The Brewers have a flat Teflon diffuser, which is covered by a quartz dome. The light is directed from the diffuser towards the spectrometer using prisms. In the spectrometer, gratings are rotated by stepper motors to select the wavelength. A low-noise photomultiplier detector (PMT) is used to measure the photon counts. The most important corrections, which need to be applied after raw data measurements are corrections for dark counts, dead time and stray light, temperature correction and cosine error correction .

Generally, the corrections for dark counts and dead time are done using common practices described by the manufacturer , while corrections for stray light, temperature dependence and non-ideal angular response are more operator dependent. A usual way to correct for stray light is to consider that all counts which are measured at wavelengths shorter than 292 or 293 nm are stray light, and can be subtracted from the counts measured at other wavelengths . The temperature dependence of a Brewer is assumed to be linear, and the latest studies have shown that the sensitivity of some instruments changes by up to 5 % when the internal temperature of the Brewer changes between 10 and 50 ∘C .

Three different type of Brewers were used in the study. The MK II and IV-type Brewers are single monochromators, while the MK III-type Brewers have a double monochromator, which improves the quality of the measurements at short wavelengths by reducing the error due to stray light . MK-II Brewers measures from 285 to 325 nm, whereas MK-IV and MK-III extends the range to 363 or 365 nm.

Three Brewers from FMI and four Brewers from Agencia Estatal de Meteorología, Spain (AEMET), the serial numbers and characteristics of which are shown in Table  were investigated. The slit functions were very similar until 0.5 nm from the central wavelength (Fig. ) and the full widths at half maximum (FWHM) varied between 0.5 and 0.68 nm. The differences in the order of several magnitudes outside the central region were due to the difference in stray light rejection by single and double Brewers.

The reference spectroradiometer of the study was the portable reference spectroradiometer from the World Calibration Center for UV (WCC-UV) at the Physikalisch Meteorologisches Observatorium Davos, World Radiation Center (PMOD/WRC). This portable reference spectroradiometer is referred as QASUME, which comes from “Quality Assurance of Spectral UV Measurements in Europe”. It is a double monochromator spectroradiometer, the solar UV irradiance measurements of which are traceable to the primary spectral irradiance standard of the Physikalisch Technische Bundesanstalt (PTB), Germany, through transfer standard lamps . The global entrance optic of QASUME has a shaped Teflon diffuser with an angular response very close to the desired cosine response. The global irradiance measurements of QASUME are not corrected for the remaining cosine error, resulting in an average uncertainty of 1.2 % in clear sky situations . The expanded relative uncertainty (coverage factor k=2) of solar UV irradiance measurements with QASUME for solar zenith angles smaller than 75∘ is 3.1 % . For measurements from 2002 to 2014 the expanded relative uncertainty was 4.6 % .

For Brewer no. 214, the angular response measurements were performed in the dark room at Sodankylä , in which the ambient temperature was kept constant at 23 ∘C. The measurements were performed in 2014 during the QASUME site audit, and the standard cosine measurement device of the PMOD-WRC was used. A 250 W halogen lamp was seated in a holder, which could be moved to different zenith angles. Four azimuth angles (north = 0∘, east = 90∘, south = 180∘, west = 270∘) were measured for zenith angles from 0∘ up to 85∘ and back to 0∘, in steps of 5 or 10∘. The angular responses obtained at 310 nm, normalised to the ideal cosine response, are shown in Fig.  for the four azimuth angles for Brewer no. 214. The deviation from 1 is the cosine error of the instrument.

The angular response of Brewer no. 037 was measured in the old laboratory of the FMI Arctic Research Centre in Sodankylä in 2000. There were same instrumentations in the laboratory than in , but the laboratory was located in a different building. A 1 kW DXW lamp was used, and similarly to the characterisation of Brewer no. 214, the four azimuth angles were measured and the lamp holder was moved in steps of 5 or 10∘. The angular response of Brewer no. 107 was measured in the laboratory of the Swedish Meteorological Hydrological Institute in 1996 following similar measurement procedures.

The angular responses of the AEMET's Brewers were measured during the first Regional Brewer Calibration Center – Europe (RBCC-E) Campaign in Huelva in 2005 with a portable device developed within the European Commission funded project QASUME. A detailed uncertainty analysis of the laboratory measurements using the angular response measurement device is presented in .

For the cosine error correction algorithm, the mean of the four azimuth angles at one measured wavelength was calculated and used as the angular response of the instrument (Fig. ). From Fig. b it can be seen that the cosine error of most Brewers exceeded 10 % at angles higher than 60–70∘. The angular response of the Brewer no. 117 differed from the others at 85∘, which was due to relatively increased inhomogeneity among the measurements over the four planes, for such high measurement angles. However, laboratory measurements at such angles become more uncertain due to the low measurement signals .

To correct the measured irradiances it is essential to know the correction factor for the angular response of a spectroradiometer for a particular global irradiance measurement (cglob). This correction factor depends on the distribution of sky radiance and is a function of solar zenith angle (θ), azimuth angle (ϕ) and wavelength (λ).

