ALTSched: Improved Scheduling for Time-domain Science with LSST

Telescope scheduling is the task of determining the best sequence of observations (pointings and filter choices) for a survey system. Because it is computationally intractable to optimize over all possible multiyear sequences of observations, schedulers use heuristics to pick the best observation at a given time. A greedy scheduler selects the next observation by choosing whichever one maximizes a scalar merit function, which serves as a proxy for the scientific goals of the telescope. This sort of bottom-up approach for scheduling is not guaranteed to produce a schedule for which the sum of merit over all observations is maximized. As an alternative to greedy schedulers, we introduce ALTSched, which takes a top-down approach to scheduling. Instead of considering only the next observation, ALTSched makes global decisions about which area of sky and which filter to observe in, and then refines these decisions into a sequence of observations taken along the meridian to minimize airmass. We implement ALTSched for the Large Synoptic Survey Telescope (LSST) and show that it equals or outperforms the baseline greedy scheduler in essentially all quantitative performance metrics. Due to its simplicity, our implementation is considerably faster than OpSim, the simulated greedy scheduler currently used by the LSST Project: a full 10-yr survey can be simulated in 4 minutes, as opposed to tens of hours for OpSim. LSST’s hardware is fixed, so improving the scheduling algorithm is one of the only remaining ways to optimize LSST’s performance. We see ALTSched as a prototype scheduler that gives a lower bound on the performance achievable by LSST.


INTRODUCTION
A succession of ground-based wide-field survey telescopes (SDSS (Eisenstein et al. 2011), PanSTARRS (Chambers et al. 2016), PTF (Rau et al. 2009), ZTF (Bellm 2014;Smith et al. 2014), ATLAS (Tonry et al. 2018)...) has provided an ever-increasing ability to scan the sky and carry out "static" science with co-added images and "timedomain" science by searching for variability with frame subtraction methods.The LSST project (Ivezic et al. 2008) will provide unprecedented survey capability, reaching 24th magnitude in 30 seconds of integration while covering the entire accessible sky every few nights.With the LSST project well into the construction phase, the system's engineering characteristics (aperture, field size, slew rate, quantum efficiency...) are fixed.Apart from actively engineering the weather in Chile, the only remaining opportunities to extract optimum performance from the LSST system are in the scheduler and in the processing pipeline that extracts knowledge from the survey's image stream.
In this paper we demonstrate a way to schedule LSST observations that increases its overall scientific effectiveness, as compared to previous scheduling schemes.Our approach is fundamentally different from local greedy optimizers, including the "feature-based scheduler" Naghib et al. (2018) that uses machine-learning methods to determine parameters for a greedy optimizer scheduler.The LSST project's baseline scheduler, implemented in a software suite called OpSim, successfully meets the LSST minimum scientific requirements.But we if there are ways to increase the scientific utility of the LSST system, we should consider them as viable alternatives for survey operations.
The literature on scheduling algorithms for synoptic wide-field ground-based telescopes is mostly limited to discussions of how telescope-specific problems were overcome, and there has been little theoretical analysis of what constitutes a globally optimal schedule.In this paper, we describe some properties of a schedule that determine how well, if the schedule were executed, the resulting survey would enable a range of scientific goals.We discuss the tensions between the properties, and examine how a scheduler can control which properties to prioritize.
The total merit of a schedule is some function of the schedule's achieved value on each property, whose form depends on the scientific priorities of the survey's designers.In the space whose dimensions are these desirable schedule properties and where points represent schedules, there is a volume of accessible space determined by the instrument hardware and by basic physics.An ideal scheduler will only produce schedules that lie on the surface of this volume (the Pareto frontier), and will take as inputs parameters that directly control where on the surface the resulting schedule lies.This gives the survey designers complete control over which Pareto-optimal schedule to execute.In this paper, we argue that a scheduler that produces a class of schedules called "deterministic" schedules approaches this ideal for ground-based wide-field survey telescopes with fast readouts.We then describe our implementation of a deterministic scheduler for the Large Synoptic Survey Telescope (LSST) and compare its performance to that of the existing state-of-the-art.

