Concentric 3rd and 4th order rainbows at Dresden-Langebrück, Aug 11th, 2020

17:27:54 CEST, Pentax K-5, smc Pentax-DA fisheye 10-17 mm at f = 10 mm, f/8, 1/1600 s at ISO 80, artifact-minimized processing by separate unsharp masking of color differences

Original image

Green minus blue color channel difference unsharp masked

Color differences unsharp masked

Taking photographs of tertiary and quaternary rainbows is difficult, as usually you won’t see what you are aiming at – theory predicts that the tertiary is just at the threshold of visibility [1], and for the quaternary the situation is even worse. I myself have been lucky only once before [2], (though having tried for almost a decade now), if one does not count experiments using artificial sprays [3]. This cannot be attributed to a lack of opportunities. Over the years, I experienced several promising situations, but afterwards no higher order rainbows could be extracted from the photographs by image processing. One problem is that cloud structures in the background mess up the unsharp mask filter. But maybe also my timing was just wrong and these rainbows did not appear when I expect them to do, because I misjudged the shower and illumination geometry.

Anyway, on Aug 11^th, 2020, my instincts were right. Just before I finished work in Dresden-Langebrück, a moderate shower moved in from the northeast. The sunlight was somewhat dim, which resulted in an unspectacular primary rainbow from 17:15 CEST onwards. I could see some larger drops glint in the sunlight. I went to my car on the parking lot during the next minutes, and lost sight to the east, so I do not exactly know what happened later on this side of the sky. The rain was still ongoing, but not heavy, there were people cycling around without looking too much disturbed. I’m not entirely sure if it was still the same rain shower, or another one which had meanwhile moved in. I took three photos at the parking lot (17:23-24 CEST), and then drove about 100 m to a spot with a good view towards the west. There was nothing to be seen with the naked eye, just the sun, some glinting raindrops, a cloud, and blue sky below. A major problem is that raindrops will fall on the front lens, especially when using a fisheye objective that cannot be shielded and has to be held out of the car window to make proper use of its field of view. So one needs to be fast, otherwise the images will be spoiled by artifacts from the drops. That is why I did not do any image stacks, just a threefold exposure bracket per shot for safety (to have at least one picture with a useful exposure). In between, I had to wipe the lens dry. All of the ten shots I took there from 17:26-30 showed the tertiary rainbow after some processing, and it also appeared on the earlier pictures from the parking lot, as far as the view permitted. I was rather overwhelmed to see more than the complete upper half of the tertiary against the blue sky and cloud background when I first applied an unsharp mask to the images. The quaternary rainbow (located just outside the tertiary) can also be detected beyond any doubts, especially on the left side. I found no supernumerary arcs and no traces of the seventh order [4] (which I had missed to pay attention to during the observation, I also did not use any polarizers, and as mentioned, I did not record image stacks).

17:30:00 CEST, Pentax K-5, smc Pentax-DA fisheye 10-17 mm at f = 17 mm, f/14, 1/400 s at ISO 80, artifact-minimized processing by separate unsharp masking of color differences

Original image

Green minus blue color channel difference unsharp masked

Color differences unsharp masked

It is well known that aerodynamic flattening of larger raindrops has an impact on rainbows through the so-called Möbius shifts, but so far the consequences for higher order rainbows have only been studied theoretically. Are there any new insights from this observation? This, of course, requires an image calibration, i.e. the assignment of scattering coordinates (scattering angle and clock angle, i.e. the sun-centered azimuth, which I will count clockwise from the rainbow’s top here) to the individual pixels. The position of the sun is easily calculated from the time the respective photograph was taken (17:27:54 CEST for the top image, checked with a radio-controlled watch) and the location (51.13° N, 13.83° E), which gives in this case an elevation of 27.6° and an azimuth of 259.5°. The pixel coordinates of the sun’s center could also be reliably determined from an image version developed as dark as possible from the RAW file in Photoshop. The projection of this specific lens I had measured eight years ago (when I newly got it), but I did also a cross-check with recent starfield pictures (abundantly available from my attempts to catch Perseid meteors the following nights). In order to determine the relevant Euler angles (elevation and azimuth of the camera’s optical axis, and the rotation of the camera sensor chip around this axis), I still needed another reference mark. Luckily there was a telecommunication tower at the horizon which I could identify and then calculate under which elevation and azimuth it is seen from the observing location.

