CASA, the Common Astronomy Software Applications for Radio Astronomy

CASA accepts a variety of configuration options, specified either through configuration files or command line arguments. A configuration file config.py can be edited prior to starting a CASA session, which can be used to set reference data paths, pipeline installation, options for display of GUIs, and more. In addition, a startup.py file can be used to customize the CASA shell, which provides the environment for interactive Python-based data analysis using CASA tasks and tools, as described in Section 2.3.2. The startup.py file allows setting paths, importing Python packages, and executing other Python code to prepare the session.

CASA tools are Python objects that use C++ functions to perform operations on the data. The collection of tools contains a large part of the functionality of CASA, but each tool in itself performs a separate low-level operation. This means that procedures to achieve a typical data analysis objective, such as performing a bandpass calibration, typically need many calls to different tools. To make CASA more user-friendly, higher-level CASA tasks are offered, which are constructed based on the tools and cover a comprehensive set of use-cases for data processing and analysis. A CASA task performs a well defined step in the processing of the data, such as loading, plotting, flagging, calibrating, imaging, or analyzing the data. Each task has a well defined purpose and set of input parameters, which may have a two-layer hierarchy containing sub-parameters.

While CASA tools are Python class objects with methods, CASA tasks can be called as a function with one or more arguments specified, which allows for robust scripting and pipeline use in Python. Where possible and useful, parameters have a specified default value. CASA tasks can also be controlled by setting global parameters. These can be inspected in the terminal via the inp command, before executing the task by typing go. Throughout this paper, we will refer to the names of CASA tasks and tools in bold , and task parameters in italic. For an overview of the tasks and tools that are available in CASA, see CASA Docs. ¹²

CASA also contains a variety of applications with Graphical User Interfaces (GUIs) to examine visibility data, image products, and meta-data. Two of the most widely used GUIs are plotms , for diagnostic plotting of visibility and calibration-table data, and the CASA viewer , for visualizing image products. More information on data analysis and the use of GUIs in CASA is provided in Section 3.6.

Each CASA version comes with a minimal repository of binary data that is required for CASA to function properly. In particular the CASA Measures system, which handles physical quantities with a reference frame and performs reference frame conversions, requires valid Earth Orientation Parameters (EOPs) and solar system object ephemerides. These are contained in the "Measures tables." The EOP tables are updated daily by ASTRON, ¹³ based on the geodetic information from services like the International Earth Rotation and Reference Systems Service (IERS). ¹⁴ The ephemeris tables are copied over from the Horizons online solar system data and ephemeris computation service at the Jet Propulsion Laboratory (JPL). ¹⁵ Other data that is updated less frequently is also stored in the CASA data repository, such as observatory-specific beam models, correction tables for Jy/K conversion, and files that specify the antenna array configurations for the CASA simulator.

The CASA data repository, and the runtime data therein, is necessarily updated frequently compared to the much sparser schedule by which new CASA versions are released. Operations on recently observed data may need an up-to-date data repository to work correctly. Upon starting a CASA session, it may therefore be necessary to update either the Measures tables or the entire data repository with the most recent version. Routines for handling these external data dependencies are readily available in CASA.

Calibration is the process of correcting the signals measured with an interferometer or single-dish radio telescope for instrumental and environmental propagation factors that corrupted the desired astronomical signal, and transform the data from instrumental units to absolute standard units. This is necessary for making an accurate image of the sky, which can be related to images made by other telescopes and theoretical predictions. In CASA, the calibration process consists of determining a series of complex correction factors and applying these corrections to the visibility measurements of objects of scientific interest. Calibration solutions are typically derived from observations of well-characterized calibrator sources, often radio-bright and unresolved quasars for which a simple visibility model may be assumed, and occasionally solar system objects for absolute flux calibration. In cases where a source in the target-field is bright enough, calibration solutions can often also be obtained or improved through the iterative process of self-calibration, using this target source as the de facto calibrator source (e.g., Pearson & Readhead 1984; Wilkinson 1989; Cornwell & Fomalont 1999).

