The Impact of Scanning Pattern Strategies on Uniform Sky ... - JACH

Doc No: SC2/ANA/S210/008 

Vers: 0.2 

Category: Report 

Doc Type: L A TEX 

State: Draft 

Author: Jennifer Balfour, Douglas Scott & Edward Chapin 

Date: 2006/11/16 

<strong>The</strong> <strong>Impact</strong> <strong>of</strong> <strong>Scanning</strong> <strong>Pattern</strong> <strong>Strategies</strong> on Uniform 

Sky Coverage <strong>of</strong> Large Maps 

Summary 

This document discusses the impact <strong>of</strong> the SCUBA-2 (Holland et al. 2006) array design on 

the ability to fully sample large maps with the proposed scanning strategies. In particular we 

study how the sampling <strong>of</strong> maps depends on the orientation <strong>of</strong> the SCUBA-2 array relative to 

the scan pattern, also considering the effect <strong>of</strong> the gaps between sub-arrays and the possibility 

<strong>of</strong> blocks <strong>of</strong> non-functioning bolometers. Pseudo-code for scanning strategies are added in 

appendices. 

1 SCUBA-2 : A submillimetre camera for the JCMT 

SCUBA-2, due to arrive at the James Clerk Maxwell Telescope on Mauna Kea early in 2007, 

will be used to map large areas <strong>of</strong> the submillimetre sky at 450 µm and 850 µm. One <strong>of</strong> the 

primary science goals <strong>of</strong> SCUBA-2 is to map the sky in an efficient manner, using filled arrays 

<strong>of</strong> bolometers and scanning patterns designed to cover large areas in minimum time. With 

an 8 × 8 arcminute field <strong>of</strong> view and over 10,000 bolometers in two arrays, SCUBA-2 will be 

able to map the sky up to 1000 faster than the original SCUBA instrument. For example, a 

proposed 180 ◦ × 2 ◦ Galactic Plane survey will take 50 hours, as opposed to over 5 years with 

SCUBA. Wide-field surveys performed by SCUBA-2 will be invaluable to the next generation 

<strong>of</strong> submm interferometers such as ALMA. In addition, SCUBA-2’s high sensitivity at multiple 

wavelengths will allow observers to exploit periods <strong>of</strong> good weather at the JCMT. In order to 

do so, s<strong>of</strong>tware is being developed which will provide the user with the best scanning solutions 

to cover the chosen area <strong>of</strong> sky to the required depth, while ensuring adequate coverage <strong>of</strong> 

the entire map. 

2 SCUBA-2 Array Design 

SCUBA-2 is comprised <strong>of</strong> 2 bolometer arrays, one operating at 450 µm, and one at 850 µm. 

Each <strong>of</strong> these arrays is divided into 4 subarrays <strong>of</strong> 32 by 40 detectors, arranged in a “pinwheel” 

formation (Figure 1).

SCUBA-2: <strong>The</strong> <strong>Impact</strong> <strong>of</strong> <strong>Scanning</strong> <strong>Pattern</strong> <strong>Strategies</strong> on Uniform Sky Coverage <strong>of</strong> Large 

Maps 4 SCANNING PATTERNS 

Figure 1: SCUBA-2 subarray layout. Each sub-array consists <strong>of</strong> 32 rows <strong>of</strong> 40 detectors with a gap 

corresponding to the extent <strong>of</strong> approximately 4 detectors. 

<strong>The</strong> sub-arrays themselves are filled areas <strong>of</strong> multiplexed transition edge sensors, which provide 

a fully sampled image at 850 µm, and an undersampled image at 450 µm. Due to the 

complex fabrication process <strong>of</strong> the high number <strong>of</strong> bolometers, it is estimated that perhaps 

10–20% <strong>of</strong> the bolometers will not function correctly. Possible causes are manufacturing defects 

<strong>of</strong> individual bolometers, shorts in patches <strong>of</strong> the sub-arrays, and wiring failures in the 

multiplexing which could cause an entire row or column <strong>of</strong> bolometers to malfunction. 

Several different scan patterns are being considered. 11 <strong>The</strong>se scanning patterns will be used 

to map out areas larger than the field <strong>of</strong> view <strong>of</strong> the arrays, while also providing fully sampled 

450 µm array data, filling the gaps between the sub-arrays, and covering areas missed by 

dead and faulty bolometers. 

3 SCUBA-2 Operating Modes 

SCUBA-2 will have 3 primary modes <strong>of</strong> operation, STARE, 6 DREAM, 7 and SCAN 4, 5 . STARE 

is a simple ‘point-and-shoot’ mode in which the camera stares at a specified area <strong>of</strong> sky for a 

period <strong>of</strong> time. In DREAM (Dutch REal-time Acquisition Mode) the secondary mirror (SMU) 

is rapidly moved in a star-like pattern such that each bolometer observes multiple points on 

the sky. SCAN patterns such as Pong, Boustrophedon, and Lissajous are used to map large 

areas. Each scanning pattern determines a solution for x(t) and y(t) with which to drive the 

telescope while continuously collecting data at 200 Hz. In addition to these basic imaging 

and survey modes, SCUBA-2 will include imaging polarimetry with POL-2 8 and medium 

resolution spectroscopy with FTS-2. 2 

4 <strong>Scanning</strong> <strong>Pattern</strong>s 

Due to the shape and layout <strong>of</strong> the sub-arrays, scanning patterns must be chosen which not 

only overcome the challenges <strong>of</strong> observing in the submillimetre, but also provide acceptable 

(i.e. ∼close to uniform) sampling across an entire map, despite the ‘gaps’ between the subarrays. 

Several factors must be considered when designing a scanning pattern, the following 

being some <strong>of</strong> the most crucial 11 : 

1. Move the telescope across the sky as quickly as possible, taking data at a rate which is 

Page 2 <strong>of</strong> 41



much faster than the changes in the atmosphere. 

2. Maximize the number <strong>of</strong> different detectors which collect data from each sky pixel in order 

to correct for instrumental variations including faulty bolometers, correlated noise, 

and gain fluctations. 

3. Cover each sky pixel in different directions while avoiding periodic patterns in order 

to separate astronomical structure from temporal variations (i.e. varying atmospheric 

emission and gain fluctuations). 

In addition, a scanning pattern should strive to provide uniform sky coverage. Given the 

design <strong>of</strong> the array and the wide variety <strong>of</strong> map shapes and areas a user might choose to scan 

(as well as the likelihood <strong>of</strong> non-functioning bolometers), absolutely uniform coverage will 

rarely be possible. However, the scanning pattern should be designed to maximize sampling 

<strong>of</strong> every pixel while covering an area in a minimal time. 

<strong>The</strong> simplest scanning strategy to map an area larger than the field-<strong>of</strong>-view <strong>of</strong> the array 

is a ‘boustrophedon’ or raster pattern, such as that previously implemented by the original 

SCUBA instrument. 3 In the simple example below this pattern is used to map a rectangular 

region at an angle parallel to one side <strong>of</strong> the region. Additional details regarding the 

BOUSTROPHEDON scanning pattern are given in Appendix A. 

Figure 2: Simple Boustrophedon scanning pattern. <strong>The</strong> diagram indicates the path <strong>of</strong> the centre <strong>of</strong> the 

array as it scans across the sky. Successive sweeps should be no farther apart than the full array size 

to ensure that every region is covered, and it is probably desirable for the spacing to be less than half 

the field-<strong>of</strong>-view <strong>of</strong> the array in order to provide overlap in the two directions. 

<strong>The</strong> Boustrophedon pattern violates the third <strong>of</strong> the guidelines above: the pixels are revisited 

in a periodic fashion, and in addition, each pixel is visited in only two directions (which are 

just 180 ◦ from each other). 