Technical sidenote: The two coordinates of the reference mark provide me indeed with one condition more than the number of available degrees of freedom, so I can check the overall consistency of the calibration. And here it initially turned out to be not very convincing. What went wrong? When checking my hidden assumptions I found that I had pinned the piercing point of the optical axis to the precise center of the CMOS sensor (in terms of pixel coordinates). This may not be realistic, and, moreover, in a Pentax K-5 camera the sensor can move several millimeters to compensate for shaking. Even with the shaking compensation turned off there is no guarantee that it will find and stay in the optical center position (plus, there are decentering errors of the lens). From the working principle of the calibration procedure I expect the decentering error to be of quadratic order, and it may turn out to be negligible for longer focal lengths. But it matters for a fisheye lens. So I used the amounts of decentering in X and Y as further degrees of freedom and achieved consistent results for a shift of 26 pixels (0.12 mm) in the horizontal (and zero in the vertical). However, a further reference mark would be needed for a truly unique determination. Then there would still remain the assumption of a rotationally symmetric lens, but this seems to be acceptable as indicated by the recent starfield test.

From an equilateral projection in scattering coordinates it can then be deduced that the tertiary does not bend significantly over the recorded range of clock angles, which also holds for the quaternary as far as it peaks out of the noise. So seemingly they appeared as perfectly concentric circles here!

Equilateral projection in scattering coordinates

This is somewhat surprising, as theory predicts that these rainbows are also subject to Möbius shifts of various amounts along their circumference, which should become noticeable if larger (more distorted) drops are involved. Interestingly, for this sun elevation the Möbius shifts will move the tops of  the 3^rd and 4^th order rainbows towards each other. They might even overlap for an effective drop radius of 0.5 mm [5]. However, in nature, in most cases the drop size distribution (DSD) will cover a broad range of sizes, also including small drops. Because the shape distortion sets in (at least) quadratically with rising drop radius, it is likely to see some of the traditional concentric sphere-drop rainbows shine through in the full mixture of size dependent rainbows. This I already noted in simulations using broad Marshall-Palmer DSDs (i.e. a simple decaying exponential). As mentioned, there was no heavy rainfall going on during the observation on Aug 11^th, so a dominance of the less distorted smaller sizes can be reasonably assumed – regrettably there are no direct measurements of the DSD. Lee and Laven [1] argue that broad DSDs tend to wipe out the tertiary’s top and leave only the sides, but their analysis was based on much lower sun elevations than occurring here.

So in order to see how much the concentricity of the tertiary and quaternary rainbows will be affected in this specific case I did some simulations for the proper sun elevation and a (guessed) DSD which contains mostly small and moderate sized drops: A Marshall-Palmer with decaying parameter (Lambda) of 4 mm^-1, as previously used [5]. There are two complementary simulation methods which I can apply: 1) GO: Geometric optic raytracing (including polarization, but neglecting interference and diffraction) for all rainbow orders up to the 7^th, based on a Beard-Chuang cosine series drop shape model, with optional (2,0) quadrupole mode oscillations and Gaussian tilts of the symmetry axis from the vertical, and 2) DMK: Debye series calculations for spherical drops of various sizes, superimposed in intensity after being shifted in scattering angle by the appropriate Möbius value (depending on drop size, rainbow order, and clock angle, following Können [6]). These calculation include only rainbow orders up to the 5^th. The Möbius shifts themselves are taken from a look-up table comprising earlier raytracing results. These were calculated from a simpler shape model (two conjoined half spheroids fitted to Beard-Chuang shapes) and do not include drop oscillations or tilts for the higher-order rainbows yet. However, this second method has the advantage of showing if supernumerary arcs can be expected under the given conditions.

I removed the most disturbing directly transmitted light (sometimes referred to as “zero order glow”) as well as the less important contributions from external reflection and the lowest two rainbow orders from the simulation, and show the resulting clear higher-order rainbows in the same sunward projection (and for the same sun elevation) as in the top image. As a reference, I also let simulations run for spherical drops with the same DSD. These, of course, turn out perfectly concentric (in scattering coordinates, not necessarily in the projected image). After having switched on drop distortions, it is reassuring to see that both methods agree in keeping the upper halves of the tertiary and quaternary well separated and still nearly concentric. However, a tendency to blur these parts can be noted, due to the contribution of larger drops. Two more pieces of information can be extracted here: Introducing moderate axis tilts and (2,0) oscillations (both their amplitude distributions set to the “standard values” used in [7]) does not lead to visible changes in the result (GO), and supernumeraries do not appear, neither for flattened nor spherical drops (DMK).

