In cloud-free conditions, UV irradiance at ground level depends mostly on the SZA (θs); the ground albedo; the total column content of ozone; the vertical profile of ozone, temperature, pressure, density and volume mixing ratio for gases as a function of altitude; aerosol optical depth (AOD); the Ångström coefficient; and aerosol type and the elevation of the ground above sea level. As the method shall be used operationally, the sources of these inputs have been selected to allow estimation of UV irradiance at any location and any time.

The Copernicus Atmosphere Monitoring Service of the European Commission provides aerosol properties together with physically consistent TOC for any place and any time after 2003. Along with TOC, the AOD at 550 nm, Ångström coefficient and aerosol type are collected from this source of data following exactly the path of the McClear model (Lefèvre et al., 2013). θs is given by the SG2 algorithm for the Sun position and angles (Blanc and Wald, 2012). Ground elevation is extracted from the Shuttle Radar Topography Mission database and has been downloaded from the website http://srtm.csi.cgiar.org/SELECTION/inputCoord.asp.

The albedo is the ratio of the upwelling to downwelling flux at the surface and is the integral of the bidirectional reflectance distribution function (BRDF), which depends on the surface-type, its roughness and the wavelength of the impinging radiation. A few institutes provide BRDF products in the UV range or Lambertian equivalent reflectivity (e.g., Herman and Celarier, 1997) of the Earth's surface with a coarse resolution of 0.5 or 1∘. In absence of the ideal solution – BRDF parameters in the UV range available worldwide with a grid cell of 0.05∘ or better – an approximate solution has been adopted by using the so-called shortwave [250, 5000] nm BRDF parameters proposed by Blanc et al. (2014). The US National Aeronautics and Space Administration (NASA) provides worldwide maps of the BRDF parameters that are derived from the MODIS (Moderate Resolution Imaging Spectroradiometer) instrument (Schaaf et al., 2002). Blanc et al. (2014) have created a series of maps of the MODIS BRDF parameters for each calendar month for the shortwave albedo with no missing values at a spatial resolution of 0.05∘. In addition, these authors proposed a method for computing the albedo simultaneously for direct and global irradiances. These maps and this method are those used by the McClear model (Lefèvre et al., 2013). As a first approximation, the UV albedo is assumed to be spectrally constant and equal to the shortwave albedo. This assumption may result in biases depending on the surface. For example, in the case of snow surface, Varotsos et al. (2014) reported from many aircraft measurements that spectral albedo exhibits a tendency to decrease with increasing wavelength of about 0.7 in UV to about 0.4 in the near-infrared independently of the sky conditions. Therefore, the albedo integrated over the spectrum becomes less than 0.7, resulting in the underestimation in UV albedo, hence in a lesser contribution to diffuse UV irradiance, and therefore in the underestimation of the global UV.

Let λ be the wavelength, Gλ the global spectral irradiance at the surface and Bλ the direct normal spectral irradiance, i.e., the irradiance received from the direction of the Sun at the surface on a plane normal to the Sun's rays. The irradiance in a given interval [λ1, λ2] is given by Gλ1λ2=∫λ1λ2Gλdλ. Similar expressions may be obtained for irradiances in UV-A, UV-B, UV or over KBj. For example, the UV irradiance GUV and the direct normal irradiance BUV or the irradiance GKB4 in KB4 are given by GUV=∫280400Gλdλ,BUV=∫280400Bλdλ,GKB4=∫307328Gλdλ, where λ is expressed in nm.

The KBs do not fit the UV spectral ranges exactly. For example, the UV-B is covered by KB3 and a part of KB4. One solution to estimate the irradiance in a UV interval is the use of weighted sums based on the overlap between KBj and this interval. Another technique is adopted here whose concept is to determine several 1 nm spectral bands NBi whose transmissivities are correlated to those of the KBj and are then used in a linear interpolation process to compute the UV irradiance. A similar approach has been used by Wandji Nyamsi et al. (2015a) for the calculation of the photosynthetically active radiation with better results than a weighted sum.

