HaloCam has been operated in Munich (Germany) since September 2013, where it provides continuous measurements including contributions to the ML-CIRRUS campaign in March and April 2014 . It was installed in Cabauw (the Netherlands) only during the ACCEPT campaign (Analysis of the Composition of Clouds with Extended Polarization Techniques, ) in October and November 2014. A first visual evaluation of halo display frequency during ACCEPT (10 October until 14 November 2014) was performed. The results are displayed in Fig.  as a Venn diagram . The occurrence of each different halo type is visualized by a circle. The radius of each circle scales with the total observation time for the respective halo type. Cross sections between the circles indicate instances where two or three halo displays were visible at the same time. The observation time is given in hours. The total time of HaloCam observations, which were collected during daytime only, amounts to about 344 h. With about 30 h, halo displays were observed in almost 9 % of the time. The presence of cirrus clouds within the HaloCam field of view was evaluated visually and amounts to about 110 h. Thus, about 27 % of the cirrus clouds produced a visible halo display. The 22∘ halo (complete or partial) occurred in 16.2 %, the sundogs in 19 % and the upper tangent arcs in 7.8 % of the time when cirrus clouds were present. Circumscribed halos were not observed during the campaign due to the low solar elevations.

As illustrated in Fig. , sundogs were observed more often than 22∘ halos, for about 21 vs. 18 h. Thus, sundogs occurred in 70 % and 22∘ halos in 60 % of the total halo observation time (30 h). Upper tangent arcs occurred in total for about 9 h (30 %) and were accompanied most of the time by 22∘ halos and sundogs. Thus, the majority of the halo displays were produced by oriented ice crystals.

Compared to the findings of , the relative fraction of 22∘ halos is roughly similar with 50 %, but sundogs with 12 % and upper/lower tangent arcs with about 15 % were far less frequent than observed during ACCEPT. The AKM observed the left and right sundogs with a relative frequency of 18 % each, compared to 36 % for the 22∘ halos. Although the frequency of simultaneous occurrence of the left and right sundog is unknown (from the AKM database), one can deduce that the relative frequency of sundogs is at least 18 % and thus larger than the result of . The reasons for the differences in the observed halo frequencies could be manifold: one main reason might be that a statistical evaluation over 6 weeks is compared to a database of 10 and 30 years (AKM). It is possible that the observation time during ACCEPT was not long enough to yield representative results for the frequency of the different halo displays. Another factor could be the observation site. The mountains in the east of Salt Lake City, the observation site of , could obscure the sun during periods with low solar elevation which are favorable for the formation of sundogs. So it is possible that on average fewer sundogs could have been observed in Salt Lake City than in Cabauw, which is surrounded by a rather flat landscape. Additionally, differences in the dominating weather patterns forming cirrus clouds in Salt Lake City and Cabauw could have an impact on halo formation as discussed in . For the AKM and the HaloCam dataset, information about dominating weather patterns for different halo displays is not available. Furthermore, the observation period during the ACCEPT campaign from October until mid-November was dominated by low solar elevations, which implies a higher chance for observing sundogs. Long-term observations have to be evaluated to obtain representative results of the frequency of the different halo types. To evaluate the large HaloCam dataset that has been collected for more than 2.5 years, an automated algorithm was developed for the detection of 22∘ halos. The following sections describe how the HaloCam images are processed and which features are extracted for an automated halo detection.

Halo displays are single scattering phenomena and thus are directly linked to the optical properties of the ice crystals producing them. The ice crystal phase function predicts the scattering angle Θ of the 22∘ halo relative to the sun. Thus, the analysis of the HaloCam images can be simplified significantly by mapping the image pixels to scattering angles. This means the camera has to be calibrated in order to determine the parameters for mapping the camera pixels to the real world spherical coordinate system. For this mapping the intrinsic camera parameters have to be determined, which are the focal lengths (fx, fy) and image center coordinates (cx, cy), as well as the distortion coefficients of the camera lens.

