Polarized radiation can be described by what is known as the “Stokes vector” S . It describes its total intensity, as well as its polarization state. 1S=IQUV=Ex2+Ey2Ex2-Ey22ExEycos⁡δ2ExEysin⁡δ, where Ex and Ey are the strength of the electromagnetic radiation in the two transversal directions. δ is the phase shift between these two components. I describes the total intensity, Q and U the intensity of the linear polarized contribution, and V that of circular polarization. As a result, the first component has to be larger than or equal to the sum of the others. In atmospheric radiative transfer, the contribution of circular polarization is about 3 orders of magnitude smaller compared to linear polarization e.g.,, so it can be ignored here (V≈0). This leads to the definition of the degree of linear polarization (DoLP) η: 2I≥Q2+U2,3⇒η=Q2+U2I.

The polarimetric and radiometric calibration of the sky radiance channels were recently performed at Laboratoire d'Optique Atmosphérique (LOA) in Lille, France. To produce linear polarized light, a combination of an Ulbricht sphere and the so-called POLBOX was used . Figure  depicts the calibration setup see also.

The POLBOX acts as a linear polarizer for the unpolarized light coming from the sphere. It consists of two glass plates that can be tilted up to 65∘ relative to the optical axis. According to the Fresnel equations, the total attenuation exerted by a glass plate differs for radiation polarized in the incident plane (I∥) and perpendicular to it (I⟂). Therefore, the DoLP η of the transmitted light is higher than that of the incident light. This degree of linear polarization hereby depends on to the tilting angle of the glass plate α. The Ulbricht sphere used here does not have to be radiometrically calibrated, but its intensity needs to be constant over the time of the calibration. 4ηα,n=I∥-I⟂I∥+I⟂5=1-n2cos⁡2α-1-1n2sin⁡2α1+n2cos⁡2α+1-1n2sin⁡2α

The output DoLP of the POLBOX can be determined with a high accuracy, as the plate angle can be set with high precision. gives an uncertainty in the DoLP of ∼0.0015; even gives 0.00128. The entire assembly can be rotated around its optical axis, therefore changing the polarization plane of the transmitted light. When using two plates and tilting the second by the same angle α but in the opposite direction, a divergent ray of light hitting the first plate at angle α+δα will hit the second plate at an angle α-δα. This compensates for linear terms of error in the DoLP due to divergent light. It can be shown that the DoLP after the second plate ηtot is given by ηtotα,n6=2ηα1+η2α+Oδα27≈1-n4cos⁡4α-cos⁡4α′1+n4cos⁡4α+cos⁡4α′+4n2cos⁡2αcos⁡2α′.

Here, α is the angle between the incident light and the normal of the glass plate; α′ is the same but for the refracted light inside the glass. α′ can be calculated using the Snellius law. 8sin⁡α=sin⁡α′⋅n9⇒cos⁡α′=1-1n2sin⁡2α

The refractive index of air is assumed to be 1. The plates are fabricated from Schott SF-11-type glass. Its data sheet provides coefficients for the Sellmeier equation (Eq. ), to calculate the refractive index n:

10nλ=1+∑iBiλ2λ2-Ci, with B1=1.73759695,C1=0.01318870700µm2,B2=0.313747346,C2=0.0623068142µm2,B3=1.898781010,C3=155.2362900µm2.

The POLBOX has a maximum tilt angle of α=65∘. The resulting DoLP is roughly 58 % at the SSARA polarized wavelength of 501.5 nm.

In the Stokes–Müller formalism, interactions with optical components or the atmosphere are described by left multiplication of the Stokes vector of the incoming radiation Sin with the appropriate real 4×4 Müller matrices (M1 to Mn): 11Sout=Mn⋯M1⋅Sin.

In this context, a linear polarizer can be described as a linear diattenuator, meaning its attenuation differs for the two directions of polarization. The Müller matrix LD for a linear diattenuator rotated by an arbitrary angle ϑ is given in as LDϑ=1212abcos⁡2ϑbsin⁡2ϑ0bcos⁡2ϑacos⁡22ϑ+csin⁡22ϑa-ccos⁡2ϑsin⁡2ϑ0bsin⁡2ϑa-ccos⁡2ϑsin⁡2ϑasin⁡22ϑ+ccos⁡22ϑ0000c, with a=k0+k1, b=k0-k1, and c=2k0k1. k0 and k1 are the intensity transmission values for the filter in the directions parallel and perpendicular to its orientation, respectively. ϑ is the angle between the polarization direction of the incoming radiation and the filter. Since a photodiode can only measure the total intensity of the light (first component of Stokes vector), the measurement operator 〈M| projects only the first row of the matrix. Mathematically, it can be described as a transposed vector 1,0,0,0: 13I=〈M|LD|S〉14=12[a⋅I0+b⋅cos⁡2⋅Δϑ⋅Q0+b⋅sin⁡2⋅Δϑ⋅U0].

