Broadband radiometers, which are based on thermopile sensors, use the temperature increase in the illuminated receiver area compared to a shaded reference area as the primary measure. A simplified drawing and the radiative budget of a broadband radiometer are illustrated in Fig. . Based on the Seebeck effect, which describes the thermopile sensitivity α (unit: V K-1), the resulting temperature difference ΔT between the sensor surface temperature Ts and the reference area temperature Tref generates a detectable voltage Uth, which is used to compute the irradiance: 1Uth=α⋅Ts-Tref=α⋅ΔT.

In most cases, the temperature of the reference area is measured with a standard temperature sensor, e.g., Pt-100 or thermistor. The sensor itself absorbs the incoming irradiance Fin and also emits radiation Fout in the thermal–infrared wavelength range. In the case of pyrgeometers, the emission is a major issue. From a simplified energy budget of the thermopile sensor, the total effect of the net irradiance Fnet,dyn on the sensor can be described by 2Fnet,dyn=Fin-Fout=C⋅∂Ts∂t+K⋅ΔT, with C the heat capacity of the sensor surface (unit: J m-2 K-1) and K the thermopile thermal conductance (unit: W m-2 K-1).

Assuming thermal equilibrium, ∂Ts/∂t=0, Eq. () reduces to the net irradiance Fnet,stat in static conditions: 3Fnet,stat=Fin-Fout=K⋅ΔT. Following the simplified radiative budget of a broadband radiometer as shown in Fig. , the standard formulation of the calibration equation of broadband radiometers in radiative equilibrium is postulated e.g.,. The incoming and outgoing irradiances Fin and Fout at the sensor surface can be expressed independently by 4Fin=τd,sol⋅Fsol+τd,ir⋅Fir+ϵd,ir⋅σ⋅Td4+ρd⋅Fout,5Fout=ϵs,ir⋅σ⋅Ts4+ρs⋅Fin, with τd,sol the solar transmissivity of the dome, Td the temperature of the dome, and τd,ir and ϵd,ir the thermal–infrared transmissivity and emissivity of the dome. The sensor is characterized by the sensor emissivity ϵs,ir and temperature Ts. σ is the Stefan–Boltzmann constant. The sensor reflects the incoming irradiance Fin with the reflectivity ρs, while the dome can reflect the outgoing irradiance Fout with a reflectivity ρd. For simplicity, all transmissivities, emissivities, and reflectivities are broadband quantities; spectral dependencies of the materials are not considered. Assuming that ρs⋅ρd≪1 and using the following assumption for the polynomial of fourth order to replace Ts in Eq. () by Tref , 6Ts4=Tref+1α⋅Uth4≈Tref4+4⋅Tref3⋅1α⋅Uth, the equations finally can be resolved for the thermal–infrared and solar incident irradiance: 7Fir=Uth⋅Kα⋅ϵs,ir⋅τd,ir⋅1+4⋅σ⋅Tref3K⋅ϵd,ir/τd,ir+1⋅ϵs,ir⋅τd,ir+σ⋅Tref4+ϵd,irτd,ir⋅σ⋅Tref4-Td4-Fsol⋅τd,solτd,ir,8Fsol=Uth⋅Kα⋅ϵs,ir⋅τd,sol⋅1+4⋅σ⋅Tref3K⋅ϵd,ir+τd,ir⋅ϵs,ir⋅τd,sol+ϵd,irτd,sol⋅σ⋅Tref4-Td4+τd,irτd,sol⋅σ⋅Tref4-Fir.

The last term in Eqs. () and (), the so-called longwave and shortwave leakage, is only a function of the incident irradiance and the ratio between solar and thermal–infrared transmissivity of the dome, which are determined by material properties of the dome and/or filter coating. There is evidence for the existence of such errors due to spectral imperfections of the dome, and possible corrections were developed e.g., . By careful selection of the dome material and coating (low τd,sol and high τd,ir for pyrgeometers and high τd,sol and low τd,ir for pyranometers), this error can be minimized or even neglected as confirmed by long-term comparison of different shaded and illuminated pyrgeometers . For pyranometers, the longwave leakage is correlated with the net thermal–infrared irradiance measured by a pyrgeometer (third term in Eq. ) and can thus hardly be distinguished from the thermal dome offset (second term in Eq. ).