In order to overcome this, the region can be scanned at two different angles, neither <strong>of</strong> which 

are parallel to a side <strong>of</strong> the rectangle. In Fig. 3 below, the boustrophedon pattern is again 

used to scan a rectangular region, but the time taken to cross the width <strong>of</strong> the pattern and 

turn around varies between sweeps across the sky. <strong>The</strong> pattern could then be repeated at an 

angle approximately 90 ◦ different from the first pattern (right panel <strong>of</strong> Fig. 5). 

For SCUBA-2, a scanning strategy called Pong, similar to the ‘boxscan’ used by SHARC-II 

on the CSO, 1 is proposed. In this pattern, a rectangular region is filled by ‘bouncing’ the 

telescope <strong>of</strong>f the limits <strong>of</strong> the box, covering each pixel in multiple directions and without periodic 

revisitation. Similarly to a Lissajous pattern, the number <strong>of</strong> ‘nodes’ (where the pattern 




Figure 3: Boustrophedon scanning at an angle (45 ◦ here). <strong>The</strong> right panel shows how a repeat <strong>of</strong> the 

scan with a 90 ◦ rotation provides cross-linking. 

bounces <strong>of</strong> the edges <strong>of</strong> the box) in the x and y directions must not share common factors, 

and one must be even while the other is odd. This ensures that the pattern continues until 

the entire box is filled, with relatively even coverage over the entire area (with a shallower 

border, about half the array wide). Additional details regarding the Pong scanning pattern 

are given in Appendix B. 

Figure 4: ‘Straight’ Pong scan with a map width ≃4000 arcsec, height ≃3200 arcsec, and sweep spacing 

<strong>of</strong> 500 arcsec. This spacing is measured between the parallel scan lines as indicated by the bold arrow. 

Two possible Pong solutions are proposed. In the first, the map is scanned in straight line 

segments between the vertices, with abrupt 90 ◦ turns at the boundaries <strong>of</strong> the box (Fig. 4). 

This strategy ensures that the sweeps across the sky are parallel and have constant spacing. 

Pseudo-code for this ‘straight’ Pong scanning pattern can be found in Appendix C. Note 

that in principle, an angle other than 45 ◦ can be selected for the paths connecting the Pong 

vertices. In this case, the grid would be composed <strong>of</strong> rhombuses as opposed to squares, and 

the vertical and horizontal vertex spacing would not be equal. For simplicity, all Pong scans 




considered in this document will be those which fill the box dimensions with 45 ◦ angles, providing 

orthogonal cross-linking. 

In the event that abrubt changes in velocity at the turnaround points are not preferable, one 

could simple change the velocity in some way near the turnarounds. An alternative is that 

the corners can be ‘rounded-<strong>of</strong>f’ by approximating the Pong scan with a Fourier expanded 

Lissajous scan instead (Fig. 5). In this case, x(t) and y(t) are approximations <strong>of</strong> triangle waves 

with 5 terms each. Pseudo-code for this ‘straight’ Pong scanning pattern can be found in 

Appendix E. <strong>The</strong> specific formulae are : 

x(t) = 8α x 

π 2 

y(t) = 8α y 

π 2 

∑ 

n=1,3,5,7,9 

∑ 

n=1,3,5,7,9 

(−1) n−1 

2 

n 2 

(−1) n−1 

2 

n 2 

( ) 2πnt 

sin ; (1) 

β x 

( ) 2πnt 

sin . (2) 

β y 

Where α x and α y are the amplitudes <strong>of</strong> the x and y motion, respectively, and β x and β y are 

the periods <strong>of</strong> the x and y motion, respectively. 

Figure 5: ‘Curvy’ Pong scan created by approximating triangle waves with 5 terms in x(t) and y(t). 

This approach results in almost straight paths across the central region <strong>of</strong> the scan while 

avoiding sharp corners at the vertices. It can be seen from Fig. 6 that with an approximation 

using 5 terms, the scans are almost identical in the path that they take to cover the map, 

with the exception <strong>of</strong> the outer edges. 

In both Boustrophedon and Pong scanning, several factors must be considered to ensure that 

the scanning area is fully sampled, and that there is not a wide variation in sky coverage. 

<strong>The</strong> distance between successive sweeps across the sky should be no farther apart than half 

the total field <strong>of</strong> view <strong>of</strong> the array to ensure that each pixel is visited in at least two directions 

and by multiple passes <strong>of</strong> the array. Also, the angle <strong>of</strong> the scan relative to the array itself 

should be chosen such that the gap between sub-arrays (and undersampling at 450 µm) is 



Maps 5 IDENTIFYING ‘BAD ANGLES’ IN SINGLE STRAIGHT SCANS 

Figure 6: ‘Straight’ and ‘Curvy’ Pong patterns compared. 

filled on a single sweep. <strong>Scanning</strong> at an angle parallel or close to parallel to the gap between 

sub-arrays will cause there to be undersampled ‘stripes’ <strong>of</strong> lower or zero sampling on each 

sweep across the sky. 

5 Identifying ‘Bad Angles’ in single straight scans 

For SCUBA 3 there were ‘Nyquist angles’ chosen for scanning, such that successive rows filled 

in gaps between bolometers. However, the large number <strong>of</strong> rows (and columns) in SCUBA- 

2 means there are many ‘good angles’, and hence it is more appropriate to be concerned 

with avoiding ‘bad angles’ than with selecting ‘good angles’. To determine the impact <strong>of</strong> 

the chosen angle, the SCUBA-2 simulator was used to first map out the hits-per-pixel for 

simple straight scans across the sky at various angles, with four subarrays in their correct 

configuration and with the assumption <strong>of</strong> no malfunctioning bolometers. In this scenario, the 

telescope accelerates to a maximum velocity <strong>of</strong> 600 arcsec s −1 , at which point the lowest pixel 

hitcount within a region surrounding the gap was measured. For these tests the pixel size 

is 6 arcsec and a ‘hit’ is counted simply through determining the pixel in which the centre <strong>of</strong> 

each bolometer lies at every 200 Hz sample. 

One can see from Fig. 7 that when the angle is 0◦ relative to the symmetry axis, there are 

zero hits in the gap and a great deal <strong>of</strong> structure around the edges. Figs. 8, 9, 10 and 11 

show the effect <strong>of</strong> changing the scan angle to 10 ◦ , 20 ◦ , 30 ◦ , and 40 ◦ . One can see the gap 

filling up and the structure smoothing out as the angle is increased. 

NOTE : All the maps in this study were created from data generated by the SCUBA-2 simulator, 

using a pixel size <strong>of</strong> 6 arcseconds. 

In order to determine the effect <strong>of</strong> the scan angle on hits-per-pixel, the central region <strong>of</strong> the 

scan was measured for minimum, median, and maximum hits-per-pixel. <strong>The</strong> area chosen 

for these measurements included the path <strong>of</strong> all four sub-arrays and excluded the start and 

endpoints <strong>of</strong> the scan as shown in Fig. 12. Of course the detailed results will depend on the 




Figure 7: Hits per pixel for a scan at 0 ◦ (relative to one <strong>of</strong> the array edges) for a 1000 arcsec scan length 

with 600 arcsec s −1 scan velocity. 

Figure 8: Hits per pixel for a scan at 10.0 ◦ . 

specific choice for this area; we need to chose something and our choice (Fig. 12) minimizes 

effects from the ‘edges’, so we can focus on the ‘well-sampled’ centre <strong>of</strong> the mapped region. 

