An Independent Simulation of Imaging Characteristics of a Millimetre
Array, with and without a single Large Element (LE), and a LE pointing
correction algorithm
Abstract
As part of the MMA imaging group's investigations into the MMA imaging characteristics, a study has been made of imaging characteristics of a Millimeter Array consisting of a large number of 8m antennas, in which short spacing information is obtained either from the 8m array elements operating in single dish mode, or from a separate 20m dish. This study overlaps with some of the work of earlier studies (refs (1), (2), (3) and other references therein) and gives an independent confirmation of some of that work. The current study examines in more detail the relative importance of different components of the pointing error model, and presents an algorithm (PHFIT) which may lead to a relaxation of pointing requirements of the individual antennas.
The conclusions are summarized as:
1. Where similar parameters of noise and pointing errors have been used there is
good agreement with earlier work (refs (1),(2))
2. Independent investigation of different pointing error components (global,
differential systematic, random) shows that the dominant components regarding
degradation of dynamic range after single dish and MMA interferometric data have
been combined are:
(a) For single dish data, using either the 8m elements in total power or a
separate 20m large element (LE), global map pointing offsets.
(b) For interferometric measurements, the differential systematic pointing
error, i.e. the mean pointing error in the relative offset of one field of the
mosaic with respect to its neighbours, is the dominant source of error.
A given uncorrected pointing error in short spacing data, derived either from a 20m
or an 8m antenna element, is more harmful than the same pointing affecting
interferometric data, by a factor typically of 4 to 6, depending on circumstances.
If not corrected, a given global pointing error on 20m single dish is much more
harmful than the same angular pointing error on an 8m dish. Purely random errors,
such as tracking errors, are less important.
3. The effect of the global pointing error of single dish observations, either with
the 8m dishes or a separate 20m dish providing the short-spacing data, can
effectively be eliminated by making use of the overlap in the UV plane of regions
common to both single dish and interferometric data. This technique is very
effective, and probably should be applied as a standard correction to most combined
single dish and interferometric data sets. In the presence of a poor signal-to-noise ratio, the technique degrades more rapidly with a smaller single dish and its
smaller area of UV plane overlap with interferometric data, than with a large single
dish element.
4. If the phase overlap correction technique is applied, the 20m single dish always
gives greater dynamic range in the combined MMA image than a corresponding 8m dish
in total power mode. However, the difference is often marginal.
Introduction
Some extensive investigations into mosaicing and the imaging properties of the proposed MMA have already been carried out (refs (1),(2),(3)). Work reported here supplements these earlier investigations, examining in more detail the comparison between arrays with and without an extra single large element (LE), in the presence of different types of pointing error and in the presence of noise. Simulations were carried out using the same model source as the earlier investigations, but with software developed independently. Because of the importance of results derived from simulations to the design specifications of the MMA, it was thought useful to verify some aspects of the earlier simulations with independent software and starting with different initial assumptions.
Description of the Simulations
The simulations described here are equivalent to an ideal linear mosaic analysis of data. The model source, shown in Fig (1), is the same as was used in earlier simulations. Here, the model is represented by a 128 by 128 grid, sampled at 1" intervals, i.e. a total extent of ~2.1 arc min. The assumed wavelength is 1.3 mm. The region of emission extends over ~1.5' within this field. Observations were simulated using a MMA with 8m diameter antenna elements, with interferometer spacings range from 9m to 70m. Short spacing information is provided either by the 8m elements operating independently, not simultaneously, in total power mode, or by a separate 20m diameter single dish. The 8m and 20m elements were given a gaussian illumination taper truncated at -12 dB at the edge of each dish. The effect of varying the degree of taper was investigated, but only very minor variations on the results of the simulations were found. The beamwidth (fwhp) of an 8m dish at 1.3mm is ~40", and of a 20m dish ~16". The polar diagram of the interferometer dishes was truncated in the simulations at the first null, a radius of 50". The nominal mosaic pointing centres of the interferometer array were spaced at 16" intervals, with some tests varying this spacing. The data were well sampled in the UV plane, and the synthesized beam was assumed to be identical at all parts of the mosaiced field. For some tests various amounts of noise was added. In most tests, the same UV plane rms noise level was used for single dish data as for the interferometric data.
