ALMA Calibration

Bryan Butler, Stephane Guilloteau, Al Wootten, Ewine van Dishoeck

2003-Aug-07



1  Introduction

This document outlines the various quantities that will need to be measured or taken into account 
in order to collect and calibrate ALMA data.  It is an update of Chapter 3 of the old Project 
Book, and supersedes it.  Traditionally, "calibration" has often been thought of as strictly a post-
processing exercise in radio interferometry, essentially only involving things done to data after it 
has been collected.  In this document we take a broader view, and include all quantities that must 
be measured before correlation of the antenna signals as well.  These are still formally 
"calibrations", since they are measurements of instrumental parameters – they are generally just 
measured less frequently.  They are, however, no less important than the post-processing 
calibrations.  In addition, we address some topics that are not even direct measurements, but are 
rather things that affect the measurement of our desired quantities.  An example is the relativistic 
deflection of radio waves in the gravitational potential of the sun, which is not really a directly 
measured or calibrated quantity (except indirectly), but does certainly affect our ability to 
properly calculate delay, which in turn affects our ability to calibrate, for example, antenna 
station locations.

Accounting for all of these types of calibrations, measurements and effects is critical for ALMA 
to achieve its full  potential.  We must understand what effects must be accounted for, and how 
we will measure and/or correct for them during the data collection and post-processing.  Never 
before has a radio astronomical instrument been built with such a detailed understanding of the 
site and its impact on the telescope.  With this knowledge in hand, we can optimize the full 
measurement and calibration strategy to produce the maximum scientific output for ALMA.

In addition to simply describing the different types of calibrations, measurements and effects 
anticipated for ALMA, we also provide some specifications on the accuracy to which the 
measured quantities must be determined or effects must be accounted for.  As for other 
interferometric arrays, the necessary or possible calibrations can be split into several types (see, 
e.g., Fomalont & Perley 1999; Thompson, Moran, & Swenson 2001).  For our purposes we 
divide the calibrations into two main types:

?	Precalibrations.  These are quantities that must be known when the data is collected, and 
are used to make corrections to the signals or data at that time.  They are by definition 
irreversible.  They generally vary relatively slowly (longer than a single observation, and 
typically days to weeks.  They are often fundamental instrumental quantities, collected 
during special (and relatively infrequent) calibration observations.  In some instances, 
however, they vary more rapidly, and may have to be measured during the observation.

?	Postcalibrations.  These are quantities that can be measured before, during, or after the 
actual observation, which can be applied to the visibilities or total power data at any point 
in the data postprocessing.  This is the majority of what is normally considered 
"calibration" historically.

Note that a special case of postcalibration is selfcalibration.  If the target source is strong enough, 
then the technique of selfcalibration (see, e.g., Cornwell & Fomalont 1999) can be used to derive 
some calibration quantities.  We do not treat the topic of selfcalibration in this document, 
because we cannot assume that in all cases it can be used for ALMA observations.

The calibration requirements in this document all derive from a series of meetings, discussions, 
and memos on the requirements necessary for the types of scientific experiments anticipated to 
be done with ALMA, and incorporate input from the ALMA Science IPT, other ALMA IPTs, the 
ASAC, and the community at large.  They also incorporate a sense of reality, when the desired 
accuracy is simply not achievable or is too expensive.  These scientific experiments are 
described briefly in the Requirements section, and examples given where appropriate.

In some cases, achieving the required accuracy requires the use of special measurement devices, 
either on the antenna, or separate from it.  These devices are described where necessary, and 
requirements on their ability to measure the necessary quantity are given.

Finally, some of these calibration techniques require development and testing, as they are not 
commonly implemented on existing arrays.  In these cases, there will be an attempt to test the 
technique on an existing array or antenna (30-m, OVRO, BIMA, PdBI, VLA, etc…), or on the 
prototype antennas and interferometer at the ATF.  In some cases, testing on the early ALMA 
array itself will be required.

2  Requirements

The calibration requirements for ALMA derive from the types of experiments foreseen, and their 
required accuracies.  In general, the astronomers want things "as good as possible", and we 
should always strive to make this so, but often it is simply not feasible to make things as accurate 
as desired by the astronomers.  In some cases a direct cost increase is implied by improving a 
particular characteristic of the array, and that cost may be deemed as prohibitive.  In others, it 
simply may not be technically feasible to reach the desired levels.  We note here also that some 
desired final accuracy parameters are very hard to turn into specific calibration requirements, 
since ALMA will be such a complex instrument.  In many cases simulations of the behavior can 
help, but in others we merely make educated guesses and try to make things as well as we can.  
This is not completely satisfying, but is the situation we are in.  The various different 
requirements on calibration accuracy will now be described.

2.1 Antenna Characteristics

2.1.1 Pointing

Accurate pointing, or at least knowing where the antenna was pointed as a function of time, is 
important for most mosaiced observations.  It is anticipated that a very large fraction of ALMA  
observations will be mosaiced – as much as 50% overall or even more in the compact 
configurations.  It is also important for single-field submillimeter observations, given the very 
small primary beam at those frequencies.  High frequency mosaics would benefit from better 
pointing than it is deemed possible to obtain with a 12m diameter aperture.  Note that single-field 
observations at the lower frequencies, mosaiced observations at ?<120 GHz and mosaics which 
do not require high fidelity ("fidelity", though its precise meaning can be tricky, is essentially the 
ability to faithfully reproduce the actual sky brightness at all places in the image) at even higher 
frequencies will not require the best pointing.  

In an ideal world, we would know the types of experiments to be performed exactly, along with 
specific accuracy requirements, and would specify a pointing requirement that allowed those 
experiments to be done (derived via simulation).  Unfortunately, this is not the case, since the it 
is difficult for the astronomers to come to a consensus on the types of experiments which are 
most important to ALMA, and the derived requirement depends strongly on the types of 
experiments.  In addition, it is difficult to define a single imaging metric which is applicable to 
all types of astronomical sources (see, e.g., the wide variety of imaging metrics reported in Pety 
et al. 2001b).  We are not completely lost, however, as we do have some guidance for what the 
errors might be for at least some sources, and we have at least a general idea of the image quality 
that we want out (the disagreement is in the details).  In general, it is agreed that much science 
demands an image fidelity (by some definition) of around 100 (i.e., errors in the image are 1%), 
and that this must be true for large, complex sources.

In addressing the general problem of required pointing accuracy for high fidelity mosaicing, 
Cornwell, Holdaway, & Uson (1993) derived a pointing requirement of 1/16th of the primary 
beam FWHM, which translates to 0.85 arcsec for a 12m diameter primary aperture at 0.8 mm 
wavelength.   This was for a 40 element array (the old MMA), and to obtain an image fidelity (as 
they define it) of 100 on a particular simulated image.  Holdaway (1997a) performed a more 
detailed analysis, showing the effects of pointing errors ranging from totally random to totally 
systematic.  He found that with 12m antennas, a 1.1 arcsecond pointing error resulted in a 
median image fidelity of < 20 at 0.8 mm.  Morita (2001) performed a similar study, and found 
that with 0.6 arcsecond pointing errors on 12m antennas, the image fidelity in a 7x7 mosaic was 
limited to <30 for  ?>200 GHz.  Adding an array of smaller antennas (the ACA) improved this 
by about a factor of two.  Similar, but much more detailed, studies were performed in Pety et al. 
(2001a, 2001b), who obtained qualitatively similar results.  All of these studies indicate that the 
old adage of "as good as you can make it" is certainly true – to quote Sebastian von Hoerner (in 
VLA Test Memo 129, 1981): "In summary, the astronomical demands are not so well-suited for 
deriving a well defined specification for the pointing error … Thus, it comes back to wanting the 
pointing error just as small as possible, and then to live with what one can get."  This is as true 
for ALMA as it was for the VLA.  In discussions with the prototype antenna vendors, it became 
clear that 0.6 arcsec was an acceptable error to them, and that is what has been adopted.  It 
should not be thought that this number has been derived rigorously – it is as good as we can get, 
for as much as we want to spend.  There is one detail here – the prototype antenna vendors were 
not happy to supply an antenna that could "blind point" (point to any direction with only the aid 
of a global pointing model) to 0.6 arcseconds.  They will instead provide an antenna that blind 
points to 2.0 arcseconds, and then requires a periodic local pointing model update to reach 0.6 
arcseconds.  In terms of the calibration, we must then make sure that we can derive the quantities 
of the global pointing model to allow for the 2.0 arcsecond pointing, and the quantities of the 
local pointing model (and the frequency of updating it) which allow for the 0.6 arcsecond 
pointing.  If these requirements cannot be met, we are faced with the prospect of developing 
active metrology.

While we do not divide up the 0.6 arcsec pointing error specification into various systematic and 
random terms here (but see Table 3.4.3-2 of the ALMA Antenna RFP), we note that the effects 
of any pointing error budget with various systematic and random terms could be translated to an 
estimated image quality, given the image (see, e.g., Pety et al. 2001b). We also note here that 
random errors have less effect than systematic errors, while pointing out that it may be possible 
to correct for some of the effects of the systematic errors.  With only minor exceptions, pointing 
calibration must be performed prior to astronomical observations or the data are irretrievably 
corrupted (but see the note on pointing selfcalibration below). This also means that we cannot 
generally interpolate pointing solutions backwards in time. This makes pointing calibration 
critical.

2.1.2 Surface Setting and Primary Beam

In order to maximize the forward gain of the antenna, the surface errors must be minimized.  In 
addition, surface errors can be the limiting factor in mosaicing of large complex sources.  
Cornwell et al. (1993) derived a requirement of ?/40 for the surface accuracy, driven by 
obtaining a fidelity of 100 on large (mosaicing necessary) complex images.  For the MMA at 350 
GHz, this turned into a surface rms of 22 ?m.  More detailed analysis of the MMA case showed 
that 25 ?m was probably sufficient for frequencies up to 650 GHz (Holdaway 1992).  This 
surface error was considered reasonable given the 8m diameter of the MMA antennas (see, e.g., 
Lamb 1993).  Even given the move to 12m diameter antennas for ALMA, it was believed that 
this surface error was achievable (this after discussions with the prototype antenna vendors).  
The antenna RFPs therefore contained specifications of 25 ?m with goals of 20 ?m.  This, like 
the pointing requirement, is a matter of getting as good as we can get for the price we want to 
pay (if we thought it were possible and we could afford it, we might ask for true ?/40 at 950 
GHz, which would be ~8 ?m).  The difficulty is then in making measurements and adjustments 
to the surface panels to meet that requirement.

For accurate mosaicing, it is also necessary that the antenna voltage response be precisely known 
on each of the elements of the array.  Cornwell et al. (1993) determine that the primary beam 
must be known to 6%, in order to reach image fidelity of 100 (in their definition), and in order 
that errors from this be smaller than those from pointing.  We adopt 6% as the requirement on 
this calibration measurement, and further stipulate that it applies out to the point in the beam 
where the response is 10% of the peak.  Note that this is the power pattern – the requirement on 
knowledge of the voltage pattern is 3%.

2.1.3 Subreflector and Feed Positioning

Having the subreflector or feeds positioned incorrectly will result in a loss of efficiency and in 
primary beam abnormalities that may be very hard to correct in mosaicing.  Accounting only for 
the loss of efficiency, and requiring that the loss is < 1%, Butler (2003) calculated the allowable 
subreflector and feed offsets shown in Table 1 for the ALMA antennas.  When the antenna 
design is finalized, this calculation should be repeated.