Fig. 13 shows the minimum, median, and maximum hits-per-pixel as a function <strong>of</strong> angle, 

clearly indicating the severe drop-<strong>of</strong>f in the minimum sampling rate as the angle approaches 

4 ◦ . From examining the pixel count in the region surrounding the less sampled section caused 

by the gap, it is clear that below an angle <strong>of</strong> 4 ◦ , some pixels are not visited even once in a 

single path across the sky. Because gaps exist in both the horizontal and vertical directions, 

there is no guarantee that these missed pixels would be visited when the scanning pattern 

returns on the second scan direction, depending on the choice <strong>of</strong> overlap between successive 

scans. As a result, a scanning pattern chosen with minimum overlap (less than half the full 

array width), and scan angles <strong>of</strong> less than 4 ◦ from being parallel to the gaps may yield some 

pixels in the map which have zero hits. 

This value can be understood geometrically from the ratio <strong>of</strong> the gap size to the full array size 

(i.e. tan −1 (gapwidth / arraywidth) where the array width is twice the length <strong>of</strong> the side <strong>of</strong> a 

sub-array plus the width <strong>of</strong> the gap. Because <strong>of</strong> the non-symmetrical ‘pinwheel’ orientation 

<strong>of</strong> the sub-arrays, the arraywidth is either 2×32 + 4 or 2×40 + 4 detectors, depending on the 

direction <strong>of</strong> the scan). In either case, the answer is ∼4 ◦ . 

As the angle is increased above 4 ◦ , the gap is quickly filled in by the coverage provided by multiple 

bolometers. <strong>The</strong> minimum pixel sampling increases until reaching a maximum value at 

∼7 ◦ , and remains relatively constant to ∼33 ◦ . Beyond this angle, the area with the minimum 

hits-per-pixel becomes the undersampled outer edges <strong>of</strong> the scan path. At this point, the 

overlap <strong>of</strong> the scans across the sky becomes the primary factor in determining the minimum 

pixel count, as the area around the gap is well covered by the sub-array geometry. 






<strong>The</strong> median sampling (middle curve in Fig. 13) gradually decreases as the angle is increased 

from 0 ◦ to 45 ◦ . This is due to the rise in undersampling that occurs towards the outer edges <strong>of</strong> 

the scan. As the sub-array layout is tilted towards a 45 ◦ angle, the outer edge is sampled by 

a smaller area (the corner <strong>of</strong> the sub-arrays). However, as expected, the maximum sampling 

(upper curve in Fig. 13 remains relatively constant, because for all angles there exist pixels 

which are ‘hit’ at almost every sample interval. 

When scanning at 450 µm, additional care must be taken to ensure that the scan angle fills in 

the undersampled pixels caused by having the same pixel size on the sky, but about half the 

beamsize. While the gap will have essentially the same effect on scans at 850 µm and 450 µm, 

there are more potential ‘bad’ angles for 450 µm scanning. For example, a scan at 45 ◦ would 

fail to compensate for the undersampling, because the centres <strong>of</strong> the detectors would pass 

directly over the same map pixels, leaving undersampled areas between the detectors. Crosslinking 

solutions such as Pong should be able to eliminate this problem, and in particular a 

scan which does not proceed in a straight line (for example, the ‘Curvy’ Pong solution) will 

not suffer from these additional ‘bad’ angles. 

<strong>The</strong> basic conclusion from this part <strong>of</strong> our study is that one should avoid having a substantial 

fraction <strong>of</strong> data for a map being gathered at scan angles below about 7 ◦ , but otherwise the 

choice <strong>of</strong> angle makes no dramatic difference to the sampling. 





Figure 12: Area measured for hits-per-pixel 

Figure 13: Minimum (bottom line), median (central line), and maximum (top line) pixel sampling as a 

function <strong>of</strong> scan angle for a single scan path, all measured within the box delineated in Fig. 12. 



Maps 6 THE IMPACT OF FAULTY BOLOMETERS 

6 <strong>The</strong> impact <strong>of</strong> faulty bolometers 

It is guessed that SCUBA-2 may have 100–200 ‘dead’ bolometers per sub-array. Such bolometers 

may record incorrect values <strong>of</strong> flux, or may be unable to collect any data at all. In any 

case, malfunctioning bolometers must be flagged so as not to be used in any map-making 

procedure. <strong>The</strong> scanning patterns chosen should be able to provide a fully sampled and reasonably 

uniform image at both 450µm and 850µm, regardless (within reason) <strong>of</strong> the layout <strong>of</strong> 

the bad bolometers. 

Unlike the original SCUBA, which employed separated feedhorn-coupled arrays (Fig. 14), 

SCUBA-2 will have 4 filled sub-arrays <strong>of</strong> detectors (Fig. 15) which can provide a fully sampled 

image (at 850µm) without requiring ‘jiggling’ <strong>of</strong> the secondary mirror. <strong>The</strong> highly sensitive 

transition-edge sensors used to detect sub-mm radiation must be kept at temperatures below 

1 K. In order have such a close-packed array <strong>of</strong> detectors, multiplexing is used to share common 

wiring between detectors in a single column or row <strong>of</strong> a sub-array. As a result <strong>of</strong> this 

multiplexing, the possibility exists <strong>of</strong> losing an entire row (40 detectors) or column (32 detectors) 

<strong>of</strong> bolometers to a wiring or electronic fault. Blocks <strong>of</strong> contiguous detectors could also 

be ‘dead’ due to fabrication problems. On top <strong>of</strong> that it is likely that some isolated individual 

bolometers will not meet performance requirements. In perhaps the worst case scenario, 

groups <strong>of</strong> bolometers near the gaps will be affected, causing the undersampling effect <strong>of</strong> the 

gap to have an impact at a wider range <strong>of</strong> angles. 

In order to examine the increase in angle required to compensate for dead rows <strong>of</strong> bolometers, 

simulations were run with the outer rows and columns <strong>of</strong> the sub-arrays flagged as ‘bad’. In 

the first simulation, only one sub-array had bad bolometers added to it, effectively reducing 

the field <strong>of</strong> view <strong>of</strong> the sub-array by one row/column <strong>of</strong> bolometers on all four sides. This 

creates a gap in the full array which, for half <strong>of</strong> its length, is one bolometer wider than the 

regular gap. In this case, the three uncompromised subarrays are able to provide enough 

coverage that the effect <strong>of</strong> the gap is not significantly worse, as we show in Fig. 16. 

In the next simulation, the edges <strong>of</strong> all four sub-arrays were flagged as bad, essentially increasing 

the total width <strong>of</strong> the gaps from 4 to 6 bolometers for their entire length. This may 

be regarded as a worst case scenario, since it seems unlikely that isolated groups <strong>of</strong> dead 

bolometers within a sub-array could be as large as the gap between the sub-arrays. 

As expected, the minimum hits-per-pixel over the gap increases in a similar fashion to scans 

without dead bolometers, however the angles at which some areas are totally missed include 

those within about 5 ◦ <strong>of</strong> the direction parallel to the gap. This indicates that the angles 

Figure 14: SCUBA feedhorn-coupled array. 

Figure 15: SCUBA-2 filled array. 




Figure 16: Simulation with bad bolometers along all four edges <strong>of</strong> one sub-array. Minimum, median, 

and maximum hits are plotted as a function <strong>of</strong> scan angle, again measured within the box shown in 

Fig. 12. 

Figure 17: Simulation with bad bolometers along all four edges <strong>of</strong> every sub-array. Minimum, median, 

and maximum hits are plotted as a function <strong>of</strong> scan angle. 




Figure 18: Black indicates approximate area 

flagged as ‘bad’ for simulation in Figs 20 and 

22. 

Figure 19: Black indicates approximate area 

flagged as ‘bad’ for simulation in Figs 21 and 

23. 

considered to be ‘bad’ will need to be modified if multiple rows/columns along the gap are 

faulty, although the change is not dramatic. <strong>The</strong> changes to the median and maximum in 

each case are fairly insignificant. 

