The effect of night sky background on source reconstruction in atmospheric Cherenkov telescope arrays and implications for image cleaning

Gamma-ray astronomy through imaging atmospheric Cherenkov telescope (IACTs) has enabled the study of numerous astrophysical sources in the GeV-TeV domain. The presence of night sky background (NSB) in IACT images affects the reconstruction of the gamma-ray being observed and is usually dealt through image cleaning. In this work, we propose that the image cleaning approach might be adapted to the reconstruction method and NSB levels. In particular, we study the effect of NSB on source reconstruction for gamma-rays observed through IACT arrays through Monte Carlo simulations. We find that the presence of NSB can have a smoothing effect on the shower images enabling better source reconstruction under certain circumstances. We also find that there is an optimum cleaning threshold depending on the energy of the initial gamma-ray. This threshold shows very little dependence on the NSB level or core position. Based on these results, we propose and test a hybrid image cleaning method for source reconstruction. We find that this hybrid image cleaning method results in better source reconstruction when observations are made in high NSB conditions. Whereas at low NSB levels, the application of a single image cleaning threshold yields better results. We finally conclude that this implies that different image cleaning methods might be needed to best carry out source reconstructions at different NSB levels.


Introduction
The imaging atmospheric Cherenkov telescope (IACT) technique for observing gamma-rays from about a hundred GeV to a few tens of TeV has proven to be very successful. The first observations were made by Whipple in 1989 and involved a single IACT of 10 metres in diameter (Weekes et al 1989). The current generation of IACT arrays, MAGIC, HESS, VERITAS have allowed stereoscopic observations of these high energy gamma-rays and raised the number of TeV sources to above 250 (Weekes et al 2002, Lorenz and Martinez 2005, Aharonian et al 2006, Wakely and Horan 2008. Moreover, the future CTA observatory will be at least 10 times more sensitive than the current generation of IACT arrays (Actis et al 2011).
The IACT technique consists of observing the gamma-rays indirectly through the showers they produce in the atmosphere (Weekes 2005). The electrons and positrons in these showers have speeds higher than the speed of light in the atmosphere and produce Cherenkov light. An IACT obtains an image of the shower thought the Cherenkov light collected by its mirror. When observations are made through an array of IACTs, stereoscopic images of the same shower can be obtained. These shower images are subsequently analysed to identify whether the shower was produced by a gamma-ray or a hadron, as well as reconstruct the source and energy of the gamma-ray.
However, even on very dark nights, faint diffuse light from the sky also gets added to the Cherenkov images of the showers. This light, also known as the night sky background (NSB), is the result of multiple sources such as the night airglow due to chemical reactions in the upper atmosphere, zodiacal light from the scattering of sunlight from interplanetary dust, diffuse galactic light due to unresolved stars, diffuse extragalactic light, back-scattered light from the ground among others (Leinert et al 1998, Preuss et al 2002. Many of these components vary with time as well as terrestrial and celestial coordinates, making the NSB a complex phenomena to study and predict. Since the NSB gets added to the Cherenkov images observed by IACTs, these need to be cleaned to obtain a maximum signal from the shower itself and reject the maximum NSB contamination in each image. Traditionally, images are cleaned through a tail-cut method involving two thresholds and then subsequently analysed to reconstruct the gamma-ray parameters and reject hadronic background.
In this work, we show that in certain circumstances, the presence of NSB in images can have a beneficial smoothing effect. This effect can be exploited to improve the reconstruction of shower parameters, especially in the presence of high NSB levels. This has the potential to improve the sensitivity for observations near the galactic plane or in other situations where NSB levels might be higher. In addition, this might also be useful for observations in faint and moderate moonlight By faint and moderate, we mean that the sky brightness due to the moonlight is around a few to a dozen times the brightness of the NSB from a dark sky (Ahnen et al 2017, Archambault et al 2017. Observations under brighter moonlight involve hardware solutions for IACTs such as changing the gain of photomultipliers and the use of UV bandpass filters. In this work, we focus only on the regular mode of observations of IACTs. One notes that the full Moon sky can be as bright as 100 times the dark sky NSB. In general, moonlight period observations allow not only a greater duty cycle for IACT observatories, they are important to a number of scientific programs. These include the monitoring of flaring sources such as Active Galactic Nuclei (AGNs), potential observations of Gamma-ray Bursts (GRBs), multi-messenger observations of neutrino or gravitational wave sources, among others (Ahnen et al 2017, Archambault et al 2017, Büchele 2021, Hoischen et al 2022.
In this work, we focus on the source position reconstruction of the original gamma-ray and carry out a detailed study through Monte Carlo simulations for various photon energies, levels of NSB, shower core positions and image cleaning cuts. We determine an optimum image cleaning threshold and also study the cases in which the NSB can improve the reconstruction of the photon source. We also evaluate the impact on source reconstruction when images are cleaned for only part of the process and find that this type of hybrid approach can improve results in the presence of relatively high NSB levels.
This article is organised in the following way. Section 2 gives the details of the simulation and analysis methods used in this study. The simulation methods are described in section 2.1. We also described the configuration of the IACT array and NSB values we used in this section. This is followed by a description of the analysis methods in section 2.2. In particular, we describe the various approaches used for image cleaning in section 2.2.3. The results are discussed in section 3, followed by the conclusions in section 4.