The MeasurementSet structure in CASA is designed to permit interferometric data to be calibrated following the Measurement Equation (Sault et al. 1996a; Hamaker et al. 1996; Noordam 1996). The Measurement Equation is based on the fact that the visibilities measured by an interferometer have been corrupted by a sequence of multiplicative factors, arising from the atmosphere, antennas, electronics, correlator, and downstream signal-processing. For visibility calibration in CASA, the general Measurement Equation (Sault et al. 1996a; Hamaker et al. 1996) can be written as:

$$\begin{eqnarray}&&{{\boldsymbol{V}}}_{{ij}}={J}_{{ij}}\,{{\boldsymbol{V}}}_{{ij}}^{\,\mathrm{TRUE}},\end{eqnarray} \tag{ 1 }$$

where V _ij represents the observed visibility, which is a vector of complex numbers that represents the amplitude and phase of the raw correlations formed among the sampled dual-polarization wavefronts, as received by a pair of antennas (i and j) at each time sample and per spectral channel. ${{\boldsymbol{V}}}_{{ij}}^{\mathrm{TRUE}}$ represents the corresponding true visibilities, in a nominally perfect polarization basis (typically either linear or circular), that are to be recovered by the calibration process, and which are proportional to combinations of the true Stokes parameters of the source visibility function (Stokes 1852). J_ij is an operator that represents an accumulation of all corruption factors affecting the correlations on baseline i − j. As written here, J_ij is a Mueller matrix for baseline i − j, formed in most cases from the outer product of a pair of (single-index) antenna-based Jones matrices, J_i :

$$\begin{eqnarray}&&{J}_{{ij}}={J}_{i}\otimes \,{J}_{j}.\end{eqnarray} \tag{ 2 }$$

Each antenna-based Jones matrix, J_i , models the propagation of a signal from a radiation source through to the voltage output of antenna i, and characterizes the net effect on the resulting correlations (Jones 1941; Mueller 1948; Heiles et al. 2001). Factoring J_ij in Equation (1) into the most common series of recognized effects, we have:

$$\begin{eqnarray}&&{{\boldsymbol{V}}}_{{ij}}={M}_{{ij}}\,{K}_{{ij}}\,{B}_{{ij}}\,{G}_{{ij}}\,{D}_{{ij}}\,{E}_{{ij}}\,{P}_{{ij}}\,{T}_{{ij}}\,{F}_{{ij}}\,{{\boldsymbol{V}}}_{{ij}}^{\,\mathrm{TRUE}},\end{eqnarray} \tag{ 3 }$$

with the individual terms described in detail below. For just the antenna-based calibration terms, we can write:

$$\begin{eqnarray}&&{J}_{i}={K}_{i}\,{B}_{i}\,{G}_{i}\,{D}_{i}\,{E}_{i}\,{P}_{i}\,{T}_{i}\,{F}_{i}.\end{eqnarray} \tag{ 4 }$$

The order of terms in Equation (4) reflects the order in which the incoming signal encounters the corrupting effects (from right to left), and in general, terms cannot be arbitrarily reordered, i.e., they do not all algebraically commute. In practice, not all terms need to be considered in all cases, and some (or even most) may be ignored depending upon their relative importance with respect to dynamic range requirements and scientific goals. In aggregate, the calibration terms in the Measurement Equation describe a net effective (imperfect) polarization basis, characterizing the departure from the ideal intended polarization basis of the instrument. The internal algebra and properties of particular calibration terms in some cases depend on the nominal basis, as will the heuristics engaged to solve for or calculate them, and thus also their implementations in CASA. However, the form of the equation as expressed here represents an approximation to the sequence of physical effects encountered by the observed wave front, and is therefore polarization basis-agnostic.

