[evlatests] D*P contributions to total intensity

[George Moellenbrock]

> 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