Table 1. Allowable Feed and Subreflector Position Errors
Type of Error
Positioning Requirement
Requirement @ 650 GHz
Feed Axial Offset
0.9 ?
 280 ?m
Feed Lateral Offset
10.0 ?
3.2 mm
Subreflector Axial Offset (Focus)
0.09 ?
28 ?m
Subreflector Lateral Offset
0.45 ?
140 ?m
Subreflector Rotation (Tilt)
?/.63m (rad)
1.7 arcminutes

2.1.4 Motion

In order to calibrate the fluctuations in the atmospheric delay, it is necessary to be able to switch 
to nearby calibrator sources on short timescales.  Combining our knowledge of the distribution of 
such calibrator sources on the sky with what is realistically achievable in terms of slew speed and 
settle times for a 12-m antenna produced the current specification that the antenna be able to 
slew 1.5 degrees and settle down to within 3 arcseconds of the new desired pointing location in 
1.5 seconds.

In addition, the antenna must be able to slew quickly across relatively large sources to allow for 
cancellation of atmospheric fluctuations in single dish data (D'Addario 2003).  For a small 
number of antennas, wobbling subreflectors are required for this same purpose (Pety et al. 
2001a).

2.2 Amplitude

For the amplitude calibration, a well defined scientific goal can be elucidated and set as the 
requirement.  In numerous meetings and discussions, the scientific community has made clear its 
desire to reach 1% flux density accuracy (most recently reiterated in the report of the spring 2003 
ASAC meeting), which means that we must be able to determine the overall flux density scale 
(and apply it to the visibilities and total power measurements) to 1%.  In addition, the capability 
of achieving a dynamic range in images of 10000 or higher means that we must track the 
amplitude fluctuations to better than 1% (Yun et al. 1998).  A later study by Moreno & 
Guilloteau (2002) showed that it is impractical to achieve 1% at the submm wavelengths, and so 
a requirement of 3% has been adopted for ?>300 GHz.

2.3 Delay

Because ALMA is a connected element interferometer, with correlation happening real-time, the 
delay must be calculated correctly at the time of observing.  If delay is not calculated properly, 
we will never be able to properly calibrate ALMA data.  The delay involves several components,  
including source position, earth orbit and orientation, atmosphere (both bulk and fluctuating, and 
both neutral and ionized), station location, antenna structure delay, and electronic delay.  All of 
these components must be properly calculated or measured.  This is very important, as the delay 
is applied to the voltages before correlation, and hence the effect of an incorrect delay cannot be 
undone in many cases.

There is an additional fluctuating portion of delay due to the atmosphere, antenna, and 
electronics.  Delay fluctuations limit resolution, limit the dynamic range of images, introduce 
artifacts, and reduce sensitivity by decorrelation. Without effective calibration of these 
fluctuations, the maximum usable ALMA baseline would generally be about 300 m. Amplitude 
errors would limit image dynamic range and skew the flux density scale.  Therefore, high fidelity 
imaging of any kind (mosaiced or not) requires accurate delay calibration.

The phase calibration working group report (Woody et al. 1995) considered three cases at 300 
GHz: high quality imaging with 8o total phase errors, median conditions with 19o deg phase 
errors, and poor imaging with 48o phase errors (these phase errors are antenna based – baseline 
based errors are ?2 larger). The phase errors have a budget which includes the atmosphere, the 
antenna, and the electronics.  Converted to path lengths, these correspond to 23, 52, and 134 ?m, 
or, again, converted to delays (in time), these correspond to 77, 172, and 446 fs (femtoseconds!).  
This has been updated based on extension of ALMA to higher frequencies, and a recognition that 
when the atmosphere is less dry we will not be observing at the higher frequencies.  D'Addario 
et al. (2002), further recognizing that the delay can have both a fluctuating and systematic offset 
portion, specified that the requirements on the delay setting and measurement were:

error (fs)
?a
?s
?e
total
systematic
8.4
4.8
6.9
11.9
random
38.5
22.2
31.4
54.5

where ?a is the atmospheric delay, ?s is the antenna structure specific delay, and ?e is the 
electronic delay.

We will now refine slightly these delay measurement and setting requirements.  We define the 
total system delay at the ith antenna as:
	 						(n)
where ?g is the geometric delay, ?a,?s, and ?e are as above, and ?f is a fudge factor subsuming the 
errors in the other quantities.  We specify the following requirements on the different parts of the 
delay:

error (fs)
?g
?a
?s
?e
total
systematic
5
10
7
7
15
random
0
35*(1.25 + PWV)
50
30
 

Where PWV is the precipitable water vapor in mm.  We now discuss in more detail some of 
these requirements and requirements on other related measurement quantities.

2.3.1 Source Location

In order to accurately calculate the geometric delay, it is necessary to know where the desired 
source (or interferometric phase pointing center) is.  We require that the phase center can be 
pointed to a fraction of a synthesized beam width for the most spread out configuration.  Given 
14 km baselines (the longest baselines are 18 km, but the equivalent baseline length to give the 
resolution of the most spread out configuration is ~15 km – Otárola & Holdaway 2002), and 950 
GHz frequency, the synthesized beam FWHM is ~4 milliarcseconds.  It seems reasonable to 
require that the phase center be pointed with an accuracy of 0.5 milliarcsecond.  In order to do 
this, we propose that only J2000.0 coordinates be used, with proper precession and nutation and 
timing (Sovers et al. 1998).  An additional correction that will be necessary is the relativistic 
bending of the radio waves in the gravitational potential of the sun (Sovers et al. 1998; Wade 
1976).  The use of CALC (Ryan & Ma 1979) or a similar package to calculate the geometric 
delay will automatically include this correction. 

2.3.2 Atmospheric Delay

The Earth's atmosphere provides an additional component of delay between the two antennas.  
There are two possible parts of the atmospheric delay term that will have to be accounted for, the 
neutral atmosphere (mostly the troposphere), and the ionosphere.

2.3.2.1 Neutral Atmosphere

For the most part, the differential delay between the two antennas due to the non-fluctuating part 
of the neutral atmosphere is the same, and hence can be ignored.  However, on the longest 
baselines, the difference in elevation between two antennas can be of the order of 10 arcminutes, 
resulting in a differential delay that is significant (at 10 degrees elevation, the difference can be 
as large as 400 picoseconds).  The bulk atmospheric delay must therefore be calculated 
separately for each antenna.

In addition, there is a fluctuating part of the delay, caused by fluctuations of water vapor in the 
troposphere.  The calibration of this fluctuating tropospheric delay is often called "phase 
calibration", but we do not use that term here.  The current specification on the delay correction 
for the fluctuating atmospheric contribution for the WVR systems is: 10 * (1 + PWV) ?m, where 
PWV is the precipitable water vapor column in mm (Hills & Richer 2000; Hills et al. 2001).  
Note that this is in disagreement with the value adopted in Chapter 7 of the Project Book, which 
lists the allowable atmospheric phase path error as 38.5 fs, regardless of conditions, i.e., leaving 
off the PWV component (see Table 7.4).

There is a further fluctuation due to the fluctuations in density and temperature of the dry (non-
water) part of the neutral atmosphere.  It is likely that this term is small, but we do not know this 
definitively yet, and so allow for some small additional dry component fluctuations in the overall 
error budget for atmospheric fluctuations.  

2.3.2.2 Ionosphere

An effect that is often ignored in millimeter interferometry is the delay induced by the 
ionosphere.  Given the extreme accuracy required by ALMA, we may not be able to afford to 
completely ignore it.  The zenith ionospheric delay at 8.5 GHz is typically of order 1 ns.  Since 
the ionospheric delay goes like ?-2, this would result in about 100 ps at 30 GHz.  It is not the total 
delay we are interested in, but the differential delay on 18 km baselines.  While this should be 
smaller than the total delay, it still can be significant (experience at the VLA tells us this), and 
note that the delay is larger at lower elevations (see the discussion in Sovers et al. 1998).  A 
recent study by Hales et al. (2003) concluded that fast phase fluctuations seen in the site phase 
interferometer are due to ionospheric fluctuations.  The magnitude of these fluctuations can be as 
large as 30o rms at 11 GHz.  Scaling this to 30 GHz gives fluctuations of order 5o.  Not overly 
large, but beginning to be of some concern.  The scaling with baseline length and elevation is not 
well constrained.  So, we do not have enough information at present to tell whether we will 
definitely have to monitor the ionosphere, but indications are that we might.  For this reason, we 
allocate a small additional amount to the error budget for atmospheric fluctuations to account for 
this.

2.3.3 Antenna Location

The relative positions of the antennas must be determined accurately so the geometrical delay 
can be correctly calculated and supplied to the correlator.  Residual delays due to incorrect 
antenna locations will result in phase errors which change across the observing band and 
differential phase errors between two different sources on the sky (for example, a calibrator and 
target source).  For ALMA, with an 8 GHz bandwidth, a reasonable limit of 1/3 radian phase 
difference across the band requires a baseline accuracy of about 1 mm.  The differential phase 
error between two sources on the sky can be written as (Sramek 1981; Wright 2002):
	 										(1)
where ?? is the phase error in turns, ?x is the baseline error in wavelengths, and ?? is the 
separation of the sources on the sky in radians.  We take 10o as the maximum allowable phase 
error at the highest observing frequency (950 GHz).  We are more conservative with baseline 
errors than with atmospheric errors because they will be partially systematic.  For a source 
separation of 5o, if the maximum error is constrained to be 10o at 950 GHz (equivalent to 9 ?m 
of path), then the error in the baseline must be less than about 90 ?m.  This means that the error 
on an individual antenna location determination must be less than about 65 ?m, if the errors add 
in quadrature (not true for any given baseline, but averaged over the array, this is a reasonable 
assumption).  Note that the antenna location determination is needed as a precalibration, as it is 
used to determine the geometric delays applied to the antenna signals before correlation, but it 
can also be used in postcalibration for the baseline determination (u,v,w can be recalculated 
based on better antenna location [baseline] calibration).

Efforts will be required to properly understand how the effective antenna location changes with 
time. On the longest baselines, earth tides and polar motion will have significant effects, and we 
will need to borrow the techniques of VLBI for antenna location determination (Clark 1973; 
Sovers et al. 1998).  Once again, the use of CALC (Ryan & Ma 1979) or a similar package for 
the calculation of the geometric delay model will automatically include these terms.

Note that the ability to determine the antenna location to a particular accuracy also provides a 
requirement for the accuracy of timing across the array.  The required timing accuracy from this 
one consideration is:
 									(2)
where ?e is the rotation rate of the Earth, and Bmax is the maximum baseline.  For 65 ?m 
accuracy over 20 km baselines this implies a timekeeping accuracy of ~ 45 ?s.  This is easily met 
by the current design (D'Addario 2000).

2.3.4 Antenna Structure Delay

There is an additional component of delay introduced by the structure of the antenna itself, and 
its change with azimuth and elevation.  These can be thought of as modifications to the antenna 
location, in many ways.  There is a full discussion of this in the U.S. Antenna RFP, section 3.4.5.

Repeatable (and thus perhaps predictable) parts of this delay include the change of main reflector 
shape with elevation or azimuth (e.g., nodal point changes with tilt), subreflector position with 
elevation (see, e.g., what is done with axial displacement ["focus"] as a function of elevation at 
the DSN antennas in Jacobs & Rius 1990) axis non-intersection (Butler 2003; Wright 2002; 
Sovers et al. 1998; Wade 1974), the illumination offset (Holdaway 2001a), bearing runout, and 
bearing alignment.  The specification on this component in the Antenna RFP is 20 ?m, 
equivalent to 67 fs.  However, most of this repeatable component can be accurately predicted, if 
measured properly, so we do not allocate 67 fs to this term.  We estimate that only about one 
tenth of the repeatable residual delay will result in true systematic delay offsets, so allocate 7 fs 
to this term.  