GO sphere reference simulation

GO distorted drops, no oscillations, no tilts

GO distorted drops, with oscillations and tilts

DMK sphere reference simulation

DMK distorted drops, no oscillations, no tilts

The latter result illustrates that in broad DSDs supernumeraries need the stabilizing “Fraser mechanism” [8] to become visible: If, with increasing drop size, the Möbius shifts grow in the opposite direction than the supernumeraries’ convergence towards the Descartes angle due to their shrinking angular width, there will be a certain critical drop size at which these effects compensate. Because of the resulting position stability against changes of drop radii, the supernumeraries of drops around the critical size will peak out from the unstructured background of superimposed non-aligned supernumeraries of other sizes. Traditionally, this argument is invoked for the primary rainbow (with a critical drop radius of about 0.25 mm for the first supernumerary), but it holds likewise for all other orders [9]. If the Möbius shifts have the wrong sign (as for the tertiary and quaternary bows at the sun elevation of my observation) or are set to zero (as in the sphere reference simulations), there exists no compensation point and the averaging of all supernumeraries results in a more or less uniform intensity gradient.

The GO simulations reproduce also the 7^th order rainbow, but, under the assumed conditions, do not predict any amplification effects for it caused by drop distortions or oscillations. In fact, it is not even recognizable in the simulation pictures shown here, but can be extracted by a larger intensity-to-RGB-value scaling factor (or higher gamma value).

In conclusion, the observed concentric tertiary and quaternary rainbows without supernumeraries can be consistently interpreted in the current theoretical framework of broad raindrop size distributions and drop shapes with aerodynamically plausible amounts of flattening and oscillations. Even though shape distortions have a larger influence on higher orders, they do not forbid that traditionally shaped rainbows are formed, if enough small drops are present. Of course, any observations of genuine non-spherical drop effects such as higher order twinned bows are highly welcome as they would allow for a more challenging test of the simulation models.

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A multi-split rainbow from south-east China, August 12th, 2014

 

Yulong river, Guangxi province, August 12^th, 2014, 17:18 Chinese standard time

Twinned rainbows are rare sightings, in the sense that one may see on average only one per year in Central Europe even when paying close attention. Much rarer still, and maybe restricted to regions closer to the equator, are multi-split rainbows. Only few cases have been documented so far [1, 2, 3], though more snapshots can be found on image sharing platforms labeled as “triple rainbow” etc. It is always a very favorable situation if an archivist and analyst like myself can establish direct communication with a skilful observer, who recorded details of a rainbow display that provide some insight beyond the pretty pictures.

In April 2019 I emailed Mr. Ji Yun, who manages a Facebook group dedicated to atmospheric optical phenomena in China, asking about a spectacular photograph of a multi-split rainbow which had been shared there. He kindly relayed my request to Mr. Liu Hai-Cheng, the original observer. Mr. Liu agreed to answer a long list of questions and I also received two sets of photographs from August 12^th, 2014, one from his Sony NEX-5C camera (equipped with a Nikon AF 28mm f/2.8 lens) and the other from his cell phone (Coolpad 8720L). The camera clock’s time stamps were calibrated with respect to the actual local time by comparing camera and cell phone pictures, and assuming the cell phone clock to be synchronized over the network. All time data are given here in Chinese standard time (UTC+8h).

Mr. Liu observed this rainbow rarity in the beautiful landscape of the karst mountains near the Yulong bridge (Yangshuo County, Guilin City, Guangxi province, about 400 km northwest of Hong Kong, 24.8° N, 110.4° E) during a boat trip on the Yulong river. He remembers that it was very hot that afternoon. It began to rain before he passed through the tunnel of the bridge (at about 16:50), with some heavier rain lasting for about 25 minutes. There was no lightning, thunder or strong wind.

Judging from the photos, the rainbow appeared at about 17:10 within 30 s or less. Already on the early photographs there are hints of the unusual splitting of the primary:

17:11

However, Mr. Liu’s visual impression was that the splitting became prominent only later, after the (seemingly ordinary) primary and secondary bow had appeared successively. He also noted that the visibility of the split branches changed over time, while the main primary could always be seen clearly.

Towards the end of the shower, the display reached its peak quality. The following pictures cover the full right-hand side of the rainbow and some of the left. They are presented without additional filtering to allow for a better assessment of the natural contrast conditions.

17:18 (unfiltered version of the title image)

17:18

17:19

For a deeper analysis, I chose the title picture, recorded at 17:18. In the contrast-enhanced version, three primary branches are directly visible, with the most intense one in the center. The secondary rainbow, as far as it is included in the frame, does not exhibit any anomalies. This is a typical feature in (almost) all split rainbow observations known so far. My goal was now to transform the photograph into the scattering angle vs. clock angle coordinate system (in equirectangular projection), as I did on previous occasions [1, 4]. The scattering angle is the angular distance from the sun, and the clock angle the azimuth around the rainbow’s circumference, with the 0° position corresponding to its top.

The sun’s position is easily obtained from standard astronomy software (giving an elevation of 25.4°, and azimuth of 275.4°). Additionally, the precise focal length of the lens (in pixel units) and distortion characteristics need to be known, as well as the camera pointing direction in elevation and azimuth, and the angle describing the rotation of the sensor’s pixel grid with respect to the vertical.