RELATED WORK
The scheduling principles we describe in this paper apply to any ground-based telescope able to image a large sky area per unit time.Such telescopes include LSST (Ivezic et al. 2008), Palomar Transient Factory (PTF) (Rau et al. 2009), Zwicky Transient Facility (ZTF) (Smith et al. 2014), Pan-STARRS1 (Chambers et al. 2016), SkyMapper (Keller et al. 2007), the Dark Energy Survey (DES) (The Dark Energy Survey Collaboration 2005), and others.Scheduling software has been developed for each of these telescopes, but often only limited information about these schedulers can be found in the literature.Broadly speaking, most schedulers operate by choosing each observation as the night progresses based on where the telescope is currently pointing and on current or predicted observing conditions.By choosing observations that maximize scientific merit at each point in the night, these scheduling algorithms seek to maximize the scientific merit of the entire resulting schedule.Algorithms that follow such a strategy are called greedy algorithms, since they greedily take the most meritorious option at each decision point instead of strategizing about what will lead to the most merit in the long term.In general, greedy algorithms are guaranteed to maximize merit in the long term in only the simplest of problems, and telescope scheduling is not one of those problems.To see why, consider a scenario where every field North of the telescope has a merit score of, say, 5, and every field South of the telescope has a merit score of 4. Fields directly overhead have low merit.If the telescope happens to be pointing in the South and long slews are penalized, then a greedy algorithm will call for observing in the South the entire night, even though the globally optimal policy would be to slew to the North at the very beginning and observe there for the rest of the night.
The PTF scheduler is a greedy algorithm that operates under four modes, where the two main modes are a 5-day cadence (5DC) and a dynamic cadence (DyC) (Law et al. 2009).In the 5DC, the telescope takes two 60s exposures separated by 60 minutes on average every 5 days.The DyC observes a smaller number of fields with a higher cadence, with the exact observing strategy to be reviewed by the PTF collaboration every 6 months.The scheduler itself chooses fields by calculating, for each candidate observation, values for several criteria: the sun altitude, moon sky brightness and phase, expected slew time, and airmass.Candidates get a point for each criterion for which they exceed a threshold value.A smooth function of the time since the candidate was last observed is added to this value to get a final score.The scheduler then chooses the candidate with the highest score.For the 5DC, "the scheduler is required to optimize the field selection so that each field is observed twice during the night for asteroid detection" (Law et al. 2009).We were unable to find information about what this optimization was or whether it succeeded at achieving all 60-minute revisits.We were also unable to determine if the scoring system described achieved a constant 5-day cadence or whether the distribution of inter-visit gaps followed a potentially less desirable distribution.
ZTF, unlike PTF, schedules an entire night of observations at a time.It chooses a schedule by solving an integer program designed to maximize observing efficiency subject to constraints on when and how often each field needs to be observed (Bellm et al. 2019).The Las Cumbres Observatory schedules its telescopes in a similar way (Saunders & Lampoudi 2014).For LSST, this approach is challenging for a few reasons: the overhead for each observation varies considerably (a few seconds to 2 minutes); in order to achieve a more uniform sky coverage, LSST may not use fixed fields at all; and the large number of fields and observations per night may render the integer program intractable to solve repeatedly during a night when weather changes.
Another telescope, Pan-STARRS1, conducted a 3π steradian survey while following two different observing strategies: the Design Reference Mission (DRM) from 2009 to early 2012, and the Modified Design Reference Mission (MDRM) from early 2012 to 2014.The DRM (Chambers & Denneau 2008) specifies that observations in the 3π steradian survey should be carried out in pairs, separated by about 30 minutes, with each field observed at a cadence of about 5 days over two months.Images in the z and y bands are taken mostly near twilight.Although the DRM schedule is described in almost sufficient detail to implement in code directly, it was instead implemented using an early version of the LSST scheduler, which uses a greedy algorithm.We did not obtain the final algorithm that was used to schedule Pan-STARRS1.
Lastly, LSST's scheduler is part of a package called the Operations Simulator (OpSim) (Delgado & Reuter 2016).OpSim uses a greedy algorithm to choose fields based on a proposal system, where abstractions of different scientific proposals give a score for each candidate field, and the scheduler chooses the field with the highest combined score.Results from OpSim are presented in more depth in Section 9. Several alternatives to OpSim have also been proposed.Naghib et al. (2016) frame the scheduling problem as a Markovian decision process.Ridgway (2015) proposes scheduling LSST by dividing observing into blocks that are observed (and then reobserved) in an optimal manner.Ridgway's proposal is similar in spirit to the algorithm developed and described in this paper.
Most classically scheduled non-survey telescopes operate on a proposal system, where scientists submit observing proposals that are then scheduled on the instrument by a central scheduler.The literature on scheduling algorithms for such telescopes is much more developed, and Colomé et al. (2012) survey various algorithms that are currently in use to solve this sort of scheduling problem.Few-night observing run scheduling decisions are typically made by the astronomer, based on scientific priorities, with dynamic adjustments for weather and seeing conditions.

DESIRABLE PROPERTIES OF A SCHEDULE
A schedule for a particular survey telescope is a feasible sequence of observations to be carried out, where each observation is assigned, in addition to a pointing, an exposure time, rotator angle, and photometric filter.The scientific merit of a schedule can be assessed by calculating summary statistics, such as "fraction of supernovae that are sampled often enough to classify their type with high confidence" or "fraction of galaxies for which data quality is high enough to measure shape for weak lensing".For the purposes of this paper, summary statistics like these are too difficult to calculate and too numerous to be of use.We therefore distill these "science metrics" into much simpler "technical metrics", or simple properties of the final schedule.We contend that a schedule that performs well on these properties is likely to perform well on the more complicated science metrics as well.We divide these properties into two groups: those that depend on the time and rough location of the pointings, and those that depend on the sky tiling from which the exact pointings are drawn.
Properties that depend on the gross distribution of pointings in RA/Dec/time are: • Typical single-visit 5σ depth, • Open-shutter fraction, • Final co-added depth in each band, • Typical histogram of inter-visit times, and • Typical histogram of per-band inter-visit times.
Properties that depend on the sky tiling are: • RMS fluctuation in sky maps of co-added depth, • Typical distribution over position angles at each sky pixel, and • Typical distribution over CCDs used to image each sky pixel.
The properties in the first group are highly inter-connected with each other.The properties in the second group can be jointly optimized by carefully choosing a sky tiling and rotator angle strategy, and are mostly independent of the properties in the first group.We examine each of these properties in the following two sections and explore the tradeoffs between them.Along the way, we introduce tunable parameters that allow the survey designers to control these tradeoffs directly.

POINTING HISTORY PROPERTIES
Consider a multi-year schedule, and for simplicity, assume that the survey footprint extends over all right ascensions between two declinations.We consider the five properties from the first group in order.

