key: cord-0554945-bbjdoeak
authors: Malik, Umang; Sharp, Rob; Martini, Paul; Davis, Tamara M.; Tucker, Brad E.; Yu, Zhefu; Penton, Andrew; Lewis, Geraint F.; Calcino, Josh
title: Observational window effects on multi-object Reverberation Mapping
date: 2022-03-09
journal: nan
DOI: nan
sha: 84319982817e53207b4517815647ecea6bfc95cd
doc_id: 554945
cord_uid: bbjdoeak

Contemporary reverberation mapping campaigns are employing wide-area photometric data and high-multiplex spectroscopy to efficiently monitor hundreds of active galactic nuclei (AGN). However, the interaction of the window function(s) imposed by the observation cadence with the reverberation lag and AGN variability time scales (intrinsic to each source over a range of luminosities) impact our ability to recover these fundamental physical properties. Time dilation effects due to the sample source redshift distribution introduces added complexity. We present comprehensive analysis of the implications of observational cadence, seasonal gaps and campaign baseline duration (i.e., the survey window function) for reverberation lag recovery. We find the presence of a significant seasonal gap dominates the efficacy of any given campaign strategy for lag recovery across the parameter space, particularly for those sources with lags above $sim$100 days in the observed-frame. Using the OzDES survey as a baseline, we consider the implications of this analysis for the 4MOST/TiDES campaign providing concurrent follow-up of the LSST deep-drilling fields, as well as upcoming programs. We conclude the success of such surveys will be critically limited by the seasonal visibility of some potential field choices, but show significant improvement from extending the baseline. Optimising the sample selection to fit the window function will improve survey efficacy.

The innermost regions of active galactic nuclei (AGN) provide a crucial window for the understanding of supermassive black hole (SMBH) physics and galaxy evolution. Within the local Universe, this zone can be resolved to within the gravitational sphere of influence through the use of the highest spatial resolution instruments (e.g., Gebhardt et al. 2000; Greene et al. 2010; Gebhardt et al. 2011; Kuo et al. 2011; Event Horizon Telescope Collaboration et al. 2019) . Beyond the local Universe the achievable angular resolution becomes inadequate to resolve the environments of even the most massive black hole systems. We must turn to indirect techniques to probe the structure of such systems.

Reverberation Mapping (RM) provides a powerful opportunity to probe the structure of even the most distant of these compact systems. By resolving the innermost regions of AGN in the time-domain, the variable continuum emission from an accretion disk triggers a reverberation response in the surrounding environment and this sig-★ umang.malik@anu.edu.au † rob.sharp@anu.edu.au nal can be inverted to reveal otherwise unresolvable physical detail. The broad line region (BLR) produces an emission-line spectrum in which the flux of the emission lines reverberates in response to the variability of the ionising continuum emission (Blandford & McKee 1982; Peterson & Horne 2004) . The delay due to light travel-time between the continuum emission from the compact core and the emission from the BLR, , allows the radius of the BLR ( BLR ∼ ) to be inferred. The velocity dispersion of the BLR (Δ ) can be estimated from the width of the broadened emission lines. The mass of the central black hole ( BH ) can then be measured using the virial theorem:

where is the virial coefficient; a dimensionless scale factor that accounts for the geometry, orientation, and kinematics of the BLR (Woo et al. 2015) . This technique has been used to measure the masses of over 100 SMBHs, establishing RM as the primary method of SMBH mass measurement at > 0.1 at present. The tight correlation found between the AGN luminosity and the radius of the BLR ( − relation; e.g., Bentz et al. 2009 ) has facilitated the use of indirect virial BH mass estimators using single-epoch spectroscopy (e.g., Shen et al. 2011) . Watson et al. (2011) proposed using the − relation to standardise AGN for use in cosmology. Although RM is a powerful technique with many applications, its full potential is yet to be reached. Few measurements have been made beyond relatively low redshifts ( > 0.5), preventing the − relations for the Mg and C emission lines from being better constrained. This has also meant that the subset of AGN for which most RM measurements have been made are limited to local AGN, which are not representative of the general AGN population (Shen et al. 2008; Shen 2013) . It is necessary to probe the full range of AGN luminosities to improve virial BH mass estimates for the broader population.

Time-domain photometric and spectroscopic monitoring is required to successfully conduct RM of the BLR, since the technique relies on tracing the stochastic variability of AGN. To mitigate the bias towards local AGN, recent 'industrial-scale' RM campaigns have probed new regions of the AGN luminosity-redshift parameter space, with a particular focus at high-redshifts. This includes the Australian Dark Energy Survey (OzDES) RM Program (King et al. 2015) and Sloan Digital Sky Survey RM Project (SDSS-RM, Shen et al. 2015) , which began in 2013 and 2014, respectively. Developments with multi-object spectroscopy have enabled these surveys to monitor large AGN samples. The success of lag recovery is highly dependent on the observational window, therefore the cadence, season length and baseline must be optimised to achieve the goals of the survey.

