

13. Fourth IRAM Millimeter Interferometry School 2004: Atmospheric phase correction 13 Phase structure function Characterization of fluctuations by the structure function D v (d) = ≈ v d 2 ~  d   (velocity) Phase fluctuations are induced by fluctuations of the refractive index due to water vapor eddies in the turbulent atmosphere D n (d) ~  d   (refractive index) On large scales (d ≫ height of turbulent layer, “thin screen”, 2D) D  (d) ~  d   (phase, 2D) On smaller scales: 3D description, “thick screen” D  (d) ~  d   (phase, 3D) For the rms phase noise   = ( D  (d)) 1/2 power law spectra are expected with exponents between 1/3 and 5/6 (On scales d > L: uncorrelated, D  (d) ≈ const)

14. Fourth IRAM Millimeter Interferometry School 2004: Atmospheric phase correction 14 Results 22GHz correction (III) Turbulent conditions, 4.4mm pwv, A-configuration    exp(-   2 /2) 150.97 300.87 500.68 700.47 1000.22 2000.002 Decorrelation factors

15. Fourth IRAM Millimeter Interferometry School 2004: Atmospheric phase correction 15 Results 22GHz correction (IV) @ 3mm

16. Fourth IRAM Millimeter Interferometry School 2004: Atmospheric phase correction 16 Results: Statistics (II) pwv < 5mm pwv > 5mm

17. Fourth IRAM Millimeter Interferometry School 2004: Atmospheric phase correction 17 Results 22GHz correction (I) Turbulent conditions, 4.4mm pwv, A-configuration

18. Fourth IRAM Millimeter Interferometry School 2004: Atmospheric phase correction 18 Results 22GHz correction (II) Stable conditions, 3.7mm pwv, A-configuration

19. Fourth IRAM Millimeter Interferometry School 2004: Atmospheric phase correction 19 Results 22GHz correction (IV) Stable conditions, 3.7mm pwv, A-configuration