Different methods exist for the geometric calibration. Here, we use the method described by , which is based on , to estimate the intrinsic camera parameters as well as the radial and tangential distortion parameters of the lens. This method requires several pictures of a planar pattern, for example, a chessboard pattern with known dimensions, taken with different orientations. The calibration method using a chessboard pattern was implemented in OpenCV by and is described in detail by . Using the distortion coefficients and intrinsic parameters, the camera pixels can be undistorted and mapped to the world coordinate system. Thereby a zenith (ϑ) and azimuth angle (φ) relative to the image center can be assigned to each pixel. Since the image center is pointing to the center of the sun, the relative zenith angle (ϑ) corresponds to the scattering angle Θ in this case.

An overlay of the scattering angle grid onto a HaloCam picture is shown in Fig. a with representative contour lines at ϑ=22, 35 and 46∘. From the scattering angle grid the horizontal and vertical FOV can be calculated to ∼93.4 and ∼70.2∘, respectively. HaloCam images are recorded with a resolution of 1280×960 quadratic pixels, which results in an angular resolution of ∼0.07∘ for both the horizontal and the vertical direction. Figure b shows the relative azimuth angle grid, which is chosen such that the image is separated into six segments. For further analysis and feature extraction, each of these segments is averaged azimuthally.

For processing, the HaloCam images can be decomposed into their red, green and blue color channels. The brightness I of each pixel, provided in digital numbers (DN), can then be represented as a data array with 1280×960 elements. As an example the HaloCam image of Fig.  is used to demonstrate how the images are processed in case of a 22∘ halo. Figure  depicts the brightness distributions of the red, green and blue channel as a function of the scattering angle, averaged azimuthally over the uppermost image segment (no. 4 in Fig. b). The shaded areas around the lines in Fig.  represent twice the standard deviation of the averaged image region.

For analyzing the HaloCam observations several features can be extracted from the brightness distribution across the 22∘ halo, which will be explained in the following. The angular position of the 22∘ halo maximum (ϑhalo, max) is found by searching for the maximum brightness in the interval (21.0∘, 23.5∘). Then the angular position of the halo minimum (ϑhalo, min) is determined by looking for the minimum brightness in the interval (18.0∘, ϑhalo, max). Another important feature is the brightness contrast of the halo. In previous publications the so-called “halo ratio” (HR) was introduced as a measure for the brightness contrast of the 22∘ and 46∘ halo in the scattering phase function. In analogy, here, the halo ratio is defined as the brightness I at the scattering angle of the halo maximum ϑhalo, max divided by the brightness at the scattering angle of the minimum ϑhalo, min: HR=I(ϑhalo, max)/I(ϑhalo, min).

As an example, the values for I(ϑhalo, max) and I(ϑhalo, min) are displayed in Fig.  by the blue triangles pointing up (max) and down (min), respectively. For clear-sky conditions and homogeneous cloud cover, the brightness distribution decreases from the sun towards larger scattering angles, as shown in the example in Figs.  and . If HR<1 the brightness at the scattering angle of the halo maximum (I(ϑhalo, max)) is smaller than for the minimum (I(ϑhalo, min)), which is representative for a monotonically decreasing, featureless curve in this scattering angle region. This is the case for clear-sky conditions or homogeneous cloud cover without a halo. For HR=1 the brightnesses at the halo maximum and minimum are the same, causing a slight plateau in the brightness distribution. A distinct halo peak occurs for the condition HR>1. Thus, we assume HR=1 as lower threshold for the visibility of a halo.

For the example of Fig.  the 22∘ halo features are compiled in Table , which evaluated for the uppermost image segment. The scattering angle of the halo minimum (ϑhalo, min) is smallest for the red channel and largest for the blue channel, which is responsible for the reddish inner edge and the slightly blueish outer edge of the 22∘ halo visible in Fig. . It should be noted that in many cases the 22∘ halo appears rather white apart from a slightly reddish inner edge . The differences between scattering angles for the three colors are smaller for ϑhalo, max, with a slightly larger value for the blue channel. The halo ratio amounts to about 1.15 averaged over all three channels and is largest for the green and smallest for the blue channel.

The angular position of the 22∘ halo brightness peak (ϑhalo, max) can also be used to estimate the positioning accuracy of HaloCam relative to the sun. Figure  shows a histogram of ϑhalo, max for 1289 randomly selected HaloCam pictures showing a 22∘ halo in the uppermost image segment. This segment was chosen since it contains the most pronounced halos. For a faint halo the peak in the brightness distribution is rather flat, causing a larger uncertainty in finding the angular position of the peak. The mean value of ϑhalo, max amounts to 21.9∘ with a 2σ standard deviation of 0.5∘, which is a rough estimate of HaloCam's pointing accuracy. Since ϑhalo, max and ϑhalo, min are searched for within an angular interval, the pointing accuracy of ±0.5∘ is sufficient to detect the halo.