Nonrepeatable (and thus probably not predictable) parts of this delay include the change of main 
reflector shape, subreflector position, or feed position with temperature changes (causing thermal 
deformation of the structure) or wind or other acceleration forces, and bearing nonrepeatability.  
In fact, some of these nonrepeatable errors may be predictable, but we cannot count on it.  The 
total of all of these effects will likely only be answered by experimentation over time with the 
final ALMA antennas.  In order to be somewhat conservative, we adopt the Antenna RFP value 
for the nonrepeatable residual delay as the allowable random antenna structure delay error.  This 
value is 15 ?m, or 50 fs.  Note that both this and the repeatable delay error are specified for 
motions within a 2o solid angle on the sky, and the nonrepeatable portion is specified for 
timescales less than 3 minutes.

2.3.5 Electronics Delay

Finally, there is a component of delay introduced because of the electronics between the feeds on 
the antennas and the samplers.  This will need to be measured for each antenna, receiver 
cartridge, and polarization.  The determination of this delay should be accurate to a fraction of 
the other error and offset terms, if possible.  We adopt the numbers recommended in Chapter 7 
of the Project Book.

2.4 Bandpass

ALMA will always operate as a spectral line instrument, in the sense that multiple frequency 
channels will always be present in the product of the correlator.  As such, knowledge and 
calibration of the shape of the frequency response of the instrument and atmosphere is very 
important, and can limit not only the spectral dynamic range (in the case of observations of 
spectral lines), but also the image fidelity (in the case of continuum imaging, since bandpass 
variations are effectively like amplitude variations if not calibrated properly).

The required spectral dynamic range is driven by a few types of experiments that need to observe 
weak spectral transitions against a strong continuum.  These include observations of rare isotopes 
(extremely optically thin) against quasars (potentially many Janskys of flux density), and HDO 
emission (weak emission feature on limbs, even weaker in absorption against the disk) in the 
atmosphere of Mars (extremely strong continuum).  In these cases, a dynamic range of 10000 to 
1 is required.  For most observations, however, a dynamic range of 1000 to 1 is sufficient.

2.5	Polarization

Because measurements of the polarization of sources yield intrinsic information on the physical 
parameters of those sources, it is a requirement that ALMA be able to produce visibilities 
simultaneously in all four Stokes parameters (or some equivalent, from which they can be 
derived) when asked for by the observer.  Measurement of the polarization morphology of 
isolated protostars, circumstellar disks, or or protostellar clusters requires that the polarization 
percentage be measured to 0.1% in amplitude, and that the position angle of the linear 
polarization be measured to an accuracy of 6o.  With these accuracies, magnetic field geometries 
in protostellar outflows can also be measured, providing a test of the mechanism of generating 
these outflows.

2.6	Sideband Gain Ratio

Since some receivers will be double sideband (DSB), they will have emission from an unwanted 
part of the spectrum in the single-dish signal.  In order to reach the overall 1/3% amplitude 
calibration requirement in the single-dish data, it will be required to determine the sideband gain 
ratio to a small fraction of this – 0.1%.

2.7	Summary

   Pointing				2.0 arcseconds blind; 0.6 arcseconds offset
   Primary Beam          		6% in power out to 10% point
   Feed Setting	 			280 ?m vertical; 3.2 mm lateral
   Subreflector Setting	 		28 ?m vertical; 140 ?m lateral; 1.7 arcminutes rotational
   Antenna Motion			1.5 degrees in 1.5 seconds with settle to 3 arcseconds;
					XXX more here
   Amplitude                		1% for ?<300 GHz; 3% for ?>300 GHz
   Antenna Location			65 ?m
   Geometric Delay                              5 fs systematic
   Atmospheric Delay			10 fs systematic; 40 (1.25 + PWV) fs fluctuating
   Antenna Delay                                 7 fs systematic; 50 fs fluctuating
   Electronic Delay			7 fs systematic; 30 fs fluctuating
   Bandpass                  		1000:1 in most cases; 10000:1 in a few select cases
   Polarization              		0.1% in amplitude; 6o in position angle
   Sideband Gain Ratio                        0.1%

3  Precalibrations

Precalibrations are those calibrations to determine quantities necessary to make adjustments to 
the instrument which affect the signal prior to correlation.  In almost all cases, these adjustments 
are irreversible.  The quantities may vary slowly, in which case they will generally be 
determined in infrequent special calibration observations, or they may vary more quickly, in 
which case they will need to be determined by measurements during the observation.

3.1 Antenna Characteristics

3.1.1 Pointing

The antenna pointing will be affected by several slowly varying terms such as systematic 
imperfections of the antenna and the pad, gravitational forces, and thermal loading from the Sun.  
Depending upon the strategy of the astronomical observations, much of the effect of these slowly 
varying quantities can be removed by frequent offset pointing observations.  Some active 
corrections might also be incorporated into the antenna design. Thermistors have been employed 
on existing antennas to monitor solar thermal response (Lamb & Forster 1993; Lamb & Woody 
1998a; Bayley et al. 1994), and their use, along with tiltmeters and other metrology devices 
(lasers, e.g.) in the ALMA antennas has been proposed and investigated (Anderson 1999; 
Plathner et al. 1999; Woody & Lamb 1999; Kingsley et al. 1999; Plathner 1999; Lugten et al. 
1999; Lugten 1998; Payne 1997). The antenna contract stipulates that the contractor supply any 
extra metrology devices deemed necessary to meet the pointing specification, and allow for the 
addition of other metrology devices by the ALMA project, including laser/quadrant detectors, 
tiltmeters, temperature probes, and laser/retroreflector systems. 

In addition to these slowly varying systematic pointing errors, there will also be highly random 
pointing errors caused by wind loading and dynamic refraction (often called "anomalous 
refraction", but we dislike that term, and adopt the one proposed by Pety et al. 2001b [note that 
Barry Clark once proposed that this be called "nomalous refraction"]).  Finally, at some level 
there will be a limit to the mechanical repeatability of the antenna pointing. At this level, we are 
left with completely random pointing errors which cannot be calibrated. If these purely random 
errors are too large, they will spoil the imaging characteristics of ALMA and will not be 
correctable. If they are small to moderate in magnitude (i.e., < 0.5 arcsec), we can tolerate them 
quite well as these random errors are the least damaging of any pointing errors.

Our goal in determining the pointing parameters of the antenna is to remove all of the systematic 
pointing errors (as best we can), leaving only the small purely random errors.  We do this in three 
ways: through use of a good global pointing model; by using local offset pointing determination; 
and by determining refraction (both static and possibly dynamic) offsets.

3.1.1.1 Global Pointing Model

The first step in removing the systematic pointing errors is to determine the systematic 
imperfections of the antenna and pad and the effects of gravity.  Most radio telescopes 
periodically undergo a pointing routine which samples the sky with pointing measurements on 
about a hundred astronomical sources taken at night to minimize thermal and wind pointing
errors.  ALMA will take about 60 minutes to perform 100 pointing calibration observations 
across the sky (each one consisting of one or more pointing patterns around a strong quasar – see 
below).  The results of these pointing measurements (beam offsets in azimuth and elevation) are 
then used to fit about 10 parameters in a pointing equation which accounts for various physical 
terms, such as misalignment of optical axes or four fold sag due to the antenna base being 
supported at four locations.  Some experimenting will go into determining the optimal form for 
the ALMA pointing equation (Mangum 2001).  In particular, the precision of the refraction 
correction is essential (see below).  We note here that the pointing calibration observations (the 
pointing patterns) can be performed either in total power or interferometric mode, but the 
interferometric mode is significantly more sensitive, so we anticipate using it in almost all 
pointing calibration observations.

One point of note is that the determination of the axis non-intersection term is difficult unless the 
antennas are commanded to go "over the top" during the pointing parameter determination, at 
least on some sources (Butler 2003).  This is a requirement for the ALMA antennas, and it will 
need to be utilized for the determination of this parameter.

ALMA will probably rely heavily upon a lower frequency (probably 90 GHz or below) system 
for determining the pointing model.  The wider beam at this low frequency and the high 
sensitivity and denser grid of bright astronomical sources will facilitate pointing measurements, 
even after a reconfiguration.  However, precise pointing offsets among the different frequencies 
will also need to be determined (often referred to as the "collimation offsets").

One question is how many antennas will be required to do an antenna pointing model 
determination?  Given the current operational model of frequent antenna moves (roughly one per 
day), this question is quite important.  After an antenna is moved, it will be required to determine 
its pointing model parameters, and this will require some number of other antennas to be used for 
this.  In practice, some number around 3-6 are used in current arrays, but this number remains to 
be determined.

3.1.1.2 Local Offset Pointing

The requirement for the blind pointing after application of the pointing model is 2 arcseconds 
rms.  To achieve the precision relative pointing specification of 0.6 arcseconds, frequent local 
offset pointing calibration will be required, to remove local deviations from the pointing model, 
or systematic slowly varying effects due to wind and/or thermal gradients.  This type of local 
offset pointing has been used very successfully at many observatories for many years now (for 
the example of the VLA, see Rupen 1997).

Holdaway (1996) and Lucas (1997) have both demonstrated the feasibility of performing this 
kind of local offset pointing with ALMA – adequate SNR can be obtained with sufficient speed.  
The minimum calibrator flux density, and hence the typical minimum distance to a pointing 
calibrator for pointing calibration, is a function of both the collecting area and the number of 
elements in the array.

A key question concerning the efficiency of these offset pointing calibrations is: what are the 
differential pointing errors as a function of distance between calibration and target sources?  This 
depends upon the direction of wind, the sun angle, the flux density of the calibration source, 
etc…, and probably needs to be answered experimentally.  Also, we must understand the 
stability of the pointing offsets among the different frequency bands, as we would prefer to do 
the pointing calibration at 90 GHz, but if the collimation offsets are changing this will not be 
possible.  This understanding will only come after the antennas are complete and we can test for 
these particular issues.

It might be possible to minimize the differential pointing errors by performing a local average on 
a few nearby strong (0.5 Jy) quasars (Moreno & Guilloteau 2002).  Simulations should be 
performed to check the impact of dynamic refraction in such a mode, and to determine the 
optimal strategy for this kind of "local pointing model" (how often to update, how many pointing 
calibrators, position relative to target source, necessary flux density, etc…).

Both the pointing model solution and the offset interferometric pointing will require an  
extensive, up-to-date catalog of pointing calibrator sources, and the observing schedule program 
should allow for automation of the choice of a pointing calibrator and the pointing calibration 
strategy.

Measurements of the wind at the site (see, e.g., Sakamoto et al. 2000) indicate that a great deal of 
the power in the wind is often in a constant speed and direction, and this wind at constant speed 
and direction will often be corrected by the local offset pointing calibrations.  In that case, it is 
only the gusts about this mean speed and direction which will result in differential pointing 
errors between the calibration and the target source.

We do not know currently how often we will need to perform offset pointing.  Experience on the 
VLA is that under benign conditions every hour or so is sufficient.  Around sunrise or sunset, it 
must be done every 15 minutes or so.  Only experience with the real ALMA will be the guide as 
to how often the offset pointing must be done on the array.

3.1.1.3 Pointing Pattern

When observing the calibration sources during either a global or local offset pointing calibration 
observation, a number of positions in the beam must be observed – these comprise the "pointing 
pattern."  There have been many implemented pointing patterns at radio telescopes.  The most 
common is the "5-point", where a point at the center, and at the 4 half-power points (N and S in 
elevation, E and W in azimuth), is taken.  In addition, one or more "off" positions might be 
measured (but only if observing in total power [single dish] mode).  A variant of this is the one 
used on VLBA antennas, called the "Craig Cross" (named after Craig Walker, its inventor).  This 
is a 10-point pattern which attempts to make a better baseline subtraction, especially in the 
elevation direction – useful for total power observing.  The IRAM 30-m has used scans in 
azimuth and elevation instead of discrete pointings (Greve et al. 1996).  The OVRO pointing 
pattern is a series of rotating triangles at the half-power point, repeated until the pointing error 
reaches a specified level (Scott 1993).  Scanning circles have been employed at Arecibo (Olmi & 
& Davis 1998).  We do not know which of these patterns will be employed for ALMA at this 
time.

