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DefinitionWe present here a new statistical approach to measure and characterize light pollution. The objective is to define an indicator which is not limited to clear sky measurements and does not require a precise calibration of a photometer. The key attributes of the indicator are the following:

It requires the automated acquisition of a large number of zenithal NSB measures when the Sun is below (-18^{circ }) and the Moon below (-5^{circ });

The acquisitions must at least cover a period of 6 months in order to record a wide range of possible weather conditions from perfectly clear to totally overcast skies. The objective is to obtain a significant sample of every type of cloud conditions (e.g. cloud density and ceiling altitude) as well as a good characterization of the average clear sky ;

It is based on the analysis of the zenithal NSB measure dispersion which is directly linked to the level of light pollution a site experiences.

As presented above in Fig. 2, the NSB density histograms, which are assembled from a large number of NSB measures, display a higher density zone which denotes a characteristic clear sky level that we name nominal NSB in the scope of the indicator calculation. On both sides of the clear sky level (above and below), NSB measures are distributed in a way that reflect the zenithal night sky luminance in cloudy conditions: NSB measures above the clear sky level mean that the light pollution is amplified by clouds while those below the clear sky level indicate a darker environment where clouds mask light pollution from distant sources as well as natural light sources. The calculation of the indicator is based on the evaluation of the NSB measure dispersion on both sides of the nominal NSB (i.e. characteristic clear sky level). Since there can have strong variations of artificial light emitted into the environment at the beginning and end of the night (decrease then increase of human activity, extinction of public lighting, etc.), the range of NSB measures retained for calculating the indicator is restricted to a portion in the middle of the night, typically 2 h.Figure 8 shows a typical NSB density histogram for a site which is quite severely impacted by light pollution. It covers a 2 h time range between 23:00 UTC and 01:00 UTC and one can easily see that the zone above the nominal NSB is much higher and denser than the one below, i.e. cloud conditions create more often a brighter environment than a darker one and with a greater amplitude.Figure 8NSB density histogram where the nominal NSB that represents the most common clear sky conditions is identified. It delimits two areas, the NSB bright dispersion above the nominal NSB and the NSB dark dispersion below.Full size imageBased on the determination of the nominal NSB, a quantitative indicator, called NDR for NSB Dispersion Ratio, is calculated in the following way:$$begin{aligned} NDR = (N_b cdot MAD_b) / (N_d cdot MAD_d) end{aligned}$$where (N_b) is the number of measures above the nominal NSB (brighter sky), (N_d) is the number of measures below the nominal NSB (darker sky), (MAD_b) is the median absolute deviation of the measures in the bright dispersion zone (above the nominal NSB) and (MAD_d) is the median absolute deviation of the measures in the dark dispersion zone (below the nominal NSB). The median absolute deviation is a statistical tool used to measure the variability of a data set, which is exactly what we try to achieve with the two NSB extensions above and below the nominal NSB. It is formally defined as (MAD = median(|X_i – tilde{{mathbf {X}}}|)) where (X_i) in our case represents an NSB value and (tilde{{mathbf {X}}}) is (median(X_i)). The median absolute deviation is a better choice than the usual standard deviation to measure the spread of NSB measures since the data does not follow a normal distribution.In order to make the determination of the NSB Dispersion Ratio stronger from a statistical standpoint, we use a bootstrapping with replacement resampling method on the set of night portions used to compute the indicator. Assuming we have N night portions at our disposal, we randomly select a sample of N items in this set of night portions knowing that a given item can appear multiple times in the sample (hence the bootstrapping with replacement). The NDR value is then computed for the considered sample. This process is repeated 1000 times and the average NDR value if eventually computed. This average value represents the actual NDR indicator of the considered site.The NDR indicator takes into account both the number of NSB values on each side of the nominal NSB and the dispersion of these values. This is what makes it relevant as an indicator of light pollution which encompasses all kinds of meteorological conditions experienced at a particular site. On that aspect, it is therefore not an astronomical light pollution indicator since it is not focused on clear sky conditions. On the opposite, it requires to have a significant number of NSB measures in all sorts of cloudy conditions so that a valid NDR indicator can be derived.A key aspect of the NDR calculation methodology is to determine the level of the nominal NSB, i.e. the typical clear sky level, since it will be used to differentiate the NSB measures that go in each of the two sets to calculate the bright and dark dispersions. As we have seen earlier in the article, such a determination can be biased by natural light sources that raise or lower clear sky NSB at different times of the night. This can result into a “blurry” high density zone which makes the determination of the nominal NSB difficult or even impossible depending on the observation period. Based on the quantitative estimate of the different natural light sources presented above, the most important bias to address is the contribution of the Galactic plane. This contribution must be eliminated for all the NSB measures which are used to calculate the NDR indicator. In order to do that, Noxi, the Ninox processing software developed by DarkSkyLab, calculates for each NSB measure the corresponding Galactic plane and star fields contribution using the galactic coordinates of the zenith and integrating the combined flux of all stars in the field of view using the UCAC4 astrometry and photometry star catalogue. It is not possible to cancel the contribution of the airglow due to its unpredictable nature, but since it only appears in rare occasions, it is not seen as a problem and is ignored. Regarding the contribution of the zodiacal light, it is considered as minimal at the zenith and it is also ignored.As an example, Fig. 9 shows on the left an NSB density histogram where the Galactic plane bias has not been corrected in the data, and on the right the same data but with the Galactic plane bias corrected. It is easy to see that in the latter the nominal NSB is much easier to determine, providing a more accurate reference level to calculate the NDR indicator. Once the Galactic plane bias has been corrected, the nominal NSB is determined as the highest density zone of the NSB histogram. It must be noted that, as of today, all the NSB measures are corrected from the Galactic plane bias without regards to the presence of clouds or high levels of light pollution. This results into an additional source of inaccuracy that will be addressed in the future through the implementation of two heuristics within the Noxi software:

1.

A first heuristic will determine if a night portion is considered as having a clear sky or not so that the Galactic bias correction is applied only if the sky is clear. In order to do that, we have developed an indicator called the NSS (for Night Sky Stability). To determine the NSS for a full night of measures or just a night portion, we fit the NSB curve with a degree 10 polynomial and we then compute the difference between each NSB measure and it polynomial counterpart. As a result, we obtain a set of residuals. The variance of all the residuals defines the NSS for the considered NSB dataset. Below a given value, the sky is considered as clear knowing that the NSS indicator has been calibrated on several NSB data sets for which the corresponding weather conditions are known;

2.

A second heuristic will allow us to weight the Galactic bias correction to be applied to NSB measures according to their value. For non-polluted skies with high values of NSB, the full Galactic bias correction will be applied while below a certain NSB threshold (for instance 21 mag(_{mathrm{SQM}})/arcsec(^{2}) which corresponds roughly to the brightest parts of the Milky Way) no correction will be applied.

Figure 9NSB density histograms of the same data set with no correction of the Galactic plane bias applied on the left and a full correction applied on the right.Full size imageThe NDR indicator is unitless since it is the ratio of two quantities with the same unit (mag(_{mathrm{SQM}})/arcsec(^{2})). For the data set presented in Fig. 9, the NDR value which is obtained is 25 (which is justified by the fact that the bright extension in the density histogram is much higher and denser than the dark extension). This denotes a quite high level of light pollution despite the fact that the nominal NSB is at a level of 21.6 mag(_{mathrm{SQM}})/arcsec(^{2}). This highlights the fact that there is not always a strict correlation between the typical clear sky NSB obtained for a given site and its NDR indicator, i.e. the presence of clouds decreases the NSB more than we could have expected just by knowing the clear sky NSB. On that respect, the NDR ratio brings more information that the clear sky NSB alone.In addition to provide an indicator which is representative of light pollution in all possible atmospheric conditions, the NDR provides a tool to compare locations in a more meaningful way than just using a set of standalone NSB evaluations. First it is not dependent of an inter-calibration between different systems and second its statistical nature makes it more robust when it comes to perform comparisons.NDR into practiceThe NDR indicator has been calculated for several different sites by DarkSkyLab during various projects in France that involved NSB measuring sessions in the field. To demonstrate some of the results that have been obtained, Fig. 10 provides the density histograms of 4 different sites which have quite different light pollution profiles.Figure 10NSB density histograms of 4 different sites used to compute the NDR indicator. The nominal NSB (which corresponds to the most common clear sky conditions) is noted with a white tick mark next to the vertical axis. Relative levels of the bright and dark dispersion terms (((N_b cdot MAD_b)) and ((N_d cdot MAD_d))) are noted respectively with an orange tick mark and a green tick mark. The computed values of the NDR indicator and nominal NSB are provided in the top-left corner of each figure.Full size imageTo build these diagrams, only the measures acquired during a few hours in the middle of the nights have been used to ensure the maximum stability of the NSB curves and avoid lighting extinctions that create large gaps in NSB profiles. The Galactic plane bias is corrected on all plots and the same NSB scale is used in order to perform comparisons between the 4 sites. One can notice that the number of measures and nights for the 4 sites are quite different. However, they are all sufficient to derive a meaningful value of the NDR indicator using the bootstrapping with replacement resampling method described above, but it is clear that the more NSB measures used, the more accurate the NDR indicator.Table 2 summarizes the NDR indicators as well as the nominal NSB for the 4 sites which are sorted in the order of decreasing NDR indicator values.Table 2 Summary of the nominal NSB and NDR indicators of the 4 different sites.Full size tableOne can see that the NDR indicator values are not strictly correlated to the nominal NSB values, e.g. despite the fact that the nominal NSB of site (a) is slightly better than the one of site (b), the NDR indicator value is much larger for site (a) than for site (b). This can be explained if we consider the specificities of each site:

