An optimised tiling pattern for multi-object spectroscopic surveys: application to the 4MOST survey
E. Tempel, T. Tuvikene, M. M. Muru, R. S. Stoica, T. Bensby, C. Chiappini, N. Christlieb, M.-R. L. Cioni, J. Comparat, S. Feltzing, I. Hook, A. Koch, G. Kordopatis, M. Krumpe, J. Loveday, I. Minchev, P. Norberg, B. F. Roukema, J. G. Sorce, J. Storm, E. Swann, E. N. Taylor, G. Traven, C. J. Walcher, R. S. de Jong
MMNRAS , 1–20 (2020) Preprint 8 July 2020 Compiled using MNRAS L A TEX style file v3.0
An optimised tiling pattern for multi-ob ject spectroscopicsurveys: application to the 4MOST survey
E. Tempel, (cid:63) T. Tuvikene, M. M. Muru, R. S. Stoica, T. Bensby, C. Chiappini, N. Christlieb, M.-R. L. Cioni, J. Comparat, S. Feltzing, I. Hook, A. Koch, G. Kordopatis, M. Krumpe, J. Loveday, I. Minchev, P. Norberg, B. F. Roukema, , J. G. Sorce, , J. Storm, E. Swann, E. N. Taylor, G. Traven, C. J. Walcher, and R. S. de Jong Tartu Observatory, University of Tartu, Observatooriumi 1, 61602 T˜oravere, Estonia Universit´e de Lorraine, CNRS, IECL, 54000 Nancy, France Lund Observatory, Department of Astronomy and Theoretical Physics, Box 43, SE-221 00 Lund, Sweden Leibniz-Institut f¨ur Astrophysik Potsdam (AIP), An der Sternwarte 16, D-14482 Potsdam, Germany Zentrum f¨ur Astronomie der Universit¨at Heidelberg, Landessternwarte, K¨onigstuhl 12, 69117 Heidelberg, Germany Max-Planck-Institut f¨ur Extraterrestrische Physik (MPE), Giessenbachstr., D-85748 Garching, Germany Physics Department, Lancaster University, Lancaster LA1 4YB, UK Zentrum f¨ur Astronomie der Universit¨at Heidelberg, Astronomisches Rechen-Institut, M¨onchhofstr. 12, 69120 Heidelberg, Germany Universit´e Cˆote d’Azur, Observatoire de la Cˆote d’Azur, CNRS, Laboratoire Lagrange, France Astronomy Centre, University of Sussex, Falmer, Brighton BN1 9QH, UK Institute for Computational Cosmology and Centre for Extragalactic Astronomy, Department of Physics, Durham University,South Road, Durham DH1 3LE, UK Institute of Astronomy, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University, Grudziadzka 5,87-100 Toru´n, Poland Univ Lyon, Ens de Lyon, Univ Lyon1, CNRS, Centre de Recherche Astrophysique de Lyon UMR5574, F–69007, Lyon, France Institute of Cosmology and Gravitation, University of Portsmouth, Burnaby Road, Portsmouth PO1 3FX, UK Centre for Astrophysics and Supercomputing, Swinburne University of Technology, Hawthorn, VIC 3122, Australia
ABSTRACT
Large multi-object spectroscopic surveys require automated algorithms to optimisetheir observing strategy. One of the most ambitious upcoming spectroscopic surveys isthe 4MOST survey. The 4MOST survey facility is a fibre-fed spectroscopic instrumenton the VISTA telescope with a large enough field of view to survey a large fraction ofthe southern sky within a few years. Several Galactic and extragalactic surveys will becarried out simultaneously, so the combined target density will strongly vary. In thispaper, we describe a new tiling algorithm that can naturally deal with the large targetdensity variations on the sky and which automatically handles the different exposuretimes of targets. The tiling pattern is modelled as a marked point process, which ischaracterised by a probability density that integrates the requirements imposed bythe 4MOST survey. The optimal tilling pattern with respect to the defined model isestimated by the tiles configuration that maximises the proposed probability density.In order to achieve this maximisation a simulated annealing algorithm is implemented.The algorithm automatically finds an optimal tiling pattern and assigns a tentativesky brightness condition and exposure time for each tile, while minimising the totalexecution time that is needed to observe the list of targets in the combined input cata-logue of all surveys. Hence, the algorithm maximises the long-term observing efficiencyand provides an optimal tiling solution for the survey. While designed for the 4MOSTsurvey, the algorithm is flexible and can with simple modifications be applied to anyother multi-object spectroscopic survey.
Key words: surveys – methods: miscellaneous – techniques: miscellaneous (cid:63)
E-mail: [email protected] © a r X i v : . [ a s t r o - ph . I M ] J u l E. Tempel et al.
An integral part of the preparation of any multi-object spec-troscopic survey is the construction of the tiling pattern (theset of centres and orientations on the sky of the observa-tional field, “tiles”) – we need to know where to point thetelescope and for how long each tile should be observed.In general, there are two approaches for finding an optimaltiling pattern. In the first approach, the tiling pattern isconstructed on the fly, and the job of the tiling algorithmis to find the next telescope pointing, while taking into ac-count already observed fields and targets. An example is theheuristic Greedy algorithm (Robotham et al. 2010) that isused in the Galaxy And Mass Assembly (GAMA) survey(Driver et al. 2009; Liske et al. 2015) and will be used in theTaipan survey (da Cunha et al. 2017).In the second approach, the tiling pattern is constructedbefore the survey starts, and is usually used to cover a givensky area uniformly. This approach is successfully used inthe Two Degree Field Galaxy Redshift Survey (2dFGRS,Colless et al. 2001), the Sloan Digital Sky Survey (SDSS,Blanton et al. 2003), the Six-degree Field (6dF) Galaxy Sur-vey (Jones et al. 2004) and the WiggleZ survey (Drinkwateret al. 2010). For these surveys, an adaptive tiling algorithmis used, where the uniform distribution of field centres issuccessively altered to more closely follow the target dis-tribution. This algorithm is effective in providing uniformtargeting completeness over the sky.The Greedy algorithm (Robotham et al. 2010) worksvery well for dense surveys, where the same sky region isvisited several times. In contrast, an adaptive tiling algo-rithm is used when a given sky region needs to be coveredwith a minimum number of fields. In the 4MOST survey(de Jong et al. 2019; Walcher et al. 2019), both of these as-pects must be optimized, so a new algorithm needs to bedeveloped.The 4MOST survey is a spectroscopic survey that willobserve millions of targets covering almost the entire south-ern sky. The 4MOST survey consists of many sub-surveyscovering different areas in the sky, which have very dif-ferent number densities of targets. Fig. 1 shows the es-timated exposure time in the sky based on the current4MOST mock catalogues and the present survey strategy(Guiglion et al. 2019). In Fig. 1 we have combined the tar-gets from the ten 4MOST consortium surveys: the MilkyWay Halo Low-Resolution Survey (Helmi et al. 2019), theMilky Way Halo High-Resolution Survey (Christlieb et al.2019), the Milky Way Disc and Bulge Low-Resolution Sur-vey (4MIDABLE-LR, Chiappini et al. 2019), the Milky WayDisc and Bulge High-Resolution Survey (4MIDABLE-HR,Bensby et al. 2019), the eROSITA Galaxy Cluster RedshiftSurvey (Finoguenov et al. 2019), the Active Galactic NucleiSurvey (Merloni et al. 2019), the Wide-Area VISTA Extra-galactic Survey (WAVES, Driver et al. 2019), the CosmologyRedshift Survey (CRS, Richard et al. 2019), the One Thou-sand and One Magellanic Fields Survey (1001MC, Cioniet al. 2019), and the Time-Domain Extra-galactic Survey(TiDES, Swann et al. 2019). The mock catalogues are basedeither on Gaia catalogues (Gaia Collaboration et al. 2016,2018) or on the Galaxia model of the Galaxy (Sharma et al.2011), or on MultiDark simulations augmented with modelsof galaxies and clusters (Klypin et al. 2016; Comparat et al. 2019), or on TAO mocks (Bernyk et al. 2016), or on GAL-FORM mocks (Cole et al. 2000; Lagos et al. 2012). Theyrepresent reasonably well each survey individually. Futurework on mock catalogues should accurately reproduce thecross-correlation between surveys. In the 4MOST surveys,most of the targets that will be observed are known fromprevious surveys and selected beforehand. The only excep-tion is a small fraction of transients from the TiDES surveythat will be selected based on live LSST observations. Sincethe number of transients is small and their spatial distri-bution is not clustered, we will ignore these targets in thecurrent paper and we assume that all targets and their esti-mated exposure times are known.The 4MOST field of view covers approximately foursquare degrees and is hexagonally shaped. It is covered by1624 low-resolution (LR) and 812 high-resolution (HR) spec-trograph fibres. On average, there are 391 LR and 196 HRspectrograph fibres per square degree. Fibres are placed witha regular pattern in the field of view and have some rangeof movement that allows them to be aligned to targets ofinterest (see Fig. 2). Tempel et al. (2020) gives a detailedoverview of the capabilities and efficiency of the probabilisticfibre-to-target assignment algorithm developed specificallyfor the 4MOST survey. In order to apply the probabilistictargeting algorithm, we need a predefined tiling pattern thatis optimised for the input targets and takes the constraintsand requirements of the 4MOST facility and surveys intoaccount. The most significant factor in determining the sci-entific impact of the 4MOST will be efficiency – maximisingfibre occupancy and minimising observational overheads.The aim of this paper is to find an optimal tiling so-lution that increases survey efficiency. We propose an al-gorithm based on marked point processes. The idea is tomodel the tiling pattern as a marked point process, whereeach tile is considered as a free object, whose parameters(location, exposure time, etc.) need to be determined. Amathematically similar approach is successfully used to de-tect galaxy filaments (Tempel et al. 2014, 2016) and galaxygroups (Tempel et al. 2018) in spectroscopic galaxy surveys.Although the detection of cosmic web elements and findingthe optimal tiling pattern are seemingly very different ap-plications, mathematically both applications are pattern de-tection problems that can be tackled with the marked pointprocess approach.This paper is organised as follows. In Section 2 we de-scribe the tiling challenge and define the inputs for, and theoutputs of, the proposed tiling algorithm. In Section 3 wedescribe the marked point process framework that we useto solve the optimal tiling problem. In Section 4 we illus-trate the proposed algorithm with examples. Conclusionsare drawn in Section 5.
The challenge we are facing in 4MOST survey preparationis how to most efficiently observe all required targets in theinput catalogues, while maximising the fibre usage and min-imising the total time (including overhead time) requiredto successfully observe the given set of targets. This can beconsidered as a tiling pattern optimisation problem. In thecurrent paper we define a tile as a single science exposure.
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MNRAS000 , 1–20 (2020) ptimal tiling algorithm
360 300 240 180 120 60 00 030 3060 6010 20 50 100 200 500 1000Exposure time per field (min)
Figure 1.
Required exposure-time map in equatorial coordinates for low-resolution (LR) targets, based on current 4MOST mockcatalogues, both Galactic and extragalactic. The required exposure times and associated target densities vary significantly from one skyregion to another. Exposure times have been calculated using the 4MOST Exposure Time Calculator assuming a fixed median seeingof 0.8 arcsec and airmass 1.2 for all targets. The targets are limited to declination between − and + degrees. The footprints of thedifferent sub-surveys in 4MOST are clearly visible. The same set of targets is used in the examples presented in Section 4. -1.0-0.50.00.51.0 -1.0 -0.5 0.0 0.5 1.0 Y ( deg ) X (deg) LRHR Figure 2.
