Accuracy of the Surface Measurements

Studies of the holography procedure, Scott and Ryle (1977), Kesteven (1994), Butler (1999) have shown that the main factors which define the rms surface accuracy are:

• System Noise

  The rule of thumb is :

  
  

  $$\sigma \sim \frac{\lambda N}{3 \pi SNR}$$
  
  

  for an N by N map, with a boresight signal to noise of SNR.

  We can etimate SNR from figure 5 which shows the phase rms as a function of the visibility amplitude.

  $\sigma_{phase} \sim 5$ (degrees) at Vis = 10.0, which implies a noise amplitude of 0.6 units (rms). The boresight SNR is thus:

  
  

  $$SNR = 10000 / 0.6 \sim 15000.$$
  
  

  implying a surface rms of 0.02 mm. We need to refine this estimate, and account for the variation of the illumination over the aperture - in effect, the signal-to-noise is rather worse at the edge of the antenna.

  The complex holography images, after the FFT and the normalisation, will have a pixel noise rms, $\sigma = 0.0005$ (surface current units). Since the illumination falls from 1.0 at the origin to $\sim$ 0.1 at the edge, the phase which we map to a surface error will have an rms error of 0.3 degrees at the antenna edge. This means a surface measurement error of 0.04mm (rms).

• Pointing Errors

  Random pointing offsets amount to random errors in the gridding operation, and lead to a surface error:

  
  

  $$\sigma \sim \frac{\theta_{rms} D}{12}$$
  
  

  We derive $\theta_{rms} < 3$ arcsecs from the hourly pointing checks, indicating a contribution to the surface error of

  
  

  $$\sigma < 0.1 mm$$
  
  

  The hourly checks show a systematic trend in the pointing, suggesting a small error in the satellite ephemerides. This could be accommodated at the gridding stage, although it has not been done in this experiment. Our estimate for the pointing rms is also close to the repeatability error in the pointing checks.

• Phase Errors

  We rely on the boresight calibrations to provide the fundamental phase reference. Errors in that calibration translate directly to surface error :

  
  

  $$\sigma = \frac{\sigma_{phase} \lambda} {12 \pi}$$
  
  

  We estimate $\sigma_{phase} < 5$ degrees from the calibration plots and the duplicate scans; this translates to :

  
  

  $$\sigma < 0.05 mm$$
  
  

All this suggests that the surface accuracy is $\sim$ 0.1mm, set primarily by the antenna tracking and the satellite ephemerides accuracy.

Detailed examination of the two images suggests that this estimate is realistic over most of the antenna, but much caution is required at the outer edges - the error in rings 16 and 17 does appear to be somewhat higher.