Knowledge of the relative angular sensitivities of the optical receivers is the basis of assessing the uncertainties and correcting atmospheric measurements of spectral actinic flux densities. Angle-dependent sensitivity measurements were carried out in the laboratory with a goniometric setup on an optical bench, where the receivers including their aircraft flanges were positioned at different incident angles relative to a stabilized point light source (1000 W tungsten halogen lamp). Polar angles of incidence ϑ were defined here as usual in geometric optics and are indicated in Fig.  for a 2π receiver. Azimuth angles φ=0∘ refer to fixed positions on the receiver base flanges which correspond to the flight directions of the aircraft-installed receivers. Pictures of the goniometric setup are shown in Fig. S2.

Angle-dependent measurements of lamp spectra were made in a range for 0∘≤ϑ≤115∘. By extending the range beyond 90∘, the cross-talk for each receiver was investigated, including the shading effects of the aircraft-specific flanges. Azimuth angles were changed in 45∘ steps in a range 0∘≤φ≤360∘.

Following the notation introduced by , relative angular sensitivities Zp were determined by normalizing background-corrected signal spectra S with those obtained at normal incidence (ϑ=0∘): 3Zp(λ,ϑ,φ)=S(λ,ϑ,φ)S(λ,ϑ=0,φ). For an ideal receiver, Zp=1 for all wavelengths at ϑ≤90∘ and Zp=0 for ϑ>90∘.

The index of Zp indicates the use of a point light source in front of which the receiver was rotated. For a point light source, the problem is that the flux density strongly depends on distance following an inverse square law. As a consequence, for actinic radiation receivers with vertical extensions, the concept of an equivalent plane receiver is used for calibrations with irradiance standard lamps: the lamp position is adjusted for a receiver-specific distance Δz with respect to the quartz dome tip. Typical Δz values range around 20 mm for an incident angle ϑ=0∘, as indicated in Fig. . They have to be determined experimentally for each receiver to ensure accurate calibrations . In this work, Δz values were also determined for ϑ=90∘, which turned out to be smaller by 8–15 mm. The polar-angle-dependent differences correspond to small but significant signal changes that can affect the angle-dependent Zp measurements at the lamp distances used. Enhanced distances z between the lamp and receiver would be favourable to avoid this problem, but greater distances also result in smaller signals, dependent on lamp power, wavelength and the detector used.

To avoid uncertainties caused by the potentially ϑ-dependent Δz, the laboratory procedure was revised. Angle-dependent measurements were performed at two lamp distances of z=400 mm and z=800 mm with respect to the equivalent plane at ϑ=0. The final Zp values were then determined by a two-point extrapolation towards an inverse distance of zero; i.e. they correspond to a hypothetical infinite distance z. The influence of distance on the measured Zp was generally small but not negligible, at least for two of the employed receivers. More details on the experimental approach and a formal derivation of the two-point method are given in Sect. S2.1 in the Supplement.

Contour plots of the final Zp values are shown in Fig.  for the HALO top and bottom receivers for a wavelength of 400 nm as an example. Corresponding plots for the Zeppelin receivers are shown in Fig. S4. An azimuthal equal-area projection was chosen to correctly reproduce the solid-angle contributions for different polar angles relevant for actinic flux density measurements; i.e. the areas increase with the sine of the polar angle, consistent with Eq. () (dω=sin⁡(ϑ)dϑdφ). Because of the rotational symmetry of the receivers, dependencies on azimuth angles are typically minor (<5 %). Cross-talk effects are not visible in Fig. . Similar plots for the opposite hemispheres are not shown because the values are mostly zero, except for narrow ≤15∘ bands close to the horizon. Instead, Fig.  shows azimuthal mean Zp values for the HALO top and bottom receivers for selected wavelengths where the cross-talk to the other hemisphere becomes visible. This cross-talk quickly diminishes above 90∘ and vanishes at around 105∘. The Zp dependencies on polar angle and the wavelength dependence are slightly different for the different receivers but can differ by up to 15 % at greater polar angles. The properties of the 2π receivers investigated here are similar to those shown in previous work using the same type of receivers . Corresponding plots for the Zeppelin receivers are shown in Fig. S5.

