We now describe the results of simulations where artificially generated antenna signals are analysed with the different methods illustrated above (Sect. 3 (click here)). In doing this we assume first a Gaussian beam, as we did in Sect. 2 (click here), and will then investigate the case of a non-Gaussian antenna beam.
In this section we shall assume that the antenna beam is Gaussian
and we shall almost always consider
relatively small target displacements (
, 
) because when
the pointing errors are much larger other pointing techniques
(e.g., cross-scan) are more suitable than conical scanning.
Before actually showing the results of the individual techniques,
we shall estimate the root-mean-square (rms) noise, 
, used
in the aforementioned simulations.
Assuming a 0.25 Jy pointing source, an antenna gain of 10 K/Jy
(Arecibo approximate gain in L-band), and an integration
time as short as 
s,
one gets 
mK, if 
MHz
and 
K. For a 
K source this implies a
SNR 
. However, in the simulations discussed below we shall use lower
SNRs to show the performance of the various retrieving methods in less
favourable conditions.
Comparison of the true and calculated pointing errors
is evaluated by computing the residual pointing error
(in units of full beamwidth) defined as:
The simulated noise is obtained using a random number generator and
then added to the antenna signal. The retrieving procedure is
carried out with this noisy signal and then repeated several times,
adding a different random noise at each run. Finally, a mean value of
the residual pointing error, 
, is obtained.
In Fig. 3 (click here) we show an example of application of the
standard (non-linearized) fitting method to the antenna signal
of the source, 
, as described in the first part of
Sect. 3.1 (click here). When compared to Fig. 4 (click here)
(see Sect. 4.1.2 (click here)) it is evident that the data to be fit
when the antenna signal, rather than the normalised
signal difference, is used, are much less noisy. As one can see,
in Fig. 3 (click here) the 1 
noise added to the signal can
barely be seen.
Figure 3: Example of fit to the antenna signal, during 6-second scans,
utilising the 
method
(solid lines) for three different target positions: 
(filled
triangles), 
(filled squares), and 
(filled circles).
The radius of the conical scan has been
held constant, 
. The data shown are for s,
and 
. A 
noise is superimposed
on the signal. The 0.25 Jy calibration source is assumed to have
K 
Figure 4: Top: Example of fit to the
normalized signal difference, 
. (solid line)
and 
(dashed line), 
s,
and . The true values for the pointing
errors in this simulation are , whereas
the estimates from the fit (smooth solid line) are 
and 
.
Bottom: Same as before, but with true values
, whereas
the estimates from the fit are 
and 
It is also important to note that if only
the peak-to-peak response were measured, the conical scan output could
give rise to ambiguos results in terms of the target position, because
the modulation amplitude is zero for both zero and very large
pointing errors, and rises to a maximum in between. This is shown in
Fig. 3 (click here), where the peak-to-peak amplitudes of the
and response curves do not
differ greatly. There is however no ambiguity when the off-source
zero level is known (see Sect. 2 (click here)), as in Fig. 3 (click here).
When Eqs. (7 (click here)) or (9 (click here)) are used then,
as we have seen in Sect. 3.1 (click here), the resulting linearized
standard method makes maximum use of the available signal. However,
the appropriate integration time in the radiometer equation is only one half
of the conical scan period. In fact, two independent parameters,
and 
, are to be determined and it can be shown that the weighted
mean value of the variance of the fit is directly
proportional to the number of parameters to be estimated
(Richter 1995). As a consequence, the noise contribution
must be calculated as if we were observing with an "effective integration
time'' which is half the duration of a single continuous conical scan.
In principle, sources with flux density 
Jy
can be used as pointing calibrators; assuming a system equivalent flux
density of 
Jy and a bandwidth of
100 MHz, then the rms flux density becomes 
mJy,
with a 4 s effective integration time (see also below). This gives a 800:1
SNR on boresight. However, depending on the
frequency of observation and the RFI
situation, the bandwidth to be used could also be as small as 2 MHz.
But, even in this case, we still get a SNR of about 100 : 1.
In practice, atmospheric anomalous refraction (Olmi 1995) and, to a lesser extent, gain instability are more likely than noise to limit the pointing accuracy; this is a strong argument for keeping the conical scan period short. At Arecibo, a triplet of conical scans, consisting of OFF-SOURCE, CALIBRATION, BORESIGHT, CON-SCAN, CORRECTION, CON-SCAN, CORRECTION, CON-SCAN, BORESIGHT, OFF-SOURCE, CALIBRATION, should take less than a minute, if the conical scans take 8 s and the remaining steps 2 s each, with 2 s slew time between steps. However, given the uncertainty concerning the real magnitude of the anomalous refraction effects at Arecibo, extensive tests will be needed in the field.