Neglecting this leakage effect, Eqs. () and () can be reduced to the commonly known formulas . For the thermal–infrared irradiance measured by pyrgeometer it is 9Fir=A1⋅Uth⋅1+A2⋅σ⋅Tref3+σ⋅Tref4-A3⋅σ⋅Td4-Tref4,A1=Kα⋅ϵs,ir⋅τd,ir,A2=4K⋅(ϵd,ir/τd,ir+1)⋅ϵs,ir⋅τd,ir,A3=ϵd,irτd,ir, with the parameters A1, A2, and A3 summarizing the instrument characteristics. If the temperature dependence of the thermopile sensitivity, A2⋅σ⋅Tref3, is compensated for electronically within the radiometer, the first term of Eq. () can be further reduced, leading to the formulation by : 10Fir=1air⋅Uth+σ⋅Tref4-bir⋅σ⋅Td4-Tref4,1air=A1⋅1+A2⋅σ⋅Tref3,bir=A3, with air the adjusted pyrgeometer thermopile sensitivity (unit: V (W m-2)-1) and bir the pyrgeometer dome factor A3. The last term, also known as the window heating offset, corrects for a thermal imbalance between the dome and sensor surface mainly caused by solar radiative heating of the dome in static conditions. The dome factor bir in Eq. () theoretically defines the ratio of thermal–infrared emissivity ϵd,ir to transmissivity of the dome τd,ir. showed that the dome factor, experimentally determined from a black-body calibration of the instrument, yields significantly higher values than expected from theory. Only by using data obtained in thermal equilibrium is the theory fulfilled. This indicates that the commonly used higher dome factor implies non-equilibrium effects. Optimizing the thermal design of the radiometer can reduce the window heating offset such that a dome temperature measurement can be omitted and no dome factor is needed .

Applying a similar transformation, the solar irradiance measured by pyranometers in Eq. () reduces to 11Fsol=1asol⋅Uth-bsol⋅σ⋅Td4-Tref4,1asol=Kα⋅ϵs,ir⋅τd,sol⋅1+4⋅σ⋅Tref3K⋅(ϵd,ir+τd,ir)⋅(ϵs,ir⋅τd,sol),bsol=ϵd,irτd,sol. The adjusted pyranometer thermopile sensitivity asol (unit: V (W m-2)-1) includes the weak temperature dependence of the thermopile as defined in theory by Eq. (), which can often be compensated for by the construction of the radiometer or determined in extended laboratory calibrations.

The static pyranometer thermal dome effect is scaled with the dome factor bsol and the temperature difference between the dome and sensor. This effect is often called the zero or dark offset since it is mainly caused by radiative cooling of the dome and is best visualized as a negative offset during night measurements in the absence of solar irradiance. A second dome with high thermal conductivity, e.g., quartz, in good thermal contact with the instrument housing can reduce this error to a few watts per square meter (W m-2) . Ventilation of the dome can further reduce the zero offset . If the thermal dome effect cannot be neglected, available correction methods are applied. These have been developed based on either an additional dome temperature measurement or the simultaneous measurement of the net thermal–infrared irradiance by a pyrgeometer e.g., .

The assumption of thermal equilibrium is valid for standard ground-based measurements with slowly varying environmental conditions. However, if the radiometers are subject to fast temperature changes like during airborne measurements, e.g., during ascents and descents, the slow adjustment of the sensor temperature (first term in Eq. ) needs to be considered. An offset voltage ΔUs and offset irradiance ΔFs will be generated by the thermal lag between the reference and the sensor, which is initiated by the thermal conductance and capacity of the sensor.

Replacing the sensor temperature Ts in Eq. () by the reference temperature, Ts=Tref+ΔT, the thermal reaction of the sensor to an outside temperature change can be described by 12Fnet,dyn=C⋅∂Tref∂t+∂ΔT∂t+K⋅ΔT. The assumption that ΔT changes much less than Tref leads to the dynamic sensor thermal offset ΔFs defined as the difference between the net irradiance in static Fnet,stat and dynamic conditions Fnet,dyn: 13ΔFs=Fnet,stat-Fnet,dyn=-C⋅∂Tref∂t. This error correction term for dynamic temperature changes is proportional to the time derivative of the reference temperature. ΔFs often is called “zero offset B” or “zero offset due to temperature change” and is mostly specified by the instrument manufacturer for a fixed temperature change of 5 K h-1. However, during airborne observation, especially during ascents and descents, faster temperature changes of the order of Kelvin per minute (K min-1) occur.

