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# Basic Information on gpdof


Purpose: Compute Degrees-of-freedom for calibration solutions.
Categories: calibration

GPDOF estimates the likely error in a calibration solution, by
calculating the errors in the "Degrees-of-Freedom" (dofs) of an
interferometer. This is useful for calibration of high-precision
circular polarization observations.

What GPDOF does is to determine the average calibration solution
of several input solutions. Then, assuming that this average
solution is "ideal", GPDoF determines the error in each of the
input calibration solutions, using the degrees-of-freedom
notation. Finally, GPDoF calculates the std and standard error
of the derived dofs, which you can use to estimate the errors in
the Stokes V of a target observation.

The input calibration solutions should have been processed using
the method outlined in the "Circular Polarization User's Guide"
(see References below). Obviously, the more calibrators you
have, the more accurate the dof estimates will be.

Explanation:

The degrees-of-freedom (dofs) are linear combinations of leakage
(delta) or gain (gamma) errors. The advantages in the dofs over
other measures of calibration consistency (eg. rms scatter in
the leakages) is that the dofs actually describe how the
calibration error would affect a target observation.  Thus, for
the dofs delta-+, delta--, and gamma--, and equatorially-mounted
antennas:

Stokes V error = delta-+ * Stokes I +
delta-- * Stokes Q +
gamma-- * Stokes U

For the ATCA, the parallactic angle \chi comes, in so it
isn't quite so elegant:

Stokes V error = delta-+ * I +
delta-- * ( Q.sin(2\chi) + U.cos(2\chi)) +
gamma-- * ( U.sin(2\chi) + Q.cos(2\chi))

Note that these are the instantaneous calibration errors in
a target source; for a synthesis observation, where \chi
varies, the resultant error in the Stokes V image will be
more complicated.

GPDoF estimates the std and standard error in the dofs. These
parameters be combined with the above equation for Stokes V
Error to estimate the likely level of leakage errors in a target
observation. If you are just using the solution from a single
calibrator, calculate using the std. If you are going to use the
average calibration solution (using GPComb), then calculate
using the standard error.

For each dof, GPDoF has an "error" estimate, which measures the
variations in the calibration solutions which are NOT
attributable to a dof. The physical significance of these
quantities isn't even clear to the author. In general, however,
if the "error" term is dominant, then un-dof-like errors are
dominating the calibration eg. pure random errors, or bizzarro
systematic errors. In this situation, the relationship between
the dofs and the error in the target observation may break down.

Um, I did mention that there are actually seven dofs? GPDoF only
computes the three which affect calibration of circular
polarization.  There are some good reasons for this (some of the
others can't easily be computed, or are time-variable), besides
the obvious one that I'm too lazy.

Note that GPDoF is entirely ATCAcentric.

Note that the leakage errors derived from GPDoF are not the only
errors in a circular polarization observation! See Equation 1 of
Rayner et al, 2000. A task to estimate the effect of the dofs on
a synthesis observation will hopefully become available
soon. Actually, forget the "soon"...

And finally, if you have consistent systematic errors in your
solutions (eg. you haven't used xyref,polref in gpcal), then
GPDoF will severely underestimate the leakage errors.

References:

For an explanation of degrees of freedom, see:

Sault, R J, "The Hamaker-Bregman-Sault Measurement Equation",
pp 657--699, in Syhthesis Imaging in Radio Astronomy II, 1999,
eds. Taylor, Carilli,and Perley.

or

Sault, Hamaker and Bregman, "Understanding radio polarimetry II.
Instrumental calibration of an interferometer array", 1996
A&AS, v117, pp149-159.
ftp://ftp.atnf.csiro.au/pub/people/rsault/papers/polar2.ps.gz

For an overview of high-precision circular polarization
calibration with the ATCA, see

Rayner, "Circular Polarization User's Guide", ATNF Technical
Document Series, 2000, 39.3/102,
http://www.atnf.csiro.au/people/drayner/Publications/
index.html#Circular_Polarization_Users_Guide

For a summary of the ATCA circular polarization error budget,
see Equation 1 of

Rayner, Norris, Sault, "Radio circular polarization of active
galaxies", 2000, v319, pp484-496
http://www.atnf.csiro.au/people/drayner/Publications/mnr3854.pdf

Key: cal
The data-sets containing the nominally correct polarization
calibration. No default.

Key: select
Normal uv selection. Only antenna-based selection is supported.


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