Brightness distribution and visibility

In principle (and approximately), the visibility measured with aradio interferometer is the two-dimensional Fourier transform ofthe brightness distribution:

V (u, v) =

∫ +∞

−∞

∫I(x , y)e−i2π(ux+vy)dudv

Consequently, the brightness distribution should be possible toobtain by the inverse Fourier transformation:

I(x , y) =

∫ +∞

−∞

∫V (u, v)ei2π(ux+vy)dudv

Fourier transform pairs (1)I(x, y) |V (u, v)|

δ function uniform distribution

Gaussian Gaussian

(figures: Amy Miodyszewski& David Wilner)

Fourier transform pairs (2)I(x, y) |V (u, v)|

Gaussian ellipse Gaussian ellipse

disk Bessel function

(kuvat: Amy Miodyszewski& David Wilner)

Aperture synthesis in practice (1)

I The visibility, V (u, v), is a complex function: V = |V |eiΦ

I Because the product PnIν (beam pattern × brightness) isreal, V (−u,−v) = V ∗(u, v)(V is hermitean - the order of antenna does not matter)

I V (u, v) is only known in a limited region and for discreteset of points in the (u, v) plane.

I Therefore, we cannot calculate the brightness distributionby the inverse Fourier transform.

Sampling function and dirty image

I By defining a sampling function g(u, v) so that g 6= 0 onlyin points (uk , vk ) where V has been measured, the productgV is defined in the whole (u, v) plane.

I The inverse Fourier transform of the product gV can becalculated:

ID =

∫ +∞

−∞

∫g(u, v)V (u, v)ei2π(ux+vy)dudv

I The resulting function ID(x , y) is called dirty image

Synthetic beam (1)

I According to the convolution theorem of the Fouriertransform, the dirty image is the convolution of the trueimage (of the brightness distribution), I(x , y), and thesynthetic beam, Psyn:

ID = I(x , y)⊗ Psyn ,

where

Psyn ≡∫ +∞

−∞g(u, v)ei2π(ux+vy)dudv

Synthetic beam (2)

I The sampling function, g(u, v), can be written in terms ofthe δ function:

g(u, v) =ΣN−1

k=0 wkδ(u − uk , v − vk )

ΣN−1k=0 wk

,

where wk are weight coefficients.I The normalization

∫ ∫g(u, v)dudv = 1 implies that

Psyn(0,0) = 1 (as for the normalized beam pattern of asingle-dish telescope).

I The half-power width of the synthetic beam (“main” beam)is ∆x ∼ 1/umax, ∆y ∼ 1/vmax

I The “side lobes” extend across the the whole sky.

Example 1

synthetic beam uv coverage

I(x, y) ID(x, y)

(figures: Amy Miodyszewski& David Wilner)

Missing information

Incomplete uv coverage:

I “Central hole”: the total flux of an extended flux is notrecovered because of the missing zero spacings:V (0,0) =

∫source I(x , y)dxdy

I The largest scales that can be imaged: ∆α < 1/umin,∆δ < 1/vmin

I Angular resolution: ∆α ∼ 1/umax, ∆δ ∼ 1/vmax

I The gaps in the uv plane create side lobes.

Example 2

Brightness distribution: two pointsources at the offsets (∆α,∆δ) =(0′′, 0′′) and (10′′, 10′′), δ = 80◦

from the phase centreMeasurement H = −2h → +2h

Antenna positions:0,90,170,210,225 m in the East-West directionBaselines: B=15, 40, 55, 80, 90,120, 135, 170, 210 and 225mWavelength: λ = 1cm

uv coverage

Dirty image Synthetic beam Cleaned image

Example 3

Brightness distribution:Gaussian surface den-sity, FWHM (full widthat half-maximum) 10′′,(∆α,∆δ) = (0′′, 0′′), emitsspectral line emission, totalflux density 1 Jy at the linepeakMeasurement near themeridian H ∼ 0h

