Ice Crystal Characterization in Cirrus Clouds II: Radiometric Characterization of HaloCam for the Quantitative Analysis of Halo Displays

This study presents the weather-proof RGB camera HaloCamRAW, which is part of the automated halo observation system HaloCam and designed for the quantitative analysis of halo displays. We present a procedure for both the geometric and absolute radiometric characterization of HaloCamRAW and demonstrate its application in a case study. The geometric calibration was performed using a chessboard pattern to estimate camera matrix and distortion coefficients. For the radiometric characterization of HaloCamRAW, dark signal and vignetting effect were determined to correct the measured signal. Further5 more, the spectral response of the RGB sensor and the linearity of its radiometric response were characterized. The absolute radiometric response was determined by cross-calibrating HaloCamRAW against the completely characterized specMACS imager. For a typical measurement signal the relative (absolute) radiometric uncertainty amounts to 2.8% (5.0%), 2.4% (5.8%), and 3.3% (11.8%) for the Red, Green, and Blue channel, respectively. The absolute radiometric uncertainty estimate is larger mainly due to the inhomogeneity of the scene used for cross-calibration and the absolute radiometric uncertainty of specMACS. 10 Geometric and radiometric characterization of HaloCamRAW were applied to a scene with a 22° halo observed on 21 April 2016. The observed radiance distribution and 22° halo ratio compared well with radiative transfer simulations assuming a range of ice crystal habits and surface roughness. This application demonstrates the potential of developing a retrieval method for ice crystal properties, such as ice crystal size, shape and surface roughness using calibrated HaloCamRAW observations together with radiative transfer simulations. 15

The HaloCam system was installed in September 2013 on the rooftop platform of MIM (LMU) in Munich with HaloCam JPG only and was extended in September 2015 by HaloCam RAW . The MIM rooftop platform hosts a cloudnet site (Illingworth 25 et al., 2007) featuring operational measurements by a MIRA-35 cloud radar (Görsdorf et al., 2015), a CHM15kx ceilometer (Wiegner et al., 2014), and a RPG-HATPRO microwave radiometer. Further operational measurements are performed by a CIMEL sun photometer, which is part of the AERONET (Aerosol Robotic Network) network (Holben et al., 1998) as well as with the institute's own sun photometer SSARA (Sun-Sky Automatic Radiometer, Toledano et al. (2009Toledano et al. ( , 2011). HaloCam observations ideally complement these measurements to retrieve information about ice crystal properties. 30

Geometric calibration
Halo displays are single scattering phenomena. Thus, their relative position to the sun is directly linked to the scattering phase function of the ice crystals producing them: the phase function of smooth hexagonal solid columns, for example, predicts a 22°and 46°halo at a scattering angle (Θ) of 22°and 46°, respectively (e.g. Yang et al. (2013)). To uniquely identify halo displays in terms of their relative position to the sun and to allow a quantitative analysis of HaloCam RAW images a geometric calibration is necessary which determines the transformation between the image pixels and scattering angles. For the geometric calibration of HaloCam RAW several pictures are taken of a chessboard pattern from different angles and orientations. These pictures are used to estimate the intrinsic camera parameters as well as the radial and tangential distortion parameters of the 5 lens. This method was already used in Forster et al. (2017). It is based on Zhang (2000) and Heikkilä and Silvén (1997) and was implemented in OpenCV by Itseez (2015) with a detailed reference in Bradski and Kaehler (2008).
Using the distortion coefficients and intrinsic parameters, the camera pixels can be undistorted and mapped to the spherical world coordinate system. Projected onto the image plane, a zenith (ϑ) and azimuth angle (ϕ) can be assigned to each pixel relative to the center of the sun. In this case the relative zenith angle ϑ corresponds to the scattering angle Θ. 10 An overlay of the relative zenith (ϑ) and azimuth (ϕ) for the HaloCam RAW red channel (R-channel) is displayed in Fig. 3 with representative contour lines at ϑ = 22°, 35°and 46°. The geometric calibration was performed for the raw image (cf.   nd ∼65°, respectively. With a resolution of 608×968 quadratic pixels the angular resolution of each of the 4 color channels amounts to about 0.1°. Note that the individual channels are extracted from the Bayer pattern without interpolation (cf. Fig. 2).
The HaloCam RAW image is separated into segments using the relative azimuth angle ϕ. Figure 3b indicates the 5 azimuth segments, each 30 • wide. For further analysis the radiance within each image segment is averaged in the azimuthal direction 5 (ϕ), i.e. along the 22°halo. The angular width of the segments allows to reduce measurement noise and at the same time maintains the necessary spatial resolution to resolve brightness fluctuations of the 22°halo. Upper tangent arcs would be covered by segment no. 3, while sundogs would be visible below segments no. 1 and 5. Thus, if halo displays are visible, segments no. 1, 2, 4, and 5 are expected to contain only features of the 22°and 46°halo. Tilting the camera upward by 26°, as shown in Fig. 3a allows to observe of the more frequent upper part of the 22°and 46°halo and is therefore more suitable for a 10 quantitative analysis.