In order to observe a given set of targets, we need a tilingpattern induced by the survey’s input catalogues.Each tile has a fixed sky coordinate (field centre) andinstrument position angle. Several tiles with the same skycoordinates and position angle can be combined into oneobserving block (OB). Within one OB, tiles can have differ-ent exposure times and each tile has its own fibre-to-targetconfiguration.An algorithm that defines an optimal tiling pattern forthe 4MOST survey should take into account the followingaspects:(i) Each tile has an individual exposure time that takes into account the requested exposure times of targets in thefield of view. The exposure time is attached to each tile, as-suming that it is observed in a fixed sky brightness condition(i.e., bright, grey or dark). This separates tiles into B/G/Dgroups .(ii) Tiles can be combined into OBs, which allows reduc-ing the overhead time associated with telescope movementand field acquisition. One OB can contain one or many tileswith the same sky brightness condition (B/G/D) observedduring one telescope pointing. The duration of one OB islimited by the total exposure time (approximately one hourper OB).(iii) In 4MOST there are two resolution modes – high andlow resolution. The fibre pattern for each of them is fixedand both of them are used simultaneously (see Fig. 2). Eachsub-survey specifies whether they want to use the high orthe low resolution. The optimal tiling algorithm then aimsto optimise both high- and low-resolution observations atthe same time.(iv) Some sub-surveys can have specific requirements thataffect the tiling pattern. For example, if a region in the skyis covered by several OBs (due to repeatability and/or highdensity of targets), then the centres of the OBs should pref-erentially avoid each other. Such a strategy will help to mit-igate fixed fibre patterns. It will also tend to reduce visu-ally striking contributions of the shape of the 4MOST fieldof view in the selection functions. Additionally, some sub-surveys require contiguous coverage of the sky, which trans-lates to no gaps between tiles, while other sub-surveys wishto cover largest possible sky area and gaps between tiles arenot a problem. An optimal tiling algorithm should be able to Separation into B/G/D (bright/grey/dark observing condi-tions) groups is somewhat arbitrary. In general, any number ofgroups can be used if it is necessary and if it helps to increasesurvey efficiency without over-complicating the problem.MNRAS , 1–20 (2020)
E. Tempel et al. take into account these various scientific requirements fromthe different surveys.In general, to find the optimal tiling is a complicatedproblem that is interlinked with many other aspects of sur-vey optimisation, including, for example, fibre-to-target as-signment algorithm and the OB scheduling algorithm. Thelatter affects the division of tiles between different skybrightness conditions and the tentative exposure times ofthe tiles. For multiplex surveys such as the 4MOST survey,it is computationally unfeasible to solve all problems simul-taneously. In the current paper, to reduce the complexity ofthe survey optimisation, the optimal tiling problem is solvedindependently with clearly defined inputs and outputs. Theinput data for the proposed tiling algorithm are described inSection 2.1 and the output data are defined in Section 2.2.The main aim of the proposed tiling algorithm is to findan optimal set of tiles that is required to observe a given setof targets. A probabilistic fibre to target assignment algo-rithm that uses the tiles as an input is described in Tempelet al. (2020). For the proposed algorithm, it is not importantin which order the tiles are observed. The latter is a schedul-ing problem, which will be solved independently from thetiling and the fibre-to-target assignment algorithms.
The input data for the tiling algorithm are the following. Wehave a fixed set of targets, where for each target we have thefollowing parameters: • RA, Dec : coordinates on the celestial sphere (“skyplane”). • Low or high resolution : a flag that specifies whetherthe target should be observed with an LR or HR spectro-graph fibre. • T Bexp , T Gexp , T Dexp : required exposure times of the targetin bright, grey and dark sky conditions, respectively. • f compl : the probability that the target should be suc-cessfully observed. To fulfil the survey science goals, somesub-surveys require only a fraction of targets from their in-put catalogues.During the 4MOST five-year survey, approximately 32%of the observing time is bright, 21% is grey and 47% is dark .The generated tiling pattern (total exposure time for D/G/Bconditions) should roughly follow these fractions.Because of the telescope and instrument design, themaximum exposure time for a single exposure is limited. Inthe current paper, we assume that the maximum exposuretime is 30 minutes. Additionally, the total time (summedexposure times plus overheads) for a single OB is typicallyaround one hour. In the current paper, we adopt a maximum The required exposure time during real observations also de-pends on the sky transparency and seeing conditions. In this pa-per we ignore this effect and assume an average conditions every-where. The fraction of bright, grey and dark time depends on the thethresholds of sky brightness levels adopted. The current estimatesare based on the ESO definitions for bright, grey and dark skyconditions ( ). OB length of 75 minutes, which is also feasible. This allowsthe observation of two 30 min science exposures in a singleOB. Because of the overhead time for each exposure/tile andfor each OB , several tiles/exposures are combined into oneOB, which helps to reduce the total overhead time. Addi-tionally, since overhead is constant (it does not depend onexposure times), this choice should reduce the total numberof tiles, which tends to yield longer exposures. The tiling algorithm should find the tiling pattern that al-lows optimal observation of the given set of targets. Theoutput of the tiling algorithm is the list of OBs, where foreach OB we have the following parameters: • RA, Dec : the coordinates of the centre of an OB. • Position angle : an angle determining the rotation ofthe field (hexagon) in the sky. • B/G/D flag : a flag specifying whether the OB shouldbe preferentially observed during bright, grey or dark skyconditions. • List of tiles and exposure times : one or severaltiles/exposures. The tiling algorithm should give the numberof tiles for each OB. Tiles in one OB can have different ex-posure times. The algorithm should determine the expectedexposure time for each tile. The sum of these exposures plusthe overhead time is the total time for a single OB.The distribution of OBs/tiles in the sky and exposuretimes per tiles should allow the observation of the requiredset of targets in the input catalogue, while minimising un-used/wasted observational time (e.g., empty fibres, overex-posure). In general, the optimal tiling solution allows to suc-cessfully observe the required set of targets with a minimumamount of time.
In the next Section, we describe the mathematical frame-work of the proposed optimal tiling algorithm and provideall the necessary details. To help understand the general con-cept of the algorithm, here we present a general outline ofthe process.In the proposed algorithm, we model the tiling patternas a marked point process (see Sections 3.1 and 3.2), wherethe number of tiles, together with the location and exposuretime of each tile, are free parameters. An optimal tiling pat-tern is defined via an energy function: the global minimumof this energy function defines the optimal tiling pattern.We define the energy function as a sum of individual com-ponents, where each optimises a certain aspect of the tilingpattern. The most important energy function component iscomputed using a statistical fibre-to-target assignment algo-rithm. This allows us to compare the generated tiling pattern In the current paper, each OB has an overhead of 3.5 min-utes and each exposure/tile has an additional overhead of 4.4minutes ( ).These overhead times are current estimates and might change be-fore the 4MOST survey starts. MNRAS000
In the next Section, we describe the mathematical frame-work of the proposed optimal tiling algorithm and provideall the necessary details. To help understand the general con-cept of the algorithm, here we present a general outline ofthe process.In the proposed algorithm, we model the tiling patternas a marked point process (see Sections 3.1 and 3.2), wherethe number of tiles, together with the location and exposuretime of each tile, are free parameters. An optimal tiling pat-tern is defined via an energy function: the global minimumof this energy function defines the optimal tiling pattern.We define the energy function as a sum of individual com-ponents, where each optimises a certain aspect of the tilingpattern. The most important energy function component iscomputed using a statistical fibre-to-target assignment algo-rithm. This allows us to compare the generated tiling pattern In the current paper, each OB has an overhead of 3.5 min-utes and each exposure/tile has an additional overhead of 4.4minutes ( ).These overhead times are current estimates and might change be-fore the 4MOST survey starts. MNRAS000 , 1–20 (2020) ptimal tiling algorithm with the targets in the input catalogue, in order to estimatethe time that would still be needed in order to observe the re-quired targets in the input catalogue (“missing” time) and toestimate the time that remains unused due to empty fibres.Additional energy function components are used to minimisethe total overhead time and to divide the tiles between pre-defined sky conditions, while minimising the total time thatis necessary in order to observe all required targets in theinput catalogue. We also define an energy function with com-ponents that allow us to define the interactions between tilesin a way that potentially minimises the impact of the fixedtiling pattern on the final selection function of the survey.The optimisation challenge we are facing involves a largenumber of parameters, whereas the number of free param-eters (the number of tiles) is itself a free parameter. Theproposed algorithm finds itself the number of tiles. The min-imisation of the energy function is achieved via a simulatedannealing algorithm, which is a global optimisation methodthat avoids local minima. The key assumption of our proposed algorithm is that thetiling pattern is a configuration of random interacting ob-jects driven by the probability density of a marked pointprocess. The solution of the optimal tiling pattern is givenby the construction and manipulation of such a probabil-ity density. The probability density we propose takes intoaccount all the observational constraints and requirementsfrom all surveys. Statistical inference using this probabilitydensity is done using Markov-chain Monte Carlo (MCMC)techniques. Such a probability density can be written as p ( y | θ ) ∝ exp [− U ( y | θ )] , (1)where U ( y | θ ) is the energy function, y is the pattern of ob-jects (tiles in the sky) and θ is the vector of model parame-ters. The marked point processes driven by probability den-sities in Eq. (1) are known in the literature as a Gibbs pointprocesses. The energy function U ( y | θ ) can be further writtenas the sum of several components that take into account dif-ferent aspects of the optimisation problem (see Section 3.2).The Bayesian framework allows the introduction of theknowledge regarding the parameters via a posterior distri-bution p ( θ ) . This allows writing the joint distribution of thetiling pattern and the model parameters: p ( y , θ ) = p ( y | θ ) p ( θ ) . (2)A joint tiling pattern and parameter estimator is given bythe maximum of the probability density (2): ( ˆ y , ˆ θ ) = arg max Ω × Θ p ( y , θ ) = arg max Ω × Θ p ( y | θ ) p ( θ ) , (3)where Ω is the pattern configuration space and Θ representsthe parameter space. The estimator given by (3) can be com-puted using a simulated annealing algorithm (van Lieshout1994; Stoica et al. 2005).For simplicity and in order to reduce the computationalcost, most of the model parameters θ are fixed during the Monte Carlo simulation. In the current paper, the estimationof these parameters is done using an educated guess and viatrial and error, whenever necessary. The free parameters ofthe model are described in Section 3.2 and the parametervalues used in the current