The detection is performed by a classification algorithm which is trained to predict whether a HaloCam picture belongs to the class “22∘ halo” or “no 22∘ halo”. For such a binary classification a decision tree can be used to create a model which predicts the class of a data sample. Details on decision trees are explained in Appendix . One major issue of decision trees is their tendency to overfit by growing arbitrarily complex trees depending on the complexity of the data. In this study we use the random forest classifier as described by , which improves the issue of overfitting significantly by growing an ensemble of decision trees. A description of the random forest classifier used in this study is provided in Appendix . In principle, other classification algorithms could be used, like artificial neural networks. The reasons why the random forest classifier was chosen are as follows. Apart from being robust against overfitting it does not require much preprocessing of the input data like scaling or normalizing. During the training of the individual trees the out-of-bag (OOB) samples (i.e., the samples which were not in the training subsets) are used as test data, and classification error estimates (e.g., out-of-bag error) can be calculated simultaneously . In contrast to an artificial neural network, the basic structure and the internal threshold tests of the decision trees are simple to understand and can be explained by boolean logic. Henceforward, the algorithm applied to the classification of 22∘ halos will be called HaloForest.

The features used here for the classification are the 22∘ halo ratio, the scattering angle position of the halo minimum and maximum, and the scattering angle confining the halo peak ϑhalo, end, which are shown in Fig.  together with the slope of the regression line in black (solid). The halo peak is confined by ϑhalo, end (dash-dotted line), which represents the scattering angle with the same brightness level as ϑhalo, min in the scattering angle interval (ϑhalo, max, 35∘]. This feature is used to ensure that the brightness for angles larger than ϑhalo, max is decreasing again. The slope of the regression line serves as an estimate for the brightness gradient around the sun. For clear-sky images this gradient is steeper than for overcast cases. As a measure of the separation of color in the halo, the scattering angle difference between the blue and red channel for the halo minimum (Δϑhalo, min) and maximum (Δϑhalo, max) are calculated, which are defined as Δϑhalo, max=ϑhalo, max, blue-ϑhalo, max, red,Δϑhalo, min=ϑhalo, min, blue-ϑhalo, min, red. Furthermore, the standard deviation of the brightness averaged over the image segment is used as a proxy for the inhomogeneity of the scene. These eight features are calculated for each of the six image segments separately.

In order to get an impression of typical values of the training features for the two classes, Fig. a–c show two-dimensional scatter plots of selected feature pairs for the upper image segment (no. 4). Features which belong to the class “22∘ halo” are displayed in blue, whereas the features of the class “no 22∘ halo” are represented by gray scatter points. Figure a shows the distribution of the scattering angle of the halo maximum vs. minimum. The scattering angles of the halo maximum ϑhalo, max are confined to a smaller interval for “22∘ halo” compared to “no 22∘ halo”. However, the two classes share many data points in this projection, so more features are needed to generate decision boundaries in a higher, here eight-dimensional, space. Figure b depicts the scattering angle difference between the blue minus the red channel for the halo maximum (Δϑhalo, max) vs. minimum (Δϑhalo, max), which is positive for the “22∘ halo” class since the inner edge (smaller ϑ) of the 22∘ halo is slightly red. The HR, which is shown in Fig. c, takes values between 1 and ∼1.3 for “22∘ halos”. Images with a low mean standard deviation of the image segment indicate rather homogeneous scenes which are present most of the time when a 22∘ halo is visible. Figure a–c visualize that the two classes, “22∘ halo” and “no 22∘ halo”, can not be separated easily since the values of the features overlap. The lower panels of Fig. d–f display the regions which are detected as “22∘ halo” (blue) and “no 22∘ halo” (gray) by the trained algorithm.

For each of the six image segments an individual classifier was trained using a dataset of visually classified HaloCam images which were chosen randomly from the dataset. The performance of the classifiers was tested using a random selection of 30 % of the dataset which was excluded from training. This procedure was repeated 100 times to get statistically significant results for the performance of the classifier.