A similar behavior is expected for the dome, leading to a slow adjustment of the dome temperature Td. Due to the different thermal properties of the dome and sensor, the dynamic thermal offsets in both parts do not compensate. The dynamic dome effect ΔFd can then be expressed as 14ΔFd=σ⋅Tref4-Tref+ΔTd4. As indicated by , the temperature difference of the dome ΔTd depends linearly on the temporal change in Tref: 15ΔTd=γ⋅∂Tref∂t, with the coefficient γ (unit: s) characterizing the relationship. Assuming ΔTd≪Tref and approximating the fourth-order polynomial similar to Eq. (), the dome effect reduces to 16ΔFd=4⋅σ⋅γ⋅Tref3⋅∂Tref∂t. Adding both effects (Eqs.  and ), the total dynamic thermal offset ΔF is described by 17ΔF=ΔFs+ΔFd=4⋅σ⋅γ⋅Tref3-C⋅∂Tref∂t=β⋅∂Tref∂t. Based on the initial assumption that Td and Ts can be related linearly to Tref, Eq. () indicates that the thermal offset can be linearly parameterized by the change rate of the sensor reference temperature, providing the dynamic thermal offset correction coefficient β (unit: W m-2 K-1 s). During data post-processing, such a parameterization can be applied to correct the irradiance measurements in high dynamic conditions.

For the measurements of upward and downward broadband irradiance, F↓ and F↑, separated into the solar and thermal–infrared spectral range, BACARDI combines two sets of Kipp & Zonen pyranometers (CMP22) and pyrgeometers (CGR4). The CMP22 pyranometers detect radiation in the wavelength range of 0.2–3.6 µm, which covers almost the entire solar spectral range . The CGR4 pyrgeometer is sensitive to wavelengths between 4.5 and 42 µm, covering a large fraction of thermal–infrared radiation . Both radiometers use thermopile sensors, providing a sensitivity in the range of 10 µV (W m-2)-1. The radiometric calibration of the radiometers, which refers to the entire solar and thermal–infrared spectral range is repeated regularly by the manufacturer a few months in advance of a HALO measurement campaign. The radiometers are calibrated as a secondary standard (Class A) through comparison with a reference instrument traceable to the World Radiation Center. For the pyranometers, this comparison is done in the laboratory, and for the pyrgeometers, the comparison is performed outside under mainly cloud-free conditions during nighttime. Additionally, for both radiometers the temperature dependence of the thermopile sensitivity is determined within a climate chamber for the temperature range of -40 to 50 ∘C. Calibration uncertainties typically range below 1 % for the CMP22 pyranometers and 4 % for the CGR4 pyrgeometers (2 times the standard deviation confidence level). The temperature dependence of the thermopiles does not exceed 0.5 % for a wide temperature range (-30 to 50 ∘C). To track the sensor temperature, each radiometer is equipped by the manufacturer with a platinum (Pt-100) resistance thermometer.

The respective quartz and silicon domes function as wavelength bandpass filters and are characterized by a cosine response, which is less than 1 % off from theory over the entire 180∘ field of view . The optimized thermal design of both radiometers reduces the window heating offset to less than 4 W m-2 and makes them suited for aircraft operation. However, the time response of the radiometers needs to be considered for airborne measurements. As specified by the manufacturer, the CMP22 pyranometer typically reacts quicker with a 1/e (63 % adjustment) response time of about 2 s, while the CGR4 pyrgeometer is characterized by a response time on the order of 6 s.

The CMP22 and CGR4 radiometers do not contain any internal signal conditioning and only provide a low-voltage (in the range of 10 µV (W m-2)-1) thermopile signal and a four-wire Pt-100 temperature signal.

