VLA > Mosaicking and OTF

# Mosaicking and OTF

## Mosaics

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 larger than the field of view of the telescope and thus is made up of more than one observed field. There are two different ways of making up such a patch: one is to combine together fields from individual pointings of the telescope, the other is to combine 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.

These two different ways are referred to as a discrete or pointed mosaic, and on-the-fly mapping or OTF(M), respectively. The former is typically used for smaller and non-rectangular patterns or when significant time needs to be spent per sky area to obtain sufficient sensitivity and image fidelity, the latter 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 which are outlined below. 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 normal 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.

### Considerations for Discrete Mosaics

To set up a discrete mosaic for VLA observing, you need to first work out:

1. What area of sky do you want to cover?
• Compute the total mosaic area $A_{\rm mos}$ 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.
2. 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) $\theta_p$ of the VLA at a representative frequency $v_{\rm obs}$, usually the center of your observing band, using the formula
$$\theta_P \approx 42^\prime \frac{\rm 1 \: GHz}{\nu_{\rm obs}}$$ (see the Field of View section in the VLA OSS document)
Example
I am observing in L-band 1-2 GHz, so νobs = 1.5GHz and θP $\approx$ 28'
(Note that the formula given above for $\theta_P$ 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) )
• Compute the mosaic beam area ΩB from the FWHM, using the formula (see the Gaussian Beam Pattern Sensitivity subsection below for more details) $$\Omega_B = 0.5665 \theta^2_P$$
Example
For my θP = 28' = 0.47° the equivalent mosaic beam area is ΩB = 0.123 square degrees
• Compute the number of independent/effective beam areas in mosaic using the formula

$$N_{\rm eff}=\frac{A_{\rm mos}}{\Omega_B}$$
Example
For my 25 square degree mosaic with ΩB = 0.123 square degrees I have 203 effective beams.
3. How much integration time do I need? What is my Survey Speed (SS)?
• Compute the integration time per "beam" tcalc using the VLA Exposure calculator
Example
I wish to reach RMS 0.05 mJy. For 600MHz useable bandwidth at 1.5GHz in B-configuration robust weighting dual pol I need 3m31s on-sky.
• Compute the total integration time ttotal 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.)
Example
For 3m31s per beam and 203 beams I need 11h54m total over the mosaic.
• Compute the Survey Speed (SS) by taking the mosaic area and dividing by the total integration time (SS = Amos/ ttotal). This is equivalent to simply computing directly SS = ΩB / tcalc also!
Example
For ttotal=11h54m (11.9 hours) total over the 25 square degree mosaic the implied survey speed is 2.10 square degrees per hour (or equivalently SS = 2.10 square arc-minutes per second).
4. What mosaicking pattern do 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 θhex long rows and θrow ≈ √3 θhex/2 between rows. (Typically a value of θhex = θP/√2 is sufficient, but consider using θhex = θP/√3 if uniformity is a strong concern.)
Example
For our θP = 28' we get θhex = 19.8' for the spacing along rows and θrow = 17'9" (1029") between rows. We note that 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.
5. How many pointings will this mosaic take for this pattern?
• 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 θrow = 17'9" (1029") so there should be 17.5 spacings, and we will therefore schedule 18 rows. Our 300'-length rows will have spacing between pointings of θhex = 19.8', so we will observe 16 pointings in the short rows. The beginning and end pointings in each of these short rows will be 15 x 19.8' = 284' apart. The short rows will alternate with longer rows of 17 pointings (16 x 19.8' = 306'8" 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.
6. 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 11h54m is spread among 297 pointings, so each pointing should get 2m24s 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 - see below.
7. Calculate approximate duration (excluding calibration) for the mosaic.
• The VLA slew and settling time for short distance (sub-degree) moves is 6-7 seconds.
Example
For 2m24s integrations we add 7s so we can have observations of 2m31s. The total time for 297 pointings is 12h28m.
• Follow the Exposure and Overhead guidelines in the Guide to Proposing for the VLA.
Example
For ease of scheduling, we will break our ~12.5h mosaic into three parts, each with 4h10m 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 about every 28m (10 visits) to observe our gain calibrator. The total time for each scheduling block comes to 5 hours. This amounts to a 20% overhead, which is about average for VLA's low frequencies.
9. If project is approved, when it comes time to observe, make schedule in 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