To precisely determine these quantities, a rather extensive calibration must be carried out. Here I had to try some reasonable guessing: There is a nominal focal length in mm, the sensor data (pixel pitch) can be looked up, as well as some distortion information for this specific lens. From aerial pictures showing the river and individual mountains, the viewing direction can be estimated. The appearance of the water surface gives some clues about the camera rotation. In combination, all these estimations allow for a plausible transformation:

Assuming this reconstruction to be not too far off, it is immediately obvious that the bright central branch does indeed fit to the conventional primary rainbow locus at a constant scattering angle of about 138°. As expected, the secondary ends up at about 129°, also as a straight line. The lower branch (i.e. at higher scattering angles) can in principle be explained by aerodynamically flattened raindrops, following a long tradition in rainbow physics [5, 6, 7, 8, 4]. However, the upper branch penetrating into Alexander’s dark band requires elongated raindrops, whose existence cannot be accounted for by aerodynamics alone. Electrostatic fields [9] can elongate raindrops, but in the absence of any lightning activity it is speculative if any higher fields were present. Elongated shapes do also occur as transitory states during oscillations of larger drops in the appropriate (axisymmetric) modes [10].

The problematic element in this explanation is, however, that in the case of the rainbow we deal with a large number of contributing raindrops and a temporal average due to the finite exposure time. So we need an argument why contributions from transitory states are not simply wiped out. The resonance frequencies of the individual drops depend on their size, so no singular event such as an acoustic shock wave from thunder (if there had been any at all) can synchronize the oscillations. The only plausible idea for a formation of stable rainbow branches by drop oscillations in a stochastic ensemble might be that the two extremal states of the oscillation (flattened and elongated) are encountered with a higher probability than intermediate ones, as the momentary velocity decreases to zero at the turning points of any classical oscillation. Admittedly, this requires a rather narrow distribution of amplitudes throughout the ensemble (at least in the dominant drop size range), as otherwise the branches will be wiped out again due to the spread in extremal axis ratios. To my knowledge, there is not enough data on the statistical properties of oscillations in large ensembles of natural raindrops published yet to draw a definitive conclusion here.

Some further details of this observation are worth to be noted: The three branches of the primary bow appear each in a distinct fashion: The lowest is broad and rather diffuse, the middle one is bright and shows the features of a typical primary rainbow, the top one is narrow with a sharp uppermost outer rim. Moreover, it gives the impression of having developed a downward sub-branch in the –10°…+5° clock angle interval, resulting in a four-fold split bow there.

Rainbows certainly go on fascinating people all over the world, and rightfully so: Even in the 21^st century, some outstanding displays occur from time to time that still challenge our understanding. Maybe those in hotter climates with intense rain showers have better chances of catching such rarities. In any case, we have to go out and take a look and a picture at the right time.

Third and fourth order rainbow

In the late afternoon of July the 7th 2017 there were strong thunderstorms in the area around Berlin, Germany. AKM member Andreas Möller was driving trough heavy rainfall, when suddenly the sun came out. His report:

On my way home, I could observe a beautiful bright primary and secondary rainbow. It was still raining heavily and my intension was to observe the area towards the sun. Therefore I turned into a side street and stopped in front of an old industrial area.

Location details:

• Ferdinand-Schultze-Straße 18, 13055 Berlin, Germany
• Weather: Strong rain
• Sun altitude: ~19°
• Date: 2017-07-07
• Time: 19:00 – 19:08 CEST

https://kachelmannwetter.com/de/regenradar/stadt-berlin/20170707-1705z.html

I took a lot of pictures in hope to get the third and fourth order rainbow. My equipment was a Nikon D750 with a Tamron SP 15-30mm f/2.8 at 15mm. The rain was strong and I had problems cleaning my lens from waterdrops. Later at home, I started to process the pictures directly. Amazingly, I could discover the third and fourth order in almost all of the pictures I took.

The image processing did clearly point out a colorful third and fourth order rainbow.

Processing details:

1. stack of 8 frames
2. unsharp masking (USM)
3. contrast and light adaption with Photoshop
4. unsharp masking with Photoshop

Here is another image processed out of a single RAW file. (USM)

Gloridescent clouds

As “Gloridescence” I define colored clouds in the antisolar area, where there is no visible connection to a glory.

The first observation of colored clouds at the antisolar point was made by Stefan Rubach on Mt. Großer Arber at Jan. 26, 2007. We suspected fragments of a glory, but we were not sure.