Table  shows the confusion matrix for the classifier of the segments directly above (no. 4) and below the sun (no. 1) which represent the two extreme cases of the performance of the six different classifiers: the upper part of the 22∘ halo has a higher brightness contrast compared to the lower part which is often obstructed by the horizon. For the training of HaloForest 1289 samples with a 22∘ halo and 5181 samples without 22∘ halo were used for the uppermost segment (no. 4). The lowermost segment (no. 1) was trained with 296 and 3370 samples of the classes 22∘ halo and no 22∘ halo, respectively. The lines of the confusion matrix indicate the true class labels of the samples (“22∘ halo” and “no 22∘ halo”), whereas the columns contain the predicted class labels. The number of true positive and negative (in bold) as well as false positive and negative classifications are evaluated and provided with a 2σ standard deviation. The correct classification of “22∘ halo” is maximum for the uppermost image segment (no. 4) with about 98 % and minimum for the lowermost segment with about 89 %. The correct classification of “no 22∘ halo” is overall higher than 99 %, so the HaloForest algorithm seems to be able to separate the two classes well. The performance of the other four segments ranges between the results of the upper and lowermost segments.

HaloForest is used to evaluate the dataset HaloCam collected in Munich between January 2014 and June 2016. To ensure a high classification accuracy, only the classifiers for the upper image segments (3, 4 and 5) were used (cf. Table ). A HaloCam image was assigned to the class “22∘ halo” if at least one of the image segments 3, 4, or 5 predicts a 22∘ halo. Applying a probability threshold of p>0.5, 22∘ halos were detected in 152 h. Relative to the total observation time during daylight of 7345 h, 22∘ halos occurred about 2.1 % of the time. As an additional test, the classification accuracy of HaloForest was checked for 470 randomly chosen HaloCam images for the “22∘ halo” and “no 22∘ halo” class within this long-term observation period in Munich. The confusion matrix for this test is provided in Table  for the image segments no. 3, 4 and 5 together. More than 88 % of the 22∘ halos are classified correctly and less than 12 % are classified incorrectly as 22∘ halos.

Images were incorrectly classified as 22∘ halo predominantly due to small bright clouds or contrails in a blue sky or structures in overcast conditions which happen to cause a peak in the averaged brightness distribution at a scattering angle of 22∘.

Based on these results we investigated the fraction of cirrus clouds which produced a halo in Munich during this time period. The total frequency of occurrence of cirrus clouds was determined by independent data of co-located CHM15kx ceilometer observations . To guarantee consistent observational conditions, only ceilometer measurements in the absence of low-level clouds were considered. Proprietary software of the ceilometer automatically provides up to three cloud base heights with a temporal resolution of 15 s. The detection is based on the fact that in case of clouds backscatter signals are significantly larger than the background noise.

The sensitivity of the ceilometer is sufficient to even detect clouds near the tropopause during daytime. Since ceilometers, however, do not provide depolarization information, the discrimination between water and ice clouds was made by means of the cloud base temperature Tbase. state that cirrus cloud base temperatures ranged between -30 and -40∘C during the 10-year observation period at the FARS observation site. As a temperature threshold is not an unambiguous criterion for the existence of ice clouds, we have calculated the frequency of occurrence for three different temperatures: -20, -30 and -40∘C. If Tbase is lower than the given temperature threshold, the cloud is considered a “cirrus cloud”. The temperature profiles were obtained from routine radiosonde ascents of the German Weather Service at Oberschleißheim (WMO station code 10868), which is located about 13 km north of the HaloCam site. During the time period from January 2014 until June 2016 a fraction of 5.6 % cirrus clouds was detected for a cloud base temperature of Tbase<-20∘C. Towards lower cloud base temperatures the amount of detected cirrus clouds decreases to 3.5 % for Tbase<-30∘C and 1.9 % for Tbase<-40∘C.