Approach used
As mentioned in the introduction, the IACT technique uses the Cherenkov light emitted by the atmospheric showers produced by astrophysical gamma-rays. The images thus obtained are analysed to reconstruct the parameters of the original gamma-ray and reject hadronic background.
In this section, we give an overview of how we simulated atmospheric showers, the NSB and an IACT system for this study. We also describe the image cleaning and source reconstruction method used. In parallel, we give a brief description of the IACT technique. For a more in-depth discussion of the IACT technique, one may consult (de Naurois and Mazin 2015).

Shower and Cherenkov light simulation
High energy gamma-rays produce extensive particle showers in the atmosphere. A gamma-ray enters the atmosphere and produces an electron-positron pair. In turn, these particles interact with the nuclei of the atmosphere and produce bremsstrahlung photons which undergo further pair-production.
As a result, a cascade of high energy particles is formed. When the average energy of electrons falls below the critical energy of air (86 MeV), the number of particles in the shower starts diminishing due to ionisation. In the meantime, the charged particles in the shower have speeds greater than the speed of light in the atmosphere. As a result, they produce Cherenkov photons.
We carried out Monte Carlo simulations with the COsmic Ray SImulations for Kascade (CORSIKA version 6.020) package (Heck et al 1998) to represent this process. CORSIKA uses the Electron and Gamma Shower EGS4 package (Nelson et al 1985) to carry out detailed simulations of the interactions and transport of electrons and gamma-rays in electromagnetic showers. Minor process such as muon pair production and muon bremsstrahlung are also taken into account by CORSIKA. For the current study, we simulated 1000 showers with zenith angle 0 • at 500 GeV and 1 TeV, each.
We also used CORSIKA to simulate the emission and absorption of individual Cherenkov photons through the atmosphere. We used the U. S. Standard parameterisation of the atmosphere (Heck et al 1998) along with the standard extinction coefficients supplied with CORSIKA.
A note on the choice of energies: The 1 TeV energy was chosen since it is in the middle of the most sensitive energy range for current-day IACT arrays (see for instance Sinnis 2009 for sensitivity curves of various gamma-ray telescopes). The analysis tends to be less efficient at both lower and higher energies.
At lower energies, the analysis of images becomes less efficient since the showers are smaller. As a result, the images have fewer photo-electrons and the effect of fluctuations becomes important. At high energies (above ∼10 TeV), shower images get cut off by the field of view of the telescope which results in poorer reconstruction of gamma-parameters.
The use of 1 TeV showers allows one to limit these effects and focus on the impact of NSB levels and image cleaning approach. We also worked with 500 GeV showers for comparison's sake. One notes, that shower reconstruction and gamma-hadron discrimination remain good at 500 GeV, as well.