paper are given in Table 1. Themodel parameters θ can be estimated if the tiling patternis available following Stoica et al. (2017) and the referencestherein. Let W be a spatial observation window of Lebesgue mea-sure ν ( W ) . In the current paper, W is a finite region in thesky plane (sky area reachable by the 4MOST facility). Asimple point process on W is a finite random configurationof points x i ∈ W , i = , . . . , n such that x i (cid:44) x j whenever i (cid:44) j , where n is the number of points in a point process.Characteristics or marks can be attached to the points viaa probability distribution. A finite random configuration ofmarked points is a marked point process if the distributionof only the locations is a simple point process. For furtherreading on marked point processes we recommend the mono-graphs by van Lieshout (2000) and Møller & Waagepetersen(2004). In the current paper, tiles are considered as markedpoints and they are modelled as a marked point process.The generating object (a marked point) of the tilingpattern is given by a tile y = ( α, δ, PA , i BGD , T exp ) . The tilecentre coordinates are given by α, δ ∈ W , where α, δ are rightascension and declination in the sky. The mark is representedby the following parameters: PA is a position angle, i BGD isthe sky condition flag and T exp gives the exposure time ofthe tile. To find the optimal tiling pattern means to find theset of tiles y = y , y , . . . , y N tiles that are needed to observe agiven set of targets t = t , t , . . . , t N tar . While the number oftargets N tar and parameters of targets (see Section 2.1) areknown, the number of tiles N tiles and parameters of each tile( α, δ, PA , i BGD , T exp ) are the subject of optimal tiling patterndetermination described in Section 2.The optimal tiling estimator is defined by the tiling con-figuration that maximises the probability density in Eq. (1)as it minimises the corresponding energy function U ( y | θ ) .For the problem at hand, the energy function U ( y | θ ) is con-structed as follows: U ( y | θ ) = U targets ( y | θ ) + U overhead ( y | θ ) + U tiles ( y | θ ) + U BGD ( y | θ ) , (4)where each component in the energy function takes into ac-count different aspects in the optimal tiling challenge. Theenergy function U ( y | θ ) is calculated for a given set of tiles y and using a fixed set of targets t . Each component of theenergy function is described below in detail. The function U targets ( y | θ ) takes into account the exposure times of tar-gets and is used to minimise the summed exposure timeof tiles that is needed to observe a given set of targets t ; U overhead ( y | θ ) minimises the overhead associated with eachOB and individual exposures; U tiles ( y | θ ) is used to opti-mise the placement of tiles with respect to each other; and U BGD ( y | θ ) is introduced to comply with the available frac-tion of observational time in bright, grey or dark sky condi-tions.The definition of the terms of each energy function com- MNRAS , 1–20 (2020)
E. Tempel et al. ponent (see below) together with the values of the parame-ters lead to a locally stable model. This means that the con-tribution to the general energy function of a new tile to anexisting configuration is bounded below. This property im-plies the integrability of the model. The local stability is alsorequired in order to obtain the required convergence proper-ties for the simulation algorithm of the model (van Lieshout2000; van Lieshout & Stoica 2003; Møller & Waagepetersen2004). In the following, we describe the implementation de-tails of each of the energy function components and of theMCMC simulation method. U targets ( y | θ ) This is the most important component of the energy func-tion. This component ensures that the targets in the in-put catalogue are observed efficiently. The energy function U targets ( y | θ ) is based on all targets and all tiles in the sky.While minimizing this energy we find the best tiling thatallows optimal observation of a given set of targets. This isdefined as U targets ( y | θ ) = A ( FoV ) ∬ S U s targets ( y | θ ) d s , (5) U s targets ( y | θ ) = (cid:2) c miss T s miss + c wasted T s wasted (cid:3) for { t ∈ t : (cid:107) t − s (cid:107) < s max } , (6)where S ∈ W is the region in the sky where targets are lo-cated. The inverse of the normalization constant in front ofthe surface integral, A ( FoV ) , is the area of one 4MOST fieldof view. This gives the energy (missing and wasted time)as an average quantity per one field of view. For a region s in the sky, the function U s targets is estimated based on tar-gets closer than s max = . deg from the centre of region s .In a circle with radius 0.1 deg there are on average 12 LRand 6 HR spectrograph fibres. For simplicity, the integral inEq. (5) is estimated as a sum over HEALPix pixels (G´orskiet al. 1999) in the ( H = , N = ) member of the HEALPixfamily of equal-area projections from the sky to the plane(Calabretta & Roukema 2007). We use the HEALPix Nsideparameter of 1024, which gives around 1300 pixels in one4MOST field of view. The s max defines the smoothing scalefor the U s targets energy function component.In Eq. (6), T s miss is the exposure time that is missing inorder to observe all targets in region s and T s wasted is the ob-servational time that is not used for science targets. Wastedtime counts the time that is not used at all for science targets(e.g., empty fibres) and counts the time over which sciencetargets were over-exposed. Positive constants c miss ≥ and c wasted ≥ can be used to fine tune the balance betweenmissing and wasted observations.In the current paper, the missing and wasted time isestimated as: T s miss = (cid:213) X ∈[ LR , HR ] c X (cid:104) T s req , X − T s obs , X (cid:105) , (7) https://healpix.jpl.nasa.gov T s wasted = (cid:213) X ∈[ LR , HR ] c X (cid:104) T s over − exp , X + T s not − used , X (cid:105) , (8)where T s req is the required exposure time in a region s inorder to observe all targets in this region; T s obs is the expo-sure time that was actually used to observe the targets inthis region. The term T s over − exp takes into account the over-exposure of targets and T s not − used gives the time that was notused for science targets (time lost because of empty fibres).The missing and wasted time is calculated separately forlow and high resolution. The parameters c LR and c HR canbe used to control the relative importance of LR and HRtargets and fibres. In the current paper, we set c LR = / and c HR = / , so that they reflect the number density ofLR and HR spectrograph fibres.To calculate the quantities T s req , T s obs , T s over − exp and T s not − used , we have to assign fibres to targets. Tempel et al.(2020) describes a probabilistic fibre-to-target assignmentalgorithm. However, this algorithm is computationally ex-pensive and in practice it cannot be used in the proposedtiling algorithm. To generate an optimal tiling pattern, wewill use a simplified (and computationally faster) version ofthe fibre-to-target assignment algorithm. The simplified ver-sion described below does not assign real fibres to targets;it is only used to statistically mimic the fibre-to-target as-signment. Hence, the simplified targeting used in the currentpaper does not replace the need for a probabilistic target-ing such as the one proposed in Tempel et al. (2020). Inthe current paper, the calculation of T s req , T s obs , T s over − exp and T s not − used is performed as described below. The calculation ofthese quantities is the same for LR and HR targets, exceptthat the fibre density is different for LR and HR fibres. Calculation of T s req . To estimate the required exposuretime in a region s , we assume that any fibre can be placedon any target in that region. The T s req for LR targets is esti-mated as T s req , LR = N s LR , fib (cid:213) t ∈ t LR s T Dexp ( t ) · f compl ( t ) , (9)where the summation is over LR targets t that belong toregion s , in the sense that they are closer than s max to thecentre of region s in the sky. The set of LR targets that be-long to region s is designated as t LR s . The parameter N s LR , fib gives the average number of LR fibres in region s . It is esti-mated as N s LR , fib = c sci fib · ρ LRfib · A ( s ) , (10)where ρ LRfib defines the average LR fibre density in one field ofview, A ( s ) gives the area of region s and parameter c sci fib ∈[ . . . ] defines the fraction of fibres that are available forscience targets .The calculation of T s req for HR targets is the same, ex-cept that we use HR targets and HR fibre density ρ HRfib .In the 4MOST facility, the average LR fibre density is ρ LRfib =
391 sq deg − and for HR it is ρ HRfib =
196 sq deg − . The fraction is less than one, because some fibres are used forcalibration or are allocated as sky fibres.MNRAS000
196 sq deg − . The fraction is less than one, because some fibres are used forcalibration or are allocated as sky fibres.MNRAS000 , 1–20 (2020) ptimal tiling algorithm In Eq. (9), we use exposure times for the dark sky con-dition. Hence, Eq. (9) gives the minimum exposure time re-quired to observe all targets in the dark sky condition, as-suming perfect fibre-to-target assignment without any loss.Effectively, the perfect fibre-to-target assignment is onlyused to estimate the missing time, T s miss . The simplified fibre-to-target assignment described below takes into account thesky conditions associated with each OB and different skyconditions are also included to calculate the T s obs . Calculation of T s obs . The observed exposure time T s obs counts the time that is used to observe targets in a region s .It is estimated as T s obs , LR = N s LR , fib (cid:213) t ∈ t LR s min [ . , f obs ( t )] · T Dexp ( t ) · f compl ( t ) , (11)where notations are the same as in Eq. (9) and f obs ( t ) givesthe completion fraction for target t . If the summed expo-sure time of a target t is equal or larger than the requestedexposure time, then effectively f obs ( t ) = . .The completion fraction f obs ( t ) for target t is estimatedusing the simplified fibre-to-target assignment. For that westart with all tiles that cover the region s . The set of thesetiles is designated y s . We assume that all fibres in region s can be used for all targets in that region. This is an ap-proximation, but since region s is relatively small, it is astatistically unbiased approach. Since the exposure times oftiles y ∈ y s can be different, the assignment of fibres/tilesto targets itself is an optimisation problem. In the currentpaper, we do not solve this extra optimisation problem, forreasons of minimising the computational time. We insteaduse a simple scheme that provides an efficient enough solu-tion. The simplified fibre-to-target assignment for LR targetsin region s is done as follows.(i) Initialize all tiles y ∈ y s in the region s . All tileshave the same fixed number of LR fibres for science tar-gets, N LR , fib . Initially, for all tiles y ∈ y s , N allocfib ( y ) = , whichis the number of allocated fibres in tile y .(ii) Select all LR targets t ∈ t LR s in the region s . Sort alltargets t ∈ t LR s for descending order of target exposure time T Dexp ( t ) . For each target, set the completion fraction to zero, f obs ( t ) = .(iii) For each target, set the overexposure fraction to zero, f over − exp ( t ) = .(iv) Loop over targets t in the region s , starting from thetarget with the longest exposure time T Dexp ( t ) . For each target,allocate fibres as follows.(a) For target t , calculate completion fraction when ob-serving with field y ∈ y s . Completion fraction for target t in field y is estimated as f y ( t ) = T y exp / T BGDexp , where T y exp istile exposure time and for the target exposure time T BGDexp we use the sky condition that matches with the tile con-dition i y BGD .(b) Look through all tiles that are available in the re-gion s . Tile y is available for target t , if N allocfib ( y ) < N LR , fib and target t is not yet observed with tile y .(c) If there exists a tile y for which f obs ( t ) + f y ( t ) > . ,then assign target t to the tile y , where f obs ( t ) + f y ( t ) is thelowest. For that tile, increase the fibre allocation N allocfib ( y ) by f compl . For the target t , set f obs ( t ) = . . For target t , set T e x p ( m i n )
11 10 9 8 7 6 5 4 3 2 1 1 2 3 4 5 61 2 3 7 81 2 4 5 9 10T obss T reqs - T obss T over_exps T not_useds N fib T e x p ( m i n )
10 9 8 7 6 5 4 3 2 1 1 2 3 4 5 61 2 7 81 2 9 10 N fib Figure 3.