3.1.1.4 Refraction

The bending of the radio waves in the atmosphere of the Earth (refraction) must be accounted for 
when calculating the instantaneous pointing direction to a given source.  This is an explicit 
additive term to the elevation offset in the pointing determination.  The bulk of the refraction is 
due to the dry component of the atmosphere, and is relatively well determined given surface 
measurements of temperature and pressure.  We follow the recommendation of Pety et al. 
(2001b) and call this the "static refraction."  There is an additional rapidly fluctuating refraction 
component which is due to the turbulent water vapor above each antenna.  Again, we follow the 
recommendation of Pety et al. (2001b) and call this the "dynamic refraction."

3.1.1.4.1 Static Refraction

This is the refraction of the radio wave due to the dry and non-fluctuating wet atmosphere.  It 
causes a deflection in the direction of elevation, i.e., at low elevations, a source appears to be at a 
lower elevation than it really is.  There have been many studies of this effect for radio telescopes 
(see the discussion in Mangum 2001 and references therein).  Note that it requires an accurate 
measurement of the surface temperature, pressure, and humidity.  Mangum (2001) give the 
following measurement requirements: ambient pressure: 0.5 mb; ambient temperature: 0.1 C; 
relative humidity: 0.5%; wind speed: 0.5 m/s; wind direction: 5o.  Note also that the difference in 
elevation at the different antenna locations can provide a relatively large offset in the refraction 
correction, so that correction will have to be calculated for each of the antennas separately.  For 
example, two antennas separated by 16 km will have an elevation pointing difference of roughly 
10 arcminutes, and at 10 degrees elevation, this results in a refraction correction difference of 
about 4 arcseconds (out of 190 arcseconds total refraction correction).  Since this is larger than 
even the blind pointing requirement, it will have to be accounted for.

3.1.1.4.2 Dynamic Refraction

This fluctuation in the pointing is a natural consequence of the fluctuation water vapor above the 
antenna.  It is akin to the fluctuations in the dry atmosphere which cause so-called "Angle of 
Arrival" fluctuations on optical/IR telescopes, because the bulk of the effect of the fluctuating 
refraction is a simple change in the effective direction between the propagating plane wave and 
the antenna optical axis.

The effect of dynamic refraction on the pointing will usually not be a severe problem, but will 
sometimes limit the pointing (Holdaway & Woody 1998a; Lamb & Woody1998b; Holdaway 
1997b; Butler 1997). Since the refractive pointing is random on time scales of the antenna 
crossing time of the atmosphere (~ 1 s), we merely need to have many statistically different short 
measurements for each of the points of a pointing pattern to ensure that we are not applying an 
erroneous pointing position when we collect data on the target source.  Note that this averaging 
process contradicts the idea of performing "fast" pointing calibration every 30 min or so. Also, it 
is in apparent contradiction with Lucas (1997) which shows a 1:1 correlation between the rms 
pointing error and the anomalous refraction, although the pointing was measured on 1 min 
timescales.  

There is the possibility of correcting for the dynamic refraction with additional devices, or at 
least measuring what the offset due to anomalous refraction is (Lamb & Woody 1998b; Hills & 
Richer 2000).  These are still under study.

3.1.1.5 Pointing with Optical/IR Telescope

The ALMA prototype antennas, and perhaps the production antennas, will be outfitted to 
perform infrared or optical offset pointing, with a small telescope looking through a hole in the 
antenna primary surface (Mangum 2000).  Offset interferometric pointing should work well 
enough.  However, optical/IR pointing will increase the overall efficiency of the instrument, may 
improve the pointing of the antennas and may help characterize mount components in the 
pointing equation. The infrared pointing will be largely immune to dynamic refraction effects 
(since it is such a small aperture, normal optical/IR angle-of-arrival [AOA] fluctuations will be 
minimal).  We have not determined whether an optical/IR pointing telescope will be needed on 
all ALMA antennas yet, this is still under study and discussion.  It is clear that such a telescope 
will be of great benefit during the verification and commissioning of the antennas, however.  At 
the very least each antenna should allow for the mounting of such a telescope, even if there is not 
one present on all antennas at all times.

3.1.2 Antenna Surface and Primary Beam

In order to maximize the forward gain of the antennas, and to make the beams as well behaved as 
possible, it is desirable to measure the deviations of the surface from the optimal surface, and to 
correct for them (by adjusting the surface panels).  We will use the technique of interferometric 
holography to do this (Scott & Ryle 1977).  

The requirement on the holography raster size can be derived from the fact that there should be 
roughly four resolution elements across the smallest panel dimension (Bennet & Godwin 1977).  
For the current prototype designs, this is roughly 0.5 meters.  This implies that a 128 x 128 raster 
will be required for holography (if a power of two is required – otherwise it would require a
96 x 96 raster).  If holography is done at 90 GHz, and if the oversampling factor is 10%, then this 
is a total angular size of roughly 1.8o.  Given that we would like to do holographic measurements 
on a timescale that is short relative to any thermal deformations, this implies sample times at 
each raster point of about 0.4 seconds (2 hours per raster, with a boresight every 15 seconds).

The surface measurement error should be at least a factor of two less than the desired surface 
accuracy.  Given a 20 ?m surface accuracy goal (see the antenna RFPs), this implies a surface 
measurement accuracy of at least 10 ?m.

The rms surface measurement error arising from measurement at wavelength ? of a source with 
SNR S over an N x N raster can be written (Butler 1999):

        ? N
? ~ -------	.
      5 ? S

For a frequency of 90 GHz, a required rms of 10 ?m, and a raster size of N = 128, this gives a 
required SNR of S > 2700.  Given a sample dwell time of 0.4 seconds, this implies a source flux 
density of F > 45 Jy (assuming a single baseline and 17 mJy noise in 0.4 seconds on that single 
baseline).  The required flux density can be reduced if many reference antennas are used, but at 
an operational penalty (these antennas are not available for normal observing).  There are not 
very many sources with flux density of this magnitude – planets are an exception, but are often 
resolved (and hence require a correction based on that fact, which increases the uncertainty since 
the sky brightness distribution is never perfectly known).  The required SNR is decreased with 
shorter wavelength, but the noise goes up in almost the same proportion.  We may, therefore, be 
required to have a transmitting beacon on a nearby mountain top (Cerro Chascon or Cerro 
Chajnantor, perhaps).  If this is the case, care must be taken to properly apply the second order 
(and possibly higher) near-field corrections.

The rms surface measurement error on antennas with dish diameter D arising from random 
pointing errors ?rms is (Butler 1999):

        ?rms D
? ~ ----------	.
           12

Given 0.6? rms pointing errors, this yields a measurement rms of 3 ?m.  Not unreasonable, but it 
must be kept in mind when calculating the entire measurement error budget.

Amplitude fluctuations (which can come from electronics or atmospheric scintillation) of 
magnitude fa cause an rms measurement error of (Butler 1999):

        ? fa
? ~ -------	.
       15 ?

For an amplitude fluctuation of 10-3, this results in a measurement error of 0.1 ?m, which should 
be negligible in the error budget.  Note that this assumes that we track the amplitude 
scintillations to that level – if left uncorrected, the level might be larger than this by an order of 
magnitude (still a small part of the entire budget).

Phase fluctuations of magnitude ?rms (or, equivalently, path length errors of ?rms) result in a 
measurement error of (Butler 1999): 

        ? ?rms                   ?rms
? ~ ----------	~  --------   .
         12 ?                6


We expect to correct the phase fluctuations to ?rms <   ?m.  For 
PWV of 1 mm, this is ?rms ~ 29 ?m, or a measurement error of ~ 5 ?m.  We will have to use 
good weather conditions when performing holography measurements.

There are other issues to be concerned with – bulk reflector shape change (with elevation, 
mostly), when using astronomical sources; ground scatter if using a beacon; unmodelled EM 
clutter from the structure itself, etc…, but most of them are of less concern than the above points.

Holography measurements also provide us with a measurement of the primary beam shape of the 
antennas.  However, a holography raster which is optimized to measure the small-scale 
deviations in the surface (with large spatial extent on the sky) is not optimal for measuring the 
larger scale deformations which affect more seriously the inner part of the primary beam – that 
part which is most important for mosaicing.  In order to measure the inner part of the beam 
accurately, a different raster should be used, the extent of which is only out to a few FHWM or 
so, and heavily oversampled.

Note that the primary beam patterns must be known as a function of both frequency and 
polarization.  The measurements should be done to cover this parameter space.  In addition, it 
may be desirable to have information on how the primary beam pattern changes as a function of 
elevation or other conditions.  The full parameter space may be too wide to sample fully, but we 
should gain some experience during early operations on how finely to sample it.

Holography measurements also provide us with a measurement of subreflector and feed 
positioning errors.

3.1.3 Subreflector and Feed Positioning

Subreflector and feed positioning errors reduce the on-axis gain of the antennas, and distort the 
primary beam shape in bad ways.  Errors in feed positioning are most easily measured by doing 
holography.  However, a much more efficient way of getting the subreflector positioning errors 
is to do scans while looking at a strong source.  Since we will have a subreflector positioning 
mechanism which can be commanded by software, it is easy to do not only the traditional so-
called "focus curves" (which measure the optimal axial position of the subreflector), but in 
addition to do "tilt curves" (which measure the optimal rotational angle of the subreflector) and 
"offset curves" (which measure the optimal lateral position of the subreflector).

3.2 Amplitude

The amplitude calibration contains no required component of precalibration.  If quantities are 
available at the time of observation, then the visibilities and single dish powers can be 
immediately put onto a realistic amplitude scale, but this is not necessary, and it can all be done 
after the fact (in postcalibration).

3.3 Delay

3.3.1 Source Location and Geometric Delay

As mentioned above in the requirements section, we must be able to accurately specify source 
locations (or, phase pointing locations) accurately.  In order to do this, we should use only 
J2000.0 coordinates, with proper precession and nutation.  In addition, we should account for all 
modifiers of geometric delay that we can reasonably include in the model.  We recommend that a 
package similar to CALC (Ryan & Ma 1979) be used to calculate geometric delay, which should 
include polar motion, earth tides, and other effects.  We will have to obtain Earth Orientation 
Parameters from some source – we recommend the USNO for this purpose (similar to what is 
done with VLBA, VLA, and ATCA currently).  These parameters are changed on roughly month 
long timescales, and we should implement a scheme to automatically incorporate them into the 
geometric delay model if possible (as ATCA has done).

3.3.2 Antenna Location and Structure

In order to accurately calculate the geometric delay before correlating the voltages from two 
antennas, their locations must be known.  The process of locating these antenna positions is often 
called "baseline calibration" because it allows for accurate specification of the baseline as well, 
but we prefer to simply call it "antenna location calibration" since that more accurately describes 
what we are measuring/calibrating.

The antenna locations may be measured by determining the delay on each baseline for on order 
of a hundred observations of point sources sampling the entire sky.  Individual delays can be fit 
across the spectrum, as in VLBI.  The complete set of delays is used to solve for the three 
dimensional locations of all antennas relative to a reference antenna (or any reference  location). 
Fomalont & Perley (1999) and Thompson et al. (2001) give good general overviews of this 
measurement.  Wright (2002) has given a good description of this process on the BIMA array.  
Sovers et al. (1998) give a description of the technique used in VLBI.

The observing strategy is similar to that for the pointing model determination, and should take 
about an hour to complete.  Signal to noise is not an issue for 65 ?m accuracy, and the 1 hour 
time scale is set more by the minimum time to sample many sources around the sky.  
Atmospheric phase fluctuations may affect the baseline delays, so ideally the observing 
conditions should be excellent.  In poor conditions, the delays can probably still be determined 
based on the statistics of many differential measurements, as the atmosphere should tend towards 
a zero mean in differential measurements.  A timescale longer than 1 hour is required for 
baseline measurements to reach the outer scale of the atmosphere and sample several 
atmospheric screen patterns.  Local rotation of the baseline plane is possible otherwise.  
Experience in the early phases of operations will be required.