If F denotes the actual and F′ the measured irradiance cglob(θ,ϕ,λ)=Fglob(θ,ϕ,λ)Fglob′(θ,ϕ,λ), where the subscript glob corresponds to global irradiance. Both F and F′ are functions of θ, ϕ and λ; however, for the sake of clarity in the equations below, we omit the dependence on θ, ϕ and λ. As global irradiance includes direct (dir) and diffuse (diff) components, Eq. () can be rewritten as cglob=Fdiff+FdirFdiff′+Fdir′. By dividing the numerator and denominator of Eq. () with Fdiff, and rearranging the terms by including Fdir in the first addend of the denominator, Eq. () becomes cglob=(Fdir/Fdiff+1)(Fdir′/Fdir⋅Fdir/Fdiff+Fdiff′/Fdiff).

From Eq. () it can be seen that in order to calculate the cosine error correction factor three components are needed:

Fdir′/Fdir, the ratio between measured and actual direct irradiance, i.e. angular response of the spectroradiometer,

From the definition of the cosine error we get, that the ratio between the measured and actual direct irradiance is the ratio of the angular response of the diffuser (C(θ,λ)) and the cosine of the solar zenith angle (θ), Fdir′Fdir=C(θ,λ)cos⁡(θ).

The ratio between the measured and actual diffuse radiation is Fdiff′Fdiff=∫L(θ,ϕ,λ)⋅C(θ,λ)dΩ∫L(θ,ϕ,λ)⋅cos⁡(θ)dΩ, where the integration is performed for the upper hemisphere. As the exact distribution of sky radiance is not known during the measurements, isotropic diffuse radiation is assumed and L(θ,ϕ,λ) becomes a function of wavelength L(λ). Then, Eq. () can be simplified to Fdiff′Fdiff=L(λ)⋅∫C(θ,λ)dΩL(λ)⋅∫cos⁡(θ)dΩ.

As ∫cos⁡(θ)dΩ=π, the Eq. () becomes Fdiff′Fdiff=∫C(θ,λ)dΩπ.

Using the definition of the solid angle, dΩ=sin⁡θdθdϕ, the Eq. () can be written as Fdiff′Fdiff=∫02π∫0π/2C(θ,λ)sin⁡θdθdϕπ.

As C(θ,λ) is assumed to be independent of azimuth angle, Eq. () is simplified to Fdiff′Fdiff=2ππ∫0π/2C(θ,λ)sin⁡θdθ=2∫0π/2C(θ,λ)sin⁡θdθ.

The only unknown component in Eq. () is the ratio between actual direct and diffuse irradiance, Fdir/Fdiff. It is calculated by using a radiative transfer model and lookup tables. The libRadtran package and UVspec disort version 1.4 (http://www.libradtran.org, last access: 20 August 2018) was used. The steps to retrieve the Fdir/Fdiff ratio are the following: (1) the measured spectral irradiances are corrected using the assumption that all radiation is diffuse, i.e. integrating Eq. () over all SZAs. (2) The corrected irradiances are used to find the corresponding cloud optical depth from a lookup table. A six-dimensional lookup table was precalculated assuming that UV irradiance can be expressed as a function of wavelength, solar zenith angle, cloud optical depth, ozone absorption, aerosols and albedo. As all other parameters are known, the cloud optical depth τcloud(λ), can be found as a function of wavelength from the table. The calculation of the lookup tables is explained in more details in Sect. . Once τcloud(λ) is found, (3) the radiative transfer model is used to derive the direct-to-diffuse ratio as a function of wavelength.

When Fdir/Fdiff is obtained and the angular response of the diffuser, C(sza,λ) is known, Eq. () can be used to calculate the cosine error correction factor for each wavelength. The ratio Fdiff′Fdiff was calculated using Eq. () and it is shown for the studied Brewers in Table .

Data from the Brewer comparison campaign held in Huelva (37.10∘ N, 6.73∘ W), Spain, from 26 May to 4 June 2015, were used. Measurement were performed on the roof of El Arenosillo Atmospheric Sounding Station of the Instituto Nacional de Tecnica Aeroespacial (INTA), which altitude is 50 m above sea level. The near surroundings is characterised by pine forest. The roof was above the top of the trees. The sea side of the Atlantic Ocean was at 1 km from the station in the South. The horizon of the measurement site was free to SZA 85∘.

During the campaign, comparisons of spectral global solar irradiance measurements were done between the 21 spectrophotometers participating in the 10th Regional Brewer Calibration Center – Europe (RBCC-E) campaign and the travelling reference spectroradiometer QASUME. The UV comparison days were 2–4 June. Synchronous UV measurements were performed from sun rise to sun set every 30 min. The start of the UV scans were simultaneous and the measurement wavelength and time step were 0.5 nm and 3 s. With this set up all instruments were measuring the irradiance of the same wavelength at the same time, avoiding differences linked with rapid changes of the radiation field during one scan. During the campaign, the operators of the instruments submitted the data, which were processed using their own calibration and UV processing algorithms. These algorithms differed, for example, by how the temperature dependence or angular dependence was taken into account. For most Brewers, no temperature or cosine error correction was performed. In addition to irradiances submitted by the operators, the spectral UV irradiances were calculated using the standard UV processing of the COST Action 1207, EUBREWNET and a calibration performed with a common lamp during the campaign .

