Correction of a lunar irradiance model for aerosol optical depth retrieval and comparison with star photometer

. The emergence of Moon photometers is allowing measurements of lunar irradiance over the world and increasing the potential to derive aerosol optical depth (AOD) at night-time, that is very relevant in polar areas. Actually, new photometers implement the latest technological advances that permit lunar irradiance measurements together with classical Sun photometry measurements. However, a proper use of these instruments for AOD retrieval requires accurate time-dependent knowledge of the extraterrestrial lunar irradiance over time, due to its fast change throughout the Moon’s cycle. This paper uses the 5 RIMO model (an implementation of the ROLO model) to estimate the AOD at night-time assuming that the calibration of the solar channels can be transferred to the Moon by a vicarious method. However, the obtained AOD values using a Cimel CE318-T Sun/sky/Moon photometer for 98 pristine nights with low and stable AOD at the Izaña Observatory (Tenerife, Spain) are not in agreement with the expected (low and stable) AOD values, estimated by linear interpolations from daytime values obtained during the previous evening and the following morning. Actually, AOD calculated using RIMO shows negative 10 values and with a marked cycle dependent on the optical airmass. The differences between the AOD obtained using RIMO and the expected values are assumed to be associated with inaccuracies in the RIMO model, and these differences are used to calculate the RIMO correction factor (RCF). The RCF is a proposed correction factor that, multiplied by RIMO value, gives an effective extraterrestrial lunar irradiance that provides the expected AOD values. The RCF varies with the Moon phase angle (MPA) and with wavelength, ranging from 1.01 to 1.14, which reveals an overall underestimation of RIMO to the lunar 15 irradiance. These obtained RCF values are modeled for each photometer wavelength to a second order polynomial as function of MPA. The AOD derived by this proposed method is compared with the independent AOD measurements obtained by a star photometer at Granada (Spain) for two years. The mean of the Moon-star AOD differences are between -0.015 and -0.005 and the standard deviation between 0.03 and 0.04 (which is reduced to about 0.01 if one month of data affected by instrumental issues is not included in the analysis), for 440, 500, 675 ad 870 nm; however, for 380 nm, the mean and standard deviation of 20 these differences are higher. The Moon-star AOD differences are also analyzed as a function of MPA, showing no signiﬁcant dependence. Different methods are proposed in the literature for calibration Acknowledgements. The authors are grateful to the Spanish Ministry of Science, Innovation and Universities for the support through the 465 ePOLAAR project (RTI2018-097864-B-I00). This work was also supported by the Spanish Ministry of Economy and Competitiveness through projects CGL2016-81092-R, and CGL2017-90884-REDT; by the Andalusia Regional Government through project P18-RT-3820; and by the European Union’s Horizon 2020 research and innovation program through ACTRIS-IMP (grant agreement No 871115). We thank Emilio Cuevas and their staff for establishing and maintaining the Izaña station used in this investigation.