The light entering the instrument behind the POLBOX is taken to be polarized only in the positive Q direction. This means the Stokes vector is given by I0,ηtotI0,0,0T, with ηtot again being the degree of linear polarization produced by the POLBOX. Also, the sensor has a certain radiometric response C, so the measurement vector becomes 〈M|=C,0,0,0. 15Sϑ=C2a⋅I0+b⋅cos⁡2ϑ-ϑ0⋅η⋅I016=12a′⋅I0+b′⋅cos⁡2ϑ-ϑ0⋅η⋅I017=12A′+η⋅B′⋅cos⁡2ϑ-ϑ0

It can be seen that the polarimetric (described by a and b) and radiometric response (C) of the instrument–filter combination cannot be determined separately. Therefore, we introduce a′=C⋅a and b′=C⋅b. Also, since the total intensity of the incoming light is unknown, we define A′=a′⋅I0 and B′=b′⋅I0. Measuring the signal S at varying rotation angles ϑ of the POLBOX, the parameters A′, B′ and ϑ0 can be obtained by performing a Levenberg–Marquardt (LM) fit using Eq. () as a model. k0 and k1 cannot be determined independently, but it is possible to derive the diattenuation D as 18D=k0-k1k0+k1=ba=b′a′=B′A′.

It is independent of the intensity of the incoming radiation (I0), as long as it is stable over the time of the calibration. The LM fit also gives estimations for the uncertainties in A′, B′, and ϑ0. For determining the response a′, we use LOA's SphereX, a radiometrically calibrated Ulbricht sphere. As it provides unpolarized light with known intensity, the measured signal is given by 19S=C2a⋅I0=a′2I020⇒a′=2⋅SI0.

For the SSARA calibration on 2 February 2017, the fit of Eq. () to the measurements can be seen in Fig. . The determined values and their uncertainties are shown in Table . It should be noted that the sensor head was placed on its right side, therefore adding roughly 90∘ to the filter orientation.

What remains after this calibration is the collective rotation of all channels in the sensor head, which also includes rotations stemming from the mount. When only the degree of linear polarization is of interest, this is not relevant. However, this global rotation has to be known to determine the polarization angle, which influences how the polarized radiation is divided between the Q and U components. As outlined in , this could be done by using known features of the Rayleigh background (e.g., U=0 in the principal plane).

Even unpolarized channels can have a polarization sensitivity. However, channels 3, 7, and 11 are assumed to have no polarization dependence, meaning the filters fully transmit light regardless of the polarization state. In future calibration sessions, the validity of this assumption could be investigated.

To determine the potential error arising from neglecting the imperfections of the filters and their orientation, a polarized radiance all-sky panorama was simulated for 500 nm using the MYSTIC 3-D Monte Carlo solver of the libRadtran package . To get the maximum error corresponding to the highest possible degree of linear polarization, a pure Rayleigh atmosphere was used as model input, without aerosol or clouds. Scattering processes by these would “destroy” polarization. The ground is non-reflective for the same reason, and the Sun is at a zenith angle of 30∘. The simulation is used to generate synthetic measurements in the three polarized SSARA channels, taking into account the filter characteristics from Table . From these, the Stokes vector is reconstructed, once assuming perfect polarizers (D=1) at exact angles (90, 45, and 180∘) and again with the actual filter characteristics in Table . Their relative difference in the total radiance and the degree of linear polarization is displayed in Figs.  and , respectively. The relative error in total radiance varies between -1.1 % and +0.8 %, and the relative error in DoLP varies between -2.4 % and +1.5 % (relative, not in absolute value). Due to the relative rotation of the polarizers, the pattern is not symmetrical. To evaluate the remaining difference induced by the uncertainties in ϑ and D shown in Table , the rotation angle and diattenuation of the channel 15 polarizer are perturbed by 0.07∘ and 0.002, for Figs.  and , respectively. For the total radiance, the remaining relative error is below 0.1 %; for the DoLP, it is 0.2 %, so about a factor of 10 smaller than without the calibration.