The observational window function (both photometric and spectroscopic) of any observing campaign imposes structure on the light curve sampling for a RM survey. Long baseline time-series photometry is readily obtainable through contemporary wide-field surveys (e.g., Dark Energy Survey, LSST). Obtaining concurrent spectroscopy is more challenging, due to the large multiplex of a wide field-of-view required. Facility choice is furthered restricted by the need for a large telescope aperture, essential to achieve sufficient signal-to-noise in the spectroscopic data for practical integration times. Improved facility access, and broad appeal across a range of survey applications, drives many surveys to equatorial fields to facilitate access to the survey program from both hemispheres. Observing equatorial fields immediately restricts field visibility and imposes a significant seasonal gap on the data.

The RM technique relies on recovering the lag between observed continuum and emission-line light curves using cross-correlation, which is highly dependent on how the light curves are sampled. The effect of the observational window on lag recovery has been studied in a limited capacity. White & Peterson (1994) investigated the minimum required sampling interval (i.e., cadence) for accurate recovery, and Welsh (1999) identified improvement with extending the duration of light curves. However both these studies were limited to studying the recovery for local AGN with lags less than a month long, on an individual source basis. For contemporary large-scale surveys, King et al. (2015) and Shen et al. (2015) conducted simulations of the OzDES and SDSS-RM surveys, respectively. The simulations tested some program alterations and extensions, which highlighted improvements with extending the season length and baseline, but this was not used to alter the implemented survey design. Recent RM results from these programs (Grier et al. 2017; Hoormann et al. 2019; Grier et al. 2019; Homayouni et al. 2020; Yu et al. 2021; Penton et al. 2022) show how the observational window presents challenges for recovery of these high-AGN lags, such as aliasing due to seasonal gaps. In addition, lag recovery depends on the signal-to-noise of the flux measurements and observed variability of the AGN. After com-bining these factors we see how complex lag recovery can be, and the need for a comprehensive study of how it can be improved.

There are factors in the success of RM which can be controlled, and those that can not. It is possible that the observed variability of a source is simply unfavourable (e.g., monotonic behaviour, Kaspi et al. 2007) , or that some sources do not reverberate ideally (e.g., BLR 'holiday', Goad et al. 2016; Dehghanian et al. 2019) . As more RM surveys are completed, a better understanding of such phenomena can be achieved. The signal-to-noise of spectroscopic measurements, particularly at high redshifts, also presents a challenge. Presently RM surveys must focus on the factors that can be controlled: the survey window. In this work, we present a detailed investigation of each component of the global window function which describes the survey sampling. In Section 2 we describe the AGN light curve model, window functions, and lag recovery methods. Section 3 presents simulations designed to probe the 3-dimensional parameter space of cadence, season length and baseline to see how the accuracy and precision of the lag recovery changes. We then extend this to a subsample of simulated AGN with a range of expected lags before conducting high-resolution simulations for a grid of AGN probing a wide region of redshift-luminosity parameter space. We simulate the efficacy of a future planned 'industrial-scale' RM survey and potential extensions that could improve its success. We discuss the implications and caveats of our work in Section 4 and summarise our key results in Section 5.

The DES deep survey targeted legacy supernovae fields across a wide range in declination, with all regions followed up with multi-object spectroscopy conducted by OzDES. We model the resulting observational window function for RM using this contemporary survey as an initial baseline. The OzDES window function is largely representative of the window functions of current and planned large-scale RM surveys (e.g., Shen et al. 2015; Kollmeier et al. 2017; Brandt et al. 2018; Swann et al. 2019; The MSE Science Team et al. 2019 ). The custom window functions we apply probe the window function parameter space by modelling surveys that could be performed by contemporary facilities. To conduct our simulations, we created mock AGN light curves to which we could apply our custom window functions. In this section we introduce the OzDES survey, used as our base model survey, and describe how we create mock light curves. We then explain the custom window functions, and the lag recovery methods used for this work. Expected lag (observed-frame, days) multi-object spectroscopy for the SNe program was provided by the OzDES program. OzDES performed the spectroscopic follow-up of the deep field with the 2dF multi-object fibre positioning system and the AAOmega spectrograph (Sharp et al. 2006 ) on the 3.9m Anglo-Australian Telescope (AAT), with a monthly cadence over the same 6-year baseline (Yuan et al. 2015; Childress et al. 2017; Lidman et al. 2020) . The primary scientific goals of OzDES were to follow-up supernovae discovered by DES, by using their spectra to determine host-galaxy redshifts and for classification, and to conduct RM of AGN.

