Calibration¶

Calibration is necessary to remove instrumental effects on the measured visibilities prior to imaging. The process of calibration can generally be divided into two categories:

1. Direction independent calibration
2. Direction dependent calibration

Direction independent calibration attempts to remove instrumental effects arising from the electronics between the antenna and the correlator. For example amplifiers and cables can add to the amplitude and phase of the voltage respectively. Whereas direction dependent calibration attempts to remove instrumental effects arising from the antenna beam pattern and propagation through the ionosphere.

Both types of calibration can be viewed as trying to solve for best Jones matrices $J$ in the following equation

$$V_{ij}^\text{measured} = J_i V_{ij}^\text{model} J_j^* + \text{noise},$$

where $V_{ij}$ is a 2x2 matrix containing the xx, xy, yx, and yy correlations between antenna $i$ and antenna $j$, and $J_i$ is the Jones matrix associated with antenna $i$. In direction independent calibration the Jones matrices are a function of frequency only. In direction dependent calibration the Jones matrices are a function of frequency and direction.

Iterative Least Squares¶

TTCal uses the method of iterative least squares independently described by Mitchell et al. 2008 and Salvini & Wijnholds 2014 to solve for the Jones matrices. Given a set of model visibilities we seek to minimize

$$\sum_{ij} \left\| V_{ij}^\text{measured} - J_i V_{ij}^\text{model} J_j^* \right\|^2$$

If $J_i$ is assumed to be constant, the least squares solution for $J_j^*$ is

$$J_j^* = \frac{\sum_i \left(J_i V_{ij}^\text{model}\right)^* V_{ij}^\text{measured}} {\sum_i \left\|J_i V_{ij}^\text{model}\right\|^2}.$$

This can be computed rapidly but iterating will tend to oscillate around the solution to the original optimization problem. These oscillations need to be damped for rapid convergence. TTCal damps these oscillations by using Runge-Kutta steps.

Gain Calibration¶

With gain calibration TTCal will solve for a set of diagonal Jones matrices (one per antenna and frequency channel). The diagonal terms of the Jones matrix represent the complex gain (amplitude and phase) of each antenna.

See the cookbook for an example gain calibration on an LWA dataset.