In order to reduce errors during antenna location determination, only sources with well 
determined locations should be used (Feissel-Vernier 2003; Johnston & de Vegt 1999).  In 
addition, since source structure will introduce further uncertainty in the antenna location 
determination (Sovers et al. 1998), sources with very simple structure should be chosen, 
otherwise we must use a more complicated self-calibration procedure, obtaining both the 
structure and the location.

Of some concern is the time scale over which we can expect the antenna positions to remain 
fixed to within 65 ?m. Permafrost has been reported on the ALMA site (see, e.g., Snyder et al. 
2000), which enables an entire class of soil movements.  We can probably expect some amount 
of soil creep, especially after earthquakes (see Otárola et al. 2002). We will gain experience 
concerning the frequency of baseline calibration, or the possibilities to correct for some of the  
changes using on site measurements, once ALMA begins to be operational at Chajnantor.

One question is how many antennas will be required to do an antenna location determination?   
Given the current operational model of frequent antenna moves (roughly one per day), this 
question is quite important.  After an antenna is moved, it will be required to determine its 
location (or, equivalently, its delay), and this will require some number of other antennas to be 
used for this.  In theory, this could be done with a single antenna with accurately known position, 
but in practice, several are always used.  But whether this means 3 or 6 or some other number 
remains to be determined.  We want as many as possible to increase the SNR during the 
observations, but as few as possible because it takes functional antennas away from normal 
observing.

3.3.2.1 Subreflector Position Change with Elevation and Other Conditions

The positioning of the subreflector (axial, lateral, and rotational) which maximizes antenna gain 
will change as a function of elevation and ambient conditions (notably temperature).  There are 
two possibilities for determining this optimal position and its effect on the delay.  First, we 
should measure, very carefully, for each antenna, what the optimal position is as a function of 
elevation.  If this optimal position is stable enough over time, with given conditions, then we 
may simply be able to use a look-up table (as a function of, say, elevation and temperature) to 
find this optimal position and the delay term.  If not, then we must go to the second possibility, 
which is much more time consuming, but might be required in the end – periodic measurements 
of the optimal position may be required during observation.

3.3.2.2 Axis Non-intersection Determination

The fact that the elevation and azimuth axes do not intersect exactly (nor are the angles between 
them 90o) provides an apparent offset of the antenna position (Butler 2003; Wade 1974; Wright 
2002).  This axis non-intersection term can be determined during pointing runs as long as "over 
the top" pointing is allowed on the antennas (Wade 1974), which it will be for the ALMA 
antennas.  This will be done as part of the normal pointing model determination runs for 
antennas, but may not have to be done every time, as it should vary slowly (on the VLA, for 
instance, it is only done once every 16 months or so).  Only experience will tell us how often we 
need to remeasure and update these quantities.

3.3.2.3 Illumination Offset

One additional modification to the antenna position is a result of the fact that the receivers are 
not illuminated perfectly on-axis.  This effect, and its measurement and correction are described 
in Holdaway (2001a).  Although we should attempt to measure this and correct for it at the time 
of observation, much of the introduced error can be corrected in postcalibration by adjusting the 
effective (u,v,w) coordinates of the visibilities and modifying the primary beam pattern.

3.3.3 Atmospheric Delay

3.3.3.1 Neutral Delay

Although the differential neutral atmospheric delay from antenna to antenna can be determined 
by postcalibration, we would like to get it as close as possible to begin with, before correlation.  
If the CALC (Ryan and Ma 1979) package is used to calculate the delays, it includes the 
provision for calculating neutral atmospheric delay, however it is widely accepted that the 
atmospheric model used therein is not very accurate.  It would be much better to use the ATM 
model (Pardo et al. 2001) to calculate the expected neutral atmospheric delay.  This model will 
require some inputs – measurements where available (surface meteorological parameters, for 
example), or estimates where not (atmospheric profiles, if not actually measured).  The surface 
meteorological quantity measurement requirements given in section 3.1.1.4.1 above should 
suffice for this purpose.

3.3.3.2 Ionospheric Delay

Similar to the neutral delay, we would like an estimate of the bulk ionospheric delay above each 
antenna if it is available.  This would require some knowledge of the total electron content above 
each antenna, which is probably not feasible.  We therefore do not expect to input this as a 
normal part of the precalibration quantities (though the provision should be there in the 
software).

3.3.4 Electronic Delay

In addition to the geometric delay introduced from other factors, there is an additional 
component introduced from the path of the signal through the electronics for each antenna, 
frontend, and backend.  This electronic delay must be determined each time an antenna is moved 
to a new pad, and if it varies as a function of time on a pad, must be tracked also.  Once the 
antenna positions are well-determined, it is relatively easy to measure the electronic delay for an 
antenna because it introduces a phase slope across the spectrum (Thompson et al. 2001).  All that 
is necessary is the observation of a strong source, and measurement of this phase slope (it could 
also be done by looking at the amplitude loss from the delay error but looking at the phase slope 
is much more accurate).  The delay will have to be determined separately for each receiver band, 
and also for the cross polarizations.  Because of this, the observed source should have strong 
linear polarization (it is unimportant at what angle, it just needs to be strong).

Only time and experiment will tell us whether the electronic delays remain stable over time.  We 
expect that they should (D'Addario 2000), but should test this in the early days of ALMA to gain 
practical experience with it.

3.4 Bandpass

There is no component of bandpass calibration that requires a precalibration.  While it might be 
nice to be able to do this calibration real time in order to see realistic spectra, we do not find it 
compelling enough to cause us to include this in precalibration quantities.  If it turns out, in 
practice, that some parts of the bandpass are very stable (the instrumental response, for example), 
then it may be possible to measure them infrequently and store them as templates to be used 
when looking at spectra in real time.  We know, however, that there are parts of the bandpass 
that change with time more frequently (the atmosphere, for example), so we cannot assume that 
we can precalibrate the bandpass entirely.

3.5 Polarization

There is no component of polarization calibration that requires a precalibration.  While it might 
be nice to be able to do this calibration real time in order to see realistic polarization results, we 
do not find it compelling enough to cause us to include this in precalibration quantities.  If it 
turns out, in practice, that some parts of the polarization calibration are very stable (the 
instrumental response, for example), then it may be possible to measure them infrequently and 
store them as templates to be used when looking at results in real time.  We know, however, that 
there are parts of the polarization calibration that change with time more frequently (the 
atmosphere, for example), so we cannot assume that we can precalibrate the polarization entirely.

4  Postcalibrations

4.1 Antenna Characteristics

4.1.1 Pointing Selfcalibration

It has been proposed that it might be possible to derive pointing offsets from the data themselves, 
in mosaiced observations – so-called "pointing selfcalibration" (Wright 1997).  While it might 
work in theory, this technique has never been used in practice.  Further, it may not work well at 
all unless there is at least one bright point-like source within the target area. Once the antenna 
pointing offsets have been determined, it is simple and not too CPU-expensive to apply the mean 
array offset as a function of time to mosaic or single pointing data and use that in the imaging.  
However, if there is significant scatter among the antennas' pointing positions at each time, 
imaging wide field sources considering the correct pointing data may be prohibitively expensive 
(Holdaway 1993).

4.1.2 Primary Beam

While we expect to measure the primary beams of the antennas as a precalibration (as part of 
setting the surface to its optimal shape), it may be true that a better measurement of the primary 
beam pattern is obtained after observations are completed.  In this case, the newest (appropriate) 
measurement should be used.

4.2 Amplitude

4.2.1 Instrumental Amplitude Calibration

Because of the fast fluctuations in the atmosphere, it is necessary to track the changes in system 
temperature caused by them.  This monitoring must be accurate to at least the level that the 
amplitudes are desired to be calibrated to in the end, i.e., 1% at frequencies < 300 GHz; 3% at 
higher frequencies.  The currently used technique of tracking these fluctuations, the ambient load 
chopper wheel method, is accurate to about 5% (see Ulich & Haas 1976; Kutner 1978).  Since 
the requirement is 1 or 3%, we cannot rely on this technique to provide sufficient accuracy.  We 
have investigated two alternatives to this scheme: one in which a dual-load system shines 
through a hole in the subreflector (Bock et al. 1998; Mangum 2002); and one in which an 
ambient load shines through a semi-transparent vane into the feeds (Guilloteau 2002).  Both of 
these systems could theoretically reach 1% amplitude tracking accuracy, but initial tests have 
shown that it will be extremely difficult to reach this level in practice, due to several problems 
(Bock & Welch 2003; Martin-Pintado et al. 2003).  Currently, a scheme in which a wire grid is 
used instead of a semi-transparent vane, and two ambient loads (of different temperatures), is 
under investigation (Houde et al. 2001; presentations at ALMA Week 2003).  A dielectric 
beamsplitter has also been suggested (James Lamb's presentation at ALMA Week 2003).  
Whatever the final system, it is clear that some device where multiple temperatures are allowed 
to be sampled by the receivers is required (Guilloteau et al. 2003).

With any of these systems, it may be necessary to obtain independent estimates of either the 
atmospheric opacity, or the atmospheric equivalent radiating temperature (or, even better, the 
profile of temperature through the atmosphere), or both.  At frequencies at which the atmosphere 
is opaque at low elevation angles and partially transparent at high elevation angles, it will be 
possible to solve for the sky temperature and the opacity with a sky tip. However, at frequencies 
at which the atmospheric transmission is excellent, a sky tip will only give the product of the 
opacity and the sky temperature, and the temperature must be assumed to calculate the opacity.
Currently, atmospheric models are not sufficiently accurate to measure the opacity and sky 
temperature at a partially opaque frequency and accurately estimate the opacity at another 
frequency. This estimate would have to come either from a model of atmospheric opacity, or 
from emperical results.  The best model currently available is the ATM model 
(http://www.submm.caltech.edu/~pardo/atm.html).  The expected accuracy of the ATM model is 
about 2%. The Water Vapor Radiometry (WVR) system should provide a measure of emission 
and absorption at its operational frequency of 183  GHz. Actual measurement of the sky 
temperature and water vapor profiles via radiosonde are currently underway, along with 
modeling of the data. Continuous radiosonde monitoring,  however, seems unwieldy and 
expensive. A more cost effective solution would be to float a tethered balloon over the site 
several times a day. Unfortunately, these systems (tethersondes) are  notoriously unreliable and 
hard to make work under all expected site conditions.  The temperature, pressure, and water 
vapor information would also be useful for the radiometric phase correction schemes.

The best results from atmospheric models may require use of a dedicated sounding device, like 
an FTS, in conjunction with an atmospheric temperature sensor based on the strong Oxygen 
lines.  This is currently under investigation.

4.2.2 Flux Density Calibration Using Astronomical Sources

In theory, with accurate monitoring of the system temperature, and precise knowledge of other 
antenna parameters (aperture efficiency, most importantly), an accurate conversion from 
measured correlated power (or correlation coefficient) can be made.  Unfortunately, in practice, 
it is extremely difficult to know the quantities to enough precision to rely completely on this.  
Therefore, we are probably going to have to rely on the traditional method of flux density 
calibration, which relies on astronomical sources of relatively well determined flux density, 
either a priori, or by accurate bootstrapping or measurement at the time of observation.
It is therefore important to find astronomical sources that could serve as flux density standards.  
The ideal standard should have a predictable flux density as function of time,  frequency, and  
polarization from first principles. Next are standards whose time and frequency dependence can 
be predicted, but which still require one initial measurement. Finally, sources which are not time 
variable but still need detailed measurement of their spectrum could be used as flux density 
standards. Planets are the closest to the first category of standards, but the estimated accuracy of 
the flux densities of the planets is only currently about 5-10% (with a few exceptions), so active 
research is required in this field.

