Total Power Map to Visibilities (TP2VIS): Joint Deconvolution of ALMA 12 m, 7 m, and Total Power Array Data

A common technique of the after-deconvolution combination (category 3a) applies low- and high-pass filters to single-dish and cleaned interferometer maps (or images), respectively. The maps are then added in the Fourier domain and transformed back to a final map. It is now called the Feather technique. Several implementations exist (e.g., the Imerge task in Aips; Immerge task in Miriad; Herbstmeier et al. 1996; Weiß et al. 2001). The Feather task⁷ in Casa is one more recent example. It adopts the low-pass filter, $L(u,v)={FT}\{{B}_{\mathrm{TP}}(l,m)\}$, and a high-pass filter, $H(u,v)=1-L(u,v)$, where the Fourier transform (FT) of the TP primary beam ${B}_{\mathrm{TP}}(l,m)$ is used.⁸ Feather's choice of H(u, v) extends to uv ≈ 0 which interferometer data does not cover. To avoid this, Blagrave et al. (2017) used different filters with non-zero values only where the data exist. Since multiplications (or divisions) in the Fourier domain are equivalent to convolutions (deconvolutions) in the image domain, it is possible to implement the same technique both in the Fourier and image domains. Faridani et al. (2018) implemented the Feather technique in the image domain (category 3b).

The TP map can be used as an initial model for deconvolution of interferometer data (category 2a). Practically, such a scheme deconvolves single-dish and interferometer data jointly (Cornwell 1988). The Clean/Tclean tasks in Casa can take a single-dish map as an input. Dirienzo et al. (2015) employed this technique and produced a combined map. They further applied Feather to ensure that the large-scale components precisely represented the TP data.

Pseudo visibilities can be generated from a single-dish map. They can be added to interferometer visibilities before deconvolution (category 1a; Vogel et al. 1984). Once TP visibilities are generated from the TP map, they are readily fed into existing deconvolvers (e.g., Clean/Tclean). This technique has also been used, though there are some discussions on how to optimize the distribution and weights of the pseudo visibilities (e.g., Rodríguez-Fernández 2008; Kurono et al. 2009; Pety & Rodríguez-Fernández 2010; Koda et al. 2011). For example, Koda et al. (2011) generated a Gaussian visibility distribution and set the visibility weight so that the single-dish primary beam is represented. One of the advantages of pseudo visibilities techniques is that the visibility weight can be easily controlled within single-dish data, as well as with the interferometer data.

The same technique can be implemented in the image domain (category 1b; Stanimirovic et al. 1999). The single-dish and interferometer dirty beams, as well as their maps, are added with a choice of L(u, v) and H(u, v). The combined dirty map can then be deconvolved with the combined dirty beam. This is equivalent to the pseudo visibility techniques, because a FT of a sum of two functions is equal to the sum of the FT of the two functions. The weights can be changed by adjusting L(u, v) and H(u, v). Such implementation is being developed in Casa (U. Rau et al. 2019, in preparation).

Tools are needed for the combination of ALMA data in the platform of the Common Astronomy Software Applications (Casa; McMullin et al. 2007). Here, we introduce a new package in Casa that converts ALMA's TP map into pseudo visibilities (dubbed Tp2vis). It generates visibilities as if they were observed by a virtual interferometer with short spacings down to zero spacial frequency. Together with ALMA 12 m+7 m data, the Tp2vis visibilities can be fed into standard deconvolvers for joint deconvolution. The technique follows the one developed by Koda et al. (2011) on the Miriad platform (Sault et al. 1995), but with improvements.

The Tp2vis package provides four functions in the CASA platform. The key function, Tp2vis, generates a CASA measurement set (MS) –visibilities with weights–from a TP map. The weights are calculated based on the root-mean-square (rms) noise from the TP map. This MS is ready for joint deconvolution, though in some cases one may wish to manipulate the weights using the supplementary function, Tp2viswt. An accessory function, tp2vispl, plots the weights of ALMA 12 m, 7 m, and TP visibilities for comparison. Another function, tp2vistweak, attempts to fix the problem of beam-size mismatch in the image space after deconvolution (Section 3.2).

This step is a preparation for the virtual interferometer observations. When the TP telescopes observe the sky, the TP telescope beam, Ω_TP, is convolved with the brightness distribution in the sky. Therefore, the first step is to deconvolve the TP map with Ω_TP and to obtain the true sky brightness distribution in Jy/beam.

We approximate Ω_TP with a Gaussian with a FWHM of 56.6'' at 115.2 GHz. We assume that it scales linearly with 1/frequency.

When an interferometer observes the sky, the brightness distribution is attenuated by its primary beam (PB). Hence, the sky brightness distribution from Step A is multiplied by the PB pattern of the virtual interferometer (Ω_vir). In case of mosaic observations, this must be done at each pointing position.

In our virtual observations, the coordinates of pointing centers can be set arbitrarily within the TP map. We typically take them from the ALMA 12 m data, so that the 12 m and TP visibilities have the same sky coverage. The pointing coordinates are listed in an ASCII file and passed to the Tp2vis function with the "ptg =" argument.

In parallel with steps A and B, a visibility distribution for the virtual observations (hence a MS) is prepared. We generate a two-dimensional Gaussian distribution of visibilities in the uv space, so that its Fourier transformation, i.e., the synthesized beam Ω_syn, reproduces the Gaussian beam of the TP telescopes (Ω_TP). Adopting a constant weight for all TP visibilities (Section 4), this distribution would represent the sensitivity distribution within the TP beam. The width of the Gaussian visibility distribution is determined from those of the Gaussian beam. The standard deviations of the Gaussian beam in the sky σ and the Gaussian distribution in the uv space σ_F are related as σ_F = 1/(2πσ).