Introduction
Atmospheric aerosols interact with radiation by scattering and absorption mechanisms and with clouds mainly by acting as cloud condensation nuclei, which modify the cloud properties like cloud lifetime or droplet size (Boucher et al., 2013). These 25 issues make the aerosol direct and indirect effects to play a crucial role in the Earth's energy budget, being its impact still subject to large uncertainties (IPCC, 2014) due to the large aerosol diversity in size, chemical composition or spatial distribution.
These current uncertainties in climate models point out the need to monitor aerosol properties and motivate the study of their interaction mechanisms with the Earth-Atmosphere system (Myhre et al., 2013). In addition, the impact of aerosols is important in several fields such as: air quality and human health (Davidson et al., 2005); marine and land ecosystems (Koren et al., 2006;30 Ravelo-Pérez et al., 2016); primary productivity (Jickells et al., 2005), precipitation (Twomey, 1977;Stevens and Feingold, 2009); solar energy production (Neher et al., 2017); or air traffic (Flentje et al., 2010), among others.
Most of the aerosol studies focused on the aerosol role in the climate change field are based on daytime measurements.
However, the knowledge of aerosol properties at night-time is also important, especially in polar areas, where a lack of aerosol observations over winter still exists (Herber et al., 2002;Mazzola et al., 2012;Graßl and Ritter, 2019). In addition, a large 35 fraction of aerosols at night-time remains in the residual layer, which may even act as a source for aerosol formation into the boundary layer the next day (Sun et al., 2013;Liu et al., 2020). Moreover, the lack of ultraviolet (UV) radiation at night-time should reduce the events of new particle formation at night since it is reported that solar UV radiation helps to induce some nucleation events (Petäjä et al., 2009). Finally, the direct effect of aerosols on solar radiation at night-time is avoided, but the aerosols presented at night-time can profoundly modify the longwave balance by means of the change in cloud properties and 40 the impact on the longwave radiation absorbed by clouds, which is back-emitted to the Earth's surface.
Two of the most important/used aerosol properties in climate-change studies and modelling are: the aerosol optical depth (AOD), which represents the light extinction in the atmospheric column caused by aerosols; and the so called Ångström Exponent (AE; Angström 1961) which quantifies the AOD spectral variation. These two parameters provide information about the aerosol load and the particle size predominance, respectively. Moreover, AOD values are useful to estimate other aerosol 45 properties in combination with other measurements (e.g., sky radiance and lidar signal) or even without them (Lopatin et al., 2013;Román et al., 2017Román et al., , 2018Benavent-Oltra et al., 2019). However, ground-based AOD values are usually obtained by solar radiation extinction measurements. In-situ instrumentation is useful to obtain aerosol properties at night-time but they are usually representative only of the aerosol at ground level, with the exception of airborne in-situ measurements (e.g., Remer et al., 1997). Some remote sensing techniques used to derive the aerosol properties at night-time 50 are the Raman lidar systems (Ansmann et al., 1990), which provide AOD but also vertically-resolved extinction profiles; and the star photometers, which derive the AOD from star light extinction measurements (Pérez-Ramírez et al., 2008a;Baibakov et al., 2015). The availability of star photometers is very scarce, existing approximately only five star photometers at present in the world operating for aerosol monitoring . Recent technical advances allow accurate measurements of direct lunar irradiance (Berkoff et al., 2011;Barreto et al., 2013), therefore the emerging Moon photometry technique appears as a plausible and operative alternative for AOD calculation at night-time. One disadvantage of Moon photometry is that lunar irradiance is only recorded from first to third Moon quarter, which implies a lack of data during half of the Moon cycle.
Some Moon photometers are capable to take measurements of solar and lunar direct irradiances, like the CE318-T Sun/sky/Moon photometer (Cimel Electronique S.A.S.), which is the standard instrument in AERONET (AErosol RObotic NETwork;Holben et al., 1998). This fact allows the well established calibration of the solar channels in the AERONET protocols to be transferred 60 to the Moon (Barreto et al., 2016;Li et al., 2016). The main difference between Sun and Moon photometry is that the extraterrestrial lunar irradiance quickly varies even in the course of one night while the extraterrestrial solar irradiance is more stable, with a smooth variation over the year. This remarks the need for knowledge of accurate extraterrestrial lunar irradiance values and their temporal variations. To this end, some models are used, being ROLO (RObotic Lunar Observatory, Kieffer and Stone (2005)) the most widely used in the literature. Here we make use of one implementation of the ROLO named as RIMO (ROLO Implementation for Moon's Observation; Barreto et al., 2019). The irradiance from these models is usually assumed as true for the AOD calculation, however different authors reported some uncertainties and biases in these models (e.g., Viticchie et al. 2013;Lacherade et al. 2014;Barreto et al. 2017;Geogdzhayev and Marshak 2018).
In this framework, the main objective of this work is to evaluate the RIMO accuracy from the differences between the expected AOD in a pristine environment (where AOD is assumed to be low and stable) and the AOD derived by the RIMO 70 with the CE318-T in the same place. The purpose behind this evaluation is to find a correction of the RIMO model that provides an effective lunar extraterrestrial irradiance, which will be assumed as true, useful at least to derive accurate AOD values in the CE318-T bands using the operative Sun-Moon calibration transfer technique. In addition, this paper aims at studying the performance of the AOD obtained with Moon photometry using the proposed RIMO correction, through a comparison with the AOD from a star photometer.

75
This paper is structured as follows: Section 2 introduces the sites and instrumentation used in this paper; Section 3 presents the development of the proposed correction on the RIMO lunar irradiance model, while the comparison of the AOD derived using this correction and the one obtained by a star photometer is shown in Section 4. Finally, Section 5 summarizes the main conclusions of this work.
2 Sites, Instrumentation and data 80