Schematic overview of the simplified fibre-to-targetassignment for the 4MOST surveys. The left panels show LR tar-gets in a small sky area ordered based on the requested exposuretimes. On average, this sky area includes 4.3 LR fibres (maximum N fib value in the right-hand side panels). The column widths givethe target completion fractions f compl . The column heights are therequested exposure times. The right panels show the distribu-tion of these targets divided between three exposures with 30, 20,and 10 min, respectively. The division of targets between threeexposures is done according to the algorithm described in Sec-tion 3.2.1. The upper panels show nearly perfect fibre-to-targetassignment. The lower panels show a fibre-to-target assignmentfor targets, where the target distribution does not allow for idealallocation and some fibres are left empty. Different energy func-tion components are shown with different colours and patterns. the over-exposure fraction f over − exp ( t ) = f obs ( t ) + f y ( t ) − . .Go to the next target in the region s .(d) If the condition in (c) is not met then allocate target t to a tile, where f y ( t ) is the largest. For that tile, increasethe fibre allocation N allocfib ( y ) by f compl . For the target t , set f obs ( t ) = f obs ( t ) + f y ( t ) .(e) If for any tile ( y ∈ y s ) N allocfib ( y ) is larger than thenumber of available fibres N fib , remove this tile from theavailable tile list.(f) Go to the point (b) and add target t to another tile y . (g) If there are no tiles available (for example when therequested target exposure time is larger than the totalexposure time in this region), go to the next target in theregion s .Since for each target we estimate the target completion frac-tion f obs ( t ) , Eq. (11) allows us to estimate the actual ob-served time for dark sky conditions, and combining this withEq. (9), allows us to directly estimate the missing observa-tional time T s miss , see Eq. (7).The simplified fibre-to-target assignment is illustratedin Fig. 3. The upper panels show an almost perfect fibreallocation. The lower panels show target allocation in thecase of a target distribution that does not allow an optimal MNRAS , 1–20 (2020)
E. Tempel et al. fibre allocation. Regardless of the tile exposure times, someof the fibres are always empty, while long-exposure targetsare not fully observed. This situation can only be improvedby changing the target distribution in the sky. The proposedtiling algorithm tries to minimize the time that is missingand the time that is not used.After the simplified fibre-to-target assignment, we havefor each target t ∈ t LR s the completion fraction f obs ( t ) andfor each tile y ∈ y s we have the number of allocated fibres N allocfib ( y ) . Additionally, if target t was over-exposed ( f obs ( t ) > . ) we have the over-exposed fraction f over − exp ( t ) . Calculation of T s over − exp . Over-exposed time in a region s is estimated using the over-exposed fraction of each target T s over − exp , LR = N s LR , fib (cid:213) t ∈ t LR s f over − exp ( t ) · T Dexp ( t ) · f compl ( t ) , (12) Calculation of T s not − used . The total time for empty fibres isestimated as T s not − used = N s LR , fib (cid:213) y ∈ y s (cid:110) max (cid:104) , N s LR , fib − N allocfib ( y ) (cid:105)(cid:111) · T y exp , (13)where T y exp is the exposure time of tile y . U overhead ( y | θ ) The energy function component for overheads, U overhead ( y | θ ) , is introduced to reduce the total amount ofoverhead time that is associated with each observation. Foreach science exposure, there is additional T tileoverhead = . minoverhead (including calibration). Hence, for short exposuresthe fractional overhead is larger than for long exposures. Atthe same time, there are many targets that require short ex-posures, hence, short exposures are the optimal in some skyregions. In addition to the overhead associated with eachexposure, there is additional overhead T OBoverhead = . minassociated with each OB. This mainly covers the time forthe telescope to move from one sky region to the other andthereafter acquiring the necessary guide stars. To reduce thesummed T OBoverhead , 4MOST will combine several exposuresinto one OB.The energy function that minimises the overhead timeis defined as: U overhead ( y | θ ) = c overhead (cid:104) N tile T tileoverhead + N OB T OBoverhead (cid:105) , (14)where N tile is the total number of tiles (exposures) and N OB is the number of individual OBs in the tiling solution. Theparameter c overhead can be used to fine-tune the importanceof overhead energy component in the optimisation process. U tiles ( y | θ ) One 4MOST field of view is a hexagon. If we have to coversky only once, then the optimal tiling is a beehive pattern.If some sky regions should be observed many times, thenthe optimal pattern should follow the target distribution inthe sky. In intermediate cases, where the sky should be cov-ered only twice, the optimal tiling is a beehive pattern thatcovers the sky twice. Since there is a small overlap between neighbouring tiles (because of the curved spherical surfaceof the sky), the two beehive patterns should be shifted withrespect to each other, which minimises the number of over-laps in any sky location. To encourage this kind of pattern,the energy function U tiles ( y | θ ) is defined as U tiles ( y | θ ) = c tiles · N OB (cid:213) i = (cid:110) R lim − min (cid:104) R lim , d ( y OB i , y OB k (cid:44) i : k = . . . N OB ) (cid:105)(cid:111) , (15)where d ( y OB i , y OB k ) is the angular distance between two OBcentres and R lim is the limiting radius. If the distance be-tween OBs is larger than R lim then there is no penalty in theenergy function. If the distance between two OBs is smallerthan R lim , we add a small penalty to the total energy U ( y | θ ) .Optimal R lim should be close to the radius of one field ofview, which also minimises the gaps between tiles. The in-teraction between tiles defined with Eq. (15) is known as anearest neighbour interaction in the point processes appli-cations (see e.g. van Lieshout 2000).If required by the surveys, a similar scheme can be usedto force gaps between OBs in some specific sky areas. In thiscase the U tiles ( y | θ ) should be defined individually for eachsky region. In the current paper, for simplicity, we only testthe energy function given with Eq. (15). U BGD ( y | θ ) The energy function component U targets ( y | θ ) depends onwhether a given sky region is observed during bright, greyor dark sky brightness conditions. Since observations duringdark time are generally preferred, the previously introducedenergy function components highly prefer observations dur-ing dark time. However, the fraction of total bright, grey anddark time is fixed. To take that into account, we introducean energy function component U BGD ( y | θ ) that somewhatbalances the total time between B/G/D.This energy function component is defined as U BGD ( y | θ ) = c B N Btile + c G N Gtile + c D N Dtile , (16)where N B / G / Dtile is the number of tiles with B/G/D flag and c B / G / D are constants. In practice, c D > c G > c B , whichslightly encourages bright and grey time tiles over dark timetiles. The parameters c B / G / D should be chosen so that thefraction of the total bright, grey and dark time is as ex-pected. To simulate marked point processes, several techniquescan be used: spatial birth-and-death processes, Metropolis-Hastings (MH) algorithms, reversible jump dynamics or ex-act simulation techniques (Geyer & Møller 1994; Green 1995;Geyer 1999; Kendall & Møller 2000; van Lieshout 2000;van Lieshout & Stoica 2006; van Lieshout 2019).In the current paper, we need to sample from the law p ( y | θ ) . This is done by using an iterative Monte Carlo al-gorithm. In our case the model parameters θ are fixed andconditional on θ , and the object pattern is sampled from p ( y | θ ) using an MH algorithm (Geyer & Møller 1994; Geyer MNRAS000
E. Tempel et al. fibre allocation. Regardless of the tile exposure times, someof the fibres are always empty, while long-exposure targetsare not fully observed. This situation can only be improvedby changing the target distribution in the sky. The proposedtiling algorithm tries to minimize the time that is missingand the time that is not used.After the simplified fibre-to-target assignment, we havefor each target t ∈ t LR s the completion fraction f obs ( t ) andfor each tile y ∈ y s we have the number of allocated fibres N allocfib ( y ) . Additionally, if target t was over-exposed ( f obs ( t ) > . ) we have the over-exposed fraction f over − exp ( t ) . Calculation of T s over − exp . Over-exposed time in a region s is estimated using the over-exposed fraction of each target T s over − exp , LR = N s LR , fib (cid:213) t ∈ t LR s f over − exp ( t ) · T Dexp ( t ) · f compl ( t ) , (12) Calculation of T s not − used . The total time for empty fibres isestimated as T s not − used = N s LR , fib (cid:213) y ∈ y s (cid:110) max (cid:104) , N s LR , fib − N allocfib ( y ) (cid:105)(cid:111) · T y exp , (13)where T y exp is the exposure time of tile y . U overhead ( y | θ ) The energy function component for overheads, U overhead ( y | θ ) , is introduced to reduce the total amount ofoverhead time that is associated with each observation. Foreach science exposure, there is additional T tileoverhead = . minoverhead (including calibration). Hence, for short exposuresthe fractional overhead is larger than for long exposures. Atthe same time, there are many targets that require short ex-posures, hence, short exposures are the optimal in some skyregions. In addition to the overhead associated with eachexposure, there is additional overhead T OBoverhead = . minassociated with each OB. This mainly covers the time forthe telescope to move from one sky region to the other andthereafter acquiring the necessary guide stars. To reduce thesummed T OBoverhead , 4MOST will combine several exposuresinto one OB.The energy function that minimises the overhead timeis defined as: U overhead ( y | θ ) = c overhead (cid:104) N tile T tileoverhead + N OB T OBoverhead (cid:105) , (14)where N tile is the total number of tiles (exposures) and N OB is the number of individual OBs in the tiling solution. Theparameter c overhead can be used to fine-tune the importanceof overhead energy component in the optimisation process. U tiles ( y | θ ) One 4MOST field of view is a hexagon. If we have to coversky only once, then the optimal tiling is a beehive pattern.If some sky regions should be observed many times, thenthe optimal pattern should follow the target distribution inthe sky. In intermediate cases, where the sky should be cov-ered only twice, the optimal tiling is a beehive pattern thatcovers the sky twice. Since there is a small overlap between neighbouring tiles (because of the curved spherical surfaceof the sky), the two beehive patterns should be shifted withrespect to each other, which minimises the number of over-laps in any sky location. To encourage this kind of pattern,the energy function U tiles ( y | θ ) is defined as U tiles ( y | θ ) = c tiles · N OB (cid:213) i = (cid:110) R lim − min (cid:104) R lim , d ( y OB i , y OB k (cid:44) i : k = . . . N OB ) (cid:105)(cid:111) , (15)where d ( y OB i , y OB k ) is the angular distance between two OBcentres and R lim is the limiting radius. If the distance be-tween OBs is larger than R lim then there is no penalty in theenergy function. If the distance between two OBs is smallerthan R lim , we add a small penalty to the total energy U ( y | θ ) .Optimal R lim should be close to the radius of one field ofview, which also minimises the gaps between tiles. The in-teraction between tiles defined with Eq. (15) is known as anearest neighbour interaction in the point processes appli-cations (see e.g. van Lieshout 2000).If required by the surveys, a similar scheme can be usedto force gaps between OBs in some specific sky areas. In thiscase the U tiles ( y | θ ) should be defined individually for eachsky region. In the current paper, for simplicity, we only testthe energy function given with Eq. (15). U BGD ( y | θ ) The energy function component U targets ( y | θ ) depends onwhether a given sky region is observed during bright, greyor dark sky brightness conditions. Since observations duringdark time are generally preferred, the previously introducedenergy function components highly prefer observations dur-ing dark time. However, the fraction of total bright, grey anddark time is fixed. To take that into account, we introducean energy function component U BGD ( y | θ ) that somewhatbalances the total time between B/G/D.This energy function component is defined as U BGD ( y | θ ) = c B N Btile + c G N Gtile + c D N Dtile , (16)where N B / G / Dtile is the number of tiles with B/G/D flag and c B / G / D are constants. In practice, c D > c G > c B , whichslightly encourages bright and grey time tiles over dark timetiles. The parameters c B / G / D should be chosen so that thefraction of the total bright, grey and dark time is as ex-pected. To simulate marked point processes, several techniquescan be used: spatial birth-and-death processes, Metropolis-Hastings (MH) algorithms, reversible jump dynamics or ex-act simulation techniques (Geyer & Møller 1994; Green 1995;Geyer 1999; Kendall & Møller 2000; van Lieshout 2000;van Lieshout & Stoica 2006; van Lieshout 2019).In the current paper, we need to sample from the law p ( y | θ ) . This is done by using an iterative Monte Carlo al-gorithm. In our case the model parameters θ are fixed andconditional on θ , and the object pattern is sampled from p ( y | θ ) using an MH algorithm (Geyer & Møller 1994; Geyer MNRAS000 , 1–20 (2020) ptimal tiling algorithm D e c ( deg ) RA (deg)-20-15-10-5 Figure 4.
Optimal tiling pattern in the case of one visit. Each tileis shown as a blue hexagon. Uniformly distributed targets (greydots) are restricted in the right ascension range . . . deg anddeclination range − · · · + deg. For clarity, only 10 per centof the targets are shown. Stitches in the tiling pattern are due tothe hexagons along the neighbouring edges being rotated by 90degrees. i ) Birth: with a probability p b a new object ζ , sampledfrom the birth rate b ( y , ζ ) , is proposed to be added to thepresent configuration y . The new configuration y (cid:48) = y ∪ ζ isaccepted with the probability min (cid:26) , p d p b d ( y ∪ ζ, ζ ) b ( y , ζ ) p ( y ∪ ζ ) p ( y ) (cid:27) . (17)( ii ) Death: with a probability p d an object ζ from thecurrent configuration y is proposed to be eliminated accord-ing to the death proposal d ( y , ζ ) . The probability of accept-ing the new configuration y \ ζ (the set of objects y omittingthe object ζ ) is computed by inverting the ratio (17).( iii ) Change: with a probability p c we randomly choosean object ζ old in the configuration y and propose to slightlychange its parameters using uniform proposals. The new ob-ject obtained is ζ new . The new configuration y (cid:48) = y \ ζ old ∪ ζ new is accepted with the probability min { , p ( y (cid:48))/ p ( y )} .For the death rate, we adopt the uniform choice d ( y , ζ ) = / n ( y ) , where n ( y ) is the number of objects in the config-uration. For the birth proposal above, we have a mixtureproposal with two types of sub-moves: • Random: with a probability p rndb a new random tile (anew OB with just one tile) is added to the configuration. Thetile centre is chosen uniformly in the sky (in the observedwindow W ). For the new tile, we assign a random positionangle and a random exposure time between T min and T max .For the new tile we also attach a B/G/D flag, where theprior for the B/G/D flag is the fraction of time available inbright, grey or dark sky conditions. • Tile in an OB: with a probability p OBb = . − p rndb we choose an existing tile ζ (cid:48) and add a new tile ζ to this OB.The tile coordinates, position angle and B/G/D flag becomethe same as for the existing OB. The exposure time for thenew tile is chosen randomly between T min and T max .The birth rate for the combined birth move is b ( y , ζ ) = p rndb { ζ ∈ W } ν ( W ) + p OBb ˜ b ( y , ζ ) , (18) ˜ b ( y , ζ ) = n ( y ) (cid:213) ζ (cid:48)∈ y { ζ ∈ b ( ζ (cid:48) , r )} ν [ b ( ζ (cid:48) , r ) ∩ W ] , (19)where ν ( W ) is the Lebesgue measure (sky area) of the ob-served window W , b ( ζ (cid:48) , r ) is a ball centred in ζ (cid:48) with radius r in the sky, and {·} is the indicator function. For simplicity,we set the area of the ball ν [ b ( ζ (cid:48) , r )] = and ν ( W ) = N expected ,where N expected is the expected number of tiles in the con-verged solution. The actual number of tiles may differ fromthe expected number and is mainly determined by the en-ergy function U targets .For the change move above, we adopt the following sub-moves: • Position in the sky: with a probability p posc we slightlyshift an OB centre (where the selected tile is in) and positionangle with respect to the OBs original values. • Exposure time: with a probability p expc we slightlychange tile exposure time with respect to the original expo-sure time. • Change B/G/D flag: with a probability p BGDc we pro-pose to change the B/G/D flag for a selected OB. • Combine close tiles into the same OB: with aprobability p OBc we select randomly a tile and if there isanother tile close to the selected tile, we join both tiles intoone OB.The previously introduced birth, death and changemoves define a Markov chain transition kernel whichis φ − irreducible, Harris recurrent and geometric ergodic(van Lieshout 2000; Møller & Waagepetersen 2004; Stoicaet al. 2005).In order to maximise p ( y , θ ) , the previously describedsampling mechanism is integrated into a simulated anneal-ing (SA) algorithm. The SA is an iterative algorithm thatsamples from p ( y , θ ) / T , while T goes slowly to zero. The fol-lowing ingredients are needed in order to ensure convergenceof the SA algorithm: high value of the initial temperature,a convergent sampling algorithm for the probability densityand an appropriate cooling schedule (van Lieshout 2000; Sto-ica et al. 2005). We adopt the polynomial cooling schedule,where the temperature is lowered as T k + = α T k , (20)where k is a time step in a simulation and < α < . definesthe speed of temperature decrease. The initial temperaturefor the simulation is set to T . The temperature is loweredafter every N moves , which allows the system to reach a near-equilibrium state. In practice, the N moves should be severaltimes greater than the number of tiles in the configuration.Altogether we change the temperature N cycles time. Hence,the total number of moves in our algorithm is N moves · N cycles .The MH algorithm described above does not requireany tiling initialisation. The MCMC algorithm starts with MNRAS , 1–20 (2020) E. Tempel et al. D e c ( deg ) RA (deg)No interactions-20-15-10-5 D e c ( deg ) RA (deg)Repulsive interactions-20-15-10-5 Figure 5.