As part of the OzDES RM program, 771 AGN have been monitored, at redshifts 0.12 < < 4.5 with apparent magnitudes 17.2 < AB < 22.3. The distributions of the redshifts and magnitudes of these targets are shown in Figure 1 , which also shows the expected observed-frame lags for our sample. The sample was screened for sources that fell within DECam chip gaps, or with emission lines compromised by the transition between the red and blue arms of AAOmega (see King et al. 2015; Tie et al. 2017 , for more details). The SDSS-RM sample follows a similar distribution (Shen et al. 2015) .

We continued photometric observations with DECam in 2019, and spectroscopic observations with the AAT in 2020. Due to COVID-19 disruptions closing the observatory in 2020, we continued the photometric observations with DECam in 2021. This will partially bridge the gap to the Legacy Survey of Space and Time (LSST) conducted with the Vera Rubin Observatory, which will also be observing the ELIAS, XMM-LSS and Chandra DFS, as three of its four deep-drilling fields. This will also provide continuity to the spectroscopic follow-up of the LSST deep-drilling fields by SDSS-V (Kollmeier et al. 2017 ) and the Time-Domain Extragalactic Survey (TiDES; Swann et al. 2019 ), which we discuss in §3.3.

The damped random walk (DRW) model developed by Kelly et al. (2009) was used to create simulated continuum light curves. It has been shown to provide an adequate representation of the intrinsically stochastic behaviour exhibited by AGN, as seen in their continuum light curves (Kozłowski et al. 2010; MacLeod et al. 2010 ). The DRW model has been widely used for RM simulations (Shen et al. 2015; King et al. 2015; Li et al. 2019; Penton et al. 2022) . Although AGN variability has been observed to deviate from the DRW on short time-scales (Mushotzky et al. 2011; Kasliwal et al. 2015; Smith et al. 2018) , this has been found to not affect lag measurements (Yu et al. 2020) .

Following Penton et al. (2022) , the required inputs are the source redshift and apparent magnitudes, from which each of the model parameters are derived. For each simulated source, the expected restframe lag was estimated using the − relations for the H (0< <0.62) (Bentz et al. 2013) , Mg (0.62< <1.78) (Trakhtenbrot & Netzer 2012) and C ( >1.78) (Hoormann et al. 2019 ) emission lines, and multiplied by (1 + ) to obtain the input observed-frame lag, input . Unless otherwise stated, the light curves, input lags and timescales we refer to are in the observed-frame.

The simulated AGN light curves were generated with daily cadence of the photometric and spectroscopic light curves, then appropriately down-sampled by the window function. The photometric and spectroscopic fluxes have 3% and 10% observational uncertainties applied, respectively, approximately representative of the DES and OzDES survey data (Dark Energy Survey Collaboration et al. 2016; Lidman et al. 2020 ).

The dominant observational effect on lag recovery is expected to be the survey window function. For our simulations we apply custom window functions to model observed light curves. We apply a custom window function that samples the original simulated light curves with a desired baseline, season length, and photometric or spectroscopic cadence (for the continuum and emission line light curves, respectively). To be representative of the uneven sampling of surveys, we permit a scatter range for the cadence (e.g., 14 ± 4 days), drawing the dates from a uniform distribution.

One of the most widely used lag recovery techniques is the interpolated cross-correlation function (ICCF, Gaskell & Peterson 1987) . In order to focus on the effects of the window function over the critical variables, we chose not to use the more complex JAVELIN (Zu et al. 2011) We use the ICCF method as described by Peterson et al. (1998) , and summarised here. The recovered lag and its uncertainty are measured using the flux randomisation and random subset sampling (FR/RSS) method. Since the ICCF method does not directly use the flux measurement uncertainties when computing the cross correlation function (CCF), flux randomisation accounts for the uncertainties by shifting the original flux values by Gaussian deviates based on the standard deviation on the fluxes. The random subset sampling accounts for the effect of any one flux measurement on the lag measurements, by randomly selecting light curve points with replacement (resulting in a sample ∼37% smaller than the original set). Both the continuum and emission-line light curve are altered by this FR/RSS process. A CCF is computed with the continuum light curve and the linearly interpolated emission-line light curve, which are cross correlated as a function of time lag. The light curve is linearly interpolated at a specified time lag spacing. Another CCF is computed using the linearly interpolated continuum light curve and the emission-line light curve. The final CCF is the average of these two CCF's. The lag is the time at which the peak of the final CCF occurs.

A cross-correlation peak distribution (CCPD) is obtained with the lags found from 1000 Monte Carlo realisations of FR/RSS. The recovered lag is taken to be the median of the CCPD, and the uncertainties are the upper and lower limits which contain 68.27% of the realisations (equivalent to 1 uncertainties of a normal distribution). An example is shown in Figure 2 . The centroid is also commonly used, as recommended by Peterson & Horne (2004) , however in this work the peak lag was found to be more accurate and consistent than the centroid lag, when dealing with various CCF shapes (which arise due to varied light curve sampling). For example, as shown in Figure 2 , the centroid was often unable to be computed due to the broad, flat shape of the CCF. It can also be seen that the centroid lag tends toward the centre of the lag prior range.