Antennas: 0,90,150,400 m,East-WestWavelength: λ = 1.3cmBaselines: B=90, 150, 60,400, 310, 250 mλ/B ≈ 29′′, 17′′, 43′′, 7′′, 8′′

and 10′′

Spectra with different baselines

Visibility weighting (1)

I Sampling function g(u, v):

g(u, v) =ΣN−1

k=0 wkδ(u − uk , v − vk )

ΣN−1k=0 wk

I The weight coefficients wk affect the synthetic beam andthe side lobes

I Natural weighting: wk = 1/σ2k

(σ is the RMS noise)I Uniform weighting: wk = 1/ρ(uk , vk )

(ρ is the density of measurement points in theneighbourhood of (u, v))

Visibility weighting (2)

I In addition, long baselines can be given smaller weights inorder to create a larger synthetic beam. This is calledtapering or the uv plane

I The weight function wk can be then written as

wk = TkDk ,

Tk is the tapering function of the uv planeDk is density weighting function of the measured points

I The angular resolution can be “worsened” using thefunction Tk , for example to help comparison with someprevious data, or to achieve a better signal-to-noise ratio.

Visibility weighting (2)

I The most common tapering function is Gaussian:T (r) = EXP(−r2/2σ2)

I In CASA (and Miriad) the desired angular resolution in the imageplane is given (FWHM of the synthesized beam)

I In AIPS the tapering function is given in the uv planeRelationship: FWHM[radians]= 0.37/σ[wavelengths] orFWHM[′′]= 0.77λ[cm]/σ[km]

Density weighting function Dk (1)

I Natural weighting is good for detecting weak compactsources (the best S/N for a point source)

I Uniform weighting gives a better image quality if the sourcehas both large scale and small scale structure

I A hybrid scheme, robust weighting, is a compromise of thetwo trying to attain a narrow main beam and weak sidelobes.

Density weighting function Dk (2)

natural uniform

ATCA 1933-400, contours 5% and 50%

Tapering function Tk

natural FWHM 10′′ uniform FWHM 10′′

ATCA 1933-400, contours 5% and 50%

Interferometric mapping (1)

I In aperture synthesis the antennas track a certain point inthe sky.

I Because of the earth rotation complex source visibilitiesare measured with a large number of projected baselines.

I A science target and a calibration source alternate as thephase centre.

I The calibrator is typically a strong point source, a planet, orits satellite

Interferometric mapping (2)

I Flux density calibrator (primary calibrator): stable,unpolarized source with well known flux density

I Phase calibrator (secondary calibrator): a strong pointsource lying close to the science target. Used to correctthe gain ja phase variations.

I Bandpass calibrator: Very strong point source used todetermine the response of the receiver as a function offrequency (bandpass function)

Calibration (1)

I The amplitude and phase variations in each receiver (inthe antennas), owing to instrumental effects and theatmosphere, are described by a complex gain G:

G(t , ν) = A(t)B(ν)ei2πτ(t)(ν−ν0)

I A(t) is the complex “antenna gain”I B(ν) is the bandpass functionI τ(t) delay termI ν0 centgral frequency of the band

Calibration (2)

I In addition, a part of X (L) polarized radiation leaks into theY (R) receiver, and vice versa (instrumental polarization)

I The terms A(t) and τ(t) exhibit rapid changes, B(ν) andinstrumental polarization change slowly

I In calibration G(t , ν) is solved using point sources.I The solutions are interpolated to the science target.

Calibration example (1)

Flux density calibrator 1934-634 - original visbility data

amplitude phase

-flag the beginning

Calibration example (2)

Flux density calibrator 1934-634 - original and corrected data

phase phase

-For corrected data the phases of the XX and YY polarizationproducts are close to zero

Calibration example (3)

Flux density calibrator 1934-634 - original and corrected data

imag vs. real imag vs. real

-For corrected point source data the real parts of XX and YYpolarization products are close to the true flux density(note the scale on the y axis at right)