Sites
The RIMO correction proposed in this paper is based on photometer data recorded at the Izaña Meteorological Observatory (IZO; 28.309ºN; 16.499ºW; 2401 m a.s.l.) in the Canary Islands (Tenerife, Spain), which is managed by the Spanish Meteorological Agency (AEMET). This high-mountain observatory is representative most of the time of the subtropical free troposphere over the North Atlantic, because of its location in the descending branch of the Hadley's cell (Rodríguez et al., 85 2009;Cuevas et al., 2019). Pristine skies, dry atmospheric conditions and atmospheric stability prevail throughout the year, as a consequence of the quasi-permanent temperature inversion layer, normally located below the Izaña's level. This situation prevents the vertical transport of anthropogenic pollution from lower levels (Rodríguez et al., 2009).
In terms of AOD, pristine conditions are prevalent in this station, with AOD at 500 nm below 0.1 and AE above 0.6 (Guirado-Fuentes, 2015). Relatively high AOD conditions due to the Saharan dust transport from North Africa sources to the Atlantic 90 Ocean above the trade wind inversion are prevalent in summer (Basart et al., 2009;Rodríguez et al., 2011), associated typically with the presence of coarse particles (AE below 0.25) and AOD at 500 nm above 0.1 (Basart et al., 2009;García et al., 2012;Guirado-Fuentes, 2015). These privileged conditions make Izaña Observatory a suitable place for calibration and validation activities (Toledano et al., 2018). Notwithstanding, Izaña is a calibration site for the GAW-PFR (Global Atmosphere Watch precision-filter radiometer) and AERONET networks , holding a comprehensive measurement programme 95 for atmospheric composition monitoring, being designated by the WMO (World Meteorological Organisation) as a CIMO (Commission for Instruments and Methods of Observation) testbed for aerosols and water vapour remote sensing instruments (WMO, 2014). More details about monitoring programmes at Izaña can be found in Cuevas et al. (2017).
The star photometer measurements of this paper were carried out at the University of Granada (UGR) experimental station, which is the main station of the three belonging to AGORA (Andalusian Global ObseRvatory of the Atmosphere). This station 100 is located at the Andalusian Institute for Earth System Research/IISTA-CEAMA (37.164 • N; 3.605 • W; 680 m a.s.l.). The UGR station operates many remote sensing instruments in the framework of the ACTRIS (Aerosols, Clouds and Trace Gases, www.actris.eu) infrastructure, being the star photometry data at UGR the only available of this type in ACTRIS. The UGR experimental site is located in city of Granada (Spain), which is a medium-size city (535000 inhabitants in all metropolitan area) at south-eastern Spain. The region presents a continental-Mediterranean climate and the city is located in a natural basin 105 surrounded by Sierra Nevada mountains (up to 3500m a.s.l.). The city experiences a seasonal evolution of columnar aerosol types, with larger AOD in summer and lower values in winter while the opposite occurs for AE (e.g., Alados-Arboledas et al., 2003;Pérez-Ramírez et al., 2012a). The seasonal cycle in columnar aerosol properties is mostly associated with the airmass pattern (Pérez-Ramírez et al., 2016) dominated by the more frequent and intense arrival of Saharan dust during summer (e.g., Lyamani et al., 2006;Valenzuela et al., 2012;Antón et al., 2012;Román et al., 2013;Benavent-Oltra et al., 2017).
Nevertheless, the region experiences in winter long periods of air masss stagnations that increase their pollution levels to values compared with other European megacities (e.g., Casquero-Vera et al., 2019).

Instrumentation
The Sun/sky/Moon CE318-T photometer (Cimel Electronique S.A.S.) is used in this work to derive AOD at day and night-115 time. This photometer is mounted on a two-axis robot and a tracking system allows measurements of direct solar and lunar irradiance, and diffuse sky radiance at different geometries. The photometer head is mainly formed by a collimator, a filter wheel (with narrow interference filters) and two detectors. The usual nominal wavelengths of the photometer filters are 340, 380,440,500,675,870,935,1020 and 1640 nm. The detectors are a Silicon sensor to measure the wavelengths of 1020 nm and shorter, and an InGaAs sensor to measure the wavelengths equal or longer than 1020 nm; hence 1020 nm is measured by both 120 detectors. Sky at solar aureole and Moon measurements are recorded by the same detectors than Sun but with an amplification data, and the photometers #918 (from 16 th March 2016 to 25 th July 2016), #751 (from 25 th July 2016 to 26 th May 2017) and #788 (from 25 th May 2017 to 11 st October 2017) the ones operated at UGR station. These photometers were regularly calibrated following the AERONET protocols (Holben et al., 1998;Giles et al., 2019).
The star photometer EXCALIBUR (EXtinction CAmera and LumInance BackgroUnd Register; Astronómica S.L.) operated at UGR station continuously from 2006 to 2011 and during special field campaigns since 2013. A detailed description of the 130 star-photometer EXCALIBUR can be found in Pérez-Ramírez et al. (2008a, b). A brief overview is provided here. The star photometer EXCALIBUR largest innovation is the use of a CCD camera as detector attached to a commercial telescope of 30 cm diameter. A filter wheel permits the allocation of ten interference filters centered at 380, 436, 500, 532, 670, 880 and 1020 nm for aerosol studies, and an additional filter at 940 nm for precipitable water vapor measurement. In this work the 380, 436, 500, 670 and 880 nm channels are used. The one at 1020 could not be used due to technical problems. AOD in these spectral 135 bands will be assumed equal to the AOD at 440, 500, 675 and 870 nm in order to compare with the CE318-T photometer. The AOD is computed from direct star irradiance using the one-star method, that is the same approach used for Sun photometry.
The one-star method needs only a relative calibration of the instrument, but requires a first calibration for the entire set of stars used (Pérez-Ramírez et al., 2011). Nevertheless, a first calibration of the stars (isolated and stable stars) is enough as the recalibration of the instruments consists only of computing wavelength dependent calibration factors that are the same for all 140 the stars. Star photometer EXCALIBUR is able to provide measurements for all filters in approximately 1-2 minutes, but to minimize the effects of atmospheric turbulence data were averaged every 30 min (Pérez-Ramírez et al., 2011). A procedure based on moving averages an outlier removal is used for cloud-screening and data quality check (Pérez-Ramírez et al., 2012b).
In addition, a visual inspection of data has been carried out to remove spurious data. Final uncertainties in AOD are 0.02 for wavelengths below 800 nm and 0.01 for wavelengths above 800 nm (Pérez-Ramírez et al., 2011). Other authors reported a 145 higher uncertainty in AOD from star photometry, about 0.02-0.03 (Baibakov et al., 2015;Barreto et al., 2019). The analysed period in this work is for coincident measurements of star and Moon photometer, and can be divided in two periods, in the