To mitigate the effects of electromagnetic noise, the wiring of the low-voltage signal is as short as possible and a signal conditioning unit is used. This unit is placed on the radiometer mounting plate inside the fuselage, where it is protected by an electromagnetic compatibility shielding metal box. The signal conditioning is based on isolated Dataforth 8B modules that are plugged into a backplane. The 8B30-02 module with an input range of ±50 mV is used for amplification of the thermopile signals up to ±5 V, with an accuracy of 0.05 %. The four-wire Pt-100 resistance is translated into a voltage (0–5 V) by the 8B35-01 module, which covers the temperature range ±100 ∘C with an uncertainty of ±0.2 ∘C.

The output voltage signals have a bandwidth of 3 Hz and are recorded by the HALO basic data acquisition system BAHAMAS BAsic HAlo Measurement And Sensor system; with a 10 Hz data rate and 18-bit resolution. A signal path calibration is performed after aircraft installation of the radiometers, which includes all wiring, connectors, electronics, and the data acquisition. For the calibration, the radiometers are replaced by either a high-precision constant voltage source (Burster 4463) to simulate the thermopile output or a high-precision resistance decade (Burster 1427) to simulate the Pt-100. Both calibration references are set to values covering the operating range of the radiometer and Pt-100 thermometer by a computer-controlled calibration routine. The calibration factors are implemented in the first post-processing step of the BACARDI raw data.

The integration of BACARDI on HALO uses the standard 10 in. × 7 in. fuselage apertures. Because HALO is equipped with four upper and six lower central apertures, some flexibility in installation depending on the layout of the actual scientific instrumentation is given. A drawing and visualization of one BACARDI sensor package is shown in Fig. . The mounting plates, to which the radiometers are attached, compensate for the mean pitch angle of the HALO aircraft, which amounts to about -3∘ in normal flight conditions. To reduce the cable length between the radiometers and the electronics (amplifier and Pt-100 conditioner), the electronics housing is attached to the mounting plate on the opposite side of the radiometers inside the fuselage.

The radiometers of BACARDI are in an aerodynamic fairing to minimize the environmental influence on the radiation measurement by, e.g., ice aggregation or water droplet impact and heating by solar radiation. To minimize aerodynamically induced temperature gradients across the instrument, a passive ventilation of the fairing is implemented to keep the instruments close to thermal equilibrium with its surrounding environment. The ventilation is designed to divert the main airflow containing droplets or particles around the radiometer housings. The fairing exhaust acts as a water drain and avoids entrapment of water inside the fairing. Thus, the design of the fairing for the upward-looking radiometers slightly differs from that for the downward-looking radiometers.

Figure  shows measurements of all four sensor temperatures compared to the ambient temperature measured on HALO. In general, the sensor temperatures are higher, especially in cold conditions, due to low heat transfer in the rather low-density air and the heat conduction from the cabin. Temperature adjustments to changes in ambient temperature (change in altitude) significantly lag in time and may lead to thermal offsets as discussed in Sect. . This is most prominently indicated by the hysteresis between ascent (upper branch) and descent (lower branch) in Fig. . However, comparing only the sensor temperatures, the differences are larger between the pyranometers and pyrgeometers than between the upper and lower setup. This indicates that temperature adjustments are rather a matter of the internal sensor housing of CGR4 and CMP22, their internal heat transfer, and the mounting order (CGR4 mounted in front of CMP22). The ventilation within the fairing is similar in the upper and lower sensor package.

To enable maintenance work, e.g., changing desiccant cartridges and signal calibration, easy access to the radiometers is considered necessary. Therefore, the upward-looking radiometers can be detached from inside the cabin without removing the fairing, whereas for the downward-looking radiometers, it is sufficient to dismount the fairing.

The calibration of the pyranometers and pyrgeometers provided by the manufacturer includes tracking changes in the thermopile sensitivity with changing instrument temperatures. For the CMP22 pyranometers, the change in the sensitivity is estimated in the range of ±0.3 % for the temperature range between -20 and 50 ∘C. Significantly lower sensitivities of up to -2 % are registered when temperatures reach -40 ∘C. For the CGR4 pyrgeometers, lower differences in the sensitivity are reported. Here, deviations do not exceed ±0.5 %, with the largest positive biases observed for the lowest and highest temperatures and slight negative offsets in between.