Single-Visit Depth
The single-visit depth determines the detection limit for fast transients.It is determined by the single-visit exposure time, t exp , and the achieved SNR, which depends on the weather, seeing, transparency, and on the observed airmass.
There are multiple ways to go about choosing t exp .Bellm (2014, §2) propose choosing t exp in order to optimize the number of transients with some fixed absolute magnitude that are discovered.Because each visit involves some overhead closed-shutter time, there is a value of t exp that maximizes the volume in which the specified transients will be detected that is probed by the survey.For surveys that are particularly interested in transients with a single absolute magnitude (SN Ia, for instance), this method is optimal.However, for surveys interested in a transients with a wide range of absolute magnitudes, t exp could instead be chosen as the shortest possible value that does not incur an unacceptable open-shutter fraction.Choosing t exp in this way prioritizes finding brighter, rarer transients, which are the ones where rapid spectroscopic followup is easiest.
Choosing t exp gives the survey designers a tunable knob to control the tradeoff between the open-shutter fraction and the volume of space explored for fast, bright transients.Numerical values for this tradeoff cannot be calculated, since the exact abundance and nature of all transients is unknown.

Open-Shutter Fraction
The open-shutter fraction is the fraction of survey time spent with the shutter open.If t exp is the exposure time for some visit, t slew is the slew time (including any filter change time), and t read is the camera readout time, then the open-shutter fraction for a single visit is o = t exp t exp + max(t read , t slew ) assuming the camera can read out while slewing.
The open-shutter fraction can be minimized by reducing the slew time (assuming the readout time is usually smaller than the slew time).The schedule can, without penalty, reduce the slew time until the focal plane footprints of consecutive visits abut.Beyond this point, there is a tradeoff between increasing open-shutter fraction and cadence properties: if consecutive visits overlap, the schedule gains rapid revisits on the order of t exp /o, but the area of sky that can be imaged per night goes down.
This gives the survey designers another knob to tune, which controls the tradeoff between increasing open-shutter time and rapid revisit cadence on the one hand, vs. increasing sky area covered per night on the other.This tradeoff is difficult to write down analytically, since it depends sensitively on the maximum speeds and accelerations of the instrument and of the dome, along with settling time and camera readout time.However, once these parameters are set, it should be straightforward to calculate for telescopes with simple field of view shapes, and can be calculated numerically otherwise.
This tradeoff is discussed from the standpoint of the sky tiling in greater detail in Section 5.

Co-added Depth
Co-added depth is maximized by increasing open-shutter fraction and observing at the highest possible SNR. 1 The former is controlled by the tradeoffs described above, and the latter is accomplished by observing along the meridian.

Inter-Visit Gap Histogram
The cadence properties of a schedule are much more complicated than the static survey properties.For simplicity, we assume throughout this section that the same sort of cadence is desired across the entire survey footprint.We choose to analyze the inter-visit gap histograms as a metric of cadence because they are simple to interpret, they capture all the information about a cadence, and because, for alternative survey strategies, the area under each histogram is conserved, making alternative histograms easy to compare.
We first introduce the idea of a "rolling cadence", which applies at many different cadence timescales.For the purposes of this paper, we define a uniform rolling cadence (here referred to simply as a rolling cadence) to be any schedule that is produced by dividing up the sky into M R disjoint regions of equal area and then exclusively observing each region in turn for a time D R , where D R is an integer number of nights.M R is called the rolling multiplicity and D R is the dwell time of the cadence.
There are several ways to create the M R sky partitions.One simple and effective way is to divide the declination range of the survey footprint into M R ranges such that the area of each region is equal.M R and D R are global properties of the survey that don't change over time.We treat sky pixels outside of the current rolling partition as if they are not currently observable.
One quantity that is fundamental for determining cadence properties is the number of visits available to each observable sky pixel in a single night.To calculate that, we first need to determine the length of the observing season, since each field will get more visits per night if the observing season is shorter.In order to maximize SNR, the schedule should generally observe on the meridian, since fields are at their minimum airmass when they cross the meridian.If the schedule observes exclusively on the meridian, the observing season is completely determined, since it begins when a field crosses the meridian just after sunset and ends when the field crosses the meridian just before sunrise.
To allow for some flexibility in the observing season length, consider a schedule that deviates from the meridian.Define ν, the "normalized airmass" as the quotient of the airmass a field is observed at and the minimum airmass that the field could ever be observed at (i.e. the airmass of the field when it is on the meridian).For a field at declination δ observed at a zenith angle θ: Say we require ν < ν max to limit degradation to SNR.This yields a nontrivially shaped segment of the alt/az sphere that is acceptable to observe in.For simplicity, we approximate this shape as the portion of the alt/az sphere between −HA max and +HA max for some HA max that satisfies sec(HA max ) = ν max The cadence that maximizes observing season starts the night observing in the West and ends the night observing in the East.The cadence that minimizes observing seasons starts in the East and ends in the West.
Schedules that observe in this manner -scanning North and South on or parallel to the meridian at some hour angle that changes slowly as the night progresses -are called "marching army" schedules.We show how the parameters of a marching army scheduler are easily interpretable and directly control tradeoffs between the global schedule properties introduced above.
Let the duration of the night be t night hours.For simplicity, assume that the survey footprint includes all right ascensions between two fixed declinations.The possible range of observing season lengths t season (in days) is then: If we choose ν max = 1.5 and let t night = 10 hours, then the permissible season length ranges from 54 to 250 days.For a telescope that stays on the meridian (ν max = 1)), the season length will be 152 days.
To see how the observing rate changes with the observing season length, consider a survey telescope that images a solid angle ω per second2 over a survey footprint of size Ω.Assuming the focal plane tiles the sky, a survey that seeks to achieve a uniform cadence everywhere can reach a maximum cadence of visits per unit time to each sky pixel within the current rolling partition that is in season (t season here is in radians).For ν max = 1.5 and t night = 10 hours as above, and using fiducial LSST values of ω = 9.8 sq.degrees ∼37 seconds = 0.26 square degrees/second and Ω = 20, 000 square degrees, this corresponds to a range of 0.70 < γ < 3.2 visits per night, where we have set M R to 1. Figure 1 shows the relationship between cadence, season length, and ν max graphically.
There is a direct tradeoff between maximizing SNR and having flexibility in observing season length: observing only on the meridian completely fixes the observing season length, while observing slightly off-meridian (i.e.moving away from the ν max = 1 line in Figure 1) allows a the scheduler to sacrifice overall survey depth in order to prioritize sampling of shorter or longer transients.γ, which determines the range of available cadences on a shorter time scale, is fixed by choosing the observing season.ν max gives the survey designers an additional knob to turn, controlling the tradeoff between maximizing SNR on the one hand and gaining flexibility in t season and γ on the other.
Assume for the moment that γ is an integer.If γ = 1, the schedule is simple: to maintain minimum slews, it will scan North and South along a particular hour angle (along the meridian if ν max = 1), and the sky will scroll by at the right speed for the telescope to expose at each sky pixel exactly once while staying at the prescribed hour angle.Consider the histogram of inter-visit times -the "property" we are trying to optimize.If M R = 1, the histogram has a single peak around 1 day.For larger M R , the histogram will have two peaks, one at 1 day and one at D R (M R − 1) + 1 days.The ratio of the areas under the 1-day peak and the D R (M R − 1) + 1 peak is D R − 1.In this way, M R and D R give the survey designers another knob to tune, which controls the location and sizes of the peaks in the inter-night visit gap histogram.
If γ < 1, the schedule can proceed in two ways.There is more observable sky area per night than the telescope can physically observe, so the schedule can either randomly choose a subset of the observable area to observe, or it can increase M R by a factor of 1/γ in order to bring γ up to 1.In the first case, the inter-visit gaps will follow an exponential distribution; in the second, they will follow the regular distribution for γ = 1 described above.
If γ > 1, the histogram of inter-visit gaps is split into two parts: gaps between visits within a single night and gaps between visits in different nights.The latter part obeys the same properties as the γ = 1 distribution described above.We now consider the intra-night visit gap distribution histogram.Because fields rise and set each night, all γ observations to a sky pixel must occur within a relatively short time.We consider the simplest case, where a single peak in the histogram is desired at some value t sep .In this case, the schedule will observe in blocks of duration γt sep , during which time a fixed set of fields are observed, in order, γ times.Since the schedule observes on or near the meridian, this will result in observing blocks consisting of a rectangle of fields extended in the North-South direction and only a few fields wide in the East-West direction.Every t sep seconds, the schedule will scan North and South, covering the entire block, before starting again at the beginning of the block.To minimize the slew time into each revisit sequence, the schedule should scan North and South an even number of times for each block.The ratio of the area under the peak at t sep to the area under the inter-night peaks in the final inter-visit gap histogram is γ − 1. Figure 2 shows schematic representations of these inter-visit gap histograms for γ = 1 and γ > 1.
t sep gives the survey designers another knob to turn to control the intra-night visit gap histogram.More complicated histograms can be constructed by changing the value of t sep from night to night.