Jan. 26, 2007: Glorydescence on Mt. Großer Arber. Photo: Stefan Rubach

On Nov. 18, 2007, I made the first observation of my own and on Mar. 1, 2010 my second observation at Mt. Wendelstein (1835m).

Nov. 18, 2007-Wendelstein

Nov. 18, 2007-Wendelstein

Nov. 18, 2007-Wendelstein

Nov. 18, 2007-Wendelstein

Mar. 1, 2010-Wendelstein

At Mt. Zugspitze (2963m) I observed these colored clouds a few times and named them „gloridescent clouds“ (and so far no one ever challenged this name).

On Apr. 25, 2015 I made my first observation of „gloridescent clouds“ at Mt. Fichtelberg (1215m). Meanwhile we received more observations, one from the valley of Neckar river, one photo by Eva Beatrix Bora from Stavanger, Norway and some from an aircraft (1 – 2). From these we conclude that:

• Just as glories become more frequent with increasing observing levels (see this article), the frequency of “Gloridescence” also increases.
• At lower altitudes (i.e. in the area of low clouds), “Gloridescence” originates mainly from underneath of stratocumulus clouds.
• At higher mountains (e.g. Zugspitze, 2963m) and on airplanes, “gloridescent clouds” are more frequent and appear mainly in deeper cloud layers or single shreds of clouds.

 

Author: Claudia Hinz, Schwarzenberg, Germany

Frequency of glories from different observation levels

Glory and Fogbow with interferences at Mt. Zugspitze

The combination of spectre of Brocken with glory and fog bow is named after the German Brocken mountain, even though it cannot be observed there too often. My colleagues from the weather station estimated a frequency of 2 or 3 observations per year at the top of the mountain. The phenomena much more frequently observed at higher mountains.

Since there is no reliable statistics about the frequency of Glories to date, I tried to obtain some tendencies from my own observations on various mountain tops.

I observed at three different mountain tops where I worked for a longer amount of time:

1. Mount Fichtelberg, Ore mountains, 1214m (similar height as Mt. Brocken)
2. Mount Wendelstein, Alps, 1838m (standalone rock)
3. Mount Zugspitze, Alps, 2963m (main mountain chain of the Alps)

Fichtelberg I observed most frequently in the early morning hours without interferences. On Mt. Wendelstein the Glories often long duration phenomena, sometimes very colorful with impressive interferences. On top of Mt. Zugspitze the Glory was visible at every solar altitude, in most cases long duration, with impressive interferences an colors.

I tried to capture the frequency of glory statistically. Since I could not look at the same time periods, the statistics is an approximation.

These observations lead to the following conclusions:

• The frequency of glories increases with altitude (at my observing sites the number of glories increased by a factor of three for every 1000m altitude)
• The higher the altitude of the observation point, the more impressive are the glories! With increasing altitude of the cloud, the size of the droplet in the clouds decreases and interferences become more frequent. Because the smaller and more uniform the droplet size, the more impressive becomes the glory (Simulation of Les Cowley). In the best case, the glory transforms into interferences of a cloud bow.
• The duration of the phenomenon increases with the altitude, too. If the local conditions allow observations well below the horizon, the glory is possible at every solar altitude.

Fichtelberg

Fichtelberg

Fichtelberg

Fichtelberg

Fichtelberg

Wendelstein

Wendelstein

Wendelstein

Wendelstein

Wendelstein

Zugspitze

Zugspitze

Zugspitze

Zugspitze

Moon Glory Zugspitze

Author: Claudia Hinz, Schwarzenberg, Germany

Olea europaea pollen corona

During the days of the 12th Light & Color in nature meeting (May 31^st-June 3^rd, 2016) in Granada, Spain, I noticed almost constantly a diffuse aureole around the sun, appearing against the background of a clear sky:

May 31^st, 18:27 CEST

June 2^nd, 11:32 CEST

June 3^rd, 19:38 CEST

previous picture unsharp masked

All photos were cropped to a common viewing angle of 15° x 15° and the color saturation was increased.

Because of the dry and often cloudless summer weather we had back then, it seems unlikely that any kind of water drops did cause the phenomenon. On the other hand, the angular radius was way too small for Bishop’s ring, which at first seemed to be a plausible option as we had observed some haze towards Africa shortly before our plane landed in Malaga on May 30^th.

No pronounced color pattern was visible to the naked eye, nor through a gray filter, but the saturation increase in the image processing revealed a typical corona structure with alternating colors. Thinking of pollen as possible scattering particles, the large amount of olive trees (olea europaea) in Andalusia immediately comes to mind. Furthermore, we witnessed ourselves that the olive trees were blooming these days when we visited a grove at Monachil in the vicinity of Granada – some of the visitors’ shirts or backpacks got covered with green dust after coming too close to the trees.

