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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 11th, 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 3rd and 4th 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 11th, 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 7th, 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 5th. 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 7th 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.

Neklid Antisolar arcs: Case closed?

In my last post I outlined several possibilities to explain the great brightness of the antisolar arc (AA) compared to the heliac arc (HA) in the Neklid display from Jan 30th, 2014. All of them were a bit off the main road of traditional halo science, but traditional arguments did not help to clarify what was observed, hence I had to look for something else.

Both the concepts of plate Parry crystals and trigonal Parry columns should yield weak traces of unrealistic (or better to say non-traditional) halos that might appear in a deeper photo analysis. Claudia Hinz provided me with a set of pictures from the display to unleash any kind of filters that would seem appropriate. Indeed it was possible to pin down traces of the Kern arc in some of the pictures after the initial application of an unsharp mask (1, 2), followed by high-pass filtering (1, 2) or, alternatively, by Blue-Red subtraction (1, 2). Note that the Kern arc was weakly present in the simulations for hexagonal, Parry-oriented plates. This, of course, must not be confused with the recently proven Kern arc explanation relying on trigonal plates in plate orientation. Finally, trigonal columns in Parry orientation are a third non-traditional crystal configuration giving rise to new halos. However, these do not yield a Kern arc.

Obviously, the Kern arc fragments in the photos are very feeble and the whole procedure reminds a bit of the search for higher order rainbows. It is mere guesswork to detect how far the arc stretches around the zenith, but doubtlessly it extends up to 90° and more in azimuth, thus being clearly distinguishable form the circumzenith arc. Nonetheless, one would feel safer with further evidence. Comparing the simulations for Parry columns and Parry plates, three more differences are discernible (apart from the changed AA/HA ratio):

1) For Parry plates, the upper suncave Parry arc does not show an uniform brightness, but appears brighter directly above the sun and loses some intensity towards the points where it joins the upper tangent arc.

2) The upper loop of the Tricker anthelic arc is suppressed for columns, but shows up for plates.

3) Some extensions of the upper Tape arcs appear between the Wegener arc and the subhelic arc.

At least the first two points can be answered in favor of the Parry plates, being visible even without strong filtering. However, I failed to detect any extended Tape arcs as “ultimate proof” so far. This might not surprise since they are, according to the simulation, comparable to the Kern arc in intensity and appear in regions of the sky where the crystal homogeneity was not as well developed as in the vicinity of the zenith.

Piecing the parts together, it seems evident that at Neklid the AA intensity was due to Parry-oriented hexagonal plates. Their traces were detectable, whereas nothing appeared that would hint on trigonal Parry columns. In contrast to this, Parry trigonals were responsible in Rovaniemi 2008. This implies that in nature at least two different mechanisms occur for AA brightening.

Finally the question remains how plates may get into a Parry falling mode. But as long as no one understands how symmetric columns do this (though we have the empirical evidence), we should be prepared for surprises. There might also be a connection to recently discussed details of the Lowitz orientation (2013 Light and Color in Nature conference, talk 5.1).

The mystery of bright antisolar arcs




(photo by Claudia Hinz)

The antisolar (or subanthelic) arc (AA) was one out of the vast range of halo species occurring during the marvelous Neklid display observed by Claudia and Wolfgang Hinz on Jan 30th, 2014. This kind of halo seems to be exceedingly rare, since it has only been documented during the very best displays, mostly observed in Antarctica. On the other hand, the heliac arc (HA) is a, however not frequent, but well-known guest in Central Europe. Both of them are reflection halos generated by Parry oriented crystals and touch each other at the vertices of their large loops. Fisheye photos towards the zenith from Neklid shows both these halos in perfect symmetry and approximately similar intensity, at least regarding the upper part of the AA.