Solving for an antenna-based factor is generally an over-determined problem, since there are only N_ant antenna-based factors for N_ant(N_ant − 1)/2 baselines, per term. The specific calibration terms recognized within the CASA calibration model are as follows:

• 1.  
  F_i : ionospheric effects, including dispersive delay and Faraday rotation (e.g., de Gasperin et al. 2018). Most relevant at low radio frequencies (≲5 GHz), and typically estimated from information on the ionospheric total electron content.
• 2.  
  T_i : tropospheric effects, such as opacity and path-length variations (e.g., Hinder & Ryle 1971), which stochastically affect visibility amplitude and phase, respectively, in a polarization-independent manner. Relevant at all radio frequencies, but most important at higher frequencies, where tropospheric variations set the timescale for calibration. Typically solved from the visibility data themselves, on a calibrator observed periodically near the science target. Estimates may also be obtained from water vapor radiometry (WVR; Rocken et al. 1991; Emrich et al. 2009; Maud et al. 2017).
• 3.  
  P_i : parallactic angle rotation, which describes the orientation of the antennas' polarization in the coordinate system of the sky. Calculated analytically from observational geometry information.
• 4.  
  E_i : effects introduced by properties of the optical components of the telescopes, including gain response as a function of elevation and the net aperture-efficiency scale.
• 5.  
  D_i : instrumental polarization response, describing the polarization leakage between feeds. This factor describes the fraction of one hand of nominally received polarization that is detected by the receptor for the other hand, and vice versa. Solved from observations on an appropriately chosen calibrator.
• 6.  
  G_i : general time- and polarization-dependent complex gain response, including phase and amplitude variations due to the signal path between the feed and the correlator, and sometimes including tropospheric and ionospheric effects when not separately factored into F_i and T_i . Solved from the visibility data themselves (like T_i ), G_i includes the scale factor for absolute flux density calibration, by referring solutions to observations of a calibrator with known flux density. In some cases, this scale reconciliation is performed on T_i solutions.
• 7.  
  B_i : the amplitude and phase "bandpass" response introduced by spectral filters and other components in the electronic transmission of the signal. This is effectively a frequency channel-dependent version of G_i . Usually assumed to be stable in time, B_i is solved from calibrator observations with sufficient signal-to-noise ratio per frequency channel.
• 8.  
  K_i : general geometrically parametrized gain phases that are time- and frequency-dependent, such as delay and delay-rate. Includes antenna-position corrections and traditional fringe-fitting.
• 9.  
  M_ij : baseline-based correlator (non-closing) errors. When used with extreme care, baseline-based solutions can account for residuals not factorable as antenna-based errors, but this usually indicates a failure to fully and adequately model and calibrate more subtle antenna-based effects, such as instrumental polarization. Once invoked, subsequent antenna-based calibration cannot be further used reliably since the baseline-based term will have absorbed antenna-based information arbitrarily.

Occasionally, specializations of some of these terms are invoked for particular purposes. For example, system temperature calibration is implemented as a variation of bandpass calibration, position angle calibration is implemented as an offset to the P_i term, and cross-hand phase as an offset to the G_i term (but located in front of D_i ). In general, the modular implementation of calibration terms in CASA is designed to permit such specializations, where warranted. Additional corrections, such as for direction-dependent widefield or wideband effects, can optionally be invoked during the imaging stage (see Section 3.5.3). ¹⁹

In CASA, solutions for the calibration corrections can be derived using various calibration tasks and tools. The calibration solutions are then stored in separate calibration tables, in a system similar to the MeasurementSet. The most widely used tasks for deriving calibration corrections are:

• 1.  
  gaincal : solves for time- and optionally polarization-dependent (but channel-independent) variations in the complex gains ("phases" and/or "amplitudes"), i.e., G_i and T_i . Also used to derive rudimentary delay corrections, as a variation of K_i .
• 2.  
  bandpass : solves for frequency-dependent complex gains, i.e., B_i .
• 3.  
  fluxscale : applies a scale factor to the G_i or T_i gain solutions from gaincal , according to the gains derived from observations of the flux-density calibrator, on the assumption that the net electronic gain is stable with time. Normally the task setjy is used to set the model visibility amplitude and phase associated with a flux density scale prior to running gaincal and fluxscale .
• 4.  
  polcal : solves for instrumental polarization calibration factors, including leakage, cross-hand phase, and position angle corrections.
• 5.  
  fringefit : solves for delay- and rate-parameterized gain phases for cases where uncertain array geometry and/or distinct clocks limit coherence in frequency and time, as is common in Very Long Baseline Interferometry (VLBI; e.g., Wiesemeyer & Nothnagel 2011). Also supports a dispersive delay term. The implementation of fringefit is described in detail by van Bemmel et al. (2022).
• 6.  
  gencal : derives various calibration solutions from ancillary information stored in the MeasurementSet or otherwise specified or retrieved, including ionosphere, system temperature, antenna position corrections, opacity, gain curves, etc.
• 7.  
  applycal : applies all specified calibration to the MeasurementSet DATA column, according to the order prescribed in the Measurement Equation (Equation (3)), and writes out the CORRECTED_DATA column for imaging.