Due to the different pointing directions of the ceilometer (towards zenith) and HaloCam (towards sun), the instruments observe different regions of the sky. This is accounted for by prescreening the data for 1 h time intervals when the ceilometer detected a cirrus cloud. The prescreening is subject to data availability for both instruments. The subsequent analysis of cirrus fraction and halo frequency of occurrence is based on the full temporal resolution of 15 and 10 s, respectively. Relative to the amount of detected cirrus clouds about 25 % occurred together with a 22∘ halo for the image segments 3, 4 and 5. This fraction does not change much for the different cloud base temperatures (26.4 % for Tbase<-20∘C and 24.5 % for Tbase<-40∘C) since the fraction of detected clouds decreases together with the detected halos for lower temperatures. According to the confusion matrix in Table , 88.8 % of the detected “22∘ halos” are real halos, while 2.8 % of the “no 22∘ halos” are actually “22∘ halos”. Correcting the result for the estimated false classifications, the fraction of “halo-producing” cirrus clouds amounts to about 25%×88.8%+75%×2.8%≈24%. The comparison of the ceilometer and HaloCam data implies that about 25 % of the cirrus clouds contain some fraction of smooth, hexagonal ice crystals. observed a fraction of 37.3 % cirrus clouds which produced a 22∘ halo within 1 h time intervals. The results most likely differ because the observations originate from different locations which might be dominated by different mechanisms for cirrus formation. It has to be noted, however, that the evaluation method is very sensitive to the sampling strategy of the observations: the fraction of halo-producing cirrus clouds increases to more than 50 % if the HaloCam observations are binned to 1 h intervals, which are counted as containing a halo regardless of their duration.

For comparison, the fraction of cirrus clouds producing a halo display was evaluated visually for the HaloCam observations during the ACCEPT campaign and amounts to about 27 % including 22∘ halos, sundogs and upper/lower tangent arcs (cf. Sect. ). This value is also lower than the result provided by who observed any of the three halo types in about 54 % of the 1 h periods with cirrus.

The current version of HaloForest discriminates only between the two classes “22∘ halo” and “no 22∘ halo”. Thus, interference with other halo types as sundogs or upper/lower tangent arcs and circumscribed halos might occur at certain solar elevations. The position of sundogs relative to the sun depends on the solar zenith angle (SZA) and can be calculated analytically as described in , , , and . The sundogs are located at scattering angles close to the 22∘ halo for large SZAs and occur at larger scattering angles for small SZAs, i.e., high solar elevations. Figure  shows the same HaloCam image with the azimuth segments as Fig. b. In addition, the minimum scattering angle of the sundogs are calculated as a function of the SZA and represented by the red and green squares. The SZAs range between 90 and 35∘ with a resolution of 1∘. The two white circles centered around the sun at scattering angles of 21.0 and 23.5∘ indicate the mask which is used to find the scattering angle of the 22∘ halo peak. For SZA≤67∘ the sundog positions are located outside this mask and cannot be misclassified as 22∘ halo (green squares). The red squares represent sundog positions which are located within this mask and might therefore be misclassified. This is the case for SZAs between 90 and 67∘. To obtain an estimate of the fraction of sundogs which are misclassified as 22∘ halo, 1000 randomly selected HaloCam images were counter-checked visually. This revealed that only six images showing sundogs without 22∘ halo in the segments (3–5) were misclassified as 22∘ halo, which is <1%. Upper tangent arcs could be detected by the uppermost image segment (no. 4) and might be misclassified as 22∘ halo. For very small SZAs (high solar elevations) the tangent arcs merge to form the circumscribed halo which could be detected in the segments 3 and 5 as well. The same procedure was repeated for these halo types: 1000 randomly selected images were checked for the presence of tangent arcs and circumscribed halos without 22∘ halo, yielding 28 images or 2.8 %. However, if only a fragment of a halo is visible in the uppermost segment, it is generally difficult to discriminate between an upper tangent arc or circumscribed halo and a 22∘ halo.

The halo classification algorithm was presented for 22∘ halos, but it is possible to include training data for other halo types as well. With the current version of HaloForest and the co-located ceilometer observations, the fraction of cirrus clouds producing a halo display was estimated to about 25 % for Munich between January 2014 and September 2016. Extending HaloForest for the detection of other halo types, such as sundogs, the fraction of halo-producing cirrus clouds could easily exceed 25 %. In principle, HaloCam could also be equipped with a wide-angle lens to observe halo displays in a larger region of the sky, however, at the expense of spatial resolution.