Figure a shows the sensor temperatures of all four radiometers and the ambient static air temperature for the EUREC4A flight on 22 January 2020 (flight ID HALO-0122). Except for takeoff and landing, the flight altitude and, thus, the ambient temperature changed only for one flight section when HALO climbed to 13.5 km altitude. At cruising altitude, minimum ambient temperatures down to -60 ∘C were observed. However, due to thermal conduction from the aircraft, the sensor temperatures remained significantly higher and did not reach -40 ∘C, which is the lower boundary of the calibration certificate. The same holds for all other flights during EUREC4A. In other environments, e.g., Arctic conditions, in which low temperatures are reached at lower altitudes with higher air densities, extending the calibration to lower temperatures needs to be considered as demonstrated by .

The effect of the temperature dependence on the sensor sensitivities is shown in Fig. b and c. The changes in the sensor temperature are well documented, and the radiometric calibration is adjusted by up to 1 % (0.1 V (W m-2)-1). Converted into irradiance, this corresponds to a maximum correction of 5 W m-2 for the downward solar irradiance during local solar noon (16:00 UTC) when the Sun is high. Fluctuations in the time series are caused by the presence of clouds, which reduce or enhance the irradiance and thus the corrected bias. Due to the lower thermal–infrared irradiances, the differences here are 1 order of magnitude lower.

The response times τr of the CMP22 and CGR4 radiometers provided by the manufacturer are evaluated by measurements during a test flight in cloud-free conditions. The response time of the upward-looking pyranometers is determined by the cross-correlation between the measured downward irradiance and the aircraft attitude angles, assuming that the aircraft attitude is recorded instantaneously by the GPS-aided inertial navigation system. A 1/e (63 % adjustment) response time of τr = 1.2 s, which is slightly lower than reported by the manufacturer, is obtained. The same response time is assumed for the downward-looking pyranometer.

The response times of the CGR4 pyrgeometers are extensively characterized by in a laboratory study with a reported τr of about 3 s. EUREC4A measurements of flight sections with sharp turns are used to validate the τr of the BACARDI pyrgeometers. During the turns, the upward-looking radiometer partly observed the warmer lower hemisphere, which caused a sudden increase in the upward irradiance. Based on a detailed analysis of this systematic change, the response time is adjusted to 3.3 s.

The inertia of the measured irradiances caused by these response times are corrected following the deconvolution method proposed by . To minimize numerical effects of the deconvolution at sharp gradients and the sensor noise, a cut-off frequency of 0.6 Hz and a moving average filter with 0.5 s window length are applied to the reconstruction of the pyranometer measurements. For the pyrgeometers, a slightly lower cut-off frequency of 0.5 Hz and a longer window length of 2 s are chosen.

BACARDI is fixed to the aircraft fuselage and does not actively align with the horizontal plane. Therefore, the measurements are affected by the aircraft attitude. Except for turns, changes in the roll and pitch angles of HALO typically do not exceed ±1∘, limiting the alignment error . Changes in the pitch angle are mostly related to flight altitude and true air speed and are up to ±3∘. For the downward solar irradiance, a post-correction following the approach by and is applied. This correction is valid only for the downward direct solar irradiance. Therefore, the relative fractions of direct and diffuse solar radiation in cloud-free conditions are estimated using radiative transfer simulations. The simulations are updated continuously based on available in-flight observations of temperature and humidity profiles. The one-dimensional (1D) plane-parallel radiative transfer solver DIScrete ORdinaTe DISORT 2.0 embedded in the library for radiative transfer is applied libRadtran;. For the conditions during ACLOUD, a 5 % uncertainty of the simulated fraction of direct radiation amounts to less than 1 % uncertainty of the corrected downward irradiance. In cloudy conditions with 100 % diffuse radiation no correction can be applied. Therefore, final BACARDI data include both uncorrected Fsol↓ to be used for cloudy conditions and a corrected product to be used in cloud-free conditions. A basic cloud mask that is based on a comparison with the expected cloud-free irradiance and the identification of enhanced variability of the downward solar irradiance within a 20 s running window is provided in the published data set.

