Let xe and xe, ∗ be the residual values for the epochs e = 1...Ne introduced in Sect. 4.3.4. These values are correlated due to space-dependent errors, thus they contain some common component αe. Formally, this is expressed by (A.1)where and are uncorrelated random values with zero expectation and with variances equal to the model epoch variances and , respectively. We also assume that , , and αe are not correlated. The model (A.1) implies that the measured variances (A.2)for xe and xe, ∗, respectively, are larger than the model predictions. We want to combine xe and xe, ∗ in a term (A.3)that has minimum variance reached for some a (here E denotes mathematical expectation substituted by the average over epochs). This is equivalent to the condition , which has the solution (A.4)yielding the best variance of Ae(A.5)or (A.6)Here, c2 = E [xexe, ∗] corresponds to the covariance between xe and xe, ∗ in the calibration file data. Note that c2 and the solution a in Eq. (A.4) depend on rsmall (the radial size containing the calibration stars), therefore σ2(Ae) has the lowest value for a rsmall that ensures maximum a2 or c2. Thus, the optimal rsmall is defined by the condition on the highest measured correlation , where the average is taken over epochs. For this specific value of rsmall, and using the measured value of , we recover the variance (A.7)of the systematic component between epochs.
The relation σ2(Ae) <De, which follows from Eq. (A.5), means that the correction never degrades the precision of the result, even for very large σe, ∗. For σe, ∗ → ∞, no improvement is expected, because a → 0.
We conclude that the best correction for the small-scale space errors to the measured values of xe is not αe but (A.8)which is determined with the precision σ(Ae) (in Sect. 4.3.4 denoted as φsmall(e)). It is important that the correction Eq. (A.8) efficiently mitigates the common systematic component αe. Taking into consideration Eq. (A.1), we find that the initial expectation of xe is E [xe] = αe. Following Eq. (A.3), we find that after the correction (A.9)The systematic error thus vanishes either in the trivial case of αe = 0 (no error) or if a = −1. The complete subtraction of systematic errors (the case of a = −1) is never possible, because . For stars in the field centre, a is usually −0.2... − 0.7. For DE0716−06, for example, a ≈ − 0.6, which means that the systematic errors in this case are reduced by half.
The computation of should be based on the calibration files related for the current mode k, and ideally, these files should be generated for each k. However, this is time consuming and, in practice, we computed the calibration file with k = 10 alone. In addition, instead of adding to xe, whose value depend on k, we added it to Xm for each m ∈ e. In the limit Ne → ∞, both procedures are equivalent. We verified numerically that with the limited number of epochs given in our project, the differences between the two procedures are negligible.
Because E [xe] = αe (see Appendix A), each frame measurement is biased by this constant value, thus E [xm] = αe. The systematic errors therefore induce the covariance between the frames m,m′ ∈ e. After the correction Eq. (A.3), the covariance between frame measurements within the epoch e is decreased to , hence , due to small-scale systematic errors. The effect of large-scale systematic errors is similar, because they are removed incompletely. The uncertainty φlong(e) of that correction gives the measure of the bias in the frame measurements, which results in the covariance cov(m,m′) = φlong(e)2. Taking into account the contribution from ϕ, the complete covariance is (B.1)The covariance matrix D (Sect. 3.3) is therefore non-diagonal and has a simple block structure with equal non-diagonal elements for m,m′ ∈ e and Dm,m′ = 0 for m,m′ ∉ e. The diagonal elements are . For instance, for a two-epoch data set with three frames each, the covariance matrix would be
The non-diagonal structure increases the computation time when many field stars are computed as targets (each one computed with reference to all other stars using its own matrix D). It introduces several nested loops and the computation using the direct inversion of D becomes too slow. Therefore we proceed differently when fitting data of the target and reference stars. The target is processed nominally, that is, using the generalised least-squares fit and the direct computation of the inverse matrix D-1. For reference stars, we used the approximate analytic expression for the inverse matrix (B.2)where is the weighted average of within the epoch e. The correcting term added to partially accounts for the change of precision between the frames. This approximation leads to equal non-diagonal elements in the inverse matrix and increases the computation speed. The expression (B.2) is used while running the reduction iterations, whereas the last cycle is performed with the direct inversion of D.
© ESO, 2014