Appendix A: Computation of Dw_i and M(n_iw_i)

In this Appendix, we compute the non-trivial elements of the covariance matrix of errors used in Sect. 3.2. Since we consider here a single spectral channel, the index iwill be omitted everywhere. Thus, the values N, $\hat{N}$, n, W, $\hat{W}$, and w in this Appendix are identical with the values denoted as N_i, $\hat N_i$, n_i, W_i, $\hat W_i$, and w_i respectively in Sect. 2.

In practice, the error in $\hat{W}$ depends on too many details of experimental techniques, atmospheric conditions and data processing methods to be analyzed in general form (Mourard et al. [1994]). The present analysis is restricted to a highly idealized situation. Namely, as well as in Sect. 3.1, the only source of measurement errors taken into account is the photon shot noise. We assume that the static fringes are detected by counting the photons in nonintersecting channels that cover in total the interval of length Z of optical path difference (OPD). If z is the OPD, which is measured here in units of $\lambda/2\pi$, and F(z) is the photon counting rate per unit z, then

$$F(z)=N(1+V\cos(z+z_0))/Z\,,$$ (A1)

where z₀ is unknown fringe phase shift and N is proportional to the total received flux.

If we define

$$C=2\int_0^ZF(z)\cos z\,{\mathrm d}z$$ (A2)

and

$$S=2\int_0^ZF(z)\sin z\,{\mathrm d}z\,,$$ (A3)

then assuming that $Z\gg 1$ and neglecting the terms that decrease as 1/Z for $Z\to\infty$we obtain that

$$N=\int_0^ZF(z)\,{\mathrm d}z$$ (A4)

and

$$V^2=(C^2+S^2)/N^2\,.$$ (A5)

An estimate for W=V² can be constructed by replacing the values in the RHS of Eq. (A5) by their estimates that can be easily deduced from Eqs. (A2)-(A4). If the interval of OPDs covered by measurements is divided in M subintervals of length $\Delta_p z$ and $\Delta_p z\ll1$ for $p=1,\ldots,M$, then an estimate $\hat{W}$ of W based upon the number of photons received in each of these subintervals is given by

$$\hat W=(\hat C^2+\hat S^2)/\hat N^2\,,$$

where $\hat N=\sum_{p=1}^{M}{\hat N}_{\rm p}$, ${\hat N}_{\rm p}$ is the number of photons received in the p-th subinterval, $\hat C=2 \sum_{p=1}^{M}{\hat N}_{\rm p}\cos z_{\rm p}$, and $\hat S=2 \sum_{p=1}^{M}{\hat N}_{\rm p}\sin z_{\rm p}$.

If we assume further that $\hat{W}$ as a function of $n_{\rm p}={\hat N}_{\rm p}-N_{\rm p}$can be linearized in the vicinity of zero and take into account that ${\tens M}(n_{\rm p})=0$and ${\tens M}(n_{\rm p} n_{\rm q})=\delta_{{\rm pq}}N_{\rm p}$, then after some algebra we obtain for the statistical characteristics of $\hat{W}$

$$\begin{eqnarray*}{\tens D}(w)&=& {\tens M} \left( \left( \sum_{p=1}^{M}\fra... ...right) \left( \sum_{q=1}^{M} n_{\rm q} \right) \right) =0\,, \end{eqnarray*}$$

which proves the validity of Eqs. (15) and (16) for i=j.