Per-Band Inter-Visit Gap Histograms
As shown above, the band-independent inter-visit gap distribution is mostly determined by M R , D R , γ, and t sep .However, there is still considerable leeway in the per-band inter-visit gap histograms.For example, if γ = 1 and D R is long, the schedule could change filters before every night, or it could use the same filter for a week at a time before switching.A full description of the per-band inter-visit gap histograms depends in a complicated way on M R , D R , γ, the number of filters n f ilt , and when and how often filters are changed compared to t sep .We consider only a few simple cases here.In addition, we ignore band-dependent atmospheric effects such as weather, the moon, and twilight.In a production scheduler, these must be accounted for, but they make the analysis more complicated, obscuring the main tradeoffs that control per-band cadence.
First, let M R = 1 and γ = 1.There is no need to change filters more often than before the beginning of each night, since there are no revisits.If the schedule cycles through the n f filters in order, each band will have an inter-visit gap histogram with a single peak at n f days.Now let γ > 1.If the schedule uses the same filter cadence (change before each night), the inter-visit gap histograms for each band will have a peak at t sep and another at n f days, where the ratio of the areas under the t sep and n f peaks is γ − 1.On the other hand, if the schedule changes filters before every revisit block, the histogram will instead have a single peak at n f γ , assuming that the schedule cycles through the n f filters at each sky pixel.This filter changing strategy requires (γ − 1)t night /γt sep filter changes per night: one for each time an observing block is revisited.This is a critical point: if an even cadence is desired, the maximum achievable cadence in a particular band is one visit every n f γ nights.Because the area under the inter-visit gap histogram is conserved, it is impossible to image the field more regularly without also leaving it unobserved for longer periods.
For M R , the filter changing strategy can be applied independently to each rolling partition.However, the situation becomes more complicated if D R is on the order of the typical inter-visit gap time or shorter.Furthermore, in real surveys, different depths are often desired in different filters, which precludes strictly cycling through the n f filters.Adding band-dependent sky brightness from the moon and from twilight introduces a time-varying favorability for observing in each filter.Lastly, it is often the case that certain bands are prioritized when the seeing happens to be especially good, making this favorability function more complicated.Despite these many caveats, we find that using the principles described above to design a filter changing strategy -and then adding a few ad-hoc rules to prevent the schedule from, say, staring at the moon in the u band all night -yields significantly more favorable per-band cadence properties than existing alternatives.

