5. Error analysis
An estimate of the standard error in the source-separation determination derived solely from the postfit residuals may well underestimate the true standard error. For example, slowly varying measurement errors may be absorbed in the astrometric analysis in such a way that they contribute to errors in the estimated values of the parameters, but do not leave traces in the residuals. Furthermore, errors in the parameters whose values are fixed instead of estimated may affect the solution significantly but have no perceptible effect on the residuals.
Elósegui et al. (1991) performed an error analysis to determine reliable standard errors for the estimates of the angular separations of 1038+528 A and B in 1981.2 and 1983.4 at 3.6 cm and 13 cm. Their analysis concerned the propagation of errors in the values assumed for the quantities used to model the geometric, atmospheric and instrumental contributions to the angular-separation determination (e.g., baseline vectors, position of reference source on the sky, tropospheric zenith delays, ionospheric delays, and amplitude calibration). We repeated this analysis for the third epoch of observation and reassessed the solar plasma contribution; Table 3 summarizes the results of this error analysis for 1990.5.
Table 3. Compendium of effects considered in estimating the standard error in the relative position of the quasars 1038+528 A and B in June 1990, at 3.6 cm and 13 cm.
In this paper we readdress some types of errors which can affect the determination of the proper motion inferred from the direct comparison of the relative positions measured at different epochs. These types of error stem from: (1) the criteria used to choose the reference points within the brightness distributions; (2) the particular software used for the data analysis; (3) the set of values used for the Earth's pole position and orientation; and (4) the non-identical observing conditions for the three epochs. We summarize below the means we used to estimate these effects at 3.6 cm:
(1) We compared the positions of the reference points in the quasars A and B determined as indicated in Sect. 3, with positions derived using two other plausible procedures: (a) a two-dimensional interpolation with the method of "bicubic splines", and (b) different pixel sizes in the mapping with the same method of choosing the reference point. For 1038+528 A, the pairwise differences for the three methods are less than 6 as in magnitude for all three epochs; for 1038+528 B, these differences are less than 29, 22 and 10 as in 1981.2, 1983.4 and 1990.5, respectively.
(2) We adapted two geodetic software packages, OCCAM (Zarraoa 1993) and CALC/SOLVK (see, for example, Herring 1990, and references therein), for differential astrometric analysis, and used them to analyze the observations of 1038+528 A and B from each of the three epochs. The pairwise differences in the determinations of the angular separation estimated using these two software packages and VLBI3 are at the as level for each epoch.
(3) We re-analysed the data from the three epochs using four different sets of consistent values for the baseline vectors, the reference source coordinates, and the Earth orientation parameters provided by Goddard Space Flight Center (namely, GLB401, GLB622, GLB718 and GLB831) (Chopo Ma priv. comm.). The angular separations of A and B for each epoch were found to agree at the as level.
(4) We studied the dependence of the angular separation determinations on the coverage of the (uv) plane. The (uv) coverages for the observations in 1981.2 and 1983.4 were quite similar, whereas that for 1990.5 was much less complete. We therefore selected a subset of the 1983.4 measurements which has a (uv) sampling similar to that in 1990.5. The difference in the angular separation determination for the 1983.4 data, estimated from this subset and from the total data set, was 11 as in right ascension and 1 as in declination.
Considering the above and looking at Table 3, we conclude that the largest source of uncertainty in the estimate of the angular separation of quasars A and B is the error in the identification of the reference points within the structures of each quasar, at 13 cm about as, and at 3.6 cm, as (Rioja 1993). This error is independent of the angular separation of the two sources; instead it is directly proportional to the interferometric beam size in each particular direction and inversely proportional to the signal-to-noise ratio in the maps (Thompson et al. 1986).
We take as reasonable estimates of the standard errors for the angular separations measured, at 3.6 cm and 13 cm, the root-sum-squares of the statistical standard deviations and the individual uncertainties listed in Table 3. These statistical standard deviations were derived from the postfit residuals and are between 7 and 10 times smaller than our above estimates of the standard errors at 3.6 cm for each of the three epochs.
© European Southern Observatory (ESO) 1997
Online publication: May 5, 1998