For ground-based measurements, the Zp data of Figs.  and  are directly applicable for the calculation of correction functions (Sect. ). On the other hand, for airborne measurements, the combined total sensitivities of the receivers installed on the top and bottom fuselage have to be considered. As an example, Fig.  shows contour plots of total relative angular sensitivities ZpT of the FLT configuration on HALO in the upper and the lower hemisphere for a wavelength of 400 nm. The ZpT values comprise the combined effects of the Zp of top and bottom receivers, geometrical restrictions of the fields of view by the aircraft and fuselage reflections. More details on field-of-view effects and fuselage reflections are given in Sect. S2.2 and S2.3. The range of incidence angles in Fig.  was extended to 0–180∘ with ϑ=0∘ and 180∘, corresponding to zenith and nadir directions, respectively. The cross-talk effects on ZpT are most pronounced towards the aircraft sides where the field-of-view restrictions were smallest because of the curved fuselage. Towards the flight direction, the cross-talk is correspondingly smaller and also influenced by the 3.3∘ tilt angle of the aircraft (Fig. ). In the rear direction, the field of view in the lower hemisphere was for this configuration restricted by a containment on the bottom fuselage. This restriction prevented cross-talk to the upper hemisphere in a rearward section visible in panel (a) of Fig.  and causes the dark area close to the horizon in panel (b) where radiation was blocked. For ϑ<80∘ and ϑ>100∘, the ZpT values correspond to those shown in Fig. . Similar plots for the two other HALO configurations FLN and FLV as well as for the Zeppelin are shown in Figs. S8, S10 and S12.

Azimuthal averages of the data in Fig.  are plotted in Fig. a. In this representation, the contributions of the top receiver ZpZ (zenith-oriented) and bottom receiver ZpN (nadir-oriented) become visible. At ϑ>80∘, total sensitivities are enhanced (on average) by up to a factor of about 1.6 at ϑ=90∘ because radiation can strike both receivers simultaneously, caused by the non-ideal field-of-view limitations. As a consequence, radiance contributions from polar angles around 90∘ have to be corrected substantially, which also applies to direct sun actinic flux densities at low sun.

In Fig. b, relative sensitivities were multiplied by sin⁡(ϑ) to account for the solid-angle contributions, consistent with the ϑ-dependent areas in the projections of Figs.  and . In the simplest case of an isotropic radiance distribution, the data shown in Fig. b would lead to an overestimation of measured actinic flux densities that correspond to the integral of the sin⁡(ϑ)×ZpT curve divided by the integral of the ideal sin⁡(ϑ) curve. In this example, the ratio is 1.045, which is suitable to correct measurements at 400 nm, albeit under the special conditions of constant radiances. In order to obtain more realistic corrections, sensitivity distributions as shown in Fig. , as well as wavelength-dependent direct sun contributions and diffuse spectral radiance distributions, are required. The latter information is usually not available under measurement conditions. Correction functions were therefore calculated based on results from a radiative transfer model.

Distributions of diffuse spectral radiances were calculated with the radiative transfer model uvspec from the libRadtran package (version 2.0.4) . The purpose was not to obtain radiance distributions for actual measurement conditions. Rather, a range of atmospheric scenarios was created that should ideally cover all realistic measurement conditions. The main model input parameters are listed in Table . The radiative transfer equation solver DISORT in pseudo-spherical geometry was utilized with 16 streams to obtain accurate spectral radiance output suitable to calculate spectral actinic flux densities by numerical integrations . Calculations were made for 12 different solar zenith angles and an arbitrary solar azimuth angle of 180∘. The radiance output was generated with a step size of 2∘ in 0–180∘ ranges for polar and azimuth angles of incidence, resulting in 8280 spectral radiance values for each wavelength. In subsequent calculations, radiances in the azimuth range 180–360∘ were produced by inversion of the 0–180∘ results. In addition, spectral actinic flux densities for total downward, diffuse downward and diffuse upward radiation were calculated for consistency checks and as an additional input for the evaluation of correction functions (Sect. ).

All model calculations were made in the wavelength range 290–660 nm using 5 nm steps below 310 and 20 nm steps above 320 nm; i.e. the total number of wavelengths was confined to 23. This is justified because, except for the UV-B range, which is affected by stratospheric ozone, a smooth change of radiance distributions with wavelength was expected. Despite this coarse-wavelength sampling, a triangular response function with a full width at half maximum (FWHM) of 1.7 nm was adopted in the model to allow for an optional comparison of the model output with measurements .