SKY TILING PROPERTIES
We now consider properties of a schedule that depend on how the sky is tiled.Telescopes are increasingly being designed to cover a large area of sky per unit time, so we assume here that each sky pixel will get a large number of visits over the course of the entire survey.We define a tiling to be a set of distinct pointings in RA/Dec with associated position angles on the sky, where the average number of fields that overlap a sky pixel is about 1. Transformations to a tiling include: • Rotation: the tiling can be rotated rigidly on the celestial sphere about the x/y/z axes.
• Dithering: the individual pointings in the tiling can be slightly offset from one another, usually by less than a single field width.
• Rotational dithering: the individual pointings in the tiling can change position angle.
We consider the three schedule properties described above: RMS fluctuation in co-added depth, distribution at each sky pixel over CCDs, and distribution at each sky pixel over position angles.To measure the first, we use as a proxy the RMS fluctuation in number of visits, since this is easier to compute.
Consider a survey that observes only at pointings drawn from some fixed sky tiling.Let the area not covered by any pointing be Ω 0 , the area covered by exactly one pointing be Ω 1 , and the area covered by exactly two pointings be Ω 2 .Assume no part of the sky is covered by three or more fields -i.e. that the total sky area Ω = Ω 0 + Ω 1 + Ω 2 .If this tiling is observed once, the mean number of times a sky pixel will be observed is and the RMS fluctuation in number of visits to a pixel is If the tiling is observed N times without rotation or dithering, the final average and standard deviation in number of visits will be N µ t and N σ t .Using this strategy, each sky pixel would be imaged on the same CCD and with the same position angle every time.Now consider a random tiling, where every night, the schedule randomly chooses a number of fields in the survey area to observe based on how many observations it expects to carry out (each with a random position angle).Let the mean number of times a sky pixel is visited be µ r .The RMS fluction in number of visits to a pixel is because the distribution of number of visits to a sky pixel is Poisson with parameter λ = µ r .With this strategy, the distribution of CCDs that are used to image each sky pixel is flat, and so is the position angle.Lastly, consider again the fixed sky tiling, but introduce a random rotation in all three axes that is much larger than a field of view.If this tiling is observed without dithering N times, then by the central limit theorem, the probability distribution of number of visits to each sky pixel is normal with mean As with the random strategy, the distributions of CCDs and position angles to each pixel are flat in expectation.
To compare these strategies numerically, we consider two tilings: one with no gaps and 20% double coverage (Tiling 1, similar to the OpSim tiling), and one with 10% gaps and 10% double coverage (Tiling 2).We let µ = 200, since LSST will get approximately this number of visits in each band to each sky pixel.This yields the results in Table 1.  1. List of tiling strategies.N = µ/µt is the number of times the tiling must be observed to reach an average of µ = 200 visits per sky pixel.σ is the standard deviation of the distribution of number of visits to a sky pixel.
Strategies where a fixed tiling is dithered are also of interest.However, these strategies are not as amenable to closed-form analysis, since the results depend on the shape of the focal plane.However, we have found empirically that, although dithering yields improvements over a completely fixed tiling, using a randomly rotated non-dithered tiling yields a lower σ than any dithering strategy we have tried.
As shown in Table 1, the randomly rotated tilings achieve the lowest standard deviation in number of visits.In this case, a tiling with double coverage but no gaps has a lower σ than one with some double coverage and some gaps, but the full relationship between the Ω i and σ is not monotonic.Changing Ω 0 and Ω 2 controls the tradeoff discussed in Section 4.2 between open-shutter fraction and rate of sky area covered, since for higher Ω 2 , the individual tiles are closer together, reducing slew time.However, changing Ω 0 and Ω 2 also changes σ in a somewhat complicated way, as shown here.Note that σ only depends on the Ω i and N , meaning that different tilings with the same Ω i will produce the same final σ.

SUMMARY OF TUNABLE KNOBS
For a marching army scheduler, we have described several parameters that can be tuned to control the final schedule's properties: • Exposure time (t exp ): longer exposure times increase open-shutter fraction at the expense of reducing γ, the number of visits available to each sky pixel per unit time.
• Maximum normalized airmass (ν max ): using a high ν max allows the schedule to lengthen or shorten the observing season (thereby changing γ) at the expense of SNR.
• Rolling cadence parameters (M R , D R ): changing these affects the histogram of inter-night visit gaps as shown in figure 2.
• Observing block length (t sep ): the observing block length determines the location of the peaks in the intra-night visit gaps histogram.
• Tiling parameters (Ω 0 , Ω 2 ): the spacing of the tiles controls the RMS fluctuation in final co-added depth, the open-shutter fraction, and γ as described in the previous section.
Because a marching army scheduler maximizes SNR for a given observing season length under our simplifying assumptions, and because the scheduler's parameters allow for direct control over the cadence properties, we conclude that a marching army scheduler allows the survey designers to efficiently explore the Pareto frontier of the schedule space.

IMPLEMENTATION
In practice, some of the assumptions we have made above do not hold: mainly, that there are no azimuth-or band-dependent sky brightness effects.In order to determine how a marching army scheduler performs under more realistic conditions, we implemented a deterministic marching army scheduler for LSST named ALTSched (Another LSST Telescope Scheduler).The values of the parameters for ALTSched are summarized in Table 2. Our rolling partitions are divided by declination, and the tiling we use is a solution to the Thomson charge-distribution problem for N = 3500.We run our simulation for 10 years of survey operations, including realistic weather and downtime models from the LSST simulated observatory control system.
Our algorithm differs slightly from the marching army cadence described above in the following ways: • LSST will be on an alt/az mount, requiring a zenith-avoidance region.Because we never stray far from the meridian, the region of sky that is affected by the zenith avoidance region consists of all RAs near a Dec equal to the telescope's latitude.In order to avoid long slews in azimuth, we observe this region of sky to the East of the meridian in scans that go East to West and West to East.
• In addition to carrying out a wide-fast-deep survey across most of the Southern sky, LSST plans to dedicate about 10% of its observing time to "special surveys".Although the exact nature of these surveys is not fully decided, they will include time spent observing single "deep drilling fields" to high depth in many filters.We allow for special surveys by removing 6% of the total survey time in 1-hour chunks on certain nights to demonstrate that our cadence can tolerate gaps in observing time.
Our implementation suffers from a few problems.One is that we ignore all azimuth-dependent sky brightness terms, including the moon.Therefore, some observations are taken while pointing straight at or very near the moon, rendering them useless.Another problem is that our filter cadence is underdeveloped, causing our final coadded distributions in each filter to be somewhat non-uniform across the sky.We believe that neither of these problems are insurmountable, but they currently reduce our performance on some of the metrics that we introduce below.