A range of other tasks and tools for calibration support are available in CASA, including plotweather to plot weather information and estimate opacities, plotbandpass for plotting bandpass information, wvrgcal for WVR-derived gains (Nikolic et al. 2012), and blcal to derive baseline-based gains.

Solving for calibration is a generalized bootstrapping process wherein each additional solved-for component is determined relative to the best available existing information for other calibration terms and the calibrator visibility model. Initially, there may be no prior calibration available (or only a few terms derived from ancillary information), and only point-like visibility models. As new calibration is derived, it may be desirable to revise terms that were solved for earlier in the process. For example, a solution for the bandpass, B_i , may be determined with a provisional time-dependent G_i solution from the same calibrator data (but derived from only a subset of frequency channels without the benefits of a prior B_i solution), and then the G_i solution will be revised and improved using the B_i solution as a prior and all frequency channels for more sensitivity. Algebraically, the prescribed order of calibration terms (4) is respected in all solves. When all relevant calibration terms have been solved-for, their aggregate is applied (with appropriate interpolation) to calibrator and science target data for imaging and deconvolution.

Insofar as the calibration derived from calibrator observations may not precisely characterize corruptions occurring in the science target data (due to time- and direction-dependence), it is often desirable to self-calibrate against the visibility model derived from the initial imaging and deconvolution process, and then re-image to better bootstrap and converge on an optimized net calibration and source model (e.g., Pearson & Readhead 1984; Wilkinson 1989; Cornwell & Fomalont 1999). In some cases, the calibrators may not be perfect point sources, and self-calibration may be used to jointly optimize their visibility model(s), and the calibration derived from them, before proceeding to the science target. The CASA implementation for visibility calibration is designed to support this generalized self-calibration ideal for the iterated revision and convergence of all calibration terms and source model estimates. In practice, the largest benefit is obtained by iterative revision of time-dependent gains, mainly due to the troposphere, and the visibility model of the source.

The concept of single-dish calibration is different from that of an interferometer. Atmospheric variability that causes phase decoherence in interferometric data is not relevant for single-dish observations, but large-scale atmospheric fluctuations that are resolved out by interferometers yield power in single-dish telescopes. Single-dish telescopes detect and quantify signals in brightness temperature T_B (Kelvin). The brightness temperature of an astronomical target can be measured through:

$$\begin{eqnarray}&&\displaystyle \frac{{T}_{\mathrm{target}}}{{T}_{\mathrm{sys}}}=\displaystyle \frac{{T}_{\mathrm{ON}}\,-\,{T}_{\mathrm{OFF}}}{{T}_{\mathrm{OFF}}},\end{eqnarray} \tag{ 5 }$$

where T_ON is the on-target measurement, and T_OFF is a measurement of the blank sky, taken at roughly the same elevation and time as the target measurement, but ideally absent of any target emission or contaminating sources at the frequencies of interest to the observer. The measured signal includes contributions from sky targets (for T_ON), cosmic background signals, atmosphere, ground, and the telescope itself, as well as noise from the instrument's electronics. T_sys is the system temperature, which is obtained through an observation of the sky together with measurements of loads with known temperatures that are placed in front of the receiver.

Typical observing modes include Position Switching, by which separate exposures are taken at discrete "ON" and "OFF" positions on the sky, or On-The-Fly mapping, where the telescope smoothly scans the field that contains the "ON" and "OFF" measurements (e.g., O'Neil 2002; Mangum et al. 2007; Sawada et al. 2008). In CASA, the calibration of single-dish data generally requires several steps, namely the application of the T_sys calibration, and of the "sky" calibration, i.e., T_OFF. The T values are a function of frequency, hence calibration must be performed in the spectral domain. In addition, in the spectral domain, single-dish data rely on accurate fitting and subtraction of the spectral baseline emission.