Data management
The University of Valladolid (UVa; Spain) is in charge of one AERONET calibration center since 2006 and, in this framework, the UVa staff developed the CAELIS software tool  with the aim of managing the data generated by AERONET photometers and for calibration and quality control purposes. This tool contains relevant information about the 155 different photometers, like the spectral response of the filters or the signal temperature correction coefficients, and also includes climatology tables of different atmospheric variables (like pressure or the abundance of several absorption gases) useful to perform the atmospheric correction in the AOD calculation. An AOD calculation algorithm has recently been implemented in CAELIS (González et al., 2020). Therefore, the day and night-time AOD data from the CE318-T measurements used in this work have been obtained from CAELIS.

AOD from Moon observations
The main advantage of Sun photometry is that solar irradiance is directly emitted by the Sun and then, the solar irradiance reaching the top of atmosphere (extraterrestrial irradiance) does not significantly change, at least along one day, being the Earth-Sun distance the main factor modulating this irradiance and causing variations about ±3% along one full year. This fact makes that the knowledge of the absolute extraterrestrial irradiance is not needed in the AOD calculation, because an equivalent 165 extraterrestrial irradiance, taking into account the Earth-Sun distance, can be obtained for a given instrument using Langleyplot or side-to-side calibration (Shaw, 1976(Shaw, , 1983Holben et al., 1998;Toledano et al., 2018;Giles et al., 2019;González et al., 2020). However, the Moon is not a self-illuminating body but a diffuse solar reflector with exceptional stability (Kieffer and Stone, 2005); hence, lunar irradiance at the top of the Earth's atmosphere significantly changes, mainly with the Moon Phase Angle (MPA), even along one single night. This fact points out the need of knowledge of the extraterrestrial lunar irradiance 170 for Moon photometry purposes. In this framework, AOD from lunar irradiance observation can be calculated following the Beer-Bouguer-Lambert law as follows (Barreto et al., 2013): where τ a and κ M are the AOD and the Moon calibration coefficient, respectively, for a nominal λ-wavelength; E M 0 and V M are the extraterrestrial lunar 1 irradiance and the photometer lunar signal at the same nominal λ-wavelength, respectively; while 175 m a , m R and m g are the optical airmass for aerosols, Rayleigh scattering and gaseous absorption, respectively, using the Moon Zenith Angle (MZA) instead of Solar Zenith Angle (SZA). Finally, τ R and τ g represent the optical depth of Rayleigh scattering and gaseous absorption, respectively. More details about these calculation in CAELIS can be found in González et al. (2020).

Extraterrestrial Lunar Irradiance
As already mentioned, the knowledge of the extraterrestrial lunar irradiance is necessary in Moon photometry. To this end, 180 the RIMO model has been implemented in CAELIS. RIMO (http://testbed.aemet.es/rimoapp), which is described in detail in Barreto et al. (2019), is an implemetation of the ROLO model (Kieffer and Stone, 2005), which is mainly based on empirical relationships between the lunar irradiance measured at 32 channels by two CCD devices, both mounted in a telescope, and 1 hereafter the superscripts M and S will make reference to Moon or Sun respectively. 6 https://doi.org/10.5194/amt-2020-293 Preprint. Discussion started: 10 August 2020 c Author(s) 2020. CC BY 4.0 License. the different geometrical factors of the Moon-observer positions. RIMO firstly calculates the reflectance of the Moon's disk following the next equation (Eq. (12) in Barreto et al., 2019): where A is the Moon's reflectance at one of the 32 k-wavelengths of the ROLO model; the a, b, c, d and p values are the coefficients shown in Kieffer and Stone (2005); g r and g d are the absolute value of MPA in radians and in degrees, respectively; Φ is the selenographic longitude of Sun (in radians); θ and φ are the selenographic latitude and longitude of the observer, respectively, both in degrees .