For the correction, the offset angles of BACARDI, Θ0 for the roll and Φ0 for the pitch angle, characterizing the relative alignment of the radiometer with respect to the inertial navigation system of HALO are determined from measurements during test flights. In cloud-free conditions, flight sections in different flight directions are compared to simulations of the theoretical downward solar irradiances. By minimizing the differences between corrected and simulated Fsol↓, the best-fitting pair of Θ0 and Φ0 is derived. For the installation of BACARDI during EUREC4A, two test flights are analyzed, one performed before the campaign in the vicinity of Oberpfaffenhofen, Germany, and one during the campaign based in Barbados. In both cases, Θ0=+0.3∘ and Φ0=+2.5∘ are obtained, so it can be assumed that the offset angles are stable once BACARDI is installed on HALO. To account for the limitations of the attitude correction, the downward solar irradiance is filtered before publishing the data set. Data are assumed to be valid when the attitude correction factors are less than 25 %. For larger correction factors, roll and pitch angles need to be smaller than 5∘. The excluded data correspond to turns with large roll angles or conditions with low Sun.

The dynamic thermal offset correction is most relevant when the temperature environment changes rapidly, such as during ascents and descents. Also, the aircraft flight velocity and the air density change the airflow around the sensors and control the adjustment of the thermal equilibrium. Figure  shows corrected and uncorrected profiles of all four irradiance components for an ascent up to 10 km altitude measured right after the start of the research flight on 7 February 2020 (flight ID HALO-0207). To interpret this profile, it needs to be considered that during such an ascent, HALO also covers a horizontal distance of about 200 km during which the atmospheric conditions may change. However, to estimate the effect of the dynamic thermal offset correction on the measurements, this case is well suited. Flight sections that do not comply at all with the required conditions, e.g., flight maneuvers of HALO, have been removed from the corrected data.

The ascent is characterized by an apparent cloud layer with cloud top at about 2 km as indicated by the increase in Fsol↓ and the decrease in Fir↓ at this altitude. Above this cloud, cloud-free conditions above the aircraft prevail. The upward irradiances, solar and thermal–infrared, are both affected by the changing cloud situations below HALO. The general increase in reflected solar radiation above the low-level cloud layer is covered by Fsol↑, while Fir↑ drops only for a limited period. Afterwards, the low-level cloud layer likely became thinner along the flight track, resulting in a cloud-top temperature that is more similar to the surface temperature.

This general pattern is shown by both uncorrected and corrected data. Differences, as provided in Fig. c and f, increase with altitude and are related to the temperature profile. Between 2 and 3 km altitude, a temperature inversion was present and the thermal offsets became smaller. In general, the uncorrected data underestimate the solar irradiance and overestimate the thermal–infrared irradiance. This is due to the inverse correlation of temperature change and dynamic thermal offset for the CGR4 and CMP22 radiometers as discussed in Sect. . While both CGR4 pyrgeometers show an almost synchronized pattern (almost identical β), the dynamic thermal offset correction differs for both pyranometers (higher β for Fsol↑). Therefore, the upward solar irradiance is more affected than the downward irradiance. As the dynamic thermal offset is independent of the absolute magnitude of the irradiance, this behavior might also be valid for other conditions with higher surface reflectivity or the presence of more reflective clouds.

Profiles of broadband solar and thermal–infrared irradiance are often used to study and quantify the impact of water vapor, clouds, and aerosol particles on atmospheric heating rates. To calculate atmospheric heating rate profiles, the upward and downward irradiances are combined into the net irradiance Fnet, independently for both spectral ranges: 18Fnet=F↓-F↑.

For the measurement case shown above, the net irradiance profiles for corrected and uncorrected data and their differences are shown in Fig. . Differences between corrected and uncorrected data are below 8 W m-2 for the solar irradiance and below 5 W m-2 for the thermal–infrared irradiance. As the upward and downward radiometers are almost equally affected by the temperature change during the ascent, the dynamic thermal correction mostly cancels out for Fnet. Only the upper and lower pyranometer show slight differences. This implies that the uncorrected irradiances can also be used to estimate Fnet. The dynamic thermal offset correction only becomes relevant for the profiles of Fnet,sol at higher altitudes, where temperature changes are quicker. The net thermal–infrared irradiance, Fnet,ir, significantly differs only for low altitudes below the cloud layer.