Cervières (a) is a small village in the Haut-Forez area, France, which is surrounded by large cities (Lyon, Saint-Etienne and Clermont-Ferrand at a distance between 50 to 80 km) and a closer mid-size city (Roanne at 30 km). At the top of that, the town of Noirétable and a large highway rest area are just 2 km away without any nocturnal extinction applied (as opposed to the village of Cervières itself for which public lighting is turned off from 23:00 to 05:00 local time). These conditions are favourable to the presence of a constant light pollution background which has a negative impact on the zenithal NSB measures in most cloudy conditions (distant large cities for high elevation clouds and Noirétable and the highway rest area for lower elevation clouds). Only rare cloud conditions actually protect the site from the effect of mid-distance light sources. In clear sky conditions, however, the fact that there is no close light sources provides reasonably good NSB levels;

The Copernic Association Observatory (b) is located 6 km from the large town of Gap in the mountain area of Hautes-Alpes in the south of France. There is no significant short distant light sources but in many cloud conditions the contribution of Gap has a very negative impact on the zenithal luminance. However, due to the fact that the observatory is at a higher altitude on the hills surrounding the city of Gap, there are cloud conditions that make the site darker. In clear sky conditions, the proximity of Gap does not permit a quality better than that of a rural sky;

The Astrièves Observatory (c) is located near the center of the small town of Gresse-en-Vercors in the Parc Naturel Régional du Vercors. There is a full nocturnal extinction of the village for a large part of the night resulting in a good sky quality in clear sky conditions. The large city of Grenoble is at a distance of 30 km in a valley at the north-east, and the two locations are separated by a few mountains which efficiently help masking the light pollution as soon as the cloud ceiling is below a certain altitude, resulting into a dark environment. On the opposite, high elevation clouds reflect the light from Grenoble and increase the zenithal luminance;

Eourres (d) is a small and isolated village located 20 km west of Sisteron in the department of Hautes-Alpes, France, which is surrounded by mountains. There is no significant light sources closer than those of Sisteron and this results into a very good night sky quality with, most of the times, a very dark environment in cloudy conditions.