Telescope simulation
We used the SCITAR simulation tool which takes the Cherenkov photon output of CORSIKA and allows the simulation of an IACT array with up to 100 telescopes (Sajjad 2007). For each telescope, if a Cherenkov photon is incident of the telescope's parabolic mirror, it is reflected on the focal plane and added to the shower image if its arrival position lies within the camera radius. The position, orientation, mirror size and focal length, and camera size of each telescope can be set independently. The losses due to the mirror reflectivity and quantum efficiency of photomultipliers were taken into account through CORSIKA. The values used correspond to the re-coated mirrors of the Whipple telescope in September 1993 and measured quantum efficiency for the Hamamatsu R1398HA photomultipliers.
Configuration of the simulated IACT system: We simulated a HESS-like array with four telescopes with parabolic mirrors of diameter 12.5 metres and focal length 15 m each 1 . The altitude of observation was set at 1800 m above sea level and the telescopes were placed on the corners of a 120 m square. Each of the telescopes was pointed towards the zenith. For each telescope, the camera is placed in the focal plane and has a diameter of 1.4 m which gives a field of view of 5.4 • . The camera is further subdivided into 0.1 • × 0.1 • square pixels or photomultipliers. This is comparable with the 0.1 • diameter pixels of the MAGIC telescope and slightly smaller than 0.16 • pixel size in HESS I.

NSB simulation
The NSB is simulated for a time window of 16 nanoseconds and added to the shower image obtained by the previously described system. The time interval of 16 ns is chosen as it is the signal integration time window for the HESS observatory (Funk et al 2004, Hinton 2009). One notes that a similar time cut is not applied to the arrival times of the Cherenkov photons from the simulated gamma-ray shower on the telescope. However, simulations show that most of the Cherenkov photons from gamma-ray showers arrive in the first 16 ns window.
The simulation of the NSB uses the NSB rates measured by Benn and Ellison (1998) for the La Palma sky on moonless nights, away from the galactic and ecliptic planes and at solar minimum. Benn and Ellison (1998) found a median zenith brightness of the night sky of 22.0, 22.7, 21.9 and 21.0 mag arcsec −2 in the U, B, V, and R filters, respectively. We refer to this as the baseline NSB level in the current study.
This baseline NSB level corresponds to observations in ideal conditions. NSB rates tend to be higher closer to the galactic plane or on moonlit nights. We therefore also simulated 3 and 6 times the baseline NSB level (hereafter 3× NSB and 6× NSB). Such an approach was also used by Wood et al (2016) while studying telescope designs for the CTA observatory. One notes several other studies where NSB levels of up to 3, 4, or 4.5 the baseline levels have been considered (Preuss et al 2002, Bernlöhr et al 2013, Hassan et al 2017, Parsons and Ohm 2020. The NSB generated follows a Poissonian distribution with an average most probable value of 1.44 photo-electrons per pixel in a time window of 16 ns. This gives an NSB rate of 90 Mhz per pixel. This is comparable to the value of 85±13 MHz per pixel used by Preuss et al (2002) or 100 Mhz per pixel used by Parsons and Ohm (2020) and Actis et al (2011) in various HESS and CTA simulations. It should be noted that these values are given as an indication of the simulated NSB level. They are not used at the trigger level. A brief description of the trigger used is given in section 2.2.2.

Image analysis
The images from the simulations can be analysed to reconstruct shower parameters. We describe the approach for source reconstruction and image cleaning used in this study in the next sections.

Source reconstruction
As noted earlier, current-day IACT observatories consist of arrays of telescopes that allow the stereoscopic observation of showers. Therefore, we use a source reconstruction method that allows the use of images from all telescopes of the array simultaneously Here, we give a brief overview of this method. For more details one may consult (Sajjad 2007).
The shower images obtained from all telescopes of an IACT array can be superposed as shown in figure 1. Each image has an axis which also contains the source position. Then the common point of intersection of the axes gives the source position. This position can be calculated through a likelihood maximisation method with the following constraints. The axes of all images are required to pass through a common point, the source position. In addition, each axis is also made to pass through the centroid of the corresponding image. Finally, the transverse profile of each image is assumed to follow a Gaussian probability density function. The mathematical form of the likelihood function along with other details used are given in section appendix 'Likelihood method' . The result of this likelihood fit carried out for the shower images in figure 1 is also shown through the black lines.
One notes that this method has been tested for a HESS-I type array and gives results that are comparable with those obtained from other source reconstruction methods such as the Hillas parameter based technique (Hillas 1985, Aharonian et al 2006.

