[evlatests] D*P contributions to total intensity

[Rick Perley]

    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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