In addition to the all-sky cameras, the Cimel sun photometer CE318 N (Cimel Electronique, 2015) has been necessary to quantify image vignetting. The Cimel is a benchmark device for most aerosol observing networks and more specifically for the international federation of AERONET (Holben et al., 1998). Other observations, including brightness temperature (BT), measured through a narrow band infrared pyrometer (KT15.85 II, Heitronics) with spectral sensitivity peaking at about 10.6 µm, and solar irradiance, from pyranometer (SMP11, Kipp & Zonen B.V.) measurements, have been used to confirm the presence of cirrus. The instrumentation was installed at the observatory at Bayfordbury (51.7748∘ N 0.0948∘ W) operated by the University of Hertfordshire.

Two cameras were implemented, one set up for daytime, one for night-time observations. Both used the third version of the AllSky device, called the AllSky-340 (Santa Barbara Instrument Group, SBIG). Its optical system consists of a Kodak KAI-0340 CCD sensor and a Fujinon FE185C046HA-1 lens. The KAI-0340 is a VGA resolution (640×480 active pixels) 1/3′′ format CCD with 7.4 µm square pixels. The lens is of fisheye type, with a focal length of 1.4 mm and a focal ratio range of f/1.4 to f/16, fixed open at f/1.4 in the night camera and closed down to f/16 in the daytime camera. This combination gives a field of view of 185∘×144∘47′ and average resolution of 18′ per pixel. Assuming the camera is aimed at the zenith, a maximum of 95.0 % of the area of the sky can be imaged, with 2.5 % cut-off at the top and bottom. The areas around the edges are most affected by light pollution and not suitable for any measurements, so this loss is not too detrimental. During this study the cameras were located on the roof of a two-storey building (about 8 m above ground), where the elevated vantage point from the roof gives a clear view of the skies, uninterrupted by most of the surrounding features such as trees. The cameras are connected to a computer via an RS-232 serial cable. Using a serial–USB adapter at the PC end, the maximum download speed of 471.8 kbps can be achieved. At this rate the average download time of an image is 16.5 s for the grey camera. The colour one allows a download time of 48 s (Campbell, 2010).

The all-sky cameras are illustrated in Fig. 1. Both cameras are inside aluminium enclosures with acrylic domes protecting the fisheye lens. A specially developed occulting disk was mounted on the daytime camera to prevent stray light from affecting the imaging. It consists of an opaque disk, 10 cm in diameter, connected to an L-shaped rigid arm whose long and short sections are 60 and 20 cm, respectively. The long arm is mounted on a stepper motor that can adjust its elevation angle, in turn attached to a ring which fits around the camera enclosure and can be rotated using a second stepper motor. Time, date and location provided by a GPS module (GPS-622R, RF Solutions) are fed to a microcontroller (PIC18F46K22) that calculates the sun's position (azimuth and elevation) when the sun is above the horizon. The controller operates the motors via two ST L6472 stepper motor drivers. At sunset the disk is positioned slightly below the horizon, and the microcontroller then waits for the sun to rise again.

The colour camera has a Bayer mosaic used for filtering the red, blue and green wavelengths. The MATLAB built-in function “demosaic” is applied to the raw image data to reconstruct the full colour image. The fisheye lens enables sky observation over a field of view (FOV) as large as 187∘59′ along the ENE–WSW direction and as large as 142∘38′ along SSE–NNW. The FOV of the daytime camera along ENE–WSW is slightly larger, 189∘53′. The difference is due to the different alignment of the lens relative to the sensor. This is also reflected in the position of the true zenith (x0, y0) with respect to the centre of the camera plane, which for both cameras does not coincide with the true zenith. An offset of 14 pixels in the x direction and 12 pixels in the y direction that corresponds to about 4 and 3.5∘ respectively has been measured for the night-time camera and about 1 and 9.5∘ for the daytime camera. The fisheye lens employed in both cameras uses the “f-theta”, or equidistant projection system, which means that the distance in pixels from the true zenith of the object projected onto the camera plane is simply a scaling factor f (here 3.365 pixels per degree) multiplied by the zenith angle z of the object expressed in degrees (see Sect. 2.2.1). The output images are available either in JPEG or FITS format, but the latter is preferred because of a larger dynamic range and the absence of “digital development” correction.