Radiometric characterization
Each sensor pixel is a semiconductor device which converts light into electrical charge and can be treated as an independent radiometric sensor. The charge collected on a pixel is converted to a voltage and then to a digital value by the A/D converters, which introduces noise at each step. The signal measured by the sensor can be expressed as with S d the dark signal, S 0 the radiometric signal, and the measurement noise N , as defined in Ewald et al. (2016). The measurement noise N is the sum of the radiometric signal noise N 0 and the dark signal noise N d In the following sections the components of the measured signal S will be characterized and their sensitivity on the camera settings and ambient conditions will be investigated. The dark signal measurements were performed in the optics laboratory of the Meteorological Institute at LMU on 16 July 2015. The measurements at the Large Integrating Sphere (LIS) and the spectral characterization of the sensor were performed at the Calibration Home Base (CHB) (Gege et al., 2009 are provided with a 1σ confidence interval.

Dark signal
The dark signal S d is defined as the signal which can be measured when no light is entering the camera, i.e. the shutter is 10 closed. This implies S 0 = 0 and Eq. (1) becomes For an averaged dark image S the remaining noise approaches zero N d → 0 and the dark signal S d can directly be measured. The dark signal consists of the dark current s dc , which is caused by thermally generated electrons and holes within the semiconductor material of the sensor, and the read-out offset of the A/D converters S read . The dark current s dc depends on the 15 temperature T and the exposure time t expos S d (T ) = s dc (T ) t expos + S read .
Thermal electrons are generated randomly over time with an increasing rate as the temperature rises. Since HaloCam RAW has no external shutter, the dark signal during operation has to be estimated from the laboratory characterization. The following experiments were performed in a dark room and the camera lens was covered with an opaque cloth.
20 Figure 4 displays the dark signal S d averaged over 100 images for an exposure time of t expos = 2.0 ms and a device temperature of 45 • C for the R-channel. The temporally (over 100 images) and spatially (over all pixels) averaged dark signal amounts to about S d = (16.7 ± 0.2) DN (Digital Number). For this number of averaged images, the dark signal in Fig. 4 does not show a significant spatial pattern. The same is true for the G1-, G2-, and B-channel. Therefore, a spatially averaged dark signal will be used for the following analysis and later image processing. Figure 5 shows the dependency of the dark signal on the dark signal appears to be independent of the exposure time. For larger exposure times, which are shaded in gray in Fig. 5, 30 the dark signal as well as its standard deviation increase slightly. This behavior is most likely a combination of the increasing dark current signal due to a longer exposure time and an increase of the read noise signal S read caused by the A/D converters.   To investigate the temperature sensitivity of the dark signal, measurements were performed with the camera set up inside a climate chamber (Weiss 2 , SB11/160/40) in a dark room and with the camera lens covered. The temperature inside the climate chamber can be adjusted between −40 • C to 180 • C with increments of 0.1 • C. The estimated accuracy is about 0.05 K. For the dark measurements with HaloCam RAW the temperature was increased from 10 • C to 50 • C in steps of 5 • C. Within this temperature range the averaged dark signal varied less than 0.5 DN. To obtain an estimate for the temporal drift of the dark

