Measurements of the spectrally resolved sky radiance are important for many atmospheric remote-sensing applications in atmospheric chemistry and physics. They are also useful for the quantification of the energy yield of photovoltaic cells or doses of harmful UV radiation. Possible applications include the following:

Improvement of aerosol retrievals from multi-axis differential optical absorption spectroscopy (MAX-DOAS) observations (Hönninger et al., 2002): if the measured radiances are absolutely calibrated, they do not have to be normalised by zenith sky observations. This allows e.g. to better constrain the total aerosol optical depth (AOD).

Absolutely calibrated measurements of the sky radiance are usually elaborate and complex, because the calibration is performed in a laboratory using e.g. calibrated light sources and reflecting surfaces. Also the errors of the radiometric calibration can be quite large: typical uncertainties of the calibration procedures for atmospheric radiance measurements are reported to range from 3 to 10 % (Pissulla et al., 2009). From comparison exercises during field campaigns even larger differences between individual instruments have been reported (up to 33 %; see Pissulla et al., 2009). Here it is interesting to note that measurements of the solar irradiance usually show much better agreement (e.g. Wuttke et al., 2006).

In this study we present a new method for the absolute radiometric calibration of instruments that observe spectrally resolved scattered sunlight, which is completely independent from laboratory measurements. Instead it is based on the comparison of measurements of scattered sunlight under well-defined conditions with radiative transfer simulations. In particular the dependence of the observed zenith sky radiance on the solar zenith angle (SZA) is investigated. Here it is interesting to note that typical (MAX-)DOAS instruments have rather small fields of view (usually about 1∘), but also measurements with larger fields of view could in principle be used as long as the correct field of view is considered in the radiative transfer simulations.

Continuous measurements during a couple of hours on one day are sufficient for the application of our method.

The new calibration method is particularly suitable for DOAS measurements (Platt and Stutz, 2008). DOAS measurements of direct or scattered sunlight in the UV, visible and near-IR spectral range are widely used to identify and quantify spectral absorption features of atmospheric trace gases. DOAS instruments are usually not radiometrically calibrated, because the DOAS technique is sensitive to differential absorption structures and does not require absolute calibration. But some of the data products derived from DOAS observations (see above) can be improved if radiometrically calibrated DOAS instruments are used. Our new method can also be applied to other measurements of scattered sunlight (e.g. spectroradiometer or sun photometer measurements). Here it should be noted that our method cannot be applied to irradiance measurements, because for such measurements the relative dependence of the observed irradiance on SZA hardly changes with the aerosol load and properties (except for very high aerosol loads).

The paper is organised as follows: in Sect. 2 the data sets used in our study (MAX-DOAS measurements and radiative transfer simulations) are introduced. Section 3 presents the application of the method together with an estimation of the uncertainties. Also the effects of (neglecting) polarisation and rotational Raman scattering are discussed. Section 4 presents the conclusions.

In Fig. 5 the measured radiances (blue lines) are compared to simulated radiances for different AODs (ranging from 0 to 0.5). Measurements and simulation results are displayed as a function of the SZA, but with separate y axes because of their different units (either in counts s-1 or in W m-2 nm-1 sr-1). The maxima of both scales are chosen such that measurements and model results are roughly in agreement. It is obvious that not only the absolute values of the simulations but also the curvature of the SZA dependence varies with AOD. This dependence is exploited by our method. The corresponding figures for all selected wavelengths are shown in the Appendix (Fig. A3).

The simulated radiances over the considered SZA range are fitted to the measurements by optimising a scaling factor S until a minimum of the sum of the squared differences between both data sets is reached: ∑i=SZAmin⁡SZAmax⁡RAOD,λ,i-S(λ)⋅Dλ,i2=>min⁡.