As mentioned before, POLBOX has an uncertainty of between 0.0015 and 0.00128 in DoLP. Simple Gaussian error propagation can be used to determine the resulting uncertainties in the calibration. Our calibration fits measurements to Eq. (), where the DoLP produced by the POLBOX is represented by η. As it only affects the amplitude of the cosine, the uncertainty of η propagates to the retrieved value of B′, and, according to Eq. (), in turn to D. At an assumed DoLP of ∼0.58, an absolute uncertainty of 0.0015 corresponds to a relative uncertainty of 0.26 %. For the D values given in Table , this leads to an absolute uncertainty of about 0.0025. This is on the same scale as the uncertainties we determined from the fit. Therefore, it can be assumed that higher accuracy can only be achieved using a light source with a better known DoLP.

SSARA should be set up perfectly perpendicular to the local tangential plane, facing exactly south. However, often, this is possible only to within a few degrees. Also, SSARA is designed to be portable, so the setup procedure has to be performed regularly. Therefore, it is useful to be able to quickly install the instrument in roughly the right orientation and determine the exact alignment by correlating the positions of the mount motors with the known Sun position for times with accurate Sun tracking.

To determine the actual orientation of the mount from several known Sun positions, we cannot directly fit the Euler angles using conventional real 3×3 rotation matrices, as this approach suffers from what is known as “gimbal lock”. This results from singularities in spherical coordinate systems, caused by directional “flips”, for instance, when crossing the zenith. Conventional minimization methods are not applicable in such highly non-linear cases. However, the fit can be performed using quaternions, as rotations here are always smooth and free of singularities. The mathematical fundamentals of quaternions are given in Appendix . To perform the mount calibration, several coordinate systems are defined that can be transformed into one another by rotation. Translation is ignored, as the Earth–Sun distance is much larger than the replacements in the instrument and mount. The coordinate systems used are similar to those defined in . Figure  sketches the coordinate systems used for SSARA:

East–north–up (ENU) uses the local horizon coordinate system on the tangential plane containing the observation position. Elevation and azimuth of the Sun (ϑs, ϕs) can be calculated for this system. The x axis points towards east, y axis towards north, and z axis towards zenith.

In an ENU spherical coordinate system, the azimuth ϕ is zero in the north and increases towards the east, as one would expect. The polar angle is zero in the nadir and increases towards the zenith. Rotations between the coordinate systems are described by quaternions, where BqA is a quaternion-rotating coordinate system A to B.

For direct measurements with the quadrant sensor uniformly illuminated, the Sun and viewing vector in the ENU system are assumed to be equal (to within the accuracy of the Sun tracker). The Sun position in the ENU system is determined with the pyEphem Python package . It can calculate planetary positions to a precision satisfactory for our purpose using the VSOP87 model . To obtain the viewing vector rv of the instrument, the unit vector in z direction (ez) in the SH system has to be transformed as follows: 21rv=ENUqSHrSH=ENUqSHez.

The optimal rotation quaternion can be found by minimizing the distance between viewing vectors rv and Sun vector rs: 22S=1N∑i=0Nrs-rv=1N∑i=0Nrs-ENUqSHez.

However, ENUqSH is composed of several rotations: 23ENUqSH=ENUqMNTMNTqGMBGMBqSH.

GMBqSH is defined as a 180∘ rotation around the local y axis to obtain the sensor head coordinate system. The active component of the mount acts on MNTqGMB. It contains the rotation angles of the azimuth and elevation motors (Δϕ and Δϑ), as well as the zero-point offset angles of the motors (ϑ0 and ϕ0). ϕ0 is zero due to our definition of the MNT system (it is effectively absorbed into ENUqMNT), but ϑ0 has to be determined. Both offset angles are constant over time and do not change for instrument realignment. Furthermore, the non-perpendicularity δ between the two motors is considered. MNTqGMB24=qϕ,exqδ,ezqϑ,eyq-δ,ez=qϕ0+Δϕ,ex25qδ,ezqϑ0+Δϑ,eyq-δ,ez=qΔϕ,ex26qδ,ezqϑ0,eyqΔϑ,eyq-δ,ez

ENUqMNT is unknown and contains the tilt and rotation of the mount. It changes every time the instrument is moved, involving a new calibration. The minimization now has six variables (four components of ENUqMNT, δ, and ϑ0), and one constraint (ENUqMNT has to be normed). This can be achieved using the sequential least squares programming (SLSQP) algorithm .

For the A-LIFE data, the fitting determines a non-perpendicularity of the motors δ of 0.95∘ and an elevation offset ϑ0 of -6.46∘. The rotation quaternion ENUqMNT is reconstructed to (0.704, -0.044, -0.707, 0.043). While the non-perpendicularity and the elevation offset are constant over time, the rotation quaternion will change every time the instrument is moved.