A number of atmospheric scenarios were devised to simulate realistic measurement conditions. An atmospheric scenario was defined by a cloud case, a ground albedo case and an aerosol case. For each scenario, calculations were made for up to 11 altitudes (Table ). The total ozone column was fixed at a typical value of 300 DU for the majority of the model calculations. For selected altitudes of 1 and 10 km, additional calculations were made for 200 and 400 DU to examine the influence of ozone columns. The ground elevation was set to mean sea level except for additional clear-sky calculations at a ground elevation of 1 km and heights above ground of 0 and 1 km. Atmospheric pressure and temperature profiles were not varied. Their influence is presumed to be insignificant compared to that related to the different atmospheric scenarios.

Four cloud cases were distinguished: (i) clear-sky, no clouds (Cl); (ii) an optically thin, high-level cirrostratus layer (Cs); (iii) an optically thick medium-level altostratus layer (As); and (iv) an optically thick low-level stratus layer (St). In the model, clouds were idealized as homogeneous layers. The idea was to reproduce conditions with HALO flying below, within or above clouds at different altitudes and the Zeppelin always flying below any clouds. Cloud micro- and macrophysical properties, as well as cloud optical depth (COD), are listed in Table . These data represent typical values adopted from the literature . More details on the implementation of clouds in the model are given in Sect. S3.1.

Five ground albedo cases were considered: (i–iii) a wavelength-dependent ground albedo A typical for vegetated ground, scaled to match values of 0.02, 0.04 and 0.07 at 470 nm; (iv) a high, wavelength-independent, ground albedo of 0.8 representing snow cover; and (v) a spectral ground albedo of open water. The applied ground albedos are based on literature data . A470=0.04 is considered a standard ground albedo. The theoretical case A=0 was included for test purposes but will not be used for the calculation of correction functions. More details on the ground albedos are given in Sect. S3.2.

Three aerosol cases were implemented based on the default aerosol defined in libRadtran. The properties were varied using the option to scale aerosol optical depth (AOD) to user-defined values at selected wavelengths, in this case at 550 nm. AODs for other wavelengths were scaled accordingly, resulting in the following aerosol cases: (i) AOD550=0.03, (ii) AOD550=0.2 and (iii) AOD550=1.5. These cases cover typical atmospheric properties from very clean oceanic to strongly polluted urban continental conditions. AOD550=0.2 is regarded as the standard aerosol optical depth. The theoretical case AOD =0 was included but will also not be used to calculate correction functions. More details on the aerosol optical depth are given in Sect. S3.3.

An overview of scenarios used for the platforms HALO and Zeppelin, as well as for the ground station, is given in Table S1 in the Supplement. Not all possible combinations of cloud, albedo and aerosol cases were implemented as atmospheric scenarios. For HALO, cruise flight altitudes below 200 m are unrealistic. The 200 m cloud top height of the St layer was therefore chosen so that HALO is always above this cloud type for which the influence of different ground albedos was not evaluated. For the Zeppelin, the St cloud case was neglected because visual flight rules do not permit in-cloud flights. Rare cases where the Zeppelin could be flying above low-lying clouds or ground fog are reasonably represented by scenarios with a high, wavelength-independent ground albedo of 0.8. Altitudes below 50 m were also not considered for the Zeppelin because of the ground-shading effect of the airship itself. For ground-based measurements, all scenarios for an altitude of 0 km were taken into account, except the St cloud case because radiance distributions turned out to be sufficiently similar for St and As cloud cases at ground level. Multiple cloud layers were also not considered. Such conditions are supposed to be covered by in-cloud scenarios and combinations of Cs or As cloud cases with a high ground albedo of 0.8.

Examples of modelled diffuse radiance distributions Lλ(ϑ,φ) for the upper and lower hemisphere under clear-sky conditions are shown in Fig.  for an altitude of 5 km, a solar zenith angle of 40∘ and a wavelength of 400 nm. In this example, the relative contributions of direct, diffuse downward and diffuse upward radiation to the total spectral actinic flux density are 0.52, 0.26 and 0.22, respectively. For the same scenario, Fig.  shows azimuthal averaged spectral radiances for different wavelengths, normalized to their maximum values for better comparability. In both hemispheres, these radiances are strongly enhanced at polar angles close to the horizon, except for 300 nm where the downward radiances are almost independent of polar angles.