Vignetting correction 5
The wide-angle lens of HaloCam RAW causes a decreasing radiometric signal S 0 for raypaths further away from the optical axis of the lens. This illumination falloff towards the edges of the sensor is called vignetting. There are two different types of vignetting: 1. Optical vignetting occurs when the ray bundle, which forms the image, is truncated by two or more physical structures in different planes (Bass et al., 2010). Typically, one is the nominal aperture and another is the edge of a (multiple element) 10 lens. This kind of vignetting naturally occurs in all lenses and typically affects peripheral light rays, far off the optical axis.
2. Natural vignetting describes the effect that for off-axis image points the illumination is usually lower than for the image point on the optical axis (Bass et al., 2010).
Optical vignetting can be diminished by reducing the entrance pupil, i.e. the aperture by increasing the f -number. According to Bass et al. (2010) the f -number is defined by The f -number of HaloCam RAW 's Kowa lens can be adjusted mechanically between 1.8 and 11 by a screw. A fixed value of f -number = 8 was chosen for all measurements and the calibration. For the observation of halo displays close to the sun this 5 represents a good trade-off between a small aperture and short exposure times.
To obtain a model for the non-uniformity of the sensor response as a function of the pixel location, flat-field measurements were performed using the large integrating sphere (LIS) at CHB as a uniform light source. Several measurements were performed with the same exposure time. To minimize the impact of inhomogeneities in the brightness of the integrating sphere, images were recorded at 6 different orientations by rotating the camera around its own axis, i.e. with the center of the camera roughly 10 pointing to the center of the sphere. For each orientation 40 images were recorded, dark signal corrected and averaged. The measurements, which were averaged over the rotation angles of the camera relative to the sphere, are shown in Fig. 6a with the signal normalized to 1. The spherical patches visible in the figure are due to a hole in the sphere, which allows for injecting a laser as light source for specific experiments. The hole appears at different locations on the image due to the different rotation angles of the camera. Owing to the large field of view of the camera, the edge of the two hemispheric components of the LIS is 15 visible. In order to fit a model to the flat-field measurements, these two regions were masked out as displayed in Fig. 6b. The flat-field model correcting for the vignetting effect was determined by fitting a 2-dimensional (2D) second order polynomial to the averaged and masked measurements.
where r 2 = |x − x 0 | 2 is the distance of the location x from the image center x 0 in pixel units. The result is depicted in Fig. 6c 20 for the R-channel with the following parameterization: with y 0 = 297.2 and x 0 = 473.8. Finally, Fig. 6d shows the relative difference between the flat-field model and the measurements in percent averaged over all 6×40 images. The fluctuations of the signal difference are due to inhomogeneities of the integrating sphere. However, these inhomogeneities are not relevant for the image processing procedure since the flat-25 field model is used to correct the camera measurements. The average difference between model and measurement amounts to (0.0 ± 0.5) % for the R-channel with similar values for the remaining channels. Correcting for the vignetting the flat-field corrected signal S F is defined by with the radiometric signal S 0 and the flat-field correction F . This correction is applied to the radiometric signal S 0 of the red, 30 green and blue channel separately.

Linearity of radiometric response
Similar to Ewald et al. (2016) the linearity of the CMOS sensor of HaloCam RAW was investigated by measuring a temporally stable light source using different exposure times. This experiment was performed using the LIS at CHB. Baumgartner (2013) characterized the output stability of the LIS to better than σ = 0.02% over a time range of 330 s. For a perfectly linear sensor with response R, the photoelectric signal S 0 should increase linearly with exposure time t expos and radiance L 5 S 0 = R L t expos = s n t expos , with the normalized signal s n defined by The deviation of the actually observed signal S 0 from the linear relationship of S 0 is called photo response non-linearity. The actually observed signal S 0 can be written as with the flat-field correction F and it follows that the normalized signal can be obtained by Figure 7 shows the measured radiometric signal S 0 for exposure times t expos ranging from 0.5 ms to 29.5 ms, averaged over 5 images for each exposure time. For exposure times larger than 23 ms some pixels start to get overexposed. These pixels are   Figure 8 displays the result of the spectral calibration for the red, blue and the two green channels. To obtain the spectral sensitivity curves, the raw images were averaged over the illuminated pixel region and over a set of 10 images per wavelength.
Subsequently, the dark signal was subtracted and the spectral response for each channel was normalized to 1.
An example of one of the five specMACS scans is displayed in Fig. 9 showing the upper part of a 22°halo. The first panel HaloCam RAW and weighted with the spectral response, here for the R-channel. The radiometric response for HaloCam RAW is determined by dividing s n from HaloCam RAW by the specMACS radiance L, as depicted in Fig. 9c. Here, one radiometric response R for all sensor pixels is determined by averaging over all pixels in Fig. 9c under the assumption that the photoresponse non-uniformity is already accounted for by the flat-field correction. the five evaluated scenes. These values were derived using the specMACS scan to both sides of the sun as shown in Fig. 9c.
The uncertainties are provided within a 1σ confidence interval and comprise specMACS's total radiometric uncertainty, which is computed for each pixel, and the standard deviation of the calibration factor calculated over all considered pixels.