The olive grove at Monachil with the Sierra Nevada in the background (June 2^nd, 2016)

A blooming olive twig from a tree in this grove (June 2^nd, 2016, photographed by Hironobu Iwabuchi)

In order to check this hypothesis I looked up the shape and size of olive pollen: They are almost spherical with a mean polar diameter of 20.1 µm and mean equatorial diameter of 21.5 µm. For most of the observations, the sun elevation was high enough to simply approximate the pollen as spheres of 21.5 µm in size. I calculated the resulting corona from the solar spectrum using simple diffraction theory (which at this particle sizes is justified):

Both the photograph and the simulation (right hand side) were cropped to a field of view of 10° x 10°. For the simulation, I assumed a relative spread in the pollen size (standard deviation of a Gaussian distribution divided by the mean diameter) of 15%, convoluted the result with the sun’s disk and added a gray background. It matches the photograph quite well, though the contrast of the natural corona remains lower than that of the simulation. Maybe there were other scattering particles with a broader size distribution present, which added another, rather colorless aureole “layer” on top of the pollen corona, thereby diminishing its contrast. Surprisingly, I could not find any previous reports about “olive pollen coronae”, though the phenomenon should be quite prominent during the right season in the olive-growing regions.

Three possible photographic detections of the natural fifth order rainbow from Germany

In 2014, Harald Edens reported ten cases of photographically detected natural quinary rainbows, recorded during 2009-2013 in New Mexico, USA, at altitudes of 1.8-3.2 km. These and some newer observations can also be found on his website.

So far, no reports from other locations have been published. In the German observers’ network, we analyzed many candidate photographs showing bright primary and secondary rainbows, but from most of them no reliable traces of quinary rainbows could be extracted. Such analyses are not easy, as the quinary signal is weak compared to the neighboring secondary rainbow, and processing methods such as unsharp masking can cause a leakage of colors into Alexander’s dark band. Furthermore, the processing operator will experience disturbing afterimage issues from the intense renditions of the primary and secondary on the screen after a couple of minutes.

Despite these difficulties, we now believe that we have identified three cases of genuine quinary rainbows. In cases 1 and 3, the quinary could be extracted from several photographs. Nonetheless, in order to keep this blogpost brief, we restricted ourselves to show only one image (or the results from one polarization series in case 1) per observation. We chose a straightforward processing method (= only increasing contrast and saturation, no local filtering such as unsharp masks) similar to the one applied by Harald Edens to allow for an easier comparison with his results. Alternative processing routes will be presented at a later stage.

1) April 22^nd, 2012, near Göttingen, Germany (51° 31’ N, 9° 58’ E, altitude 250 m), 19:16 CEST, sun elevation 10.2°, photographed by Frank Killich after a moderate shower

The original intention of Frank Killich was to use the primary and secondary rainbows as test objects for a home-built photopolarimetric setup made from a Canon 20D camera and a linear polarizer precisely rotatable by a stepper motor. By recording four successive images at polarizer positions of 0°, 45°, 90° and 135° with respect to the vertical, it is possible to reconstruct the first three components of the Stokes vector for each viewing direction (pixel coordinates) and color channel (red, green, blue) individually. These images can be numerically combined to reconstruct the unpolarized intensity (= the ordinary photographic result without a polarizer) and, moreover, the linearly polarized portion of the recorded light distribution (= the total intensity with the unpolarized background removed for each pixel). In the case of rainbows, this corresponds effectively to a subtraction of the radial (weak) component from the azimuthal (strong) polarization component equally all along the visible part of the circumference. As known from theory, also the quinary will be easier to detect in such a polarization contrast image.

Unpolarized intensity as calculated from the original images, f = 22 mm:

Unpolarized intensity, increased saturation and contrast:

Linearly polarized portion as calculated from the original images:

Linearly polarized portion, increased saturation and contrast:

The expected broad bands of green and blue are clearly visible in the processed linearly polarized portion picture, and might be slightly visible also in the unpolarized intensity.

The other two photographic observations were carried out without any polarizers, i.e. only the unpolarized intensity information is available in these cases.

2) March 20^th, 2013, near Pforzheim, Germany (48° 56’ N, 8° 36’ E, altitude 312 m), 16:21 CET, sun elevation 21.1°, photographed by Michael Großmann after an intense shower

Original (Canon EOS 450D, f = 22 mm):

Increased saturation and contrast:

A slight green/blue hue is visible inside the secondary at and slightly above the horizon.

3) May 15^th, 2016, Mt. Zschirnstein, Germany (50° 51’ N, 14° 11’ E, altitude 560 m), 19:57 CEST, sun elevation 6.2°, photographed by Alexander Haußmann after a moderate shower

A description of this observation (without discussing the quinary rainbow) can be found here. The images shown here are cropped from a fisheye photograph.