If one assumes that the optical properties of the atmosphere do not change over a given NBi, the integrals of Eqs. (2)–(4) may be replaced by Riemann sums over NBi. For example, if λi denotes the central wavelength of NBi, GUV can be approximated by GUV=∑i=1120Gλi. If one defines the clearness index KTi as KTi=GλiEoicos⁡(Θs), where Eoi is the irradiance at the top of atmosphere on a plane normal to the Sun's rays for NBi for a given instant t, Eq. (5) becomes GUV=cos⁡(Θs)∑i=1120EoiKTi. Similarly, the clearness index KTKBj for KBj is given by KTKBj=GKBjEoKBjcos⁡(Θs). The solar spectrum of Gueymard (2004) available in libRadtran was combined with the algorithm SG2 for computing the Sun position (Blanc and Wald, 2012) to yield Eoi and EoKBj.

For the method presented here, we assume that simple and accurate relations, e.g., affine functions, may be found between each KTKBj and a subset of several KTi, called KTk hereafter. Then it may be possible to interpolate linearly between KTk to obtain an estimate of KTi for each λi. By summing the products of Eoi and these interpolated KTi's over a given spectral interval, it is then possible to compute the corresponding irradiance. This is the principle of the resampling technique. The set of NBk is selected at the beginning of the procedure and the same set is used for all processing. The number of NBk should be as small as possible in order to decrease the amount of calculations but still large enough to allow a good accuracy.

The selection of NBk is empirically determined by means of libRadtran. The current approach is empirical with no guarantee that the selected set of NBk is the optimum. It could have been possible to use some mathematical optimization tools. This is not a straightforward process as the cost function should take into account that the number of NBk is unknown a priori.

A set of 60 000 clear-sky atmospheric states was built by means of the Monte Carlo technique in order to select NBk. Each state comprises the nine variables described above, and the value of each variable was randomly selected by taking into account their modeled marginal distribution. The distributions proposed by Lefèvre et al. (2013) and Oumbe et al. (2011) established from observations were adopted here (Table 2). More specially, the uniform distribution is chosen as a model for marginal probability for all variables except AOD, the Ångström exponent coefficient, and total column content of ozone. The chi-square law for AOD, the normal law for the Ångström exponent coefficient, and the beta law for TOC have been selected. The selection of these parametric probability density functions and their corresponding parameters have been empirically determined from the analyses of the observations made in the Aerosol Robotic Network for aerosol properties and from meteorological satellite-based ozone products (Lefèvre et al., 2013).

Each atmospheric state is input twice to libRadtran, (1) with the Kato et al. (1999) approach yielding KTKB3, KTKB4, KTKB5 and KTKB6 and (2) with detailed spectral computations providing KTi every 1 nm for the interval [283, 408] nm. Several plots were made superimposing KTKB3, KTKB4, KTKB5 and KTKB6 (in green), and KTi (in red). Figure 1 is such a graph with the following inputs: θs of 53.76∘, the midlatitude winter atmospheric profile, TOC of 470 DU, AOD of 0.78 at 1000 nm for a maritime polluted aerosol model with an Ångström exponent of 1.93, elevation of 0 m and surface albedo of 0.63. A visual inspection shows that KTKBj and KTi are approximately equal for λj in the middle of KBj, except for KB5. KTi in this band exhibits a nonlinear behavior that cannot be accounted for with a single KTk. If one selects 305, 320, 333, 346 and 386 nm as NBk (magenta crosses), then the linear interpolation (in blue) provides a fairly accurate estimate of KTi. The five NBk's were selected by a lengthy visual inspection of such plots and are reported in Table 3. The same NBk's apply for the global and direct irradiances. For each NBk, the parameters of the affine function relating KTKBj and KTk are determined by least-squares fitting technique (Table 3): KTk=akKTKBj+bk. Another set of parameters is determined in the same way for the direct irradiance. In the operational mode, given an atmospheric state, a run of libRadtran, or a fast approximation of it, yields four KTKBj, from which the five KTk's are computed using the affine functions. Then, approximate KTi*'s are computed for each nm between 280 and  400 nm using a linear interpolation and extrapolation of KTk. In cases where extrapolation provides negative values, KTi* is set to 0. Eventually, GUV is obtained by GUV=cos⁡(Θs)∑i=1120EoiKTi∗. A similar process is performed for the direct normal irradiance BUV as well as for the same quantities in UV-A and UV-B. As the method provides the spectrum KTi*, the equation may be extended to include any action spectrum S(λ), for example, GS(λ)=cos⁡(Θs)∑λ1λ2EoiSiKTi∗.