For OTF, rather than producing a mosaic from a large number of individual pointings, the VLA continuously takes data while scanning across the sky. An OTF observation typically amounts to scanning in a "back and forth" fashion over a large rectangular area of sky, one row at a time. The great benefit of OTF mosaicking is the ability to eliminate the slew-and-settle time that is required for each pointing in a discrete mosaic. Recall that the slew-and-settle time typically amounts to 7s per pointing, so OTF is particularly useful for large, shallow mosaics that require <15-25s per mosaic beam.

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.

The most important question is "Do I need to use OTF mosaicking or is standard mosaicking sufficient?". You should only use OTF mosaicking if it will be significantly more efficient than standard mosacking. This comes down to whether the required dwell time in a pointing is so short that the 7 seconds of slew-and-settle time (3-4 seconds at best for short, elevation-only moves) per field will incur a very large overhead (>30%-50%) on the observations. Thus, you should refer back to the "integration time per discrete pointing" you calculated above in Step 6. If this is shorter than 12-14 seconds, then the 6-7 seconds it will take to move and settle between pointings will incur >50% overhead. In this case, unless this extra overhead is not a problem (i.e. the overall mosaic is quite small in area) you will likely wish to use OTF. You might also be able to arrange your mosaic schedule so that the moves are in elevation, which in some cases brings the overhead down to 3-4s. If you need less than 7 seconds per pointing you almost certainly want to use OTF! From 14 seconds to 24 seconds it is a gray area - you will benefit from OTF mosaicking but the extra processing cost and added complexity probably means you should just use standard mosaicking unless you are familiar with processing this type of data in CASA.

If you decide to use OTF mosaicking, the next question is "Can I use OTF mosaicking for my observations?". In other words, will the implied scan speeds and dump rates (and thus data rates) be within the capabilities of the VLA? This comes down to whether or not you have to scan the array too fast and set the integration times tinteg out of the correlator back-end too short as to not smear the beam. We recommend that you scan such that you have at least 10 samples across the primary beam FWHM, e.g. that your required scan rate R < 0.1 θP / tinteg where the minimum allowed tinteg is set by the maximum allowed data rate (usually 25MB/s for standard observing - for bandwidths of 2GHz or less this means 0.5 sec, and correspondingly longer for 3-bit modes). Note: as of 2017A, OTF is still classified as shared-risk for which the limit is nominally 60MB/s. This means that you can use shorter integration times when choosing OTF as you are already a shared-risk observer! But don't use a shorter integration time than you really need.

Thus, you determine the implied scanning rate R = SS / θrow from the computed survey speed (SS = ΩB / tcalc from Step 3 above) and the spacing of OTF rows θrow from Step 4 required to uniformly sample the mosaic. To determine θrowrow = θP/√2), we recommend using θP 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). To determine the Survey Speed, we recommend using θP 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. For example, at S-band (using θP= 15' at 3GHz, ΩB=127.5 square arc-minutes) for a depth of 0.1 mJy (tcalc= 7.7 sec) we have a survey speed SS = 16.56 square arc-minutes per second. For a row spacing of 8' (θP/√2 with θP= 11.25' at 4GHz) we need an OTF scan rate of R = 2.07 arc-minutes per second. In terms of the primary beam FWHM at 4 GHz (11.25') we have a rate relative to the PB of R/θP = 0.18 per second. Thus, to keep RtintegP < 0.1 we need tinteg< 0.54 seconds and thus 0.5 second integrations are recommended. This gives a data rate of 24MB/s which is within the standard observing limit of 25MB/s.