The full width at half maximum (FWHM) is calculated as $\mathrm{FWHM}=2\sqrt{2\mathrm{ln}2}\sigma$ (≈2.355σ). For a FWHM of 56.6'', σ = 2.74 × 10⁻⁴ radian and σ_F = 580λ, where λ is the wavelength.

We already have the sky brightness distribution with the PB of the virtual interferometer applied (from step B), and the Gaussian visibility distribution in the uv space (step C). Now, the sky brightness distribution is Fourier-transformed into the uv space. The amplitude and phase of the transformation are then sampled at the visibility positions, which fill the AMPLITUDE and PHASE columns of the visibilities in the MS.

These visibilities are already internally consistent. The Clean/Tclean tasks with natural weighting produce a synthesized beam Ω_syn roughly equal to the TP telescope beam Ω_TP. The brightness distribution also reproduces the observed TP brightness distribution.

The Tp2vis function sets visibility weights according to the rms noise in the TP map. The rms value is set manually by the "rms =" argument [in units of Jy/beam of the TP map]. [Note that in the current implementation, the rms is set by hand, because it is not always trivial to find emission-free channels using an algorithm.] The rms is converted to the weight of individual visibilities. The equation for this conversion is in Section 4. The sensitivity-based weight is a proper representation of the quality of the TP data and is the default of Tp2vis.

In the history of radio interferometry, the weights are often manipulated. The most common examples are the uniform, robust/briggs, or taper weighting schemes. There is no reason not to manipulate them in some other ways—for example, to match the weights of TP visibilities with those of ALMA 12 m and 7 m data (their sensitivities may not match when one works on archival data). The supplementary tp2viswt function provides several options to manipulate the weights. An accessory tp2vispl function plots the weight distributions of ALMA 12 m, 7 m, and TP visibilities for comparison, so that users can decide how the TP visibilities need to be weighted with respect to 12 m and 7 m data.

The synthesized beam Ω_syn is determined by the distributions of visibility weights in uv space [Note: ${{\rm{\Omega }}}_{\mathrm{syn}}={{\rm{\Omega }}}_{\mathrm{dirty}}={{\rm{\Omega }}}_{\mathrm{dec}}$]. Its 2D pattern, B(l, m), is a Fourier transformation of the weight distribution W(u, v). With the normalization of $\iint W(u,v){dudv}=1$,

$$\begin{eqnarray}&&B(l,m)=\iint W(u,v){e}^{+2\pi i({ul}+{vm})}{dudv}.\end{eqnarray} \tag{ 1 }$$

The area of Ω_syn is the beam solid angle and is given as an integration over space,

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{\mathrm{syn}}=\iint B(l,m){dldm}=W(0,0).\end{eqnarray} \tag{ 2 }$$

It depends only on W(0, 0), the weight at (u, v) = (0, 0). Therefore, only the weight of the TP visibility matters, because the interferometer data have zero weight at (u, v) = (0, 0). Therefore, ${{\rm{\Omega }}}_{\mathrm{syn}}={W}_{\mathrm{TP}}(0,0)$ when ALMA 12 m, 7 m, and TP are combined.

The convolution beam Ω_conv is typically defined by a Gaussian fit to the central peak in the B(l, m) pattern [Note: ${{\rm{\Omega }}}_{\mathrm{conv}}={{\rm{\Omega }}}_{\mathrm{clean}}={{\rm{\Omega }}}_{\mathrm{res}}$]. Hence, it depends on the full weight distribution, W(u, v); but in practice, the outer, long-baseline part of the visibility weight distribution is often the determining factor.

In summary, Ω_syn is set at the center of the uv space, while Ω_conv is determined mainly by the outer part.

For simplicity, we use Clean as an example here, but it applies to other deconvolvers as well. The Clean task iteratively finds peaks in a dirty map, whose brightness units are Jy/Ω_syn. In each iteration Clean identifies a peak in a dirty map, fits the 2D pattern of Ω_syn to the peak and obtains its position and amplitude, subtracts it from the dirty map, and puts a δ-function at that position in a clean-component map. The Clean task then convolves the clean-component map with Ω_conv to produce a model map. The model map has brightness units of Jy/Ω_conv. Typically, Clean is not perfect and leaves emission in a residual map. The unit of the residual map remains Jy/Ω_syn. Clean naively adds the model and residual maps to produce a final map, whose unit is often denoted as Jy/beam. For the "beam" in the denominator, Ω_conv is typically adopted. Therefore,

$$\begin{eqnarray}&&\left[\displaystyle \frac{F}{(\mathrm{Jy}/{{\rm{\Omega }}}_{\mathrm{conv}})}\right]=\left[\displaystyle \frac{M}{(\mathrm{Jy}/{{\rm{\Omega }}}_{\mathrm{conv}})}\right]+\left[\displaystyle \frac{R}{(\mathrm{Jy}/{{\rm{\Omega }}}_{\mathrm{syn}})},\right].\end{eqnarray} \tag{ 3 }$$

where F, M, and R stand for the final, model, and residual maps, respectively.

The areas and patterns of Ω_syn and Ω_conv could be very different, as they depend on different parts of the weight distribution in the uv space (Section 3.1). In particular, if their areas are not equal, Ω_syn $\ne$ Ω_conv, the brightness of R in Equation (3) is calculated incorrectly. For example, if Ω_syn ≫ Ω_conv, the brightness in R is overestimated by Ω_syn/Ω_conv (≫1), leading to an overestimation of flux in F. This happens when the ALMA 12 m, 7 m, and TP visibilities are combined. In theory, the R could be zero if we could clean down to zero flux, but in practice, it is nearly impossible to get there, as the map to be cleaned has noise.