Signal noise
The measurements of the LIS can also be used to estimate the noise N of the measured signal as described in Ewald et al. (2016). The noise consists of the dark noise N d and the photon shot noise N shot . Thus, the standard deviation of the signal noise can be calculated by  10 For this analysis each sensor pixel, evaluated over five images, was used for all exposure times (0.5 to 9.5 ms). The results are shown for the R-channel here, but are very similar for the other three channels. According to Eq. (15), the variance of the signal measured by each pixel should scale linearly with the signal itself (Fig. 10a), whereas the standard deviation should scale with the square root of the signal (Fig. 10b) The total radiometric uncertainty of HaloCam RAW was estimated by applying Gaussian error propagation to the equations 5 describing the measured signal with the respective errors. Similar to the description in Ewald et al. (2016), the calculation of the total radiometric uncertainty will be outlined in the following. According to Eq. (1) the uncertainty of the radiometric signal S 0 is computed by combining the absolute uncertainties of the dark signal σ d (t expos , T ) and the instantaneous noise σ N (S 0 ) As defined by Eq. (12) the uncertainty of the normalized signal s n consists of the relative uncertainty of the photoelectric signal 10 σ S0 , the relative uncertainty of the flat-field calibration σ F , and the uncertainty due to the non-linearity of the sensor σ nonlin according to Uncertainties due to polarization of light by components of the camera or the casing were not determined for HaloCam RAW .
However, according to Ewald et al. (2016), the largest part of the polarization sensitivity of specMACS is introduced by 15 the transmission grating which adds the spectral dimension to the measurements. Since HaloCam RAW is not equipped with a grating, it is assumed that its polarization sensitivity is significantly lower than for specMACS. Direct solar radiation is unpolarized and thus the degree of polarization for scattering angles in the region of the 22°halo is expected to be less than about 5% (Hansen and Travis, 1974;Emde et al., 2010). The degree of polarization for transmitted light in the region of the 22°halo is lower than for observations of reflected light from cloud sides, especially in the rainbow scattering region, which is https://doi.org/10.5194/amt-2019-475 Preprint. Discussion started: 7 February 2020 c Author(s) 2020. CC BY 4.0 License. the focus of Ewald et al. (2016). Thus, we can conclude that even if the polarization sensitivity of the camera was significant, the error in the measured signal would be very small due to the low degree of polarization of the incoming radiation. Therefore, the contribution of the polarization sensitivity to the total measurement uncertainty is considered negligible for HaloCam RAW .
Finally, the radiometric calibration accounts for the error of the sensor response σ R , which was estimated from cross-calibration between HaloCam RAW and specMACS Table 3 provides the total relative and absolute radiometric uncertainties for the four channels of HaloCam RAW for two typical signals of 1000 DN and 3000 DN. The relative radiometric uncertainty is an estimate of the error of the normalized signal s n (Eq. (17)) which is smaller than 4% for all four channels. For larger signals the relative 2σ uncertainty is smaller since the absolute uncertainty is divided by a larger value (cf. Eq. (18)). This uncertainty is valid for signal ratios since they are 10 independent of the sensor response R. For spectral radiance measurements, however, the uncertainty increases significantly due to the contribution of the uncertainty of the estimated sensor response σ R . For the R-channel the total absolute uncertainty amounts to about 4.5% and about 5.5% for the two green channels. The uncertainty is largest for the B-channel with about 11.5%.
Since the radiometric response of HaloCam RAW was cross-calibrated against specMACS, its estimated uncertainty com-15 prises both the relative radiometric uncertainty as well as specMACS's absolute radiometric uncertainty. Additional minor sources of uncertainty are: First, the HaloCam RAW images are recorded every 10 s, so the average temporal offset between the specMACS and HaloCam RAW measurements amounts to 5 s which leads to small deviations due to cloud motion and inhomogeneities of the scene. Second, due to the temporal offset between the measurements the specMACS and HaloCam RAW scenes cannot be perfectly matched and a slight misalignment remains. Third, to compare the measurements, the specMACS 20 observations have to be convolved with the spectral response of the four channels of HaloCam RAW . For wavelengths at the edge of the spectral sensitivity of the specMACS sensor, the measurement uncertainty increases strongly introducing additional uncertainty in the estimated radiometric response for the HaloCam RAW measurements. This effect is responsible for the larger uncertainty of the blue channel, which has a spectral response centered at much shorter wavelengths. In this spectral region specMACS has a larger measurement uncertainty compared to the red and green channels.