Dynamic range was used as a measure of quality of the mosaic reconstruction. Dynamic range is defined here as the ratio of the peak brightness of the model data to the rms value of the residual map (model data - simulated reconstruction), with both model and reconstruction truncated at the same radius in the UV plane. The absolute value of the dynamic range is highly dependent on details of the model source, its extent within the total field, and the maximum interferometric baseline used for the simulated
reconstruction. However, values of this parameter are a good measure of the relative
importance of different sources of error.
Figs 1 to 5 show an example of intermediate steps of the simulations.
Fig 1 shows the original model source distribution. Fig 3 shows the source as it would
appear observed with an 8m dish, in the absence of noise and pointing errors, while Fig 4
shows the source observed with a 20m dish, also in the absence of noise and pointing
errors. Fig 2 shows the source observed with the MMA in its compact configuration, with a
maximum baseline of 70m, using only interferometric data and hence with missing short
spacing information. Fig 5 shows the combination of the interferometric data of Fig 2 and
the data of either Fig 3 or Fig 4. In the absence of noise, pointing and other errors,
the combination of interferometric data with an 8m dish and with a 20m dish is identical.
This result shown in Fig 5 is indistinguishable from the original model distribution shown
in Fig 1, apart from minor low level fine scale features caused by the truncation at a
radius in the UV plane of 70m.
In this study the effects of different pointing errors on the reconstruction are
studied, in the presence of noise. The pointing error model has the following terms;
these terms may varied independently.
(i) A systematic, global pointing offset for the entire single dish map.
(ii) A random tracking error for the single dish data.
(iii) A systematic, global pointing offset for the entire MMA mosaic.
(iv) A random tracking error occurring throughout each MMA pointing within the mosaic.
(v) A systematic random offset of each pointing of the MMA observation,
remaining constant for the duration of each individual pointing, affecting all
MMA antennas equally, but varying randomly from one pointing to the next.
This is representative of an important class of pointing error that might
affect all dishes equally, such as an uncorrected change in refraction
coefficient between a calibration measurement and the observations.
These terms are considered to be a useful representation of the variety of pointing
errors encountered in single dish or interferometric modes of observation.
The Phase Fitting algorithm
Interferometer observations made with 8m diameter antenna elements, and with a closest
element separation of 9m, in principle contain information on baselines down to 1m, albeit
with vanishing weight. Since an 8m diameter antenna in single dish mode measures data out
to a radius of 8m in the UV plane, there is at least in principle a significant overlap of
UV coverage. This is illustrated in Fig. 6. Total power observations made with a 20m
dish of course give a greater overlap in UV coverage, illustrated in Fig. 7. A Fourier
Transform of the total power map made by a single dish yields a UV map; if there is a
global pointing offset of the total power map, then this corresponds to linear phase
gradient across the UV map. Pointing errors in the antenna elements used for
interferometric measurements do not, to a good approximation, affect the phase in the UV
plane. By comparing the phase in UV space between the single dish data and the
interferometric data, it should be possible to measure the linear phase gradient resulting
from the single dish pointing error. The success of this process is limited primarily by
signal-to-noise ratio in this region of UV overlap.
In practice the pointing error measurement and correction might be made in the following way; note that steps (i) to (iv) below are equivalent to performing a cross-correlation between the sky-plane maps of the single dish and of the interferometer data, and measuring the precise position of the central peak in the cross-correlation distribution:
(i) create a phase map in UV space from the Fourier Transform of the single dish.
(ii) create a phase map in UV space from the complex vibisility functions measured by the interferometers.
(iii) subtract these two phase maps.
(iv) fit a linear inclined plane to the difference using a weighted least squares technique. The weighting will take into account the different spatial frequency response of the single dish and interferometric observations, essentially as shown by the overlap regions of Figs 6 or 7.
(v) the fitted inclined plane is subtracted from the single dish UV phase map,
thereby correcting for the global pointing offset of those measurements.