In case of Ω_syn ≪ Ω_conv, the brightness in R is underestimated. The impact of this on F may not be as noticeable as in the opposite case, as R typically has much less flux than M. This happens when the 12 m and 7 m visibilities are imaged without TP, because ${{\rm{\Omega }}}_{\mathrm{syn}}={W}_{\mathrm{TP}}(0,0)=0$ (hence, ≪Ω_conv). This flux problem did not attract much attention in the past, as single-dish data were typically not included.

One of the most intuitive weighting schemes is the one that represents the quality of TP data, namely the root-mean-square (rms) noise of the TP map. This is the default weighting scheme used by the Tp2vis function, and the Tp2viswt function with the mode = "rms" can set the weights to the sensitivity-based one. This section explains the calculation of the sensitivity-based weights in Tp2vis. Section 4.1.1 discusses the conversion of the rms noise of a TP map into the weights of individual visibilities. For a joint deconvolution, the weights of 12 m, 7 m, and TP data should be set consistently. Hence, Section 4.1.2 shows the corresponding weights of 12 m and 7 m data. [Note that CASA version 5 uses this definition of 12 m and 7 m weights by default, improving upon prior CASA versions.]

This sensitivity-based weighting is equivalent to the conventional "natural" weighting. For interferometer data, additional manipulations, such as "robust/briggs" or "uniform" weighting schemes, are often applied on top of the sensitivity-based weights. Such manipulations can also be applied to the sensitivity-based weights from Tp2vis. For simplicity, however, the discussions in this section will focus on the sensitivity-based (natural) weights.

4.1.1. Weights for TP Visibilities

With the natural weighting, the sensitivity of a map, or the rms noise of the image (${\rm{\Delta }}{S}^{i}$), is a simple summation over all visibility sensitivities (${\rm{\Delta }}{S}_{k}^{v}$),

$$\begin{eqnarray}&&{\left(\displaystyle \frac{1}{{\rm{\Delta }}{S}^{i}}\right)}^{2}=\displaystyle \sum _{k}{\left(\displaystyle \frac{1}{{\rm{\Delta }}{S}_{k}^{v}}\right)}^{2}={N}_{\mathrm{vis}}{\left(\displaystyle \frac{1}{{\rm{\Delta }}{S}^{v}}\right)}^{2},\end{eqnarray} \tag{ 4 }$$

where N_vis is the number of visibilities. All visibilities are assumed to have the same sensitivity (${\rm{\Delta }}{S}_{k}^{v}={\rm{\Delta }}{S}^{v}$), and a single rms value is assumed for the map. Hence, the weight of each TP visibility is

$$\begin{eqnarray}&&{w}_{k}\equiv {\left(\displaystyle \frac{1}{{\rm{\Delta }}{S}_{k}^{v}}\right)}^{2}=\displaystyle \frac{1}{{N}_{\mathrm{vis}}}{\left(\displaystyle \frac{1}{{\rm{\Delta }}{S}^{i}}\right)}^{2}.\end{eqnarray} \tag{ 5 }$$

4.1.2. Corresponding Weights for 12 m and 7 m Visibilities

For consistency, the 12 m and 7 m data should also use the sensitivity-based weights. The sensitivity for a single visibility k between antennas i and j is given by

$$\begin{eqnarray}&&{\rm{\Delta }}{S}_{k}^{v}={C}_{{ij}}\sqrt{\displaystyle \frac{{T}_{\mathrm{sys},i}{T}_{\mathrm{sys},j}}{B\cdot {t}_{\mathrm{vis}}}}={C}_{{ij}}\sqrt{\displaystyle \frac{{T}_{\mathrm{sys}}^{2}}{B\cdot {t}_{\mathrm{vis}}}},\end{eqnarray} \tag{ 6 }$$

where ${T}_{\mathrm{sys},i}$, ${T}_{\mathrm{sys},j}$, and B, ${t}_{\mathrm{vis}}$ are the system temperatures of i and j, bandwidth, and integration time of the visibility, respectively (Taylor et al. 1999; Thompson et al. 2007). For simplicity, we assume ${T}_{\mathrm{sys},i}={T}_{\mathrm{sys},j}={T}_{\mathrm{sys}}$. The coefficient C_ij is

$$\begin{eqnarray}&&{C}_{{ij}}=\displaystyle \frac{2{k}_{B}}{\sqrt{({\eta }_{a,i}{A}_{i})({\eta }_{a,j}{A}_{j})}}\displaystyle \frac{1}{\sqrt{2}{\eta }_{q}},\end{eqnarray} \tag{ 7 }$$

where ${\eta }_{a,i}$, ${\eta }_{a,j}$, A_i, A_j, and ${\eta }_{q}$ are the aperture efficiencies and areas of antennas i and j, and the quantum efficiency of correlator, respectively. With the natural weighting, the rms sensitivity of final map is

$$\begin{eqnarray}&&{\rm{\Delta }}{S}^{i}={C}_{{ij}}\sqrt{\displaystyle \frac{{T}_{\mathrm{sys}}^{2}}{B\cdot {t}_{\mathrm{tot}}}},\end{eqnarray} \tag{ 8 }$$

where t_tot is the total integration time ${t}_{\mathrm{tot}}={N}_{\mathrm{vis}}{t}_{\mathrm{vis}}$. The weights of 12 m and 7 m visibilities should be set to ${w}_{k}={(1/{\rm{\Delta }}{S}_{k}^{v})}^{2}$ with Equation (6).