A third test was performed to check the effect <strong>of</strong> a continguous block <strong>of</strong> ‘bad’ bolometers. In 

the following two simulations, a 10×10 area in the centre <strong>of</strong> the sub-arrays was flagged as 

‘bad’ and the hits-per-pixel were again measured. <strong>The</strong> setups are shown in Figs. 18 and 19 

and the resulting sampling curves are shown in Figs. 20 and 21. 

Let us focus on a specific scan angle, say 30 ◦ . Comparing Figs. 22 and 23 with our earlier scan 

which had no dead bolometers (Fig. 10) we can see the immediate effect <strong>of</strong> these bad regions. 

While the minimum hits-per-pixel at lower angles does not differ drastically from the results 

with no dead bolometers, at greater angles (especially above 20.0 ◦ ) the difference is more 

noticeable. In such a scenario, where blocks <strong>of</strong> dead detectors are present, the scan angles 

should be chosen such that these blocks cover the centre <strong>of</strong> each scan path, and are thus 

compensated for by the multiple sub-array geometry. Alternately, the spacing (or ‘overlap’) <strong>of</strong> 

the scan paths could be adjusted to fill in the less-sampled regions. 

<strong>The</strong> conclusion from this part <strong>of</strong> our study is that modest-sized regions <strong>of</strong> ‘bad’ bolometers will 

have no dramatic effect on the choice <strong>of</strong> scan angle. Clearly, if there are large ‘dead’ regions 

then one would have to be more careful about scan angle choice and scan overlaps (which we 

discuss in the next section), although this will be less <strong>of</strong> an issue for full Pong scan patterns. 




Figure 20: Simulation with a contiguous 10×10 block <strong>of</strong> ‘bad’ bolometers in the centre <strong>of</strong> one sub-array 

(see Fig. 18). Minimum, median, and maximum hits are plotted as a function <strong>of</strong> scan angle. 

Figure 21: Simulation with a contiguous 10×10 block <strong>of</strong> ‘bad’ bolometers in the centre <strong>of</strong> all four 

sub-arrays (see Fig. 19). Minimum, median, and maximum hits are plotted as a function <strong>of</strong> scan angle. 




Figure 22: Hits-per-pixel for a scan including one array with a 10×10 region <strong>of</strong> ‘bad’ bolometers in one 

sub-array. <strong>The</strong> scan angle here is 30.0 ◦ . This can be compared with Fig. 10, where there were no ‘bad’ 

bolometers. 

Figure 23: Hits-per-pixel for a scan including 10×10 regions <strong>of</strong> ‘bad’ bolometers in all four subarrays(see 

Fig. 19). <strong>The</strong> scan angle here is 30.0 ◦ . 



Maps 7 SPACING OF PARALLEL SCAN PATHS 

7 Spacing <strong>of</strong> parallel scan paths 

In the scanning strategies proposed for SCUBA-2, not only is it necessary to consider the 

angle <strong>of</strong> the paths across the sky, but also how close together these paths are when filling an 

area for which both dimensions exceed the field <strong>of</strong> view <strong>of</strong> the array. A simple Boustrophedon 

scan was simulated (with no bolometers flagged as ‘bad’) while varying the spacings between 

the centre <strong>of</strong> the scan paths from 100–400 arcseconds. 

<strong>The</strong>se simulations were conducted at angles <strong>of</strong> 15 ◦ , 30 ◦ , and 45 ◦ relative to the orientation 

<strong>of</strong> the sub-arrays. As the angle approaches 45 ◦ , the gradient in sampling towards the outer 

edge <strong>of</strong> each individual scan path becomes wider and more gradual, while a wider total area 

is covered in each single path. 

As seen in Figs. 24 through 35, lower angles will result in maps with more abrupt changes in 

sampling rate, caused by the overlap <strong>of</strong> individual sub-arrays. Care should be taken to avoid 

scans in which the centres <strong>of</strong> the sub-arrays are ‘lined-up’ in successive paths across the sky 

(as seen in Fig. 26), if the goal <strong>of</strong> even coverage across the map is to be achieved. Ideally, the 

<strong>of</strong>fset between scan paths should be approximately half the width <strong>of</strong> an individual sub-array 

in order to fill in the less sampled areas <strong>of</strong> the previous paths. One can see this by comparing 

the sampling for <strong>of</strong>fsets <strong>of</strong> 100 arcseconds (about half the size <strong>of</strong> a sub-array) versus the 

sampling for larger <strong>of</strong>fsets. <strong>The</strong> <strong>of</strong>fset which makes the sampling most uniform will depend 

on scan angle, but <strong>of</strong>fsets around half the sub-array width are generally good. 

Figure 24: Boustrophedon scan with an angle 

<strong>of</strong> 15 ◦ and spacing <strong>of</strong> 75 arcseconds. 






<strong>of</strong> 15 ◦ and spacing <strong>of</strong> arcseconds. 

Statistics extracted from Figs. 24–35 are shown in Figs. 36, 37, and 38, for scan angles <strong>of</strong> 

15 ◦ , 30 ◦ , and 45 ◦ , respectively. <strong>The</strong> variation in sampling as a functio fo <strong>of</strong>fset (the patterns 

apparent in Figs. 24–35) are clearly seen here. <strong>The</strong> absolute values are not important when 

comparing different <strong>of</strong>fsets (since the maps may cover different areas for different <strong>of</strong>fsets) – 

but the important discriminator is the spread among the minimum, median and maximum 

hit rates. One can see that there is little relative difference for scan spacings around 100-150 

arcseconds, and the spread is much larger for <strong>of</strong>fsets above about 200 arcseconds (i.e. roughly 

a sub-array size). 























Figure 36: Minimum (bottom line), median (middle line), and maximum (top line) hits-perpixel 

as a function <strong>of</strong> scan spacing for a 15 ◦ Boustrophedon scan. 

Figure 37: Minimum, median, and maximum hits-per-pixel as a function <strong>of</strong> scan spacing for 

a 30 ◦ Boustrophedon scan. 



Maps 8 PONG SCANNING FOR UNIFORM SKY COVERAGE 

Figure 38: Minimum, median, and maximum hits-per-pixel as a function <strong>of</strong> scan spacing for 

a 45 ◦ Boustrophedon scan. 

8 Pong scanning for uniform sky coverage 

<strong>The</strong> Pong scanning solution proposed for SCUBA-2 should take into consideration the gaps, 

faulty bolometers, and spacing <strong>of</strong> the scans (see Fig. 4), while striving to cover large maps in 

an efficient manner. <strong>The</strong> cross-linking pattern will not only aid in sky-removal, but will also 

reduce the deviation in hits-per-pixel, thus providing more uniform sky coverage. 

<strong>The</strong> Pong pattern covers the chosen map area by scanning in paths 45 ◦ from the sides <strong>of</strong> 

the ‘box’. As such, when avoiding ‘bad angles’, this 45 ◦ correction will need to be applied. 

For example, if we wish to avoid the ‘bad angle’ range <strong>of</strong> 0–4 ◦ as found in Section 5, the 

corresponding ‘bad’ Pong angles will be 41–49 ◦ (45±4 ◦ ). 

Figures 39 through 54 show the results <strong>of</strong> a similar study to that done for Boustrophedon 

scans in Section 7. It is important to notice the dynamic range <strong>of</strong> these plots, even when 

the pattern looks dramatic (one <strong>of</strong> these should be adopted as the <strong>of</strong>ficial SCUBA-2 tartan!). 