Single-dish data in CASA rely on the same MeasurementSet as interferometric data. For single-dish calibration, various CASA tasks can be used that are also suitable for interferometry, such as listobs , flagdata , gencal , and applycal . In addition, a number of dedicated single-dish tasks are available in CASA, including sdcal for calibration, sdgaincal for removing time-dependent gain variations, and sdbaseline for fitting and subtracting a baseline from single-dish spectra.

Image reconstruction in radio interferometry (Readhead & Wilkinson 1978; Sramek & Schwab 1989; Cornwell 1995; Frey & Mosoni 2009; Rau et al. 2009) is the process of solving the linear system of equations:

$$\begin{eqnarray}&&{{\boldsymbol{V}}}^{\mathrm{TRUE}}=[A]{\boldsymbol{I}},\end{eqnarray} \tag{ 6 }$$

where V ^TRUE represents visibilities that have been calibrated for direction independent effects through applycal , using the solutions from Equation (3). I is a list of parameters that model the sky brightness distribution, such as an image consisting of pixels. [A] is a measurement operator that encodes the process of how visibilities are generated when a telescope observes a sky brightness I , and is generally given by [S_dd ][F], such that

$$\begin{eqnarray}&&{{\boldsymbol{V}}}^{\mathrm{TRUE}}=[{S}_{{dd}}][F]{\boldsymbol{I}},\end{eqnarray} \tag{ 7 }$$

where [F] represents a 2D Fourier transform (Fourier 1878; Bracewell 2000) and [S_dd ] represents a 2D spatial frequency sampling function that can include direction-dependent instrumental effects. An interferometer has a finite number of array elements, which means that [A] is not invertible because of unsampled regions of the (u, v)-plane. Therefore, this system of equations must be solved iteratively, applying constraints via various choices of image parameterizations and instrumental models.

In CASA, this interferometric imaging process consists of converting a list of calibrated visibilities into a raw image, also called a "dirty" image, then "cleaning" this image to obtain an estimate of the true sky model by iterating through a few χ² minimization steps that evaluate the goodness of fit of the current model with respect to the visibilities, and subsequently constructing the final image product. Under ideal conditions, the dirty image is the true sky brightness convolved with the point-spread-function (PSF) of the instrument and added noise terms (Figure 3). The PSF typically has a complex shape determined by the (u, v)-coverage and interference pattern of the—often sparsely populated—distribution of antennas in the interferometer. Moreover, in reality, the dirty image is based only on the visibilities that are sampled by the interferometer, hence it does not contain the complete information about the true sky brightness distribution. There are several stages to forming a dirty image from interferometric data, including weighting, convolutional resampling, Fourier transformation, and normalization. To obtain the final "clean" image of the source, a model of the true sky brightness distribution is reconstructed from the dirty image and the PSF in a process called "deconvolution" (Cornwell & Braun 1989), and subsequently convolved with a Gaussian that represents the instrumental resolution specified by the main lobe of the PSF. In this Section, we explain those concepts of imaging, weighting, gridding, and deconvolution that are most sensitive to parameters that need to be specified by users in CASA. This includes an overview of how deconvolution and image restoration is implemented in the CASA imaging task tclean .

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As part of the imaging process, the visibilities can be weighted in different ways to alter the instrument's natural response function, affecting resolution, sidelobe suppression, and root-mean-square (rms) noise levels. There are three main weighting schemes for radio interferometric data: natural, uniform, and Briggs (robust) weighting (Briggs 1995).