When computing the CCF, we use the PyCCF python code (Sun et al. 2018) , setting the interpolation grid spacing to be 3 days. We set the max threshold to be 0, and disable the automatic use of the -value test to set the max threshold.

There are three critical components to the observational window function: observation cadence, the length of the seasonal gap, and the survey duration. In principle the cadence of photometric and spectroscopic data can be varied independently. However initial experimentation indicated that, under the assumption that photometric data is highly likely to be available with greater frequency than the associated spectroscopy, the cadence of said photometry has limited bearing on lag recovery. We therefore proceed with a simplified model for observational cadence, varying photometric and spectroscopic observations in lockstep.

To probe the observational window parameter space, we conducted three sets of simulations: the first keeping the baseline of the survey fixed (reflecting the 6 year OzDES survey) while varying the cadence and season length, the second keeping the cadence constant (weekly photometry and monthly spectroscopy) while varying the baseline and season length, and the third keeping the season length constant (5 months) while varying the cadence and baseline.

To model various survey cadences the light curves were downsampled sequentially over 10 levels, ranging from daily photometry and spectroscopy, to weekly photometry and monthly spectroscopy. This last down-sampling level approximately models the DES and OzDES observational cadence. More densely sampled photometry was chosen as it is more readily obtainable and has been shown to improve lag recovery (Shen et al. 2015) . The season length was extended by one month at a time, starting from 3 months up to the maximum possible season length of 12 months. The baseline was extended successively by one year, from 1 year to 10 years. Each window function was applied to 100 pairs of light-curve realisations created for a source.

To best demonstrate the interactions between the lag and various window function timescales, we simulated sources with input ∼1 year, ∼5 months, ∼1.5 years, and ∼2 months.

Our goal is to determine, across the observational parameter space, the fractional recovery of accurate lag measurements with acceptable precision. For each simulated input lag value, we consider the fractional recovery rate for each survey window function realisation. We require a quality lag recovery to be within 10% (and 100 days) of input , and ≥ 4 times the statistical uncertainty on the lag measurement provided by ICCF. Following King et al. (2015) , we use a lag prior of 0 to 3 expected . Figure 3 shows the fraction of quality recoveries for a source with a lag of input = 356 days, through the window function parameter space. For this source with a lag close to one year, there is extensive overlap between the photometry and spectroscopy, albeit from campaign seasons in different years, as shown in Figure 4 . This lag can be recovered with short seasons and low cadence, with a sufficiently long baseline. Even subtle variability features seen within short seasons will be directly observed in the emission-line light curve. The first results from OzDES (Hoormann et al. 2019 ) were for C sources with observed lags ≈ 1 year, as they could be reliably recovered when only the first four years of data was recorded.

We repeat this for sources with various input lags. Figure 5 shows the fraction of quality recoveries for a selection of sources with various observed-frame lags, through the window function parameter space. For a source with an input lag of input = 142 days, there is significant improvement in the fraction of quality recoveries as the seasonal gap is reduced. As this lag is almost 5 months in the observed frame, it is challenging to recover when the season length is equal to or shorter than the lag. The direct response of the broad-line region to the variability of the ionising continuum occurs mainly outside of the 5-month observing season, as shown in Figure 6 . Recovery of a reverberation signal from such a source would therefore rely on higher order correlation timescales, such as the variability relaxation time, as presented by (e.g., Grier et al. 2019; Homayouni et al. 2020; Yu et al. 2021) .

We see similar trends for the source with input = 500 days (Figure 5) . This is expected as this lag coincides with the second seasonal gap. However, the second order effect due to the damping timescale results in the lag being moderately recoverable even with seasonal gaps present, as information is present in the smoother long-term variations seen on timescales longer than the lag itself.

For a source with a lag of input = 71 days (and with a fixed survey baseline of 6 years) there is significant overlap between the photometric and spectroscopic window function with each campaign season. Unsurprisingly, we find increasing the observational cadence improves the precision of lag recovery, although it has limited impact on accuracy which requires closing the seasonal gap (maximising the overlap between photometry and spectroscopy with each campaign season) to achieve significant improvement. The light-echo in the emission-line light curve for this source begins around 71 days, therefore, as with the 142-day lag, we require a long enough season length to provide sufficient overlap to allow lag recovery. We see the relatively short lag can still be well recovered with the fixed cadence, if there is a sufficiently long baseline and extended season length. As the baseline or season length increase, the chance of seeing favourable structure to variability (i.e., significant large flux excursions from the mean on short timescales) increases.

