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

1.4. SOME STATISTICAL TOOLS 25
where δ( ⃗ k − ⃗ k ′ ) denotes the Dirac δ-function and the Fourier transformed autocorrelation
function is
∫
ˆξ( ⃗ k) = d n r ξ(⃗r) e i⃗k·⃗r . (1.72)
Equation (1.71) implies that the different Fourier modes (i.e. ⃗ k ′ ≠ ⃗ k) are completely
uncorrelated. This is a direct consequence of the assumed translational invariance,
or the statistical homogeneity, of the field. In real space however, f(⃗r) will
generally have extended correlations, i.e. 〈f(⃗r)f(⃗r ′ )〉 ≠ 0 for ⃗r ≠ ⃗r ′ .
Some properties of the power spectrum worth noting are:
• The Wiener-Khinchin theorem states that the power spectrum per volume and
the autocorrelation function are related by a Fourier transform: P ( ⃗ k) = V ˆξ( ⃗ k).
• Using the power spectrum, the variance of the field can be determined by taking
the inverse transform of Eq. (1.72) at ⃗r = 0 which gives
〈
f 2〉 ∫
= ξ(0) =
d n k
(2π) 3 ˆξ( ⃗ k) =
1
(2π) n V
∫
d n k P ( ⃗ k), (1.73)
thus, by integration of the power spectrum, the total variance of the field can be
derived.
• If the field can be considered to be statistically isotropic then the power spectrum
only depends on the modulus of the wave vector: P ( ⃗ k) = P (k).
• If the field f(⃗r) is incoherent, i.e. ξ(⃗r) ∝ δ(⃗r), then the power spectrum is
constant. Such fields are referred to as ’white noise’.
In real situation, one is faced with the problem of estimating the power spectrum
from a finite sample of the infinite random field (e.g. the RM distributions which are
limited by the finite radio source size even though the magnetised cluster gas extends
beyond that). One can write such samples as f s (⃗r) = W (⃗r)f(⃗r), where the function
W (⃗r) describes the sample geometry and is often referred to as the ’window function’.
The Fourier transform of such a finite sample of the field is the convolution of the
transforms ˆf and Ŵ , following the convolution theorem:
∫
ˆf s ( ⃗ d n k ′
k) =
(2π) ˆf( ⃗ n k ′ )Ŵ (⃗ k − ⃗ k ′ ). (1.74)
This means that the transform of the sample is a somewhat smoothed version of the
intrinsically incoherent ˆf( ⃗ k). The width of the smoothing function Ŵ is ∆k ∼ 1/L,
where L is the size of the sampling volume. Hence, the transform ˆf s will be coherent
over scales δk ≪ L but will be incoherent on larger scales. Thus, the act of sampling
introduces finite range correlations in the Fourier transform of the field.
1.4.3 Bayesian Statistics
A method in order to quantify uncertainties in a measurement are given by Bayesian
ideas (for a review, see D’Agostini 2003). The two crucial aspects of these ideas are
• Probability is understood as the degree of belief that an event will occur.