• 1.  
  natural weighting: the data are gridded into (u, v)-cells for imaging, with weights given by the visibility weights in the MeasurementSet. This results in a higher imaging weight for higher (u, v)-density. Natural weighting produces an image with the lowest noise, but often with lower than optimal resolution.
• 2.  
  uniform weighting: the data are first gridded to a number of cells in the (u, v)-plane, and afterwards the (u, v)-cells are re-weighted to have uniform imaging weights. Uniform weighting produces an image with higher resolution, but which has increased noise compared to natural or briggs weighting. Also available in CASA is the option of superuniform weighting, which increases the number of cells to define the (u, v)-plane patch for the weighting renormalization, and by doing so further increases the resolution and noise.
• 3.  
  briggs (robust) weighting: this is a flexible weighting scheme that provides a trade-off between resolution and sensitivity (Briggs 1995). It uses a robustness parameter that takes values between −2.0 (close to uniform weighting) and 2.0 (close to natural weighting). CASA also includes two modified versions of the Briggs weighting scheme, named briggsabs and briggsbwtaper.
• 4.  
  uvtaper: optionally, a multiplicative Gaussian taper can be applied to the spatial frequency grid, in addition to any of the above options. This effectively downweights the longer baselines, decreasing the resolution compared to the original weighting scheme it is applied on top of, and increasing surface brightness sensitivity.

After visibilities are properly weighted, the next steps in the imaging process of interferometric data consists of gridding, Fourier transformation (Fourier 1878; Bracewell 2000), and normalization of the data. Imaging weights and weighted visibilities are first resampled using gridding convolution functions onto a regular (u, v)-grid in a process called convolutional resampling (Brouw 1975). The result is Fourier-inverted using a fast-Fourier transformation, and subsequently grid-corrected to remove the image-domain effect of the gridding convolution function.

In CASA, the type and shape of the image product is determined by various parameters, including data selection, definition of the spectral mode, and the gridder algorithm. Image products can be continuum images, spectral cubes, or Stokes images, while multiple pointings can be captured into a mosaic image. The standard gridding in CASA relies on prolate spheroidal functions, along with image-domain operators to correct for direction-dependent effects. More sophisticated gridder modes can apply direction-dependent, time-variable and baseline-dependent corrections during gridding in the visibility-domain, by computing the appropriate gridding convolution kernel to use along with the imaging-weights. These gridder modes are superior for certain data sets, like widefield and wideband data, but are more computationally intensive.

Typical imaging modes in CASA include:

• 1.  
  mfs, multi-frequency synthesis imaging: standard continuum imaging, where selected data channels across a (wide) range in frequencies are mapped onto a single wideband image (Conway et al. 1990).
• 2.  
  cube imaging: Standard imaging of spectral lines, where selected data channels are mapped to a number of image channels using various interpolation schemes. Optionally, the image cube can be corrected for Doppler effects and ephemeris tracking.
• 3.  
  stokes imaging: imaging of different types of polarization products, often related to Stokes I, Q, U, V, or combinations thereof (Stokes 1852; Hamaker & Bregman 1996).
• 4.  
  mosaic imaging: continuum or spectral-line imaging of observations that consist of multiple pointings, producing fields-of-view larger than the primary beam. Options for stitched and joint mosaics are available (e.g., Ekers & Rots 1979; Cornwell 1988; Bremer 1994; Condon et al. 1994; Sault et al. 1996b).
• 5.  
  mtmfs, multi-term multi-frequency synthesis imaging: wideband image reconstruction that produces Taylor-coefficient maps, which represent smoothly varying spectral structure across the field-of-view of the image (Rau & Cornwell 2011, see also Sault & Wieringa 1994; Likhachev 2005).
• 6.  
  imaging multiple (outlier) fields: large fields imaged as a main field plus multiple (smaller) outlier fields, rather than visibility data being gridded onto one large (u, v)-grid.
• 7.  
  correction for widefield and wideband instrumental effects: high dynamic range imaging of data with a large field-of-view and large fractional bandwidth, by accounting for a number of widefield and wideband effects during gridding. These effects include sky curvature, non-coplanar baselines (Cornwell & Perley 1992), and antenna-based aperture illumination functions that change with time, frequency, and polarization (Sekhar et al.2019).

