Problem definition

2 Problem definition

The normalized cross-correlation function (the correlator output), measured by an interferometer using two antennas, antenna i and antenna j, in the limit I ≪ T_{sys_i}∕η_i, can be written as:

+ ∞ ∫∫ ∘ ---ηiηj--- √--2---2 ρOijbs= ρObs(uij,vij,wij) = I (l,m ) ---------e2πι(uijl+vijm+wij 1-l- m +ϕi-ϕj) - ∞ TsysiTsysj ∘---dl-dm-------+ ϵij (1 - l2 - m2 )

(1)

where I(l,m) is the sky surface brightness, η_i is the sensitivity and T_{sys_i} the system temperature of the antenna i in units of Kelvin/Jy and Kelvin respectively, ϵ_ij is the additive noise on the baseline i-j, and ϕ_i is the antenna based phase of the signal. The rest of the symbols have the usual meaning.

In practice however, the antenna based amplitude ( ∘ --------
ηi∕Tsysi ) and phase (ϕ_i) are potentially time varying quantities. This could be due to changes in the ionosphere, temperature variations, ground pick up, antenna blockage, noise pick up by various electronic components, background temperature, etc. Treating the quantities under the square root in the above equation as the antenna dependent amplitude gain, these antenna dependent quantities can be written as complex gains g_i = a_ie^-ιϕ_i where a_i = ∘ --------
ηi∕Tsys
i . For an unresolved source at the phase tracking center, variations in this amplitude will be indistinguishable from a variations in the ratio of η and T_sys.

In terms of g_is, we can write Eq. 1 as

ρObs = g g⋆ρ∘ + ϵ ij i j ij ij

(2)

where

∫+∞ +∫∞ √---2--2 dl dm ρ∘ij = I (l,m ) e2πι(uijl+vijm+wij 1-l -m )∘--------------- (1 - l2 - m2) -∞ -∞

(3)

The use of the word “antenna based gains” for g_is result into confusion for many and needs some clarifications. g_is are called complex “gains” since they multiply with the complex quantity ρ_ij. For an unresolved source, |g_i| represents the fraction of correlated signal and arg(g_i) represents the phase of the correlated part of the signal from the antenna with respect to the phase reference (usually the reference antenna). It is in this sense that it is referred to as “antenna based” gains. g_is are antenna based but a function of direction in the sky since, as defined here, they include T_sys which in turn includes the sky background temperature. However, here we assume that the angular scale over which g_is vary is larger than the antenna primary beam (isoplanatic case).

For an unresolved source at the phase tracking center, all terms in the exponent of ρ_ij^∘ are exactly zero. ρ_ij^∘ in this case would be proportional to the flux density of the source.

Given ρ_ij^Obs and knowing ρ_ij^∘ the goal is to determine the antenna dependent complex gains g_is.

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