# Mosaicking and OTF

This document is intended for observers planning VLA observations using multiple pointing and phase centers to create a "** mosaic**". A mosaic is an image of a patch of sky that is made up of more than one observed field.

Mosaicking should be used when the desired field of view (patch of sky to be observed) is relatively large compared to the Primary Beam at the highest observing frequency (as defined in the Field of View section of the VLA OSS). Per definition of the Primary Beam (θ_{PB}), the image sensitivity will decrease with distance from the center of the field according to a Gaussian with FWHM=θ_{PB}. Thus, the sensitivity at a distance θ_{PB} from the pointing center will be worse by a factor of two as compared with the pointing center. To achieve approximately constant sensitivity over the field of view, the field of view must be << θ_{PB}. If the loss in sensitivity at the edge of the field of view is not acceptable, the observations should be made using a mosaic rather than a single pointing.

**Note: **The Largest Angular Scale (LAS) that can be imaged by the array is *independent* of both the Primary Beam and the use of mosaicking to increase the field of view. A table of the band- and configuration-dependent LAS is presented in the Resolution section of the OSS.

## Mosaic observing with the VLA

There are two different ways of observing a patch of sky that is much larger than the telescope's instantaneous field-of-view. The standard approach, known as a** discrete** or **pointed** mosaic, is to combine together fields from individual pointings of the telescope. This method is typically used for smaller and/or non-rectangular patterns or when significant time needs to be spent per sky area to obtain sufficient sensitivity and image fidelity. The other approach, known as **On-The-Fly** mapping or **OTF**(**M**), combines data that is taken in a 'scanning' mode, where the telescope does not dwell on a position but keeps moving with respect to the sky. OTF is most useful to scan large rectangular patterns on the sky such as for shallow surveys and transient searches where at least one dimension of the mosaic is many times larger than the primary beam. Each method has their advantages, prerequisites, and limitations. Whether to choose one over the other depends on the science goal and boundary conditions such as sensitivity but also, e.g., data rate.

Important considerations are:

- size and shape of the area to cover (in primary beams)
- sensitivity of the observation over the area (amount of integration time required on any single field of the mosaic)
- image requirements (e.g., uv-coverage and largest angular size)

The VLA supports, through General Observing, mosaics that use a discrete pointing pattern. In this standard mode, the mosaic pointing centers are set up as individual fields to be observed (as if they were just a set of target sources). In data post-processing, the data that come from these groups of mosaic fields are jointly deconvolved taking into account the mosaic patterns.

Since semester 2015A, as part of our Shared Risk Observing (SRO) program, the VLA has been offering the opportunity to use OTF mosaicking to more efficiently scan large areas with small dwell times on each point. This is done by moving the telescopes while taking data (and stepping the phase centers for correlation). Special considerations must be taken in processing data taken with this mode. See the section Considerations for On-The-Fly (OTF) Mosaics below and the OPT Manual section on OTF for more details.

### Preparing for mosaic observing: Discrete or On-The-Fly?

You should only use OTF mosaicking if it will be significantly more efficient than standard mosacking. The great benefit of OTF is the ability to eliminate the settling time (after every antenna move) that is required for each pointing in a discrete mosaic. For the VLA, the settle time typically amounts to 7s per pointing. Therefore OTF is particularly useful for large, shallow mosaics that require <15-25s per mosaic beam, where the settle time per field would amount to a very large overhead (>30%-50%) on the observations.

To determine if the standard approach will work for your purposes, first determine the "required integration time per discrete pointing" using the steps outlined below. If the integration time is >25 seconds, then you should use a standard discrete mosaic. For integration times ranging from 14 seconds to 24 seconds it is a gray area - OTF mosaicking would require less overhead, but the extra processing cost and added complexity probably mean you should use standard mosaicking unless you are familiar with processing this type of data in CASA. However, if your integration time is shorter than about 14 seconds, then the time it will take to move and settle between pointings will incur >50% overhead. Unless this extra overhead is not a problem (i.e. the overall mosaic is quite small in area) you should consider using OTF.

Use the following steps to determine the individual integration times (*t _{integ}*) that would be required for discrete pointings.

**You should go through the following steps to prepare your observations, even if you already know which type of mosaic (discrete or OTF) you plan to observe.**

- What area of sky do you want to cover?
- Compute the total mosaic area [inline]A_{\rm mos}[/inline] in appropriate solid angle units, e.g. square degrees.
**Example***I want to cover 5 degrees x 5 degrees, so my area is 25 square degrees.*

- Compute the total mosaic area [inline]A_{\rm mos}[/inline] in appropriate solid angle units, e.g. square degrees.
- What is the effective primary beam size in your observing band? How many independent "beams" are in the mosaic?
- Compute the Full-Width Half-Maximum (FWHM) [inline]\theta_p[/inline] of the VLA at a representative frequency [inline]v_{\rm obs}[/inline], usually the center of your observing band, using the formula
- [display]\theta_P \approx 42^\prime \frac{\rm 1 \: GHz}{\nu_{\rm obs}}[/display] (see the Field of View section in the VLA OSS document)
**Example***I am observing in L-band 1-2 GHz, so ν*(Note that the formula given above for [inline]\theta_P[/inline] is approximate, as the beam is not perfectly linear with frequency. For more accurate beam sizes as a function of frequency, we refer the user to EVLA Memo 195 by R. Perley (2016) for the Karl G. Jansky VLA. For beam sizes for the original VLA, we refer the user to VLA Test Memo 134 by Napier & Rots (1982). )_{obs}= 1.5GHz and θ_{P }[inline]\approx[/inline] 28'.

