Appendix A: Properties of the cross-correlation function

Centering

Let c(i) be a discretized function with a prominent peak, the position of which is to be determined. As in Tonry & Davis (1979) we do so by finding the position of the maximum of a parabola fitted to the peak. Let $i_{\rm c}$ denote the pixel with the highest value; if one uses an odd number of pixels (say, 2p+1) chosen symmetrically around $i_{\rm c}$, then the position of the maximum (expressed in pixels) can be written as $i_{\rm c} + \delta$ and the pixel fraction $\delta$ is given by:  

$$\delta = Km_1\ ,\ \ \ K = \frac{4p^2+4p-3}{5p(p+1)m_0 - 15m_2}$$ (A1)

where m_l denotes the $l^{\rm th}$ moment of the cross-correlation peak (within the interval used in the fit):  

$$m_l = \sum_{j=-p}^pj^lc(j).$$ (A2)

With an even number of pixels chosen symmetrically around the highest two values, a similar expression can be obtained; if necessary both results can be combined to reduce the discretization error as discussed in David & Verschueren (1995).

Random errors

The random error on the position of a cross-correlation peak, i.e. the part of the error which is due to noise on the spectra, can be estimated in a straightforward way as the square root of the variance of the shift measured between two spectra, say S and T, which are intrinsically identical and unshifted with respect to one another. We assume the spectra to be binned on N pixels so that  

$$S(n)=S_0(n)+N_{\rm S}(n), \ \ \ T(n)=S_0(n)+N_{\rm T}(n),$$ (A3)

where the subscript 0 indicates the intrinsic part and N(n) represents the noise. We shall assume throughout that the noise on any pixel is statistically independent from the noise on all others. This assumption may not always be valid in practice, e.g. if the data have been rebinned at some stage of the data-reduction process or if they have been filtered prior to cross-correlation; in that case the results derived below will have to be modified but the argument remains essentially the same. Let us define

$$n\oplus i \equiv n+i+N \pmod{N},\nonumber$$

 

$$n\ominus i \equiv n-i+N \pmod{N}.$$ (A4)

Then the correlation function can be written as:  

$$c(i) = \frac{1}{C} \sum_{n=0}^{N-1} S(n)T(n\oplus i),\ i=-N/2,\ \dots ,\ N/2-1 \nonumber$$

$$C = \sum_{n=0}^{N-1} S_0^2(n)$$ (A5)

where we have assumed, conveniently but without loss of generality, that N is even. Similarly, we limit ourselves to the case where we can fit an odd number of pixels and we can apply Eq. (A1) (with $i_{\rm c}=0$). Moreover, for the present purpose it is sufficient to retain only terms of first order in the noise; m₁ is itself of first order since the noise-free part of the correlation function is symmetrical, so we obtain the approximation  

$$\delta \cong K_0 m_1$$ (A6)

where K₀ is given by (A1) and the subscript 0 again indicates that noise contributions have been omitted. The variance ${\sigma_{\rm P}}^{2}$ of the correlation-peak position is obtained now as the statistical average of $\delta^2$: 

$${\sigma_{\rm P}}^2 \equiv \; \langle \delta^2\rangle \; = \f... ...g^2(n,p) \left[\sigma_{\rm S}^2(n) + \sigma_{\rm T}^2(n)\right]$$ (A7)

where we used $\langle N_{\rm S,T}(n)\rangle =0$ and defined  

$$g(n,p) = \sum_{j=-p}^p jS_0(n\oplus j)$$ (A8)

$$\sigma_{\rm S,T}(n) = \langle N_{\rm S,T}^2(n) \rangle.$$ (A9)

By its definition, the random error $\epsilon$ in Sect. 5 must be identified with $\sigma_{\rm P}$.

With spectra given by Eq. (A3), both the error on the position of the cross-correlation maximum and the antisymmetric part of the correlation function originate from the noise only. Pre-eminently in this case, any relation between those two should be revealed most clearly, at least through their statistical average. We therefore calculated the average of ${\sigma_{\rm a}}^2$ defined in Eq. (11), applied with $\delta = 0$ since there is no intrinsic shift:  

$$\langle {\sigma_{\rm a}}^2\rangle = \frac{1}{C^2}\sum_{n=0}^{N-1} h(n) \left[\sigma_{\rm S}^2(n) + \sigma_{\rm T}^2(n)\right]$$ (10)

where  

$$h(n) = \frac{1}{2N}\sum_{i=0}^{N/2-1}\left[S_0(n\oplus i) - S_0(n\ominus i)\right]^2.$$ (11)