Figure  shows the remaining deviation between the fitted instrument pointing and the actual Sun position for all measurements in the A-LIFE campaign. The calibration is accurate to within 32 arcmin. The remaining inaccuracies are most likely due to the limited precision of the quadrant sensor and the way the instrument is tracking the Sun. The sensor has to pick up on brightness differences across the solar disk. Also, high aerosol loads, cirrus, or thin water clouds blur out the solar disk, resulting in an uniformly illuminated four-quadrant sensor further away from the Sun's center. If the clouds are “streaky”, this effect can occur in a certain direction. To avoid oscillation of the sensor head, the correction of pointing is damped. As a result, the instrument will most likely point to the lower left of the solar disk in the morning and the upper left in the evening. Other disruptions might occur by the instrument having to “search” the Sun after every scan. In the future, this effect should be minimized by using online fitting of the mount skewness. Furthermore, the change of the apparent solar position due to atmospheric refraction has been ignored.

Langley extrapolation is a method to enable Sun photometers to retrieve the total optical depth of the atmosphere, without the need for a radiometric calibration of the instrument in a laboratory . The basis for the extrapolation is the Bouguer–Lambert–Beer law and its logarithmic representation: 27I=I0⋅exp⁡-mτ28ln⁡I=ln⁡I0-mτ, where I and I0 are the measured and extraterrestrial irradiance, respectively. τ is the optical depth, and m the air mass factor. The latter describes the increase in the direct optical pathlength – and therefore the optical depth – from the Sun to the detector. In the simplest geometric approach, m=cos⁡-1Θ, with the solar zenith angle Θ. A more elaborate air mass model taking into account atmospheric refraction and the curvature of the Earth can be found in . Additionally, the extraterrestrial irradiance I0 has to be corrected for the seasonal variability in Sun–Earth distance .

Taking measurements at varying values of the air mass factor, and assuming the optical depth to be constant over time, the logarithm of the irradiance in Eq. () can be fitted as a linear function of m with slope τ. Extrapolating the linear fit to m=0 yields ln⁡I0. This value can then be used for reconstructing τ from measurements of I. Since only the ratio of the irradiances I and I0 is used, they can be replaced by any detector signal S that is linear in I. 29τ=1mln⁡II0=1mln⁡SS0, with S=C⋅I and τ=τR+τM+τA.

This τ is the combined value of Rayleigh (τR), trace gas (τM), aerosol (τA), and possibly cloud (τC) optical depths. The contribution from Rayleigh was determined according to , scaled with the measured air pressure. At around 500 nm, O3 and NO2 are the main contributors to the trace gas optical depth τM. Their profiles were taken from and the corresponding absorption cross-sections from . Assuming that no clouds are present, subtracting these components from the total optical depth leaves only the contribution from aerosol.

SSARA is usually calibrated once a year, either around March/April or around October/November at UFS Schneefernerhaus (2650 m) on Zugspitze. Firstly, at this height, the contamination by boundary layer aerosols is minimal. Also, early/late in the year, convective processes over the measurement site are not prominent. Therefore, temporal homogeneity of τ is found more frequently during that time. The calibration used for the data presented in this paper was done in November 2016.

17 April was chosen to illustrate the retrieval behavior during cloudy phases. Around sunrise and between roughly 11:00 and 14:15 UTC, convective clouds have been present at the measurement site. This can also be deduced from the gap in AERONET direct Sun AOD data shown in Fig. . Cirrus clouds already appeared around 10:30 UTC and persisted almost until 16:00 UTC. Figure  shows four snapshots of the cloud situation during that day. The pictures have been taken with the sky camera HaloCam installed coaxially with the SSARA sensor head.

In the early morning (until around 04:30 UTC; Fig. ), an increased AOD is retrieved. This coincides with the presence of convective clouds also visible in the top left panel of Fig. . As shown in sensitivity studies , these might lead to an overestimation of the AOD. However, it could indicate that additionally the AOD is increased, for example, due to hygroscopic growth of aerosol particles in humid air. The same can be observed in Fig.  for the convective period in the afternoon between 11:00 and 13:00 UTC. Here, it should be noted that for the corresponding scans, the residual is sometimes slightly higher, indicating a less reliable retrieval result. This is shown by the black tick marks in Figs.  and . Most of the time, the residual is below 0.1 but spikes up to 0.4. Until around 07:00 UTC, the retrieved total AOD is consistent with the values obtained from direct Sun measurements. Starting around this time, the AOD is overestimated by up to 0.1 during clear-sky periods. Small gaps in the AERONET direct measurements indicate the presence of clouds or high variability in the aerosol. Again, some deviation in the retrieval (generally overestimation) is to be expected here. Towards the evening, the optical depth seems to be underestimated. Note that perfect agreement between the values retrieved from sky radiance observation and from direct Sun observations cannot be expected. The reason for this might be spatial inhomogeneity of the aerosol properties (maritime towards ocean, anthropogenic aerosols towards city/industry). Other explanations could be measurement errors or systematic effects of the retrieval. This can also explain the differences between the results of almucantar and principal plane scans.