- Compute the mosaic beam area Ω
_{B }from the FWHM using the formula [display]\Omega_B = 0.5665~\theta^2_P[/display] (see the Gaussian Beam Pattern Sensitivity subsection below for more details)**Example***For my θ*_{P}= 28' = 0.47° the equivalent mosaic beam area is Ω_{B}= 0.123 square degrees.

- Compute the number of independent/effective beam areas in the mosaic using the formula
- [display]N_{\rm eff}=\frac{A_{\rm mos}}{\Omega_B}[/display]
**Example***For my 25 square degree mosaic with Ω*_{B}= 0.123 square degrees I have 203 effective beams.

- Compute the Full-Width Half-Maximum (FWHM) [inline]\theta_p[/inline] of the VLA at a representative frequency [inline]v_{\rm obs}[/inline], usually the center of your observing band, using the formula
- How much integration time do I need? What is my Survey Speed (SS)?
- Compute the integration time per "beam"
*t*using the VLA Exposure calculator. (Note that_{eff}*t*_{eff}is the*effective*integration time for any part of the mosaic, which is not the same as the actual integration time per discrete pointing.)**Example***I wish to reach RMS 0.05 mJy. For 600 MHz usable bandwidth at 1.5 GHz in B-configuration with robust weighting and dual polarization, I need 2m47s on-sky.*

- Compute the total integration time
*t*for the mosaic by multiplying by the number of beams. (Note: to first order, this is independent of how you actually split up the mosaic.)_{total }**Example***For 2m47s per beam and 203 beams I need 9h25m total over the mosaic.*

- Compute the Survey Speed (SS) by taking the mosaic area and dividing by the total integration time (SS =
*A*_{mos}/*t*). This is equivalent to simply computing directly SS = Ω_{total}_{B}/*t*also!_{eff}**Example***For**t*=_{total}*9h25m (9.4 hours) total over the 25 square-degree mosaic, the implied survey speed is about 2.65 square degrees per hour (or equivalently SS = 2.65 square arc-minutes per second).*

- Compute the integration time per "beam"
- What mosaicking pattern would you wish to employ? What will be the spacing between pointings?

- For discrete/pointed mosaics, we recommend to use a hexagonal mosaicking pattern with a spacing of θ
_{hex }along rows and θ_{row}≈ √3 θ/2 between rows. Typically a value of θ_{hex}_{hex }= θ_{P}/√2 is sufficient, but consider using θ_{hex }= θ_{P}/√3 if uniformity is a strong concern. You might choose to use θ_{P}computed from the*upper*frequency limit of the band, as the highest frequency corresponds to the smallest primary beam (thus the highest frequency will have the lowest uniformity in coverage).**Example***For our θ*_{P}= 28' at 1.5 GHz we get θ_{hex}= 19.8' for the spacing along rows and θ_{row}= 17'9" (1029") between rows. This will be more under-sampled at the upper band edge of 2GHz, and over-sampled at 1GHz, but for our basic observations this should be OK.

- For discrete/pointed mosaics, we recommend to use a hexagonal mosaicking pattern with a spacing of θ
- How many discrete pointings will be required to cover this mosaic?

- To fill a rectangular area, long and short rows should alternate, with 1 extra pointing in the long rows.
**Example***Our square mosaic has sides of 5 deg (300'). The spacing between rows is**θ*so there should be 17.5 spacings, and we will therefore schedule 18 rows. Our 300'-length rows will have spacing between pointings of_{row}= 17'9" (1029")*θ*The beginning and end pointings in each of these short rows will be 15 x 19.8' = 297' apart. The short rows will alternate with longer rows of 17 pointings (16 x 19.8' = 316'8" between the pointings at either end of a long row). We will have 9 short rows interspersed with 9 long rows, or 9 x 16 + 9 x 17 = 297 pointings in our mosaic._{hex}= 19.8', so we will observe 16 pointings in the short rows.

- To fill a rectangular area, long and short rows should alternate, with 1 extra pointing in the long rows.
- How much integration time should be allocated per discrete pointing?
- Divide the total mosaic integration time by the number of mosaic pointings.
**Example***Our total integration time of 9h25m is spread among 297 pointings, so each pointing should get 1m54s of integration time.*- You may wish to consider using On-The-Fly (OTF) mosaicking if your integration time per pointing is less than 24 seconds; definitely consider it if your integration time is less than 15 seconds.

- Divide the total mosaic integration time by the number of mosaic pointings.

If at this point you think you should use OTF mosaicking, see the section below on Considerations for On-The-Fly (OTF) Mosaics. Or, continue with the following steps to determine the total amount of observing time required (including overhead) for a discrete mosaic.