THE MERITS OF OBSERVING CLOSE TO THE MERIDIAN
Observing a field on the meridian ensures that the data are obtained at the minimum achievable airmass.While the merits of meridian observations are well known to experienced classical astronomers, we pause here to point out the impact of hour angle on long-term, multi-epoch surveys such as LSST.These merits include: • Observing fields at their minimum airmass provides the deepest images, overall.For the LSST passbands this is due more to the degradation of seeing with hour angle rather than atmospheric attenuation.Over the course of a ten year survey this has a substantial impact on overall performance.
• With observations taken a common airmass, differential chromatic refraction and attenuation effects are minimized, and frame subtraction artifacts are consequently suppressed.Although the LSST metric analysis framework (MAF) does not currently include a characterization of this effect, our experience (Alcock et al. 1999) suggests that this will be an important factor for suppressing false alarms in the LSST alert system.
• Slews in azimuth are expensive for LSST, due to the energy exchange needed to impart dome rotation.By keeping to the N or S meridian on alternate nights we keep azimuth slews to a minimum.
• With the LSST requirement of dual visits per field per night separated by 30-60 minutes, we can obtain these two visits at minimum airmass by in general taking the first on E of the meridian and the second one just W of the meridian.Our deterministic approach prevents the scheduler from having to chase setting fields as they sink into the West.

RESULTS
We evaluate our scheduler using the LSST Metrics Analysis Framework (MAF), and we compare our results to those of the current LSST Operations Simulator (OpSim) v3 baseline cadence, codenamed "minion 1016".We stress that we have used weather and hardware outages blackouts identical to the OpSim runs, as well as a telescope slew time calculation that matches the one used by the project.The LSST scheduler is undergoing continual development, and this comparison OpSim run has been superseded, but it was the baseline plan at the time this work was completed.Because minion 1016 and ALTSched are single simulations, there are no error bars on any of the results below.However, it is important to keep in mind that differences on the order of 1-2% between the schedulers are insignificant because of differences in the LSST software stack between when minion 1016 was run and when ALTSched was run.For example, minion 1016 does not observe on 9 nights (∼ 0.25%) even though those nights are not part of downtime, nor are they clouded out according to the current LSST stack.In addition, slew times in the minion 1016 tend to be slightly faster than the slew time calculated by the current version of OpSim, allowing minion 1016 to take about 1% more exposures than it would be able to with the current simulation software.

Survey Footprint
The minion 1016 survey has a complicated footprint with several regions.85% of all visits are designated as "universal cadence" visits, and these lie in a range of declinations from −62 degrees to +3 degrees.5% of observations are dedicated to 5 specific deep drilling fields.6% of visits lie in the "north ecliptic spur" region the follows the ecliptic as it rises above the maximum universal cadence declination of 3 degrees.2% of observations cover the south celestial pole (from dec of −62 to −90 degrees), and the remaining 2% lie in the Galactic plane (the Galactic plane is under-sampled in minion 1016).
ALTSched distinguishes between only two types of observations: universal cadence observations (94%) and deep drilling observations (6%).Deep drilling observations are taken in blocks of 1 hour, but ALTSched is agnostic as to how they are carried out (e.g. in what filters or with what rotation).Our universal cadence observations lie in a declination range of −66 degrees to +5 degrees and our deep drilling fields are the same as those used by minion 1016.Sky maps showing the total amount of survey time (including slew and readout) spent on each sky pixel are shown in Figure 3.All else being equal, one would expect a schedule that spends more survey time in a particular region to have more favorable cadence properties in that region.
Unless otherwise specified, the results below are calculated from all non-deep-drilling visits in minion 1016 and all universal cadence observations in ALTSched.

Single-Visit Properties
Both minion 1016 and ALTSched assume that for each pointing, two 15-second exposures will be carried out backto-back.The shutter in the LSST camera takes 1 second to move across the focal plane, and the camera takes 2 seconds to read out the CCDs, for a total of 4 seconds of overhead time per 30 seconds of exposure time.In this respect the timing budget of each pointing is identical for both schedulers.Table 4.Total effective time of minion 1016 and ALTSched.ALTSched makes longer slews on average than minion 1016, but the effective observing time is about the same between the two schedulers: ALTSched observes along the meridian, so the effective time per observation is higher even if fewer total observations are taken.First, we demonstrate that our scheduler achieves similar performance on the static survey as minion 1016 despite its improvements in transient sampling.

Static Survey
The "effective time" of an observation is the effective duration of the exposure had it been taken at fiducial conditions.Let m be the 5σ limiting depth achieved in the exposure, and let m f and t f be a fiducial 5σ depth and single-exposure time.Then For LSST, the fiducial 5σ depths are listed in Table 3, and the fiducial exposure time is 30 seconds.So T ef f = 30sec means that the fiducial single-visit depth was exactly reached in that exposure.
The effective time is a measure of achieved co-added depth that can be added up across different pass-bands.The total effective time of a sequence of exposures is the sum of the effective times of each exposure.The total effective times of the entire minion 1016 and ALTSched surveys are listed in Table 4.
The effective times of the two schedulers are remarkably similar, but this is only a coincidence: there are differing inefficiencies and design choices between the two schedulers whose effects on effective time happen to cancel out.