With regard to Figs.  and , it should be noted that for the modelled spectral radiances polar angles ϑ were redefined as angles of incidence with respect to the 4π aircraft assemblies, in accordance with the notation in the last sections. For the physical directions of propagation different polar angles (θ) apply: θ=180∘-ϑ. The same holds for solar zenith angles; for example, when the sun is located in the zenith (ϑ=0∘), the radiation is directed towards the nadir (θ=180∘). The use of angles of incidences has no consequences, except that polar angle integration limits interchange for the upper and the lower hemisphere in some of the equations given in the following section, Sect. .

Plots like those in Figs.  and  were produced for each atmospheric scenario, altitude, solar zenith angle and selected wavelength. They provide a quick overview of the variation of radiance distributions and actinic flux densities as a function of atmospheric conditions. In Figs. S17 and S18, a second example is shown for the As cloud case under conditions otherwise the same as in Fig. . Expectedly, the spectral actinic flux densities above the cloud layer are strongly enhanced by a factor of around 1.7, and the distributions are different for both upward and (to a minor extend) downward spectral radiances. Two further examples of radiance distributions at a lower altitude under clear-sky conditions and below the As cloud layer are shown in Figs. S19–S22. All model results are available for download for other users . More details are given in Sect. S3.5. The large number of model results naturally contains a lot of interesting information and phenomena. However, a more detailed analysis is beyond the scope of this work. Potential uncertainties of the model results were also not considered. Rather, the variability of naturally occurring radiance distributions is assumed to be represented realistically by the different atmospheric scenarios.

For solar zenith angles approaching 90∘, modelled spectral radiances will become unrealistic because diffuse radiation was calculated in plane-parallel geometry, while for direct radiation, a pseudo-spherical correction was applied in the model. On the other hand, radiance distributions were found to change smoothly on a relative scale, even at large solar zenith angles. Modelled radiance distributions for solar zenith angles of up to 85∘ are therefore considered useful, but, except for ground-based measurements, the correction procedure will be limited to solar zenith angles smaller than 80∘ anyway (Sect. ).

Regardless of the more general definition given in Eq. (), total solar spectral actinic flux density Fλ can be separated into direct and diffuse components (e.g. ): 4Fλ=Fλ,dir+Fλ,dif=Fλ,dir+∫02π∫0πLλ(ϑ,φ)sin⁡(ϑ)dϑdφ. For brevity the indication of the wavelength dependency of Fλ and Lλ variables was omitted here. Measurements can be simulated by calculating uncorrected spectral actinic flux densities Fλ,m using the receiver assemblies' relative angular sensitivities ZpT: 5Fλ,m=ZpT(ϑ∘,φ∘)Fλ,dir+∫02π∫0πZpT(ϑ,φ)Lλ(ϑ,φ)sin⁡(ϑ)dϑdφ=ZSTFλ. Angles are defined as angles of incidence, and ϑ∘ and φ∘ are corresponding solar zenith and azimuth angles, respectively (Sect. ). Accordingly, the ZpT values have to be rotated horizontally to match the actual situation, dependent on the receiver heading and the solar azimuth angle. By analogy with the hemispherical correction function ZH introduced by , the right-hand side of Eq. () defines a spherical correction function ZST=Fλ,m/Fλ for measured total spectral actinic flux densities. Because upward and downward Fλ values are determined separately, and information on their contributions is relevant, hemispherical corrections functions ZH are defined as well: 6Fλ↓=Fλ,dir+Fλ,dif↓=Fλ,dir+∫02π∫0π/2Lλ(ϑ,φ)sin⁡(ϑ)dϑdφ 7Fλ,m↓=ZpZ(ϑ∘,φ∘)Fλ,dir+∫02π∫0πZpZ(ϑ,φ)Lλ(ϑ,φ)sin⁡(ϑ)dϑdφ=ZHZFλ↓

8Fλ↑=Fλ,dif↑=∫02π∫π/2πLλ(ϑ,φ)sin⁡(ϑ)dϑdφ

9Fλ,m↑=ZpN(ϑ∘,φ∘)Fλ,dir+∫02π∫0πZpN(ϑ,φ)Lλ(ϑ,φ)sin⁡(ϑ)dϑdφ=ZHNFλ↑. Downward and upward Fλ values are indexed by downward- and upward-pointing arrows, respectively. The hemispherical correction functions ZHZ=Fλ,m↓/Fλ↓ and ZHN=Fλ,m↑/Fλ↑ refer to the zenith-oriented (Z) and nadir-oriented (N) top and bottom receivers on the upper and lower fuselage, respectively. Equations ()–() apply to conditions ϑ∘≤90∘; i.e. no cases with upward direct radiation are considered, but direct radiation unintentionally captured by the bottom receiver is included in Eq. ().