- Calculate approximate duration (excluding calibration) for the mosaic.
- The VLA slew and settling time for short distance (sub-degree) moves is 7-8 seconds.
**Example***For 1m54s integrations we add 8s so we can have observations of 2m02s. The total time for 297 pointings is 10h4m.*

- The VLA slew and settling time for short distance (sub-degree) moves is 7-8 seconds.
- Calculate schedule overheads.
- Follow the Exposure and Overhead guidelines in the Guide to Proposing for the VLA.
**Example***For ease of scheduling, we will break our ~10h mosaic into three parts, each with 3h22m of observing time. Our overheads will include: 10m allowance to get on-source at the beginning of each scheduling block, a 10m scan of a flux calibrator, and 3m after every 20m of mosaic observing (10 visits) to observe our gain calibrator. The total time for each scheduling block comes to 4h12m. This amounts to a 25% overhead, which is about average for VLA's low frequencies.*

- Follow the Exposure and Overhead guidelines in the Guide to Proposing for the VLA.
- If project is approved, when it comes time to observe, make a schedule or schedules in the OPT.
- We are working on providing some Python tools (e.g. for CASA) that will help set up mosaic observations. Stay tuned. In the meantime, you may wish to externally generate lists of sources and scans that can be uploaded into the OPT, in order to generate all pointings at once. See the Text Files section of the OPT Manual for instructions.

### Considerations for On-The-Fly (OTF) Mosaics

The use of OTF mosaicking with the VLA is the subject of ongoing development and commissioning, and as such is only available under the SRO program.

If you decide to use OTF mosaicking, after full consideration of the overheads for discrete pointings, the next question is *"Can I use OTF mosaicking for my observations?"*. The decision depends on the implied scan speeds and dump rates (and thus data rates). Limitations on the allowed correlator dump times *t _{integ}* are not just from the allowed data rates (60 MB/s for standard and shared risk observing); there are also physical limitations on how fast the data from the correlator can be handled by the back-end processing cluster. For the current restrictions on integration times, see the OSS section on Time Resolution and Data Rates. Because some use of OTF is still classified as shared-risk (as of 2020B), the data rate limit is nominally 60 MB/s. This means that you can use shorter integration times when choosing OTF as a shared-risk observer, but for efficiency you should not use a shorter integration time than you truly need.

To set up the parameters of the mosaic (e.g. for the purposes of a proposal), start by following the first several steps outlined above. But instead of computing the number of discrete pointings, you will break the OTF mosaic into a number of "rows"; the antennas will slew back and forth one row at a time at a non-sidereal scan rate (*R*). Use the following steps to determine the row spacing (*θ_{row}*; similar to the discrete mosaic case), the required scan rate (

*R*) to meet your sensitivity requirement, the integration/dump times (

*t*), and the associated data rates. Effective integration time (

_{integ}*t*) and total mosaic time are calculated as before.

_{eff}**The following steps correspond to an example that is appropriate for OTF observing.** Suppose you wish to observe 5x10 square degrees in S-band (2-4 GHz) in the B-configuration to a depth of 0.15 mJy. The effective integration time (*t _{eff}*) to reach this sensitivity (robust weighting, dual pol with 1.5 GHz of bandwidth) is 5.35 seconds.

- What is the mosaic beam area? What is my Survey Speed (SS)?
- To determine the Survey Speed (SS = Ω
_{B }/*t*), we recommend using θ_{eff}_{P}at the*middle*of the band, so that the sensitivity can be easily related to the band average of a source with a modest non-zero spectral index. (See Steps 2 and 3 from above.)**Example***At 3 GHz, the primary beam size is θ*_{P }*=14'. The equivalent mosaic beam area is Ω*_{B}= 0.031 square degrees. Therefore the survey speed is 20.75 square degrees per hour, equivalent to 20.75 square arc-minutes per second.

- What is the appropriate row spacing (θ
_{row}) to use between OTF rows? (See Step 4 from above). - To determine θ
_{row}= θ_{P}/√2, we recommend using θ_{P}computed from the*upper*frequency limit of the band, as the highest frequency corresponds to the smallest primary beam (thus the highest frequency will have the lowest uniformity in coverage). Because spacings*along*OTF rows are necessarily small to avoid beam smearing (< 0.1 θ_{P}), smaller spacings*between*OTF rows maintain better homogeneity overall.**Example***At 4 GHz, the primary beam size is θ*=10.5'. Therefore I will use a spacing between rows of θ_{P }_{row}= 7.4'.

- What is the necessary scan rate? (How quickly should the antennas slew across the sky?)
- The scan rate
*R*is calculated as [display]R=\frac{\rm SS}{\theta_{\rm row}}[/display]**Example***For a survey speed of SS = 20.75 square arcminutes per second and θ*_{row}= 7.4 arcmin, I need an OTF scan rate of R = 2.8 arcminutes per second.- Note that scans from east to west move with the sidereal motion while scans from west to east are counter-sidereal. Therefore, for the same on-the-sky angular scan rate, the east-to-west scans will require faster telescope motion. We do not recommend requesting scan rates faster than about 3 arcmin/sec, even under Shared Risk Observing. As always, you should avoid observing near the Zenith where the azimuthal rate becomes very high.

- What dump time (
*t*) should I use and what will be the corresponding data rate?_{integ}- With OTF, fast integrations are required so as not to smear the beam while scanning across the sky. We recommend to have at least 10 samples (integrations) as the antennas scan a distance equal to the FWHM of the primary beam. Therefore the integration time should comply with the formula [display]t_{\rm integ}<0.1\frac{\theta_P}{R}[/display]
**Example***For the primary beam FWHM at 4 GHz (θ*_{P }=10.5') and a scan rate of R=2.8'/s, the maximum t_{integ}to avoid beam smearing is 0.5 seconds.

