4 Applications

To illustrate the efficiency of the method an example is shown in Fig. 3: a randomly picked configuration of 64 antennas is optimized for a Gaussian distribution of FWHM, $\delta=0.7\times R$, R being the radius of the sampled uv-disc. Figure 4 shows the convergence and quality of the optimization. The standard deviation as shown on the left-hand side was computed in the same kind of grids as those used to compute the gradients. The rapid and almost monotonic decrease of the standard deviation along iterations bring to evidence the rapid convergence. On the right-hand side the distribution of antennas of the optimized configuration is close to the Gaussian expected, confirming that the configuration is optimal or at least close to it.

Figure 5 shows the resulting configurations of several optimizations. Each row corresponds to a different optimization and the profile of the model distribution is represented in dashed line. The result obtained for a uniform distribution of samples (first row) can be compared to the Reuleaux triangle obtained by Keto. It can be noticed that to one extent the shape of the configuration confirms the analyze developed in Keto (1997): it is a disturbed curve of constant width. Though, it is difficult to distinguish whether it is closer to a Reuleaux triangle than to a ring. In addition, some antennas (here 5 of the 64) are distributed in the center to compensate for the lack of weight at intermediate baseline lengths. Figure 6 shows the result of an optimization for a multi-configuration observation. Such an observation allows to get a better sensitivity on the short baselines.

Figure 7 illustrates the dependency of the configuration on the declination of the source when observing with earth rotation synthesis. To get rid of initial conditions dependency and to emphasize the general tendency of the configurations, 10 configurations have been optimized for each situation. The first two rows consider observations of a source for an hour angle interval symmetric with respect to the transit: [-1 h, +1 h]. The first one at $60\deg$ from zenith in south direction and second at  $60\deg$ from zenith in the north direction. It can be seen that for the northern source more antennas need to be at the edges of the configuration which has a slightly different shape. For asymmetric hour angle interval ([-2 h, 0 h] in the last two rows) northern and southern source observations are also different. It can also be noticed that for a southern source symmetric or asymmetric hour interval does not make any difference in the configuration. This is due to the almost circular shape of the tracks. For the northern sources the tracks are open ellipses and the degree of freedom in arranging them is lower. But once found the optimal arrangement gives much better Gaussian than for circles in the central region, i.e. for short baselines (see the profiles in Fig. 7). These examples show that the shape of the tracks has a strong impact on the configuration and that it is difficult to anticipate from a configuration optimized for a snapshot observation the way it should be modified for synthesis. This justifies the introduction of synthesis in the optimization program.

The efficiency and the flexibility of the method as illustrated by these examples make APO a well adapted tool for the array design. Interferometers have generally more stations than antennas to offer a range of resolutions and multi-configuration observations. A procedure to optimize the locations of all the stations and solve the design problem as introduced Sect. 1 is proposed:

1.

Define a set of $N_{\rm S}$ scientific goals, e.g. at 100 GHz imaging with a resolution of $1\hbox{$^{\prime\prime}$ }$, 0.5 $\hbox{$^{\prime\prime}$ }$, 0.1 $\hbox{$^{\prime\prime}$ }$ and astrometry with a resolution of 0.1mas ($N_{\rm S}$ = 4). To each of these goals corresponds a model distribution for the Fourier samples and a duration of observation to be used in the optimization (see Paper II for the case of imaging);

2.

Define a set of $N_{\rm O}$ observational situations (the durations of observation are already determined from step 1), e.g. 3 representative declinations: at zenith, at $60\deg$ toward the South and at $60\deg$ toward the North, and only symmetric observations with respect to the transit ( $N_{\rm O}=3$). Note that if computing time is not a limitation instead of taking a set of representative declinations it is possible to take a set of subsets made of several declinations covering an interval and bearing weights related to the distribution of sources over this interval;

3.

Optimize $N_{\rm S}\times N_{\rm O}$ configurations corresponding to the possible combinations going from the most compact to the most extended and taking into account the terrain constraints. At each optimization try to use as much pads as possible of the previously optimized configurations;

4.

If the total number of pads is too large merge some of them and do last step again or change the initial set of scientific purposes and start from step 1 again.

This scheme is currently used for ALMA design (Boone 2001a) in parallel with other approaches. It allows the design to be dictated by the scientific purposes and not by any a priori on the shapes of the configurations. It therefore warranties an optimal scientific return for the financial and technical effort invested in the project.