A-Projection is an example of an advanced gridding algorithm in CASA, which can correct for complex widefield (Bhatnagar et al. 2008) and wideband (Bhatnagar et al. 2013) effects (see also Tasse et al. 2013). This can be critical for sensitive widefield continuum observations, for example those performed using the lower frequency bands of the VLA (e.g., Rau et al. 2016; Schinzel et al. 2019). The aperture illumination function results in a direction-dependent complex gain that causes the primary beam to vary with time, frequency, polarization, and antenna (Jagannathan et al. 2018; Sekhar et al. 2019). These variations may be caused by pointing errors, feed leg structures that break azimuthal symmetry, parallactic angle rotation, and varying dish sizes. These variations in the primary beam are corrected during the gridding using the wideband A-Projection algorithm, which computes gridding convolution functions for each baseline as the convolution of the complex conjugates of two antenna aperture illumination functions (Bhatnagar et al. 2013). In addition, sky curvature and non-coplanar baselines result in a so-called w-term that is not zero, which means that standard 2D imaging applied to such data will produce artifacts around sources away from the phase center. A W-Projection algorithm corrects for the effects introduced by the w-term, using gridding convolution functions that are computed based on the Fourier transform of the Fresnel electro-magnetic wave propagator for a finite set of w-values (Cornwell et al. 2008). The wproject, mosaic, and awproject gridding options implement different approximations and combinations of the A- and W-Projection algorithms.

Deconvolution refers to the process of reconstructing a model of the sky brightness distribution, given a dirty image and the PSF of the instrument. This process is called deconvolution, because under ideal conditions, the dirty image can be written as the result of a convolution of the true sky brightness and the PSF of the instrument (Figure 3). The concept of deconvolution is a widely used technique in signal and image processing, and explaining the fundamentals is beyond the scope of the current paper (see Cornwell & Braun 1989). Instead, we will give an overview of how the technique of deconvolution and image reconstruction is practically implemented in CASA through the task tclean .

Tclean, CASA's powerful imaging task—The CASA task used for imaging is tclean , which is based on the CLEAN algorithm (Högbom 1974; Schwarz 1978). The tclean task takes the calibrated visibilities from the MS and applies weighting, sampling, Fourier transformation, deconvolution, and image restoration according to inputs specified by the user or pipeline. The output of tclean is a reconstructed image of the astronomical source. In addition, tclean by default also produces various other images that may be useful for subsequent processing or analysis, including images of the derived sky model and residuals, instrumental PSF, primary beam response, and sum of the data weights.

Image reconstruction in CASA follows the CLEAN method by Cotton & Schwab (see Schwab 1984), which consists of an outer loop of major cycles and an inner loop of minor cycles (Figure 4). The major cycle implements transforms between the visibility data and image domain, while the minor cycle represents the deconvolution step that operates purely in the image domain. This method implements an iterative weighted χ² minimization that solves the measurement equation, but allows for a practical trade-off between the efficiency of operating in the image domain and the accuracy that comes from frequently returning to the ungridded list of visibilities. It also allows for minor cycle algorithms to have their own internal optimization scheme.

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The minor cycle performs the deconvolution step by separating sky emission from the PSF and building up a model of the true sky brightness distribution. Different algorithms are available in CASA to identify flux components that are to be included in the sky model. These are hogbom, clark, and clarkstokes, which all identify model-components as delta-functions (Högbom 1974; Clark 1980); mem, which is based on the Maximum Entropy Method (Cornwell & Evans 1985); multiscale, which assigns model components at different spatial scales and is useful for images that include extended emission (Cornwell 2008); asp, which is an Adaptive-Scale Pixel algorithm for more flexible multi-scale source modeling (Bhatnagar & Cornwell 2004); and mtmfs or Multi-Term (Multi-Scale) Multi-Frequency Synthesis, which is a multi-scale and multi-term cleaning algorithm optimized for wideband imaging (Rau & Cornwell 2011). Deconvolution begins with a residual image that includes the astronomical signal, PSF, and noise. In this residual image, model components are identified using one of the above algorithms. After the algorithm identifies a model component, the model is updated accordingly and the effect of the PSF is removed by subtracting a scaled PSF from the image at the location of each component. Many such iterations of finding peaks and subtracting PSFs form the minor cycle (Figure 4). This deconvolution step can be performed for the whole image, or across regions of the image as defined with a mask. At the start of the next major cycle, the model of the sky brightness that is derived during the minor cycle is evaluated against the measurement equation and converted into a predicted list of model visibilities, which are then subtracted from the data to form a new residual image.