In this case we ran several Pong simulations at three Pong angles, 15 ◦ , 30 ◦ , and 45 ◦ , while 

again varying the spacing between 100 and 400 arcseconds. By comparing Figs. 55, 56, 57 

and 58 (where we show statistics for sampling rates), we can see how a well-chosen Pong 

angle <strong>of</strong> say 15 ◦ (i.e. 30 ◦ from being parallel to the gap) does better even at wider <strong>of</strong>fsets 

between the sweeps, effectively ‘filling in’ a larger area with each pass across the map. We 

can also see that the hits-per-pixel varies much more at a ‘bad’ angle (e.g. 45 ◦ , Fig 58), even 

at smaller spacings when every pixel is hit at least once. This suggests that provided one 

ignores explicitly ‘bad’ angles (Pong angles around 45 ◦ ) the Pong pattern gives fairly uniform 

coverage. It will never be exactly uniform, <strong>of</strong> course, but the contrast between minimum and 

maximum hits is very modest, particularly for pattern spacings <strong>of</strong> around half a sub-array 

width. 




Figure 39: Hits-per-pixel <strong>of</strong> the central region 

<strong>of</strong> a Pong scan at an angle <strong>of</strong> 0 ◦ (45 ◦ relative to 

the sub-array geometry) and pattern spacing 

<strong>of</strong> 75 arcseconds. 














<strong>of</strong> a Pong scan at an angle <strong>of</strong> 15 ◦ (30 ◦ relative 

to the sub-array geometry) and pattern spacing 





































<strong>of</strong> a Pong scan at an angle <strong>of</strong> 45 ◦ (parallel to 




<strong>of</strong> a Pong scan at an angle <strong>of</strong> 45 ◦ (parellel to 












as a function <strong>of</strong> scan spacing for a 0 ◦ Pong scan. 













From Sections 5 and 7 our tests suggest that scans with angles <strong>of</strong> 8–33 ◦ relative to the gaps 

between the sub-arrays, and with a spacing <strong>of</strong> approximately half the width <strong>of</strong> a subarray 

should yield a fairly uniform sampling rate across the map. So we now study four Pong scans 

to compare the uniformity <strong>of</strong> the hits-per-pixel across 4 maps made by fulfilling neither, one, 

or both <strong>of</strong> these criteria ( Figs. 59 and 60, Figs. 61–64, and Figs. 65 and 66 respectively). When 

choosing ‘bad’ and ‘good’ spacing, we consider that given a bolometer width <strong>of</strong> 6 arcseconds, 

the width <strong>of</strong> a sub-array is either 240 arcseconds (when measured across the long edge) or 192 

arcseconds (when measured across the short edge). In addition, we must bear in mind that 

the centres <strong>of</strong> two side-by-side sub-arrays are separated by a further 24 arcseconds due to 

the gaps between the arrays. For this simulation, we have chosen 108 arcseconds as a ‘good’ 

spacing (calculated using half the shorter sub-array width plus half the gap width). Scans at 

this spacing will ensure that the centre <strong>of</strong> a sub-array passes directly over the undersampled 

region caused by the gap in a previous sweep <strong>of</strong> the pattern. <strong>The</strong> ‘bad’ spacing used here 

is 264 arcseconds, a spacing which causes the sub-arrays to repeat similar paths across the 

map and leave undersampled regions in between (this value is calculated using the width <strong>of</strong> 

the longer side <strong>of</strong> the sub-array plus the width <strong>of</strong> the gap). <strong>The</strong> ‘good’ Pong angle used is 25 ◦ , 

which results in scan paths 20 ◦ from parallel to the sub-array geometry, while the ‘bad’ Pong 

angle used is 45 ◦ , which results in scan paths parallel to the sub-array geometry. Each scan 

has an approximate height and width <strong>of</strong> 1000 arcseconds. 

In order to compare the drastic difference in sampling variation among these Pong scans, we 

can take a slice across the centre <strong>of</strong> the scan (parallel to one <strong>of</strong> the gaps) and measure the 

hits-per-pixel at each point. <strong>The</strong> specific slices shown in Figs. 60, 62, 64, and 66 are taken 

across the hits-per-pixel maps such that the greatest variation in hits-per-pixel is measured 

in each case. 




Figure 59: Pong scan using a ‘bad’ angle and ‘bad’ spacing. 

Figure 60: ‘Slice’ across Pong scan in Fig. 59 showing wide variations in hits-per-pixel due to 

the ‘bad’ angle and spacing. 

Figure 61: Pong scan using a ‘good’ angle and ‘bad’ spacing. 




Figure 62: ‘Slice’ across Pong scan in Fig. 61 showing variations in hits-per-pixel due to the 

‘bad’ spacing, but with a ‘good’ angle. 

Figure 63: Pong scan using a ‘bad’ angle and ‘good’ spacing. 

Figure 64: ‘Slice’ across Pong scan in Fig. 63 showing variations in hits-per-pixel due to the 

‘bad’ angle, but with ‘good’ spacing. 




Figure 65: Pong scan using a ‘good’ angle and ‘good’ spacing. 

Figure 66: ‘Slice’ across Pong scan in Fig. 65 showing the relatively uniform coverage in the 

central region <strong>of</strong> the map when one choses a ‘good’ angle and ‘good’ spacing. 

<strong>The</strong>se simulations confirm that both the angle and the spacing play an important role in ensuring 

uniform sampling across a scanned map. It would appear from these four simulations 

that the angle has greater impact on the contrast in hits-per-pixel than the spacing. However, 

this will clearly depend on how bad a ‘bad’ spacing is – clearly the coverage would be quite 

ugly if the separation was chosen to be larger that the full array size, for example. Additionally, 

while the deviation in hits-per-pixel from the average in a scan with a ‘bad’ angle and a 

‘good’ spacing is greater than that observed with a ‘good’ angle and ‘bad’ spacing, it is should 

be noted that for the former, the values for the minimum hits-per-pixel across the majority 

<strong>of</strong> the map are significantly higher. In the case <strong>of</strong> this ‘good’ spacing, every pixel is hit approximately 

twice as many times as in the case <strong>of</strong> a ‘bad’ spacing due to the overlap <strong>of</strong> the 

array paths in the sweeps across the map. <strong>The</strong> reason for this lies in the time taken to cover 

the entire pattern. With a wider spacing (less overlap) the map is covered in a shorter time, 

resulting in lower hit counts on average across the whole map. <strong>The</strong> important feature to note 

in these simulations is not the minimum value, but the relative difference in the maximum 

and minimum hits-per-pixel as compared with the average for each scan. 

<strong>The</strong> basic conclusion from this section is that Pong scans are quite forgiving compared with 

Boustrophedon. Precisely uniform maps will be impossible to achieve, because <strong>of</strong> the gaps 

between the SCUBA-2 sub-arrays (as well as non-functioning detectors <strong>of</strong> course). However, 

maps which have variations in pixel hit rates <strong>of</strong> ∼ 20% are relatively easy to achieve with 

Pong, privided that the spacing between sweeps is about half a sub-array size (say 100–150 

arcseconds), and that one tries to avoid scanning at angles <strong>of</strong> around 45±7 ◦ relative to the 

array axes. 



Maps 9 SCANNING SOLUTIONS AT THE TELESCOPE 

9 <strong>Scanning</strong> solutions at the telescope 

When choosing a scanning strategy, it will be important to consider the gap between subarrays, 

any areas <strong>of</strong> faulty bolometers, and the scanning angle, in order to guarantee a map 

with relatively uniform sky coverage. Users <strong>of</strong> this instrument should be able to describe an 

area to be mapped and be assured that their observation will be close to the best possible 

given the array quality, map dimensions, and time constraints. 

<strong>The</strong> s<strong>of</strong>tware for SCUBA-2 should be responsible for checking the upcoming observations 

against some minimum acceptable criteria. Each scanning observation should either have 

multiple possible solutions, from which the best is chosen at the time <strong>of</strong> the observation, 

or the scheduling system should be able to change the order <strong>of</strong> observations to avoid poor 

scanning. 