An important constraint for the three correction functions is that total and hemispherical corrections are related to each other, dependent on the contributions of downward and upward actinic flux densities: 10ZSTFλ=ZHZFλ↓+ZHNFλ↑. Any final correction should comply with this equation to satisfy the general budget equation: 11Fλ=Fλ↓+Fλ↑.

For the special case of ground-based measurements of downward spectral actinic flux densities, the integration limits can be confined to the upper hemisphere if local upward radiation is negligible (low local ground albedo) or effectively shielded (extended artificial horizons): 12Fλ,m↓G=Zp(ϑ∘,φ∘)Fλ,dir+∫02π∫0π/2Zp(ϑ,φ)Lλ(ϑ,φ)sin⁡(ϑ)dϑdφ=ZHGFλ↓.

The corresponding correction functions were named ZHG=Fλ,m↓G/Fλ↓ and apply to the ground-station setup of the four receivers (Sect. ); i.e. the Zp values in Eq. () correspond to those of the individual receivers (Figs.  and S4). Other ground-based applications will be discussed in Sect. .

The ground-station ZHG of the four receivers and the three correction functions ZST, ZHZ and ZHN for the airborne platforms were calculated for the atmospheric model scenarios and altitudes summarized in Table S1. To avoid inaccuracies, numerical integrations were done after interpolating the variables to sufficiently high angular resolutions (≤1∘). The procedures were verified by comparing the numerically calculated Fλ,dif↓ and Fλ,dif↑ with the first-hand model output for these integrated quantities. The influence of different azimuthal positions of the sun was investigated by repeating the calculations after the spectral radiance distributions were rotated in φ=2∘ steps until a full circle was accomplished; i.e. all possible receiver headings with respect to the sun were tested (180 calculations). Uncertainties for each calculation were obtained based on the uncertainty estimates of the Zp variables (Sect. S2.1) and of fuselage reflectivity, if applicable (Sect. S2.3).

An example of corrections derived for a Zeppelin flight under clear-sky conditions is shown in Fig. . On this day, the airship followed a quasi-stationary circular flight pattern for about 4 h during which six height profiles were flown between about 100 and 800 m above agricultural land in the Po valley, Italy, during the PEGASOS campaign . The ZHN values show a wavelength-dependent periodic pattern induced by the altitude changes. On the other hand, the ZST and ZHZ and their uncertainties remain almost constant for a given wavelength within this flight's range of solar zenith angles.

The altitude dependence and the magnitude of the ZHN decrease with wavelength, which is explainable by increasing ground albedos over vegetated ground (Fig. S14) and decreasing diffuse sky radiance in the upper hemisphere captured by the bottom receiver. However, despite values of around 2 for the ZHN in the UV range, the ZST are merely increased by about 5 % compared to the ZHZ, which is reasonable if only small fractions of the total actinic flux densities are directed upward.

The final total, downward and upward spectral actinic flux densities are shown in Fig. , together with their total uncertainties and those resulting from the corrections. The latter are dominant for the upward component but less significant for the total and downward components. The different dependencies of the Fλ, Fλ↓ and Fλ↑ on altitude and solar zenith angle as a function of wavelength are qualitatively explainable. The increase of the Fλ↑ from 300 to 600 nm at the lowest altitudes is caused by the increasing ground albedos. On the other hand, the increase of the Fλ↑ with altitude is stronger for shorter wavelengths because of increased backscattering in the air column between the ground and the airship (Rayleigh and aerosol scattering). Increased scattering at shorter wavelengths also explains the different dependencies of the Fλ on solar zenith angles. In addition, the influence of stratospheric ozone enhances the solar zenith angle dependence for 300 nm. Expectedly, photolysis frequencies show similar patterns dependent on the wavelength range of the photolysis reactions. However, a more detailed analysis of photolysis frequencies is beyond the scope of this study.

For HALO flights, the spatial and atmospheric condition ranges were typically much greater than for the Zeppelin. An example is shown in Fig.  where HALO performed a 9 h non-stop return flight from Taiwan to Japan over the East China Sea during the EMeRGe-Asia campaign. Several flight levels between 0.5 and 12 km were operated on this day under changing, partly cloudy atmospheric conditions. Again, the ZHN turned out to be most variable and uncertain, dependent on altitude and wavelength, but generally smaller compared to the Zeppelin. Minor, short-term variations at constant altitudes indicate sporadic cloud influence. Gaps in the data record mark periods where flight manoeuvres led to attitude deviations that exceeded the HALO-specific limit of 2.5∘. Towards the end of the flight, solar zenith angles approached 80∘, resulting in increased uncertainties of the ZHN at longer wavelengths.