In this section, results of the proposed technique are compared with results from the detailed spectral calculations made by libRadtran to assess the accuracy of the proposed technique for GUVA, GUVB, BUVA and BUVB for cloud-free conditions. The errors made by using the proposed technique for calculations of the UV-A and UV-B irradiances are presented. To that extent, an additional sample of 10 000 atmospheric states has been randomly constructed following the marginal distribution variables described in Table 2. The proposed technique was applied to the outputs of libRadtran using the Kato et al. (1999) approach and the estimates were compared to the detailed calculations performed by libRadtran. Following the ISO standard (1995), the deviations were computed by subtracting measurements for each instant from the results of the method. They were summarized by the bias (mean error), the root mean square error, and their values rBias and rRMSE relative to the mean value of the measurements. In addition, the coefficient of determination (R2) is computed.

Table 4 reports the statistical indicators for the global and direct normal UV-A and UV-B irradiances. For UV-A fluxes, the bias for the global irradiance and the direct irradiance is +0.10 W m-2, i.e., +0.2 % in relative value, and -0.15 W m-2, i.e., -0.7 % in relative value respectively. The RMSE is respectively 0.12 W m-2 (0.3 %) and 0.18 W m-2 (0.8 %). For UV-B fluxes, the bias for the global irradiance and the direct irradiance is -0.04 W m-2, i.e., -1.6 % in relative value, and +0.07 W m-2, i.e., +10.1 % in relative value respectively. The corresponding RMSE is respectively 0.14 W m-2 (6.2 %) and 0.15 W m-2 (20.5 %). The coefficient of determination R2 is greater than 0.99 except for the direct normal UV-B irradiance which is 0.966. Expectedly, these indicators prove the good level of performance of the proposed technique.

Figure 2 exhibits the scatter density plot between ground-based instantaneous measurements made for each station in cloud-free conditions and estimates from the proposed method combined with inputs from CAMS. The station name is indicated at the top of each plot. Figure 2a exhibits the results for Barrow. All points are well located along the identity line. The slope of the fitting line is 0.995, i.e., very close to 1, showing a very good estimation of the measurements by the method. R2 is 0.97, meaning that all the variability in the measurements is very well explained by the estimates. The bias is low with a value of -0.2 W m-2, i.e., -1 % of the mean value of the measurements, 20.5 W m-2. The RMSE is small with a value of 1.4 W m-2, which is around 7 % of the mean. These statistical indicators for each station are reported in Table 5. The measurements are mostly between March and September. The shortwave albedo is 0.8 from the beginning of March until the middle of May. With the progressive snowmelt, this shortwave albedo decreases from mid-May down to 0.12 in mid-July. This variation corresponds well to the climatological evolution reported by Bernhard et al. (2008) and supports the choice of this approximation by the shortwave albedo.

Even if the points follows the perfect line (Fig. 2a) quite well, a set of points is seen where the method noticeably underestimates by more than 20 %. These underestimations occurs between the end of May and mid-July. During that period, the shortwave albedo was less than the effective UV albedo by a factor 0.8. The effective UV albedo is part of the version 2 dataset and was derived by comparing measured clear-sky spectra with corresponding radiative transfer model results (Bernhard et al., 2007). As a smaller albedo means a smaller contribution to the diffuse part of the irradiance, the difference between the shortwave and effective UV albedo may explain these underestimations seen in Fig. 2a.

Results for Sodankylä are shown in Fig. 2b. Cloud-free conditions occur mostly between February and September. The points lie along the identity line with a slight overestimation by the method at low irradiance and an underestimation at large irradiance. R2 is 0.98. The bias is low with a value of -0.5 W m-2, i.e., -2 % of the mean value of 20.8 W m-2. The RMSE is low with a value of 1.9 W m-2 (9 %). As for Jokioinen (Fig. 2c), all points are well located along the identity line. R2 is 0.98. The bias is low with a value of -0.5 W m-2, i.e., -2 % of the mean value of the measurements, 22.1 W m-2, as well as the RMSE with a value of 1.6 W m-2 (8 %).