This assumes that the feed arm and the Gregorian continually track the nominal source position, and that all offsets are done with turret and tertiary. Accuracy in the absence of gain instability, anomalous refraction and RFI should be consistently better than 0.01 FWHM for any standard calibrator. The sequence given above should reliably handle pointing errors up to one beamwidth, much larger than those expected to occur in practice.
Figure 5:
Grey-scale map of 
as a function of and
, in the case of a 
and . The figures on
the left show the result of a single conscan, whereas those on the right
were obtained after a second conscan was applied to the output of the first
ones (see Sect. 4.1.4 (click here)). Top: linearized standard method.
Middle: fitting 
.
Bottom: DP method. 
and 
are as in
Fig. 4 (click here). Contour levels are from -2.5 to 3.3 by 0.4,
and negative values are plotted as dashed contours.
The innermost and outermost thick contours in the top panels correspond
to 
and 
, respectively. In all other panels the
longest and outermost thick contour corresponds to , whereas
the smallest and randomly distributed thick contours correspond to
. All figures have the same grey-scale contrast, so they
can be directly compared
The results obtained by averaging 
over several runs, as
explained at the beginning of this section,
are shown in the top-left panel of Fig. 5 (click here)
() where we plot
as a function of the true pointing errors and .
In the simulation program, the integrals in Eqs. (10 (click here)) have
been substituted with the proper sums, and the other parameters are
as those of Fig. 4 (click here).
We note that the region within the 
contour
extends up to 
,
despite all caveats discussed in Sect. 3.1 (click here),
and it would be about the same also for a lower SNR (e.g.,
). However, a region with much better estimates
(
, innermost thick contour in Fig. 5 (click here))
is only found in the simulation with .
If the method of Sect. 3.2.1 (click here) is used,
one can fit the measured values of on one cycle
by using the expression
on the right hand side of Eq. (17 (click here)). This procedure leads to
estimates of and .
In Fig. 4 (click here) we show an example of such a fit. The top panel
shows the case of a small pointing error, , and
with 
assuming two different values, 0.2 and 0.5.
In the bottom panel of Fig. 4 (click here)
a similar example is shown, but for a larger
pointing error, .
In our test we have also added a 1 noise to the source signal,
as described above. One can clearly see in this case the strong
contribution of the noise (which depends on ),
which does not prevent the fit from
giving a good estimate of and , however.
The residual pointing error
is 
in the first case, and 
in the
second one.
In the middle-left panel of Fig. 5 (click here)
we plot
as a function of the true pointing errors
and , in the same conditions as those of Fig. 4 (click here)
and fixing .
Using a lower SNR (e.g., 
) one finds that the parameters
region within the contour would considerably shrink.
Furthermore, beyond the limit the residual pointing error
increases quickly and irregularly.
We can also estimate the total time needed for a complete pointing check,
which is 
, where 
is the time needed to move from one position to
the next, roughly estimated to be of order 1 s.
Therefore, in the case represented by Fig. 4 (click here) the complete test
would last approximately 2-3 minutes.
When the DP method is used,
the normalised signal difference must be multiplied by a sine-wave
of proper phase in order to retrieve the target position,
as detailed in Sect. 3.2.2 (click here).
In the simulation we have carried out, the oscillating terms present in
Eq. (18 (click here)) have been eliminated integrating both
left hand and right hand sides between 0 and 
. After performing
such an integration the factor c in Eq. (19 (click here)) becomes
equal to . The geometry of conical scanning also implies that
whereas 
will be the less noisy interval in the case
of , 
(see Figs. 1 (click here) and 4 (click here)), it will be the
noisiest part in the opposite case, i.e. , 
.
This method, as well as the -fit,
needs a complete scan cycle to
estimate the angle errors. Furthermore, when filtering
is carried out through the integration described above, the
noise components of which are added
in this process can give rise to large errors in the final estimate of
and .
As in the previous two cases, the output of the DP method is shown in
the bottom-left panel of Fig. 5 (click here). The parameters region
within the contour shows a similar behaviour to the fitting method,
although in the DP case this contour is smaller.
A performance comparison of the various methods
described above can be made using Fig. 5 (click here).
The series of figures on the right column of Fig. 5 (click here) have
been obtained by using the output from the single conscan runs
(left column in Fig. 5 (click here))
to apply a first pointing correction, and then running the conical scan again.
The rightmost column in Fig. 5 (click here) therefore shows the final
results as a function of the initial pointing errors and ,
after a pair of consecutive conical scans and pointing corrections.