The SPF is obtained by averaging the image brightness over pixels which are equidistant from the light source (sun or moon) in terms of scattering angle. To achieve this, the raw image is transformed to move the light source to the zenith, to give a light-source-centred system of coordinates. This is accomplished by mapping the original coordinates onto a sphere of unit radius and then by rotating the spherical coordinates to centre the light source at the origin of the final system. The first step consists in changing the projection from “linear” with respect to z to proportional to sin z. Given d, from Eq. (4) z is determined for each pixel (x, y) and the new coordinates (x′, y′) are calculated. The change of projection can be seen as the wrapping of the raw image around a sphere of unit radius with the constraint that the horizon, in the new coordinate system, will now be at a unit distance from the zenith. The periphery of the image which is below the horizon (see Fig. 3) is projected in the lower hemisphere, while points above the horizon appear in the upper hemisphere. The projection of the raw image of Fig. 3 in the upper hemisphere is shown in Fig. 4. The new coordinates x′ and y′ determine the Z coordinate: pixels above and below the horizon are associated with positive and negative Z, respectively. A rotation transformation T is then used to rotate the coordinates (x′, y′, Z) of a generic pixel, identified by the vector P, around the unit vector u that is perpendicular to the line linking the true zenith and the light source, by an angle θ equal to the light source zenith angle zsrc (see Fig. 4) according to T(P)=(1-cos⁡θ)(P⋅u^)u^+cos⁡θP+sin⁡θ(P×u^). With the rotated coordinates (x′′, y′′, Z′) available, the transformed image in the upper hemisphere can be obtained, and an example is shown in Fig. 5. This can be achieved by interpolating the original image into the new coordinates. Pixels that belong to those portions of the image that after rotation would be below the new horizon are in the lower hemisphere, which is not shown here. This should be accounted for if the SPF is to be calculated for all sky pixels and scattering angles available. Nevertheless that portion of the SPF is not necessary for HR calculation purposes and will not be covered here. The interpolated image is ultimately used to calculate the brightness as a function of the scattering angle by averaging over the entire azimuth angle range. Figure 6 (red dashed line) shows the end result corresponding to the raw image from Fig. 3.

To prevent the contamination of the sky image by background objects above the horizon, a mask, derived from a summer time image, when vegetation is thicker, was used. To prevent the mask edge from falling within the region of the SPF (between 18 and 22∘) from which the HR is derived, only images with zsrc<65∘ were used.

The fall-off of brightness for increasing z, associated with vignetting, takes place in nearly every digital photograph, in all optical lens systems, in particular wide-angle and ultra-wide-angle lenses like fisheye lenses (Jacobs and Wilson, 2007). Optical and natural vignetting are associated with a smaller lens opening for obliquely incident light and the cos⁡4 law of illumination falloff, respectively, both inherent to any lens design (Ray, 2002). In general, vignetting increases with the aperture and decreases with the focal length. Here it is quantified by comparing daytime image data with sun photometer data under clear sky. While the camera is affected by vignetting, the sun photometer is not; hence the ratio of the corresponding radiance measures, over a similar spectral range, can be used to quantify vignetting. The blue channel of a clear-sky daytime RGB image, with a peak wavelength of 0.46 µm, was extracted (TRUESENSE imaging, 2012), as it provides the best match to the spectral channels of the sun photometer (1.0205, 1.6385, 0.8682, 0.6764, 0.5015, 0.4403 µm) and has larger quantum efficiency and narrower spectral width than the green and red channels. A single sun photometer measurement along the solar principal plane, performed over the various wavelengths, one at a time, about 35 s apart, was compared to the corresponding image brightness from the closest camera measurement (see Fig. 7); as vignetting is assumed to be symmetric under rotations around the camera zenith, pixels such that z is greater than approximately 20∘ over the meridian containing the sun were neglected, and the mean of the sun photometer sky brightness at the visible wavelengths of 0.5015 and 0.4403 µm was smoothed and fitted according to LOWESS (locally weighted scatterplot smoothing). Analogously a polynomial was fitted to the LOWESS smoothed image data. The ratio between the two fitting polynomials (camera to sun photometer) after normalization (see Fig. 8) is what we refer to as the devignetting coefficient (DC; see Fig. 9, black curve). As the DC was expected to be monotonically decreasing with z, the shift of the maximum from the zenith to the position at roughly 8∘ was investigated. This shift was systematically observed for all data. Since the use of the occulting disk allows us to exclude stray light being the cause of it, the simplest explanation comes from observing that such an offset, corresponding to around 27 pixels, corresponds to a displacement of only 0.2 mm between the lens axis and the centre of the aperture. To form a correction function symmetric about the zenith, the original curve was “mirrored” about the zenith (see Fig. 9, red curve), then a Gaussian of the form shown in Eq. (5) was fitted to the mean of the original and the mirror curve (see Fig. 9, dashed curve). f(z)=A+B⋅e-(z/C)2 A, B and C obtained in this way are 0.74, 0.26 and 40.03, respectively. The fitting curve provides a working approximation of the actual DC. Figure 6 (black dashed line) shows the SPF corresponding to the raw image from Fig. 3 when geometric, air mass (AM), vignetting and mask corrections are included. The latter removes the contamination associated with objects in the field of view, such as trees, by excluding non-sky pixels from the azimuthal average.