Inefficiencies
The main inefficiency in minion 1016 is that it observes at high airmass: on average 1.28 compared to ALTSched's 1.11.The observed-airmass difference between the schedulers is shown in Figure 4.There is no fundamental reason why a greedy algorithm will end up observing at high airmass, and we expect that continuing efforts on OpSim will lead to a scheduler that observes at lower airmass.
Two main inefficiencies in ALTSched are a lack of moon avoidance and non-optimized slews.As mentioned above, ALTSched makes no attempt to avoid the moon, and therefore wastes observing time when pointing very close to the moon.In addition, slew times are significantly longer in ALTSched than in minion 1016: 11.1 seconds vs 7.4 seconds.3This is because ALTSched uses a very simple algorithm to determine which field to image next: when scanning North (South), visit the fields in the current scan in strict ascending (descending) order of declination.A more intelligent partitioning of the sky into scans and a more intelligent choice of path through each scan should reduce the slew time closer to the value that minion 1016 attains.

Design Choices
A major design choice that reduces ALTSched's effective time is the choice to execute many more filter changes than minion 1016: 11.4 per night vs 4.3 per night on average.As described in Section 4.5, the optimal cadence requires (γ − 1)T night /γT sep ≈ 10 filter changes per night for ALTSched.When we run ALTSched with only two filter changes per night, we get large improvements in cadence metrics over minion 1016, but not as high as when we run ALTSched with the optimal number of changes.By adding 7 2-minute filter changes per night, we lose about 2% of our total observing time.
In sum, we expect improved versions of OpSim and of ALTSched to achieve near-optimal performance on the static survey.The main trade-off to keep in mind between static and transient surveys that can't be written off as a temporary inefficiency is the number of filter changes.However, even when we use a very large number of filter changes (11/night), the available observing time goes down by only about 2%.

Transient Science
Good performance on the static survey is relatively easy to accomplish simply by pointing up and not slewing too far.The main advantages that a marching army scheduler has over greedy algorithms is in the sampling of transients.
The average slew time in ALTSched is 11 seconds, the single-exposure time 30 seconds, the visit overhead time 4 seconds, and the area of the field of view is about 10 square degrees, so ω, the solid angle that can be imaged per second, is about 0.2 square degrees per second.The survey footprint is about 20, 000 square degrees, the season length for ν max = 1 is 152 days, the night length (dark time) is about 10 hours on average, and the rolling multiplicity in ALTSched is 2, so γ, the rate of visits to a given pixel, is about 2 per night.Most observable sky pixels are in fact visited exactly twice every night in ALTSched.Therefore, we would expect the median inter-night visit gap to a particular sky pixel to be 2 days for most of the sky, since the rolling multiplicity is 2 and the dwell time is only one day.As is shown in Figure 5, a median inter-night visit gap of 2 days is achieved over most of the sky.
Although minion 1016 is not a marching army schedule, it can be analyzed similarly because it mostly observes at an hour angle that is only a function of declination (specifically, it mostly observes at the hour angle that is at 1.3 airmasses for any given declination).If it observed exclusively at this hour angle and if it, like ALTSched, observed nearly all sky pixels exactly twice per night, minion 1016 would have the same median inter-night visit gap histogram as ALTSched.However, as can be seen in Figure 4, minion 1016 does sometimes stray from 1.3 airmasses, effectively increasing T season , and it sometimes observes fields more than twice in a night.These two effects cause minion 1016 to have longer median inter-night visit gaps, as is shown in Figure 5.
Just as important as achieving a low inter-night visit gap overall is achieving low inter-night visit gaps in each band separately, since many transients are chromatic, and constraining the light curve in each filter separately is desirable.This is much harder to do with a greedy algorithm, since without computationally expensive look-ahead, a greedy scheduler cannot naturally arrange the visits so that it can cycle through filters at each sky pixel without a prohibitive number of filter changes.This is borne out in the actual results shown in Figure 6.
Although the median inter-night gaps discussed above are important, taking the median hides the regularity of the cadence.For example, a cadence could achieve a 1-day median inter-night gap by observing 51% of all observations for a year on consecutive days, and then neglecting the field entirely for many months.Such a cadence would be ineffective at characterizing month-long transients, but would achieve an excellent median inter-night gap.To compare cadences in a more hollistic way, Figure 7 shows the actual cadences for the two schedulers for an arbitrarily chosen field in its first year.ALTSched achieves a much more regular cadence for this field than minion 1016, and it goes without observing the field for long periods of time much less often.Another advantage of ALTSched is that it achieves 30-min separated color measurements by changing filters between visit and revisit observing blocks.Building this kind of functionality into a merit function is awkward, since the merit function would have to heavily prioritize changing filters when starting to revisit a block of fields while de-prioritizing filter changes otherwise.minion 1016 gets very few simultaneous color measurements as a result.This is demonstrated in Figure 8.
Even when a greedy algorithm includes a component in the merit function that prioritizes a certain type of visit, the resulting schedule does not necessarily maximize the number of those visits.For example, OpSim's merit function temporarily boosts the merit of fields that were recently visited in order to achieve asteroid-linkage revisits (revisits about 30-60 minutes later used to track asteroids).In ALTSched, we explicitly instruct the scheduler to always revisit fields a short time later.As is shown in Figure 9, ALTSched achieves more asteroid-linkage revisits than minion 1016.
The result of the improved sampling is better science.Figure 10 shows the fraction of z=0.5 type Ia SN that are detected in at least 3 filters before peak and two times in every filter post-peak (considering only the griz filters).ALTSched will deliver several times more well-sampled SN Ia than minion 1016.Although we consider only one specific transient light curve here, the benefit of improved sampling applies to a broad range of transients with characteristic timescales ranging from hours to months.We expect that, for chromatic transients that last between a few hours and a few months, ALTSched will obtain significantly better sampling on several times as many of these transients as minion 1016.Other than ALTSched, we know of no other scheduler that has been proposed for LSST that achieves transient sampling superior to minion 1016.
One case where ALTSched might underperform is on the sampling of astronomical objects that change on very long time-scales.To investigate this, we measure the parallax accuracy that will be achieved for a star with r magnitude of 24.Because ALTSched has a shorter observing season for each field, sampling a smaller fraction of the Earth's orbit around the sun, we might worry that the uncertainty in parallax measurement would be high.However, as is shown in figure 11, we find that the parallax uncertainty is not significantly higher than it is for minion 1016, which observes fields for a larger fraction of the year.We suspect this is the case for two reasons: minion 1016 observes at higher airmass, yielding lower SNR on each visit; and minion 1016 mostly observes fields at a single hour angle, as shown in Figure 4, so the effective parallax baseline isn't actually significantly longer than ALTSched's.