ALMA will have the sensitivity to use much fainter sources as flux density standards than the 
current mm arrays. This opens the possibility to use new categories of objects. Among these, the  
brightest stars could be useful, especially at the highest frequencies. Asteroids or satellites of 
giant planets could also be interesting. For none of these objects is the prediction of the flux 
density accurate enough so far. ALMA should keep close contact with the communities of 
experts to improve this situation.

Objects like MWC 349, which has a well defined spectral index, can also be used provided an 
accurate measurement is made at the lowest frequency. Even quasars like 3C286, which have not 
shown any sign of time variations, could ultimately be used (possibly, though they begin to 
become more variable at mm wavelengths, since they become more core-dominated, and the 
cores are known to fluctuate – for example, 3C48 is a very good calibrator at frequencies less 
than about 15 GHz, but becomes quite variable by 40 GHz).. A detailed discussion of flux 
density standards is given in Yun et al. (1998) and Moreno & Guilloteau (2002). 

Our current thinking is that we will have a very few true primary standard calibrators, whose 
flux densities are presumed known and stable – these may include Titan, Uranus, Mars (though it 
is rather too large to be most useful), the largest asteroids, and MWC 349, among others.  We 
will then have a set of secondary standard calibrators, whose flux densities will have to be 
monitored relatively frequently (time scale of days) against the primary standards – these may 
include asteroids with diameter greater than about 200 km (so they are roughly spherical), and a 
number of stronger quasars.  We will then have a large database of other possible calibrators, 
including quasars, stars, and other relatively strong, compact sources, which can be used to track 
the long timescale amplitude variations during each observation.  These other calibrators must be 
measured against one of the primaries or secondaries each time they are used for an observation.

While more work is clearly needed in this area, it will be important for ALMA to keep a 
database of the calibration parameters, so that improvement in the calibration accuracy can be 
made when our understanding of the flux standards improves. ALMA should also keep close 
contact with other instruments working in similar frequency ranges which could provide key 
parameters on the flux density standards.


4.2.3 Phase Decorrelation

An uncorrected antenna based phase error of 10 degrees rms will result in a 3% decrease in the 
visibility amplitude due to decorrelation. As the characteristics of the phase noise change, the 
amount of decorrelation will also change. The primary defense against decorrelation is to try to 
correct for the phase (delay) fluctuations as much as possible. However, when the phase cannot 
be fully corrected, we can estimate the magnitude of the decorrelation and correct the visibilities. 
Decorrelation could be estimated from:

?	phase calibrator data (fast switching)
?	independent phase monitor (atmospheric) with an injected LO signal (antenna mechanical 
& electronic)
?	radiometric data (atmospheric) with an injected LO signal (antenna mechanical & 
electronic)

Holdaway (1997c) has provided estimates of atmospheric coherence times.  Electronic 
decorrelation is stable with time and (to first order) baseline length independent. Although it is 
antenna dependent, this should be calibrated out by relative amplitude calibration. Atmospheric 
decorrelation is the most damaging because of its baseline length dependence, but can be 
estimated from WVR data. Mechanical decorrelation remains to be evaluated. It is antenna 
dependent, but also possibly time dependent.

4.2.4 Changes in Transparency

At millimeter wavelengths, the changes in atmospheric transparency will be very modest, under 
1% over 10 minutes about 80% of the time.  Since the same amount of water vapor results in 
much larger opacities in the submillimeter, the transparency fluctuations in the submillimeter 
over characteristic calibration time scales will be much larger, typically several percent during 
median stability conditions. Because of the lack of strong enough amplitude calibration sources 
(Moreno & Guilloteau 2002), the transmission needs to be corrected either by appropriate  
prediction or measurement (see above discussion). Because of the many ozone lines in the  
submm windows, this transparency correction will need to be made with sufficient spectral  
resolution (10 to 100 MHz), and probably implies use of models for the highest spectral 
resolution data.

4.2.5 Polarization Complications

As mentioned below under polarization calibration, if a linearly polarized calibration source is 
used to track changes in the amplitude gain or opacity, a telescope with linear feeds will produce 
parallel hand visibilities which are modulated by the linear polarization. The extra signal varies 
as a sinusoid of the parallactic angle, so the errors are systematic.  This must be accounted for.

4.2.6 Single Dish Issues

ALMA differs from any other aperture synthesis array in that, from the outset, the instrument 
will support no-compromise single-dish observing modes in addition to the more usual 
interferometric modes. Some of the issues are discussed in Emerson & Jewell (1993).  Receiver 
stability and other issues are addressed in Welch (2003) and D'Addario (2003).

Because single-dish observing is in total power, albeit switched against, for example, blank sky, 
there are extraordinary demands on instrumental gain stability.  In addition, the extra, variable 
emission from the sky comes in directly, and tends to mask the much weaker (by perhaps 4 
orders of magnitude) astronomical emission. This is in contrast to interferometry, which by the 
use of cross-correlation rather than self-correlation, is relatively immune to these factors.  
Another serious problem for single-dish observing is that of the sideband gain ratio.  Since some 
receivers will be double sideband (DSB), they will have emission from an unwanted part of the 
spectrum in the single-dish signal.  The interferometry data do not suffer from this problem 
because the unwanted sideband is rejected in the downconversion.

Astronomical calibration in single-dish mode has to be on a dish-by-dish basis; calibration 
sources need to be detectable with adequate signal-to-noise ratio by one single dish of the array. 
This is again in contrast to interferometric astronomical calibration measurements, in which the 
large collecting area of the entire array can contribute to the signal-to-noise ratio achievable in 
calibrating individual dishes of the array.  We can attempt to calibrate interferometrically, and 
apply these gains to the single-dish data, but in this case, the decorrelation must be accurately 
calibrated, as must the sideband gain ratio.

Polarization calibration of single-dish observations has its own problems.  At mm wavelengths, 
polarization measurements are conventionally made with a "widget" in front of the receiver feed. 
This "widget" introduces changes in the polarization response of the receiver - for example a 
rotating grid and screen combination can continuously rotate the incident plane of linear 
polarization.  The astronomical polarization is then detected by synchronous changes in total 
power intensity through the receiver as the sense of polarization changes.

ALMA may indeed have to provide such "widgets" for each of the antennas. However, the 
complexity and potential unreliability of such a device could be avoided if it were shown 
possible to measure polarization reliably, in single-dish mode, by cross-correlation of the signal 
from orthogonally polarized feeds. Tests of the feasibility of this techniques are planned.

4.2.6.1 Atmospheric Emission Cancellation

The emission from the atmosphere is much stronger than the emission from most astronomical 
sources, and, even worse, the atmospheric emission is variable as well.  The variable part of the 
emission is mainly due to inhomogeneously distributed water vapor, which also causes the phase 
fluctuations.  Since we have excellent statistics of the phase stability on the Chajnantor site, we 
can infer the severity of the variable atmospheric emission at any desired frequency by using a 
transmission model or FTS measurements.

For an interferometer, the atmospheric emission above two different antennas is not correlated, 
so it does not affect the visibilities.  In total power continuum observations, the variable 
atmospheric emission is a major problem which requires some sort of switching on the sky.  The 
total power spectral line case is much less demanding, as large atmospheric fluctuations can be 
tolerated, considering the much smaller channel widths and much higher thermal noise and the 
possibility of fitting an average baseline to each spectrum.  The spectral line data will have 
secondary effects, such as the bandpass changing in response to the changing atmospheric load.  
However, the spectral line observations are much easier than the continuum case, so if we can 
beat the atmosphere for continuum observations, the spectral line observations will be no 
problem.  The detailed treatment of this problem is presented in an upcoming ALMA Memo 
(Holdaway, Lugten, and Freund, in preparation).  Lucas (2000) presents a novel method for 
calibration to remove atmospheric emission which makes use of the very wide bandwidths of 
ALMA.

4.2.6.1.1 Beam Switching

Traditionally, beam switching by a nutating subreflector has been used to remove the variable 
atmospheric emission.  Our study indicates that most beam switching is non-optimal. For any 
given observation, we would like to be roughly equally limited by thermal SNR and by the 
residual variable atmospheric emission.  If the noise is dominated by the variable atmospheric
emission, we need to switch faster.  The faster we switch, the better the atmospheric cancellation, 
but the lower the duty cycle, so the thermal noise will increase.  Furthermore, the distance of the 
throw also needs to be considered.  In general, it is optimal to have the smallest throw which
gets completely off source.  However, in an unstable atmosphere, multiple short throws are 
better.  Hence, the detailed use of a nutating subreflector needs to be fine tuned to match the 
atmospheric conditions and the observing frequency.  As with fast switching, we hope that the 
observer does not have to perform the calculations to find the optimal switching strategy; the 
observer should provide high level guidelines, and the program which performs the micro-
scheduling should calculate the optimal switching strategy for the current atmospheric 
conditions.

The nutator design (see specifications in paragraph 4.2.9 of the Antenna chapter of this ALMA 
Construction Project Book) for the ALMA prototype antenna allows for a maximum throw of 
about 1.5 arcmin for symmetric beam throwing. Maximum nutating frequencies of about 10 Hz 
are planned. If it is affordable, nutators with higher peak acceleration and larger maximum 
throws would be desirable for the production antennas. The two beams should be as similar as 
possible, and as similar as possible to the "non-wobbled" beam, to reduce the level of systematic 
errors in beam switching.  All of these beams must be accurately measured and recorded.

The analysis of the On-The-Fly technique for total power continuum observations indicates that 
it will be as good or better than beam switching in all situations.  However, there is considerable 
risk involved in relying on the On-The-Fly method to cancel all atmospheric fluctuations.  For 
this reason, it is generally agreed that the prototype antennas need to have nutating subreflectors. 
Currently, this is planned according to the Payne design referred to above.

4.2.6.1.2 On-The-Fly

In On-The-Fly (OTF) observing, the antennas scan quickly across a source at constant elevation 
angle, using the off-source regions on other side of the source region to define the sky emission.   
Very large sources will need to be pieced together at some SNR expense.  The OTF technique 
promises to be quite effective at removing the atmospheric emission for three reasons:

?	each Nyquist sample on the sky is observed for a very short time, so the system noise is 
large and a larger amount of sky fluctuation noise is tolerable. (The large number of 
Nyquist samples observed in each scan compensates for this large noise per Nyquist 
sample.)
?	since more time is spent observing the OFF than an individual ON Nyquist sample, the 
atmosphere is well determined, unlike beam switching where we are differencing two noisy 
numbers.
?	since the OFF's are observed over a range of time, we can remove a second order 
polynomial trend in the atmospheric emission time series, which greatly reduces the 
residual sky emission fluctuations.

For sources which are about one beam across, the OTF observing strategy works about as well as 
beam switching.  For larger sources, OTF wins because of the relative increase in the SNR of the 
atmospheric determination and because multiple throws begin to degrade the beam switching 
SNR.

Because the entire antenna is moving, many systematic errors which plague beam switching 
(such as differences in the shapes and gains of the ON and OFF beams) are eliminated.  
However, it takes much more energy to move the entire antenna, and there is more risk in 
general with an observing strategy that attempts to move the entire antenna.

OTF will work only if we can slew and reaccelerate the antenna quickly without exciting the 
lowest resonant frequency of the antennas.  An initial analysis of this problem has been 
performed by Holdaway, Lugten, and Freund (in preparation). Using a Gaussian acceleration 
profile and an error function velocity profile, they predict the antennas will be able to turn 
around from one scan direction to the other in about 0.2 s without appreciably exciting the lowest 
resonant frequency.  This acceleration profile is a good one, but probably not an optimal one, so 
further work could help optimize the profiles for both OTF antenna motion and fast switching 
antenna motion.  In order not to excite the antenna motions, the acceleration must be very 
smoothly varying. This will put strong constraints on both the control system and on the servo 
system.