We extend these simulations to a sub-sample of OzDES sources. These sources were selected based on their expected lags, to cover the range of expected observed-frame lags for this sample. As most of our sources have expected observed-frame lags up to 1000 days ( Figure 1 ), we set this as the upper limit for these simulations. We apply window functions that vary one component at a time, using a constant number of data points. This will allow us to directly compare how each component of the window function affects the recovery of a particular lag. We added approximately 90 photometric epochs and 20 spectroscopic epochs to the base OzDES window function, by independently changing one component of the original window function. In addition to applying the base OzDES window function, we conducted three sets of simulations with the following window function changes: extending the season to 9 months, extending the baseline to 10 years, and increasing the cadence to 4-day photometry and 18-day spectroscopy. Lags were recovered for each of 100 pairs of light curve realisations created for each simulated source. The top panel of Figure 7 shows the median and 1 scatter of the recovered lags for the simulations in which only the season was extended, compared to the results with original OzDES window function. The same is shown in the middle panel of Figure 7 , for the case with the baseline extended, and in the bottom panel for case with increased cadence. The lags reported here have been recovered from ICCF directly, without any recovery criteria applied.

When sampled by the OzDES window function we see the results expected from the trends observed in §3.1. There is broad scatter in the recovered lags where the input lag is + 1/2 years long, where is an integer number of years (i.e., coincident with seasonal gaps). Input lags coinciding with the annual seasonal overlap are recovered accurately, regardless of the window function applied (at the observational data densities considered). The scatter in the lags is significantly reduced across all input lags when the season length is extended. The extension of the baseline reduces the scatter to a greater extent than increasing the cadence. Although the lags are recovered more accurately in these latter cases, a considerable number of lags remain outside the limits for a correct recovery. In these simulations we clearly see the advantage of extending the season length for a largescale survey hoping to recover lags ranging from tens to hundreds of days. 

There is little consistency within the literature for the priors placed on lag recovery in RM analyses. Whilst there is no objective way to choose a prior, they need to be as wide as possible as to avoid confirmation bias, but also not so wide as to lose the lag in noise or aliasing signals. We investigate the effect of changing the lag prior on lag recovery. Our initial simulations used 0 → 3 times the expected lag expected (which is the same as input for our simulations), with a maximum allowed prior of 1300 days (about two-thirds of a 6-year baseline). We repeat the lag estimation process using the following lag priors, in order of most broad to most narrow:

• ± 1300 days • −3 exp to 3 exp , with a lower bound of -1300 and upper bound of 1300 days. τ i n p u t ± 2 τ i n p u t ± 1 . 5 τ i n p u t ± τ i n p u t ± 0 . 7 5 τ i n p u t ± 0 . 5 τ i n p u t ± 0 . 2 5 τ i n p u t τ input = 511 days

Prior narrowing --> • exp ± 2 exp , with a lower bound of 0 and upper bound of 1300 days.

• exp ± 1.5 exp , lower bound of 0, upper bound of 1300 days.

• exp ± exp , lower bound of 0, upper bound of 1300 days.

• exp ± 0.75 exp , lower bound of 0, upper bound of 1300 days.

• exp ± 0.5 exp , lower bound of 0, upper bound of 1300 days.

• exp ± 0.25 exp , lower bound of 0, upper bound of 1300 days. Figure 8 shows the fraction of quality recoveries with the above priors, for four sources with various input lags. The sources chosen here have lags similar to the sources used for the simulations in Figure 5 . Although a prior of ±0.75 input maximises the recovered fraction in most cases, this is very narrow and would result in confirmation bias in RM studies. Overall, when a window function is applied that efficiently recovers a particular lag, it is recovered well and the recovery is mostly unaffected by the prior. This becomes more complex when a lag can be recovered with a certain window function, but less effectively with other window functions. In this case, the recovered fraction can vary greatly depending on the prior.

We now consider the extended parameter space of observations required for cosmological studies. We generate a simulated grid of AGN (similar to Shen et al. 2015; King et al. 2015; Li et al. 2019) , with 16 < (AB) < 23 over the redshift interval 0 < < 5. The redshift-luminosity range slightly exceeds the range covered by the OzDES and SDSS-RM samples but is inline with what is expected from future generation surveys (e.g., Kollmeier et al. 2017; Brandt et al. 2018; Swann et al. 2019; The MSE Science Team et al. 2019 ). There are also redshift regions where more than one emission line is visible. These regions are particularly valuable as multiple lags can be recovered independently for one source, allowing the ionisation stratification of the BLR to be studied. Future surveys conducted with the Maunakea Spectroscopic Explorer (MSE, The MSE Science Team et al. 2019) will have access to both the optical and near-infrared bands ( and ). This will allow observation of the H and Mg lines at higher redshifts due to the extended wavelength coverage, and increase the range over which two or more emission lines are visible simultaneously for a source. We conduct separate simulations for each emission-line, over the redshift range of the grid for which the emission-line will be visible with future optical and near-IR surveys. We simulate a pair of light curves for each of 100 AGN per bin, with the expected lag for the relevant emission line.