Application
With a radiometrically characterized camera it is possible to quantitatively analyze the measured radiance distribution.  solute radiometric characterization (cf. Fig. 11b). The 22°halo ratio (HR) serves as a measure of the brightness contrast of the halo display (e.g. Forster et al. (2017)) and amounts to about 1.03 in the azimuth segment indicated in yellow in Fig. 11a LibRadtran radiance simulations were performed for an aerosol optical thickness 0.15 with the "continental average" optical properties mixture from OPAC (Hess et al., 1998), for an average solar zenith angle of SZA = 42.8°, assuming an absorbing surface, and using the spectral response of the red channel (cf. Fig. 8).  observations. Since the cirrus and aerosol optical thickness as well as the ice crystal effective radius are only a rough estimate, a slight offset between observations and simulations remains. On this basis a method can be developed to retrieve ice crystal properties which best match the observations with help of radiative transfer simulations.

Conclusions
We present a procedure for both geometric and absolute radiometric characterization of HaloCam RAW . HaloCam RAW is a 5 camera designed for the quantitative analysis of halo displays and is part of the automated halo observation system HaloCam described in Forster et al. (2017). The geometric calibration was performed using a chessboard pattern with known dimensions to determine the camera matrix as well as the distortion coefficients of the RAW image with Bayer pattern. The sensor's dark signal was determined using a climate chamber in a dark room and with the camera lens covered. The photoresponse nonuniformity (i.e. the vignetting effect), the spectral response of the RGB sensor, linearity of the sensor's radiometric response 10 as well as signal noise were characterized at the Calibration Home Base (CHB) of the Remote Sensing Technology Institute at the German Aerospace Center in Oberpfaffenhofen. While the spectral response was characterized using a monochromator, the remaining effects were determined using the large integrating sphere (LIS) of the facility. The absolute radiometric response was estimated by cross-calibrating the HaloCam RAW observations against the completely characterized specMACS imager for simultaneously measured scenes of a 22°halo. Finally, the total radiometric uncertainty was determined by taking into account the aforementioned error sources as well as the radiometric uncertainty of specMACS for the cross-calibration. The polarization sensitivity of HaloCam RAW was considered negligible.
For a typical measurement signal of 1000 DN the relative radiometric uncertainty amounts to 2.4% for both green channels, 2.8% for the red channel and 3.3% for the blue channel. For a larger signal of 3000 DN the relative radiometric uncertainty ranges between 1.7% for the green channels and 2.1% blue channel. The absolute radiometric uncertainty is larger due to the 5 additional uncertainty of specMACS and amounts to 5.0% for the red channel, 5.8% for the green channels and 11.8% for the blue channel for a sensor signal of 1000 DN and is similar to a signal of 3000 DN since a large part of the additional uncertainty arises from the inhomogeneity of the scene used for cross-calibration and the absolute radiometric uncertainty of specMACS.
The geometric and radiometric characterization of HaloCam RAW were applied to a scene observed on 21 April 2016 when a 22°halo was present for about 2 hours. The observed radiance distribution was compared to radiative transfer simulations using 10 solid columns, hollow columns and plates as well as three different mixtures of severely roughened and smooth ice crystals which produced a similar 22°halo ratio as the measurements. The remaining parameters of the cirrus were kept constant: an optical thickness of 0.4 was chosen to roughly match the absolute values of the HaloCam RAW radiances and an effective crystal radius of 20 µm was assumed, which is a typical value for cirrus clouds. Although this parameter choice produces a comparable brightness contrast for the 22°halo, some ice crystal habits produce additional features in their radiance distribution, which 15 are not visible in the HaloCam RAW observations and can be excluded in this case. For example, plates produce an additional 46°halo and hollow columns feature another the radiance peak at a scattering angle of about 18°. This comparison demonstrates the potential for developing a retrieval method for ice crystal properties including shape and roughness.
The absolute radiometric characterization allows to compare HaloCam RAW observations with radiative transfer simulations.
If the HaloCam RAW observations are analyzed using ratios of radiance values, the relative radiometric uncertainty applies.

20
This characterization is an important pre-requisite to develop a quantitative retrieval of ice crystal properties, like crystal shape, size, and roughness, from observations of halo displays using HaloCam RAW . Using a long-term database of calibrated HaloCam RAW observations together with radiative transfer simulations typical ice crystal properties of halo-producing cirrus clouds can be retrieved which will be the focus of future publications. The retrieved ice crystal properties have the potential to complement space-borne retrievals by adding information about the forward scattering part of the phase function.

25
Data availability. The HaloCamRAW data used for the camera characterization and the simulation results will be provided upon request.
Author contributions. LF and AB performed the measurements for the characterization of HaloCamRAW at CHB, DLR. LF evaluated the measurements and performed the radiometric and geometric characterization of the camera. LF also performed the radiative transfer simulations and evaluation of the case study for demonstrating the application of the camera characterization. MS designed and manufactured the weather-proof housing of the camera. TK assisted with the specMACS measurements and data processing for the absolute calibration. BM