The final spectral actinic flux densities and their uncertainties are shown in Fig. . The uncertainties from the corrections are again more significant for the Fλ↑, especially at low altitudes. Flux densities and uncertainties reveal a complex dependence on altitude, solar zenith angle and cloud presence for the selected wavelengths. The variability of the Fλ↑ is strongly enhanced, and values can become as high as the Fλ↓ through cloud influence. Accordingly, the total Fλ values increase during such periods. Cloud influence on Fλ↓ is hardly visible in this specific flight but is clear in others, in particular at low altitudes. Because of wider ranges, the influence of altitude and solar zenith angles is greater than in the Zeppelin example. The minor differences between 500 and 600 nm are explainable by similar scattering properties of air, aerosols and clouds, as well as similar ocean albedos. An analysis of these data with the help of radiative transfer model calculations is currently under preparation but beyond the scope of this work. The corresponding photolysis frequencies again exhibit very similar, wavelength-dependent patterns. However, because of the greater altitude range, for some photolysis frequencies, the additional influence of temperature and pressure variations, affecting absorption cross sections and quantum yields, can become significant (Eq. ).

The correction functions ZHG for measurements of downward spectral actinic flux densities are comparable with previous results for other receivers from the same manufacturer . Except for one receiver and wavelengths >500 nm, the corrections remained below 10 %, with maximum uncertainties below 3 %. Moreover, for the four receivers used in this work, similar corrections were obtained using radiative transfer modelled and isotropic diffuse radiance distributions in the upper hemisphere. This result probably also holds for other receivers with comparable properties, which simplifies the calculations. However, this does not mean that corrections for ground-based measurements are generally straightforward or secondary. Substantial corrections and large uncertainties can result for receivers with poorer reception characteristics , and, as already mentioned in the Introduction, the basically high accuracy of radiometric calibrations can be significantly degraded by uncertainties of receiver-related corrections. This issue may even remain unnoticed unless the quality of receivers is thoroughly tested. On the other hand, as shown in Fig. S28, a constant correction factor covering all conditions can be sufficient in the UV range. This is of relevance for measurements of j(O1D) and j(NO2) with filter radiometers. If a calibration of these instruments is done by comparison with a corrected reference instrument, receiver related mean corrections are already included . In contrast, in the VIS range where significant contributions of direct radiation are possible at large solar zenith angles, further refinements can be helpful. The potential presence or absence of direct radiation increased the uncertainties of the ZHG when all atmospheric scenarios were included (Fig. S26). Therefore, uncertainties can be reduced if conditions with and without direct radiation are distinguished, either based on the measurements themselves, by the use of auxiliary instruments (sky cameras, pyrheliometers or sunshine recorders), or on a separate determination of the contribution of diffuse sky radiation. The latter is feasible using a classical shadow ring, a sun tracker or a rotating shadow band (only one receiver required). Such approaches would, for example, be useful for a more accurate determination of j(NO3) (λ≤640 nm) at low sun.

Generally, for measurements of downward spectral actinic flux densities, the cross-talk to the lower hemisphere should be minimized by sufficiently large artificial horizons, dependent on the local ground albedo, as already noted by , who estimated corrections of up to 15 % for a ground albedo of 0.9 (fresh snow) with a 150 mm diameter artificial horizon. As a consequence, the size of the artificial horizon (shadow ring) was doubled in subsequent applications of the same instrument (e.g. ).

Ground-based measurements of upward spectral actinic flux densities may be desirable as well, for example, at sites with regular snow cover. However, useful measurements of upward spectral actinic flux densities are challenging. First, downward-facing receivers capture the reflective properties of the natural or artificial ground at close range which may be different from the ground in the surrounding area. A careful selection of the location is therefore important. If no suitable location is available, an estimation of upward from measured downward flux densities is possible based on typical ground albedos in the area . Second, also a downward-facing receiver should be equipped with a large artificial horizon to prevent (i) cross-talk to the usually brighter, upper hemisphere and (ii) reception of direct solar radiation at low sun, although this is a minor problem in the UV range as mentioned in Sect. . The situation on the ground is comparable with the Zeppelin at very low altitudes, where the limited size of the extension flange produced overestimations by a factor of 2–3 in the UV range (Sect. ). Similar overestimations are expected on the ground (at low ground albedos) unless the upper hemisphere is effectively shielded. Of course, if required, a 4π correction approach like for the Zeppelin can be implemented for a single, zero height above ground.

