Mosaicking

Setting up the Mosaic

This document is intended for observers planning VLA observations using multiple pointing and phase centers to create a "mosaic".

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.

Starting with semester 2015A, as part of our Shared Risk Observing (SRO) program, we will offer the opportunity to use On-The-Fly (OTF) mosaicking to more efficiently scan large areas with small dwell times on each point. This is done by scanning 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 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 [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.

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) [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 45^\prime \frac{\rm 1 \: GHz}{\nu_{\rm obs}}[/display]

   
   

   Example

   I am observing in L-band 1-2 GHz, so ν_obs = 1.5GHz and θ_P [inline]\approx[/inline] 30'

   • Compute the mosaic beam area Ω_B from the FWHM, using the formula [display]\Omega_B = 0.5665 \theta^2_{P}[/display]

   • Example

     For my θ_P = 30' = 0.5° the equivalent mosaic beam area is Ω_B = 0.142 square degrees

   • Compute the number of independent beam areas in mosaic using[display]N_{\rm eff} = \frac{A_{\rm mos}}{\Omega_B}[/display]

     
     

     Example

     For my 25 square degree mosaic with Ω_B = 0.142 square degrees I have 176 beams.

3. How much integration time do I need? What is my survey speed (SS)?
   
   • Compute the integration time per "beam" t_calc 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 t_total for the mosaic by multiplying by the number of beams. (Note: this is independent of how you actually split up the mosaic to first order.)

     Example

     For 3m31s per beam and 176 beams I need 10h19m 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_total). This is equivalent to simply computing directly SS = Ω_B / t_calc also!
     

     Example

     For t_total=10h19m (10.32 hours) total over the 25 square degree mosaic the implied survey speed is 2.42 square degrees per hour (or equivalently SS = 2.42 square arc-minutes per second).

4. What mosaicking pattern do you wish to employ?
   • We recommend a hexagonal packed mosaic with a spacing of θ_hex = θ_P/√2 (or θ_P/√3 if uniformity is a strong concern, see below). You may wish to consider using On-the-fly (OTF) mosaicking with θ_row = θ_P/√2 if you integration time per pointing is less than 24 seconds - see below.
     

     Example

     For our θ_P = 30' we get θ_hex = 21'. We note that this will be more undersampled at the upper band edge of 2GHz, and oversampled at 1GHz, but for our basic observations this should be OK (see below).

5. How many pointings will this mosaic take for this pattern?
   • Keep in mind a hexagonal mosaic has spacing θ_hex along a row (e.g. horizontal or in RA) and √3 θ_hex/2 between rows (e.g. vertically or in Dec). Long and short rows alternate (if you want to fill a rectangular area) with 1 extra pointing in the long rows, so take that into account.

     Example

     Our square mosaic has sides of 5 deg (300') and 21' hexagonal spacing. The implied row spacing is 18'11" (1091") so there should be 16.5 spacings which we round up to 17 rows. The centers of the top and bottom rows are 291' apart. Our 300' long rows at 21' spacing take 14.28 spacings or 15 pointings in a row. The beginning and end pointings in each of these short rows are 14 x 21' = 294' apart. These will alternate with longer rows of 16 pointings (15 x 21' = 315' between end pointing centers). We put a short row in the center and 8 more short rows, plus 8 long rows interspersed. We have a total of 9 x 15 + 8 x 16 = 263 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 of 10h19m is spread among 263 pointings, each of which should get 2m21s of integration time.

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 2m21s integrations we add 7s so we can have observations of 2m28s. The total for 263 pointings is 10h48m.

8. Calculate schedule overheads.
   • Follow our guidelines
     

     Example

     For ease of scheduling, we will break our ~11h mosaic into two parts. Each will start with 10m allowance to get to first source, at least one 10m scan on a flux calibrator that might not be close to our target, we will spend 3m every 30m (12 visits) to observe our gain calibrator, and spend 5h24m on half the mosaic. Total comes to be 6h20m. SBs are allowed in increments of 30m so we can expect to schedule in two blocks of 6h30m each. Note that this comes out to a 20% overhead, which is about average at 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.

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. Note that the benefits of OTF mosaicking are the ability to eliminate the slew and setup up overheads and thus are useful for shallow very large mosaics where you want <15s integration per mosaic beam.

The most important question is "Do I need to use OTF mosaicking or is standard mosaicking sufficient?". You should only use OTFM 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 6-7 seconds of move 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 take 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 (ie. 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 OTFM! From 14 seconds to 24 seconds it is a gray area - you will benefit from OTFM 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.