Figure 11 provides a graphical representation of the NDR indicator values for the 4 sites. On the NDR scale, the value 1 indicates that the bright and dark dispersion terms (respectively ((N_b cdot MAD_b)) and ((N_d cdot MAD_d))) are equal, which means there is a balance between dark and bright conditions at the zenith on the considered site with reference to the most common clear sky level.Figure 11Summary of the NDR indicators obtained for the 4 sites. The diagram uses 1 as the pivotal value to delineate sites according to the two bright and dark dispersion terms ((N_b cdot MAD_b)) and ((N_d cdot MAD_d)).Full size imageThe NDR can theoretically vary between 0 (totally dark site) and several hundreds (extremely bright site) but in practice the best sites can reach NDR indicator values down to 0.3 in the best preserved locations and up to 200 for very large and polluted cities.Robustness of the NDR indicatorIt is important to evaluate how the NDR indicator is dependant on the number of measures used to compute it and to figure out what would be the minimum number of night sessions required to obtain a meaningful NDR indicator value at a given site. To achieve that, we have used the data from two of the four sites presented above (the two which have the largest number or recorded nights: Cervières with 424 nights and the Astrièves Observatory with 373 nights). The 1000-step bootstrapping procedure has been repeatedly executed on each data set with a regularly decreasing sample of nights: starting from the full number of nights, a decrement of 10 nights is applied at each step until only 20 nights are remaining. At every bootstrap step, each sample is composed of n nights randomly chosen among the N available ones knowing that any night can be selected several times.Figure 12 shows the NDR indicator values that have been obtained for each of the two sites as a function of the night sample considered. The 95% confidence interval is plotted against each NDR indicator value (it is preferred to the standard deviation since the NSB distribution in the data sets is not normal). In the right plot of Fig. 12, the last confidence interval for the 24 night sample is too wide to fit in the y-axis NDR range (the top value is 195).Figure 12Results of the NDR resampling on the two data sets of Cervières and Astrièves Observatory. The horizontal axis is the number of nights considered into the night sample and the vertical axis provides the NDR indicator obtained for each sampling set through a 1000-iteration bootstrapping with replacement calculation.Full size imageDiscussion on the required number of nightsWe can see in Fig. 12 that the NDR indicator and the confidence interval remain stable down to 200 nights. Below this threshold, the NDR starts to become unstable with growing confidence intervals. Based on this data, we can estimate that the minimum number of nights required to compute a robust NDR indicator is 200 (therefore between 7 and 8 months since there are periods around the full moons where there is no night portions recorded).However, depending on the measuring session objectives, the NDR indicator can be considered as accurate enough even when using a smaller number of nights. If the goal is simply to get a first estimate of the light pollution level at a given site, we can consider that 90 nights (a little more than 3 months of measures) are enough. On the opposite, if we want to perform a comparison between several sites for evaluating the impact of light pollution on a particular species, we might want to perform at least 200 nights of measurement to get a better accuracy for the NDR indicator. The experience from DarkSkyLab through many NSB measuring sessions is that 3 to 4 months of measures are required to get a meaningful density histogram, hence an accurate enough NDR indicator, so that a site can be sufficiently characterized from a light pollution perspective. Such a measuring period usually guarantees that the clear sky nominal NSB is well defined and that various cloud conditions have been observed. This estimate is sustained by the results obtained in Fig. 12.Value of the NDR indicator for ecological researchThe study of the impact of light pollution on biodiversity is currently in full expansion, amplifying a political and citizen demand for the reclamation of the night2,32,33.We identify three main contributions of the NDR indicator for ecological research. First, it overcomes the limits of an old problem of communication in terms of measurement units between disciplines and potentially limits the use of units without real meaning from a biodiversity point of view34,35. Secondly, the use of the NDR indicator limits the common biases linked to a characterization of the effects of anthropogenic light which is too limited in time and space35. Indeed, the life history traits of species are not only shaped by the intensity of light emitted into the nocturnal environment but also by its variation over time34,35,36. Currently, the characterization of light pollution is too often limited in time and space, which can lead to misinterpretation37. Thirdly, the NDR indicator provides ecological researchers with a unit of measurement that integrates a sufficiently long time step to study the impact of light pollution on the evolutionary processes at work in the life of species and particularly on population dynamics and animal behavior36,38,39.Limitations and future improvements of the NDR indicatorThe main limitation of the NDR indicator resides in the possible difficulty to identify a well defined value for the nominal NSB, i.e. the NSB value that represents the most common clear sky conditions of a given site. For the most part, this is due to the contribution of the Galactic plane to the zenithal sky brightness and, to a lower extent, to the contribution of other natural light sources (dense star fields, airglow and zodiacal light). The residual spread of NSB measures is due to changing atmospheric conditions at various time scales, but, for this particular contribution, we can expect a statistical compensation to eliminate a systematic associated bias.At the moment, the contribution of the Galactic plane and star fields is canceled into the NSB measures by calculating in the Noxi software the integrated flux of all the stars that belong to the field of view (using the UCAC4 star catalogue). However, this approach has proven some limitations, especially in the southern hemisphere where the Galactic center goes through the zenith and is particularly bright. A probable explanation for that lack of predictability is the fact that the Galactic plane contains diffuse sources such as nebulae which are not accounted for into the star catalogues and which actually cannot be ignored. To address this issue, DarkSkyLab has the project to create a brightness map of the Galactic plane with a square degree resolution or better so that the contribution of all sources can be correctly accounted for.In addition to better correcting the Galactic plane bias, an improvement must be made with regards to the NSB measures that need to be corrected. At the moment, all NSB measures are corrected from the Galactic plane bias without regards to the presence of clouds or high levels of light pollution. So a first heuristic must be implemented to only apply the bias correction to clear sky NSB measures. An other heuristic must also be developed to reduce the correction applied as a function of the NSB level.A third limitation of the NDR indicator is related to a possible lack of cloudy conditions at some sites (e.g. in the Atacama desert in Chile with more than 320 clear nights per year), the reason simply being that the NDR indicator requires the presence of clouds to differentiate the bright and dark extensions into the NSB density histograms. This means that the NDR indicator can hardly be used for such astronomy-oriented sites which experience rare cloudy conditions. More