Optimal tiling pattern in the case of two visits. Each tile is represented as a blue hexagon. Targets are restricted in thesame area as in Fig. 4. On the left-hand panel there are no interactions between tiles, U tiles = . . On the right-hand panel we use therepulsive interactions (see Section 3.2.3) where tile centres are maximally pushed apart from each other. Clearly, the resulting tilingpattern depends on the type of interactions among tiles. -4-3-2-1 l og ( p ) T exp (min)-4-3-2-1 p f compl Figure 6.
The distribution of exposure times (upper panel) and f compl values (lower panel) for the targets shown in Fig. 7. The dis-tribution is shown in arbitrary units. The minimum exposure timefor all targets was set to 10 minutes. Most of the long-exposuretargets are located in the WAVES region (see Fig. 7). In the inputcatalogue, each sub-survey has a fixed f compl value. The variety of f compl values shows that the completeness requirements in varioussurveys are very different. zero tiles and additional tiles are added to the tiling con-figuration during birth moves. The final number of tiles inthe configuration is mostly determined by the U targets energyfunction and is influenced by the expected number of tiles, N expected . As a first test, we generated Poisson-distributed targets inthe sky and ran the tiling algorithm on these points. The tar-gets were restricted in the right ascension range . . . degand declination range − . . . + deg. The number densityof targets was slightly lower than the number density of fi-bres. Hence, the expected optimal tiling pattern covers thesky only once.Fig. 4 shows the optimal tiling pattern generated usingthe algorithm described in Section 3. An ideal theoreticaltiling pattern would be a perfect honey-comb pattern. How-ever, the celestial sphere is curved, the target region in thesky is restricted and the tiles should not be located outsideof the target region, so the perfect honey-comb pattern can-not be achieved exactly. The stitches visible in Fig. 4 aredue to the orientation of tiles (hexagons) at the neighbour-ing edges of the sky area being rotated by 90 degrees. Theorientation of tiles at the field edges are determined by thesharp edge of the field and fixed hexagon orientations arethe only solution we found to produce an optimal tiling thatminimises the tile area outside of the target region. For alarge field of view, these stitches are not present and thealgorithm generates a nearly perfect honey-comb pattern inthe case of one visit to each tile.As a second test with Poisson-distributed targets, wedoubled the number of targets. Consequently, the expectedperfect tiling covers the sky twice. We use this test to showthe effect of U tiles (see Section 3.2.3) on the final tiling pat-tern. In Fig. 5 we show the tiling pattern generated in twodifferent cases. In the left panel, we show the tiling patternwhere U tiles = . , which means that there are no interac- MNRAS000
The distribution of exposure times (upper panel) and f compl values (lower panel) for the targets shown in Fig. 7. The dis-tribution is shown in arbitrary units. The minimum exposure timefor all targets was set to 10 minutes. Most of the long-exposuretargets are located in the WAVES region (see Fig. 7). In the inputcatalogue, each sub-survey has a fixed f compl value. The variety of f compl values shows that the completeness requirements in varioussurveys are very different. zero tiles and additional tiles are added to the tiling con-figuration during birth moves. The final number of tiles inthe configuration is mostly determined by the U targets energyfunction and is influenced by the expected number of tiles, N expected . As a first test, we generated Poisson-distributed targets inthe sky and ran the tiling algorithm on these points. The tar-gets were restricted in the right ascension range . . . degand declination range − . . . + deg. The number densityof targets was slightly lower than the number density of fi-bres. Hence, the expected optimal tiling pattern covers thesky only once.Fig. 4 shows the optimal tiling pattern generated usingthe algorithm described in Section 3. An ideal theoreticaltiling pattern would be a perfect honey-comb pattern. How-ever, the celestial sphere is curved, the target region in thesky is restricted and the tiles should not be located outsideof the target region, so the perfect honey-comb pattern can-not be achieved exactly. The stitches visible in Fig. 4 aredue to the orientation of tiles (hexagons) at the neighbour-ing edges of the sky area being rotated by 90 degrees. Theorientation of tiles at the field edges are determined by thesharp edge of the field and fixed hexagon orientations arethe only solution we found to produce an optimal tiling thatminimises the tile area outside of the target region. For alarge field of view, these stitches are not present and thealgorithm generates a nearly perfect honey-comb pattern inthe case of one visit to each tile.As a second test with Poisson-distributed targets, wedoubled the number of targets. Consequently, the expectedperfect tiling covers the sky twice. We use this test to showthe effect of U tiles (see Section 3.2.3) on the final tiling pat-tern. In Fig. 5 we show the tiling pattern generated in twodifferent cases. In the left panel, we show the tiling patternwhere U tiles = . , which means that there are no interac- MNRAS000 , 1–20 (2020) ptimal tiling algorithm D e c ( deg ) -25-20-15-10-5 R equ i r ed T e x p ( m i n ) WAVES regionWAVES region D e c ( deg ) -25-20-15-10-5 A ll o c a t ed T e x p ( m i n ) D e c ( deg ) R.A. (deg)-25-20-15-10-5 A ll o c a t ed T e x p ( m i n ) Figure 7.
The upper panel shows the required exposure time for different sky regions. Targets for the test region were selected fromthe 4MOST mock catalogues. The required exposure time was calculated using Eq. (9). The upper region with a high number densityof targets is the WAVES survey region (Driver et al. 2019). Middle and lower panels show the allocated exposure times (sum of tilesexposures times) for the same test region. In the middle panel, each tile is an individual OB, while in the lower panel, tiles are collectedinto OBs, to reduce the total overhead time. In both cases, the allocated exposure time traces the required exposure time shown on theupper panel very well. Fig. 8 gives the actual tiling pattern in the sky for these two cases. Fig. 9 shows how well the allocated exposuretime matches with the required exposure time, while taking the fibre-to-target assignment into account. tions between tiles. In the right-hand panel of Fig. 5 weshow the tiling pattern where the repulsive interaction oftiles is added, as described in Section 3.2.3. The parameter c tiles = . , which is relatively large to forces the tile centresmaximally apart from each other.Fig. 5 clearly shows that the final tiling pattern depends on the choice of U tiles . In the left-hand panel, in several loca-tions two tiles are put almost perfectly on top of each other.In the right-hand panel, the distance between tile centres ismaximised and the resulting pattern appears more regular.In both cases, the number of tiles is practically the sameand both patterns cover the sky twice with minimal over- MNRAS , 1–20 (2020) E. Tempel et al. -25-20-15-10-5 D e c ( deg ) -25-20-15-10-5 tilesIndividual tiles-25-20-15-10-5 D e c ( deg ) R.A. (deg)-25-20-15-10-5 in OBsTiles in OBs
Figure 8.
Tiling pattern in the sky that corresponds to the allocated exposure times shown in the middle and lower panel in Fig. 7.Each OB is shown as a blue hexagon. In the upper panel, each tile is an individual OB. In the lower panel, several tiles are collected intosingle OBs. Clearly, the flexibility of the tiling pattern depends on the number of OBs. laps and holes between tiles. Hence, U tiles has a negligibleeffect on the survey efficiency. In the proposed algorithm, U tiles can be used to influence the tiling pattern so that itmaximises the survey science goals. Depending on the sur-vey, these goals can be rather different.The tiling solutions presented in Figs. 4 and 5 eachshow just one realisation of a solution to the optimal tilingproblem. Since the MCMC algorithm involves randomness,if we run the tiling algorithm a second time with exactly thesame parameters and different random seeds, or using paral-lel computation as typically implemented (which is what wecurrently have coded), the outcome will be slightly different.For example, in Fig. 4, the stitches will appear in different lo-cations. Due to the complexity of the optimal tiling problem,it is hard to define the optimal tiling pattern. In practice,there are many optimal tiling patterns and the proposed al-gorithm only provides one numerical realisation of a solutionto the problem. Due to the high-dimensionality of the prob-lem, there are many local minima that are all approximately equal in practice and the MCMC algorithm provides one lo-cal minimum as a final solution. The optimal tiling patternis defined via the energy function in Eq. (4) and dependson its form and parameters. For different scientific applica-tions the optimal tiling pattern might be different and theproposed algorithm allows to take this into account. In this section, we test the proposed tiling algorithm in thecase of a varying number density of targets in the sky. Tar-gets are taken from the 4MOST mock catalogues, coveringthe Galactic and extragalactic consortium surveys (see Sec-tion 1). The distribution of exposure times and f compl valuesfor targets in our test region are shown in Fig. 6. The upperpanel in Fig. 7 shows the required exposure time in a testsky region. The required exposure time is estimated withEq. (9). Fig. 7 shows the footprints of individual surveys inthe sky. The upper part of this Figure shows the WAVES MNRAS000
Tiling pattern in the sky that corresponds to the allocated exposure times shown in the middle and lower panel in Fig. 7.Each OB is shown as a blue hexagon. In the upper panel, each tile is an individual OB. In the lower panel, several tiles are collected intosingle OBs. Clearly, the flexibility of the tiling pattern depends on the number of OBs. laps and holes between tiles. Hence, U tiles has a negligibleeffect on the survey efficiency. In the proposed algorithm, U tiles can be used to influence the tiling pattern so that itmaximises the survey science goals. Depending on the sur-vey, these goals can be rather different.The tiling solutions presented in Figs. 4 and 5 eachshow just one realisation of a solution to the optimal tilingproblem. Since the MCMC algorithm involves randomness,if we run the tiling algorithm a second time with exactly thesame parameters and different random seeds, or using paral-lel computation as typically implemented (which is what wecurrently have coded), the outcome will be slightly different.For example, in Fig. 4, the stitches will appear in different lo-cations. Due to the complexity of the optimal tiling problem,it is hard to define the optimal tiling pattern. In practice,there are many optimal tiling patterns and the proposed al-gorithm only provides one numerical realisation of a solutionto the problem. Due to the high-dimensionality of the prob-lem, there are many local minima that are all approximately equal in practice and the MCMC algorithm provides one lo-cal minimum as a final solution. The optimal tiling patternis defined via the energy function in Eq. (4) and dependson its form and parameters. For different scientific applica-tions the optimal tiling pattern might be different and theproposed algorithm allows to take this into account. In this section, we test the proposed tiling algorithm in thecase of a varying number density of targets in the sky. Tar-gets are taken from the 4MOST mock catalogues, coveringthe Galactic and extragalactic consortium surveys (see Sec-tion 1). The distribution of exposure times and f compl valuesfor targets in our test region are shown in Fig. 6. The upperpanel in Fig. 7 shows the required exposure time in a testsky region. The required exposure time is estimated withEq. (9). Fig. 7 shows the footprints of individual surveys inthe sky. The upper part of this Figure shows the WAVES MNRAS000 , 1–20 (2020) ptimal tiling algorithm D e c ( deg ) -25-20-15-10-5 M i ss i ng T e x p ( m i n ) T miss T miss D e c ( deg ) -25-20-15-10-5 N o t u s ed T e x p ( m i n ) T not-used T not-used D e c ( deg ) R.A. (deg)-25-20-15-10-5 U t a r ge t s U targets U targets D e c ( deg ) R.A. (deg)-25-20-15-10-5 M i ss i ng T e x p / N t il e T miss /N tile T miss /N tile D e c ( deg ) R.A. (deg)-25-20-15-10-5 N o t u s ed T e x p / N t il e T not-used /N tile T not-used /N tile D e c ( deg ) R.A. (deg)-25-20-15-10-5 U t a r ge t s / N t il e U targets /N tile U targets /N tile Figure 9.