Step (iv) has to consider the two-pi ambiguities of the phase map. A convenient
practical implementation of the above scheme would be, for step (iv), to take a Fourier
Transform using the amplitude given by the weighting function and the phase given by the
phase difference map. The offset from the origin of the peak amplitude in the resulting
transformed map - the "pointing error map" - gives the global pointing offset of the
single dish observations with respect to the interferometer phase, as required. This also
could have been derived from a direct cross-correlation between the two data sets, but
additional spatial frequency filtering has been included to enhance the signal-to-noise
ratio. The derived offset is then applied to the raw single dish data either as a shift
in the sky plane or as an added phase gradient in the UV plane. This implementation was
used for the simulations described here. Fig 8 shows a "pointing error map" made in this
way, using the weighted overlap of UV response of the interferometers and observations
with a 20m dish. A moderate degree of noise has been added to the data. The offset from
the origin of the dominant peak in this map gives the simulated single dish global
pointing error, in this case an arbitrary 10". With extremely high noise level, it may be
impossible to identify the dominant peak, and the algorithm fails. This is illustrated
later with Figs 9 - 14 and 19 - 20. Fig 9 shows a simple source, with noise added, as
might be observed by the MMA using a 20m single dish and interferometer baselines from 9m
to 30m, with no pointing errors. Fig 10 shows the effect if the single dish is mispointed
by 10" (the 20m beamwidth is 16") and remains uncorrected. Fig 11 shows the result after
applying the PHFIT algorithm to correct for the global pointing error; there are now no
residual errors attributable to the original poor pointing.
The threshold for failure of this technique, due to excessive noise, is higher for combinations using 20m single dish data than for 8m data. This is to be expected from Figs 6 and 7; the degree of deconvolution and the areas of overlap are both more favourable for observations using a 20m dish. Figs 19 and 20 illustrate this quantitively. Figs 12 - 14 show the same source as Fig 9, but using an 8m dish for the short spacing data. The noise added for the examples of Figs 9 - 14 is at the same level, but is too high for PHFIT to operate successfully with 8m data, although still within the useful range for 20m data. Fig 13 shows a small negative region close to the source, caused by the uncorrected 10" pointing error in the 8m single dish data. The dominant peak of the "pointing error map" equivalent to Fig 8 has become masked by excessive noise. In this example, PHFIT degrades the resultant image (Fig 14) and is inappropriate without further refinement. However, in the uncorrected image of Fig 13 the defects due to poor single dish pointing are nearly masked by the excessive noise level, and would be invisible if MMA data were included out to 70m or more, rather than being truncated at 30m as in this example.
Figs 19 and 20 show the residual pointing errors to be expected from the PHFIT algorithm in determining the global pointing offset of the single dish data, as a function of noise level. Fig 19 shows the fitted errors where the observations are of a single point source. In this plot, a noise rms of 1.0 along the X-axis is approximately 1.2% of the peak amplitude of the point source when observed with a maximum baseline of 70m. Beyond the last points plotted - i.e. >0.5 in rms noise level for the 8m case and >2. for a 20m dish - the algorithm breaks down, as explained above. As well as having a higher threshold tolerance to noise, the residual 20m pointing error is several times smaller than the corresponding 8m residual error. Fig 20 shows the results for the model M31S source. Here 1.0 on the noise axis corresponds to 28% of the peak amplitude of the source observed with a 70m maximum baseline. This example is of much more practical relevance than the point source example, and interestingly then the difference between 8m and 20m dishes is less pronounced. The probable reason for this change is that the spatial frequency response of the source biases the useful region of UV overlap towards lower spatial frequencies, so that the extended area of overlap with the 20m dish becomes of less significance.
The effect of these errors due to imperfect operation of the PHFIT algorithm has to be put in the correct perspective. From Fig 20, the point at which PHFIT breaks down is at a signal-to-noise ratio (Peak/rms), using all baselines out to 70m, of only 4:1. Immediately below this threshold, the residual PHFIT error is approximately 3.5" for the 8m case, and less for the
20m case. Fig 15 or Fig 17 show that a 3.5" single dish global pointing error results in
a degradation of dynamic range of noiseless data to about 100:1; i.e. the degradation in
effectiveness of PHFIT due to noise gives image defects that are more than an order of
magnitude below the noise level itself. Naturally these defects become more serious if
the data are smoothed to lower resolution, thereby reducing the effective random noise
level.
The simulations
Figs 15 - 18 show plots of dynamic range, as defined above, for different types of
pointing error.