- Use the OSS section on Time Resolution and Data Ratesto determine the data rate.
**Example***I will base my correlator tuning on the standard setup S-band for B-configuration, using 16 subbands of 128 channels each in dual polarization. For an integration time of**t*=0.5s, this yields a data rate of 22.5 MB/s which is within the standard observing limit of 60 MB/s._{integ}

- With OTF, fast integrations are required so as not to smear the beam while scanning across the sky. We recommend to have at least 10 samples (integrations) as the antennas scan a distance equal to the FWHM of the primary beam. Therefore the integration time should comply with the formula [display]t_{\rm integ}<0.1\frac{\theta_P}{R}[/display]
- How many integrations per phasecenter should I use?
- Currently we do not allow phasecenter changes in OTF mode faster than 0.6 seconds. Furthermore, we recommend keeping the phasecenter constant for at least 1s for 8-bit modes with 16 sub-bands, and for at least 4s for more than 16 sub-bands per polarization (e.g., with 3-bit continuum modes). If integration times faster than these phasecenter changes are needed, you can request multiple integrations per phasecenter in the OPT.
**Example***I want to use a dump time of 0.5s, so I will request two integrations per phasecenter. This means the phasecenter will change once per second.*

- Currently we do not allow phasecenter changes in OTF mode faster than 0.6 seconds. Furthermore, we recommend keeping the phasecenter constant for at least 1s for 8-bit modes with 16 sub-bands, and for at least 4s for more than 16 sub-bands per polarization (e.g., with 3-bit continuum modes). If integration times faster than these phasecenter changes are needed, you can request multiple integrations per phasecenter in the OPT.
- Calculate the approximate duration of a single OTF row to scan once across the length of the mosaic.
- Determine the amount of time and number of phasecenters to complete one row. An additional preparatory step, with the timing of one additional phasecenter, will be added at the beginning of each row to allow the array to accelerate from rest. The acceleration occurs from a starting position such that the antennas are at the appropriate velocity at the true start position of the OTF row. Make sure to account for the slew-and-settle time between rows, which adds about 10s per row.
**Example***The phasecenter will change on a cadence of 1 second, and the scan rate R is 2.8 arcmin/sec. Therefore every phasecenter will correspond to 2.8 arcmin of distance scanned. To scan a length of 10 degrees = 600 arcmin across the mosaic in one direction requires N=215 phasecenters. Including the starting acceleration, the scan duration will be (N+1) * 1s = 3m36s per OTF row. Adding in 10s for slew-and-settle time, this comes out to 3m46s per row.*- Note: We currently advocate OTF mosaics where the stripes are at constant Declination. You will want to switch directions for each row so as to scan back and forth (e.g., from west to east for one row, then from east to west for the next, etc.). For mosaics that are very large in the RA direction, you may wish to split the mosaic into several sections so that an individual row is not too long - scanning a long distance in RA can make it difficult to keep your observations above a reasonable elevation limit while maintaining the flexibility of dynamic scheduling. For example, a mosaic with an RA range of 0h to 3h might be broken into three sections (0-1h, 1-2h, 2-3h), with all of the different declination stripes for one RA range observed as a group before moving to the next section.

- Determine the amount of time and number of phasecenters to complete one row. An additional preparatory step, with the timing of one additional phasecenter, will be added at the beginning of each row to allow the array to accelerate from rest. The acceleration occurs from a starting position such that the antennas are at the appropriate velocity at the true start position of the OTF row. Make sure to account for the slew-and-settle time between rows, which adds about 10s per row.
- Calculate the approximate duration of the OTF mosaic, excluding calibration overheads.
- Determine the total number of OTF rows to cover the mosaic, and the total time to cover the mosaic.
**Example***With each row separated by 7.4 arcmin, I will need 41 rows to cover the 5-degree height of the mosaic. The full OTF mosaic will therefore require about 2h35m to observe.*

- Determine the total number of OTF rows to cover the mosaic, and the total time to cover the mosaic.
- Calculate schedule overheads.
- Follow the Exposure and Overhead guidelines in the Guide to Proposing for the VLA.
**Example***Our overheads will include: 10m allowance to get on-source at the beginning of each scheduling block, a 10m scan of a flux calibrator, and 3m after about every 4 rows (15m) of OTF observing (11 visits total) to observe our gain calibrator. The total time comes out to 3h28.*

- Follow the Exposure and Overhead guidelines in the Guide to Proposing for the VLA.
- If project is approved, when it comes time to observe, make a schedule or schedules in the OPT.
- You will need to generate a source list that contains both the starting and ending position of each individual OTF row. You may wish to externally generate lists of sources and scans that can be uploaded into the OPT. See the Text Files section of the OPT Manual for instructions on setting up OTF observing schedules.

### Survey Speed of the VLA for Large Continuum Mosaicked Surveys

Following the above guidelines, we can compute the survey speed SS [display]SS = \Omega_B / t_{eff} = 0.5665~\theta^2_{P} / t_{eff}[/display] of the VLA for our standard bands calculated at a given depth in RMS image sensitivity. We choose a "RMS Noise" of 0.1 mJy/beam in the Exposure Calculator (also 25 antennas, natural weighting, dual polarization, medium elevation, autumn weather, B-configuration) to compute the effective integration time, *t _{eff}*. From this, we calculate SS from θ

_{P}at band center.

