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Appendix A: Small-scale correlation in epoch residuals

Let x_e and x_{e, ∗} be the residual values for the epochs e = 1...N_e introduced in Sect. 4.3.4. These values are correlated due to space-dependent errors, thus they contain some common component α_e. 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 α_e are not correlated. The model (A.1) implies that the measured variances (A.2)for x_e and x_{e, ∗}, respectively, are larger than the model predictions. We want to combine x_e and x_{e, ∗} 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 A_e(A.5)or (A.6)Here, c² = E [x_ex_{e, ∗}] corresponds to the covariance between x_e and x_{e, ∗} in the calibration file data. Note that c² and the solution a in Eq. (A.4) depend on r_small (the radial size containing the calibration stars), therefore σ²(A_e) has the lowest value for a r_small that ensures maximum a² or c². Thus, the optimal r_small is defined by the condition on the highest measured correlation , where the average is taken over epochs. For this specific value of r_small, and using the measured value of , we recover the variance (A.7)of the systematic component between epochs.

The relation σ²(A_e) <D_e, which follows from Eq. (A.5), means that the correction never degrades the precision of the result, even for very large σ_{e, ∗}. For σ_{e, ∗} → ∞, no improvement is expected, because a → 0.

We conclude that the best correction for the small-scale space errors to the measured values of x_e is not α_e but (A.8)which is determined with the precision σ(A_e) (in Sect. 4.3.4 denoted as φ_small(e)). It is important that the correction Eq. (A.8) efficiently mitigates the common systematic component α_e. Taking into consideration Eq. (A.1), we find that the initial expectation of x_e is E [x_e] = α_e. Following Eq. (A.3), we find that after the correction (A.9)The systematic error thus vanishes either in the trivial case of α_e = 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 x_e, whose value depend on k, we added it to X_m for each m ∈ e. In the limit N_e → ∞, 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.

Appendix B: Correlation of individual frame measurements

Because E [x_e] = α_e (see Appendix A), each frame measurement is biased by this constant value, thus E [x_m] = α_e. 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 φ_long(e) of that correction gives the measure of the bias in the frame measurements, which results in the covariance cov(m,m′) = φ_long(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 D_m,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