OTF simulations of sources of various sizes indicate that the optimal slew velocity varies linearly 
with source size. For a maximum interesting source size of 1 deg, a maximum slew rate of about 
0.5 deg/s is required.  This requires a maximum antenna angular acceleration of about 12 deg/s/s. 
Since the profile is Gaussian, we do not require this maximum acceleration for very long.  These 
maximum velocities and accelerations are for an antenna with lowest resonant frequency of 6 
Hz.  An antenna which was less stiff could not utilize such large accelerations and velocities in 
OTF observing.  A stiffer antenna would permit faster turnarounds, requiring larger accelerations 
and velocities. However, the 6 Hz antenna is effectively beating the atmosphere already, so not 
much is gained from a stiffer antenna.

OTF requires that we know where the antenna is for each Nyquist beam.  At the 0.5 deg/s 
maximum slew rate, observing at 850 GHz with a half beam size of 0.001 deg will require that 
we dump the data and know where the antenna is every 2 ms.  We don't need to make the 
antenna go to any precise place at any precise time, we just need to know where the antenna was 
at a precise time.  We may not need to read the encoders every 2 ms; if the antenna position 
changes smoothly over time scales of 10 ms, we can read out the encoders more coarsely and 
interpolate.  We do not require that the encoders be accurate to within the pointing specification 
of 0.6 arcsec.

For total power OTF, we need not synchronize all the antennas in their dance across the sky.  
The antennas could be staggered to permit a more constant  utilization of electrical power.

4.2.6.1.2  1/f  Noise

In addition to atmospheric brightness fluctuations, beam switching and OTF will remove a 
portion of the receiver 1/f noise.  From the optimizations we have performed, we can set  
specifications on the 1/f noise for each observing frequency.  Even though the beam switching is 
performing the switching faster than OTF, the integration time spent on each ON is often larger 
than the integration time spent per Nyquist sample of an OTF observation, so OTF and beam 
switching are similar in their ability to switch out 1/f noise.  If these specifications cannot be 
met, we must reoptimize the OTF observing strategy, which would result in moving more 
quickly to accomplish faster switching and less time or more white noise per Nyquist sample on 
the source. This would favor both higher maximum accelerations and a stiffer antenna.  
D'Addario (2003) has discussed some of the details of 1/f noise, and compared OTF with beam-
switched measurements.

  Freq            Beam Size Source                 0.5 deg Source
  [GHz]    noise      break frequency       noise      break frequency
            [Jy]            [Hz]            [Jy]             [Hz]
   90       0.047           1.2             0.081            0.34
   230      0.088           1.2             0.25             0.29
   345      0.14            1.2             0.47             0.29
   650      0.33            1.3              1.6             0.34

 Table 1: For continuum (8 GHz bandwidth per polarization) OTF observations,
  what noise level must the 1/f noise be below, and at what frequency, for
 1/f noise to have essentially no effect on OTF observations' sensitivity?

4.2.6.2 Sideband Gain Ratio

An important issue in the single-dish calibration is the fact that for DSB receivers, there is 
emission from both sidebands which is detected.  This is not an issue for the cross correlated 
visibility data, because of the phase switching in the LO which does effective sideband rejection 
(D'Addario 2001).  Moreno & Guilloteau (2002) have studied this in a preliminary way, and find 
that this may be one of the limiting calibrations in terms of the time it takes to properly make the 
measurement.  During spectral line observations (especially surveys), it may be possible to 
deconvolve the ratio, but in normal continuum observations, this will be nearly impossible.  This 
needs more study, and some experience on the telescopes.

4.3 Delay

4.3.1 Atmospheric Delay

4.3.1.1 Neutral Atmosphere

Most of the systematic and long-term fluctuating part of the atmospheric delay is taken out by 
observing nearby calibrators on timescales of minutes or even slightly longer.  We assume that 
this will be true for ALMA, and so have no special scheme for calibrating them out.  In addition, 
if there are fast fluctuations in the dry component of the atmosphere, we assume that the fast 
switching will account for them (see below).  This, however, needs some more attention – both 
as a theoretical study, and with experience on the telescopes.

The main part of the calibration of the fluctuations in the neutral atmosphere is dealing with 
fluctuations in tropospheric water vapor.  We currently have two planned techniques for 
calibrating these fluctuations, fast switching, and water vapor radiometry.

4.3.1.1.1  Fast Switching

If a calibrator is sufficiently close and the telescope is sufficiently fast, fast switching between a 
calibrator source and a target source can effectively stop the atmospheric, electronic, and antenna 
phase fluctuations.  If fast switching is used as the phase calibration method, it makes minimum 
requirements on the system sensitivity, the slew speed and settle down time of the antennas, and 
the online and data taking systems.  Fast switching has been studied extensively in the MMA and 
ALMA memo series (see especially Holdaway 2001b; Holdaway & Pardo 2001; and references 
therein), and we are fairly confident that it will work for ALMA.

The basic criteria for fast switching to work is that the phase calibration source needs to be 
detected with sufficient SNR and the target source be observed for some amount of time within 
the coherence time and distance of the atmosphere.  This translates into a requirement that there 
be sufficiently many calibrator sources which are sufficiently bright (Holdaway, Owen, & Rupen 
1994; Foster 1994), and a  requirement on the sensitivity of the array.  In practice, this means 
that the calibrator source will typically be within a few degrees of the target source, the calibrator 
will usually be detected in less than a  second, and the entire cycle time will be about 10 s, 
though the details vary with observing frequency.  Spectral line observations will need to use 
wide bandwidth continuum observations of the calibrator.

With the current sensitivity of ALMA and our understanding of the quasar source counts and 
their dependence on frequency, we will not always be able to perform fast switching calibration 
at the target frequency, but often we will get a higher SNR phase solution by observing the 
calibrator at a low frequency (90 GHz) and scaling the solution up to the target frequency.  This 
is not to imply that fast switching will always use a lower frequency for the calibrator, but that 
sometimes it will prove to be more effective.  The optimal frequency for calibration has been 
computed by Moreno & Guilloteau (2002).  If the calibrator is observed at a lower frequency, 
and the phases scaled up, then this requires a much more accurate phase solution at the lower 
frequency.  Since the dry atmosphere is non-dispersive, this extrapolation basically relies upon 
the wet differential delay to be non-dispersive as well.  In the submillimeter, the wet differential 
delay is dispersive, which will either limit the effectiveness of fast switching or require more 
complications in the fast switching observing strategy, such as less frequent multi-frequency 
calibrator observations to help separate out the non-water vapor phase contributions.  These 
multi-frequency observations may also be required for amplitude calibration. On longer  
timescales, set by the phase stability of the electronics and mechanics, it is necessary to measure 
the instrumental phase and delay offsets between different frequencies, by observing a single 
source at both wavebands. 

The antenna movement requirement from fast switching is currently a slew of 1.5 degrees and 
settle down to 3 arcsec pointing in 1.5 seconds.

The on-line system needs to control the antennas gracefully enough to move them quickly 
without exciting the lowest resonant frequency.  Also, the quanta of integration time and scan 
length need to be sufficiently small so as not to restrict the integration time spent on the target 
source and calibrator or the time spent between sources.  Flexibility at the 100-200 ms level is 
desirable.  Fast switching data can be calibrated with existing software, but some extensions in 
spatial-temporal interpolation will be useful.

Fast switching will reduce the sensitivity of observations due to time lost observing the calibrator 
and moving the antennas, and due to decorrelation from residual phase errors.  Both effects can 
be reduced by observing in the best conditions, which often result in very low residual phase 
errors at a minimum expense in time lost to the calibration process . However, not all projects 
can be observed during the best phase conditions.  Holdaway (2001c) concludes that fast 
switching will generally result in less than a 20% decrease in sensitivity for the phase conditions 
at the Chajnantor site.

During poor phase stability conditions, fast switching won't work at the high frequencies.  Also, 
a given target field may have a dearth of calibrator sources, requiring that the field be observed 
during better phase conditions than the average field.  For reasons like these, dynamical 
scheduling is absolutely required to optimize the utility of ALMA.  We envision one or more 
phase stability monitors providing real time information to the array control center, and  
contributing to observing decisions - e.g.:

?	what project should run on the telescope?
?	do the present conditions permit the current project to continue?
?	what is the optimal calibrator for the current project in the current atmospheric conditions 
and hour angle?

The quasars which will form the bulk of the fast switching calibrators will be highly variable at 
millimeter wavelengths, and a quick survey of a few square degree region about the target source 
will sometimes be required.  ALMA has the sensitivity to perform a blind search for calibration
sources in a few minutes.  Surveys directed with lower frequency source catalogs will be even 
faster.  Whenever a potential calibrator is observed, the source information will need to go into a 
comprehensive calibrator database, which can also be used for choosing an appropriate 
calibrator.

4.3.1.1.2  Water Vapor Radiometry

The most promising alternative to fast switching is radiometric phase correction or, more 
specifically, Water Vapor Radiometry (WVR), which has a thorough treatment in the MMA and 
ALMA memo series (see especially Hills & Richer 2000; Hills et al 2001; and references 
therein).  WVR utilizes the variable emission caused by inhomogeneously distributed 
atmospheric water vapor to determine the phase fluctuations caused by water vapor. While water 
vapor is not the only source of phase errors, it is the dominant source of short time scale phase 
fluctuations.  This method has had several early successes, but the correlation between the 
radiometric fluctuations and the interferometrically measured phase fluctuations changes with 
time, and there are some times when the method does not work well at all.

The baseline plan for radiometric phase correction is a 183 GHz water vapor radiometer using an 
uncooled Schottky receiver with 5 (or 2 times 5) frequency channels. The radiometer may be of 
the Double Dicke-switch type or of the correlation type, depending on the results of the 
development phase. The optics system is designed to allow using the radiometer simultaneously 
with all observing bands, but not during some calibration operations (notably when the amplitude 
calibration widget is in use).

Details of the WVR development strategies are developed in Hills & Richer (2000) and Hills et 
al. (2001).  Active research in the interpretation of the WVR measurements is mandatory.  
Before WVR prototypes are available for on-sky tests, preferably at the ATF, but possibly on 
other interferometers (e.g. IRAM or SMA), extensive simulations should be performed using the 
ATM model. The ALMA simulator developed by the ADACE consortium incorporates such 
simulation capabilities: early delivery of the development versions of this simulator to the 
Science IPT would be valuable.

4.3.1.1.3	Paired Antenna Calibration

It is possible to use some of the antennas to observe a calibrator and the rest of the antennas to 
observe the target source.  In the smaller arrays, the configurations will naturally permit paired 
array calibration.  Carilli & Holdaway (1999) present encouraging results using the VLA. Longer 
configurations have no specific provision to optimize use of such a mode. This mode should be 
reserved as a backup solution for special projects in which the resulting sensitivity loss is not a 
major issue, but will not be the default for ALMA.

4.3.1.2  Ionosphere

It may be necessary to track fluctuations in the ionospheric delay (see discussion above).  If this 
is the case, then it is likely that we will need some sort of GPS system which can track the total 
electron content, and possibly in combination with some ionospheric model, can estimate 
ionospheric delay as a function of pointing location (see, e.g., Gradinarsky et al. 2001).

4.3.2 Calibration of the Electronic and Antenna Structure Delay with an Injected Signal

Radiometric phase correction will only correct for those phase fluctuations which are caused by 
water vapor, and will not correct for any phase errors caused by variations in the dry atmospheric 
delay, mechanical instabilities in the antenna, or instabilities in the electronics.  Therefore, 
radiometric phase correction requires some supporting observations or calibration technique to 
remove phase errors caused by these other sources.