4.1.3. For Internal Consistency

This subsection is not necessary in practice, but is included for consistency among all data columns in the MS. It might become important in future versions of CASA. Tp2vis fills the WEIGHT and SIGMA columns. Only the WEIGHT column is used by Clean/Tclean in Casa—currently, other columns do not matter. However, one could also make the EXPOSURE column (i.e., integration time of each visibility) consistent.

The sensitivity equations for TP, corresponding to Equations (6), (8), and (7), are

$$\begin{eqnarray}&&{\rm{\Delta }}{S}_{k}^{v}={C}_{\mathrm{TP}}\sqrt{\displaystyle \frac{{T}_{\mathrm{sys}}^{2}}{B\cdot {t}_{\mathrm{vis}}}}\end{eqnarray} \tag{ 9 }$$

and

$$\begin{eqnarray}&&{\rm{\Delta }}{S}^{i}={C}_{\mathrm{TP}}\sqrt{\displaystyle \frac{{T}_{\mathrm{sys}}^{2}}{B\cdot {t}_{\mathrm{tot}}}},\end{eqnarray} \tag{ 10 }$$

where

$$\begin{eqnarray}&&{C}_{\mathrm{TP}}=\displaystyle \frac{{k}_{B}}{{\eta }_{a}A}\displaystyle \frac{1}{{\eta }_{\mathrm{mb}}}\displaystyle \frac{1}{{\eta }_{q}},\end{eqnarray} \tag{ 11 }$$

and η_mb is the main beam efficiency of the TP antennas. T_sys, ${\rm{\Delta }}{S}^{i}$, and B are determined by observation. With Equation (10) we can derive ${t}_{\mathrm{tot}}$, using T_sys and ${\rm{\Delta }}{S}^{i}$ from MS and TP maps, respectively. Equations (5), (9), and (10) give ${t}_{\mathrm{vis}}={t}_{\mathrm{tot}}/{N}_{\mathrm{vis}}$.

Section 3.2 discussed the problem in flux due to the disparity between the areas of Ω_syn and Ω_conv. The beam size-based weighting scheme discussed here is an attempt to adjust the weights of TP visibilities to equalize their areas. Note again that we use the symbols, Ω_syn and Ω_conv, to represent the areas of the beams as well as to refer the beams themselves.

4.2.1. The Synthesized and Convolution Beams from Combined INT+TP Visibilities

The pattern of the synthesized beam ${{\rm{\Omega }}}_{\mathrm{syn}}$, denoted by B(l, m), is given by Equation (1). The weight is normalized as $\iint W(u,v){dudv}=1$. For interferometers (hereafter INT; e.g., ALMA 12 m and 7m) and TP, the synthesized beam patterns, ${B}_{\mathrm{INT}}(u,v)$ and ${B}_{\mathrm{TP}}(u,v)$, are derived with their normalized weights, ${W}_{\mathrm{INT}}(u,v)$ and ${W}_{\mathrm{TP}}(u,v)$, respectively.

By defining a relative weight parameter β, a combined weight distribution of INT and TP can be expressed as

$$\begin{eqnarray}&&W(u,v)=\displaystyle \frac{{W}_{\mathrm{INT}}(u,v)+\beta {W}_{\mathrm{TP}}(u,v)}{\iint [{W}_{\mathrm{INT}}(u^{\prime} ,v^{\prime} )+\beta {W}_{\mathrm{TP}}(u^{\prime} ,v^{\prime} )]{du}^{\prime} {dv}^{\prime} }\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&=\,\displaystyle \frac{1}{1+\beta }\left[{W}_{\mathrm{INT}}(u,v)+\beta {W}_{\mathrm{TP}}(u,v)\right].\end{eqnarray} \tag{ 13 }$$

The corresponding synthesized beam pattern is

$$\begin{eqnarray}&&B(l,m)=\displaystyle \frac{1}{1+\beta }\left[{B}_{\mathrm{INT}}(l,m)+\beta {B}_{\mathrm{TP}}(l,m)\right].\end{eqnarray} \tag{ 14 }$$

The area of the synthesized beam is derived from Equations (2) (13),

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{\mathrm{syn}}=W(0,0)=\displaystyle \frac{\beta }{1+\beta }{W}_{\mathrm{TP}}(0,0),\end{eqnarray} \tag{ 15 }$$

where we used ${W}_{\mathrm{INT}}(0,0)=0$ since INT has no contribution at zero spacing. Note that ${W}_{\mathrm{TP}}(0,0)$ is equal to the area of the synthesized beam of TP visibilities.

The convolution beam ${{\rm{\Omega }}}_{\mathrm{conv}}$ is typically derived by a 2D Gaussian fit to the central peak in $B(l,m)$. Using the major and minor axis diameters, b_maj and b_min, from the fit, the area of Ω_conv is

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{\mathrm{conv}}=\displaystyle \frac{\pi {b}_{\mathrm{maj}}\times {b}_{\min }}{4\mathrm{ln}2}.\end{eqnarray} \tag{ 16 }$$

4.2.2. Setting Up Weights for TP Visibilities

This weighting scheme attempts to achieve ${{\rm{\Omega }}}_{\mathrm{syn}}={{\rm{\Omega }}}_{\mathrm{conv}}$. Equation (15) gives β as

$$\begin{eqnarray}&&\beta =\displaystyle \frac{{{\rm{\Omega }}}_{\mathrm{syn}}}{{W}_{\mathrm{TP}}(0,0)-{{\rm{\Omega }}}_{\mathrm{syn}}}=\displaystyle \frac{{{\rm{\Omega }}}_{\mathrm{conv}}}{{W}_{\mathrm{TP}}(0,0)-{{\rm{\Omega }}}_{\mathrm{conv}}}.\end{eqnarray} \tag{ 17 }$$