Diagnosis plots of the tiling algorithm. The left-hand column shows the energy function U targets (lower panel) and its components T miss (upper panel) and T not − used (middle panel). In general, T miss counts the time that is missing to observe the required set of targets and T not − used counts the time that is wasted because of empty fibres. The U targets is the combination of these (see Section 3.2.1). The right-handpanels show the same energy function components divided by the number of tiles in a given sky region. This Figure shows the energyfunction components for the tiling presented in the middle panel of Fig. 7. survey where the number density of objects, as well as therequired exposure time, varies significantly even on smallscales.To generate the tiling for the selected test region, werestricted ourselves to only LR targets and all tiles had thesame sky brightness condition. We generated the tiling pat-tern for two cases. In the first case we set U overhead = . , sothat each tile was considered to be an individual OB with nopenalty from the overhead during tile generation. In the sec-ond case, we minimised the overhead associated with eachexposure and OB, we set c overhead = . , and tiles were col-lected into OBs wherever possible and efficient.The middle and lower panels in Fig. 7 show the allocatedexposure time for these two cases. In general, the allocatedtime in different sky regions is roughly the same for bothcases. In the lower part of the test region, the required ex-posure time is lower and there is less flexibility there. Hence,a slight hint of the tiling pattern of individual hexagons canbe seen in the allocated exposure time shown in the lowerpanel of Fig. 7. This is because tiles are collected into OBsand this part of the region is mostly covered only twice. Ingeneral, we can see that the allocated exposure time matchesthe required exposure times very well. In many cases, eventhe small variations in the required exposure time maps arewell traced by the allocated exposure times. The requiredand allocated exposure times are not directly matched inour algorithm. During the optimisation, we minimise themissing and wasted time (see Section 3.2.1), which auto-matically results in an excellent match between the requiredand allocated exposure times. Fig. 8 shows the actual tiling pattern for the two casesshown in the lower two panels of Fig. 7. This clearly empha-sises that if tiles are collected into OBs, then we lose someflexibility when placing tiles in the sky. This flexibility comeswith the price of increased overhead time. For this example,the summed exposure time without the overhead time forthe two cases is nearly identical, being around 860 hours.However, the overhead for the first case is 446 hours, whilefor the second case it is 339 hours. Hence, there is a balancebetween an efficient survey (minimised overhead time) andan optimal tiling pattern (flexibility of placing tiles). Thisbalance depends on the survey science goals and requiredcompleteness for a survey. The best compromise should bedetermined during the survey optimisation.Fig. 9 shows the energy function U targets and differentcomponents of U targets . Clearly, the WAVES region has thehighest energy, as it is the least efficient part of the selectedtest region, while at the same time it has the largest T missing and T not − used times. This is because of the nature of theWAVES region. The target density varies significantly atscales smaller than one 4MOST field of view. At the sametime, the exposure times of individual targets differ a lot.As a consequence, there are regions where the allocated ex-posure time is smaller than the required exposure time forsingle targets and in the same regions there are empty fibres.A similar case is illustrated in the lower panel of Fig. 3.The algorithm minimises the sum of the two components.Depending on the science goals, the relative importance ofthe two components can be altered. For example, WAVESrequires high completeness for its science case, so T missing MNRAS , 1–20 (2020) E. Tempel et al. D e c ( deg ) -25-20-15-10-5 F r a c t i on o f a ll o c a t ed f b r e s D e c ( deg ) -25-20-15-10-5 N ob s e r v ed / N r equ i r ed D e c ( deg ) R.A. (deg)-25-20-15-10-5 T ob s e r v ed / T r equ i r ed Figure 10.
Output of probabilistic fibre-to-target assignment. Using the mock targets shown in the top panel of Fig. 7 and tiling mapshown in the upper panel of Fig. 8, we ran the survey simulation using the probabilistic fibre-to-target assignment presented in Tempelet al. (2020). See Section 4.3 for more details. The upper panel shows the fraction of allocated fibres. The middle panel shows the numberof successfully observed objects divided by the number of required objects. The lower panel shows the fraction of observed exposure timeout of required exposure time. should be more important than T not − used . In the proposedalgorithm, these survey-specific requirements can be easilyincluded, while generating the final optimal tiling for the4MOST survey.Regarding the distribution of missing and not-used timein the sky and the distribution of energy function U targets ,these are more or less uniform outside and inside the WAVESregion. The proposed tiling algorithm finds the tiling that matches the required exposure times and finds a solution,where missing and not-used time is evenly distributed in thesky. Hence, the proposed tiling algorithm does not seeminglyprefer one sky region to the other. The difference betweenWAVES and other regions is due to the different target den-sities, completeness requirements and exposure time distri-butions.While the left-hand panels in Fig. 9 show the energy MNRAS000
Output of probabilistic fibre-to-target assignment. Using the mock targets shown in the top panel of Fig. 7 and tiling mapshown in the upper panel of Fig. 8, we ran the survey simulation using the probabilistic fibre-to-target assignment presented in Tempelet al. (2020). See Section 4.3 for more details. The upper panel shows the fraction of allocated fibres. The middle panel shows the numberof successfully observed objects divided by the number of required objects. The lower panel shows the fraction of observed exposure timeout of required exposure time. should be more important than T not − used . In the proposedalgorithm, these survey-specific requirements can be easilyincluded, while generating the final optimal tiling for the4MOST survey.Regarding the distribution of missing and not-used timein the sky and the distribution of energy function U targets ,these are more or less uniform outside and inside the WAVESregion. The proposed tiling algorithm finds the tiling that matches the required exposure times and finds a solution,where missing and not-used time is evenly distributed in thesky. Hence, the proposed tiling algorithm does not seeminglyprefer one sky region to the other. The difference betweenWAVES and other regions is due to the different target den-sities, completeness requirements and exposure time distri-butions.While the left-hand panels in Fig. 9 show the energy MNRAS000 , 1–20 (2020) ptimal tiling algorithm function per sky region, the right-hand panels show the sameenergy function per tile, the energy function components aredivided by the number of tiles in a given sky region. Whilethe summed energy is the lowest in the middle of the region,the energy per tile is highest there. This is because the num-ber density of objects there is relatively low and this regionis covered mainly with one layer of tiles. Since there is almostno flexibility for the tiling pattern in this region, the surveyefficiency largely depends on the match between the targetdensity and fibre density. The mismatch between these twois the reason why the energy per tile is highest there. Toconclude, the energy function maps are useful for analysingthe overall efficiency of the generated tiling pattern. How-ever, the tiling pattern should be used together with the realfibre-to-target assignment and the actual survey efficiencycan be only assessed using the full survey simulation. Thisis briefly analysed in the next section. The fibre-to-target assignment described in this paper is asimplified approach that provides only a statistical solutionand cannot be used during real observations. In reality, thegenerated tiling pattern will be used together with a more so-phisticated fibre-to-target assignment algorithm. In Tempelet al. (2020) we proposed a probabilistic fibre-to-target as-signment algorithm that takes into account survey complete-ness requirements and varying number densities of targets.In this section, we will adopt the tiling pattern generatedin Section 4.2 and use this together with the probabilisticfibre-to-target assignment described in Tempel et al. (2020).For the probabilistic fibre-to-target assignment, we use thepattern shown in the upper panel of Fig. 8 as an input.Fig. 10 shows the efficiency of the probabilistic fibre-to-target assignment algorithm. The upper panel shows thefraction of allocated fibres. This is close to unity in the lowerpart of the Figure. In the upper part, the fraction of usedfibres is on average greater than 95 %. The fraction of usedfibres is lower than 90 % only in some small regions. The lowefficiency is in regions where the completeness requirement isvery high (WAVES region) or the number density of objectsis low (middle region in the figure). In general, the adoptedtiling pattern is not visible in the completeness map and thecompleteness differences are caused by the different numberdensity of objects.The middle and lower panels in Fig. 10 show the frac-tion of successfully observed objects and of used exposuretime compared with the required number of objects andexposure times. In most of the figure, both fractions aregreater than 90 %. The greatest difference is in the WAVESregion, where the number of observed objects is close to thenumber of required objects, while the fraction of used expo-sure time is lower. This is because in the WAVES region weobserve more of the required short-exposure targets, whilesome long-exposure targets remain uncompleted (the totalexposure time is shorter than the requested exposure timefor a target). This situation can only be improved by makingthe tiling less efficient, while adding more tiles (the fractionof allocated fibres will decrease) or increasing the exposuretime of tiles (the over-exposure of short-exposure targets willincrease).
Table 1.
Parameter values in our tiling algorithm during testsimulations. The last column gives the reference to the equationor section, where the parameter is used or discussed.Parameter Value Unit Reference c miss c wasted s max c LR c HR ρ LRfib
391 Eq. (9) ρ HRfib
196 Eq. (9) c sci fib c overhead T tileoverhead T OBoverhead c tiles R lim c B c G c D p b p d T min T max
30 min Sect. 3.3 p c p rndb p OBb N expected
30 000 Sect. 3.3 and Eq. (18) p posc p expc p BGDc p OBc T α N cycles
500 Sect. 3.3 N moves
250 000 Sect. 3.3
To summarise, the simplified fibre-to-target assignmentused during the tiling pattern generation works well and isa good approximation of the probabilistic fibre-to-target al-gorithm presented in Tempel et al. (2020). Further improve-ments of the simplified fibre-to-target assignment algorithmshould take into account survey-specific requirements. Thiswill be done during the 4MOST survey optimisation phase.
To test the impact of U BGD during the tiling pattern gen-eration, we used all targets from the 4MOST mock cata-logues. The distribution of the required exposure time inthe sky is shown in Fig. 1. The tiling pattern was generatedto have roughly 50 % of dark tiles, 20 % of grey tiles and30 % of bright tiles. The regions where these tiles should belocated were not fixed beforehand. The tiling algorithm de-cides based on T BGDexp ( t ) which sky regions should be observedduring bright, grey or dark sky conditions. The fraction oftiles for each sky condition is a free parameter in the tilingalgorithm and can be tuned as necessary. Table 1 gives theparameters that were used during the test simulation.Fig. 11 shows the output tiling pattern for the full sky.In the upper panel, the footprints of different sub-surveys in MNRAS , 1–20 (2020) E. Tempel et al. D e c ( deg ) -80-70-60-50-40-30-20-10
350 All tilesAll tiles D e c ( deg ) -80-70-60-50-40-30-20-10
350 Bright tilesBright tiles D e c ( deg ) -80-70-60-50-40-30-20-10
350 Grey tilesGrey tiles D e c ( deg ) R.A. (deg)-80-70-60-50-40-30-20-10
350 Dark tilesDark tiles
Figure 11.