190
The Moon's reflectance A is calculated by RIMO using equation (2) at the 32 ROLO wavelengths and, then, each one is multiplied by a correction factor which was previously calculated by the comparison between a composite spectrum (95% soil) of Moon's reflectance based on Apollo 16 samples (soil and breccia) and the reflectance obtained with the equation (2), assuming zero libration and g = Φ = 7º (see Barreto et al. 2019 for more details). The Moon's reflectance at any different wavelength is obtained by linear interpolation of the A calculated values. Finally, in order to obtain the lunar irradiance from 195 the Moon's reflectance, some geometric factors such as the distances between the Moon, the Sun and the observer must be taken into account, as follows: where E M 0 and A are the extraterrestrial lunar irradiance and the Moon's reflectance, respectively, both at the λ-wavelength; E S 0 is the extraterrestrial solar irradiance at the λ-wavelength being obtained from Wehrli (1985) smoothed by a Gaussian  (2) are obtained from the SPICE Toolkit (http://naif.jpl.nasa.gov/naif/toolkit.html) (Acton Jr, 1996;Acton et al., 2018) developed by the NASA's Navigation and Ancillary Information Facility (NAIF). SPICE is run using the planetary and lunar ephemeris DE421 (Folkner et al., 2008) in addition to planetary constants kernel for the 205 Moon (moon_pa_de421_1900 − 2050.bpc), and lunar frames kernel (moon_080317.tp) (Seidelmann et al., 2007;Speyerer et al., 2016); the SPICE kernels pck00010.tpc and naif 0011.tls are also used for other planetary and time parameters. The NAIF pinpoint tool is used to calculate the position of the observer in each station regarding the Mean Earth body-fixed reference system (M OON _M E).

210
Once the extraterrestrial lunar irradiance is obtained from geographical and time inputs, the AOD can be calculated at nighttime using equation (1) if the calibration coefficient κ is known. Different methods are proposed in the literature for calibration purposes (calculation of κ) since the accuracy of the Langley-plot method could be affected by the fast variations of the Moon illumination. One way is the so-called Lunar Langley calibration method (Barreto et al., 2013(Barreto et al., , 2016, which is similar to a classic Langley-plot calibration but where the photometer signal is divided by the extraterrestrial lunar irradiance, as follows: Under stable atmospheric conditions, κ M can be obtained from the y-intercept of a least square fit between ln(V M /E M 0 ) and the aerosol optical airmass. However, the possible errors and uncertainties in E M 0 are propagated to the value of κ obtained by this method, although these uncertainties are partially masked in the AOD retrieval (equation (1) Another way to calculate κ M without the use of E M 0 is by the so called Gain calibration method (Barreto et al., 2016). This method, based on a vicarious calibration, consists of transferring the calibration of the solar channels to the respective Moon ones. Both CE318-T detectors are the same for solar and lunar irradiance measurements. In order to reach a higher signal range, the Moon signal is amplified, being multiplied by a gain factor, G. In fact, this factor is formed by 2 amplification steps, 225 being the first one the Sun to solar aureole gain (≈128) and the second one the solar aureole to Moon gain (≈32). The nominal value of G is therefore equal to 4096 (2 12 ). The values of G were measured with an integrating sphere in the laboratory by Barreto et al. (2016) and Li et al. (2016). These authors found experimental values for G differing less than 0.3% from the nominal value of 4096; hence, G is assumed in CAELIS as wavelength independent and with a constant value of 4096. Taking into account that the only difference between Sun and Moon measurements is this Gain factor, the Sun calibration can be 230 transferred to Moon as follows: where V S 0 is the Sun calibration coefficient and E S 0 the extraterrestrial solar irradiance (Wehrli, 1985), both at the λwavelength. The Gain calibration is simpler than Lunar Langley method because it is not dependent on the RIMO (or other lunar irradiance model) and it only requires the daytime calibration, which provides more operational character to this method.

RIMO Correction Factor
In order to evaluate the AOD obtained by the Gain calibration, the method of Barreto et al. (2017) has been followed, who assumed as a reference AOD, AOD ref , the linear temporal interpolated values using the last daytime AOD value of the previous afternoon and the first AOD of the following morning, which make sense if stable and pristine conditions were found during the night. Hence, the AOD obtained by the Gain calibration, equation (5) and (3) in equation (1), has been 240 calculated for several nights that satisfied pristine and stable conditions to be compared against AOD ref . Data from the #933 CE318-T photometer located at IZO have been selected for this purpose, since this high-elevation remote site usually presents unique atmospheric conditions with very low and stable AOD values. The morning and afternoon solar Langley-plots from this photometer have been calculated, and stable conditions have been assumed when these Langley-plots present more than 25 data, the AOD at 500 nm below 0.025 and the standard deviation below 0.006 (see Toledano et al., 2018). The nights for 245 which both the previous afternoon and the next morning solar Langley-plots fulfill the mentioned criteria, have been selected as the 'stable and pristine' nights. The AOD has been calculated for these selected nights but discarding optical airmasses larger than 6 and data under MPA absolute values above 90º. Moreover, some cloud contaminated nights have been discarded manually by visual inspection in order to warranty the AOD quality. As result, around 13500 AOD data points per wavelength, corresponding to 98 pristine and stable nights from June 2014 to March 2018 at IZO, have been selected.