W_TP(0, 0) is equal to the area of the synthesized beam of TP visibilities alone, which, by construction, is the same as that of the Gaussian primary beam of the TP antennas. Hence,

$$\begin{eqnarray}&&{W}_{\mathrm{TP}}(0,0)=\displaystyle \frac{\pi {b}_{\mathrm{TP}}^{2}}{4\mathrm{ln}2},\end{eqnarray} \tag{ 18 }$$

using the FWHM beam size of the TP antennas, ${b}_{\mathrm{TP}}$. The Ω_conv should be calculated from INT+TP visibilities, but in practice it depends primarily –and almost solely–on the longest baselines. Hence, for simplicity, we derive Ω_conv from INT visibilities alone. We can calculate β from Equations (17), (18), and (16) for INT.

Using β, we adjust the TP visibility weight distribution from W_TP to βW_TP, by scaling the weights of individual visibilities (${w}_{k}^{\mathrm{INT}}$ and ${w}_{k}^{\mathrm{TP}}$ for INT and TP visibilities in MSs, respectively).

For the following, we use the notation $\overline{W}$ for an unnormalized weight distribution. The ratio of the two terms in the numerator of Equation (13) is

$$\begin{eqnarray}&&\begin{array}{l}{W}_{\mathrm{INT}}(u,v):\beta {W}_{\mathrm{TP}}(u,v)\\ \quad =\,\displaystyle \frac{{\overline{W}}_{\mathrm{INT}}(u,v)}{\iint {\overline{W}}_{\mathrm{INT}}(u,v){dudv}}:\beta {W}_{\mathrm{TP}}(u,v)\\ \quad =\,{\overline{W}}_{\mathrm{INT}}(u,v):\beta \left[\iint {\overline{W}}_{\mathrm{INT}}(u,v){dudv}\right]{W}_{\mathrm{TP}}(u,v).\end{array}\end{eqnarray} \tag{ 19 }$$

This transformation converts the normalized ${W}_{\mathrm{INT}}(u,v)$ into the unnormalized ${\overline{W}}_{\mathrm{INT}}(u,v)$, which is what is delivered from the ALMA observatory. The term to the right of the ratio mark gives the corresponding normalization for the weights of TP visibilities. When the natural weighting is used, the unnormalized ${\overline{W}}_{\mathrm{TP}}(u,v)$ and ${w}_{k}^{\mathrm{TP}}$ are related as $\iint {\overline{W}}_{\mathrm{TP}}(u,v){dudv}={\sum }_{k}{w}_{k}^{\mathrm{TP}}={N}_{\mathrm{vis}}^{\mathrm{TP}}{w}_{k}^{\mathrm{TP}}$. Thus, for the normalized ${W}_{\mathrm{TP}}(u,v)$, we set ${w}_{k}^{\mathrm{TP}}=1/{N}_{\mathrm{vis}}^{\mathrm{TP}}$. Also, the term in the bracket is simply, $\iint {\overline{W}}_{\mathrm{INT}}(u,v){dudv}={\sum }_{k}{w}_{k}^{\mathrm{INT}}$. Therefore,

$$\begin{eqnarray}&&{w}_{k}^{\mathrm{TP}}=\displaystyle \frac{\beta }{{N}_{\mathrm{vis}}^{\mathrm{TP}}}\displaystyle \sum _{k}{w}_{k}^{\mathrm{INT}}.\end{eqnarray} \tag{ 20 }$$

This weight would satisfy ${{\rm{\Omega }}}_{\mathrm{syn}}={{\rm{\Omega }}}_{\mathrm{conv}}$ when the INT and TP visibilities are combined.

The weight distribution of TP visibilities can be controlled, either by changing the distribution of TP visibilities in uv space, and/or by adjusting the weights of individual visibilities. Tp2vis takes the former approach: generating a Gaussian visibility distribution with a constant weight for all the visibilities.

The latter approach could be useful in future. For example, one could generates a grid distribution, instead of the Gaussian distribution, and change the visibility weights as a function of uv distance so that the weight distribution follows a Gaussian. This way, one could avoid shot noise due to the randomly-distributed discrete visibilities. In the end, the Clean task would put the visibilities onto a grid. If the grid spacing is controlled consistently between Tp2vis and Clean, we can test grid-based approaches in future. Of course, the weight distribution can be realized on a grid without visibilities. An implementation of this would require an update of the Clean/Tclean task in Casa (U. Rau 2018, private communication).

6.1.1. Model Images

Two model images are used for tests (Figure 2). One is a single Gaussian emission distribution with a standard deviation of 1500 pixels in width (i.e., FWHM of ∼53'' in the definition described below), and another is a pseudo sky image that mimics small to large structures within giant molecular clouds (GMCs), i.e., dense cores and extended emission (see the Appendix). The current Clean/Tclean tasks tend to cause flux divergence near the edge of a mosaic field of view. As a workaround, the emission around the edge is tapered off outward so that it reaches zero at around the ∼80% level of the maximum primary beam coverage (see Section 6.1.4).

