[evlatests] Polarization Tests -- all is well!

[George Moellenbrock]

Folks,

For the record, perhaps I can shed a bit more light on the nature and 
impact of the FILLM problem Rick notes in his previous email.  (Perhaps 
this will be of help also in forming whatever announcement is forwarded to 
polarization observers affected by this bug.)

I apologize in advance for the length of this email; understanding the 
apparently disproportionate impact of this otherwise fairly subtle bug 
requires some patience.  But it all makes sense in the end!  (This email 
is not much longer than Rick's!)

>    Yesterday, George and Steve noted an inconsistency between data 
> filled into CASA and AIPS in the nominal sensitivities between the RL 
> and LR correlations.  Eric quickly found that for some baselines, the 
> 'R' and 'L' system temperature data were reversed by AIPS upon filling.

Actually, the precise nature of the problem was discovered in the course 
of some careful comparisons (performed in CASA and, believe it or not, 
good ol' glish!) of data filled by FILLM and the CASA equivalent.  In 
these comparisons, I discovered that the FILLM cross-hand _correlation_ 
pair for some baselines were scaled by the nominal sensitivities in 
reverse order.  The specific baselines affected are those which FILLM 
"flips" in order to ensure that the first antenna in the baseline has the 
lower number (see below for a complete description).  I supplied Eric with 
a description of my findings---mostly, but not yet completely certain that 
this wasn't a bug in CASA.  He quickly confirmed my suspicions and then 
(even quicker) corrected what was indeed a bug in FILLM.  A nice 
collaboration between CASA and AIPS, I think, as such a bug is much, much 
easier to uncover when a comparison with an independent filler is 
possible.  All of this was achieved in less than 24 hours.  Thanks are due 
Eric for responding to my query so quickly.

>    We are still trying to understand just how these reversed weights
> can have so dramatic an effect on the linear polarization data (it's not
> elementary, as the polarization programs are involved in the process).

The point is that it is _not_ merely the weights that are "reversed". 
For the affected baselines, the cross-hand correlations themselves have 
been mis-scaled.

The online nominal sensitivity estimates (~Tsys) are stored in the archive 
for both polarizations on each antenna.  Since the modcomp shutdown, it 
has been the responsiblity of fillers (FILLM and the CASA equivalent) to 
apply these factors in pairs (with sqrt) to each of the correlations. 
Now, when a baseline is "flipped" by FILLM to ensure "proper" antenna 
order, (1) all of the correlations are conjugated (i.e., moved to the 
conjugate point in the uv-plane--they are "Hermitian"; we also change the 
sign of u and v), and (2) The RL and LR correlations are exchanged.  The 
cross-hand exchange occurs because (for baseline i-j) the RL correlation 
(RiLj) becomes an LR correlation (LjRi), and vice-versa, and we have to 
maintain the canonical order for the crosshands (RL then LR).  The FILLM 
problem occurred because the baseline flip was occuring before the 
formation of the antenna- and polariziation-based nominal sensitivity 
products and their application, and this was not aware of the RL/LR 
exchange.  So RL was getting LR's nominal sensitivity scaling and LR was 
getting RL's.  Nothing more.  Unless the Tsys estimates happen to be 
identical for both polarizations on both antennas, the RL and LR 
correlations _and_ weights will be mis-scaled relative to each other. 
The parallel hands are entirely unaffected.

In my diagnosis, the clinching facts were that (1) the CASA/AIPS 
comparison was failing only for baselines flipped by AIPS FILLM (CASA, 
somewhat notoriously, but in hindsight, thankfully, doesn't do these 
flips!), and (2) that the RL mis-scaling was the inverse of the LR 
mis-scaling.  (This second point has relevance for the analysis described 
below.)  In all fairness, I should point out that the CASA filler suffered 
an almost identical pathology until recently because we reorder the 
correlations for all baselines from [RR,LL,RL,LR] to [RR,RL,LR,LL] (to 
make the matrix algebra appear more conventional; alas, with some expense 
in filling performance).  In our "correlation flip", ironically, we were 
making the corresponding mistake in the nominal sensitivity application. 
(Some degree of familiarity with this effect was, indeed, a helpful aid in 
the present diagnosis!)

NOW, the upshot for polarization calibration and imaging is that none of 
the conventional calibration methods compensate for scaling 
errors--effectively, an a priori mis-calibration--which occur _only_ in 
the cross-hands and on only _some_ of the baselines.  This is essentially 
a very selective cross-hand-only closure error---exactly what Rick 
reported for his 3C273 observations (filled with nom sens scaling turned 
ON).  Note that when Tsys is correctly applied to all correlations, the 
ordinary gain calibration methods can usually correct for (modest) errors 
in the Tsys estimates used in the initial scaling, and the cross-hands 
benefit implicitly from this recovery.  Not so here.  The present errors 
are invisible to the parallel hand calibration, and persist throughout the 
processing (likely doing unknown and unfathomable harm to the D-term 
calibration!), and showing up in the "calibrated" data.

As a point of reference, in my CASA/AIPS comparison (C-band, B-array, one 
scan on 3C48, POLCAL data from December 2007, on a baseline requiring this 
now infamous flip), I observed (pseudo-) Q and U phases (calculated from 
uncalibrated data) systematically in error by a 5-10 degrees relative to 
the same quantities formed from correctly scaled data.  Since nothing in 
the calibration method can fix this (and the D-term calibration might, in 
fact, aggravate it), I am not at all surprised by ~30 degree phase errors 
in Rick's Q and U "calibrated" visibilities for the worst offenders.

But we can actually characterize the phase errors more precisely. 
Nominally, the relevant quantity is the ratio of the cross-hand nominal 
sensitivity factors (sqrt of the ratio of the cross-polarization Tsys 
products).  Looking at the data weights, these ratios apparently can 
differ from unity by several 10s of percent.  (NB: it is not the _true_ 
Tsys that is relevant here, but rather the _reported_ Tsys, which might be 
significantly in error---indeed, Rick opted to fill simple correlation 
coefficients in earlier polarization observations to avoid the unwelcome 
effects of spurious Tsys.)  Note especially that this ratio affects RL and 
LR in the opposite sense, so the cross-hand scaling imbalance is, in fact, 
the _square_ of this ratio.

Now, the formation of Q and U involve delicately balanced sums of the 
cross-hands.  Formally:  Q = (RL+LR)/2 and U = (RL-LR)/(2i).  So, if 
RL=Q+iU and LR=Q-iU (point-like polarizaiton at the phase center), we only 
get Q back identically from the crosshands if the (RL+LR)/2 sum balances 
to make the imaginary part cancel (or, by accident, if U=0).

If the ratio described above is 'a', our "Q" estimate is, in general:

     "Q" = (a*RL + LR/a)/2 = (a+1/a)*Q/2 + i(a-1/a)*U/2

                         ( = Q  iff a=1 )

The phase of this Q estimate is, in general:

               atan( (U*(a^2-1)) / (Q*(a^2+1)) )

This is zero for a=1, as expected.  It can be arbitrarily large, even for 
modest non-unity a, if U >> Q.  E.g., for a=1.1 and (true) U/Q=6 (quite 
plausible), the phase error is 30 degrees.

Viola!

Cheers,
-George