Example 1) Given a Pong observation for a particular area <strong>of</strong> sky, two different solutions 

could be constructed at different angles, with a choice between the two made at the time <strong>of</strong> 

observation such that the better angle is used. Consider a user who has defined a rectangular 

region <strong>of</strong> the sky to be mapped. Two Pong patterns could be prepared with two different 

angles which both include the required region <strong>of</strong> sky. Provided these two angles are no less 

14 ◦ apart, we are guaranteed that at least one <strong>of</strong> the scans will have a pattern with sweeps 

which are greater than 7 ◦ from being parallel to the sub-array geometry. Compare Fig. 4 with 

Fig. 67 for examples <strong>of</strong> two possible Pong patterns which could be constructed to map an area 

approximately 3000 × 3000 arcseconds in size. Given the orientation <strong>of</strong> the region relative 

to the telescope axes at the time <strong>of</strong> the observation, the pattern which generates the better 

angle <strong>of</strong> the two would be used. 

Figure 67: ‘Straight’ Pong scan with a map width ≃4000 arcsec, height ≃3200 arcsec, and sweep 

spacing <strong>of</strong> 500 arcsec. <strong>The</strong> Pong angle here is 15 ◦ . 

Example 2) If only one solution is available the scheduling system will identify if this is a 

‘bad’ angle and reschedule the scan for a time when the sky has rotated such that the angle 

would no longer be ‘bad’. 



Maps 10 SKY ROTATIONS 

10 Sky Rotations 

One important factor to consider in larger maps is the effect <strong>of</strong> sky rotation on the scanning 

pattern. Over time, the orientation <strong>of</strong> the array relative to the area to be mapped will clearly 

change. <strong>The</strong> sky rotation will modify the results <strong>of</strong> the Pong pattern in two ways : 

1. As the sky rotates, the paths <strong>of</strong> the Pong pattern will curve slightly relative to the sky. 

On long and/or large scans, the sweeps <strong>of</strong> the Pong scan will gradually move away from 

being orthogonal, effectively ‘filling in’ the periodic variation in pixel sampling due to the 

parallel sweeps <strong>of</strong> the Pong pattern. <strong>The</strong> can be seen in Figs 68–71. In this simulation, 

the array performed a sequence <strong>of</strong> 4 Pong scans while the sky rotated. Obviously, the 

amount by which the Pong scan pattern changes relative the sky is dependent on where 

in the sky we are observing, as well as the Pong parameters such as map dimensions 

and the velocity <strong>of</strong> the scan. 

2. Over time, the sky rotates relative to the orientation <strong>of</strong> the SCUBA-2 arrays. Unless we 

are Ponging in Nasmyth coordinates, sky rotation can be used to overcome the problems 

<strong>of</strong> the ‘bad’ angles. If we find that a planned scan would be at a ‘bad’ angle, simply 

waiting until the sky rotates should be sufficient to find a time at which the orientation 

<strong>of</strong> the desired pattern relative to the arrays does not cause the angle to be ‘bad’. 

Figure 68: Pong scan pattern, showing the effects 

<strong>of</strong> sky rotation. Here, approximately one 

Pong pattern has completed. 


<strong>of</strong> sky rotation. Here, approximately two 

Pong patterns have completed. 

Figure 70: Pong scan pattern, showing the 

effects <strong>of</strong> sky rotation. Here, approximately 

three Pong patterns have completed. 


<strong>of</strong> sky rotation. Here, approximately four 

Pong patterns have completed. 



Maps A BOUSTROPHEDON SCAN DESCRIPTION 

11 Conclusions 

Throughout this study we have used the SCUBA-2 simulator to examine some <strong>of</strong> the anticipated 

scanning issues inherent in the sub-array geometry <strong>of</strong> SCUBA-2. We found that scan 

angles less that 4 ◦ from parallel to the sub-array geometry leave some regions on each path 

completely unsampled, and that angles above about 7 ◦ from parallel to the sub-array geometry 

fill the ‘gap’ between the sub-arrays and provide relatively even sampling across the scan 

path. Simulations suggest that most scanning strategies should be able to adequately compensate 

for the predicted number <strong>of</strong> ‘dead’ bolometers. A similar study should be conducted 

once the actual pattern <strong>of</strong> ‘dead’ bolometers on the arrays <strong>of</strong> SCUBA-2 is known, especially if 

the total number <strong>of</strong> unusable bolometers is much higher than predicted. We also simulated 

two proposed scanning patterns, Boustrophedon and Pong, with a variety <strong>of</strong> pattern spacings 

and angles and found that Pong should be able to provide a high level <strong>of</strong> sampling uniformity 

across a mapped region provided that care is taken in choosing the angles and spacing. 

SCUBA-2 will be an important tool for mapping large areas <strong>of</strong> the submillimetre sky with 

high sensitivity in relatively short periods <strong>of</strong> time. In order to exploit this key science goal, 

an understanding <strong>of</strong> the sub-array geometry and proposed scanning solutions is essential. 

Once SCUBA-2 is operational, the full impact <strong>of</strong> these criteria will be better understood. 

<strong>The</strong> layout <strong>of</strong> faulty bolometers at any given time, as well as the ability <strong>of</strong> the telescope to 

move in the planned patterns will need to be considered when evaluating the success <strong>of</strong> these 

proposed scanning strategies. Prior to commisioning, however, it should be possible to test 

explicit scanning patterns at the JCMT to understand whether some <strong>of</strong> the results presented 

here might change under real observing conditions. 

A 

Boustrophedon Scan Description 

This is essentially the existing SCUBA box scan – a rectangular region is filled in by a series 

<strong>of</strong> parallel scans. 

Pros: 

<strong>The</strong> orientation <strong>of</strong> the rectangle and the inclination <strong>of</strong> the scan to the rectangle are independent 

<strong>of</strong> one-another. This scan will be the preferred method in cases such as those outlined in 

the cons section for Pong. In any instance where the scan direction is highly constrained, this 

method is the most efficient for covering rectangular regions defined in arbitrary coordinate 

systems. 

Cons: 

A single map with boustrophedon will not naturally contain any cross-linking (there is no 

telescope motion in the direction transverse to the scan apart from the step and turnaround 

at the end <strong>of</strong> each row). <strong>The</strong> flexibility to choose arbitrary scan angles makes it easy to 

subsequently cover the region with another boustrophedon but with a scan angle orthogonal 

to the first pass. Even in this case, the cross-linking produced by Pong is superior since there 

is greater modulation <strong>of</strong> the time-scales over which each point in the map is visited. 

Input parameters: 

width 

= width <strong>of</strong> the rectangle in degrees 



Maps B PONG SCAN DESCRIPTION 

height 

mapangle 

scanangle 

spacing 

velocity 

= height <strong>of</strong> the rectangle in degrees 

= position angle <strong>of</strong> the box in the native coordinate system 

<strong>of</strong> the region (Nasmyth, AzEl or RADec) 

= position angle <strong>of</strong> the scan in the native coordinate system 

<strong>of</strong> the region (it may in fact be defined in some other 

coordinate system, and then converted using high-level code). 

= space between adjacent (parallel) scan lines 

Ideally this should be set to some fraction <strong>of</strong> the 

SCUBA-2 footprint such that fully-sampled maps maybe be 

constructed in a single pass. 

= target magnitude <strong>of</strong> the scan velocity excluding 

turn-arounds 

A Boustrophedon scan called ‘scanmap’ was implemented for SCUBA. Prestage et al. 10, 9 have 

documented and provided pseud-ocode for Boustrophedon-like raster mapping strategies. 