Results for Palmer are shown in Fig. 2d. One may note that the points are well aligned with low scatter along a straight line whose slope is 0.99 with a slight underestimation by the method. The bias is -0.8 W m-2 (-3 % of the mean value of 24.9 W m-2). The RMSE is 1.2 W m-2 (5 %). R2 is greater than 0.99. Cloud-free conditions occur mostly between August and April. The shortwave albedo slightly increases from 0.28 to 0.32 between August and March and then decreases until April up to 0.20. These values are small and close to those of a ground free of snow or ice. The effective UV albedo is usually greater than 0.3 with peaks up to 0.8. This difference between the shortwave and effective UV albedo may explain the slight underestimation indicated in Fig. 5.

Results for McMurdo are shown in Fig. 2e. Cloud-free conditions occur mostly between April and September. The points are aligned along the identity line. R2 is 0.99. The bias is low with a value of -0.3 W m-2 (-1 % of the mean value of 21.0 W m-2), as well as the RMSE with a value of 1.2 W m-2 (6 %). The shortwave albedo may reach 0.8 and there is no clear discrepancy between the shortwave and effective UV albedo. Nevertheless, the authors believe that the outliers may be explained by a difference in albedo.

The dependence of errors as a function of SZA was investigated. Figure 3 exhibits the ratio (top) and difference (bottom) as function of θs for each station for UV-A irradiances. For the ratio, the limits of the boxes are close from one quartile to another, meaning a very limited spread of the ratio. The deviations between maximum and minimum are approximately small. The median is similar to the mean. Regardless of the number of data (in pink color), the deviations are very close to 1 for all SZA ranges and stations except Sodankylä at high θs. As for the ratio, the similar observations are also seen in terms of the differences at the bottom of Fig. 3. The difference is very close to 0 for all the SZA ranges. The absolute value of the mean difference shows a tendency to decrease as θs increases, with the maximum being reached for low θs.

The dependence of errors as a function of TOC and albedo was also investigated (not shown). The results have revealed that there is no clear dependence of errors as a function of TOC or albedo for all stations. In addition, the absolute values of the bias (not shown) show a tendency to decrease as θs increases, with the maximum being reached for low θs, except McMurdo. In the opposite manner, the absolute values of the relative bias show a tendency to increase with θs. This could be related to the fact that low θs are reached in summer, with greater values in UV irradiance and lower values in effective UV albedo.

The UV-B band is the spectral region of UV irradiance where the ozone absorption is very strong. Figure 4 exhibits the scatter density plots between measurements of UV-B and estimates for each station. Table 6 reports the statistical indicators for UV-B. R2 is greater than 0.97 for all stations, meaning that variability in UV-B is very well reproduced by the estimates. In general, the method overestimates the UV-B irradiances. Visually, one observes that the method clearly overestimates when the irradiance is low. For the wavelength less than 320 nm, in Fig. 1, the proposed method seems to mostly overestimate when compared to the detailed spectral calculations serving as reference. This observation induces a systematic overestimation at low irradiance from the method. This mainly explains this previous observation. The absolute value of the bias is less than 0.1 W m-2. The relative bias ranges between -4 % (Palmer) and 13 % (Barrow). The rRMSE ranges between 12 % and 18 %.

Figure 5 shows the change in ratio and relative difference as a function of θs. For both ratio and differences, the spread of limits of the boxes is more or less visible. Nevertheless, the median and the mean are close. The deviations show a tendency to increase with θs for all stations, probably meaning that a systematic bias from the method is more visible at high θs as mentioned above. These results explained the main sources of errors for the dependence of errors as a function of TOC and albedo.

In addition, one notices a tendency of the bias to reach a maximum between 65 and 75∘ (not shown); this appears in the form of a plateau around 65∘ for the RMSE which decreases as θs increases, i.e., as the irradiance decreases. The relative bias increases with θs as well as the rRMSE. Both are closer in the high θs than in the small θs.