The core of the direct Sun measurement instrumentation is a Bentham NIR spectrometer: it consists of a double monochromator placed inside a thermally stabilized container, with light detection by a PbS cell. An optical fiber guides the sunlight between the entrance slit of the spectrometer and the diffusor of a 7.2∘ field-of-view (FOV) sunlight-collecting optics (telescope). The telescope is connected to an EKO Sun tracker that provides a tracking accuracy (https://eko-eu.com/products/solar-energy/sun-trackers/str-22g-sun-trackers, last access: 12 December 2018) of 0.01∘. The working wavelength range is from 0.6 to 2.3 µm, with a nominal 10 nm bandpass. The instrument characteristics are given in depth in and have remained unchanged since. No modifications have been made either to the telescope or to the spectrometer. Nevertheless, a factory defect in the assemblage of the components was detected and rectified: the lens focusing the light collected in the optic fiber into the spectrometer entrance slit was properly fixed into its barrel support for the PYR-ILIOS campaign, which was not previously the case for the IRSPERAD campaign at IZO. Another change relative to the IRSPERAD campaign was that the thermally stabilized spectrometer container was placed indoors in a thermally stabilized environment, which reduced thermal stress due to outdoor exposure and improved the stability of the spectrometer's response.

The wavelength-dependent direct transmitted solar irradiance in the atmosphere is described by the Beer–Bouguer–Lambert (BBL) law. For spectral regions where molecular absorption is negligible and only Rayleigh and aerosol scattering are present, the BBL law is written in the following form: Eλ=E0λD-2exp⁡-mRθτRλ-mAθτAλ, where E0 is the irradiance at the top of the atmosphere (TOA), m is the air mass factor (AMF) as a function of the solar zenith angle (SZA) θ and τ is the optical depth that depends on λ. D is the ratio between the Earth–Sun distance at the moment of the measurement and the mean Earth–Sun distance; subscripts R and A stand for Rayleigh and aerosol, respectively. Because the aerosol vertical profile over the measurement site at the moment of the measurement is unknown, aerosol AMF is approximated to Rayleigh AMF ; considering mA≈mR≈m, defining τ=τR+τA and taking the logarithm of Eq. (), it can be rewritten as log⁡Eλ=log⁡E0λD-2]-mθτλ.

Provided that τλ remains constant for a series of measurements of E0λ taken over a given range of mθ (spreading over a half day), the TOA value of E0λ is thus the intercept at the origin (m=0) of the least-squares regression to the data series Eλ as a function of m(θ).

Solar zenith angles (SZAs) are calculated with the NOAA Solar Position Calculator (https://www.esrl.noaa.gov/gmd/grad/solcalc/index.html, last access: 12 December 2018) that implements algorithms and are subsequently corrected for atmospheric refraction effects according to . AMFs are calculated using the Kasten and Young algorithm .

The wavelength domains for which the Langley plot method described in Sect.  is valid, i.e. atmospheric windows, were determined through the model using a procedure developed in  and also used in : using a TOA reference spectrum as input, the MODTRAN (MODerate resolution atmospheric TRANsmission) RTM (radiative transfer model) was used to simulate irradiances measured at the ground, as a function of the measurement site parameters, for a series of AMFs. The Langley plot method was applied to these simulated irradiances, and the wavelengths for which the synthetic E0 recreated the input TOA within 0.5% were kept as valuable wavelengths for the Langley plot; these set of wavelengths were grouped in contiguous windows called atmospheric windows.

The absolute calibration was performed against a primary standard of spectral irradiance, the BB3200pg black body of the PTB. It has been extensively described in  and . The spectral irradiance equation describing the black body emission is calculated using Planck's law:

EBB(λ)=εBBABBdBB2c1n2.λ51expc2nλ.λ.TBB-1, where εBB and ABB stand, respectively, for the effective emissivity and the aperture of the BB3200pg, dBB for the distance between the black body aperture and the optic centre of the telescope and n for the refractive index of air; c1 and c2 are the first and second radiation constants.

The fundamental parameter, the temperature of the cavity TBB, is known with a standard uncertainty of 0.5 K (∼0.02% for a nominal temperature of 3000 K) with a drift lower than 0.5 Kh-1 . The uncertainties on εBB and ABB are 1×10-4(0.01%) and 0.04 mm (0.03%), respectively . The distance between the black body aperture and the telescope optical active surface, the diffuser dBB, is the sum of two distances: dBB=ds+dT, where dS is the distance between the black body and the first optical surface of the telescope, the quartz plate, and dT is the distance between the quartz plate and the diffuser. The uncertainties on ds and dT are 0.05 mm and 0.5 mm, respectively; the combined uncertainty on dBB is 0.5 mm, 0.04% at the nominal distance of 1384.05 mm.