The parameters are tabulated by band below:

Band (freq) | Freq. | Bandwidth | t_{eff} (sec) | θ_{P} (arcmin) | SS (deg^{2}/hr) |
---|---|---|---|---|---|

P (230-470MHz) |
370 MHz | 200 MHz | 5940 | 135' | 1.74 |

L (1-2GHz) |
1.5 GHz | 600 MHz | 29 | 28' | 15.32 |

S (2-4GHz) |
3 GHz | 1500 MHz | 8.3 | 14' | 13.38 |

C (4-8GHz) |
6 GHz | 3.03 GHz | 4.4 | 7' | 6.31 |

X (8-12GHz) |
10 GHz | 3.50 GHz | 2.9 | 4.2' | 3.45 |

K_{U} (12-18GHz) |
15 GHz | 5.25 GHz | 3.5 | 2.8' | 1.27 |

K (18-26.5GHz) |
22 GHz | 7.20 GHz | 6.9 | 1.91' | 0.30 |

K_{A} (26.5-40GHz) |
33 GHz | 7.20 GHz | 11 | 1.27' | 0.083 |

Q (40-50GHz) |
45 GHz | 7.20 GHz | 50 | 0.93' | 0.0098 |

For C-band and higher frequencies 3-bit observing is assumed. Representative frequency, integration time, beam width, and survey speed are at approximately mid-band. You can adjust these values for different assumed sensitivity levels and bandwidths (e.g. for line sensitivity) by scaling according to the values that come out of the Exposure Calculator (e.g. SS will scale as the inverse of the integration time). These values are computed in the limit of OTF (continuous) sampling, but should be approximately valid for optimally sampled Hex mosaics also (see below). The beam widths here are approximate (see EVLA Memo 195 by R. Perley, 2016) and are narrow-band. For a wide-band mosaic, see the section below on Effective Primary Beam for a Wideband Mosaic.

## The Details: Mosaic Sensitivity

Following are some in-depth calculations of the discrete mosaic sensitivity, provided for users who wants to know the gory details of how the values are calculated. These formulas are generally applicable to mosaics made with any interferometer (e.g. ALMA, ATCA), although some allowances would need to be made in the calculations to allow for non-homogeneous array elements (e.g. with antennas of different sizes as in ALMA+ACA, CARMA).

### Gaussian Beam Pattern Sensitivity

We will be assuming a Gaussian pattern [inline]\theta_g[/inline] for the main beam response (the so-called primary beam pattern) assuming an array of homogeneous antennas. The sensitivity pattern or response to point sources at a distance θ from the pointing center on-sky is given by

[display]f(\theta) = e^{-{\frac{\theta^2}{2\theta^2_g}}}[/display]

The 2-D integral under this function gives the effective Gaussian beam area (solid angle)

[display]\Omega_g = 2\pi\theta^2_g[/display]

For purposes of mosaic coverage, the area under the primary beam *squared* is relevant ([inline]f(\theta)^2[/inline]):

[display]f(\theta)^2 = e^{-2{\frac{\theta^2}{2\theta^2_g}}} = e^{-\frac{\theta^2}{2(\theta_g/\sqrt{2})^2}}[/display]

So the effective Gaussian primary beam for a mosaic is equivalent to a Gaussian with half the area:

[display]\Omega_B = 2\pi(\frac{\theta_g}{\sqrt 2})^2 = \pi\theta^2_g = \frac{\Omega_g}{2}[/display]

It is common practice to specify the Gaussian width by the "full-width half-maximum" (FWHM) θ_{P}, where

[display]\theta_P = \sqrt{(8 \ln 2)}\; \theta_g = 2.3548~\theta_g[/display]

or

[display]\theta_g = 0.4247~\theta_P[/display]

We can reformulate the response function in terms of the FWHM via substitution:

[display]f(\theta) = e^{-\frac{\theta^2}{2\theta_g^2}} = e^{-4 \ln 2 (\frac{\theta}{\theta_P})^2} = 2^{-4 (\frac{\theta}{\theta_P})^2}[/display]

Our beam areas are

[display]\Omega_g = 2\pi\left(\frac{\theta_P}{\sqrt{8 \ln 2}}\right)^2 = 1.1331~\theta^2_P[/display]

and for the beam-squared

[display]\Omega_B = \frac{\pi}{8\ln 2} \theta^2_P = 0.5665~\theta^2_P[/display]

### Effective Primary Beam for a Wideband Mosaic

The above formulas for the Primary Beam are approximations that apply exactly in the case of a narrow-band mosaic. For a wideband (multi-frequency synthesis) mosaic, the effective primary beam depends on the frequency variation of the narrow-band beam widths and the sensitivity as a function of frequency.