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Both images have the same dimensions of 16,384² and are configured to the same setup, i.e., an image center coordinate of (RA, DEC)=(12h00m00s, −35d00m00s), a cell size of 0.015'' (hence the image size of 4.096'), and a frequency of 115 GHz with a bandwidth of 2 GHz (a single channel). We run the Simobserve task to convert the images into the CASA format with these parameters. The 0.015'' cell size is chosen to include the main structures in a 38-pointing mosaic (Section 6.1.2) and corresponds to baselines of ∼36 km when observed at 115 GHz. Therefore, the images can be properly sampled by all the ALMA configurations including the one with the longest baseline (∼16 km). The flux scale is arbitrary, and only the relative flux with respect to the peak is important in the tests. The peak flux is set to 1 for Figure 2.

6.1.2. Visibilities for 12 m and 7 m Arrays

6.1.3. Visibilities for TP Array

The TP array observes the sky as single-dish telescopes with a FWHM beam size of 56.7'' at 115 GHz. TP data are delivered as a calibrated image, or cube, with a cell size of 5.6''. Hence, we smooth the model images with a Gaussian with a FWHM of 56.7'' and regrid them on a 5.6'' cell size. We adopt the the same pointing coordinates as those of the 12 m and 7 m visibilities. Tp2vis is run on these images and generates 5,175 TP visibilities per pointing.

The model images do not have noise, and the visibility weights must be set arbitrarily. We apply the sensitivity-based weight (Section 4.1) with relative image rms sensitivities of 1 for each of the four 12 m configurations, 4.5 for the 7 m configuration, and 7 for the TP visibilities. These relative sensitivities are chosen based on typical ALMA mosaic data we have at hand (e.g., of nearby galaxies, Galactic molecular clouds), Table A-2 in the ALMA Proposer's Guide,¹¹ and the ALMA sensitivity calculator. The visibility and sensitivity distributions of the TP, 7 m, and 12 m arrays are plotted in Figure 3.

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6.1.4. Caveats

We use the Clean task for deconvolution. The termination of Clean iterations is typically controlled by a threshold from the rms noise, or by the number of iterations. The model images do not have noise, and hence we use the number of iterations for termination and set it to 2 × 10⁵. The gain is set to 0.05. The region for clean is restricted to the area above 80% of the peak of the 12m+7m+TP combined primary beam. We use the Briggs weighting with a "robust" parameter of +0.5. It results in a PSF of 1.59'' × 1.46'' with a position angle of 87.8° for both test images.

Figures 4 and 5 compare the model and cleaned images: (a) and (c) the original model maps smoothed with the PSF, (b) and (d) the cleaned maps with 12 m, 7 m, and TP visibilities, (e) the residual maps after the residual rescaling, and (f) the accuracy maps. Only relative fluxes are important for comparisons, so the fluxes of all images are normalized by a single constant so that the peak of panel (a) becomes equal to unity. Figures 4 and 5 are normalized separately. The accuracy maps show deviations from the input models and are calculated as

$$\begin{eqnarray}&&\mathrm{Accuracy}=\displaystyle \frac{(\mathrm{Cleaned}\ \mathrm{image})-(\mathrm{Smoothed}\ \mathrm{model})}{(\mathrm{Smoothed}\ \mathrm{model})}.\end{eqnarray} \tag{ 23 }$$

With this definition an accuracy map plus 1 makes a recovered flux ratio map at the 1.59'' × 1.46'' resolution. The clean is applied over nearly three orders of magnitude in flux as the residuals after clean are ≲0.3% of the peak flux (panel e). Over this wide dynamic range, the original flux is reproduced with errors mostly less than ∼5% (panel f), though some very diffuse regions show errors as high as ≳20%.

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The purpose of this paper is to introduce the new Tp2vis package. Hence, these tests are performed with simple noise-free images. Further imaging simulations with more realistic setups would help evaluating systematic errors in actual ALMA observations.

We start from a representation of a density field with the density power spectrum $P(k)\propto \left\langle {\widetilde{\delta }}^{2}\right\rangle \propto {k}^{-n}$. We assume a Gaussian random field (i.e., Fourier modes are uncorrelated) and the probability distribution function of its amplitude is drawn from a Gaussian distribution.

A.1.1. Definitions

Using the density at position ${\bf{x}}$, $\rho ({\bf{x}})$, and average density, ρ₀, the density contrast is defined as

$$\begin{eqnarray}&&\delta ({\bf{x}})\equiv \displaystyle \frac{\rho ({\bf{x}})-{\rho }_{0}}{{\rho }_{0}}.\end{eqnarray} \tag{ 24 }$$

For the discrete Fourier transformation, we set the grid coordinates

$$\begin{eqnarray}&&{\bf{x}}=D\left(\begin{array}{c}l/L\\ m/M\\ n/N\end{array}\right),{\bf{k}}=\displaystyle \frac{1}{D}\left(\begin{array}{c}u\\ v\\ w\end{array}\right),\end{eqnarray} \tag{ 25 }$$

where (l, m, n) and (u, v, w) are integers, and (L, M, N) are the numbers of grid points in x, y, and z directions, and D is the size of the image, respectively. We adopt the definition of the discrete Fourier transformation:

$$\begin{eqnarray}&&\delta (l,m,n)=\displaystyle \sum _{u,v,w}\widetilde{\delta }(u,v,w){e}^{+2\pi i\left(\tfrac{{ul}}{L}+\tfrac{{vm}}{M}+\tfrac{{wn}}{N}\right)}\left[{\rm{\Delta }}u{\rm{\Delta }}v{\rm{\Delta }}w\right],\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}&&\widetilde{\delta }(u,v,w)\,=\,\displaystyle \frac{1}{{LMN}}\displaystyle \sum _{l,m,n}\delta (l,m,n){e}^{-2\pi i\left(\tfrac{{ul}}{L}+\tfrac{{vm}}{M}+\tfrac{{wn}}{N}\right)}\left[{\rm{\Delta }}l{\rm{\Delta }}m{\rm{\Delta }}n\right].\end{eqnarray} \tag{ 27 }$$