When trying to simulate the display (solar elevation 17.5°) using HaloPoint2.0, I noticed that the AA was rendered much weaker than the HA, which of course does not match the photographic data. To obtain the Parry effects (Parry arcs, Tape arcs, HA, AA, Hastings arc, partially circumzenith arc, Tricker arc, subhelic arc) I chose a population of “normal” (i.e. symmetrically hexagonal) column crystals with a length/width ratio of c/a = 2 in the appropriate orientation. Since both HA and AA are generated by this very same crystal population, their mutual intensity ratio cannot be influenced by adding plates, singly ordered columns, or randomly oriented crystals. This mysterious issue has also been noted by a Japanese programmer who came across the Neklid pictures.

Inclusions of air or solid particles within the ice crystals are an obvious hypothesis to explain this dissenting AA/HA intensity ratio, since they cannot be accounted for in the standard simulation software. However, a look into literature reveals that there are external and internal ray paths for the HA, but only internal paths for the AA ([1] p. 34-35). That means that inclusions will diminish the AA to a greater extent than the HA. In the extreme case with the interior totally blocked, no AA can arise but a HA is still possible due to external reflection at a sloping crystal face. Hence inclusions cannot explain the bright AA from the Neklid display. Air cavities at the ends of columns which are seen quite often in crystal samples will also inhibit the AA because an internal reflection at a well defined end face is needed for its formation.

Spatial inhomogeneities in the crystal distribution might serve as explanation as long as there is only one single photo or display to deal with, especially when the air flow conditions are as special as they were at Neklid. Maybe there were just “more“ good crystals in the direction of the AA compared to where the HA is formed, either by chance or systematically due to the wind regime. But surprisingly also the observations from the South Pole (Jan 21st, 1986 (Walter Tape); Jan 11th, 1999 (Marko Riikonen), also discussed here) show an AA/HA ratio somewhere in the region of unity as far as one can guess from the printed reproductions ([1] p. 30, [2] p. 58). Parts of the AA appeared even brighter than the HA in Finnish spotlight displays. All this implies a deeper reason for the AA brightening. It seems rather unlikely that in all these cases the inhomogeneities should have worked only in favor of the AA.

Hence the crystals themselves must be responsible for AA brightening. Non-standard crystal shapes and orientations are conjectures that can be tested easily with the available simulation programs. For a first try, one can assign a Parry orientation to plates instead of columns. Changing the c/a shape ratio from 2 to 0.5 while keeping all other parameters fixed results in a much brighter AA.

It is, however, commonly accepted that due to the air drag only columns can acquire a Parry orientation ([1] p. 42). Furthermore, some halos appear in the plate-Parry simulation which have not been observed in reality, e.g. a weak Kern arc complementing the circumzenith arc. At this stage the question may arise why only due to aerodynamics any symmetric hexagonal crystal (may it even be a column) should be able to place a pair of its side faces horizontally to generate Parry halos such as the HA and AA. Cross-like clusters or tabular crystals ([1] p. 42), from whose shapes one will immediately infer that rotations around the long axis are suppressed, seem much more plausible. Surprisingly, Walter Tape’s analysis of collected crystal samples shows that Parry halos are mainly caused by ordinary, symmetric columns. Parry orientations might be a natural mode of falling for small ice crystals, though up to now the aerodynamic reasons remain unclear. Nonetheless I tested if tabular crystals would give a bright AA. This was neither the case for moderate (height/width = 0.5) nor strong aspect ratio (height/width = 0.3). The AA was in both cases even weaker than in the symmetric standard simulation with which the discussion started.