The absolute calibration coefficient R, that converts the spectrometer signal into irradiance, is given by Eq. (): R(λ)=EBB(λ,T)SBB(λ), with SBB being the signal recorded by the spectrometer and EBB, the emission of the black body, given by Eq. (). During the calibration campaign at PTB, two different temperature set points, 3016.5 and 2847.6 K, were used to build the response curve, RBB. The distance dBB was kept fixed at 1384.05 mm so that the black body aperture was seen by the entrance optics with an angular extension of 0.5∘.

The spectrometer was characterized at the laboratory of the Belgian Institute for Space Aeronomy (BIRA-IASB) for the uncertainty on the measured signal, the detector sensitivity to temperature and for the wavelength scale. The flat field of the detector was measured during the ground-based campaign at MLO and the linearity was verified during the calibrations at the PTB laboratory:

The flat field of the entrance optics was measured during the ground-based campaign. The telescope was angularly displaced from the normal Sun direction thanks to an angular fine-tuning mechanism, for a series of angular positions for two orthogonal directions. The agreement between both directions' data curves allows an insensitivity of the signal to solar depointing better than 0.05∘ to be estimated, although a finer angular sampling would be necessary to accurately determine the angular limits of this insensitivity. Given the 0.01∘ pointing accuracy of the Sun tracker, the response of the instrument is considered to be insensitive to pointing during the campaign.

A set of six FEL lamps (F102, F104, F417, F418, F545, F546) were used as relative calibration standards, to monitor a possible change of response of the spectrometer during the measurement campaign. Taking as reference the lamps' signal measured at the PTB (27 April 2016), SFjPTB(λ), four additional relative calibrations were performed:

Immediately before the start of the measurement campaign on 29 June (i=1), the signal of the six lamps, SFjMLO1, was measured on site. This first MLO relative calibration was valuable to monitor the spectrometers' response change between the calibration at PTB and the beginning of the field measurements. During this 2-month period that included the transportation of the equipment, a decrease of response varying between 1% and 3% in the 1000 to 2200 nm range was detected.

The corresponding correction factor for each relative calibration is Ki(λ)=1N∑jNSjMLOi(λ)SjPTB(λ), where N stands for the total number of lamps, j for the lamp number and i for the calibration day index. K(λ) was obtained by linear interpolation for all days of the campaign.

The PYR-ILIOS campaign took place during the first 20 days of July 2016 at the Mauna Loa Observatory (MLO) on the island of Hawaii. The MLO (19.53∘ N, 155.58∘ W) is situated at 3397 m a.s.l.; it is the leading long-term atmospheric monitoring facility on Earth, a primary calibration site for the AErosol Robotic NETwork (AERONET; https://aeronet.gsfc.nasa.gov/, last access: 12 December 2018), a global station for the Global Atmosphere Watch (GAW) of the World Meteorological Organization (WMO) and the premier site (https://www.esrl.noaa.gov/gmd/obop/mlo/programs/esrl/co2/co2.html, last access: 12 December 2018) for the measurement of the concentration of atmospheric carbon dioxide. It is considered a world reference site to accurately determine extraterrestrial constants via the Langley plot method .

From the 20-day campaign, 12 high-quality half-days, all during morning time, were kept for analysis. The selection criteria were verification of cloudless clear skies and a Langley plot correlation coefficient R2>0.9. The morning data of the days 2, 3, 5, 7, 8, 9, 10, 11, 13, 14, 16 and 17 July 2016 were kept for analysis; a subset of these selected Langley plots is shown in Fig. , for four different wavelengths.

The raw uncertainty of a spectrometer measured signal, Sxraw, regardless of its source, either solar (SS), black body (SBB) or lamp SFjPTB,SFjMLO signal, is a function of the intrinsic noise of the measured physical signal convolved by the spectrometer's transmission and detector's response. The uncertainty on a measured signal u(Sxraw) was determined in the laboratory by calculating the standard deviation for a sample of measured signals at several intensities from a 1000 W stable lamp . This uncertainty is shown in Fig. .

Additionally, all measured signals, Sxraw(λ), are affected by an uncertainty term due to the finite bandpass of the instrument, u(CΔλ), and the uncertainty on the determination of the true wavelength scale, u(Cλ) .

u(Sx(λ))2=u(Sxraw(λ))2+uCλ(Sxraw(λ),δ(λ))2+uCΔλ(Sxraw(λ),BW)2, where δ(λ) stands for the maximum deviation in the determination of the real wavelength scale of the spectrometer. δ(λ) was determined in the laboratory by measuring the deviation between the measured and the corresponding nominal peak values of a series of well-known emission rays of Xe, Ar and Kr lamps as well as of lasers and Pen-Ray lamps, δ(λ)<0.2 nm for the working wavelength range. BW stands for the spectrometer bandpass of 10.63 nm, measured in the laboratory.

The expression for a calibrated solar measurement, E(λ) is E(λ)=SS(λ).R(λ).K(λ), with SS(λ), R(λ) and K(λ) being expressed by Eqs. (), () and (), respectively.