This definition is adopted in most numerical packages (e.g., IDL, Python Numpy package, etc) and is convenient for implementation. The last terms, ${\rm{\Delta }}u{\rm{\Delta }}v{\rm{\Delta }}w$ and ΔlΔmΔn (both=1), are written explicitly because they need an evaluation later. Since the density field, δ(l, m, n), is a field of real numbers, each complex Fourier component has to satisfy

$$\begin{eqnarray}&&\widetilde{\delta }(u,v,w)={\widetilde{\delta }}^{* }(L-u,M-v,N-w).\end{eqnarray} \tag{ 28 }$$

A.1.2. Power Spectrum and Normalization

The density power spectrum is often defined in the form of a power law P(k) = C k⁻ⁿ with a normalization constant of C. To avoid the singularity at k = 0 the power spectrum is redefined here as

$$\begin{eqnarray}&&P(k)=\displaystyle \frac{C}{{\left({k}^{2}+{k}_{\min }^{2}\right)}^{n/2}},\end{eqnarray} \tag{ 29 }$$

where k_min is the minimum wavenumber (${k}_{\min }=1/D$).

The normalization constant C is calculated from the variance of the density contrast, σ² (input parameter):

$$\begin{eqnarray}\begin{array}{rcl}{\sigma }^{2} & = & \displaystyle \sum _{{k}_{u},{k}_{v},{k}_{w}}P(k)[{\rm{\Delta }}{k}_{u}{\rm{\Delta }}{k}_{v}{\rm{\Delta }}{k}_{w}]\\ & = & C\displaystyle \sum _{{k}_{u},{k}_{v},{k}_{w}}\displaystyle \frac{1}{{\left({k}^{2}+{k}_{\min }^{2}\right)}^{n/2}}[{\rm{\Delta }}{k}_{u}{\rm{\Delta }}{k}_{v}{\rm{\Delta }}{k}_{w}].\end{array}\end{eqnarray} \tag{ 30 }$$

For the purpose of analytical evaluation, we can also obtain

$$\begin{eqnarray}&&{\sigma }^{2}=\displaystyle \sum _{{k}_{u},{k}_{v},{k}_{w}}P(k){\rm{\Delta }}{k}_{u}{\rm{\Delta }}{k}_{v}{\rm{\Delta }}{k}_{w}\approx {\displaystyle \int }_{0}^{\infty }4\pi {k}^{2}{P}_{v}(k){dk}\end{eqnarray} \tag{ 31 }$$

$$\begin{eqnarray}&&\,=\,4\pi {{Ck}}_{\min }^{3-n}{\displaystyle \int }_{0}^{\infty }\displaystyle \frac{{x}^{2}}{{\left({x}^{2}+1\right)}^{n/2}}{dx}=4\pi {{Ck}}_{\min }^{3-n}\,\displaystyle \frac{\sqrt{\pi }{\rm{\Gamma }}\left(\tfrac{n-3}{2}\right)}{4{\rm{\Gamma }}\left(\tfrac{n}{2}\right)},\end{eqnarray} \tag{ 32 }$$

where the last equation is valid when n > 3 (the integral diverges when n ≤ 3). Γ is the gamma function. [Note ${\sigma }^{2}\propto {k}_{\min }^{3-n}\propto {D}^{n-3}$, and the size-line width relation of GMCs indicates n ∼ 4.] For a given set of n (>3) and σ, the normalization coefficient of the power spectrum is

$$\begin{eqnarray}&&C=\displaystyle \frac{{\sigma }^{2}}{4\pi }{k}_{\min }^{n-3}\displaystyle \frac{4{\rm{\Gamma }}\left(\tfrac{n}{2}\right)}{\sqrt{\pi }{\rm{\Gamma }}\left(\tfrac{n-3}{2}\right)}.\end{eqnarray} \tag{ 33 }$$

The above prescription generates a 3D data cube and involves a Fourier transformation in 3D. We can generate a 2D image and apply a Fourier transformation in 2D by using

$$\begin{eqnarray}&&{\delta }^{2{\rm{D}}}(l,m)\equiv \delta (l,m,n=0)\end{eqnarray} \tag{ 34 }$$

$$\begin{eqnarray}\begin{array}{rcl} & = & \displaystyle \sum _{u,v}\left[\displaystyle \sum _{w}\widetilde{\delta }(u,v,w)\right]{e}^{+2\pi i\left(\tfrac{{ul}}{L}+\tfrac{{vm}}{M}\right)}\\ & = & \displaystyle \sum _{u,v}{\widetilde{\delta }}^{2{\rm{D}}}(u,v){e}^{+2\pi i\left(\tfrac{{ul}}{L}+\tfrac{{vm}}{M}\right)},\end{array}\end{eqnarray} \tag{ 35 }$$

where

$$\begin{eqnarray}&&{\widetilde{\delta }}^{2{\rm{D}}}(u,v)=\displaystyle \sum _{w}\widetilde{\delta }(u,v,w).\end{eqnarray} \tag{ 36 }$$

A.1.3. Realizations

The power spectrum P(k) and the fluctuation of density contrast $\langle {\widetilde{\delta }}^{2}(k)\rangle$ are related as follows.

