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5 Conclusion

  The hybrid mapping procedure may fail when the observed closure phases are only a few degrees while the individual baseline phases are large quantities. This happens when one telescope is far removed from the others of the network. In the resulting unbalanced triangles the two longest baselines become nearly parallel giving essentially identical phases but with opposite sign; their contributions therefore almost cancel in the closure equation. When the source is unresolved on the remaining, short baseline, the result is a closure phase of a few degrees.

Moreover, even in the presence of large closure phase values, the hybrid mapping procedure may fail when an oversimplified initial model is used to map a complicated source structure.

In both cases, low closure phase values or large errors associated with the model, hybrid mapping fails, generating artificial structures.

We demonstrate that the general precondition to avoid false structures in the map is that the errors (or more precisely their cube) of the model should be smaller than the observed closure phases. When this condition is satisfied the equation used in self calibration becomes linear.

It is evident that the small closure phase values associated with unbalanced triangles make this upper limit quite low and therefore easy to violate. When this condition is violated, the equation used in self calibration is non-linear and we can no longer assume that the antenna corrections are a sum of individual estimates obtainable by successive iterations. This procedure, which is normally followed in VLBI data processing, and is what we call in this paper Loop "A'', fails in this case. The non-linear equation produces antenna corrections little modified by successive cycles of self calibration even if, as shown in this paper, new, better models are tried subsequently. In other words the wrong initial corrections, remaining nearly unmodified, corrupt irremediably the data.

In this case the correct procedure is simply to self calibrate again, adopting the model derived by CLEAN directly on the original data and not on the corrected ones biased by the previous wrong solution.

This is what we call Loop "B''. At each iteration Loop "B'' discards the previous accumulated corrections and starts to compute them again as a whole by using the new better model derived by CLEAN. When this is done at each iteration the use of progressive better models ensure the above given condition for linearity to be satisfied and therefore the convergence on the proper solution.


We would like to thank Guidetta Torricelli for her useful suggestions and Richard Porcas for his comments. We both acknowledge support for our research by the European Union, M.M. under contract FMGECT950012, S.A. under contract CHGECT920011.

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