Trigger condition for source reconstruction
The source is reconstructed for showers for which at least two out of the four telescopes of the array have a total Cherenkov signal from the shower of 50 photo-electrons each.

Image cleaning
In this section, we describe the methods for image cleaning used in this work.
Here, we note that the usual approach to image cleaning is to reject as much of the NSB as possible while minimising the loss of Cherenkov signal. Following that approach, thresholds of 8-10 photo-electrons would be more than enough to reject 6× NSB levels 3 . However, here we aim to find image cleaning methods that give the best source reconstruction even if this means rejecting more of the Cherenkov signal too. In other 2 The threshold of 1 photo-electron is included because CORSIKA can also generate fractional Cherenkov photon numbers. CORSIKA works by subdividing a charged particle track in sub-steps. This method results in fractional numbers of Cherenkov photons appearing in the Cherenkov photon output files. We have kept Cherenkov such fractional photo-electrons in the simulation. The 1 photo-electron threshold cuts out the fractional photo-electrons resulting from such Cherenkov photons. The 1 photo-electron cut has no effect on the NSB since the NSB generation method only generates integral photon numbers. 3 One may refer to section 2.1.3, where we found a most probable value of 1.44 photo-electrons per pixel. words, our goal is not to improve the signal to noise ratio in the image, but to find a cleaning method that best improves the source reconstruction. One also notes that for other types of analyses (e. g. energy reconstruction, gamma-hadron discrimination instead of source reconstruction, the image cleaning approach may need to be different. In particular, one notes that gamma-hadron discrimination usually relies on the differences in image morphology between the gamma and hadron shower images. Such information tends to be lost when high cleaning thresholds are applied. Therefore, the usual approach of rejecting as little of the Cherenkov signal as possible, while maximising NSB rejection would be more appropriate. Tail-cut image cleaning: The tail-cut method is currently the standard image cleaning method for IACTs (see for instance Daum et al 1997, Aharonian et al 2006, Hupfer 2008, Bernlöhr et al 2013. While there are more sophisticated methods for image cleaning such as those developed by Lessard et al (2002), Bond et al (2003), Shayduk et al (2013), the advantage of simple threshold methods is that they are easy and quick to implement and provide reasonable cleaning of the images. In the tail-cut method, two thresholds (called thr1 and thr2 in this text) are applied to the image in the following way. All pixels with photo-electron signal above the higher threshold, thr1, are kept. In addition, pixels with signals above the lower threshold, thr2, are also kept provided they are adjacent to at least one pixel with content above thr1.
In this work, we use higher and lower thresholds of 10 and 5 photo-electrons, respectively. These values are widely used for IACT image analysis. Although, in some cases, higher and lower thresholds of 8 and 4 or 6 and 3 have also been used.
Since the tail-cut method is well-established, our purpose here is not to study its effects in detail. Rather, we use it to compare the results obtained from other image cleaning approaches.
Hybrid cleaning: We also propose and test a hybrid cleaning method. This method involves finding the centroid of each shower image after applying a single threshold cut. Once the centroids have been found, the source reconstruction itself is carried out with the complete image (Cherenkov + NSB signals) without applying any cleaning method.
In order to illustrate the rationale for this cleaning method, we look at the effect of NSB on the position of centroids. Figure 2(left plot) shows that the centroid of images tends to move closer to the camera centre at higher NSB levels. This effect is more pronounced for 500 GeV showers since they tend to contain fewer photo-electrons. At 6× NSB, the centroids for both energies are within 0.2 • of the camera centre which is comparable with the pixel size of 0.1 • . This implies that the calculation of the centroid is dominated by the NSB signal. This means that the source position can not be reliably reconstructed through these centroid positions. In other words, the source will always be reconstructed at the centre of the camera, regardless of the origin of the gamma-ray. For gamma-rays simulated with 0 • zenith angle, this will seem to result in a near-perfect source reconstruction, but this will be an artefact of the method being used.
The proposed hybrid cleaning method allows us to bypass this issue since the centroid position is calculated after the application of a threshold cut. However, the reconstruction itself is made through pre-cleaning images; this allows us to study the impact of the NSB on source reconstruction.

