# Data Weights

A detailed description of visibility weighting

Visibility weight initialization and calibration has undergone several improvements in CASA 4.2.2 and CASA 4.3. This appendix briefly describes the formal weight definitions, and the changes occurring in these CASA versions.  If data sets shall be combined that were reduced with different CASA versions, the weights may need to be adjusted accordingly. This can be achieved, e.g. by running the same version of statwt on all datasets before combination. The best option, however, is to use a single CASA version for all reductions, preferrably 4.2.2 or later.

NOTE: Post-calibration weights, e.g. imaging weights or tapers are not covered here.

# The SIGMA and WEIGHT columns

Formally, in CASA 4.2.2 and later, the SIGMA column in the measurement set will reflect the per-channel noise of the DATA column as it depends on the channel bandwidth $\Delta \nu$ and the length of an integration $\Delta t$:

$SIGMA = \frac{1}{\sqrt{2\Delta \nu \Delta t}}$

The factor of $\sqrt{2}$ is for cross-correlations only and auto-correlation data follows:

$SIGMA = 1/\sqrt{\Delta \nu \Delta t}$.

SIGMA will only be updated if the time and channel widths are modified along with any DATA column manipulation, e.g. through averaging, binning, smoothing, etc. (tasks like mstransform, cvel, split, exportuvfits, etc).

The WEIGHT column reflects how much weight each CORRECTED_DATA sample should receive when data are combined (e.g., in averaging). To start with, WEIGHT is initialized from the SIGMA column via:

$WEIGHT = \frac{1}{{SIGMA}^2} = 2 \Delta \nu \Delta t$

Data calibration by applycal with calwt=True will calculate and modify the WEIGHT values but not SIGMA. Calibration applies multiplicative factors and the WEIGHT of a visibility on a baseline between antennas $i$ and $j$ is calculated via:

$WEIGHT_{ij}=\frac{c_i c_j}{{SIGMA}_{ij}^2}$

where $c_i$ and $c_j$ are the net antenna-based power calibration factors derived by applycal ($c_i=c_j$ for auto-correlation data). In the table below, we list the definitions of antenna-based $c$ for different calibration procedures and CASA versions. When more than one calibration is applied, the product of the relevant weight factors is used.

$<=$CASA 4.2.1 CASA 4.2.2/4.3 $>=$CASA 4.4 (WEIGHT_SPECTRUM)
Initialization $1.0$ $2 \Delta \nu \Delta t$ $2 \Delta \nu \Delta t$
System Temperature $\frac{1}{<\sqrt{T_{\rm sys, k}}>_{k}^{2}}$ $\frac{1}{<T_{\rm sys, k}>_k}$ $\frac{1}{T_{\rm sys, k}}$
Gain $||G||^2$ $||G||^2$ $||G||^2$
Bandpass $\frac{1}{<||B||^{-1}>_{k}^{2}}$ $<||B||^{2}>_{k}$ $<||B||^{2}>_{k}$

# Weights in CASA 4.2.1 and Earlier

The SIGMA and WEIGHT columns are initialized with values of $1.0$.  Traditionally, this convention was adequate for datasets with uniform sampling in time in frequency; a global weight scale factor would not affect calibration and imaging fidelity. In data manipulation operations (e.g., split, etc.), SIGMA was treated as a per-channel value and WEIGHT as a per-spw (all channels) weight. Combined with unit initialization, this difference in definition could lead to incongruent weight scales for different spectral windows, in particular if bandwidth and channel count varied. CASA 4.2.1 is therefore not recommended for datasets which have variety in spectral window bandwidth and channelization and for which spectral windows are to be combined in imaging.

# Weights in CASA 4.2.2

In CASA 4.2.2 the SIGMA and WEIGHT columns are properly initialized via the definition in the above equations. Both are defined as per-channel values. Also, the weight calibration factors have been subtly updated to improve robustness, as indicated in the Table.

# Weights in CASA 4.3

In CASA 4.3 frequency variations of the WEIGHT and SIGMA values are (optionally) captured in additional WEIGHT_SPECTRUM and SIGMA_SPECTRUM columns. This allows accommodation of variations of effective sensitivity on a channel by channel basis (e.g. band edges, atmospheric lines, spectral $T_{\rm sys}$ variation etc.). WEIGHT_SPECTRUM will be recognized in the applycal task as well as in mstransform and clean. Calibration solvers, however, will not yet calculate and modify WEIGHT_SPECTRUM.

# Weights in CASA 4.4 and later

Full support of WEIGHT_SPECTRUM.