[evlatests] D*P contributions to total intensity

Tue Jul 27 11:45:43 EDT 2010

Barry-

To be clear, I DO mean applying the D-terms also to the parallel
hands.   When I carry out the formal algebra (correcting an absolute
D matrix with an offset one), _and_ include reconciliation of the
ordinary gains, I get a residual Jones matrix that has (slightly!)
asymmetric off-diag elements.   I.e., it appears non-orthogonal.
And, yes, such a net calibration matrix (incl. the G reconciliation)
will not quite correct the cross-hands to for an unpolarized source
to precisely zero.

I think this is mostly a quibble about the way we factor the gains
from the general Jones matrix, and boils down to how we define
"correctly applying" the D-matrix.   I'll spend some more time
trying to work this out.

In the mean time, I do think it is worth applying linearized Ds
to the parallel hands, and I've begun taking some steps in
CASA to permit this.

-George

On Tue, Jul 27, 2010 at 8:57 AM, Barry Clark <bclark at nrao.edu> wrote:
> 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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