Smoothing effect of the NSB
We first study the smoothing effect of the NSB on shower images and its impact on source reconstruction without image cleaning. We work with simulated showers with cores right at the centre of the HESS-like four-telescope system described earlier. This allows us to study the source reconstruction in the ideal case: when the shower is observed from an equal distance (85 m) from all four telescopes. We will discuss the impact of core position later in section 3.3.4. Figure 3 shows the precision of the source reconstruction as a function of the NSB level in the absence of any cleaning. Both the 1000 and the 500 GeV showers have the poorest source reconstruction precision in the absence of NSB. The reconstruction is markedly improved for 3× NSB. At higher NSB levels, the source reconstruction deteriorates a little for 1000 GeV showers. While for the 500 GeV showers, it seems to improve a little. This slight improvement is an artefact due to the change in the centroid positions when the NSB levels are high as discussed in detail in section 2.2.3. We will come back to this point later as well.
It is useful to study these trends through the example of an individual shower at each energy. Figure 4 shows the source reconstruction for a 1000 GeV (top) and 500 GeV (bottom) shower in the presence of various NSB levels. When there is no NSB, the source reconstruction is sensitive to the presence of clumps of pixels with low signal (grey to blue pixels). This effect is especially noticeable in the 500 GeV case with the bottom image. These clumps of pixels are due to the Cherenkov light emitted from multiple scattered charged particles at the periphery of the shower. The distribution of such photo-electrons tends to be asymmetric leading to poor source reconstruction.

Pixel content
The trends discussed above are further explained through the distributions of the number of photo-electrons per pixel shown in figure 5.
The blue and orange histograms show the photo-electron content from the Cherenkov and NSB images, respectively. The Cherenkov image content shows a tall peak at the extreme left of the plot for both energies. This peak corresponds to the large number of pixels with low photo-electron content in the images (blue-grey pixels in the images shown in figure 4). As discussed in the previous section, these pixels tend to be responsible for the poor reconstruction of the source, especially when there are asymmetrical fluctuations. The rest of the distribution from the Cherenkov image in figure 5 corresponds to pixels from the core of the shower image which tends to be symmetrical.
The NSB pixel content distribution shows in the form of a Poissonian bump which roughly overlaps with low pixel content peak of the Cherenkov emission distribution. NSB photo-electrons are spread out uniformly over the camera with up to a few photo-electrons per pixel for baseline NSB. The peak of this Each plot shows the combined images obtained from the four telescopes of the simulated system. The core of both showers is at the centre of the telescope array. The reconstructed axes of the images are shown through black lines and the reconstructed source position is at the intersection of these lines. In addition, the centroid of each image is also shown through a black circle. Figure 5. The distribution of pixels with a given photo-electron signal for 1000 GeV (top and 500 GeV (bottom). The blue curves correspond to the pixel content due to the Cherenkov image. While the orange curve corresponds to pixel content due to the NSB. Three levels of NSB are shown: baseline (left), 3× NSB (centre) and 6× NSB (right). The black arrows correspond to the threshold of best precision which will be discussed in section 3.3.3. bump lies at around 1, 4 and 9 photo-electrons for baseline, 3× and 6× NSB levels, respectively. Since the NSB pixels overlap with the low pixel content peak of the Cherenkov emission distribution, it has the effect of smoothing out the fluctuations in the images and therefore tends to improve source reconstruction. However, higher levels of NSB will also overlap with the more central parts of the shower image. As a result, one expects there to be limits to this beneficial smoothing effect. One notes that while such a study is beyond the scope of this work, these plots also indicate that applying a general smoothing procedure might also be beneficial in IACT image analysis. Such a procedure might be especially beneficial when observations are made in low NSB conditions, when the NSB pixels do not completely smooth out the low pixel content peak of the Cherenkov image.

The effect of a single threshold cut
We next study the effect of applying a single threshold cut. As mentioned earlier, the threshold is varied from 0 photo-electrons (no cut) to 90 photo-electrons in order to study the effect in a general way. Figure 6 shows the precision of source reconstruction as a function of threshold cut in numbers of photo-electrons.