CONCLUSIONS, AND FUTURE DIRECTIONS
The LSST project decided early on to implement a greedy scheduling algorithm (Delgado et al. 2006(Delgado et al. , 2014;;Delgado & Reuter 2016), and the project team has been working to improve the performance of the survey scheduler ever since.One major step forward was the development of the Metrics Analysis Framework (Jones et al. 2014), which has allowed for the direct performance comparisons made above.As of the writing of this paper, the LSST project is working to reduce the typical airmass for survey observations.Altering the merit function or the logic of the scheduler to observe in repeated blocks of exposures along the meridian has also been proposed.In effect, these efforts are trying to force a pseudo-greedy algorithm to reproduce a deterministic schedule.While this is no doubt possible, we advocate instead adopting a deterministic scheduler directly; our scheduler has easily interpretable parameters that allows survey designers to efficiently explore the Pareto frontier of schedule space.Because of ALTSched's simplicity, we can simulate a 10-year survey in under 4 minutes vs. OpSim's 50 hours.Most importantly, as this paper attempts to demonstrate, determinisitic schedules admit a closed-form analysis from scheduler parameters to global survey properties in a way that greedy-optimized schedules do not.This allows survey designers to be confident that the survey they implement maximally enables their scientific goals.
Our current implementation of a deterministic meridian-scanning scheduler is not yet mature enough to be adopted for actual survey operations.The inclusion of deep drilling fields is one outstanding task, for example.But the   ALTSched proposalId 0 and (r or g or i or z): SNLots  illustration of performance gains, especially for type Ia supernova light curves, is so dramatic that we consider it well worth further exploration of this approach.
11. ACKNOWLEDGMENTS We are thankful to numerous members of the LSST project team for extensive conversations about scheduling choices and system characteristics.Stubbs and Rothchild acknowledge support from the US Department of Energy under grant DE-SC0007881, and from Harvard University.Yoachim acknowledges funding from the LSST Corporation.

Figure 1 .
Figure 1.Visit cadence vs. length of observing season for various values of νmax (t night is set to 10 hours here).The vertical line indicating νmax = 1 is for a scheduler that scans the meridian.The curve to the right of this line represents scheduling strategies that lengthen the observing season by starting each night in the West and ending in the East.The curve to the left of the νmax = 1 line represent scheduling strategies that shorten the observing season by starting each night in the East and ending in the West.

Figure 3 .
Figure 3.Total survey time (including exposure time, shutter time, intermediate readout, and slew time) to each sky pixel in ALTSched and in minion 1016.These plots only include survey time in the universal cadence proposal for minion 1016 and for non-deep-drilling fields in ALTSched.The horizontal bands in the ALTSched sky map are an artifact of how the tiling is generated.

Figure 4 .
Figure 4. Number of visits as a function of alt/az (North is up, East is right).minion 1016 exhibits an East bias, where observations are preferentially taken at high airmass in the East.ALTSched stays close to the meridian except near azimuth, where LSST's alt/az mount prevents observations directly on the meridian.

Figure 5 .
Figure 5. Histogram of inter-night visit gaps to a sky pixel in minion 1016 and in ALTSched, for visits in any filter.minion 1016 has slightly longer observing gaps because it effectively has a longer season duration and because it observes some fields more than twice per night.

Figure 6 .
Figure 6.Per-band median inter-night visit gaps for minion 1016 and ALTSched, for the u, r, and z bands.ALTSched achieves considerably better median gaps, primarily because it executes pairs of observations taken in a single night in different filters, effectively doubling the per-band cadence in each band.

Figure 7 .
Figure 7. 5σ depth vs time for an arbitrarily chosen RA/Dec for minion 1016 (left) and ALTSched (right).These plots are typical for each scheduler for the WFD region.The two plots have approximately the same number of visits (94 for minion 1016 vs. 97 for ALTSched), but ALTSched spreads the visits more uniformely over time.

Figure 8 .
Figure 8. Histogram of the fraction of nights in which a sky pixel was visited in at least two different bands.Only nights when two or more exposures were achieved at a sky pixel are included in the calculation.By observing nearly all fields in two bands instead of only one in a given night, ALTSched doubles the observing cadence in each band.

Figure 9 .
Figure 9. Fraction of visits that have no time-adjacent visit within 60 minutes.

Figure 10 .
Figure10.Fraction of z=0.5 type Ia SN that are detected in at least 3 filters before peak and two times in every filter post-peak (considering only the griz filters).The duration of the light curve is set to 45 days, with the peak occurring 15 days after the start of the event.