In this memo, I come up with best-guess values for the i.p. parameters for 
January 2006 (350 microns) and for the 450 micron runs in December 2006 
and June 2006.

Giles; Jan. 30, 2008 ... modified March 10, 2008

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(A.) Best guess for i.p. correction for the 450 micron data:


1. Start with results from John's fits to individual runs

Dec. 2006, h = 67, u_t' constrained to zero:

q'_t(%) = -0.179 +/- 0.318
q_i(%) = 0.238 +/- 0.038
u_i(%) = 0.172 +/- 0.317 

June 2007, h = 6, u_t' constrained to zero:

q'_t(%) = 0.470 +/- 0.304
q_i(%) = 0.066 +/- 0.167
u_i(%) = 0.253 +/- 0.297


2. Rotate fixed term to primed coordinate system

(using equation 27 from i.p. memo of July 3, 2007)

Dec. 2006, h = 67, u_t' constrained to zero:

q'_i(%) = 0.289
u'_i(%) = 0.052 

June 2007, h = 6, u_t' constrained to zero:

q'_i(%) = -0.012
u'_i(%) = 0.261


3. Take straight average, in primed coordinate system

q'_t(%) = 0.146
q'_i(%) = 0.139
u'_i(%) = 0.157


4. We learned that the primed fixed parameters can change from run to run
by comparing Jan. 2006 and July 2006 (see below).  There is no way to know
for sure if this was due to the change in V_null or to the change in h.  
So it makes little sense to put much faith in the average values of q'_i
and u'_i.  But its encouraging to see that they are small.

So, adopt the following best guess, for both Dec. and June:

q'_t(%) = 0.15
q'_i(%) = 0.0
u'_i(%) = 0.0


5. Translate to the unprimed coordinate system to get parameters to be 
input into sharpcombine:

Notes:

- we need to use the inverse of R, call it R^-1, to do this transformation
- R is given in equation 27 from the i.p. memo of July 3, 2007
- Note that R^-1 = R, as can be confirmed by showing that (R)(R) = I
- values of h for the two runs are given above

Dec. 2006:

q_t(%) = 0.10
u_t(%) = 0.11
q_i(%) = 0
u_i(%) = 0

June 2007:

q_t(%) = -0.15
u_t(%) = 0.03
q_i(%) = 0
u_i(%) = 0


6. Adopt a higher uncertainty in i.p. correction for the 450 micron data:  
Estimated to be a 1-sigma error of 0.2% in each Stokes' parameter.  This
should be taken into account in interpreting results.

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(B.) Best guess for i.p. for the 350 micron data from January 2006:


1. Start with values from Li et al. (Applied Optics)

(using equations 13-14 from i.p. memo of July 3, 2007)

Jan. 2006:

q'_t(%) = 0.42
u'_t(%) = 0.01
q'_i(%) = -0.41
u'_i(%) = -0.20

July 2006:

q'_t(%) = 0.29
u'_t(%) = 0.00
q'_i(%) = 0.43
u'_i(%) = -0.04


2. Since the July 2006 values agree within ~0.1% with the posted i.p. for
SHARP runs having h ~ 90 at 350 microns, we adopt the Jan 2006 values from
Li et al.  However, since we do not expect the telescope i.p. to change
from run to run, we reduce q'_t from 0.42% to the value adopted for all
subsequent 350 micron runs, which is 0.36%.  The result is:

Jan 2006:  

q'_t(%) = 0.36
q'_i(%) = -0.41
u'_i(%) = -0.20


3. Translate to the unprimed coordinate system to get parameters to be 
input into sharpcombine:

Notes:

- we need to use the inverse of R, call it R^-1, to do this transformation
- R is given in equation 27 from the i.p. memo of July 3, 2007
- Note that R^-1 = R, as can be confirmed by showing that (R)(R) = I
- h = 30 for Jan 2006 run (see memo of April 13, 2006) 

The result is:

q_t(%) = -0.18
u_t(%) = 0.31
q_i(%) = 0.03
u_i(%) = -0.46


4. Since Hua-bai's analysis did not have error-bars, we adopt a higher
uncertainty in i.p. correction for the Jan. 2006 350 micron data:  
Estimated to be a 1-sigma error of 0.2% in each Stokes' parameter.  This
should be taken into account in interpreting results.
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