In this work, the UV irradiances measured by five Brewers were calculated using the routine UV processing algorithm of FMI . The cosine error correction was applied, but the temperature correction was not applied in order not to mix the effects of different corrections. The used inputs were the raw UV files, calibrations, slit functions and angular response measurements submitted by the operators. For the cosine correction, the total ozone measured by the Brewer was used, the visibility was observed by the operators and the albedo was set to 0.03. Measurements between 06:00 and 19:00 UTC, SZAs smaller than 90∘, were analysed using the matSHIC algorithm developed within the EMRP project SolarUV (http://projects.pmodwrc.ch/env03/, last access: 20 August 2018). The program is open source, based on the study performed by , and can be obtained on request. The wavelength scale of the solar spectra are adjusted to the high resolution solar spectrum KittPeak and convolved to a nominal triangular slit function with a full width at half maximum of 1 nm. Thus, the process allows comparing solar spectra measured with instruments having different slit functions. The irradiance measurements of the five studied Brewers were compared with the irradiances measured by the QASUME. The mean differences from QASUME, and 5th and 95th percentiles were calculated. For each Brewer, the mean difference was calculated separately for datasets including irradiances measured when the SZAs were (1) less than 50∘ and (2) less than 90∘. The percentiles were calculated for the dataset including all spectra.

The QASUME visited the FMI's measurement sites at Jokioinen (60.82∘ N, 23.50∘ E) and Sodankylä (67.37∘ N, 26.63∘ E), five and three times, respectively (Table ). At Sodankylä, Brewers no. 037 and no. 214 were compared, and at Jokioinen the Brewer no. 107, except in 2002 and 2010, when Brewer no. 037 travelled to Jokioinen for the comparison. During these visits, synchronous UV measurements were performed every 30 min from sun rise to sun set, with 0.5 nm wavelength steps and 3 s wavelength increment . The Brewer spectral data were submitted using the calibration from the site and compared to the QASUME instrument using the same data protocol as for the comparison campaign in Huelva. The FMI's Brewer measurements were processed using the routine UV processing of FMI and were temperature and cosine corrected . For the cosine correction the total ozone measured by the Brewer was used, the visibility was observed by the operators and the albedo was set to 0.03.

At Sodankylä, the measurements were performed on the roof of the sounding station at the Arctic Research Centre at an altitude of 179 m above sea level. The neighbouring area is boreal sparse pine forest. In the east, there are large swamp areas, and in the west the small river Kitinen. In summer, during which the comparisons were performed, the sun hardly reaches the horizon at midnight and the smallest SZA is around 45∘.

At Jokioinen, the measurements were performed on the roof of the sounding station of the Jokioinen Observatory at an altitude of 107 m above sea level. The station is surrounded by fields and coniferous forests. During midsummer, the smallest SZA is around 40∘.

During the QASUME site audits in Finland there were clear sky, changing cloudiness and overcast conditions. The Brewer irradiance measurements were cosine error corrected and the correction varied between 9–12 and 6–12 % depending on SZA, cloudiness and wavelength for Brewers no. 037 and no. 107, respectively. The results of all site audits were studied and the mean ratios of Finnish Brewers no. 037 and no. 107 cosine error corrected irradiances compared to the QASUME irradiances are shown in Fig. . The mean differences between the Brewers and the QASUME were less than 6 % for both Brewers, no. 037 and no. 107, depending on the wavelengths. Most of the spectra (2σ) were within ±2.5 % from the mean difference showed in Fig. . The results of Brewer no. 037 were strongly affected by the stray light problem of single Brewers at wavelengths shorter than 306 nm. The Finnish Brewers overestimated the irradiance compared to the QASUME during all years except the Brewer no. 107 in 2014. A possible explanation for differences between the QASUME and the Finnish Brewers was the difference in the traceability of the irradiance scale of the instruments. The irradiance scale of the QASUME was traceable to PTB, and that of FMI's Brewer was traceable via the Aalto University, Finland, to the Swedish National Testing and Research Institutes (SP), Sweden .

Another potential reason for the systematic bias is the assumption of isotropic radiation in the cosine error correction method. has indirectly shown that for overcast conditions and UV wavelengths, this distribution can not be considered isotropic, leading to a systematic overestimation of the cosine correction applied in the Brewer instruments. For such conditions we have calculated that for the Brewers presented in this work, this overestimation was from 1.5 to 2.5 %. This may explain part of the difference between the QASUME and the cosine corrected Brewer data.