During the major cycle, a pseudo inverse of [S_dd ][F] is computed and applied to the visibilities. Operationally, weighted visibilities are convolutionally resampled onto a grid of spatial-frequency cells, inverse Fourier transformed, and normalized via user-specified inputs. The accuracy and efficiency of the image reconstruction depends on the algorithms chosen for [S_dd ] and [F], and direction-dependent instrumental effects can be accounted for via carefully constructed convolution functions in CASA (see Section 3.5.3).

Image deconvolution, and the corresponding sequence of major and minor cycles, continues until a user-specified threshold criterion has been reached across the selected region. This threshold can be specified based on a cutoff level or number of interactions, optionally combined with a mask. Such a mask can be constructed a-priori by the user, iteratively after each major cycle, or determined automatically using the auto-multithresh algorithm. The auto-multithresh option evaluates the noise and sidelobe thresholds in the residual image to set an initial mask at the start of each minor cycle and then cascades that mask down to lower signal-to-noise, taking into account the fundamental properties of the image (Kepley et al. 2020).

After deconvolution, the output sky model is restored by a Gaussian function that represents the instrumental resolution specified by the PSF main lobe, but without the sidelobes. This results in a cleaned image of the sky.

The final image can optionally be corrected for the response of the primary beam of the telescope, in case this has not already been done during the gridding stage. This will provide true values of the flux density throughout the field-of-view. Options for wideband primary-beam correction are included within the mosaic and awproject gridders.

Converting single-dish observations into an image or cube is done almost entirely in the image domain. The single-dish data should be calibrated, according to the process described in Section 3.3.3. The CASA task tsdimaging then converts the single-dish observations into an image or cube by forming and populating the image grid. The convolution kernels that can be used for gridding the image in CASA consist of a boxcar function, Gaussian function, primary beam function, prolate spheroidal wave function, and Gaussian-tapered Bessel function (Mangum et al. 2007). The tsdimaging task by default chooses reasonable values for parameters like pixel size, image dimensions, and spectral resolution, but all these can be adjusted manually in CASA.

Combining interferometric with single-dish data allows for reconstructing the flux of the astronomical source on all spatial scales, including extended emission that is resolved out even by the shortest baselines of the interferometer. Techniques that rely on joint deconvolution (Cornwell 1988; Sault et al. 1996b), either in the image domain (Stanimirovic et al. 1999; Stanimirovic 2002; Pety & Rodríguez-Fernández 2010; Junklewitz et al. 2016; Rau et al. 2019) or in the visibility domain (Kurono et al. 2009; Koda et al. 2011, 2019; Teuben 2019), have been shown to be fruitful for combining single-dish and interferometric data.

CASA contains a dedicated joint deconvolution algorithm for wideband multi-term and mosaic imaging, captured in the CASA task sdintimaging (Rau et al. 2019). Interferometer data are gridded into an image cube and corresponding PSF. The single dish image and PSF cubes are combined with the interferometer cubes in a feathering step (Cotton 2017). This feathering step is based on the CASA task feather , which Fourier transforms the single-dish and interferometric images to a gridded visibility plane and weights them by their spatial frequency response. The joint image and PSF cubes then form inputs to any deconvolution algorithm, be it multi-channel (cube), multi-frequency synthesis (mfs), or multi-term mfs (mtmfs) mode. Model images from the deconvolution algorithm are translated back to model image cubes prior to subtraction from both the single dish image cube and the interferometer data to form a new pair of residual image cubes to be feathered in the next iteration. In the case of wideband mosaic imaging, wideband primary beam corrections are always performed per channel of the image cube, followed by a multiplication by a common primary beam, prior to deconvolution.

The sdintimaging task supports joint deconvolution for spectral cubes as well as multi-term wideband imaging, operates on single pointings as well as joint mosaics, includes corrections for frequency dependent primary beams, and optionally allows the deconvolution of only single-dish images in both cube and (multi-term) mfs mode. An option has also been provided to tune the relative weighting of the single dish and interferometer data alongside the standard weighting schemes used for interferometric imaging.