Here R(AOD,λ,i) indicate the simulated radiances at wavelength λ for a given AOD, D(λ,i) are the values of the detector read-out at wavelength λ, and S(λ) is the scaling factor. The fit is performed separately for the different scenarios, wavelengths and assumed AOD. Examples of the fit results as a function of the AOD for two selected scenarios (top: HG phase function with AP =0.75; bottom: AP =0.60) and wavelengths are presented in Fig. 6. The rms shows a clear minimum for a given AOD, indicating that for this AOD the shape of the SZA dependence of the simulated radiances best fits the measurements. For this AOD the derived scaling factor represents the derived absolute calibration of the instrument. Interestingly, similar scaling factors are found for both scenarios, although the minimum rms is found at quite different AOD. This is a very important finding, because it indicates that our method is rather independent from the exact knowledge of the aerosol phase function. In principle also other quantities than the rms could be calculated, e.g. a function decsribing the systematic dependence of the difference between R and S⋅D (see Eq. 4) as a function of the SZA. But we did not use such an option in this study, because the differences between the derived scaling factors for the different scenarios were found to be rather small (a few percent; see below).

Figure 7 gives an overview on the wavelength dependence of the derived scaling factors. The different graphs show results for different SZA ranges (the lower boundary was varied while the upper boundary was fixed to 90∘). The most important conclusions from the results shown in Fig. 7 are that a.

the scaling factors are similar for the different scenarios,

The first finding indicates that the new calibration method is rather independent from the assumptions on the aerosol properties (and other assumed parameters; see Table 1), confirming the results presented in Fig. 6. The second finding indicates that the statistical uncertainty of the method is rather low.

Interestingly the spread of the results is largest for the largest SZA range (36–90∘). This finding first surprised us, because we expected that using more measurements should lead to more stable results. However, the larger variability of the AOD after ∼ 08:00 (corresponding to a SZA of 50∘; see Fig. 1) is probably the main cause for the larger differences for the larger SZA interval. Another reason might be that for lower SZA the scattering angle decreases, and thus the influence of the aerosol phase function increases. The fact that those results are most dependent on phase function (Fig. 7a) also seems to point in this direction. In the following we use the SZA range from 50 to 90∘.

In Fig. 8 the results for the different scenarios are compared in more detail. The upper panel of Fig. 8a shows ratios of the scaling factors for the different phase functions compared to the standard scenario. The derived scaling factors agree within about 10 %, with decreasing differences towards shorter wavelengths. Also shown are the fit results for the rms (second panel) and the derived AOD (lower panel). While the rms is similar for all phase functions, the AOD shows a large spread. Small AOD (similar to the simultaneous AERONET results) are found for HG phase functions with a small AP (0.60). The highest AOD is retrieved for the HG phase function with an AP of 0.85 and for both Mie phase functions. At 315 and 325 nm the retrieved AOD shows no meaningful results because of the influence of the ozone absorption.

Figure 8b presents results for different aerosol single-scattering albedos. For increasing aerosol absorption decreasing scaling factors are found, but for moderately absorbing aerosols (single scattering albedo ≥0.9) the differences are small (< 5 %). However, for a strongly absorbing aerosol (single-scattering albedo of 0.8) the derived scaling factors are about 5–10 % smaller compared to the standard scenario. Thus if during the measurements such strongly absorbing aerosols are present, the derived scaling factors would be by 5–10 % too small if less absorbing aerosols were assumed in the radiative transfer simulations. Fortunately, such cases could be easily identified by the large deviation of the simultaneously determined AODs from the AODs derived from sun photometers. Also the rms is slightly larger.

In Fig. 8c the effect of varying the surface albedo is shown. Again very small variations of the derived scaling factors, rms and AOD, are found. Interestingly, for the scenario with the SZA-dependent albedo, the largest rms is derived. This finding might indicate that the assumed SZA dependence indeed overestimates the influence of the changing solar illumination (see Sect. 2.2.3).

Figure 8d shows results for different atmospheric parameters (see Table 1). The effects of aerosol layer height, temperature and pressure profiles, stratospheric aerosols, and NO2 absorption have again a rather small effect on the derived scaling factors, rms and AOD. Here it is interesting to note that the effect of a combined change of temperature and pressure is slightly smaller than the effects of individual changes of pressure and temperature (not shown). Nevertheless, they are still small compared to other uncertainties. The scenario referred to as “Cabauw” includes at the same time several changes compared to the standard scenario: temperature and pressure profiles are taken for the day of the measurements, and also stratospheric aerosols and NO2 absorption are included. In addition, Raman scattering is considered. This scenario probably best describes the atmospheric conditions during our measurements. The effect of the combined changes of the Cabauw scenario on the scaling factors compared to the standard scenario is still small (< 5 %).

