2 Method

The method to be used consists of two successive steps which can be repeated if necessary. First all plates are reduced simultaneously using a block method described by Stock (1981). Only linear terms are included in the process. At this point higher order terms or any other plate model could also be included. Not much advantage is obtained with a more sophisticated model in the first step since what is not represented by the model will be discovered and determined in the second step. To get the process started stars common to overlapping plates have to be identified first. For this purpose provisional plate constants are derived, based on single plate solutions with the help of only a few stars. These provisional solutions are good enough to identify a large number of additional cross identifications, leading to a new and improved set of plate constants. Naturally, this step can be repeated if necessary. Star by star the coordinates coming from different plates are averaged, leading to a first catalogue.

Once the first catalogue is obtained residuals may be calculated in two ways. If a star is not in the reference catalogue the residuals are simply the differences between the individual positions and their average. If the star is contained in the reference catalogue instead we use the difference between the individual positions and the catalogue position. The reason is that a rigid overlapping scheme as it was employed for the entire AC plates can cause certain problems. If we consider for example the X-coordinates (i.e. right ascension measures) we find that their differences derived from pairs of images of stars common to two or more plates are practically constant, representing the displacement of the plate centers in right ascension from plate to plate. This means that distortion terms which are periodic with this displacement cannot be discovered from the coordinated differences. To detect these an external comparison is needed, for instance with the reference catalogue. Thus, in view of the high accuracy of the catalogue positions, higher weights are given to these residuals.

These residuals are then plotted as a vector field in a common diagram. Because of the large number of residuals and the dense coverage of the field covered by a plate, systematic patterns make themselves apparent quite easily. These patterns could in principle be represented by a polynomial or any other mathematical expression, but we prefer to average and interpolate the arrows in the vector field to obtain the desired correction function. This function is then applied to the original measured coordinates. Subsequently the first step, i.e. the block adjustment with only linear terms, is repeated, followed by averaging the positions and constructing a new vector field. A new correction function can then be applied if found to be necessary. For the interpolation we use a sliding weighted polynomial described by Stock & Abad (1988).

In order to determine a magnitude dependent field distortion the data are divided into several groups of fixed magnitude intervals and the above steps are carried out separately for each group. The correction to be applied is obtained with a three-dimensional interpolation, the dimensions being the right ascension, the declination, and the magnitude. A similar process can be used for the determination of color dependent terms.