The vignetting correction obtained this way is intended to be generic. Since the correction is nearly rotationally symmetric, and we have concluded that the residual asymmetry is the outcome of a small misalignment of the sensor, which is likely to vary between cameras, the generic, symmetric correction can be applied to sites where deriving a camera-specific correction is not possible due to the absence of a sun photometer.

The additional scattering that light undergoes for slant paths is associated with increased image brightness, evident for large z. While recent sky imaging work neglects it (Forster et al., 2017), such correction is consistent with sky observations (Volz, 1987; Patat, 2003). In this context we will be assuming single scattering approximation, as justified by co-located non-cloud-screened AERONET measurements showing optical thickness <1. To model relative AM, the ratio of the absolute optical air mass M calculated along z to the zenith air mass M0, a non-refracting homogeneous radially symmetric atmosphere was assumed. This provides realistic values of AM near the horizon, where it is less than 40 as is expected (see Table 1 in Rapp-Arraras and Domingo-Santos, 2011). Equation (7) from the same work was used. Such a functional form has already been used for atmospheres with elevated aerosol layers (Vollmer and Gedzelman, 2006). Each pixel brightness was then corrected by dividing it by the corresponding AM(z). With this correction in place, averaging of sky brightness along lines of constant scattering angle becomes possible. However, a sudden drop in the value of the SPF towards large scattering angles was observed, ascribable to the large value that the AM takes for large zenith angles and causing the image brightness to drop rapidly. By excluding such pixels, this unwanted drop is significantly reduced. Moreover, it has been shown previously through radiative transfer calculations accounting for multiple scattering that the brightness of the lower part of the halo reaches a maximum for smaller cloud optical thickness τ than the portion above the sun (Gedzelman and Vollmer, 2008). Consequently, for low solar elevations, a zenith cut-off angle that is too large can cause the HR to decrease due to multiple scattering affecting the part of the halo below the sun. By setting the cut-off at z=70∘, the SPF becomes smooth and multiple scattering effects are reduced while avoiding seasonal bias due to the solar zsrc range covered during the year.