Threshold cut in the absence of NSB
The top plot shows the evolution of the precision in the absence of any NSB. For both energies, the precision initially improves as the cut threshold is increased. After reaching the best precision value (∼40-50 photo-electrons for 1000 GeV and ∼20 photo-electrons for 500 GeV, the precision deteriorates again. One notes that the best threshold values correspond to the position of the low content peak in figure 5. These trends are easily explained in the following way. The precision of reconstruction improves as more and more pixels due to the fluctuations in the shower are cut out when the applied threshold increases. However, when the applied threshold is too high, pixels that belong to the core of the image also get cut out, resulting in poorer source reconstruction. To further illustrate this, we have also given the images of a 1000 and 500 GeV shower with various threshold cuts in figures A1 and A2 in the appendix.

Threshold cut in the presence of NSB
In contrast, when NSB is added to the images, source reconstruction deteriorates at low threshold cuts. This deterioration continues up until a certain threshold after which the precision starts improving. The improvement continues till around the same best threshold as the one discussed in the previous section. After this, the precision starts deteriorating again. We further illustrate this through the example in figure 7 which shows the source reconstruction for a 1000 GeV shower when different cleaning thresholds are applied in the presence of 3× NSB. The second image shows that the 8 photo-electron cleaning threshold still allows a large number of pixels clumps corresponding to fluctuations and/or the NSB. This results in a poorer reconstruction compared to the first image on the right without any image cleaning. When the cleaning threshold is 50 photo-electrons (the best threshold for 1000 GeV as seen in figure 6), only the core pixels of the image are kept which results in an improvement in the reconstruction. Finally, in the last image the reconstruction deteriorates again since, at high cleaning thresholds, only a few pixels from each shower remain. One may also refer to figures A1 and A2 where we give the images of a 1000 and 500 GeV shower with various threshold cuts and NSB levels.

Optimum cut for single threshold cleaning
These results also show that for each energy, there is an optimum threshold for cleaning. For 1000 GeV showers it lies in the 40-50 photo-electron range. While for 500 GeV showers it lies around 20 photo-electrons. One notes that this threshold has a slight dependence on the NSB level too (this can be seen by comparing the plots in figure 6). However, we note that the precision of source reconstruction varies very slowly around this optimum value. Therefore, the effect of this dependence is expected to be minimal.
As discussed in the previous sections, the optimum cut not only rejects NSB pixels, but also rejects Cherenkov image pixels that correspond to the fluctuations in the image. We further illustrate this by referring back to figure 5 and the discussion in section 3.2, where we had shown the distribution of both NSB and Cherenkov pixels as a function of photo-electron content. The optimum cleaning threshold is also indicated on each plot through a black arrow. In each case, the optimum cleaning threshold removes both the NSB pixels given by the orange curve and the tall peak on the left of the Cherenkov image curve. As discussed earlier, these pixels tend to be responsible for the poor reconstruction of the source. The optimum threshold allows us to keep the pixels of the image-core allowing better source reconstruction.

Dependence on core position
Since we have only looked at showers with cores at the centre of the 4 telescope system up until now, in this section, we also study the impact of core position on the reconstruction of the source in the presence of NSB.
In figure 8, we show the precision of source reconstruction as a function of cleaning threshold for various core positions (different marker and line types). Each plot corresponds to different NSB levels. The top-right figure shows a schematic diagram of the HESS-like telescope array and the four core positions studied. The origin (0, 0) metres lies at the centre of the telescope array with telescopes at (±85, 0) and (0, ±85) metres. Then the core positions studied are (0, 0), (50, 50) metres, (100, 100) and (50, 0) metres.
The results in figure 8 show the position of the core does not have a very large effect on the source reconstruction, especially for low cleaning thresholds. At higher cleaning thresholds, the difference between the 500 GeV and 1000 GeV results becomes quite important. However, for a given energy there is not much difference due to the core position. Most importantly, the optimum cleaning threshold remains the same (i. e. around 50 and 20 photo-electrons for 1000 and 500 GeV, respectively) for all core positions. This implies that one can apply such a threshold without having to consider the effect of core positions.