Note that the limitations on the allowed correlator dump times tinteg are not just from the allowed data rates (25MB/s for standard observing and 60MB/s for shared risk). There are also physical limitations on how fast the data from the correlator can be handled by the back-end processing cluster. Currently, we do not allow phasecenter changes in OTF mode faster than 0.6 seconds. However, it is possible in the OPT to request multiple integrations per phasecenter (e.g., one might request an integration time of 0.5s with two pointings per phasecenter, such that the phasecenter will change once per second). We restrict the allowed visibility integration times to a minimum tinteg of 0.5sec for up to 4GHz of correlated bandwidth, and 1 second for 4-8GHz of bandwidth. Shorter integration times are possible, but require restricted bandwidth to stay within the allowed data rates. For the current restrictions on integration times, see the OSS section on Time Resolution and Data Rates.

To set up the parameters of the mosaic (e.g. for the purposes of a proposal), carry out the steps outlined above, except for computing the number of discrete pointings.

• Instead, you will break the OTF mosaic into a number of "rows" each of which is one or more "stripes" (e.g. a row from RA 0h to 0h15m at a given Declination might be broken into three stripes 0m-5m, 5m-10m, and 10m-15m) with calibration in between. Rows should be separated as in the hexagonal mosaic case, we recommend a value around $$\theta_{\rm row} = \theta_P / \sqrt{2}$$ where θP is computed at the upper frequency limit of the band.
• We currently advocate OTF mosaics where the stripes are at constant Declination (e.g. you only need to clock the HA rate of the telescope). 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.). 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. We have confirmed that OTF functions acceptably for scan rates R up to 3 arcmin/sec, and are testing faster rates. Note that, as always, you should avoid observing near the Zenith where the azimuthal rate becomes very high.
• Integration time and total mosaic time are calculated as before. Decide stripe times based on this.
• The correlator dump time tinteg should be calculated using the scan rate considerations given above. We recommend to have RtintegP < 0.1 (i.e., at least 10 integrations covering the distance across the primary beam).
• Your schedules for basic OTF mosaicking can now be made in the OPT. Choose "On The Fly Mosaicking" as the SCAN MODE, and fill in the START SOURCE, END SOURCE positions (from a Source Catalog as usual), which will set up a single stripe between those positions. You also choose a "Number of Steps" for phase center switching, and a number of "Integrations per Step". For example, for an instrument configuration with 0.5sec integrations you might choose 135 steps with 8 integrations per step. This means the phase center will change every 4 seconds (8 integrations) and thus an OTF stripe will be 540 seconds long (9 minutes). Note that an additional preparatory step will be added to allow the array to get moving in the right direction, and so the "stripe duration" will be reported as 544 seconds. Finally, the user chooses a total Duration for the scan which must be larger than the "stripe duration" needed and including overhead to get to the start position. If you have a previous scan at or close to the START SOURCE then something like an additional 10-12 seconds should be enough. In our example a Duration of 9m15s should be enough. You then schedule the stripes (scans) needed to do your entire mosaic. Note: we recommend setting the "Integrations per Step" to a value corresponding to at least 1 second for 8-bit modes with 16 sub-bands per polarization, or at least 4 seconds for more than 16 sub-bands per polarization (e.g. 3-bit continuum modes).

Rather than entering each OTF scan into the OPT individually, you may wish to generate a scan list in an external text file that can be uploaded into the OPT, in order to generate all scans at once. See the Text Files section of the OPT Manual for instructions.

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

Following the above guidelines (Steps 1-3 in the mosaic calculations) we can compute the survey speed SS $$SS = \Omega_B / t_{calc} = 0.5665 \theta^2_{P} / t_{calc}$$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 tcalc. From this, we calculate SS from θP at band center.