We apply a representative window function of the upcoming LSST and TiDES RM survey to each AGN across the grid. The 5-year baseline planned for 4MOST/TiDES survey (Swann et al. 2019) will provide 14 ± 4 days spectroscopic sampling, combined with the VRO/LSST RM program (Brandt et al. 2018 ) which proposes to deliver a photometric cadence of ∼2 days for each of the four deepdrilling fields. LSST will be conducted using the Rubin Observatory at Cerro Pachón (-30.2 • ), and TiDES by the VISTA telescope at the Paranal Observatory (-24.6 • ). Figure 9 shows that the Chandra DFS and ELIAS fields, with the lowest declinations, are visible for the longest maximum season of ∼9 months. The equatorial COSMOS and XMM-LSS fields are visible for a maximum of 7.5 and 8 months, respectively. Using the shortest season, we define the window function of this base survey with a 5-year baseline, 7-month season length, photometric cadence of 2 days and spectroscopic cadence of 14 ± 4 days.

As previously, we define a quality lag recovery to be ≥ 4 times the statistical uncertainty of the lag measurement provided by ICCF, and within 10% and 100 days of the input lag. The fraction of quality lag recoveries, for this representative LSST/TiDES window function, are shown in Figure 10 , as a function of apparent magnitude and redshift, bounding the probable observable survey parameter space.

We see a heavily banded structure in this visibility map with high recovery fractions only achieved for sources with observed lags that are out of phase with the seasonal gaps in the window function. Our grid has a resolution of 0.25 in (AB) and 0.05 in , which is significantly higher than previous simulations. This was necessary to resolve the high-frequency structure. The highest fraction of quality recoveries are for lags in the region with expected lags between 14 and 180 days, i.e., when the lag occurs within a single campaign season. This lower limit is set by the spectroscopic cadence of 14 day, and the upper bound is the length of the season, which is ∼210 days. Beyond this timescale, it becomes challenging to reliably recover lags of longer duration until we approach lags on order of one year, at which point the photometric and spectroscopic observations come back into phase. The pattern is repeated for longer observed-frame lag times at 2 and 3 years, although these bands are fainter due to the limited baseline.

We have shown that extending the season length (closing the seasonal gaps in the window function) generates the greatest improvement in lag recovery rate, followed by extending the survey baseline ( §3.2). We repeated the simulations with three season lengths (5, 7 and 9 months) and two baselines (5 and 10 years). In Figure 11 and Figure 12 , the panels show the results of these simulations with the various window functions, for H and C lags. We do not perform simulations for Mg lags separately as there are few measurements constraining this relation (Trakhtenbrot & Netzer 2012) , with a large scatter in recent results (Homayouni et al. 2020) . Although there are few sources with lags measured with both H and Mg (e.g. Clavel Table 1 . The percentage of quality recoveries for a mock target sample from the visibility maps of each window function, done for each emission line (see Figure 11 and Figure 12 ). . 1991; Metzroth et al. 2006; Homayouni et al. 2020) , the lags are found to be similar. Therefore we extend the range of the H simulations to also cover the range of visibility of Mg with future near-IR facilities. By extending the baseline to 10 years, the survey becomes much more sensitive to lags around 1 and 2 years long. However, the seasonal gaps are still limiting a significant region of this parameter space. We see the gap in quality recoveries around ∼180 days effectively disappear once we extend the season to 9 months. Finally, when both the baseline and season are extended, we see quality lag recoveries for almost all AGN across the parameter space, with the fraction decreasing only for lags longer than 2-3 years. For these longest lags, the lag recovery could benefit from revising/reducing the lag prior ( §3.2.1) or extending the baseline further.

We defined a mock target sample using the redshifts and apparent -band magnitudes of the OzDES RM sample (see Figure 1 ). Each mock target was randomly selected from the respective redshiftluminosity bin on the simulation grid. We calculated the percentage of sources with a quality lag recovery. Table 1 shows the mean percentage for each window function simulation.