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In the version of this article initially published, there were errors in equations and notations in the Methods “Model development” subsection which arose during manuscript preparation; the errors affect presentation of the study but not the analysis, results, or code provided with the article. Clarifications to text and equations follow.In Equation (1), “N” replaces “Normal”; in Equations (2), (3), (7) and in text directly below Equations (3), (5) and (7), “ys,i,z” now replaces “Δxs,t1, t2.” In the two paragraphs below Equation (2), “t2 = 2016” and “t1 = 2001” now replace “2016” and “2001” in five instances. Further, Equations (5)–(7) have been revised as follows:$$begin{array}{ll}fleft( {x_{s,t}} right) = {{{mathrm{exp}}}} & left( {beta _0 + mathop {sum }limits_{i = 1}^{I = 5} beta _{1,i} x_{s,i,t} + mathop {sum }limits_{i = 1}^{I = 5} mathop {sum }limits_{k = i}^{K = 5} beta _{2,i,k}x_{s,i,t}x_{k,s,t}}right. \ & quad quad left. {+ mathop {sum }limits_{i = 1}^{I = 5} mathop {sum }limits_{k = 1, k neq i}^{K = 5} beta _{3,i,k}x_{s,i,t}x_{k,s,t}} right)end{array} {rm{Revised}} {rm{Eq}}. (5)$$$$begin{array}{ll}fleft( {x_{s,t}} right) \ = expleft( {beta _0 + mathop {sum }limits_{i = 1}^{I = 5} mathop {sum }limits_{j = 1}^{J = 2} beta _{0,i,j,}x_{i,s,t}^j + mathop {sum }limits_{i = 1}^{I = 5} mathop {sum }limits_{k = i + 1}^{K = 6} beta _{1,i,k}x_{i,s,t}x_{k,s,t}} right) {mathrm{Original}} {rm{Eq}}. (5)end{array}$$$$y_{s,i,z} = left{ {begin{array}{*{20}{l}} {y_{s,i,1} = left| {Delta x_{s,i}} right|,} hfill & {y_{s,i,2} = 0,} hfill & {{{{mathrm{if}}}},Delta x_{s,i} < 0} hfill \ {y_{s,i,1} = 0,} hfill & {y_{s,i,2} = Delta x_{s,i}} hfill & {{{{mathrm{otherwise}}}}} hfill end{array}} right. {rm{Revised}} {rm{Eq}}. (6)$$$$x_{i,s,} = left{ {begin{array}{*{20}{l}} {x_{1,i,s} = left| {Delta x_{i,s}} right|,} hfill & {x_{2,i,s} = 0,} hfill & {if,Delta x_{i,s} < 0} hfill \ {x_{1,i,s} = 0,} hfill & {x_{2,i,s} = Delta x_{i,s},} hfill & {otherwise} hfill end{array}} right. {rm{Original}} {rm{Eq}}. (6)$$$$omega left( {y_{s,i,z};gamma } right) = {{{mathrm{exp}}}}left( {mathop {sum }limits_{i = 1}^{I = 5} mathop {sum }limits_{z = 1}^{Z = 2} - gamma _{i,z} y_{s,i,z}} right) {rm{Revised}} {rm{Eq}}. (7)$$$$omega left( {Delta x_{s,t_1,t_2};gamma } right) = expleft( {mathop {sum }limits_{i = 1}^{I = 5} - gamma _{i,z}Delta x_{z,s,i}} right) {rm{Original}} {rm{Eq}}. (7)$$All changes have been made in the HTML and PDF versions of the article. More

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