Gain Types

This page details the various different gain types which QuartiCal can solve for and includes some tips for ensuring you get good results.

Amplitude - amplitude

This solves for amplitudes given gains of the following form (in the case of linear feeds, but the same is true for circular feeds):

\[\begin{split}\mathbf{G} = \begin{bmatrix} A^{XX} & 0 \\ 0 & A^{YY} \end{bmatrix}\end{split}\]

where \(A\) is a real-valued, positive amplitude.

Complex - complex

This solves for gains of the following form (in the case of linear feeds, but the same is true for circular feeds):

\[\begin{split}\mathbf{G} = \begin{bmatrix} g^{XX} & g^{XY} \\ g^{YX} & g^{YY} \end{bmatrix}\end{split}\]

Note

  • This term contains amplitude, phase and leakage information and may not be appropriate in the absence of a polarised model.

Crosshand Phase - crosshand_phase

This solves for the crosshand phase given gains of the following form (in the case of linear feeds, but the same is true for circular feeds):

\[\begin{split}\mathbf{G} = \begin{bmatrix} e^{i\theta} & 0 \\ 0 & 1 \end{bmatrix}\end{split}\]

where \(\theta\) is the crosshand phase.

Note

  • This term requires four correlation data.

  • This term requires an excellent polarisation model.

  • This term must be solved over the entire array rather than per antenna.

Delay - delay

This solves for the delays given gains of the following form (in the case of linear feeds, but the same is true for circular feeds):

\[\begin{split}\mathbf{G} = \begin{bmatrix} e^{i 2\pi d_{XX}(\nu - \nu_c)} & 0 \\ 0 & e^{i 2\pi d_{YY}(\nu - \nu_c)} \end{bmatrix}\end{split}\]

where \(\nu\) is the frequency of a particular channel, \(\nu_c\) is the central frequency and \(d\) is the delay.

Note

Delay solutions support the initial_estimate parameter. If specified, this will initialise the delays using the Fourier transform.

Warning

Solving for a delay is very difficult if the phases are not approximately aligned. Thus, it is recommended to to solve for residual delay errors after applying a term which will approximately align the phases. This can be accomplished using e.g. solver.terms="[G,K]" where G is a diag_complex term with long solution intervals.

Delay and Offset - delay_and_offset

This solves for the delays and offsets (means) given gains of the following form (in the case of linear feeds, but the same is true for circular feeds):

\[\begin{split}\mathbf{G} = \begin{bmatrix} e^{i (2\pi d_{XX}(\nu - \nu_c) + \theta_{XX})} & 0 \\ 0 & e^{i (2\pi d_{YY}(\nu - \nu_c) + \theta_{YY})} \end{bmatrix}\end{split}\]

where \(\nu\) is the frequency of a particular channel, \(\nu_c\) is the central frequency, \(d\) is the delay and \(\theta\) is some mean phase offset.

Note

Delay solutions support the initial_estimate parameter. If specified, this will initialise the delays using the Fourier transform.

Warning

Solving for a delay is very difficult if the phases are not approximately aligned. Thus, it is recommended to to solve for residual delay errors after applying a term which will approximately align the phases. This can be accomplished using e.g. solver.terms="[G,K]" where G is a diag_complex term with long solution intervals.

Diagonal Complex - diag_complex

This solves for gains of the following form (in the case of linear feeds, but the same is true for circular feeds):

\[\begin{split}\mathbf{G} = \begin{bmatrix} g^{XX} & 0 \\ 0 & g^{YY} \end{bmatrix}\end{split}\]

Note

  • This term contains amplitude and phase information but does not not incorporate leakage information. This makes it appropriate for the majority of use-cases.

Leakage - leakage

This solves for gains of the following form (in the case of linear feeds, but the same is true for circular feeds):

\[\begin{split}\mathbf{G} = \begin{bmatrix} 1 & d^{XY} \\ d^{YX} & 1 \end{bmatrix}\end{split}\]

where \(d\) is a complex-valued quanitity which descibes the leakage.

Phase - phase

This solves for the phase given gains of the following form (in the case of linear feeds, but the same is true for circular feeds):

\[\begin{split}\mathbf{G} = \begin{bmatrix} e^{i\theta_{XX}} & 0 \\ 0 & e^{i\theta_{YY}} \end{bmatrix}\end{split}\]

Rotation - rotation

This solves for gains of the following form (in the case of linear feeds, but the same is true for circular feeds):

\[\begin{split}\mathbf{G} = \begin{bmatrix} \cos{\theta} & -\sin{\theta} \\ \sin{\theta} & \cos{\theta} \end{bmatrix}\end{split}\]

where \(\theta\) is some unknown angle.

Note

  • This term is only applicable to four correlation data.

  • Solving for this term requires a polarised model.

Warning

This solver is highly experimental. Any problems should be reported via the issue tracker.

Rotation Measure - rotation_measure

This solves for gains of the following form (in the case of linear feeds, but the same is true for circular feeds):

\[\begin{split}\mathbf{G} = \begin{bmatrix} \cos{(\lambda^2\mathrm{RM})} & -\sin{(\lambda^2\mathrm{RM})} \\ \sin{(\lambda^2\mathrm{RM})} & \cos{(\lambda^2\mathrm{RM})} \end{bmatrix}\end{split}\]

where \(\lambda\) is the wavelength in a particular channel and \(\mathrm{RM}\) is an estimate of the rotation measure.

Note

  • This term is only applicable to four correlation data.

  • Solving for this term requires a polarised model.

Warning

This solver is highly experimental. Any problems should be reported via the issue tracker.

TEC and Offset - tec_and_offset

This solves for the differential TEC values and offsets (means) given gains of the following form (in the case of linear feeds, but the same is true for circular feeds):

\[\begin{split}\mathbf{G} = \begin{bmatrix} e^{i (2\pi t_{XX}(\nu^{-1} + \frac{\log(\nu_{min}) - \log(\nu_{max})}{\nu_{max} - \nu_{min}}) + \theta_{XX})} & 0 \\ 0 & e^{i (2\pi t_{YY}(\nu^{-1} + \frac{\log(\nu_{min}) - \log(\nu_{max})}{\nu_{max} - \nu_{min}}) + \theta_{YY})} \end{bmatrix}\end{split}\]

where \(\nu\) is the frequency of a particular channel, \(\nu_{min}\) is the smallest frequency, \(\nu_{max}\) is the largest frequency, \(t\) is the differential (not absolute) TEC and \(\theta\) is some mean phase offset.

Warning

This solver is highly experimental. Any problems should be reported via the issue tracker.