The mosaic imaging process weights the data explicitly by assigned weights (e.g., by the rms noise) and implicitly by the beam area at each frequency (because the effective integration time at each frequency is proportional to the beam area). This effect was pointed out by Condon (2015; reference 5) and can be simply calculated as the frequency-weighted mean beam area over the frequency channels [inline]k[/inline] according to the formula

[display]\bar{\Omega}_B = \frac{\Sigma_k\; w_k\; \Omega_{Bk}}{\Sigma_k\; w_k}[/display]

For uniform weights [inline]w_k[/inline] = const. and uniform frequency coverage over the band, we can approximate this sum by the integral

[display]\bar{\Omega}_B = \frac{1}{\nu_{\rm max} - \nu_{\rm min}}\; \int^{\nu_{\rm max}}_{\nu_{\rm min}}\; d\nu\; \Omega_B(\nu)[/display]

If we assume the primary beam FWHM scales inversely by frequency, then

[display]\bar{\Omega}_B = \frac{\nu_0^2}{\nu_{\rm min}\; \nu_{\rm max}}\; \Omega_B(\nu_0)[/display]

[display]\Omega_B(\nu) = \Omega_B(\nu_0)\; \left(\frac{\nu_0}{\nu}\right)^2[/display]

or equivalently [inline]\bar{\Omega}_B = \Omega_B(\bar{\nu})[/inline] where [inline]\bar{\nu} = \sqrt{\nu_{\rm min}\; \nu_{\rm max}}[/inline] is the geometric mean frequency.

### Weighted Image Sensitivity

A mosaic image can be considered to be a weighted sum of individual field image data ([inline]d_k[/inline]) corrected for the beam response ([inline]f[/inline]) at each individual pointing:

[display]F = \frac{1}{Z}\sum_{k} w(\theta_k) f^{-1}(\theta_k)\; d_k[/display]

[display]Z = \sum_{k} w (\theta_k)[/display]

where the [inline]\theta_k[/inline] are the distances to the pointing centers for the image data points *d*_{k}, [inline]w(\theta_k)[/inline] is the weight for data point *d*_{k}, and [inline]Z[/inline] is the sum-of-weights function. If the image data have equal RMS sensitivity levels σ_{k} = σ_{0} then the *optimal* weighting gives

[display]w(\theta_k)=f^2(\theta_k)[/display]

and

[display]F = \frac{1}{Z}\sum_{k}f(\theta_k)\; d_k[/display]

[display]Z = \sum_{k} f^2 (\theta_k)[/display]

This image will have the lowest possible RMS noise level, with the variance of [inline]F[/inline] given by

[display]\sigma_F^2 = \frac{1}{Z^2}\sum_k f^2(\theta_k)\sigma_0^2 = Z^{-1}\sigma_0^2[/display]

which just scales inversely with the sum-of-weights function Z. Since the equivalent integration time at a given point in the mosaic is inversely proportional to the variance (with all other things being equal) then this is given by Z:

[display]t_{eff} = Z t_0[/display]

where *t* is the integration time per field (assuming a uniformly observed mosaic).

### Discrete Mosaic Spacing Considerations

For discrete (as opposed to OTF) mosaicking, the sampling pattern and spacing of pointing centers determines the sensitivity response of the mosaic. The concept of stepping or scanning an interferometer over an area of sky to synthesize a larger image has been around for a long time, see Ekers & Rots 1979 (reference 1) for the conceptual framework.

The simplest pattern is a rectangular mosaic, with pointing centers at vertices of squares. From the perspective that the FFT of the mosaic pattern is a "synthesized beam" in uv-space that sub-samples the antenna voltage patterns, the Nyquist sampling theorem suggests that a spacing of [inline]\theta_{\rm rect} = \theta_P/2[/inline] or better is needed (e.g. Cornwell 1998, reference 2). This is the spacing of samples on the sky needed to reconstruct the low spatial frequencies on the scale of the primary beam θ_{P}. However, if the goal is merely to cover large areas of sky to survey for relatively compact sources, then the spacing limit given by Nyquist sampling of the primary beam can be loosened and wider separations can be used, as long as the dimples in the sensitivity pattern are not too deep for purposes of having a nearly uniform survey over a large area.

The hexagonal-packed mosaic is the classic mosaic observing pattern. It has long been used at the VLA (e.g. for the NVSS, see reference 3) and at the ATCA (see reference 4). This pattern is like a regular rectangular raster but with alternate rows offset by 1/2 field separation, allowing rows to be placed further apart while still getting nearly uniform sensitivity. The mosaic is thus filled by equilateral triangles, with the triangle vertices defining the pointing centers of the pattern.

The ATCA recommended value for [inline]\theta_{\rm hex} = {\theta_P}/{\sqrt 3}[/inline] is based on Nyquist arguments (see reference 4). For the NVSS survey (reference 3), the authors argued that a spacing not much wider than [inline]\theta_{\rm hex} = {\theta_P}/{\sqrt 2}[/inline] would be acceptable from a sensitivity perspective, and in fact used a spacing of approximately [inline]{\theta_P}/{1.2}[/inline]. This is sufficient (see below) to have a reasonably uniform sensitivity pattern.

For the current VLA, where we have 2:1 bandwidths possible in a given band (e.g. 1-2 GHz, 2-4 GHz), you have to consider the spacing with respect to the primary beam FWHM over the range of frequencies you are going to map together. For example, for observations from 1-2 GHz, setting a spacing of 0.71 FWHM at 1.5 GHz would give a spacing of only 0.94 FWHM at 2 GHz which gives a minimum weight of *Z*_{hexmin} = 0.586 (see below), but also with significant oversampling of 0.47 FWHM at 1 GHz. You may wish to err on the side of caution in these cases if having more variable sensitivity at the upper band edge is expected to be an issue for you. Note that we have not quantified any imaging consequences from this (e.g. for spectral index maps) so for now these are just some general guidelines.