Fig 15 shows the dynamic range achieved in this study with uncorrected global pointing errors in the single dish data, but with no errors in the interferometer data. Here, as earlier studies showed, for a given pointing error the larger single dish actually gives worse dynamic range.
Fig 16 shows the equivalent data where there are no single dish pointing errors, but with interferometer antenna pointing errors. Here the larger single dish gives a slightly higher dynamic range. This is to be expected, since corrupted interferometer data is given lower weight at radii smaller than 20m in the UV plane. The dynamic range is degraded about 6 times less seriously from interferometer pointing errors, than from single dish errors, for the same pointing inaccuracy.
As a comparison with earlier work, Fig 17 shows dynamic range achieved in this study as a function of pointing error, using an 8m dish for short spacing information, compared with the results shown by Holdaway in ref. (1). A slightly worse dynamic range is achieved in the current work, but considering the different assumptions of both independent simulations and the differing pointing models, the agreement is considered to be extremely good.
Fig 18 shows the result of simulations with short spacing data provided with an 8m
antenna and a 20m antenna, using the following parameters:
Single dish data: 6" global pointing offset,
1" random tracking errors.
PHFIT pointing correction algorithm applied.
Sampling in sky plane better than Nyquist rate.
Interferometer data: 6" global pointing offset, identical for each pointing of the mosaic,
1" random tracking error,
a randomly different systematic pointing offset for each pointing of the mosaic away from its nominal position, varying from 0" to 6".
Adequate sampling in UV plane, for all pointings of the mosaic.
With these parameters, even with a pointing error as high as 6", a dynamic range of a
little over 500:1 is achieved with a 20m single dish, and a little under 500:1 with the 8m
dish providing short spacing data. For pointing errors as small as 1", a dynamic range of
several thousand is obtained.
Limitations
These simulations have only explored a tiny volume of the potentially important
parameter space, and like all simulations involve several important initial assumptions
and approximations. One important area that has been neglected is the effect of shadowing
between adjacent dishes. With the assumed minimum baseline used here of 9m, shadowing
would begin at elevations below 63 degrees. Increasing the minimum baseline would
minimize shadowing, but decrease the effectiveness of an 8m element in providing all the
necessary short-spacing data in the presence of noise and other errors. This aspect of
the MMA design needs study.
Conclusions
The following conclusions are drawn:
1. Where similar parameters of noise and pointing errors have been used, the
predicted dynamic range vs. pointing errors from this study is in good agreement
with earlier work (ref (1)).
2. Independent investigation of different pointing error components (global,
differential systematic, random) shows that the dominant components regarding
degradation of dynamic range are:
(a) For single dish data, using either the 8m elements in total power or a
separate 20m large element (LE), global map pointing offsets dominate the
effective reduction in dynamic range.
(b) For interferometric measurements, the differential systematic pointing
error, i.e. the pointing error of one field of the mosaic with respect to its
neighbours, is the dominant source of error.
A given uncorrected pointing error in short spacing data, derived either from a 20m
or an 8m antenna element, is more harmful than the same pointing affecting
interferometric data, by a factor typically of 4 to 6, depending on circumstances.
If not corrected, a given global pointing error on 20m single dish is much more
harmful than the same angular pointing error on an 8m dish. This had already been
shown in the earlier studies.
3. The effect of the global pointing error of single dish observations, either with
the 8m dishes or a separate 20m dish providing the short-spacing data, can
effectively be eliminated by making use of the overlap in the UV plane of regions
common to both single dish and interferometric data. This technique is very
effective, and probably should be applied as a standard correction to most combined
single dish and interferometric data sets. In the presence of a poor signal-to-noise ratio, the technique degrades more rapidly with a smaller single dish and its
smaller area of UV plane overlap with interferometric data, than with a large single
dish element.
4. if the phase overlap correction technique is applied, the 20m single dish always
gives greater dynamic range in the combined MMA image than a corresponding 8m dish
in total power mode. However, the difference is often marginal.
References:
(1) M. Holdaway (1990), "Imaging characteristics of a homogeneous millimeter array", NRAO Millimeter Array Memo No. 61.
(2) R. Braun (1988), "Mosaicing with high dynamic range", NRAO Millimeter Array Memo No. 46.
(3) "The Millimeter Array", Proposal to the National Science Foundation submitted by Associated Universities, Inc., July 1990, Chapter IV.