Consequently, the atmospheric heating rates, defined as the vertical change in net irradiance, 19∂T∂t=1ρ⋅cp⋅∂Fnet∂z, also show only a minor impact of the radiometer dynamic thermal offsets. In Eq. () ρ represents the air density and cp the specific heat capacity of the air. From the example profile, heating rates are calculated for a 50 m layer thickness, showing the strongest heating rates of down to -4 K h-1 at the top of the low-level cloud layer. However, the differences between corrected and uncorrected data are less than ±0.2 K h-1 for the entire profile. These results demonstrate that for this specific application of BACARDI, which is based on differences of upper and lower broadband radiometer measurements, a dynamic thermal offset correction could be neglected. This might not be valid if the mounting position of BACARDI on HALO changes for other missions, which deviate from the instrument configurations presented by .

The HALO flights of EUREC4A have mostly been performed at flight altitudes above 10 km and often under cloud-free conditions above HALO, causing high values of downward solar irradiance. In such conditions the solar leakage of the pyrgeometer dome interference filter can produce an overestimation of the thermal–infrared irradiance . For cloud-free ground-based measurements, identified an overestimation of up to 10 W m-2 depending on the amount of downward solar irradiance.

This bias was investigated for BACARDI using radiative transfer simulations of Fir↓, which are reliable and can serve as a benchmark for measurement above 10 km and under cloud-free conditions. The simulations have been performed along the HALO track, considering the time of day, the geographical position, and flight altitude of HALO with a temporal resolution of at least 30 s. The radiative transfer solver DISORT 2.0 and the lowtran parameterization of molecular absorption embedded in libRadtran are applied . In the simulations, the cloud-free atmosphere is defined by merged temperature and humidity profiles from the Barbados Cloud Observatory radiosondes BCO; and the frequent dropsonde measurements from HALO .

Filtered for cloud-free conditions, Fig.  shows the difference between measured and simulated Fir↓ as a function of the measured downward solar irradiance Fsol↓. The data indicate a trend to overestimate Fir↓ for increasing Fsol↓. For values of Fsol↓ above 1000 W m-2, typical for times around solar noon, the bias ranges up to 10 W m-2, comparable to the findings of . A linear regression suggests an increase in the bias by 1 W m-2 for each 100 W m-2 increase in Fsol↓. However, the data show a large variability and the regression suggests a negative bias for the absence of solar radiation. This may be attributed to remaining uncertainties of the radiative transfer simulations and the pyrgeometer sensitivity due to changes in water vapor concentrations above HALO , a permanent bias of the radiometer calibration, and a static thermal offset.

During EUREC4A, HALO frequently flew a circular flight pattern that aimed to quantify the large-scale vertical motion, an eminent parameter characterizing the dynamic state of the atmosphere . The typical circle had a diameter of roughly 220 km, which corresponds to a permanent roll angle Φ of HALO between 2∘ and 3∘. Therefore, the correction of the downward solar irradiance Fsol↓ for horizontal misalignment, as described in Sect. , becomes more important. At the same time, a circular flight pattern provides observations over the full range of relative solar azimuth angles and is thus an ideal test bed for evaluating the performance of the solar irradiance measurements.

The accuracy of the attitude correction is tested against radiative transfer simulations that are introduced in Sect. . For the simulations of Fsol↓ the absorption by ozone, which becomes relevant for the typical flight altitude of 10 km, is determined by satellite estimates of the atmospheric ozone column. The sea surface albedo is parameterized on the basis of using the 10 m wind speed obtained from the lowermost wind speed value of the HALO dropsondes. For the high flight altitudes above 10 km and often under cloud-free conditions above HALO, the simulations of Fsol↓ are reliable and can serve as a benchmark due to the implementation of frequent radiosonde and dropsonde observations.