The correction functions ZST, ZHZ and ZHN for the Zeppelin and HALO typically produce changes no greater than 5 %–10 %. An exception are the ZHN values at low altitudes and low ground albedos, which can become significantly greater. The results are comparable with corrections applied in the literature for other airborne platforms. A direct comparison with previous work is difficult because the corrections are specific for each experimental setup and the individual receivers employed.

used a prototype of the quartz dome receivers employed since then to measure j(NO2) with filter radiometers (370±40 nm) on board a Lockheed C-130. The diameter of the base flanges was limited to 200 mm, and the authors optimized the total angular sensitivity with circular rims at the flange edges acting as artificial horizons. The performance of the 4π reception characteristics was tested in-flight by dedicated circular flight patterns with roll angles of 30∘ at different solar zenith angles, which merely resulted in small variations of the total radiometer signals. From these test flights, uncertainties of the total j(NO2) caused by the 4π receiver imperfections of 1.5 % and 6 % were estimated for solar zenith angles below and above 75∘, respectively. For downward and upward contributions under horizontal flight conditions, altitude-dependent correction factors in a range 1.00–1.04 and 0.69–1.01 were derived, respectively, with uncertainties of 2 % and 5 %–12 % at solar zenith angles ≤75∘. These factors, which correspond to reciprocal values of the ZHZ and ZHN defined in this work, were derived based on radiative transfer calculations, including the polar angle dependence of diffuse radiances, though confined to clear-sky conditions. In qualitative agreement with the results presented here, the corrections for the upward component increased with decreasing altitude, leading to a minimum factor of 0.69 (ZHN=1.45) close to the ground.

employed a similar setup as on a Douglas DC-8 for spectral actinic flux density measurements in a range 280–420 nm. No wavelength dependencies of angular sensitivities were detected and the effects of receiver imperfections were calculated assuming isotropic radiance distributions of diffuse sky radiation in both hemispheres. Average corrections of 1.036 and 1.027 which correspond to the ZST were derived for the UV-B and UV-A range, respectively, independent of measurement conditions, with an estimated uncertainty of 4 %. Because the work focused on photolysis frequencies from total spectral actinic flux densities, no separation of upward and downward components was done. In a follow-up study by , the DC-8 inlet configuration was modified and equipped with larger 300 mm artificial horizons (including rims), which resulted in close to ideal angular responses in both hemispheres. Consequently, no corrections were applied for total, downward and upward spectral actinic flux densities, and the remaining uncertainty was estimated to be 1.5 %. The distinction of upward and downward contributions was confined to conditions where aircraft pitch or roll angles did not exceed ±5∘. A second, similar setup as on the DC-8 was installed on a Lockheed P-3B aircraft , and in-flight intercomparisons of the two instruments confirmed good agreements of j(O1D) and j(NO2) from total spectral actinic flux densities within 2 % .

made clear-sky spectroradiometer measurements on a Falcon-20E aircraft in a range 280–420 nm. Similar to HALO, the smaller size of the aircraft did not allow for extended artificial horizons, and the upward- and downward-looking receivers were tilted in the flight direction by ±5∘ to compensate for the typical pitch angle. The overall angular sensitivity of the receiver assembly was comparable with that described in this work. The consequences of the non-ideal 4π behaviour were investigated by radiative transfer calculations, including spectral radiance distributions under the measurement conditions. The deviations for total spectral actinic flux densities ranged from +1.4 % (0.1 km) to +3.6 % (12 km) at solar zenith angles <23∘ under clear-sky conditions. From these calculations, a maximum 4 % overestimation (ZST=1.04) was derived, but no corrections were applied. Upward and downward components were not distinguished.