The effect of hybrid cleaning and comparison with other approaches
As discussed earlier, the source reconstruction method we are using depends on the centroid positions of the images which can be changed significantly in the presence of high levels of NSB. Therefore, we also tested a hybrid cleaning method which involves calculating the centroid with a single-threshold cut, but carrying out the rest of the source reconstruction without image cleaning (see a detailed description in section 2.2.3). Given the results in the previous section, we choose to calculate the centroid position after applying the optimum image cleaning cut for each simulated energy.
We have, therefore, compared the source reconstruction results for the following image cleaning approaches.
• No cleaning: It might give good reconstruction due to the smoothing effect of NSB, but at high NSB, the position of the centroid is not calculated accurately. • Optimum threshold cleaning We apply a single threshold cut. It might result in cutting out too much of the Cherenkov image since the optimum thresholds are quite high (20 and 50 photo-electrons for 500 and 1000 GeV respectively).     Figure 9 shows the precision of source reconstruction as a function of NSB for different image cleaning approaches. For low NSB levels, the hybrid cleaning method gives similar results to the no-cleaning approach. However, at high NSB levels, the hybrid method gives clearly better results than the no-cleaning approach. This is true for 6× NSB at both energies as well as 3× NSB for 500 GeV. As discussed earlier, it also circumvents the problem due to poor centroid calculation at high NSB levels.
We also note that at high NSB, the results from the hybrid method are better than those from the tail-cut and best-threshold methods as well. This shows that a high NSB levels, the reconstruction method benefits from the smoothing effects of the NSB than it does from the application of a cut.
At lower NSB levels, the tail-cut and single threshold methods give better results than the hybrid and no-cleaning approaches. The latter two keep the asymmetrically distributed clumps of pixels that deteriorate the fit (see section 3.1). While the best threshold and tail-cut methods keep the central core of pixels in each image. The best threshold method gives the best result at low NSB levels since it only keeps the core pixels which are enough for this source reconstruction method.
We further illustrate these trends by taking the ratio of the precisions obtained from the tail-cut (left) with the precision from the hybrid method in figure 10. This shows that while the precision from the tail-cut method is better when there is no NSB, the hybrid method starts giving slightly better results at baseline NSB. For instance, for 1000 GeV, the tail-cut and hybrid methods gives a precision of 0.028 • . and the hybrid method gives a precision of 0.0024 • giving a ratio of 1.14. At higher NSB levels, the improvement with the hybrid method is more evident. For 3× NSB, the precision obtained from the tail-cut and hybrid methods is 0.025 • and 0.016 • , respectively, giving a ratio of 1.5. The ratio further improves to 2.14 at 6× NSB with precisions of 0.025 and 0.012 from the tail-cut and hybrid methods, respectively. For the 500 GeV showers, the ratio at higher NSB level is even greater. In addition, the right figure shows similar trends for the ratio of the precision from the single threshold and hybrid methods.

Summary and conclusions
We have carried out a detailed analysis to study the effect of NSB on source reconstruction through IACT array images. For this, we carried out Monte Carlo simulations of 500 and 1000 GeV showers and their Cherenkov emission through the CORSIKA package. We also simulated a four-telescope HESS-I like IACT array through the SCITAR simulation tool which uses the Cherenkov photon output of CORSIKA to obtain shower images. Three different levels of NSB (baseline, 3×baseline and 6×baseline) were also added to the shower images.
These simulations allowed us to study the effect of the NSB on source reconstruction through stereoscopic images. We used a source reconstruction method that uses the information from all telescopes simultaneously. Since the effect of NSB on parameter reconstruction is closely related to the need for image cleaning in IACT analysis, this also allowed us to study different approaches for image cleaning. Our main conclusions are the following.
• Optimum cleaning threshold: When a single threshold image cleaning approach is used, there is an optimum threshold for source reconstruction for each energy. We found this threshold to be around 20 photo-electrons for 500 GeV showers and 50 photo-electrons for 1000 GeV showers. This threshold allows us to remove both NSB and shower fluctuations pixels from the image and shows very little dependence on the NSB level and shower core position. We note that while we know the energy of the shower from simulations, that is not the case with observations. In order to apply an energy dependent optimum cleaning, it may be beneficial to consider an iterative reconstruction method. Where the gamma-ray source and energy are first reconstructed with a standard image cleaning cut such as the tail-cut method or a single threshold adapted to the energy range of observations. Once the energy has been calculated, the source position may be calculated a second time with the optimum cleaning threshold for that energy. • Smoothing effect of the NSB: Fluctuations in the shower tend to lead to asymmetrical images which deteriorate the source reconstruction. The presence of NSB can have a smoothing effect on these fluctuations and therefore improve source reconstruction. • Centroid dependent source reconstruction methods: However, if the source reconstruction method uses the centroid position of each shower image, the presence of high levels of NSB can also have a negative effect. This is because the calculation of the centroid position can be significantly affected by the presence of the NSB. • Hybrid cleaning method:These two points imply that for such source reconstruction methods, a hybrid image cleaning method might be preferable when NSB levels are greater than the baseline. This is especially true for high NSB levels such as 3× NSB and 6× NSB. With such a method, the centroids would be calculated with cleaned images while the source reconstruction would be done without cleaning. • Cleaning approaches for different NSB levels: When different approaches for image cleaning are compared, we see that at lower NSB levels, using a single threshold or tail-cut cleaning method works better since the pixels due to shower fluctuations can be cut out. Whereas at high NSB levels (above baseline NSB) a hybrid image cleaning approach yields better results, due to the smoothing effect of the NSB.