Tiling pattern for all 4MOST mock catalogues. The distribution of targets in the mock catalogues is shown in Fig. 1. Theupper panel shows all tiles that are necessary to observe the required set of targets from the mock catalogues. The lower panels show thetiles for bright, grey and dark sky conditions. The division between different sky conditions works as expected. The Milky Way and theMagellanic Clouds are mostly observed during bright time, while the extragalactic sky is mostly observed during dark time.MNRAS000
Tiling pattern for all 4MOST mock catalogues. The distribution of targets in the mock catalogues is shown in Fig. 1. Theupper panel shows all tiles that are necessary to observe the required set of targets from the mock catalogues. The lower panels show thetiles for bright, grey and dark sky conditions. The division between different sky conditions works as expected. The Milky Way and theMagellanic Clouds are mostly observed during bright time, while the extragalactic sky is mostly observed during dark time.MNRAS000 , 1–20 (2020) ptimal tiling algorithm Table 2.
Summary of algorithm performance tests. We applied the tiling algorithm to the 4MOST mock catalogues. We ran the algorithmwith the default parameters and with parameters where certain optimisation options were disabled. In each test, the same set of targetswas expected to be completed with the same completion criteria. The table below gives the summary statistics for each generated tiling.As expected, the test with all optimisation enabled gives better results than tests with disabled optimisation options.Test name N tile N OB Mean T exp Mean T OB Sum of T exp Sum of T OB Obs. frac. Extra T obs Extra T total min min hours hours % % %(1) (2) (3) (4) (5) (6) (7) (8) (9)Default a b c d e f a Default tiling with all optimisation options enabled. b Position angle of each tile is kept fixed during the MCMC run. Initial position angle for each tile is randomly determined. c Exposure time of each tile is fixed to 17.7 min, which allows three exposures during single OB. d Exposure time and position angle of tiles are kept fixed during the MCMC run. e Overhead fraction is not minimised during the optimal tiling generation. f Exposure time and position angle of tiles are kept fixed and overhead fraction is not minimised during the MCMC run. ( ) Number of tiles in the final tiling configuration after the MCMC run. ( ) Number of OBs in the final tiling configuration after the MCMC run. ( ) Mean exposure time of tiles in the final tiling configuration. ( ) Mean OB length (including overheads) in the final tiling configuration. Maximum OB length is 70 min. ( ) Sum of exposure times of all tiles in the final tiling configuration. Total observational time. ( ) Sum of exposure times and overheads associated with each tile and OB. Total telescope time with overheads. ( ) Fraction of total time that is spent for observations. ( ) Extra observational time (without overheads) that is needed compared with the Default tiling. ( ) Extra total telescope time (with overheads) that is needed compared with the Default tiling.
In this section we analyse how well the algorithm performscompared with slightly less optimised tiling solutions. Ingeneral, it is not straightforward to compare the proposedalgorithm with other available methods. The main reasonis that different algorithms optimise different aspects and itis not straightforward to define a common merit function(often called a “metric” ) that can be easily compared.In Section 4.2 we presented two tiling solutions with andwithout an overhead minimisation (see Figs. 7 and 8). While Not to be confused with the differential geometry sense of “met-ric” that is fundamental to the spacetime of modern astronomy. both of them required approximately the same amount ofsummed exposure time, the tiling solution without the over-head optimisation requires about 30% more time for over-heads. In this section we extend this analysis using the4MOST mock catalogues and compare the proposed algo-rithm performance against itself.We ran the algorithm several times. During each testrun, we disabled one or many optimisation options. Thisallows us to estimate the effect of these optimisation options.During these tests, we either fixed the position angle of eachtile, fixed the tiles exposure times, disabled the overheadminimisation or applied several of them together. Table 2gives the summary statistics for these test runs. As expected,the tiling with all optimisation options enabled provides thebest results. With some optimisation disabled, the final tilingconfiguration requires up to eight per cent more telescopetime.During all test runs we used the same 4MOST mocktarget catalogues and the generated tiling uses exactly thesame completion criteria. Hence, all these test runs shouldprovide roughly the same scientific outcome. Although thegenerated tiling solutions are all slightly different, each oneof these solutions constitutes an optimal solution given theparametrisation used in the optimisation process. The finaltiling solution is mostly determined by the underlying tar-get density. The disabled optimisation options have only asecond order effect on the final solution. Using a naive tiling To estimate the real scientific merit of the generated tiling con-figurations requires full simulation of the 4MOST observations.The generated tiling pattern alone does not allow the estimationof the real scientific merit directly.MNRAS , 1–20 (2020) E. Tempel et al. that does not follow the underlying target density would givea significantly worse solution.To conclude, the optimal tiling solution is mostly drivenby the underlying target density. The MCMC optimisationof the tile position angles, exposure times and minimisationof the overall overhead fraction gives up to an eight percent improvement compared with the slightly less optimisedtiling solutions.
In this paper, we propose a tiling algorithm for multi-objectspectroscopic surveys that is based on marked point pro-cesses. In the algorithm, the optimal tiling pattern is mod-elled as a marked point process where each tile is consideredas a marked point or object. Finding the optimal tiling so-lutions is equivalent to finding the set of tiles with exposuretimes that is required to efficiently observe the targets givenin the input catalogue. The optimisation problem is solvedusing a Metropolis-Hastings algorithm with simulated an-nealing.The proposed algorithm finds an optimal tiling patterngiven an input target catalogue. The algorithm finds the op-timal tiling solution in the sky regions that are observedonce or several times. Simultaneously, the algorithm findsan efficient solution in regions that should be visited multi-ple times. We found that the optimal tiling pattern selectedby the algorithm follows the underlying target density verywell. Hence, the algorithm can be used simultaneously forsurveys that require multiple visits and for surveys that needuniform sky coverage.The proposed algorithm does not assume a fixed expo-sure time per observation. Assuming that the required ex-posure time per target is available in the input cataloguedata files, the algorithm determines a tentative exposuretime for each tile, while taking the overhead time per eachobservation into account. In general, the algorithm allows tominimise the total time that is needed to successfully andefficiently observe the objects given in the input target cat-alogue. Additionally, the algorithm can divide the tiles be-tween different sky conditions, assuming that the exposuretime per target as a function of sky condition is available.Finding an optimal tiling solution requires a clear def-inition of a merit function that should be maximised. Inthe proposed algorithm, the merit function is defined viaan energy function, where the energy function takes differ-ent aspects of the optimal tiling problem into account. Theenergy function defined in this paper optimises the fibre-to-target assignment, minimises the total overhead time, in-cludes interactions between tiles, and forces the tiles to bedivided between predefined sky conditions. The balance be-tween these components can be fine tuned in the algorithm,based on the input catalogue and the survey science goals.The proposed algorithm is tested using the currentmock catalogues of the 4MOST consortium surveys, cov-ering the Milky Way and extragalactic sky. We show thatthe generated optimal tiling pattern matches the estimatedrequired exposure time as a function of sky coordinates verywell. The optimal tiling pattern follows the edges of differentsub-survey patches in the sky, allowing the generation of anefficient tiling that takes the target density variations in the sky naturally into account. The generated tiling pattern isused together with the probabilistic fibre-to-target assign-ment algorithm proposed in Tempel et al. (2020), showingvery high fibre-usage efficiency and survey completeness. Ingeneral, the optimal tiling algorithm proposed in this paperis an input for the probabilistic fibre-to-target assignmentalgorithm described in Tempel et al. (2020).The marked point process framework behind the pro-posed tiling algorithm is very flexible and allows the redef-inition of the described energy function components or theintroduction of new components. For example, the interac-tion between tiles can be used to minimise the gaps betweenindividual tiles and to construct a tiling pattern that uni-formly covers a contiguous area of sky. When necessary, theinteraction energy can also be used to force gaps betweentiles in order to cover larger sky areas with the same num-ber of tiles. Depending on the survey science case, an appro-priate interaction energy for an optimal tiling pattern canbe chosen. The exact definition of the interaction energy forthe 4MOST survey will be determined during the 4MOSTsurvey optimisation phase.The proposed algorithm generates a tiling pattern thatis needed to most efficiently observe the given set of tar-gets in the input catalogue. However, the tiling algorithmdoes not determine when each tile should be observed. Nei-ther does it constrain how much time is available for the4MOST survey. To solve this problem, one needs a schedul-ing algorithm that determines which tiles should be observedand when they should be observed. The scheduling problemcan be solved independently of the tiling challenge. A goodscheduling algorithm for the 4MOST survey will be devel-oped during the 4MOST survey preparation and is not partof the algorithm proposed in this paper.Regarding the division of tiles between various prede-fined sky conditions, in the algorithm the fraction of timethat is available during dark, grey or bright sky conditionsis currently considered. In reality, the division between pre-defined sky conditions should also take the distribution oftiles in the sky into account. This is necessary, since certainsky regions are only visible during the summer or winter pe-riods and the algorithm should generate tiles with varioussky conditions everywhere in the sky. This shows one possi-ble improvement for the proposed tiling algorithm that stillneeds to be studied. For the 4MOST survey, the need forthis improvement will be assessed in combination with thescheduling algorithm. A simple solution is to observe sometiles during better sky conditions than those assigned by thealgorithm and to scale the exposure time per tile accordingly.A more optimal but time-consuming solution is to fine-tunethe tiling algorithm parameters so that the produced dis-tribution of tiles with predefined sky conditions follows thefraction of available time in different sky regions.Table 1 gives the free parameters of the tiling algorithmthat should be determined for an optimal tiling solution.Many of these parameters affect the speed and convergenceof the algorithm and have only a minor impact on the fi-nal tiling solution. However, some of the parameters havea direct impact on the optimal tiling solution and shouldbe determined while taking the input target catalogue andsurvey science goals into account. One example is the param-eters c LR and c HR that determine the importance betweenthe numbers of low-resolution and high-resolution targets. MNRAS000