Original (Pentax K-5, f= 17 mm, cropped):

Increased saturation and contrast:

Again, a slight green/blue hue appears close to the horizon.

At this point it is of course not possible to draw any statistical conclusions about the frequency of detectable quinary rainbows. However, it seems worthwile that every rainbow observer re-examines his photographical treasure trove for previously overlooked rarities, even if no polarizer enhancement was involved during photographing.

Crepuscular rays from above and below

Yesterday there were observations of spread Crepuscular rays over Germany. The satellite image shows the origin of the long shadows: a powerful squall line over northwest Germany. The length of the shadows is about 400km – this is enormous!

Near Pforzheim in Baden-Württemberg Michael Großmann observed rays passing from the setting sun to the antisolar point. Rene Winter was in the district Gotha, Thuringia and saw crepuscular rays that were unusual intensively. Laura Kranich in Kiel wasn’t far away from the thunderstorms and had intense Crepuscular rays, too. There were single beams that ran across the entire sky.

Photo: Michael Großmann

Photo: Michael Großmann

Photo: Michael Großmann

Photo: Michael Großmann

Photo: Rene Winter

Photo: Laura Kranich

Photo: Laura Kranich

Photo: Laura Kranich

Crepuscular rays are rays of sunlight that appear to radiate from the point in the sky where the sun is located. These rays, which stream through gaps in clouds (particularly stratocumulus) or between other objects, are columns of sunlit air separated by darker cloud-shadowed regions. Despite seeming to converge at a point, the rays are in fact near-parallel shafts of sunlight, and their apparent convergence is a perspective effect (similar, for example, to the way that parallel railway lines seem to converge at a point in the distance).

The name comes from their frequent occurrences during twilight hours (those around dawn and dusk), when the contrasts between light and dark are the most obvious. Crepuscular comes from the Latin word “crepusculum”, meaning twilight.

Three quarters of a double rainbow, plus an accidental snapshot of a tertiary, Mt. Zschirnstein, Germany, May 15th, 2016

Over the past two decades it has become a tradition among my friends to carry out a bicycle tour to the Elbe Sandstone Mountains (“Saxon Switzerland“) at the Pentecost weekend. We then often pay a visit to a table hill named “Großer Zschirnstein“ (561 m), which features a remarkable cliff of 70 m in height at its south-eastern edge.

Almost 15 years ago, on the evening of June 3rd, 2001, we had the opportunity to observe from there a rainbow extending well below the horizon almost down towards its bottom. Unfortunately, we only had a compact camera without a fisheye lens at hand back then, so the old photos show only some sections of the whole phenomenon.

This year, on May 15th, we were finally granted the proverbial second chance. I already anticipated some rainbow potential in the “Icelandic” weather that day. In the early afternoon, there had already been a rain shower while the sun was shining, but as we had not yet ascended the mountain and the sun was still high in the sky, there was no chance for a rainbow observation.

Some minutes after reaching the plateau in the evening, we had to retreat to the shelter when a rather strong shower of hail and rain set in. To the west a stripe of clear sky widened, and sunshine seemed at hand soon. It took longer than expected, as the clouds were moving rather slow. On the left side, a small rainbow fragment suddenly appeared at the horizon, resulting from sunlit drops a few kilometers off. It was a rather unusual observation to see this rainbow streak vanish and reappear again, as its sight was repeatedly obstructed by scudding (and non-illuminated) mist around the Zschirnstein massif:

(19:42 CEST, f = 88 mm, Pentax K-5)

Finally the great moment came: Sunshine was reaching the Zschirnstein while the shower, now mostly composed of rain instead of hail, still continued. Within a few minutes we could enjoy this marvelous view:

(19:56 CEST, f = 10 mm / fisheye)

Unfortunately there was no safe way to access a viewpoint which would have allowed to study the missing quarter, as this would have required some careful climbing around the sandstone rocks for which I already felt too excited at that moment. The fisheye picture can hardly express how huge both rainbows looked like, and how beautiful the raindrop clusters glittered as they drifted around the cliff some 10 m further down. These are certainly the moments that make you understand that famous “double rainbow enthusiasm”, thought not everyone is as outgoing as other people on the internet. Maybe we also stayed a bit calmer because the strong and cold wind added a rather painful component to the taking of photographs and videos.

Later the right part of the primary close to the horizon became especially bright:

(19:59 CEST, f = 80 mm)

This photo has been processed in a way that no color channel reaches saturation, which is a necessary prerequisite for analyzing possible kinks in the rainbow. In this case, the red rim looks as if would bend inside a bit below the horizon, but this might only be an illusion due to the intensity gradient.