The visibility maps created from our simulations show banding structure in AGN luminosity-redshift space for lag recovery efficacy. This identifies regions of this parameter space where reverberation mapping is more likely to be successful, and use this to inform survey planning. This would require understanding the biases introduced by the specific window function, as certain lags, and therefore certain luminosity-redshift bins, will be over-represented when constraining new − relations. We defined a 'visible' mock sample comprised of sources selected from a redshift-luminosity bin with an overall fraction of quality recoveries ≥ 0.5. The − relation was constrained using the full mock sample, and again using only the 'visible' mock sample, using the Bivariate Correlated Errors and Intrinsic Scatter (BCES) code from Nemmen et al. (2012) . This is shown for each window function of the C simulations in Figure 13 . The scatter in the recovered lags from the mock samples is reduced significantly as the season length and baseline are extended. The dynamic range in luminosity of the 'visible' mock sample is limited when the season length and baseline are short. As a result of statistical leverage, the slope of the recovered − relation is shallower. Although the scatter in the recovered lags is greater for the full mock sample, the slope recovered for the − relation is consistent within the fitting error. Fraction of quality recoveries Figure 10 . The fraction of quality recoveries across the AGN redshift-luminosity space, with light curves sampled with a representative LSST and 4MOST survey window function. The time-lag contours indicated by the coloured lines show constant expected observed-frame lag through this parameter space, calculated using published − relations for each emission line (Bentz et al. 2013; Trakhtenbrot & Netzer 2012; Hoormann et al. 2019) . The upper limit on the prior is 1500 days, sources with input lags above this have been masked. For reference, the black vertical lines indicate the redshift at which the respective emission lines are visible at a wavelength of 1 micron (10000 Å); representing the transition between visible light and infrared for spectroscopic capabilities.

It is evident that the window function has a significant effect on the Reverberation Mapping lag recovery over a range of AGN intrinsic and observed properties.

• Cadence: Improvement in the precision of lag recovery gained by increasing the observational cadence is an expected result, as previous studies have also shown (White & Peterson 1994; Shen et al. 2015; King et al. 2015) . This is particularly pronounced for sources with shorter lag times (for which low observational cadence is not readily compensated through simply extending the survey baseline). However, for sources with longer observational lags, the baseline is critical even if only modestly sampled. This naturally suggests a multi-tiered observational strategy when resources are restricted (typically the case for spectroscopy), similar to that used by SDSS-RM (Shen et al. 2015) .

• Baseline: The effect of the baseline on the ability to recover long lags had been predicted (Shen et al. 2015; King et al. 2015) , although current surveys are only now reaching sufficient baselines to confirm such assumptions (Shen et al. 2019; Hoormann et al. 2019) .

• Season length: While the limits imposed by seasonal gaps had been recognised previously (King et al. 2015) , their impact had not been fully appreciated until recent industrial-scale surveys delivered their early results, which required implementing alias removal techniques (Grier et al. 2019; Homayouni et al. 2020 ).

This has motivated our work to investigate the interplay of each of these observational window function components in more detail. A large portion of sources observed by current industrial-scale RM surveys have expected observed-frame lags in the sub-year range, as seen in Figure 1 , leaving them vulnerable to the impact of seasonal gaps. This is a cause of concern for current and future surveys, which have or are expected to have significant seasonal gaps, as it limits the number of lags which can be reliably recovered. 

From the population simulations conducted with the LSST/TiDES window function, we were able to predict the efficacy of this RM survey. The TiDES sample comprises 700 AGN at < 21 and < 2.5 (Swann et al. 2019) . Focusing on this region of the parameter space, we see the season length imposes critical limitations on the areas where the survey would be able to successfully recover quality lags. We demonstrated the vast improvement apparent with altered window functions where the baseline or seasons were extended. As LSST is planned to operate for 10 years, it is feasible to extend the TiDES program to cover the same baseline. However, the field visibility can not be changed, therefore the efficacy in certain redshift-luminosity ranges will be restricted for the equatorial fields. Our simulations show that significant seasonal gaps may not cause as much issue for sources with long relaxation times. However, this would require long survey baselines, of at least 6 years; beyond what is currently planned.

• Recovering the − relation: Optimising surveys ultimately depends on the science goals. For RM studies this would be to not only recover as many lags as possible for a sample, but also ensure measurements are made across a range of luminosity, to anchor both Multi-line quality lag recovery probability Redshift Figure 14 . Probability of quality lag recoveries for both emission lines, for each window function. This was generated by multiplying the visibility maps shown in Figure 11 and Figure 12 . The vertical black lines indicate the maximum redshift range over which either H or Mg can be seen at visible wavelengths (0 < 2.65). With infrared capabilities, H can be observed up to ∼2.5, and Mg to = 5 (as shown in Figure 10 ); therefore all three lines can be seen from 1.5 < < 2.5. ends of the − relationship, for each line. The lower luminosity ends of the relations for Mg and C still lack measurements, as the expected observed-frame lags are short. The window functions of recent surveys have not been optimised to recover these short lags. As shown in Figure 13 , the baseline length is important for this, particularly when seasonal gaps are present.

• Multi-line measurements: Other science goals include recovering lags for multiple emission lines, to study the ionisation stratification of the BLR (Dietrich et al. 1993; Peterson & Horne 2004) . Considering the banding structure we see in the visibility maps when there are significant seasonal gaps, we need to see if there are regions of the parameter space that have high likelihood of lag recovery while two (or more) lines are visible. As H and Mg have similar expected lags, the bands of high efficacy overlap for redshift regions where both lines can be seen. However for Mg and C , although both lines can be seen over a wider range in redshift (Figure 10) , there is less overlap between the bands of high efficacy on the visibility maps. The visibility map showing the probability of quality lag recoveries with both lines is shown in Figure 14 , for each window function. For Mg and C , the number of redshift-luminosity bins with a high probability of providing multi-line measurements greatly increases with near-IR coverage.

