Hertz-mode Data Analysis for SHARP

Giles ... Sept. 21, 2005

This deals with the possibility of developing a data analysis mode for
SHARP that is analogous to what we used for Hertz.  In this approach, we
treat SHARP as two 12 x 12 arrays.  The idea would be to develop
Hertz-mode in time for the January run.

SHARP alignment adjustments can produce very good alignment (0.1 pixel)  
for the center pixels, but somewhat worse (0.5 pixel) near the edges.  As
discussed below, the misalignment near the edges will make it problematic
to do any project that is based on the idea of taking small amounts of
data (e.g. one polarimetric cycle) at a large number of positions.  In
such cases, pointing errors will dominate and will preclude achieving
background limited signal-to-noise.  For this reason, the basic Hertz-mode
will not work well for a GMC survey, and we would have to simply put off
the GMC survey until a later run.  This problem and possible solutions to
this problem are discussed in (3a) below.

As for Hertz, the analysis will be done using pairs of pixels.  (When one
of the detectors in a pair is bad, the other pixel will not be used.)

The basic idea is the same as is discussed in papers by Dowell and Kirby.  
The only difference is that we have a rotating field, but if we assume
that the i.p. can be described by two fixed contributions, one from the
telescope and one from SHARP, then it is fairly straightforward to adapt
Kirby's method to SHARP:

(1) ANALYSIS OF ONE CYCLE:

Start with raw data files for theta = 0, 22.5, 45, 67.5 

(a) for each pixel-pair compute PF_theta,i and TF_theta,i 

   this step is from Dowell et al.
   need f (e.g., use Dowell et al. method)

(b) compute PF_theta and TF_theta

   this step is from Dowell et al.
   errors in PF_theta and TF_theta are computed as in Dowell et al.

(c) compute Q_inst, U_inst, and I

   Q_inst = PF_45 - PF_0  ... also propagate errors
   U_inst = PF_67.5 - PF_22.5  ... also propagate errors
   I = SUM-OVER-THETA (TF_theta)
   sigma_I derived from scatter in TF_theta (ignore sky rotation)

(2) COMBINING CYCLES:

(a) combine all of the measurements of I and sigma_I into one map

   this map will be on a 2" grid
   we need to know the RGM to ~5% accuracy
   we need to know how to correct for tau to ~5% accuracy
   we need to know how to correct for loading gain compression (~5% acc.)
   techniques for doing this step are available (e.g., Harper used "COMB")
   techniques for propagating errors to final map should also be available  

(b) compute q and u from each cycle

   first correct Q_inst, U_inst for RGM, tau, loading gain compress. (~5%)
   interpolate I from above map
   correct for IP(instr) and IP(tel)
   rotate to q_sky, u_sky

(c) make q, u maps, using the same algorithm used for I maps
  
   will usually want to downgrade resolution to 4" or 8" grid (e.g., bin)

(3) OPEN QUESTIONS:

(a) effect of pixel-pair mis-alignment

To first order, ~2.5" pixel misalignment will merely degrade resolution
(perhaps we go from 9" -> 10") at edges.  But there is a bigger problem.  
Flux gradients, combined with pixel-pair misalignment, combined with
pointing drifts within a cycle, will cause a fake polarization.  It will
integrate down as it will be uncorrelated from file to file.  It will
probably be of order 10% per cycle.  For faint sources this will not be a
big problem, as the errors will be 10% per cycle anyway.  For these
sources we require 100 cycles (~ 10 hours) for a good measurement. For the
proposed survey of Orion, however, this is a killer.  It degrades our
signal-to-noise by a factor of 5.

Here are some possible solutions to the edge pixel-pair misalignment, 
which would probably have to wait until after the January run:

Finish development of a sharpsolve-type method

Hardware fix: the biggest problem is relative rotation between H and V
subarrays.  This might be correctable by tipping the combiner.  But we
have not studied this carefully.

Correct for this effect: For some sources, the TF_theta map can tell us
the pointing for that particular value of theta, and how it might be
different from the pointing at other values of theta.  Then we can correct
for the fake polarization caused by this pointing drift.

Improve the pointing: Martin says it may be possible to obtain extremely 
stable pointing using a new pointing system being developed at CSO

(b) calculating f

f may need to be known to high accuracy, just as in past (Hertz).  One
possibility is to use the method of Dowell et al. to calculate f;  
another is to incorporate a variation of John's (sharpsolve/sharpstokes)  
method to calculate f in to Hertz mode analysis.

(c) determining RGM

One method is outlined in Dowell et al.  It works to 10%, which is almost
good enough.  There may be alternatives based on short sharpsolve
integrations.