**Recommendation:** For most cases where structure on large angular scales is not being imaged, a hex-pattern mosaic with relatively loose spacing of 0.70 - 0.85 FWHM is probably sufficient. If good imaging of large-scale low surface brightness emission is the goal, then a mosaic sampled at the Nyquist spacings or better should be used. In most cases, you can reasonably get away with setting the spacing by the FWHM at the center of your observing band, leaving the mosaic at the upper end of the band less well-sampled while the mosaic at the lower band edge will be better sampled. If for some reason you require excellent sampling over the whole band, then set the spacing using the FWHM at the highest frequency to be safe.

#### Example: Discrete Hexagonal Mosaic

Each mosaic pointing center has 6 nearest neighbors (hence the hexagonal pattern), with a distance to each given by θ_{hex}. For a pixel at a pointing center, counting that point and the 6 neighboring centers, the sum-of-weights is given by

[display]Z_{\rm hexmax} = 1 + 6f^2(\theta_{\rm hex})[/display]

The worst response is at the center of one of the equilateral triangle tiles. The nearest 3 vertices are at distances given by [inline]\theta_{\rm hex}/{\sqrt 3}[/inline] giving

[display]Z_{\rm hexmin} = 3 f^2 (\frac{\theta_{\rm hex}}{\sqrt 3})[/display]

If NVSS-style image-plane mosaicking is used, then next sets of vertices out will likely not be included in the image due to an imposed cutoff (see reference 3). Assuming a moderately liberal spacing of [inline]\theta_{\rm hex} = \theta_P/{\sqrt{2}}[/inline] we get:

[display]\theta_{\rm hex} = \frac{\theta_P}{\sqrt 2}[/display]

[display]f(\theta_{\rm hex}) = 0.25[/display]

[display]f(\theta_{\rm hex}/\sqrt 3) = 0.63[/display]

and

[display]Z_{\rm hexmax} = 1.375\; \; \; \; \; \; \; \;Z_{\rm hexmin} = 1.191[/display]

Thus the lowest points in the mosaic weighting pattern are at 0.87 of the maximum (for the NVSS choice, they are at 0.81 of the maximum). Note that the locations at the pointing centers get an equivalent integration time of 1.375 times the individual pointing integrations.

A large hexagonal mosaic of N x M rows and columns will cover a total area of approximately

[display]A_{\rm hex} = N \times \theta_{\rm hex} \times M \times \frac{\sqrt 3}{2}\theta_{\rm hex} = \frac{\sqrt{3}}{2} N_{\rm pt}\theta^2_{\rm hex}[/display]

For a sampling of [inline]\theta_{\rm hex}= \theta_P/\sqrt{2}[/inline] we get

[display]A_{\rm hex} = \frac{\sqrt3}{4}N_{\rm pt}\theta^2_P = 0.7644 N_{\rm pt}\Omega_B[/display]

for the area under the squared beam defined above. If we observe the mosaic for a total time *T* with each of the *N*_{pt} pointings getting the same integration time *t*_{int}

[display]t_{int} = \frac{T}{N_{\rm pt}}[/display]

then

[display]A_{\rm hex}t_{\rm int} = 0.7644 T \Omega_B[/display]

or using the effective integration time per point in the mosaic

[display]A_{\rm hex}t_{\rm eff} = 0.7644 Z\;T \Omega_B[/display]

For our hexagonal mosaic, the minimum weight is *Z*_{hexmin} = 1.191 so

[display]A_{\rm hex} t_{\rm hexmin} = 0.91 T \Omega_B[/display]

For practical purposes, as we will see later on, mosaics in general follow the relation that

[display]A_{\rm mos} t_{\rm eff} \approx T \Omega_B[/display]

which can be used to compute the effective integration time on-sky to put into the exposure calculator for RMS sensitivity.

**Recommendation:** For most cases where you are using a hexagonal (or rectangular or OTF) mosaic with close to the optimal sampling and want the average sensitivity (not the max or min specifically), you can simply use the following to calculate the total integration time needed

[display]T \approx t_{\rm eff}\frac{A_{\rm mos}}{\Omega_B}[/display]

after getting *t*_{eff} from the VLA Exposure Calculator for your needed sensitivity and chosen bandwidth etc.

#### Example: Discrete Rectangular Mosaic

The use of rectangular mosaics has been deprecated in favor of hexagonal packed mosaics, but there are cases where they are expedient to set up, and they provide an illustrative case leading in to the discussion of on-the-fly mosaics.