We compare the results of these simulations to recent lag recoveries from the pioneering 'industrial-scale' OzDES and SDSS-RM surveys ( Homayouni et al. 2020; Yu et al. 2021 ). In Figure 15 , the sources with published high-quality lags from OzDES and SDSS-RM are overlaid on the visibility maps for the respective emission lines. We define a 'visible' sample, as previously done for the mock sample, comprising sources which lay on redshift-luminosity bins with a fraction of quality recoveries ≥0.5. The − relations recovered using the full and 'visible' published samples are shown in Figure 16 . A significantly shallower slope for the Mg − relation was found by Homayouni et al. (2020) , which they suggest could be attributed to the larger intrinsic scatter in the recovered lags. The 'visible' samples do not have less scatter, although the number of data points are limited. We note that all the published lags were recovered using JAVELIN (Zu et al. 2011) , which could cause some discrepancy with our predictions, which were performed using ICCF. There are many Mg and C lags which lie in the seasonal gaps. These sources have been observed to vary smoothly on multi-year timescales, allowing the lag to be recovered with little to no overlap between the continuum and emission-line light curves. Our use of the DRW light curve model may not appropriately represent this variability over longer timescales. Although the seasonal gap lag recoveries are judged to be reliable, we should expect to see a similar number of recoveries that do lie on regions of high 'visibility', which is not the case for the current sample of Mg and C lags. It is possible that this issue will be alleviated with future results from the longer baseline data from these surveys, which should also help to confirm seasonal gap lags. If the issue remains, it confirms the need for future surveys to ensure the seasonal gaps are reduced as much as possible, to see if this is the cause for the lack of high quality lag measurements from multi-object RM surveys in the regions we predict to be 'visible'. (Grier et al. 2017; Hoormann et al. 2019; Grier et al. 2019; Homayouni et al. 2020; Yu et al. 2021 ) overlaid on the respective visibility map for each emission line. The 'visible' subsample, lying on bins with a fraction of quality recoveries ≥0.5, is in red.

We acknowledge our simulations are idealised, and the following effects should be appreciated. The broad-line region emission response has been observed to demonstrate non-reverberating behaviour, such as the BLR 'holiday' (Goad et al. 2016; Dehghanian et al. 2019) . The light curve model we used does not include these deviations, as they are still poorly understood. Recently Stone et al. (2022) have found that the DRW damping timescale, , constrained with 20-year light curves, continues to increase as the baseline of the light curves increases. They also observe smooth, long-term variations in some sources, leading them to suggest that the current DRW model may be insufficient for these sources. The significant seasonal gaps in the window function of the data may also preclude proper convergence of the DRW model parameters. A large sample of emission-line light curves is required to better model reverberation in AGN light curves in future work.

We present results of comprehensive simulations which demonstrate the impact of each component of the global observational window function on reverberation mapping lag recovery. From individual sources to a population of AGN, we have seen the extent of the restrictions imposed by significant seasonal gaps in the data. Reverberation mapping surveys should prioritise observing fields with longer seasonal visibility, not only to improve lag recovery for more sources, but also to mitigate signal aliasing. We note that seasonal gaps may not affect lag recovery as much for sources with long variability timescales, but long baseline monitoring is required in such cases.

There are complex interactions between the window function, emission-line lag and the relaxation time of the variability, which are further complicated by the redshift time dilation effects. As industrial-scale RM surveys target large AGN samples over wide redshift and luminosity distributions, it will be difficult to find a 'one-size-fits-all' solution to survey design. Careful optimisation of targets lists of sources to be monitored will likely deliver significant survey success rates. Future surveys should consider a multi-tiered observing strategy to ensure the window function is optimised for a more diverse range of sources. Extending the wavelength range of spectrographs to the near-IR will be a critical requirement for joint emission-line lag analyses. 

Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series. p

PyCCF: Python Cross Correlation Function for reverberation mapping studies

The Messenger

We acknowledge parts of this research was carried out on the traditional lands of the Ngunnawal and Ngambri peoples. We pay our respects to their elders past, present, and emerging.This analysis used NumPy (Harris et al. 2020) , Astropy (Astropy Collaboration et al. 2013Collaboration et al. , 2018, SciPy (Virtanen et al. 2020) , and thorsky (https://github.com/jrthorstensen/ thorsky). Plots were made using matplotlib (Hunter 2007) and seaborn (Waskom 2021) . This work has made use of the SAO/NASA Astrophysics Data System Bibliographic Services.

The simulation data will be made available upon reasonable request to the corresponding author.