In this paper, we propose a tiling algorithm for multi-objectspectroscopic surveys that is based on marked point pro-cesses. In the algorithm, the optimal tiling pattern is mod-elled as a marked point process where each tile is consideredas a marked point or object. Finding the optimal tiling so-lutions is equivalent to finding the set of tiles with exposuretimes that is required to efficiently observe the targets givenin the input catalogue. The optimisation problem is solvedusing a Metropolis-Hastings algorithm with simulated an-nealing.The proposed algorithm finds an optimal tiling patterngiven an input target catalogue. The algorithm finds the op-timal tiling solution in the sky regions that are observedonce or several times. Simultaneously, the algorithm findsan efficient solution in regions that should be visited multi-ple times. We found that the optimal tiling pattern selectedby the algorithm follows the underlying target density verywell. Hence, the algorithm can be used simultaneously forsurveys that require multiple visits and for surveys that needuniform sky coverage.The proposed algorithm does not assume a fixed expo-sure time per observation. Assuming that the required ex-posure time per target is available in the input cataloguedata files, the algorithm determines a tentative exposuretime for each tile, while taking the overhead time per eachobservation into account. In general, the algorithm allows tominimise the total time that is needed to successfully andefficiently observe the objects given in the input target cat-alogue. Additionally, the algorithm can divide the tiles be-tween different sky conditions, assuming that the exposuretime per target as a function of sky condition is available.Finding an optimal tiling solution requires a clear def-inition of a merit function that should be maximised. Inthe proposed algorithm, the merit function is defined viaan energy function, where the energy function takes differ-ent aspects of the optimal tiling problem into account. Theenergy function defined in this paper optimises the fibre-to-target assignment, minimises the total overhead time, in-cludes interactions between tiles, and forces the tiles to bedivided between predefined sky conditions. The balance be-tween these components can be fine tuned in the algorithm,based on the input catalogue and the survey science goals.The proposed algorithm is tested using the currentmock catalogues of the 4MOST consortium surveys, cov-ering the Milky Way and extragalactic sky. We show thatthe generated optimal tiling pattern matches the estimatedrequired exposure time as a function of sky coordinates verywell. The optimal tiling pattern follows the edges of differentsub-survey patches in the sky, allowing the generation of anefficient tiling that takes the target density variations in the sky naturally into account. The generated tiling pattern isused together with the probabilistic fibre-to-target assign-ment algorithm proposed in Tempel et al. (2020), showingvery high fibre-usage efficiency and survey completeness. Ingeneral, the optimal tiling algorithm proposed in this paperis an input for the probabilistic fibre-to-target assignmentalgorithm described in Tempel et al. (2020).The marked point process framework behind the pro-posed tiling algorithm is very flexible and allows the redef-inition of the described energy function components or theintroduction of new components. For example, the interac-tion between tiles can be used to minimise the gaps betweenindividual tiles and to construct a tiling pattern that uni-formly covers a contiguous area of sky. When necessary, theinteraction energy can also be used to force gaps betweentiles in order to cover larger sky areas with the same num-ber of tiles. Depending on the survey science case, an appro-priate interaction energy for an optimal tiling pattern canbe chosen. The exact definition of the interaction energy forthe 4MOST survey will be determined during the 4MOSTsurvey optimisation phase.The proposed algorithm generates a tiling pattern thatis needed to most efficiently observe the given set of tar-gets in the input catalogue. However, the tiling algorithmdoes not determine when each tile should be observed. Nei-ther does it constrain how much time is available for the4MOST survey. To solve this problem, one needs a schedul-ing algorithm that determines which tiles should be observedand when they should be observed. The scheduling problemcan be solved independently of the tiling challenge. A goodscheduling algorithm for the 4MOST survey will be devel-oped during the 4MOST survey preparation and is not partof the algorithm proposed in this paper.Regarding the division of tiles between various prede-fined sky conditions, in the algorithm the fraction of timethat is available during dark, grey or bright sky conditionsis currently considered. In reality, the division between pre-defined sky conditions should also take the distribution oftiles in the sky into account. This is necessary, since certainsky regions are only visible during the summer or winter pe-riods and the algorithm should generate tiles with varioussky conditions everywhere in the sky. This shows one possi-ble improvement for the proposed tiling algorithm that stillneeds to be studied. For the 4MOST survey, the need forthis improvement will be assessed in combination with thescheduling algorithm. A simple solution is to observe sometiles during better sky conditions than those assigned by thealgorithm and to scale the exposure time per tile accordingly.A more optimal but time-consuming solution is to fine-tunethe tiling algorithm parameters so that the produced dis-tribution of tiles with predefined sky conditions follows thefraction of available time in different sky regions.Table 1 gives the free parameters of the tiling algorithmthat should be determined for an optimal tiling solution.Many of these parameters affect the speed and convergenceof the algorithm and have only a minor impact on the fi-nal tiling solution. However, some of the parameters havea direct impact on the optimal tiling solution and shouldbe determined while taking the input target catalogue andsurvey science goals into account. One example is the param-eters c LR and c HR that determine the importance betweenthe numbers of low-resolution and high-resolution targets. MNRAS000 , 1–20 (2020) ptimal tiling algorithm In this paper, constant values are assumed across the sky.However, if a sky region is dominated by LR targets, thenthe optimisation should take that into account. This can beachieved by defining different c LR and c HR values in differentparts of the sky. These optimisations depend very stronglyon the input target catalogue and will be included in thealgorithm during the 4MOST survey preparation.Computationally, the proposed algorithm is somewhatdemanding. In the 4MOST survey, we have approximately40 000 individual tiles and Markov-chain Monte Carlo(MCMC) sampling and the optimisation of large numberof tiles take some time. Additionally, during each MCMCstep we have to perform the fibre-to-target assignment. Inthe proposed algorithm we use a statistical fibre-to-targetassignment, which helps to improve the speed of the algo-rithm significantly. Despite that, to find an optimal tilingpattern for the full 4MOST survey (about 50 million targetsand 40 thousand tiles) takes currently up to a few days using24 cores on a shared memory machine. The algorithm scalesreasonably well using OpenMP parallelisation. It is not yettested, how well the algorithm scales using MPI paralleli-sation. The computation time can be potentially reducedby using better optimisation and parallelisation. During realobservations, the tiling algorithm should be run at the begin-ning of the survey, in which case a few days of computationaltime is not a problem. However, during the execution of thesurvey, one might want to rerun the tiling algorithm to op-timise the tiling that better matches the remaining targets,or because the input target catalogue has been updated.In these cases, one does not have to run the tiling algorithmfrom scratch. The MCMC sampling of the tiling pattern canbe initialised using the previous tiling solution. This will sig-nificantly reduce the computational cost and allow the tilingpattern to be updated during the survey within a reasonableamount of computational time.To conclude, the tiling algorithm presented in this paperis a new approach to solving the optimal tiling challenge formulti-object spectroscopic surveys. The current algorithmis a proposed solution for the 4MOST survey and in combi-nation with the probabilistic fibre-to-target assignment pre-sented by Tempel et al. (2020) solves two major challengesfaced during the 4MOST survey preparation. With appro-priate modifications, the algorithm that we propose can bealso applied to other forthcoming multi-object spectroscopicsurveys. ACKNOWLEDGEMENTS
This work has made use of the development effort for4MOST, an instrument under construction by the 4MOSTConsortium ( ) forthe European Southern Observatory (ESO). Part of thiswork was supported by institutional research fundingIUT40-2 of the Estonian Ministry of Education and Re-search. We acknowledge the support by the Centre of Ex-cellence “Dark side of the Universe” (TK133) and by thegrant MOBTP86 financed by the European Union throughthe European Regional Development Fund. Part of this workwas supported by the “A next-generation worldwide quan-tum sensor network with optical atomic clocks” project,which is carried out within the TEAM IV programme of the Foundation for Polish Science co-financed by the Eu-ropean Union under the European Regional DevelopmentFund. This work has been supported by the Polish MNiSWgrant DIR/WK/2018/12. AK and NC acknowledge fundingby the DFG – Project-ID 138713538 – SFB 881 (“The MilkyWay System”), subprojects A03, A05, A09, A11. MRC ac-knowledges funding from the European Research Council(ERC) under the European Union’s Horizon 2020 researchand innovation programme (grant agreement no. 682115).GT was supported by the grant The New Milky Way fromthe Knut and Alice Wallenberg foundation, by the grant2016-03412 from the Swedish Research Council, and by theSwedish strategic research programme eSSENCE.
DATA AVAILABILITY
The 4MOST mock catalogues used in this article are subjectto the data access policies of the 4MOST consortium. Thesoftware code will be shared on reasonable request to thecorresponding author.
REFERENCES
Bensby T., et al., 2019, The Messenger, 175, 35( arXiv:1903.02470 )Bernyk M., et al., 2016, ApJS, 223, 9 ( arXiv:1403.5270 )Blanton M. R., Lin H., Lupton R. H., Maley F. M., Young N.,Zehavi I., Loveday J., 2003, AJ, 125, 2276 ( arXiv:astro-ph/0105535 )Calabretta M. R., Roukema B. F., 2007, MNRAS, 381, 865( arXiv:astro-ph/0412607 )Chiappini C., et al., 2019, The Messenger, 175, 30( arXiv:1903.02469 )Christlieb N., et al., 2019, The Messenger, 175, 26( arXiv:1903.02468 )Cioni M.-. R. L., et al., 2019, The Messenger, 175, 54( arXiv:1903.02475 )Cole S., Lacey C. G., Baugh C. M., Frenk C. S., 2000, MNRAS,319, 168 ( arXiv:astro-ph/0007281 )Colless M., et al., 2001, MNRAS, 328, 1039 ( arXiv:astro-ph/0106498 )Comparat J., et al., 2019, MNRAS, 487, 2005 ( arXiv:1901.10866 )Drinkwater M. J., et al., 2010, MNRAS, 401, 1429( arXiv:0911.4246 )Driver S. P., et al., 2009, Astronomy and Geophysics, 50, 5.12( arXiv:0910.5123 )Driver S. P., et al., 2019, The Messenger, 175, 46( arXiv:1903.02473 )Finoguenov A., et al., 2019, The Messenger, 175, 39( arXiv:1903.02471 )Gaia Collaboration et al., 2016, A&A, 595, A1( arXiv:1609.04153 )Gaia Collaboration et al., 2018, A&A, 616, A1( arXiv:1804.09365 )Geyer C. J., 1999, in Stochastic geometry, likelihood and compu-tation, ed. O. Barndorff-Nielsen, W. S. Kendall, & M. N. M.van Lieshout (Boca Raton: CRC Press/Chapman and Hall),79Geyer C. J., Møller J., 1994, Scan. J. Stat., 21, 359G´orski K. M., Wandelt B. D., Hivon E., Hansen F. K., BandayA. J., 1999, preprint ( arXiv:astro-ph/9905275 )Green P. J., 1995, Biometrica, 82, 711Guiglion G., et al., 2019, The Messenger, 175, 17( arXiv:1903.02466 )MNRAS , 1–20 (2020) E. Tempel et al.
Helmi A., et al., 2019, The Messenger, 175, 23( arXiv:1903.02467 )Jones D. H., et al., 2004, MNRAS, 355, 747 ( arXiv:astro-ph/0403501 )Kendall W. S., Møller J., 2000, Adv. Appl. Prob., 32, 844Klypin A., Yepes G., Gottl¨ober S., Prada F., Heß S., 2016, MN-RAS, 457, 4340 ( arXiv:1411.4001 )Lagos C. d. P., Bayet E., Baugh C. M., Lacey C. G., BellT. A., Fanidakis N., Geach J. E., 2012, MNRAS, 426, 2142( arXiv:1204.0795 )Liske J., et al., 2015, MNRAS, 452, 2087 ( arXiv:1506.08222 )Merloni A., et al., 2019, The Messenger, 175, 42( arXiv:1903.02472 )Møller J., Waagepetersen R. P., 2004, Statistical inferenceand simulation for spatial point processes. Chapman andHall/CRCRichard J., et al., 2019, The Messenger, 175, 50( arXiv:1903.02474 )Robotham A., et al., 2010, Publ. Astron. Soc. Australia, 27, 76( arXiv:0910.5121 )Sharma S., Bland-Hawthorn J., Johnston K. V., Binney J., 2011,ApJ, 730, 3 ( arXiv:1101.3561 )Stoica R. S., Gregori P., Mateu J., 2005, Stochastic Processes andtheir Applications, 115, 1860Stoica R. S., Philippe A., Gregori P., Mateu J., 2017, Statisticsand Computing, 27, 1225 ( arXiv:1507.04228 )Swann E., et al., 2019, The Messenger, 175, 58( arXiv:1903.02476 )Tempel E., Stoica R. S., Mart´ınez V. J., Liivam¨agi L. J., CastellanG., Saar E., 2014, MNRAS, 438, 3465 ( arXiv:1308.2533 )Tempel E., Stoica R. S., Kipper R., Saar E., 2016, Astronomyand Computing, 16, 17 ( arXiv:1603.08957 )Tempel E., Kruuse M., Kipper R., Tuvikene T., Sorce J. G., StoicaR. S., 2018, A&A, 618, A81 ( arXiv:1806.04469 )Tempel E., et al., 2020, arXiv e-prints, p. arXiv:2001.09348( arXiv:2001.09348 )Walcher C. J., et al., 2019, The Messenger, 175, 12( arXiv:1903.02465 )da Cunha E., et al., 2017, Publ. Astron. Soc. Australia, 34, e047( arXiv:1706.01246 )de Jong R. S., et al., 2019, The Messenger, 175, 3( arXiv:1903.02464 )van Lieshout M. N. M., 1994, Advances in Applied Probability,26, 281van Lieshout M. N. M., 2000, Markov point processes and theirapplications. Imperial College Press/World Scientific Publish-ingvan Lieshout M. N. M., 2019, Theory of Spatial Statistics. AConcise Introduction. Chapman and Hall/CRCvan Lieshout M. N. M., Stoica R. S., 2003, Statistica Neerlandica,57, 177van Lieshout M. N. M., Stoica R. S., 2006, Computational Statis-tics and Data Analysis, 51, 679This paper has been typeset from a TEX/L A TEX file prepared bythe author. MNRAS000