B 

Pong Scan Description 

Description: 

<strong>The</strong> telescope covers a rectangular region by scanning at angles inclined 45 ◦ to the fundamental 

axes <strong>of</strong> the rectangle, and when a side is encountered the angular velocity is reflected. By 

choosing the starting position and spacing <strong>of</strong> nodes (points <strong>of</strong> reflection) carefully, the telescope 

covers the entire map, and the path taken is closed (i.e. eventually repeats itself). Note 

that in principle one can generalize this to scanning angles other than 45 ◦ (resulting in ‘cells’ 

which are diamonds rather than squares), but we do not consider this further here, since it 

would required a different algorithm than given in the pseudo-code below. 

Pros: 

This method automatically produces the best cross-linking <strong>of</strong> scans currently planned for 

implementation on the JCMT. Every point on the map is visited on different time-scales, 

with the telescope scanning in orthogonal directions. This scan should generally be the best 

method for SCUBA-2. 

Cons: 

Since the scan angle is fixed with respect to the axes <strong>of</strong> the rectangular region, under certain 

special circumstances, Pong will not be very efficient. If, for example, it is discovered that 

the JCMT has a problem with motion in certain directions (such as induced vibrations in the 

carousel and membrane as was seen with AzTEC while performing azimuthal scans) the scan 

angle, and hence box orientation could be severely constrained. In such cases, the rectangles 

can not be tiled in such a way as to optimally cover huge areas (such as for SASSy), since 

they will have random orientations once projected onto the sky. 

Input parameters: 

width 

height 

angle 

= width <strong>of</strong> the rectangle in degrees 

= height <strong>of</strong> the rectangle in degrees 

= position angle <strong>of</strong> the box in the native coordinate system 



Maps C PSEUDO-CODE FOR ‘STRAIGHT’ PONG SCAN PATTERN 

spacing 

velocity 

<strong>of</strong> the region (Nasmyth, AzEl or RADec - note that Galactic 

coordinates and RADec have a static transformation between them, 

so a box defined in a high-level description in Galactic 

coordinates may be transformed to an RADec description before 

calling the low-level Pong scan routine). 

= space between adjacent (parallel) scan lines in the Pong 

pattern. Ideally this should be set to some fraction <strong>of</strong> the 

SCUBA-2 footprint, such that fully-sampled maps maybe be 

constructed in a single pass. 

= target magnitude <strong>of</strong> the scan velocity excluding 

turn-arounds 

C 

Pseudo-code for ‘Straight’ Pong Scan <strong>Pattern</strong> 

; Determine number <strong>of</strong> vertices (reflection points) along each side <strong>of</strong> the 

; box which satisfies the common-factors criterion and the requested 

; size / spacing 

; Find out how far apart the vertices are along the axes 

vert_spacing = ( 2.0 * spacing ) / sqrt ( 2.0 ) 

; Determine how many vertices (minimum) there must be along 

; each axis to cover the required area 

x_numvert = ceil ( width / vert_spacing ) 

y_numvert = ceil ( height / vert_spacing ) 

; Determine which is lower and check to make sure that one is 

; even while the other is odd 

if ( x_numvert >= y_numvert ) begin 

most = x_numvert 

least = y_numvert 

else begin 

most = y_numvert 

least = x_numvert 

end 

; If both are odd or both are even, increment the lesser <strong>of</strong> 

; the two, and update which is least 

if ( ( x_numvert % 2 ) == (y_numvert % 2 ) ) begin 

least += 1; 




else begin 



end 

end 




; Check for common factors between the two, and adjust as 

; necessary until x_numvert and y_numvert do not share any 

; common factors 

for ( i = 3; i



ygrid[i] = (i - y_ngridseg/2)*grid_space 

end 

; Initialization <strong>of</strong> the scan pattern 

x_init = 0 

y_init = 0 

x_<strong>of</strong>f = x_init 

y_<strong>of</strong>f = y_init 

; starting grid coordinates 

; current grid <strong>of</strong>fsets 

grid[0,0] = xgrid[x_init] ; starting tanplane <strong>of</strong>fsets <strong>of</strong> the scan 

grid[0,1] = ygrid[y_init] 

steps = 0 

; count number <strong>of</strong> steps taken in the pattern 

; Loop over line segments and calculate list <strong>of</strong> endpoints for each 

; reflection in order 

for i=0 to n_seg begin 

; increment steps to the next boundary reflection (whichever comes 

; first - along the sides or top/bottom <strong>of</strong> the rectangle) 

if( (x_ngridseg-x_<strong>of</strong>f)


Maps D PSEUDO-CODE FOR ‘CURVY’ PONG SCAN PATTERN 

if( even(y_refl) ) begin 

grid[i,0] = ygrid[y_<strong>of</strong>f] 

end 

else begin 

grid[i,0] = ygrid[y_ngridseg-y_<strong>of</strong>f] 

end 

end 

; even reflections 

; odd reflections 

; On exit, "grid" contains the ordered list <strong>of</strong> tanplane <strong>of</strong>fsets for the 

; individual line segments for the Pong pattern. At this stage, the rotation 

; <strong>of</strong> the box through "angle" may be applied to grid. 

for ( j=0; j< numvertices; j++ ) begin 

tx = grid[j][0] * cos(angle) - grid[j][1] * sin(angle); 

ty = grid[j][0] * sin(angle) + grid[j][1] * cos(angle); 

grid[j][0] = tx; 

grid[j][1] = ty; 

end 

Since this first rotation occurs in the native coordinate system <strong>of</strong> the box (Nasmyth/AzEl/RADec), 

a further rotation must be applied such that the scan end points are defined in a coordinate 

system useful for the TCS (probably AzEL tangent-plane <strong>of</strong>fsets). As the scan is executed, 

the telescope slew speed in the tangent-plane should be ‘velocity’. 

D 

Pseudo-code for ‘Curvy’ Pong Scan <strong>Pattern</strong> 

<strong>The</strong> ‘Curvy’ Pong pattern is based on the ‘boxscan’ used by SHARC-II and allows for an 

approximation <strong>of</strong> a Pong pattern while avoiding sharp turnarounds at the vertices. <strong>The</strong> path 

<strong>of</strong> this ‘Curvy’ Pong pattern is described by two triangle wave approximations. With x(t) and 

y(t) driven by Fourier expansions (approximately 5 terms) <strong>of</strong> a triangle wave, the pattern 

approaches straight lines across the central region <strong>of</strong> the scan while rounding <strong>of</strong>f the corners. 

This approach may be necessary if it is found that abrupt acceleration changes result in 

pointing problems. Note, with only one term, this pattern describes a Lissajous pattern, 

while with an infinite number <strong>of</strong> terms it recreates the ‘Straight’ Pong pattern as above. 