250
The differences between the AOD obtained by the Gain calibration and the reference values (∆AOD G−r ) have been calculated following the next equation: where ∆τ G−r , τ Gain and τ ref are ∆AOD G−r , the AOD from the Gain calibration and the interpolated AOD used as reference (AOD ref ), respectively, for the λ-wavelength. Figure 1a shows the obtained ∆AOD G−r values as a function of the 255 MPA at IZO for the 98 chosen stable nights and for all photometer channels. These differences point out negative values in the calculated AOD with Gain method and RIMO model, and the existence of a fictitious nocturnal cycle, symmetrical with the optical airmass, which could be associated in Sun photometry to a deficient calibration (Cachorro et al., 2004(Cachorro et al., , 2008Guirado et al., 2014). However, in Moon photometry this cycle, as equation (1) 265 where E M 0−RIM O is the extraterrestrial lunar irradiance from RIMO (the one used in CAElis) and E M 0−ref is the extraterrestrial lunar irradiance that provides the AOD ref if the Gain calibration is applied, both for the λ-wavelength. A correction factor that transforms RIMO irradiance into the reference irradiance, named RIMO Correction Factor (RCF), is defined as the ratio between the extraterrestrial lunar irradiance assumed as reference and the obtained by RIMO. RCF can be derived for each λ-wavelength from equation (7) as: https://doi.org/10.5194/amt-2020-293 Preprint. Discussion started: 10 August 2020 c Author(s) 2020. CC BY 4.0 License.
The RCF values have been calculated by equation (8) using the data of Figure 1a, and they are shown in Figure 1b. The UV channels present high dispersion, while the longer wavelengths point out a decay in RCF close to the full Moon. The other channels show less dependence on MPA and, excluding the UV channels, the extraterrestrial lunar irradiance from RIMO underestimates the assumed as reference between 1% and 14% (between 3% and 12% for MPA absolute values between 5º 275 and 70º). This last result is in agreement with the differences reported by Lacherade et al. (2014), who found that ROLO underestimates around 6-12% (in the same MPA range than in this paper) for wavelengths between 505 and 844 nm, using as reference an imagery absolute calibrated system on board two PLEIADES satellites. Geogdzhayev and Marshak (2018) observed that ROLO underestimates, within 10%, the irradiance at six wavelengths between 443 and 780 nm using EPIC (Earth Polychromatic Imaging Camera) images, calibrated using MODIS (Moderate Resolution Imaging Spectroradiometer) data.  (2)) with a MPA dependence fitted to a quadratic equation of the absolute value of the Moon phase angle (C = A c g 2 + B c ; g= Moon phase angle) compared with the reflectance obtained with photometer measurements; however, these authors did not consider the use 290 of the solar spectrum of Wehrli (1985), which was used by Kieffer and Stone (2005) to derive the Moon reflectance of ROLO, neither asymmetries on the phase angle dependence of ROLO reflectance correction. Considering the results reported in the literature, RCF values of Figure 1b have been fitted by a least square method to a 2nd order polynomial as a function of MPA: where g is the MPA, and a, b and c the fitting coefficients at λ-wavelength. The obtained coefficients are shown in Table 1 for 295 the different wavelengths; the RCF values produced by these coefficients are also shown in Figure 1c, indicating RCF values between 1.03-1.14 for all MPA range except for the UV channel, for which the fit indicates much larger dependence on MPA.
The coefficients for 340 nm channel have been calculated only using data with MPA absolute values lower or equal to 55º since the AOD at 340 nm is too noisy due to the low lunar signal, especially far from the full Moon. The discrepancies in the RCF value for 1020 nm between Silicon and InGaAs (1020i) channels (see Figure 1b) are also marked in the fitting coefficients 300 which point out a RCF overestimation of InGaAs over Silicon around 0.03. The median (MD) and standard deviation (SD) of the RCF fitting residuals (RCF resid ) are also in Table 1, showing the worst fit for UV channels followed by 440 nm and the InGaAs channels; the InGaAs channels present a median and standard deviation in the RCF residuals around -0.001 and 0.013, https://doi.org/10.5194/amt-2020-293 Preprint. Discussion started: 10 August 2020 c Author(s) 2020. CC BY 4.0 License.
respectively, which could explain part of the mentioned discrepancies between the RCF values at 1020 nm in both Silicon and InGaAs channels. On the other hand, the lowest deviation (around 0.01) is reached for the 675, 870 and 935 nm.