A quantification of the temporal fluctuations of the infrared brightness temperature (BT) is expressed in terms of the fluctuation coefficient (FC). The FC is obtained here through detrended fluctuation analysis (DFA) of the BT and is expressed as the exponent of the DFA function following a simple power law (Brocard et al., 2011). It allows the presence of clouds to be detected, unless very thin optically. In fact BT fluctuates significantly under optically thick clouds, while under clear sky it follows the relatively slow-changing water vapour diurnal cycle. Therefore clear-sky FC allows a threshold (DFA threshold) to be set for the transition from clear to cloudy sky. This threshold was chosen empirically to be 0.02 on the basis of the DFA output calculated, unlike in Brocard et al. (2011), every 5 min, over data sampled with 1 s resolution and for time intervals ranging between 20 and 60 s. The intervals were chosen as the time range over which the DFA function was relatively stable and the slope of the function (the FC) was as sensitive as possible to the presence of clouds. The DFA algorithm was applied to 1 year of data, and the corresponding FC time series was averaged over clear-sky periods to set the DFA threshold. This was obtained after manual cloud screening of images associated with minima of the fluctuation coefficient. Furthermore, we wish to point out that like in Brocard et al. (2011) the initial analysis step of cumulative summation (integration) of the time series was not carried out. We note that cumulative summation of a self-affine series shifts the FC by +1 (Heneghan and McDarby, 2000), so the threshold applied here would have a value close to 1 if a standard DFA procedure was followed. Separately, a departure from modelled clear-sky BT due to the presence of relatively optically thick cirrus can provide a BT threshold (Ci threshold) that was used for assessing cloud phase. A simple analytical model of downwelling thermal radiation under clear skies was implemented for this purpose. The model uses ground-level air temperature and integrated water vapour path (retrieved locally from GNSS delays) as input parameters to estimate clear-sky BT at the central wavelength of the pyrometer (Dandini, 2016). In order to establish via BT whether warm or cold clouds are present in the field of view of our instrument, an estimate of the departure of BT from clear-sky BT due to cirrus was calculated. We set the maximum possible departure from clear-sky BT attributable to cirrus as the one when the cloud is warmest and optically thick. Warm liquid clouds would certainly determine a departure from clear-sky BT larger than the one due to such cirrus and hence would lend themselves to be discriminated. By assuming an optically thick cirrus (emissivity of 1) at a temperature of -38 ∘C, representative of the upper temperature limit of ice clouds (Heymsfield et al., 2017), an estimate of the irradiance due to the direct emission of cirrus corrected for the atmospheric attenuation and emission was obtained and converted to brightness temperature by inverting Planck's law (see Figs. 11b and 13b, black dashed line). The Ci threshold follows the diurnal water vapour cycle corrected for the attenuated contribution of such thick cirrus. Above this threshold we can expect optically thick clouds warmer than -38 ∘C, i.e. theoretically not cirrus. Hence, HR time series corresponding to two test cases discussed in the coming section are complemented by simultaneous comparisons of BT with the Ci threshold and the FC with the DFA threshold, as well as broadband downwelling irradiance I. Validation of the presented Ci threshold method is beyond the scope of the present study but will be the subject of a future publication.

On 6 July between 08:00 and 10:00, cirrus occurs as all-sky camera, BT observations and FC measures all show (see Fig. 11). The FC is nearly always above the DFA threshold, while the BT is below the Ci threshold. Over this time window, on qualitative grounds, the behaviour of the BT and the solar irradiance, which in general grows and decreases with τ, respectively, suggests that the HR increases with the optical thickness τ (see Fig. 11b and c). This is consistent with the results of simulations (Kokhanovsky, 2008), based on ray tracing techniques incorporating physical optics (Mishchenko and Macke, 1998) and neglecting molecular and aerosol scattering, that show a linear increase of halo brightness with increasing τ up to τ=3 and a decrease for τ>3 due to multiple scattering. Between about 08:00 and 08:20, the HR, mostly <1, shows a maximum and a minimum at about 08:06 and 08:12, respectively, when a relatively faint 22∘ halo is visible. Maxima with HR >1, on the other hand, are measured at 08:33 and 09:06, when bright halo is observed, and at 09:30, when lower halo brightness is associated with decreased HR. Similarly, the attenuated halo brightness at about 8:51 corresponds to an HR minimum. Between 09:36 and 10:00 the HR drops below 1, and cirrus gets thinner. The small local maxima observed over the same time window at about 09:38, 09:45 and 09:54 are ascribable to cirrus optical depth variations. From 10:00, as cirrus disperse, cumuli start entering the field of view of the camera. A local HR maximum is then measured at 10:03, whereas HR values are >1 at 10:16, 10:33 and 11:00, while scattered cumuli over a mostly clear background increasingly occur. The FC testifies to the presence of clouds, whereas the BT becomes larger than the Ci threshold only at about 10:54, probably because of the sparse nature of the cumuli preventing significant direct radiation from falling within the field of view of the radiometer. However, dips of the solar irradiance I at approximately 10:24, 10:32 and 10:42, indicating increased optical depth, are also observed. While on an overcast day global irradiance depends primarily on diffuse irradiance (Kaskaoutis et al., 2008), on a partly cloudy day clouds crossing the sun path are the main factor to determine variations in the signal of the pyranometer. In particular the ratio of diffuse to global irradiance has been estimated to be 0.15, 1 and >0.15 under clear, overcast and partly cloudy sky, respectively (Duchon and O'Malley, 1999; Orsini et al., 2002). These relatively large HR values are artefacts caused by the image becoming brighter at larger scattering angles, possibly due to bright cloud edges. Such cases should be screened out from the analysis as the HR parameter applies to ice clouds only. This example shows that while a relatively large HR (HR >1) can be an indication of the presence of halo-producing cirrus, this does not always have to be the case.