An initial idea was to periodically inject a stable signal, perhaps derived from the LO, into the 
feed to calibrate the electronic contributions to the phase errors.  If the calibration signal is 
injected from the subreflector, then this calibration system will also track the mechanical phase 
drifts due to focus and surface changes of the antenna.  If the calibration signal is derived from 
the LO, and the LO itself has phase instabilities, they will either cancel or be doubled, depending 
upon the relative phase of the LO and the injected signal.  In fact, by alternating the relative 
parity of the injected signal and the LO, we can solve for both phase errors in the LO and in the 
rest of the electronics and the antenna up to the subreflector.  However, phase changes due to  
mechanical deformations in the antenna mount (i.e. baseline fluctuations, for example due to 
wind-induced tilt of the fork) are not accounted for in such a system.  Fluctuations in the dry 
atmosphere will also be unaccounted for.

The on-line system would need to control the details of the injected signal.  Information about 
the injected signal would need to be recorded with the data, and an option for determining and 
correcting for the electronic phase errors in real time should exist.

The injected signal calibration scheme is still an area of research in this phase of the project.  
Implementation decision can only be taken once a prototype is build and tested at the ATF.

4.4 Bandpass

When observing spectral lines, we would like to always be in the situation where the noise in the 
channel is limited by thermal noise rather than bandpass calibration.  A majority of the spectral 
line observations made with ALMA will probably have no problem meeting this condition. 

In rough terms, the dynamic range DR of a single spectral channel which is limited by errors in 
continuum subtraction caused by bandpass errors will be: 

 								(1)

where Sline is the flux density of the peak of the line, Nant is the number of antennas, Scont is the 
continuum flux density of the source, and ?s is the rms error in the bandpass.  For the spectral 
line observations to be limited by thermal noise and not by bandpass errors, and assuming the 
bandpass errors are themselves due to thermal noise in the observations of the bandpass 
calibrator, we have the condition that 

 

where Scal is the flux density of the bandpass calibrator, tcal is the time spent on the bandpass 
calibrator, and tline is the time spent on the line source.

Because in some cases, the continuum flux density of the line source is quite large (planets and 
strong quasars, for example – see discussion in requirements section), we cannot simply rely on 
the technique of choosing a bandpass calibrator which is much stronger than our line source.  
Injection of a strong coherent (so that the interferometric mode can be used) noise signal which 
is flat over the observed frequency range would be an ideal solution to both the planet problem 
and to the galactic confusing line problem.  However, such a system has not yet been 
successfully developed so far, and the requirements (flatness of frequency response and time 
dependence) are extremely difficult to meet.

We still always have the capability of calibrating the bandpass by observing a strong source for a 
sufficiently long time.  Because the noise increases with frequency, while the calibrator source 
strength usually decreases, it will be advantageous for SNR considerations to split the bandpass 
calibration into several separate pieces: wideband at the observing frequency (WO), wideband at 
the calibration frequency (WC), and narrowband at the calibration frequency (NC).  The final 
bandpass (narrowband at the observing frequency, or NO) is then given by:
NO = NC x WO / WC.

At millimeter and submillimeter wavelengths, the atmosphere will also contribute to the 
bandpass for wide bandwidth observations, so we must either perform an independent 
determination of the bandpass astronomically to solve for the atmospheric bandpass component, 
or we would measure the precipitable water vapor from opacity measurements made at a fiducial 
frequency and determine the atmospheric contribution to the bandpass through the use of an 
atmospheric transmission model.  Currently, the atmospheric transmission models are probably 
not good enough for this sort of work, but ALMA would provide enough data for an ad hoc 
model or to improve the theoretical models.  We will be concerned with changes in the
atmospheric component of the bandpass on reasonably short time scales and among the different 
antennas. Prediction of the magnitude of such changes is feasible using the ATM model.  It is  
essential to note that because of a large number of Ozone lines in the sub-mm domain, the 
bandpass requires modeling at quite high resolution, some of the lines having large opacity 
changes on scales of a few MHz only.  

For pure continuum observations, masking these regions of the frequency spectrum may be 
needed in software to improve the amplitude calibration.  In some amplitude calibration 
techniques, the calibrated visibility is derived from the measured correlation coefficient and the 
calibrated auto-correlation spectra (i.e., Tsys as function of frequency).  For Single-Sideband 
systems, this should provide an accurate bandpass calibration.  The situation for Double-
Sideband systems is more complex because of the emission from both bands, and should be 
investigated in details.

There is an implicit specification placed on the system design that the electronic bandpass be 
either stable or that it vary linearly with time to something like 10000:1 to 100000:1.  If the 
bandpass changes are mainly linear, we can remove them through interpolation if we calibrate 
often enough.

A significant difficulty in the stability of the bandpass is the existence of standing waves in the 
system.  Standing waves may be in emission (which affect bandpass normalization) or in 
transmission (which directly affect the astronomical signal).  Any modification of the optical 
layout of the antenna will modify the standing wave pattern.  This will happen in the dual-load 
calibration system when switching from load to signal for example.  Standing waves will usually 
average out as ?Nant since every antenna has a different standing wave pattern.  An issue which 
remains to be quantified is what design of the calibration system will minimize the standing 
wave problems.

4.5 Polarization

ALMA will use linearly polarized feeds because they have a wider usable bandwidth than 
circularly polarized feeds, and can provide complete coverage of all millimeter wavelength 
atmospheric windows with a reasonable number of receivers.  Cotton (1998) treated the problem 
of polarization calibration for the MMA in detail.  A more recent treatment for ALMA can be 
found in Appendix C to the Report of the ALMA Science Advisory Committee March 2000 
Meeting.  The general problem of calibrating polarization in interferometry has been described 
fully by Hamaker, Bregman, & Sault (1998), who develop a Jones matrix approach (for an 
earlier similar approach, see Schwab 1979).  The main detail that we must be concerned with 
here is that the measurement of linear polarization is corrupted by contamination from Stokes I.  
For linearly polarized feeds, this corruption is in the form of a gain stability term (as opposed to 
circularly polarized feeds, where the corruption arises from a leakage term).  Another point of 
note is that it is not easy to distinguish circular polarization from the instrumental polarization 
terms when using linearly polarized feeds.

Because the linear polarization is entangled with the total intensity, there are times when all four 
cross correlations per baseline will need to be performed, which will probably result in halving 
the bandwidth and cutting the sensitivity by root two.  We consider several cases which could 
come up with ALMA to demonstrate when we may need to consider all four cross correlations 
and when we may use approximations to make use of just the two parallel hand cross 
correlations:

?	Amplitude calibration is performed by knowing precisely the gains and system 
temperatures of the antennas (not by looking at an astronomical source), and phase (delay) 
calibration is performed on a quasar (or a combination of radiometric [WVR] plus a 
quasar).  The quasars will generally be a few percent linearly polarized, but may be as 
much as 10-20% polarized, and hence Stokes Q and U will influence the parallel hand 
visibilities.  These sources have almost no circular polarization.  For a point source, the 
linear polarization of the calibrator will not affect the phase, only the amplitude.  We 
further consider two subcases:
o	Total intensity imaging with no polarization in the target source.  Many 
millimeter spectral line sources will have little or no linear polarization.  Nothing 
special needs to take place, as the parallel hands will basically contain Stokes I.
o	Total intensity imaging with appreciable linear polarization in the target source. 
The linear polarization in the target source will corrupt the parallel hand 
visibilities in a systematic way.  However, when the XX and YY visibilities are 
added together, the linear polarization corruptions cancel out.  This is acceptable 
for low to moderate dynamic range total intensity observations, but may not be 
sufficient for high dynamic range total intensity observations, as residual gain 
errors will limit the cancellation of the linear polarization and adding the XX and 
YY correlations results in a condition in which gain errors no longer close, 
limiting the use of self-calibration.  High dynamic range total intensity imaging of 
a source with appreciable linear polarization may require full polarization 
calibration and imaging.  
o	Polarization imaging.  A bright calibration source must be observed to determine 
the instrumental polarization leakage or "D" terms.  If the calibrator has known 
(or zero) linear polarization and no circular polarization, the D terms can be 
determined in a single snapshot.  If the calibrator has unknown linear polarization, 
the calibrator must be observed through sufficient parallactic angle coverage to 
permit separation of the calibrator and the D terms.  Application of the D terms 
will permit the polarization imaging.
?	Amplitude calibration is performed by assuming the flux density of some astronomical 
sources is known precisely, and using measurements of that source to set the voltage to flux 
density conversion scaling.  If the source of flux density is not polarized, there is no 
problem.  If it is linearly polarized, then the parallel hand visibilities will vary 
systematically with parallactic angle, the XX and YY visibilities varying in opposite 
senses.  There are several options:
o	For total intensity observations of a target source at low to moderate SNR, the 
array-wide XX and YY gain ratios can be determined and corrected for.
o	High SNR total intensity observations will require accounting for the different 
parallactic angles of each antenna, which will result in imperfect cancellation 
when using the array-wide gain ratios.  In this case, the full polarization 
calibration will need to be performed on the quasar, even if there is no interest in 
polarization.  Full polarization observations of the source are only needed if 
desired.

In all cases in which the cross hand visibilities are explicitly used, the X-Y phase offset must be 
monitored for each antenna.  As there is no simple way to determine the X-Y phase offset 
astronomically, ALMA could inject a tone into the feeds, as the AT does.  Cotton (1998) points 
out that it is difficult to generate a millimeter RF tone, and that injecting an IF tone further 
downstream in the electronics is simpler, though not as good instrumentally (it does not calibrate 
the portion of the offset which occurs before the IF).  On the other hand, we could derive an RF 
signal from the LO and inject it into the feeds for the X-Y phase calibration.

The choice of a flux density calibrator may also interact with the polarization calibration.  
Unresolved asteroids which are not azimuthally symmetric will have some time dependent linear 
polarization, which will complicate the flux density calibration.  If stars are used for a flux 
standard, they may display some circular polarization, which would require that another source 
be used for the D term calibration.

As stated above, the full polarization calibration requires good coverage in parallactic angle to 
separate the constant instrumental polarization (D term) signal from the sinusoidally varying 
astronomical polarization signal.  This causes some concern since ALMA is envisioned to be 
predominantly a near-transit instrument with real time imaging capability.  If instrumental 
polarization calibration is required for many observations, it may be prudent to keep a database 
of the instrumental polarization solutions at the various frequencies and bandwidths and rely 
upon that whenever possible.  Unlike the VLA, the ATNF compact array shows essentially no 
time variability in the instrumental polarization (less than 1:10000 over 12 hours, with variations 
of 0.1% over months).  Given the constraints of ALMA, time constant instrumental polarization 
is certainly an important design goal.  An analysis of the cause of the VLA unstability would also 
be useful. 

One way around the complication of good parallactic angle coverage is to use sources of known 
polarization (one special case of which is totally unpolarized sources).  Holdaway, Carilli, and 
Owen (1992) have demonstrated that it is possible to solve for the instrumental polarization for a 
single snapshot, (i.e., a single parallactic angle) if the source polarization is known in advance. 
So, it would be beneficial to ALMA observing to identify bright, compact sources with known 
polarization or no polarization for use as polarization calibrators.  Unfortunately, such sources 
are currently completely unknown at millimeter wavelengths – all quasars have variable 
polarization angle.  Some study of what level of polarization is acceptable as "unpolarized" is 
warranted.

6  Miscellaneous

6.1 Solar Observing

Calibration of solar observations will involve special care.  When observing the sun, a special 
attenuator will be placed in front of the receiver package.  Because of this, it will be necessary to 
calibrate the properties of that attenuator.  Normal calibrator-source-calibrator sequences will be 
of little use, because there are few calibrator sources which are strong enough to be detected 
through the attenuator, and if they are observed without the attenuator in place, there will be an 
unknown amplitude and phase effect from the attenuator which must be calibrated.


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