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Astron. Astrophys. 337, 69-79 (1998)

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We must be careful with the interpretation of multi-epoch data, because analysis methods can strongly affect the reliability of our results. It is very important to know which features to believe, and how accurate our measurements of very small changes are. To investigate these problems we made mapping and modelfitting tests in which we generated simulated data using the program FAKE in the Caltech VLBI data analysis package with the same u-v coverages as the real observations. These `fake' data contained realistic additive Gaussian noise, random antenna-based phase errors and time-varying amplitude calibration errors of order of 10%.

A.1. Reliability of features in CLEAN maps

Fake data for the 1st and 5th epoch were made using the same 9 gaussian components model, which was fitted to the real 5th epoch data (see Table 2). These simulated data were each mapped separately with a point component as a starting model. The final maps had different extended structures in the northern and southern components, they also contained apparent bridge emission between these three main components, which was not part of the model. These detected errors were presumably due to inadequacies and differences in u-v coverage at each epoch combined with differences in the details of mapping at each `fake' epoch (choice of windows etc). This test leads us to the conclusion that the details of diffuse and bridge structures seen in the CLEAN maps made from the real data (see Fig. 1a) are not reliable. We also conclude that using maps made separately at each epoch is an unreliable way of detecting changes in source structure.

A.2. Estimating modelfitting component position errors

As described in Sect. 4.1 we can set accurate limits on component motions by gaussian modelfitting to each epoch. We carried out `fake' simulations to answer two questions about this procedure. The first was to determine the size of the remaining random errors due to different u-v coverages and data reduction at each epoch under the null hypothesis of no component motion. The second question was to investigate if this method introduced systematic errors by biasing the results in the sense that real changes would be removed or reduced by initially trying to force the data at each epoch to agree with the same 5th epoch starting model.

In our first test in order to make the simulation as realistic as possible we attempted to take account of the fact that the real source structure in 0710+439 is almost certainly more complex than can be represented by 9 Gaussian components. This complexity is demonstrated by the fact the final agreement factors (see Sect. 4.1) of our models are further from unity than would be expected purely from random noise. It is conceivable that this extra complexity might interact with differences in u-v coverage to give apparent changes in the centroid position when a 9 component model is fitted at each epoch, even if no position changes actually occur in the source. To test the above possibility we created a 18 component model by replacing each component in the original 9 component model by two slightly shifted gaussians. The final model was such that the agreement factors on fitting a 9 component model to the corresponding `fake' data was roughly similar to that obtained with the real data.

Having chosen a suitable 18 component model we created `fake' data for the 1st and 5th epochs. Using a procedure as similar as possible to that used to analyse the real data we then fitted a 9 component model at each epoch and compared the separations between the gaussians we obtained. The size of the apparent changes gave us an estimate of the residual random error. On doing this test we found an apparent change of A2-C2 separation of 8.09 µas and in the A2-B2 separation of 30.9 µas between the 1st and the 5th epochs. The values are much smaller than the changes in component separation which we detected from the real data (see Fig. 3) and comparable with the variances we estimated from linear regression analysis (see Sect. 4.2).

A.3. Biasing due to cross self-calibration

In our final test we sought to determine if our modelfitting procedure introduced a systematic error due to initially self-calibrating all models against the same 5th epoch starting model. It is possible that real changes might be reduced or removed by initially trying to force all epochs to agree with the 5th epoch model. To quantify this effect we simulated the case of a 200 µas shift of A2 (and then C2) position between epochs 1 and 5. Applying our standard modelfitting procedure (see Sect. 4.1) we determined an estimated motion of just under 200 µas. The largest negative bias found in our test was only 9 µas. We conclude that this biasing mechanism has a negligible effect on the estimate in the shift of A2 seen in the real data.

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© European Southern Observatory (ESO) 1998

Online publication: August 6, 1998