305
Unifying equation (1), (3), (5) and (9) with the coefficients of Table 1, the AOD can be calculated using the Gain calibration method as: which is the final way used by CAELIS to derive AOD at night-time, adding the RCF values and using the Gain method to transfer Sun to Moon calibration. Finally, in order to see how the residuals in the RCF fitting are propagated to the AOD, the 310 median and standard deviation of the residuals between the AOD from equation (10)  that the 340 nm channel (at least for MPA absolute values above 55º), and possibly 380 nm, should not be used due to the high dispersion, which caused by the low signal to noise ratio of these channels. In addition, the AOD from InGaAs channels should be carefully used since they present the highest deviation (apart from the UV channels). An example of the AOD at night-time obtained by the proposed method using CAELIS is shown in the Figure 8 of González et al. (2020), where the AOD continuity from day to night-time can be appreciated for different sites and MPA values.

Moon vs star photometer
In order to evaluate the performance of the AOD calculated by the method developed in section 3, the AOD from a Moon photometer has been compared with the AOD measured by a star photometer. To this end, the AOD from the different Moon photometers at UGR station in 2016-2017 has been obtained from CAELIS. These AOD data have been previously cloudscreened using the criteria explained in González et al. (2020), which is similar to the one used at daytime by Giles et al. 325 (2019). The applied criteria are mainly based on: the recorded signal must be higher than a threshold value in some infrared channel (to warranty the correct pointing at the Moon); the AOD variation in a triplet observation must be below a threshold; the temporal variation of AOD at 500 nm must be smooth (below 0.01 per minute); among others. The AOD negative values, or below a established threshold, have been not discarded in this work, since this kind of criteria are usually based on a threshold marked by the AOD uncertainty, but in this case the uncertainty is still not well known. Finally the cloud-free AOD values   intensive field campaigns. All channels show correlation between both AOD data sources, being the correlation coefficient, shown in Table 2, higher than 0.96 (0.97 if only 2016 is considered) except for 380 nm, which presents a lower value around 0.71. Table 2 also shows the slope and y-intercept of the linear fits shown in Figure 3, both ranging from 0.975 (440 nm) to 1.038 (870 nm) and from -0.012 (870 nm) to -0.004 (500 nm), respectively, for the wavelengths between 440 and 870 nm; these results reveal that the obtained fitted lines are close to the 1:1 line. Table 2  AE computations, which is particularly critical for low AOD (Cachorro et al., 2008).

370
In order to quantify the discrepancies between the AOD retrieved by Moon and star photometers, Figure 4 shows frequency histograms of relative differences in AOD, assuming the star photometer as reference. Figure 4 reveals that in general the differences are centred around zero and normally distributed. The influence of the negative AOD at 380 from Moon observations can be observed in the negative tail showed by the differences in this channel distribution. The percentage of AOD absolute difference values below 0.02 are 27%, 47%, 45%, 57% and 63% for 380, 440, 500, 675 and 870 nm, respectively; 375 these percentages rise up to 46%, 65%, 63%, 69% and 75% for differences below 0.03. Table 2 shows the mean, median and standard deviation of the differences given in