$$\begin{eqnarray}&&{\sigma }^{2}=\displaystyle \frac{1}{{LMN}}\displaystyle \sum _{l,m,n}{\delta }^{2}(l,m,n)\left[{\rm{\Delta }}l{\rm{\Delta }}m{\rm{\Delta }}n\right]\end{eqnarray} \tag{ 37 }$$

$$\begin{eqnarray}&&\begin{array}{l}=\,\displaystyle \frac{1}{{LMN}}\displaystyle \sum _{u,v,w}\displaystyle \sum _{u^{\prime} ,v^{\prime} ,w^{\prime} }\widetilde{\delta }(u,v,w)\widetilde{{\delta }^{* }}(u^{\prime} ,v^{\prime} ,w^{\prime} )\\ \quad \times \,\left\{\displaystyle \sum _{l,m,n}{e}^{+2\pi i\left(\tfrac{(u-u^{\prime} )l}{L}+\tfrac{(v-v^{\prime} )m}{M}+\tfrac{(w-w^{\prime} )n}{N}\right)}\right\}[...][...][...]\end{array}\end{eqnarray} \tag{ 38 }$$

$$\begin{eqnarray}&&=\,\displaystyle \sum _{u,v,w}\widetilde{\delta }(u,v,w)\widetilde{{\delta }^{* }}(u,v,w)\left[{\rm{\Delta }}u{\rm{\Delta }}v{\rm{\Delta }}w\right]\end{eqnarray} \tag{ 39 }$$

$$\begin{eqnarray}&&=\,\displaystyle \sum _{{k}_{u},{k}_{v},{k}_{w}}P(k)\left[{\rm{\Delta }}{k}_{u}{\rm{\Delta }}{k}_{v}{\rm{\Delta }}{k}_{w}\right].\end{eqnarray} \tag{ 40 }$$

The {...} term is equal to $\left\{L{\delta }_{u,u^{\prime} }^{K}\cdot M{\delta }_{v,v^{\prime} }^{K}\cdot N{\delta }_{w,w^{\prime} }^{K}\right\}$, where δ^K is the Kronecker delta. The bracket terms at the end, $\left[{\rm{\Delta }}l{\rm{\Delta }}m{\rm{\Delta }}n\right]$, $\left[{\rm{\Delta }}u{\rm{\Delta }}v{\rm{\Delta }}w\right]$, etc, are expressed as [...] to save space. From Equations (39) and (40),

$$\begin{eqnarray}&&\widetilde{\delta }(u,v,w)\widetilde{{\delta }^{* }}(u,v,w)\left[{\rm{\Delta }}u{\rm{\Delta }}v{\rm{\Delta }}w\right]=P(k)\left[{\rm{\Delta }}{k}_{u}{\rm{\Delta }}{k}_{v}{\rm{\Delta }}{k}_{w}\right]\end{eqnarray} \tag{ 41 }$$

or

$$\begin{eqnarray}&&\left\langle {\widetilde{\delta }}^{2}(k)\right\rangle =P(k){d}^{3}{\bf{k}}=P(k)/{D}^{3}.\end{eqnarray} \tag{ 42 }$$

Note that $\left[{\rm{\Delta }}u{\rm{\Delta }}v{\rm{\Delta }}w\right]=1$ and $\left[{\rm{\Delta }}{k}_{u}{\rm{\Delta }}{k}_{v}{\rm{\Delta }}{k}_{w}\right]=1/{D}^{3}$.

The Fourier components $\widetilde{\delta }(k){\rm{s}}$ are sampled using $\langle {\widetilde{\delta }}^{2}(k)\rangle$ as the variance. A random number is generated for each ${\bf{k}}$ using a Gaussian distribution with a standard deviation of $\sqrt{\langle {\widetilde{\delta }}^{2}(k)\rangle }=\sqrt{P(k){d}^{3}{\bf{k}}}=\sqrt{P(k)/{D}^{3}}$. The phase of the complex number $\widetilde{\delta }(k)$ is also drawn from a uniform distribution in [0, 2π), and the symmetry condition, Equation (28), is taken into account. Once all the Fourier components are generated, they are Fourier-transformed to the image plane using Equation (26). We could also draw the random numbers without the symmetry condition and take the real or imaginary part after the Fourier transformation. In this case, the amplitude should be multiplied by $\sqrt{2}$.

Figures 6(a), (b) show the model images generated with a power spectrum index of n = 4, an average density fluctuation of σ = 0.3, and an image size of 16,384². Both the real and imaginary images are shown. The images have periodic boundaries, and we spatially shifted them so that significant features come near the image centers.

Zoom In Zoom Out Reset image size

Figures 6(a), (b) qualitatively resemble some parts of GMCs. However, by construction, they don't have "dense cores", whose environs are often observed with ALMA. We could develop dense cores by running hydrodynamics simulations with gravity, starting from these images as the initial condition, but this is obviously beyond the scope of this paper. Here, to mimic their high brightness, we simply convert the flux ${f}^{\mathrm{old}}$ (or density $\rho ({\bf{x}})$) into new flux f^new using the following equation:

$$\begin{eqnarray}{f}^{\mathrm{new}}=\left\{\begin{array}{ll}\exp \left[{f}^{\mathrm{old}}-\beta \right] & ({f}^{\mathrm{old}}\gt \beta )\\ 0 & ({otherwise}).\end{array}\right.\end{eqnarray} \tag{ 43 }$$

We adopt β = 0.7. This conversion is applied three times to make the images resemble GMCs with some dense cores. The final images are displayed in Figures 6(c), (d) in a log scale. They have larger dynamic ranges than the original ones, and thus are more challenging to reproduce with interferometer imaging.