In Fig. 8e we show the effect of polarisation and rotational Raman scattering. The influence of rotational Raman scattering on the scaling factors is very small (< 2 %). The largest deviations occur – as expected – for wavelengths close to strong spectral variations of the solar irradiance (see Fig. 2). In contrast, the neglect of polarisation has a rather strong effect on the fit results (see also Mishchenko et al., 1994): much lower scaling factors and much higher rms and AOD compared to the standard scenario are derived.

Figure 9 presents an example of a calibrated radiance spectrum measured at 06:54 (SZA =61∘) based on the scaling factors for the Cabauw scenario. Unfortunately, we have no possibility for a direct comparison to another radiance spectrum measured at the same location and time. But in a publication by Seckmeyer et al. (2009) we found radiance spectra measured in zenith direction under similar atmospheric conditions (clear sky; SZA =62∘) on 2 May 2007 in Hanover, Germany. Unfortunately, the raw data of those spectra are no longer available. Thus we graphically copied a radiance spectrum (for a SZA of 62∘) from the original figure in Seckmeyer et al. (2009) and overlaid it on our calibrated spectrum. Note that the measurement in Hanover was scaled by a factor of 0.97 to account for the effect of the slightly different viewing geometries (exact zenith view, compared to 85∘ elevation angle of our measurement). For wavelengths > 335 nm our radiance spectrum agrees very well with the radiance spectrum measured at Hanover. Note that the sun–earth distance during the measurements in Hanover was almost the same as in our measurements (0.3 % difference). Also the AOD and the ozone column were similar: the AOD at 550 nm derived from MODIS TERRA is about 0.15 (obtained via the NASA Giovanni website: http://giovanni.sci.gsfc.nasa.gov/giovanni/). The ozone column density derived from OMI was about 330 DU (obtained from the TEMIS website: http://temis.nl/index.php).

We presented a new method for the calibration of UV–vis instruments that measure the spectrally resolved sky radiance, for example zenith sky DOAS instruments or MAX- DOAS instruments. Our method does not rely on laboratory measurements but is based on the comparison of the solar zenith angle dependence of the measured zenith sky radiance with results from radiative transfer simulations. The prerequisites for the application of our method are that the sky be clear and the aerosol optical depth be constant and low for a period of a few hours. At best, even further aerosol properties like the single-scattering albedo or the asymmetry parameter are known. This would allow the number of scenarios used for the calibration to be reduced. For observations at short wavelengths, also the thickness of the ozone layer should be constant (and known) during the measurements. We selected measurements during a period of about 4 h covering a SZA range between 50 and 90∘.

Apart from being a simple and cheap procedure, another advantage of our method is that the calibration is derived directly from the atmospheric spectra; thus the occurrence of potential changes of the instrument during the transport from the laboratory to the field can be neglected. The radiometric calibration can be determined for individual wavelengths; in this study we selected wavelengths in intervals of 10 nm between 315 and 455 nm (the wavelength range of our instrument). The calibration function was found to be spectrally smooth and can therefore be interpolated (if the intervals are not chosen too large). Alternatively, also additional wavelengths in between the chosen wavelengths could be used.

From our method also the aerosol optical depth for the selected wavelengths is determined. The comparison of the derived AOD with AOD derived from AERONET observations can be used to assess the quality of the calibration results. In particular, scenarios yielding unrealistic AOD values can be removed. If we do not take into account scenarios which yield unrealistically high AOD, the variation of the derived calibration results is further reduced. From the sensitivity studies based on different scenarios, we estimate the uncertainty of our method to be < 7 % (including the uncertainties of the solar irradiance measurement). Here it should be noted that for larger AOD larger uncertainties should probably be expected. This aspect should be investigated in future studies.

Our results clearly indicate that the radiative transfer simulations have to consider polarisation. In contrast, the effect of rotational Raman scattering can be neglected.

Our new method is of importance to many applications, including the determination of actinic fluxes, the characterisation of aerosol and cloud properties or the quantification of the shortwave radiative flux at the position of the instrument. Another interesting application might be the quantification of the yield of photovoltaic cells. For wavelengths below about 330 nm it is essential that the ozone column density during the measurements be constant and known. The accuracy of our method in the UV-B spectral range should be further explored in future studies based on measurements under constant ozone layer thickness during the period of the measurements.