385
The median and standard deviation are -0.03 and 0.06 when this period is removed. Regarding AE differences, the mean and median are below 0.10 for all data, indicating a lack of significant over or underestimation, but the standard deviation is around 0.4, revealing a high dispersion.
We have also investigated whether the performance of the AOD depends on the MPA, due to the influence of this parameter on the incoming lunar irradiance and in the RCF values (see Section 3.3). Figure 5 shows the Moon-star AOD differences as expected AOD can be mainly caused by two issues: 1) bad calibration coefficients of the photometer or 2) lack of accuracy in the RIMO values. The calibration used in this work has been based on transferring the calibration of the solar channels (well established) to the Moon channel by a vicarious method, based on the fact that the photometer takes the Moon observations with the same sensor than Sun measurements but with a two-step electronic amplification of 4096 in the signal. In principle, 405 nothing suggests that AOD errors could come from the calibration, while other works in the literature pointed out discrepancies in the ROLO model. This fact has motivated us to assume the lack of accuracy on RIMO as the responsible of the observed differences, and these differences have been used to determine the RIMO accuracy.
Detailed analyses of the differences between expected AOD and the AOD derived by RIMO have shown a bias revealing an underestimation of RIMO to the real extraterrestrial lunar irradiance about 1-14% for visible and IR channels, which in 410 addition depends on the Moon phase angle (MPA); this result agrees with other works in the literature. The mentioned bias has been modeled as a function of MPA by a 2nd order polynomial (for each wavelength). These proposed polynomials represent the named RIMO correction factor (RCF), since if a RIMO irradiance output is multiplied by this factor, then the derived AOD from the corrected irradiance will be closer to the expected AOD. The obtained RCF values are at least useful for the retrieval of AOD from Moon observations. Differences around 0.03 in the RCF values has been found for the same wavelength (1020 415 nm) using two different detectors (Silicon and InGaAs); this result has apparently no physical sense, since the lunar irradiance cannot take two different values for one single wavelength. Consequently, this result must be caused by the uncertainty of the measurements and the method itself, indicating that the uncertainty of the estimated extraterrestrial lunar irradiance with RCF might be about 3%, at least for 1020 nm. The obtained results at 340 nm have been too noisy hence the use of this channel is not recommended. This new methodology based on the modeled RCF to correct RIMO for AOD calculation has been implemented 420 in CAELIS, achieving a night-time AOD calculation in near-real time for all photometers managed by this tool in an operational way. This is possible because the used calibration only needs from the routine Sun calibration (the so called Gain calibration method).
The RIMO-corrected AODs have been evaluated versus alternative and independent measurements from a star photometer. This instrument was deployed at Granada, a different location than the one used for the proposed RCF calculation. To 425 our knowledge this is the first long-term AOD comparison between Moon and star observations. The obtained results for wavelengths between 440 and 870 nm have pointed out a good agreement between both databases, being the absolute mean difference below 0.01, except for 440 nm which is below 0.02. This indicates only a slight underestimation of AOD from Moon to star observations (used as reference) at 440 nm, but within the uncertainty of the star photometer (about 0.02-0.03).
The standard deviation of the Moon-star AOD differences for the mentioned wavelengths is about 0.03-0.04, but if some prob-430 lematic periods in the star photometer data are neglected, these values are reduced approximately to 0.01, which leads to an uncertainty in AOD from Moon observations between 0.019 (870 nm) and 0.028 (500 nm). However, these uncertainties could be lower because part of the observed differences could be caused by: detected technical problems in star photometer filter wheel; the differences in the effective wavelengths used in both instruments and in the way to correct atmospheric gaseous scattering and absorption at these wavelengths; the inhomogeneity of aerosol spatial distribution, since both instruments point 435 to different targets, which also affects to the time interval used to the averages (clouds can block the Moon but not the pointed any dependence on MPA in the Moon-star AOD comparison.This is an important result because it indicates that the proposed correction is able to remove any influence of the Moon cycle on the AOD.
The used night-time cloud-screening is in general the same that is used for daytime but without rejecting AOD values below a given threshold. In spite of providing apparently good results, the night-time cloud-screening is still in development and it could change or add other specific criteria in the future due to the particularities of night-time measurements. The development 445 of new cloud-screening criteria is out of the scope of this paper but, in the future, it could be based on the consideration of temporal smoothness in individual wavelengths or in the addition of a threshold value for the minimum acceptable AOD and for the minimum acceptable recorded signal per channel; this could help to warranty the AOD quality, especially in the noisier channels like 380 nm. Recently, AERONET is also providing AOD values from Sun/sky/Moon photometers with its own method 3 , but this product is still labeled as provisional at present, hence a direct comparison between the AOD from the 450 proposed and the AERONET methods has not been considered.
To sum up, this work provides more evidences about the reported underestimation of RIMO/ROLO model to the real extraterrestrial lunar irradiance and points to the need for a correction of this model or the development of a new extraterrestrial lunar irradiance model, at least for accurate AOD calculation purposes. Meanwhile, at least until a more accurate lunar irradiance model is available, the proposed correction can help in providing AOD retrievals with the Moon. Moreover, additional 455 studies using different Moon photometers or using alternative and independent night-time instrumentation, like lidar or star photometers, are highly recommended to characterize the AOD uncertainty of the proposed method.
Data availability. The used data are available from the authors upon request.
Author contributions. RR, RG and CT designed and developed the main concepts and ideas behind this work and wrote the paper with input from all authors. RG, AB and RR implemented the RIMO model in CAELIS. RR, JABO and DPR carried out the routine and calibration measurements of the star photometer. DPR computed and processed the AOD data from the star photometer. VEC, FJO, LAA and AMdF aided in interpreting the results and worked on the manuscript. All authors were involved in helpful discussions and contributed to the manuscript.
Competing interests. The authors declare that they have no conflict of interest.
ePOLAAR project (RTI2018-097864-B-I00). This work was also supported by the Spanish Ministry of Economy and Competitiveness through projects CGL2016-81092-R, and CGL2017-90884-REDT; by the Andalusia Regional Government through project P18-RT-3820; and by the European Union's Horizon 2020 research and innovation program through ACTRIS-IMP (grant agreement No 871115). We thank Emilio Cuevas and their staff for establishing and maintaining the Izaña station used in this investigation.   (8)), the number of used data (N), and median (Md) and standard deviation (SD) of the residuals in RIMO Correction Factor (RCF) and aerosol optical depth (AOD) for different photometer wavelengths.The fitting values at 340 nm has been obtained without MPA absolute values above 55º.