Investigations of Faraday Rotation Maps of Extended Radio Sources ...

82 CHAPTER 4. PACMAN – A NEW RM MAP MAKING ALGORITHM
where ⃗ ∇RM(⃗x) and ⃗ ∇ϕ 0 (⃗x) are the gradients of RM and ϕ 0 . A map pair, which was
constructed from a set of independent random polarisation angle maps ϕ(k), will give
a V ≥ −1. A map pair without any correlated noise will give a V ≈ 0. Hence, the
statistic V is suitable to detect especially correlated small scale pixel noise.
The denominator in Eq. (4.9) is the normalisation and enables the comparison of V
for RM maps of different sources, since V is proportional to the fraction of gradients
which are artefacts. Since, the comparision of the quality between maps calculated for
the same source using the two different algorithms is desired, it is useful to introduce
the unnormalised quantity
∫
Ṽ = d 2 x ∇RM(⃗x) ⃗ · ⃗∇ϕ 0 (⃗x). (4.10)
The quantity Ṽ gives the absolute measure of correlation between the gradient alignments
of RM and ϕ 0 . The smaller this value is the smaller is the total level of correlated
noise in the maps.
4.4.2 Error Under or Over Estimation
One problem, one is faced with is the possibility that the measurement errors are under
or overestimated. Such a hypothesis can be tested by performing a reduced χ 2 ν test
which is considered to be a measure for the goodness of each least squares fit and
calculates as follows
] 2
χ 2 ν ij
= 1 f∑
[ϕ obsij (k) − (RM ij λ 2 k + ϕ0 ij )
ν
σ 2 = s2 ij
, (4.11)
k=1
k ij
〈σ kij 〉 k
where ν = f −n c is the number of degrees of freedom and f is the number of frequencies
used and n c is the number of model parameters (here n c = 2). The parameter s 2 is
the variance of the fit and 〈σ kij 〉 k is the weighted average of the individual variances:
⎡
〈σ kij 〉 k = ⎣ 1 f
f∑
k=1
⎤−1
1
⎦
σk 2 ij
(4.12)
If one believes in the assumption of Gaussian noise, that the data are not corrupted
and that the linear fit is an appropriate model then any statistical deviation from unity
of the map average χ 2 ν of this value indicates an under or over error estimation. For
a χ 2 ν > 1, the errors have been underestimated and for a χ 2 ν < 1, they have been
overestimated.
Unfortunately this analysis does not reveal in its simple form at which frequency
the errors might be under or overestimated. However, one can test for the influence of
single frequencies by leaving out the appropriate frequency for the calculation of the
RM map and comparing the resulting χ 2 ν ij
values with the original one.
It is advisable to evaluate the reduced χ 2 ν ij
-maps in order to locate regions of too
high or too low values for χ 2 ν by eye.