On 7 July between 12:00 and 15:00 the sky is mainly characterized by the presence of cirrus, except between about 12:00 and 12:30 when sparse cumuli are seen, around 12:45 when relatively opaque altocumuli occur, between around 13:30 and 13:48 and about 14:15 when cumuli overlap with the cirrus background (see Fig. 12). Correspondingly, the FC is always above the DFA threshold, while the BT is mostly below the Ci threshold, although larger values are observed at about 12:20, 12:27, 12:42, 12:48, 13:40, 14:15 and 14:18 when, as expected, the solar irradiance I drops. The HR peaks measured at about 12:06 and 12:18, in similarity with the previous case, appear to be associated with bright cloud edges. Between 12:30 and 12:38, when bright halo is visible, the HR is >1 and then decreases until about 12:42 by which time the halo is no longer visible, possibly due to multiple scattering associated with the increased τ, as demonstrated by the decreased solar irradiance at that point (Fig. 13). With halo-producing cirrus present again from 12:50, the HR increases, except for a local minimum at about 13:00, until about 13:18, when it becomes larger than 1 and the halo is correspondingly sharper. The HR then drops fairly steadily, except for a local maximum at about 13:30, while the halo, still partly visible, fades gradually away to eventually disappear by about 13:36. This is when, we hypothesize, cloud τ has become <1 and Rayleigh and aerosol scattering contribute to the halo contrast reduction. A faint halo present from about 13:45 gets quite sharp by 14:00 as the HR again becomes larger than 1. The halo brightness then stays constant for about 10 min before the HR reaches another minimum at about 14:12, when cumuli are seen by the camera. As the halo sharpness increases, the HR also increases fairly uniformly until about 14:28, when a maximum of 1.14 is reached. We speculate, based on previous findings (Forster et al., 2017; Kokhanovsky, 2008), that this is when τ, probably somewhere between 1 and 3, becomes significantly larger than Rayleigh and aerosol optical thickness without exceeding the optical depth beyond which multiple scattering becomes dominant. The HR then drops to 1 in less than 20 min, while the halo is still fairly bright, and continues to go down until 15:00, when the halo is eventually no longer visible. Peaks at about 14:40 and 14:54 are due to rapid variations in sky brightness, associated with cumuli like those observed near 14:38, which, in analogy with the minima seen at 14:12 and 14:48, cause the SPF to vary significantly as the cumuli transit over the cirrus. With an average BT roughly 10 ∘C higher than on the previous day, this is a case of relatively warm halo-producing cloud; yet the relatively large HR indicates that the cloud is dominated by ice.

Overall the HR correlates well with the fluctuating halo visibility observed throughout the periods examined. Manual inspection of the all-sky images allows us to state that when the HR >1 the halo is visible 95 % of the time (true positives), while halo visibility associated with HR <1 represents only 10 % of the occurrences (false negatives). This relatively minor fraction may be associated with locally larger values of optical thickness, as the partial cirrus thickening observed in the sky images at 13:00 and 13:24 on the 7 July suggests. According to previous radiative transfer calculations (Forster et al., 2017), we conjecture that in order to observe the absolute HR maxima measured for the two cases discussed here, a certain minimum fraction of smooth hexagonal ice columns had to be present.