We note that:
(i) Fig. 5 (click here) shows the
almost perfectly circular contour plots of the linearised
standard method. This is quite a distinct feature
from the fitting and DP methods, especially at very large 
values where the distribution of in the 
plane
becomes highly chaotic.
(ii) in the high SNR case shown in Fig. 5 (click here) the area of
the parameter region within is more extended if the fitting
or DP methods are used. This fact makes them, rather than the standard
method, more suitable for use at large (
) pointing errors.
However, it is important to note that in the case of the standard
method, after a second conical scan the region within 
is more
uniform and extended than in the fitting or DP methods.
(iii) we find that in a lower SNR case ()
the performance of the three methods
decreases considerably, although to a much lesser extent in the case
of the standard method, if just one conical scan is carried out.
The performance of the three methods as a function of the SNR is shown in
Table 1 (click here), where the average value of is calculated over a region
having a radius 
, in the case of the single-conscan
simulations.
| Method | SNR=2 | SNR=4 | SNR=8 | SNR=16 |
| Linearized standard | 0.12 | 0.09 | 0.08 | 0.07 |
|
Fitting | 16.6 | 0.85 | 0.17 | 0.05 |
| Dot product | 10.0 | 0.91 | 0.22 | 0.07 |
calculated over an area with 
, for
different SNRs and for the three methods discussed in the text
| Method | Single conscan | Double conscan |
| Linearized standard |  |  |
|
Fitting |  |  |
| Dot product |  | |
, calculated
in the area within the contour of Fig. 5 (click here)
Regarding the accuracy of the residual pointing errors estimated above,
Table 2 (click here) shows the average, , of the
standard deviations calculated for each individual point within the
contour of Fig. 5 (click here).
The results indicate that the linearized standard method has
(for 
) both lower residual pointing errors and smaller
uncertainty compared to the other two methods.
There is also a general decrease in the uncertainty after a second conscan
has been carried out.
We conclude that in high SNR situations all methods can be used.
However, one should expect better performance with the standard method
at small () initial pointing errors, and with the fitting and DP
methods at large (
) pointing errors.
This is a consequence of the approximations used in Sect. 3.1 (click here), which
make the linearized standard method less accurate when the initial
pointing offset is large. Given the short time required at Arecibo to carry
out a full conical scan, this is not however a strong limitation as a
second conical scan can rapidly improve the pointing error estimation.
With other less sensitive instruments the alternative method by
Eldred (1994) may prove to be effective. On the other
hand, as we mentioned in Sect. 4.1 (click here), the conical scan method should
be used only when the pointing error is relatively small.
A different situation arises when the simulation is carried out by
generating the antenna signal assuming a non-Gaussian beam.
In this section we shall use the following shape for the beam of the
antenna, which still has a circular cross-section in a plane
perpendicular to the beam axis:
Data generated using Eq. (21 (click here)), with the usual addition of
noise, can then be analysed with the methods
discussed in Sect. 4.1 (click here).
Figure 6:
Same as Fig. 5 (click here), for the
data being generated using the sinc2 antenna beam described in the
text and applying a single conscan only. The column on the left shows the
results obtained with , whereas the panels on the right-hand
column correspond to 
.
The two different values of show the increasingly
deteriorating effects of the sidelobes on the residual pointing error.
Top: standard method. Middle: fitting .
Bottom: DP method. The three innermost thick contours in the right-top
panel all correspond to (see also Fig. 7 (click here))
Figure 6 (click here) shows the results
of the application of the DP, linearized standard and
-fitting methods to the data generated
assuming the sinc2 cross-section defined above and applying a
single conscan only.
The simulation has been carried out for two different values
of , 0.2 and 0.6. The purpose was to show the increasingly
deteriorating effects of the sidelobes and the larger radius of the
conical scan on the residual pointing error, .
A comparison with the results of Sect. 4.1 (click here) shows that the
calculated estimates of and are much less reliable when
a relatively large conscan radius and a Gaussian
beam shape are used to fit data obtained with a beam whose
(symmetric) profile is not exactly known and has also sidelobes
(see the extent of the 
contour in
Fig. 6 (click here)).
However, the performance of the three methods is only slightly
sensitive to changes in the beam shape when the conscan radius is
small (at least when a Gaussian and a sinc2 beam are considered),
as shown by the comparison of Figs. 5 (click here) and
6 (click here). Another and more general solution when
the beam profile is expected to deviate strongly from a Gaussian
would be to measure the beam shape, and then use the appropriate function
for the numerical least-squares, as shown in Fig. 2 (click here).
Figure 7:
Same as Fig. 6 (click here), linearized standard method only.