Implications and future directions
As discussed earlier, while the baseline NSB corresponds to dark sky conditions, there are many instances when NSB levels exceed these values. Preuss et al (2002)  We suggest that in such conditions, one may consider a hybrid image cleaning method to improve source reconstruction. Although one would need to carry out further studies to quantify this, one expects that this might also result in an improvement of the shower axis and energy reconstruction since both indirectly depend on the knowledge of the source position. As discussed earlier, one does not expect the gamma-hadron discrimination to improve by using high cleaning thresholds directly. Such high thresholds tend to remove all but the core image pixels in IACT images while gamma-hadron discrimination relies on the differences in the images away from the core. However, some gamma-hadron discrimination methods also make use of the knowledge of the source position (see for instance Sajjad 2007 ). In those cases, it might be beneficial to investigate the following approach. As a first step, the source would be reconstructed using the hybrid cleaning approach. This would be followed by gamma-hadron discrimination carried out with more traditional image cleaning methods but using the source position reconstructed from the hybrid cleaning approach.
We also note that the optimum cleaning threshold might vary depending on a number of additional factors not considered in this study. In future work, one would also need to explore the effect of other parameters such as the zenith angle of the source, on the optimum threshold. If the optimum threshold shows a strong dependence on the source zenith angle, the method might require the use of different threshold values for different source positions. The effect of various observational altitudes, atmosphere profiles as well as IACT array configuration (number and size of telescopes, pixel size, . . .) might also affect the optimum threshold.
Finally, we also note that the general ideas presented in this work also have the potential to be extended to a machine learning or boosted decision tree approach. For instance, a classifier algorithm could be trained to identify pixels that would be useful for source reconstruction or not. Optimum image cleaning cuts might also be determined through such methods. Such an approach could also be potentially used to identify instances where a two-step or hybrid approach image cleaning approach might be more useful than a single step one.
More generally, this work also shows that the image cleaning methods can be tailored to the analysis methods being used. In the future, one might study the impact of NSB on source reconstruction through other analysis methods. One might also study the effects of the NSB and image cleaning methods on the reconstruction of the core and energy, as well as gamma-hadron separation.

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors. where N ij is the photo-electron signal of the ith pixel from the jth telescope and its distance from the image axis is given by d ij = |(ycj−yo)(xij−xo)−(yij−yo)(xcj−xo)| √ (xc−xo) 2 +(yc−yo) 2 . The coordinates of the pixel are given by (x ij , y ij ) and the coordinates of the centroid of the image are given by (x cj , y cj ). When this function is maximised with x o and y o as free parameters, one obtains the reconstructed position as well as the reconstructed shower image axes.

Shower images
In figures A1 and A2, we give examples of individual images from 1000 and 500 GeV showers with various NSB levels and image cleaning thresholds. These examples illustrate the gradual change in the images and shower reconstruction as the cleaning threshold is increased.