; Determine number <strong>of</strong> vertices (reflection points) along each side <strong>of</strong> the 

; box which satisfies the common-factors criterion and the requested 

; size / spacing 

; Find out how far apart the vertices are along the axes 

vert_spacing = ( 2.0 * spacing ) / sqrt ( 2.0 ) 

; Determine how many vertices (minimum) there must be along 

; each axis to cover the required area 

x_numvert = ceil ( width / vert_spacing ) 

y_numvert = ceil ( height / vert_spacing ) 




; Determine which is lower and check to make sure that one is 

; even while the other is odd 




else begin 



end 

; If both are odd or both are even, increment the lesser <strong>of</strong> 

; the two, and update which is least 

if ( ( x_numvert % 2 ) == (y_numvert % 2 ) ) begin 

least += 1; 




else begin 



end 

end 

; Check for common factors between the two, and adjust as 

; necessary until x_numvert and y_numvert do not share any 

; common factors 

for ( i = 3; i



period = x_numvert * y_numvert * vert_spacing * 2 / vavg 

; Determine the total number <strong>of</strong> positions required to be 

; recorded 

pongcount = period / sample_interval 

; Calculate the amplitudes <strong>of</strong> x(t) and y(t) 

amp_x = x_numvert * vert_spacing / 2 

amp_y = y_numvert * vert_spacing / 2 

; Get the positions at each time 

t_count = 0 

for ( i = 0 to pongcount ) begin 

end 

; Calculate the grid positions from 5 terms 

tx = ( ( 8.0 * amp_x ) / ( pi * pi ) ) * 

( sin ( 2.0 * pi * t_count / peri_x ) - 

( 1.0/9.0 * sin ( 6.0 * pi * t_count / peri_x ) ) + 

( 1.0/25.0 * sin ( 10.0 * pi * t_count / peri_x ) ) - 

( 1.0/49.0 * sin ( 14.0 * pi * t_count / peri_x ) ) + 

( 1.0/81.0 * sin ( 18.0 * pi * t_count / peri_x ) ) ) 

ty = ( ( 8.0 * amp_y ) / ( pi * pi ) ) * 

( sin ( 2.0 * pi * t_count / peri_y ) - 

( 1.0/9.0 * sin ( 6.0 * pi * t_count / peri_y ) ) + \ 

( 1.0/25.0 * sin ( 10.0 * pi * t_count / peri_y ) ) - 

( 1.0/49.0 * sin ( 14.0 * pi * t_count / peri_y ) ) + 

( 1.0/81.0 * sin ( 18.0 * pi * t_count / peri_y ) ) ) 

; Apply the rotation angle 

grid[i][0] = tx * cos(angle) - ty * sin(angle) 

grid[i][1] = tx * sin(angle) + ty * cos(angle) 

t_count++ 



Maps E VELOCITIES AND ACCELERATIONS OF ‘CURVY’ PONG 

E 

Velocities and Accelerations <strong>of</strong> ‘Curvy’ Pong 

Here we give an explicit example <strong>of</strong> a ‘Curvy’ Pong pattern and examine the velocity and 

acceleration components. 

Figure 72: ‘Curvy’ Pong used to examine velocities and accelerations. 

We choose the following parameters, based on values determined from a simulated Pong 

observation. α x and α y are the amplitudes <strong>of</strong> the x and y motion, respectively, and β x and 

β y are the periods <strong>of</strong> the x and y motion, respectively. t ranges from 0 to 17 (just over one 

full period <strong>of</strong> the complete pattern). <strong>The</strong> original Pong simulation had width ≃2400 arcsec, 

height ≃1600 arcsec, and sweep spacing <strong>of</strong> 600 arcsec. 

α x = 1272.792206; α y = 848.528137; β x = 8.475; β y = 5.65; (3) 

a = 8α x 

π 2 ; 

b = 2π 

β x 

; 

c = 8α y 

π 2 ; 

d = 2π 

β y 

; (4) 

<strong>The</strong>n we have the triangle wave expansions (plotted in Fig. 73) : 

[ 

x(t) = a sin(bt) − 1 9 sin(3bt) + 1 

25 sin(5bt) − 1 

49 sin(7bt) + 1 ] 

81 sin(9bt) ; (5) 

[ 

y(t) = c sin(dt) − 1 9 sin(3dt) + 1 

25 sin(5dt) − 1 49 sin(7dt) + 1 ] 

81 sin(9dt) . (6) 




Figure 73: x(t) (bottom) and y(t) (top) <strong>of</strong>fset by 7000 for clarity. 

<strong>The</strong> velocity and acceleration components are : 

ẋ(t) = ab 

[cos(bt) − 1 3 cos(3bt) + 1 5 cos(5bt) − 1 7 cos(7bt) − 1 ] 

9 cos(9bt) ; (7) 

ẏ(t) = cd 

[cos(dt) − 1 3 cos(3dt) + 1 5 cos(5dt) − 1 7 cos(7dt) − 1 ] 

9 cos(9dt) ; (8) 

ẍ(t) = ab 2 [− sin(bt) + sin(3bt) − sin(5bt) + sin(7bt) − sin(9bt)] ; (9) 

ÿ(t) = cd 2 [− sin(dt) + sin(3dt) − sin(5dt) + sin(7dt) − sin(9dt)] . (10) 

<strong>The</strong>se are plotted in Figs. 74 and 75. 

We can also define the magnitude <strong>of</strong> the velocity as 

v(t) = √ [ẋ 2 (t) + ẏ 2 (t)] (11) 

and similarly for the magnitude <strong>of</strong> the acceleration |a(t)| as plotted in Figs. 76 and 77. 

<strong>The</strong> acceleration components are mildy oscillating, with spikes <strong>of</strong> up to about 4000 arcsec 

s −2 . <strong>The</strong> magnitudes <strong>of</strong> the total velocity and total acceleration vary widely depending on the 

motion in the x and y directions at any given time. Clearly, the final result will depend on the 

individual x and y velocities, as well as the number <strong>of</strong> terms used in the Fourier expansion. 




Figure 74: ẋ(t) (bottom) and ẏ(t) (top) <strong>of</strong>fset by 7000 for clarity. 

Figure 75: ẍ(t) (bottom) and ÿ(t) (top) <strong>of</strong>fset by 7000 for clarity. 




Figure 76: |v(t)|. 

Figure 77: |a(t)|. 



Maps 

REFERENCES 

References 

[1] C. Borys. Box Scan for SHARC-II. CSO telescope notes, URL: 

http://www.submm.caltech.edu/∼sharc/operating/boxscan.htm, 2003. 

[2] B. G. Gom. SCUBA-2 FTS Science Case. SCUBA-2 Project SC2/FTS/SCE/001, 2003. 

[3] W. S. Holland, E. I. Robson, W. K. Gear, C. R. Cunningham, J. F. Lightfoot, T. Jenness, 

R. J. Ivison, J. A. Stevens, P. A. R. Ade, M. J. Griffin, W. D. Duncan, J. A. Murphy, and 

D. A. Naylor. SCUBA: a common-user submillimetre camera operating on the James 

Clerk Maxwell Telescope. Monthly Notices <strong>of</strong> the Royal Astronomical Society, 303:659– 

672, March 1999. 

[4] Wayne S. Holland. Functional and performance requirements for SCUBA-2. SCUBA-2 

Project SC2/SRE/SC200/002. 

[5] Wayne S. Holland. Science requirements. SCUBA-2 Project SC2/SRE/SC200/001. 

[6] B. D. Kelly. SCUBA-2 modes and data processing. SCUBA-2 Project SC2/ANA/S100/028, 

2001. 

[7] B. D. Kelly and W. D. Duncan. Application <strong>of</strong> DREAM Observing to SCUBA-2. SCUBA-2 

Project SC2/ANA/S100/006, 2001. 

[8] P.Bastien, T.Jenness, and J.Molnar. A Polarimeter for SCUBA-2. Astronomical Polarimetry: 

Current Status and Future Directions ASP Congerence Series, 343, 69, 2005. 

[9] R. Prestage, F. J. Oliveira, J. A. Bailey, P. Friberg, and S. Kenderdine. Route-finding 

algorithm for the optimal control <strong>of</strong> telescope slews. In H. Lewis, editor, Proc. SPIE Vol. 

3112, p. 76-87, Telescope Control Systems II, Hilton Lewis; Ed., pages 76–87, September 

1997. 

[10] Richard Prestage. Discussion <strong>of</strong> the JCMT Cell and Rastering Facilities. Joint Astronomy 

Centre James Clerk Maxwell Telescope TCS/DN/001, 1996. 

[11] Alex van Engelen and Douglas Scott. SCAN Mode <strong>Strategies</strong> for SCUBA-2. SCUBA-2 

Project SC2/ANA/S210/005, 2005.

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