This figure shows the innermost portion of Fig. 6 (click here);
a different grey-scale is used (levels are from -2.5 to 0.2 by 0.4)
to hightlight the presence of "ripples'' in the distribution of the
residual pointing errors in the case with ,
due to the effect of sidelobes in the antenna beam. On the left, the innermost
thick contour corresponds to , and the outermost is . On the
right, the three innermost thick contours all correspond to , whereas
the outermost is again
The most interesting feature in Fig. 6 (click here) is the appearance
of "ripples'' in the distribution of the residual pointing errors for the
standard method, in the case with , due to the presence of
sidelobes in the antenna beam.
This is better shown in Fig. 7 (click here) which plots
, for the standard method only, in the innermost part of
the 
plane. We have found that higher sidelobes would increase
the residual pointing error but only at relatively large target displacements
(
).
In the proposed conical scan method the feed is moved relative to the antenna optical axis, so a number of coma-related effects may be introduced, as compared with other pointing methods where the feed is kept aligned with the antenna structure. We briefly discuss how they may affect the accuracy of conical scan below, highlighting the Arecibo case, and also compare the conical scan technique with the standard radio astronomy method of making cross-scans.
The lateral or axial displacement of the feed in a Cassegrain or Gregorian antenna system causes a boresight beam shift with a consequent loss of axial gain and beam degradation. At Arecibo the displacement of the feed horn off axis causes a modest decrease in performance, but much less than that experienced in some other shaped reflector systems (Kildal et al. 1993), limiting the field of view.
At 1.4 GHz and above we expect a gain loss 
at 1 FWHM off axis (Kildal et al. 1993). This causes a
negligible fluctuation in the signal level
compared to the expected variation of the antenna temperature during a
typical conical scan with and 
(see
Figs. 2 (click here) and 3 (click here)). This will be easily verified through
a comparison with other methods during the commissioning phase of the
conical scan method. We would expect this effect to be even more negligible
in classical paraboloid systems with much more uniform subreflectors.
The formation of a standing-wave pattern caused by the interference of waves moving in parallel direction between the aperture plane of the feed horn and the apex of the main reflector may cause a baseline ripple in the observed spectra. However, at Arecibo we do not expect this to be a problem for several reasons.
First, the estimated peak to peak amplitude is expected to be less than 10 mK (or about 1 mJy), i.e. much less than a "typical'' 0.25 Jy pointing source. Second, the conical scan method will be used in continuum mode, and thus the typical "wavelength'' of the baseline ripple for the 132.6 m spacing between focus and reflector, about 1.1 MHz, will be usually significantly less than the measurement bandwidth. And finally, a value of beta can be selected to minimize the baseline ripple effect.
This may be a more important constraint for smaller antennas with wider ripple and lower SNR than Arecibo.
The cross-polarization at Arecibo comes from the offset nature of the dual reflector feed system, which introduces systematic cross-polarization for linear polarization, and from the platform blockage which introduces some random cross-polarization effects, at very low level. If the feed is displaced laterally from the focal point, and a dual circular polarization feed is used, then the right-hand beam points to a slightly different position in the sky than the left-hand beam.
However, the Gregorian design is expected to give cross-polarization levels in the aperture that are -27 dB or lower for frequencies above 1 GHz, and therefore the on-axis beam squint is expected to be small. To measure any possible variation off-axis during the conical scan, both polarizations will be measured separately, to determine pointing differences. The beam squint will also be checked using alternative methods such as cross scans.
The proposed conical scan technique has great advantages over the standard
radio astronomy pointing technique of making orthogonal scans on the nominal
source position in the case of large (
m) telescopes, as it
does not require to move the massive (and slowly slewing) structure of the
antenna, but only the feed support. Even when the mechanical nutation of
a single feed is not included in the design, the large telescopes currently
being planned or under construction, such as the NRAO "Green Bank
Telescope'' or the U.S./Mexican "Large Millimeter Telescope'', have the
potential to implement conical scan by displacing the subreflector or another
mirror along the optical path.
When the level of the degradations discussed above can be appropriately controlled or removed, the conical scan method has also several other advantages, such as (see also Abichandani & Ohlson 1981): (i) it uses all of the data all of the time, whereas the other schemes require larger overhead, making a more efficient use of the observing time; (ii) all pointing schemes have a degree of sensitivity to the asymmetry of the antenna beam profile, however the conical scan method has the least sensitivity as it averages around the entire scan; (iii) the standard cross-scan technique for pointing may cause unacceptable interruptions in data flow when the telescope is tracking a unique event or artificial satellite (Eldred 1994).
On the other hand, the standard pointing methods have the undoubted advantage that only the pointing parameters are involved in the retrieving technique, and that the computation requirement may be somewhat less than for some of the conical scan retrieving methods.