The primary’s right foot above the horizon remained still visible for a rather long time, as the shower withdrew in this direction:

(20:19 MESZ, f = 50 mm)

But the story does not end here. When going through the pictures later at home, I suddenly realized that I had missed to look for higher order rainbows, or to deliberately take some pictures in the appropriate directions. I was a bit disappointed about my inattentiveness, since this had been my best rainbow display in years and, moreover, I had not been hindered by the limited field of view from a window in a city building. I am often forced to decide between the sunward or antisolar hemisphere when observing rainbows from there.

Luckily I had taken two pictures (an exposure bracket) towards the sun just at the moment when the three-quarter rainbows started to evolve. The reason for this was only the lighting atmosphere – it was the moment when the sun rays had first reached the Zschirnstein plateau. As I deduced later from the movement direction of the shower, there had been rather good conditions for the formation of tertiary and quaternary rainbows when the picture pair was taken. So I decided to apply the strong filtering procedures which are needed to extract higher-order rainbows from photographs. The shorter exposure just gave noise in the interesting region. However, in the longer exposed version something interesting popped up.

Original image:

(19:54 MESZ, f = 17 mm / fisheye)

Processed image:

Slightly to the right above the stone pillar, a red-green stripe in the color ordering of the tertiary rainbow can be discerned. For an unambiguous identification it would, however, be necessary to calibrate the picture in order to assign scattering coordinates to the photo’s pixel matrix. Though I had previously calibrated the projection of the lens for the used focal length (the upper end of the zoom range), I would need two reference marks with known elevation and azimuth which are included in this specific photograph to complete the analysis. On the horizon, no distinct remote references could be found. This means that I would have to reconstruct my precise position on the plateau to minimize parallax errors, and then to record a starfield image from there at night, enabling me finally to use the stone pillar or nearby trees as references. Unfortunately, it would take an inconvenient amount of time to access the spot again and the effort for such a trip would be a bit over-the-top for the sole purpose of calibrating a photograph.

But there was still a piece of hope: From the shorter exposed version (-2 EV),  I could estimate the position of the sun quite accurately, as there is only a small overexposed area around it. This allowed me at least to draw lines of constant angular distance from the sun into the photograph in order to decide if the colored stripe appeared at the correct position or not. Using the previously measured spectral sensor response of my camera, and estimating the temperature of the water drops to be around 5°C, I derived the following values for the Descartes angles of the tertiary and quaternary rainbows: 41.7° / 43.7° (red, 620 nm), 40.6° / 45.1° (green, 530 nm), and 39.3° / 46.8° (blue, 460 nm). In the following animation, these angular distances from the estimated position of the sun have been marked by their respective colors:

The colored stripe seems to fit reasonably well to the Descartes angles of the tertiary rainbow, especially when taking into account that the positions of maximal intensity are shifted a bit inward from the Descartes angles for the tertiary (and outward for the quaternary) due to wave-optical effects. This shift was also noted in the analysis of the very first photograph of a tertiary rainbow. Further contributions form distorted drop shapes are of minor importance here, as the sun elevation is small and we are looking at the rainbow’s sides. Therefore the effective cross section of the drops should remain nearly circular, even if they are squeezed in the vertical. I leave it to the readers to decide if also traces of the quaternary might be visible among the color noise slightly to the left above the stone pillar.

Addendum: A short video clip from the observation can be found here.

Reflected Sunlight Dewbow

Reflected Sunlight Dewbow. Photos: Jérémie Gaillard

On August 19, 2010, Jérémie Gaillard made an interesting discovery when looking at the surface of the lake Etang de l´Alleu which is located in the French community of Saint-Arnoult-en-Yvelines. The water was covered with pollen, on which droplets of dew had formed. In these droplets two colourful rainbows were visible. Dewbows can be understood as the lower part of a rainbow projected onto a horizontal plane. When a dewbow is fully developed, a semi-circle which opens towards the sides should be visible, the apex of which is situated at the lower end of the observer´s shadow. Equivalent to normal rainbows, primary and secondary dewbow should run parallely, but in Jérémie Gaillard´s observation they did not.

Instead, the second colourful bow fragment is a reflected sunlight dewbow. The surface of the water acts as a large mirror reflecting the sun. The reflected image of the sun now acts as a second source of light, which is situated as far below the horizon as the sun is above it. (angle of incidence = emergent angle). So the antisolar point for the reflection of the sun is above the horizon. This reflected antisolar point, which is located the double of the real sun´s elevation above the antisolar point, is the centre of the two rainbow circles for the reflected sunlight. So the additional rainbows are displaced upwards by the double sun elevation compared to the primary and secondary rainbow, making a rather unfamiliar appearance in the open nature.

Author: Claudia Hinz