Trigonal plates have been brought into discussion as possible crystal shapes being responsible for the Kern arc (see also [1] p. 102). Out of curiosity I tested how Parry oriented trigonal columns would affect the AA/HA intensity ratio. In contrast to symmetric hexagonal columns two different cases exist here, depending on whether the top or bottom face is oriented horizontally. As seen from the results, a sufficiently bright AA can be simulated using trigonal Parry columns with horizontal bottom faces, but the upper suncave Parry arc and the lower lateral Tape arcs at the horizon disappear. Obviously they have to, since a trigonal crystal in this orientation does not provide the necessary faces for their formation. On the other hand, the simulation predicts unrealistic arcs like the loop within the circumzenith arc. Choosing a trigonal Parry population with top faces horizontal will diminish the loop of the HA and wipe out the upper part of the AA as well as the upper lateral Tape arcs and add an unrealistic halo that sweeps away from the supralateral arc.

Is it possible to generate a realistic simulation of the Neklid picture with such crystals? Clearly this will require to add a second Parry population of symmetric hexagonal prisms. Doing so, a reasonable compromise can be achieved. In this case the hexagonal crystals produce the Parry arc, whereas the trigonal ones are responsible for the AA. Due to the triangular portion being small, the unrealistic halos become insignificant. However, the fact that a further degree of freedom (mixing ratio trigonal/hexagonal) has to be added to the set of initial simulation parameters is somehow dissatisfying.

The question lies at hand if this result might also be obtained by choosing a single Parry population of intermediate shapes between the symmetric hexagonal and trigonal extremes. This idea is further motivated through pictures of sampled crystals that, though being labeled „trigonal“, show in fact non-symmetric hexagonal shapes. The simulation for these shapes does indeed predict an enhanced AA compared to symmetric hexagons, but the lower lateral Tape arcs and the upper suncave Parry arc still appear too weak. This means that an additional set of symmetric hexagonal crystals is needed again to render these halos at the proper intensity.

Moreover, quite prominent unrealistic halos like the loop crossing the circumzenith arc appear in the simulation. If this assumption for the Parry crystal shape was right, this arc should be visible in an unsharp mask processing of the photos. Its absence hints that these crystals did not play a dominant role in the Neklid display. One could argue that the unrealistic halos may depend strongly on the actual crystal shape and might be washed out in a natural mixture of different “trigonalities“. However, the simulation tests indicate that even in this case the unrealistic halos remain rather strong, as long as one still wishes to maintain an AA at sufficient intensity.

As a conclusion, it can be stated that the intensity ratio between the heliac arc and the antisolar arc in the Neklid display as well as in Antarctic and Finnish observations has raised basic questions about the shapes of the responsible crystals. Simulations with symmetric hexagonal Parry columns, i.e. the standard shapes, render the AA to weak compared to the HA. Inclusions in the crystals and spatial inhomogeneities of the crystal distribution can be ruled out as the cause of this deviation. Plates in Parry orientation or a mixture of Parry oriented trigonal columns with horizontal bottom faces and hexagonal columns both result in a more realistic AA/HA intensity ratio. However, they introduce traces of unrealistic halos and are rather uncommon hypotheses: Plate crystals are not supposed to fall like this, and the existence of “true” trigonal crystals is doubtful. Moreover, the trigonal crystals need an accompanying set of standard Parry crystals to generate other halos like the upper suncave Parry arc.

So all in all the mystery of bright antisolar arcs cannot be regarded as solved at this stage. Since this halo species is very rare in free nature, it might be helpful to test perspex crystal models of different shapes in Michael Großmann’s “Halomator“ laboratory setup. Though the refractive index in perspex is higher than in ice, the basic relations between HA and AA stay the same. However the big challenge remains to collect and document crystals during such a display, e.g. with the methods described by Reinhard Nitze.


[1]     W. Tape, Atmospheric Halos (American Geophysical Union, 1994)

[2]     W. Tape, J. Moilanen, Atmospheric Halos and the Search for Angle x (American Geophysical Union, 2006)


I missed an important piece of information from Finland 2008: The idea of trigonal crystals making Parry halos was already pointed out by Marko Riikonen in an analysis of the Rovaniemi searchlight display. In that case, even one of the halos that I termed “unrealistic“ was observed in reality, thus strongly supporting the trigonal interpretation.