The uncertainties associated with the factors in Eq. () were calculated using the law of propagation of uncertainties (LPU) and are represented in Fig. . The similarity of shapes of the curves of the individual uncertainties reflects the convolution of the measured signals by the spectrometer's response. The largest contribution to the calibrated solar signal comes from the uncertainty on the absolute calibration which is dominated by the uncertainty u(SBB) of the measured signal, SBB, of the black body, whereas the uncertainty on the emission of the black body, u(EBB), is known within 0.2% for the totality of the wavelength range.

The uncertainty in the determination of the TOA irradiance via the Langley plot method, u(E0LP), corresponds to the uncertainty on the determination of the intercept at origin, P0, when applying a linear regression on Eq. (). The uncertainty on the measured E in the Langley plot method logarithmic space, u(log⁡(E)), and the uncertainty in the u(E0LP) irradiance value are given by

u2(log⁡(E))=∂log⁡(E)∂E2.u2(E)=u(E)E2u2E0LP=∂exp⁡(P0)∂P02.u2(P)=exp⁡(P0)2.u2(P0), where E0LP=exp⁡(P0) gives the irradiance TOA value. The uncertainty in P0 was estimated using two independent methods.

A Monte Carlo method was employed. Given a measured Langley plot dataset consisting of (mi,log⁡(Ei)) points, a new synthetic dataset (mi*,log⁡(Ei*))) is created, where each log⁡(Ei*) is affected by a random normal distributed quantity, with a standard uncertainty given by Eq. (), and each mi is affected by an uncertainty defined in Sect. . The standard deviation in the distribution of the N>>1 retrieved P0 values corresponds to u(P0), with u(E0LP) given by Eq. ().

An ideal sunlight-collecting optic device should ideally have an acceptance angle equal to that of the solar disk seen on Earth, ∼0.5∘. In practice the FOV is much larger than 0.5∘ such that Sun- and sky-scattered radiation enters the FOV of the sunlight-collecting optics, affecting the direct normal Sun measurement. Circumsolar radiation is strongly dependent on aerosols' size and their abundance, increasing with AMF and decreasing with wavelength due to Rayleigh scattering . The estimation of circumsolar radiation was done with the aid of the LibRadtran RTM. LibRadtran computes the radiance field of the Sun- and sky-scattered radiation. The integral of this radiance field over the solid angle of the acceptance cone of the entrance optics is the amount of circumsolar irradiance (CSI) measured by the spectrometer in excess of the normal direct Sun irradiance (DNI) . For standard clear-sky atmospheric conditions observed at MLO and for typical aerosol charges values measured during the mission, the quantification of CSI is shown in Fig. . Given the uncertainty budget, the impact of the circumsolar radiation can be considered negligible.

As referred to in Sect. , the absence of knowledge of the vertical profile of the relevant species, namely aerosols, is a limiting factor for accurately calculating the AMF. The uncertainty in the AMF calculation is based on the approach of , who considered that mA, due to the presence of stratospheric aerosols, could take the form mA=k1.mR+k2.mO3, with k1+k2=1 and mO3 standing for the ozone air mass. Assuming a rectangular distribution of mA delimited by k1=1 and a k1=0.2, the standard deviation of mA can be calculated as u(mA)=∣mA(k1=1)-mA(k1=0.2)∣.123, to be used as input for the determination of the Langley plot parameters' uncertainty (Sect. ).

The possible bias introduced at the Langley plot's intercept at origin by a realistic non-constant aerosol concentration during the measurement was estimated considering a measured aerosol optical depth (AOD) profile. For a given measured Langley plot consisting of (mi,log⁡(Ei)) and regression parameters E0LP and τ, a synthetic Langley plot (mi,log⁡(Ei*)) is determined. The synthetic Ei* values are calculated with the expression Ei*=E0LP.exp⁡(-miτi*), where τi*=τi*(λ,ti)=τAOD(λ,ti)+τR(λ,ti); τAOD(λ,ti) stands for the real diurnal aerosol optical depth profile measured with AERONET (available at λ=870, 1020, 1640 nm) and τR(λ,ti) the Rayleigh optical depth calculated according to . This bias at the intercept at origin, expressed as a ratio, E0LP/E0*, averaged over the selected days is -0.2%, +0.4% and +0.1% for 870, 1020 and 1640 nm, respectively. The signal of the bias replicates the signal of the AOD morning trend measured at MLO, and the larger negative bias at 1020 nm relative to 1640 nm is due to the more pronounced AOD negative trend at 1640 nm. Assuming that the interval ∣E0LP-E0*∣ comprises the true value of the intercept at origin, E0, within a rectangular distribution, the corresponding uncertainty u(E0AOD)=∣E0LP-E0*∣23 amounts to 0.06% at 870 and 1640 nm and 0.1% at 1020 nm, which is added quadratically to the uncertainty on E0LP (Sect. ) to determine the uncertainty on E0. u(E0AOD) is interpolated linearly to the working wavelength range.