Figure  compares downward and upward solar irradiance, Fsol↓ and Fsol↑, measured by BACARDI during the entire flight of 7 February 2020 (flight ID HALO-0207) with along-track simulations for cloud-free conditions. To illustrate the effect of the attitude correction for the downward irradiance, data with (black line) and without attitude correction (blue line) are plotted. The uncorrected Fsol↓ shows oscillations of different frequency that are superposed to the diurnal cycle. The slow oscillations (between 0.5 and 1 h) are associated with the circular flight pattern and caused by a combination of a permanent roll angle of about 3∘ and changes in latitude (solar zenith angle). Oscillations with higher frequencies between 0.5 and 1 Hz, e.g., most obvious between 17:00 and 19:30 UTC, result from variations of the roll and pitch angle due to turbulence and the aircraft autopilot. The post-correction of Fsol↓ does remove most of these fast and slow oscillations. This confirms that the roll and pitch angle offsets are determined with sufficient accuracy and that the sensor response time of BACARDI is corrected so that the oscillations are synchronized in time with the aircraft attitude. Subsequently, the remaining slow oscillations are in phase with the simulations and are only caused by the changes in the local solar zenith angle (latitude).

A statistically more robust comparison of measured and simulated Fsol↓ is performed by merging 12 EUREC4A flights and filtering the data for cloud-free conditions and reliable attitude corrections. Data are assumed to be valid when the attitude correction factors are less than 25 %. For larger correction factors, roll and pitch angles need to be smaller than 5∘. A one-to-one comparison is shown in Fig. a. From the total number of almost 3 million individual measurement samples, 97.6 % agree with the simulations within an uncertainty range of less than 5 %. This indicates that the general performance of BACARDI including the thermal and attitude correction is stable over the entire campaign. A linear regression of all reliable filtered data shows only a slight deviation from the 1 : 1 slope with a correlation coefficient of 0.999. The absolute differences are limited by the applied filter but illustrate that for high solar irradiances, the outliers of the measurements tend to be lower than the simulations. This might be caused by a remaining contamination of the filtered data by clouds above the aircraft, which are not considered in the cloud-free simulations. For low values of Fsol↓ the measurements are slightly overestimated. These measurements correspond to conditions of high solar zenith angle, when the attitude correction becomes more critical. At the same time, the angular response of the CMP22 pyranometer is known to slightly deviate from an ideal cosine response at high solar zenith angles.

Making use of the circular flight pattern, a potential asymmetrical cosine response of the pyranometer inlet is investigated in Fig. b. The ratio of the corrected observations and simulations is analyzed as a function of solar zenith angle θ0 and relative heading of HALO with respect to the solar azimuth. Only a subset of seven flights are used. Other flights are excluded because they either did not contain the circle flight pattern or show evidence of contamination by higher clouds like cirrus. Up to solar zenith angles of approximately 75∘, the observed Fsol↓ is within 5 % of the simulated values. The good agreement of the majority of the data points is regarded as an indicator that the attitude correction is independent of the flight direction over a wider range of illumination conditions (0–1200 W m-2) and solar zenith angles (0–75∘).

For solar zenith angles larger than 75∘ a slight directional dependence, relative to the position of the Sun with respect to the orientation of BACARDI (HALO), is obvious. Fsol↓ is overestimated by BACARDI between 30 and 210∘ relative solar azimuth and underestimated if the Sun is in the opposite directions. These effects may result from different factors, which cannot be disentangled here. It might indicate slightly incorrect offset angles determined for the attitude correction, an azimuthal dependence of the cosine response of the pyranometer, or reflection by the aircraft fuselage and the tail-plane fin at high zenith angles. Therefore, it is advisable to use the data at θ0>75∘ with some amount of caution.

The upward solar irradiance as well as the upward and downward terrestrial irradiance cannot be corrected for the aircraft attitude. However, these components are characterized by a nearly isotropic radiation field compared to the downward radiation, and the effects of a misalignment are minimal for a nearly level sensor . To limit the remaining uncertainties due to the aircraft movement, measurements with roll and pitch angles exceeding ±4∘ were removed from the data set. The time series of Fsol↑ shown in Fig. b indicates that the flight track covers an area with a generally low cloud cover and some patches of low-level stratiform clouds. The cloud-free areas correspond to the low values of Fsol↑, which form a baseline at about 80 W m-2. Only these cloud-free measurements can be compared to radiative transfer simulations (red line). The measurements match the simulated baseline and also follow their slight diurnal change with higher values observed at solar noon. The agreement of observed and simulated Fsol↑ indicates that the measurements in conditions like EUREC4A are reliable, even without any attitude correction. For observations over higher reflecting surfaces like sea ice, this needs to be confirmed.