For the rectangular mosaic each point is surrounded by eight immediate neighbors with the 4 nearest separated by θ_{rect} in the cardinal directions and next 4 by [inline]\sqrt{2}\theta_{\rm rect}[/inline] on the diagonals. Thus,

[display]Z_{\rm rectmax} = 1 + 4 f^2(\theta_{\rm rect}) + 4 f^2(\sqrt{2}\theta_{\rm rect}) = 2.25[/display]

for the optimal *θ*_{rect} = 0.5 *θ _{P}*.The weight minima have 4 nearest neighbors at

*θ*

_{rect}/√2

[display]Z_{\rm rectmin} = 4 f^2(\frac{\theta_{\rm rect}}{\sqrt 2}) = 2[/display]

Thus our rectangular mosaic has dimples at 0.89 of the maximum response. The maximum effective integration time is 2.25 times the per pointing integration time. Our mosaic of N rows by M columns covers an area of approximately

[display]A_{\rm rect} = N \times \theta_{\rm rec} \times M \times \theta_{\rm rect} = N_{\rm pt}\theta^2_{\rm rect}[/display]

so going through the same calculation as for the hexagonal mosaic

[display]A_{\rm rect} = \frac{1}{4}N_{\rm pt}\theta^2_{P} = 0.4413 N_{\rm pt}\Omega_B[/display]

and

[display]A_{\rm rect}t_{\rm eff} = 0.4413\; Z\; T\; \Omega_B[/display]

For our rectangular mosaic, the minimum weight is *Z*_{rectmin} = 2 so

[display]A_{\rm rect}t_{\rm rectmin} = 0.88 T \Omega_B[/display]

### The Continuum Limit and On-the-Fly Mosaic Sensitivity

The following are some in-depth calculations of the OTF mosaic sensitivity.

We first calculate the sensitivity in the continuum limit, where the array scans the sky over a given area A in a time T in as uniform a manner as possible. In this case, except near the edges, each point along the row has the same weight, and our sums in the previous derivations become integrals. The image at a given position on the sky ([inline]{\rm\bf x}_0[/inline]) amounts to a weighted integration of the field data ([inline]D(x)[/inline]) over all nearby sky positions ([inline]{\rm\bf x}[/inline]) corrected by the beam response ([inline]f[/inline]), and keeping in mind that the beam scans across the sky over time:

[display]F({\rm x_0}) = \frac{1}{Z}\int dt\; w({\rm\bf x-x_0})\; f^{-1}({\rm \bf x - x_0})\; D(x)[/display]

[display]{\rm\bf x} = {\rm\bf x}(t)[/display]

with normalization

[display]Z = \int dt\; w({\rm\bf x})[/display]

and as before

[display]w({\rm\bf x}) = f^2(x) = e^{-\frac{x^2}{\theta^2_g}}[/display]

We are sweeping at a constant rate so the areal (solid angle) rate is

[display]\dot{\Omega} = \frac{d\Omega}{dt} = \frac{dx\; dy}{dt} = \frac{A_{\rm mos}}{T}[/display]

where *A*_{mos} is the total area of the mosaic and *T* the total integration time as before. Thus, we can recast the integrals

[display]F({\rm\bf x}_0) = \frac{1}{Z}\int\int\frac{dx\; dy}{\dot{\Omega}}\; w({\rm\bf x} - {\rm\bf x}_0)\; f^{-1}({\rm\bf x} - {\rm\bf x}_0)\; D({\rm\bf x})[/display]

[display] = \frac{1}{Z\dot{\Omega}}\int\int dx\;dy\; f({\rm\bf x} - {\rm\bf x}_0)\; D({\rm\bf x})[/display]

and more critically the normalization (which is constant over the uniform part of the mosaic) is related to area of the squared beam

[display]Z = \int\int dt \; w({\rm\bf x}) = \int\int\frac{dx\; dy}{\dot{\Omega}}f^2({\rm\bf x}) = \frac{T}{A_{\rm mos}}\Omega_B [/display]

As before, we can compute the RMS sensitivity

[display]\sigma^2_F = \frac{1}{Z^2}\int\int\frac{dx\; dy}{\dot{\Omega}}f^2({\rm\bf x})\; \sigma^2_D = \frac{\sigma_D^2}{Z}[/display]

where σ^{2}_{D} is the sensitivity of the data per unit time, so again we have the relation

[display]Z = \frac{\sigma^2_D}{\sigma^2_F} = t_{\rm eff}[/display]

Thus, in general for a uniformly scanned continuous mosaic, we have the survey area time product relation

[display]A_{\rm mos}t_{\rm eff} = T \Omega_B[/display]

which is what we found approximately for our hexagonal and rectangular discrete mosaics above. The effective integration time per point on sky is given by dividing the total time by the effective number of mosaic beams

[display]t_{\rm eff} = \frac{T}{N_B}\; \; \; \; \; \; \; \; N_B = \frac{A_{\rm mos}}{\Omega_B}[/display]

You can use the standard radiometer calculation (e.g. with the VLA Sensitivity Calculator) to compute the expected RMS on-sky for this effective integration time.

## References

1. "Short Spacing Synthesis from a Primary Beam Scanned Interferometer", Ekers & Rots 1979, IAU Colloq. 49: Image Formation from Coherence Functions in Astronomy, 76, 61

2. "Radio-interferometric imaging of very large objects", Cornwell 1988, A&A, 202, 316

3. "The NRAO VLA Sky Survey", Condon et al. 1998, AJ 115, 1693.

4. "Mosaicing Observing Strategies", MIRIAD Users Guide, http://www.atnf.csiro.au/computing/software/miriad/userguide/node168.html

5. "An Analysis of the VLASS Proposal" Condon 2015, astro-ph > arXiv:1502.05616