The next question, assuming the previous answer was affirmative, 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 t_integ 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 θ_P / t_integ where the minimum allowed t_integ 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: in 2015A OTFM is classified as shared-risk for which the limit is nominally 60MB/s. This means that you can use shorter integration times as by choosing OTFM 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 / θ_row from the computed survey speed (SS = Ω_B / t_calc from Step 3 above) and the spacing of OTF rows θ_row from Step 4 required to uniformly sample the mosaic (we recommend θ_row = θ_P/√2 where θ_P is computed at the upper frequency limit of the band). For example at S-band (θ_P= 15' at 3GHz, Ω_B=127.5 square arc-minutes) for a depth of 0.1 mJy (t_calc= 7.7 sec) we have a survey speed SS = 16.56 square arc-minutes per second. Note that we calculate this at the band center so that the sensitivity can be related to the band average of a source with a modest non-zero spectral index. For a row spacing of 8' (θ_P/√2 with θ_P= 11.25' at 4GHz, calcualted at the upper end of the band) we need a 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/θ_P = 0.18 per second. Thus, to keep Rt_integ/θ_P < 0.1 we need t_integ< 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 t_integ 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 restrict the allowed visibility integration times to a minimum t_integ of 0.5sec for <4GHz of correlated bandwidth, and 1 second for 4-8GHz of bandwidth. These restrictions may be dropped in the future as we improve the processing capabilities of the cluster.

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 "stripes" each of which is one or more "sub-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 [display]\theta_{\rm row} = \theta_P / \sqrt{2}[/display] where θ_P 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). Furthermore, you will want to scan "counter-sidereal" (e.g. from lower to higher RA) to minimize the telescope tracking rates. For example, a drift scan is just an OTF stripe upward in RA with a span in RA equal to the observing time in LST, and this does not move the telescope at all. We have confirmed that OTFM functions acceptably for scan rates R up to 2 arcmin/sec, and are testing faster rates (we do not currently recommend requesting scan rates above 2 arcmin/sec, even under Shared Risk observing). 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 t_integshould be calculated using the scan rate considerations given above.

• Your schedules for basic OTF mosaicking can now be made in the OPT. You 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 a 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).

Survey Speed of the Jansky VLA for Large Continuum Mosaicked Surveys

Following the above guidelines (Steps 1-3 in the mosaic calculations) we can compute the survey speed SS [display]SS = \Omega_B / t_{calc} = 0.5665 \theta^2_{P} / t_{calc}[/display]of the JVLA 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 t_{calc}. From this and the θ_P at band center we calculate SS and R.

The parameters are tabulated by band below:

Band (freq)	Bandwidth	tcalc (sec)	θP (arcmin)	SS (deg2/hr)	R (arcmin/sec)
P (230-470MHz)	200 MHz	8553	122'	0.98	0.01
L (1-2GHz)	600 MHz	37	30'	13.90	0.87
S (2-4GHz)	1500 MHz	7.7	15'	16.55	2.08
C (4-8GHz)	3.03 GHz	4.4	7.5'	7.21	1.81
X (8-12GHz)	3.50 GHz	3.9	4.5'	2.96	1.11
KU (12-18GHz)	5.25 GHz	3.5	3.0'	1.45	0.82
K (18-26.5GHz)	7.20 GHz	7.0	2.05'	0.34	0.28
KA (26.5-40GHz)	7.20 GHz	9.5	1.45'	0.12	0.15
Q (40-50GHz)	7.20 GHz	50	1.00'	0.011	0.017

For C-band and above 3-bit observing is assumed. Beam width and sensitivities are at mid-band. R is calculated at upper band edge. 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 Details: Mosaic Sensitivity

The following are some in-depth calculations of the discrete mosaic sensitivity. These are provided in case the user wants to know the gory details of how these are calculated. These 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 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 square of the primary beam is relevant, which is the same as that of a Gaussian with half the area

[display]\Omega_B = 2\pi(\frac{\theta^2_g}{\sqrt 2})^2 = \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 = \frac{2\pi}{8\ln 2} \theta^2_P = 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]

 

Weighted Image Sensitivity

A mosaic image can be considered to be a weighted sum of individual field image data corrected for the individual pointing responses:

[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 θ_k are the distances to the pointing centers for the image data points d_k. If the image data have equal rms sensitivity levels σ_k = σ₀ 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 variance

[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 on-the-fly) 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 of the FFT of the mosaic pattern as 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 the Nyquist sampling of primary beam scales 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 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 tiled by equilateral triangles defined by the vertices 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 1), the authors argued that a spacing not much wider than [inline]\theta_{\rm hex} = {\theta_P}/{\sqrt 2}[/inline] would be OK 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 rules of thumb.

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 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 has the same weight, and our sums in the previous derivations become integrals:

[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{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}) = \frac{1}{Z\dot{\Omega}}\int\int\frac{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/node178.html