performed spectral actinic flux density measurements in a range 305–700 nm on a Partenavia P68-B in an altitude range below about 3 km. These authors used a stabilization system that kept the receivers horizontal within ±0.2∘ as long as pitch or roll angles did not exceed ±6∘. This system was originally designed for an accurate distinction of upward and downward spectral irradiances . The size of the artificial horizons was limited by the stabilization system to a diameter of about 130 mm. Consequently, the mutual cross-talk was significant and corrected for separately for the upward- and downward-looking receivers by adopting the concept of hemispherical correction functions using isotropic diffuse radiance distributions . The wavelength and altitude dependence was investigated for clear-sky and cloudy conditions. For the downward component, a maximum correction of around 1.08 (i.e. ZHZ) was obtained in the VIS range for an altitude of 2 km, above a highly reflective cloud. For the upward component, a maximum correction of around 1.35 (i.e. ZHN) was reported in the UV range for an altitude of 1 km under clear-sky conditions using a surface albedo of 0.08. The final corrections were made along the flight tracks by attributing measurement conditions to the modelled scenarios. The uncertainty of these corrections was estimated to be 2 %.

made spectroradiometer measurements on a modified Lockheed WP-3 aircraft covering a wavelength range 280–690 nm. The setup followed that of using a 300 mm artificial horizon with an outer rim. A correction function corresponding to the ZST was estimated for isotropic radiation, ranging between about 0.99 for 300 nm to 0.95 for 600 nm. These corrections were applied independent of measurement conditions which was accounted for by an additional 3 % error. Upward and downward components were not distinguished.

Generally, on bigger aircraft, the base flanges that form artificial horizons can be larger without imposing aerodynamic issues. Under these circumstances, negligible corrections within small uncertainties can be achieved, as demonstrated by . Moreover, a combination of two virtually ideal 2π receivers is expected to perform independent of aircraft attitude, as long as only total actinic flux densities are of interest . On the other hand, even with two perfect hemispheric receivers, a distinction of upward and downward flux densities requires close to horizontal flight conditions or an active stabilization .

For HALO, the mutual cross-talk of the receivers and aircraft-specific field-of-view effects were more significant than in most previous studies, which motivated the extended correction approach of this work. The effort is justified because of the large number of HALO flights for which corrections are required, including further campaigns scheduled in the future. For the Zeppelin, mainly the cross-talk of the downward-facing receiver to the upper hemisphere was significant and produced enhanced ZHN under conditions with low ground albedo. The distinct dependence of the ZHN on the parameter Φm was instructive to derive the parametrization concept, which also proved to be useful for HALO. The main advantage of the parametrizations is that no potentially uncertain or unavailable information on the atmospheric state is required. Moreover, because different wavelengths are treated separately, it is irrelevant whether or not the wavelength dependencies of ground albedos and aerosol optical depths in the model scenarios are realistic for the measurement conditions.

The use of isotropic radiance distributions for the calculation of the corrections led to slightly different results and cannot be recommended in general because the extent of the differences depends on receiver properties and atmospheric conditions. The computational effort to derive the corrections is slightly lower, but a wide range of conditions with different contributions of direct sun should be covered anyway. Moreover, under below-cloud conditions, the assumption of isotropic radiances is clearly unrealistic for the upper hemisphere. Analytical expressions exist for the polar angle dependence of radiances under overcast conditions that can be easily implemented instead of constant radiances (e.g. ).

For the determination of total actinic flux densities and photolysis frequencies alone, the strict limitations with regard to aircraft attitudes of 2.5 or 5∘ can be relaxed in order to increase data coverage. Uncertainties for total actinic flux densities could be determined for greater maximum attitudes, or alternatively, corrections and uncertainties could be calculated as a function of attitude. However, as is evident from the example flights shown in Figs.  and , the current attitude limitations are not critical for Zeppelin and HALO measurements.

The application of the parametrizations was limited to conditions with solar zenith angles ≤80∘ because corrections for different atmospheric conditions become too variable when direct sunlight can strike both receivers simultaneously. This limitation affected a minor fraction of research flights on both HALO and the Zeppelin, but occasionally conditions with very low sun or day-to-night transitions were encountered. A reliable correction under such conditions would require an estimate of the contribution of direct sunlight (ideally based on the measurements themselves) and accurate radiative transfer model calculations at low sun, including solar zenith angles >90∘. As mentioned in Sect. , the currently applied radiative transfer model in plane-parallel geometry will not give reliable results at low sun. The libRadtran package offers solutions in spherical geometry with advanced Monte Carlo solvers, but these calculations are computationally more demanding. Moreover, a concept of how to practically combine the model results with the measurements to derive useful corrections has not been developed so far but may be worthwhile if twilight conditions become of greater interest, for example, for an accurate determination of j(NO3).