The parameters are tabulated by band below:

Band (freq)Freq.Bandwidthtcalc (sec)θP (arcmin)SS (deg2/hr)
P (230-470MHz) 370 MHz 200 MHz 5940 135' 1.74
L (1-2GHz) 1.5 GHz 600 MHz 37 28' 12.00
S (2-4GHz) 3 GHz 1500 MHz 7.7 14' 14.42
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
KU (12-18GHz) 15 GHz 5.25 GHz 3.5 2.8' 1.27
K (18-26.5GHz) 22 GHz 7.20 GHz 7.0 1.91' 0.26
KA (26.5-40GHz) 33 GHz 7.20 GHz 11 1.27' 0.083
Q (40-50GHz) 45 GHz 7.20 GHz 50 0.93' 0.045

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 $\theta_g$ 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

$$f(\theta) = e^{-{\frac{\theta^2}{2\theta^2_g}}}$$

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

$$\Omega_g = 2\pi\theta^2_g$$

For purposes of mosaic coverage, the area under the primary beam squared is relevant ($f(\theta)^2$):

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

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

$$\Omega_B = 2\pi(\frac{\theta_g}{\sqrt 2})^2 = \pi\theta^2_g = \frac{\Omega_g}{2}$$

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

$$\theta_P = \sqrt{(8 \ln 2)}\; \theta_g = 2.3548 \theta_g$$

or

$$\theta_g = 0.4247 \theta_P$$

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

$$f(\theta) = e^{-\frac{\theta^2}{2\theta_g^2}} = e^{-4 \ln 2 (\frac{\theta}{\theta_P})^2} = 2^{-4 (\frac{\theta}{\theta_P})^2}$$

Our beam areas are

$$\Omega_g = 2\pi\left(\frac{\theta_P}{\sqrt{8 \ln 2}}\right)^2 = 1.1331 \theta^2_P$$

and for the beam-squared

$$\Omega_B = \frac{\pi}{8\ln 2} \theta^2_P = 0.5665 \theta^2_P$$

### 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 $k$ according to the formula

$$\bar{\Omega}_B = \frac{\Sigma_k\; w_k\; \Omega_{Bk}}{\Sigma_k\; w_k}$$

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

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

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

$$\bar{\Omega}_B = \frac{\nu_0^2}{\nu_{\rm min}\; \nu_{\rm max}}\; \Omega_B(\nu_0)$$

$$\Omega_B(\nu) = \Omega_B(\nu_0)\; \left(\frac{\nu_0}{\nu}\right)^2$$

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

### Weighted Image Sensitivity

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

$$F = \frac{1}{Z}\sum_{k} w(\theta_k) f^{-1}(\theta_k)\; d_k$$

$$Z = \sum_{k} w (\theta_k)$$

where the $\theta_k$ are the distances to the pointing centers for the image data points dk, $w(\theta_k)$ is the weight for data point dk, and $Z$ is the sum-of-weights function. If the image data have equal RMS sensitivity levels σk = σ0 then the optimal weighting gives

$$w(\theta_k)=f^2(\theta_k)$$

and

$$F = \frac{1}{Z}\sum_{k}f(\theta_k)\; d_k$$

$$Z = \sum_{k} f^2 (\theta_k)$$

This image will have the lowest possible RMS noise level, with the variance of $F$ given by

$$\sigma_F^2 = \frac{1}{Z^2}\sum_k f^2(\theta_k)\sigma_0^2 = Z^{-1}\sigma_0^2$$

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:

$$t_{eff} = Z t_0$$

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 $\theta_{\rm rect} = \theta_P/2$ 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 θP. 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 $\theta_{\rm hex} = {\theta_P}/{\sqrt 3}$ is based on Nyquist arguments (see reference 4). For the NVSS survey (reference 1), the authors argued that a spacing not much wider than $\theta_{\rm hex} = {\theta_P}/{\sqrt 2}$ would be acceptable from a sensitivity perspective, and in fact used a spacing of approximately ${\theta_P}/{1.2}$. 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 Zhexmin = 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 θhex. For a pixel at a pointing center, counting that point and the 6 neighboring centers, the sum-of-weights is given by

$$Z_{\rm hexmax} = 1 + 6f^2(\theta_{\rm hex})$$

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

$$Z_{\rm hexmin} = 3 f^2 (\frac{\theta_{\rm hex}}{\sqrt 3})$$

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 $\theta_{\rm hex} = \theta_P/{\sqrt{2}}$ we get:

$$\theta_{\rm hex} = \frac{\theta_P}{\sqrt 2}$$

$$f(\theta_{\rm hex}) = 0.25$$

$$f(\theta_{\rm hex}/\sqrt 3) = 0.63$$

and

$$Z_{\rm hexmax} = 1.375\; \; \; \; \; \; \; \;Z_{\rm hexmin} = 1.191$$

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

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

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

$$A_{\rm hex} = \frac{\sqrt3}{4}N_{\rm pt}\theta^2_P = 0.7644 N_{\rm pt}\Omega_B$$

for the area under the squared beam defined above. If we observe the mosaic for a total time T with each of the Npt pointings getting the same integration time tint

$$t_{int} = \frac{T}{N_{\rm pt}}$$

then

$$A_{\rm hex}t_{\rm int} = 0.7644 T \Omega_B$$

or using the effective integration time per point in the mosaic

$$A_{\rm hex}t_{\rm eff} = 0.7644 Z\;T \Omega_B$$

For our hexagonal mosaic, the minimum weight is Zhexmin = 1.191 so

$$A_{\rm hex} t_{\rm hexmin} = 0.91 T \Omega_B$$

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

$$A_{\rm mos} t_{\rm eff} \approx T \Omega_B$$

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

$$T \approx t_{\rm eff}\frac{A_{\rm mos}}{\Omega_B}$$

after getting teff 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 θrect in the cardinal directions and next 4 by $\sqrt{2}\theta_{\rm rect}$ on the diagonals. Thus,

$$Z_{\rm rectmax} = 1 + 4 f^2(\theta_{\rm rect}) + 4 f^2(\sqrt{2}\theta_{\rm rect}) = 2.25$$

for the optimal θrect = 0.5 θP.The weight minima have 4 nearest neighbors at θrect/√2

$$Z_{\rm rectmin} = 4 f^2(\frac{\theta_{\rm rect}}{\sqrt 2}) = 2$$

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

$$A_{\rm rect} = N \times \theta_{\rm rec} \times M \times \theta_{\rm rect} = N_{\rm pt}\theta^2_{\rm rect}$$

so going through the same calculation as for the hexagonal mosaic

$$A_{\rm rect} = \frac{1}{4}N_{\rm pt}\theta^2_{P} = 0.4413 N_{\rm pt}\Omega_B$$

and

$$A_{\rm rect}t_{\rm eff} = 0.4413\; Z\; T\; \Omega_B$$

For our rectangular mosaic, the minimum weight is Zrectmin = 2 so

$$A_{\rm rect}t_{\rm rectmin} = 0.88 T \Omega_B$$

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 (${\rm\bf x}_0$) amounts to a weighted integration of the field data ($D(x)$) over all nearby sky positions (${\rm\bf x}$) corrected by the beam response ($f$), and keeping in mind that the beam scans across the sky over time:

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

$${\rm\bf x} = {\rm\bf x}(t)$$

with normalization

$$Z = \int dt\; w({\rm\bf x})$$

and as before

$$w({\rm\bf x}) = f^2(x) = e^{-\frac{x^2}{\theta^2_g}}$$

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

$$\dot{\Omega} = \frac{d\Omega}{dt} = \frac{dx\; dy}{dt} = \frac{A_{\rm mos}}{T}$$

where Amos is the total area of the mosaic and T the total integration time as before. Thus, we can recast the integrals

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

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

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

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

As before, we can compute the RMS sensitivity

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

where σ2D is the sensitivity of the data per unit time, so again we have the relation

$$Z = \frac{\sigma^2_D}{\sigma^2_F} = t_{\rm eff}$$

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

$$A_{\rm mos}t_{\rm eff} = T \Omega_B$$

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

$$t_{\rm eff} = \frac{T}{N_B}\; \; \; \; \; \; \; \; N_B = \frac{A_{\rm mos}}{\Omega_B}$$

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.