In Fig. a and b, the AOD is separated into fine and coarse modes. Over the entire day, the aerosol optical depth is dominated by the fine mode. This compares well to the AERONET inversion data points. The contribution of the coarse mode is larger compared to AERONET. It should be noted here that – in contrast to the AERONET inversion – we do not use the total AOD from direct Sun observations as a constraint for our minimization because the method is designed to be employed in cloudy situations, where such measurements are not available.

The retrieved effective radius of the fine mode (Fig. c) is mostly consistent over the entire day, including the cloudy period in the afternoon. This insensitivity of the effective radius to the presence of clouds was also observed in the numerical studies . However, the increased values in the morning and evening should be noted. This seems to be a systematic pattern; the reason for this is still unknown. When compared to AERONET, our fine-mode effective radii are somewhat smaller but within the 0.1 µm limit. An underestimation is also observed for the coarse mode (Fig. d). Here, the AERONET inversion suggests the presence of large particles with an effective radius of around 2 µm between 07:00 and 10:00 UTC. The values we obtain are smaller. Although previous sensitivity studies have shown that our retrieval has the tendency to underestimate the size of large coarse-mode particles, independent measurements would be needed to further investigate the discrepancy.

The retrieved real part of the refractive index changes rapidly for fine-mode particles (Fig. e). High values can be observed in the aforementioned times with clouds present. This behavior is again consistent with the results of the numerical studies, where clouds induce an overestimation of the index of refraction. The results for the coarse mode (Fig. f) are smoother in general. The retrieved value mostly stays close to the prior of 1.5, which might be caused by a low sensitivity to this parameter. The refractive index derived from AERONET ranges from 1.33 to 1.48. At around 07:00 UTC, there is an obvious discrepancy between values obtained from hybrid and almucantar scans. The values below 1.35 between 08:30 and 10:00 UTC seem unrealistic, as all expected aerosol types have a higher refractive index.

20 April was a clear-sky day. Starting in the late morning (07:00 UTC, 10:00 LT), the AOD increased. This can be attributed to the arrival of a Saharan dust outbreak over Cyprus from the west. Figure  shows the attenuated backscatter at 1064 nm of the PollyXT lidar. An aerosol layer is visible between roughly 2 and 5 km, beginning with thin filaments at around 04:00 UTC, and increasing in thickness towards noon. PollyXT also provides measurements of the particle linear depolarization ratio (PLDR) at 532 nm that can be used to discriminate between types of aerosol . In this layer, PLDR values around 25 % are observed and clearly identify the aerosol as desert dust .

With the exception of the early morning and evening, the AOD derived from the inversion of SSARA sky radiance measurements is overestimated by sometimes more than 0.1, when compared with the values obtained from direct Sun observations from SSARA and AERONET (see Fig. ). Additionally, the results from almucantar and principal plane differ significantly, with neither of them preferable to the other. Judging from the residual, the results are all equally trustworthy, barring one exception at approximately 13:00 UTC.

An increase in the coarse-mode AOD is clearly visible in Fig. b, starting at around 07:00 UTC. The retrieved values agree well with the AERONET inversion results. This increase is consistent with the arrival of Saharan dust which contains larger particles. Consequently, the overestimation of the total AOD retrieved from SSARA sky radiance measurements has to be caused by the fine mode (see Fig. a). Also it is not consistently retrieved from principal plane and almucantar scan patterns. Again, the deviation in the retrieved total AOD from the direct Sun observations is due to the fact that this value is not used as a constraint in the inversion.

The effective radius of the fine mode (see Fig. c) is stable over most of the day, only increasing in the morning and the evening again. AERONET finds larger fine-mode particles, again by up to about 0.1 µm. Apart from the single outlier at 13:00 UTC, coarse-mode effective radius is retrieved quite consistently over the entire day (Fig. d). Also, it agrees well with the AERONET inversion results. For values around 1.5 µm, the retrieval proved to be reliable in the sensitivity studies. Here, an increase in morning and evening is visible as well.

For the real part of the refractive index (Fig. e and f), most measurements indicate a value of around 1.5 for both fine and coarse modes. This agrees well with the AERONET inversion, which produces only slightly lower values. However, since this is also the prior and large discrepancies between values derived from the two scan patterns are visible, this might also be the result of lacking sensitivity to this parameter.