[evlatests] D*P contributions to total intensity

[Barry Clark]

But if the application of the D terms is done correctly, it does
reduce things to orthogonal polarizations, though not necessarily
R and L.  Application of the D terms means that cross terms for an
unpolarized source are zero, which is the very definition of a set
of orthogonal polarizations.  So the sum of the two parallel hands
is I in that system.  But the calibration is important; the two
parallel hands need to give the same value for an unpolarized source.
I agree if what you were trying to say is that adding the two parallel
hands before applying the D terms gives the wrong answer, as you have
always claimed, attributing to this the lack of dynamic range.  But
my claim is that what matters is that the D terms are applied correctly,
not just to the crossed hand products, and then it doesn't much matter
what you use for a reference for the D terms.  There is a modest
advantage in convenience of calibration to use the D terms that reduce
to R and L, and a modest advantage, as Walter pointed out, that the beam
dependence of polarization then has more nearly circular symmetry, but
there is no fundamental advantage to reducing to R and L.

Rick Perley wrote:
>     We need to separate calibration from imaging in this discussion.  
> The point I was trying to make (originally) was that for *real* antennas 
> with imperfect feeds (D .ne. 0) which do not happen to be orthogonal, we 
> can't simply add the RR and LL visibilities (no matter how well 
> calibrated) and expect to get a good value for the I visibility.  There 
> is no doubt that if the antennas polarization ellipses are strictly 
> orthogonal, (both R and L ellipses have equal ellipticity and are 
> orthogonal), and the parallel hand visibilities are correctly 
> calibrated, then adding the two will give the correct I visibility, 
> independent of what Q and U are. 
> 
>     But that's not real life.  Our antennas aren't orthogonal in their 
> polarization states.  So when we add RR and LL to get the I visibility, 
> the Q and U visibilities will not cancel, and we'll end up with a 
> polluted estimate.  In those situations where the I visibility is low 
> due to visibility cancellation in the interferometer pattern, the Q and 
> U visibilities need  not suffer the same cancellation, and can (and, I 
> claim, will!) contribute mightily to the output.  To get the true I, we 
> have to account for this leakage of Q and U into I.  This is best done 
> by proper inversion of the complex Mueller matrix -- requiring (I fear) 
> knowledge of the true ellipticities. 
> 
>     What remains to be shown (at a minimum) is if we utilize the 
> 'relative' D's in place of the 'real' ones, will that be sufficient?  I 
> doubt it (at least for some super-precise high-fidelity imaging).  
> 
>     I think we're all in agreement that some interative method will be 
> needed to separate Gs from Ds. 
> 
> George Moellenbrock wrote:
>>> I think you just agreed with me, in the end, that if we have a system
>>> of two orthogonal polarizations, and the observation is properly
>>> calibrated, I is given by the sum of the two "parallel hand" products.
>>>     
>> Right.  Rick replaced Barry's a's and b's with sines and cosines
>> of feed-related angles, and then assumed a quite peculiar polarization
>> state for the incoming radiation (pure +V) that simplifies (obscures?)
>> the range of cancellations that do indeed occur when summing the parallel
>> hands to get Stokes I for an orthogonal basis.   I find this a poor
>> choice for demonstrating anything general, not least because it
>> forbids calibrating the L gain (for which the model is zero!)....
>>
>> That 2 orthogonal states sum nicely to Stokes I was never at
>> issue.  I think the critical question is whether or not applying
>> _relative_ non-orthogonal D-terms obtained from the cross-hands
>> in the linear approximation to the parallel hands brings us to a
>> globally orthogonal basis (presumably impure at the level of the
>> unknown D-term offset) that would yield so neatly to symmetry.
>>
>> Barry asserted as much, but I think this can be shown to be
>> strictly true only if you started out in an orthogonal basis.
>> Applying offset (relative) Ds as if absolute introduces an
>> on-diag gain residual that you should reconcile with the existing
>> gain calibration (which, incidentally, also saw the
>> non-orthogonality, and is surreptitiously "storing" some
>> of that info).    When you do this, you find that the
>> gain correction is polarization-dependent (different in "R" and
>> "L") when the original basis was non-orthogonal and when
>> factored from the putatively orthogonal residual instr. pol
>> terms, you get a strictly non-orthogonal
>> D-term residual.   I.e., the resulting net basis is apparenly
>> not orthogonal.  The level of non-orthogonality is small--
>> something like a factor 1/(1-D*c) applied to c, where c is the
>> symmetric offset in the Ds from absolute, but remember that c can be
>> as large as any of the actual Ds (currently 0.1 or worse
>> in the worst cases), depending on how the Ds are referenced.
>> Nonetheless, I think it would be profitable to optionally
>> turn on the general correction even for relative Ds.  It would
>> be interesting to see how well things actually balance.
>>
>>   
>>> polarized.  However, I think calibration can be done at the required
>>> level if, as George suggested, you iterate between calibration and D
>>> term determination for a cycle or two.
>>>     
>> Yes, iteration will help in the G/D decoupling exercise.  But note
>> that this is required even when the Ds are nominally absolute
>> (e.g., VLBA), so long as we insist on pretending they are not
>> coupled at the outset.  You'll want to revise G using the general
>> D so that the net calibration is consistent with the data and model.
>> That we get only relative Ds for the VLA is merely an added
>> complication--another term to project out somehow if
>> we want to reach the highest precision.  Since it is desirable
>> to correct both the parallel- _and_ cross-hand data  well, I remain
>> inclined to invest more effort in the general solution and not
>> have to count on carefully-but-only-approx-balanced terms to
>> cancel in (weighted!) sums that occur downstream.
>>
>> Incidentally, Barry also suggested using the symmetry of the
>> primary beam polarization as a constraint.  I don't see
>> how this does anything other than constrain the primary
>> beam's contribution.  In the linear approximation, the
>> on-axis and off-axis (zero at the on-axis point) can be
>> applied serially and the net effect is that both are subtracted
>> from the cross-hands, to first order.  But one doesn't constrain
>> the other---in fact, one (off-axis) is solved relative to the
>> other, I believe.   If you want to apply both of these generally
>> (non-linearized), you have to worry about what shows
>> up on the diagonal in their product, in a manner similar
>> to what is described above when correcting an absolute
>> D with a relative one.  But for every pixel....
>>
>> -George
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