Simulating rainbows in their atmospheric environment

Stanley David GedzelmanDepartment of Earth and Atmospheric Sciences and NOAA CREST Center, City College of New York, New York,

New York 10031, USA, and Department of Optics, University of Granada, 18071 Granada, Spain(stan@sci.ccny.cuny.edu)

Received 6 May 2008; accepted 15 July 2008;posted 11 August 2008 (Doc. ID 95814); published 31 October 2008

Light and color of geometric optics rainbows are simulated in their atmospheric environment. Sunlightpasses through a molecular atmosphere with ozone and an aerosol layer near the ground to strike acuboidal rain shaft below an overhanging cuboidal cloud. The rainbows are treated as singly scatteredsunbeams that are depleted as they pass through the atmosphere and rain shaft. They appear in a settingilluminated by scattered light from behind the observer, from the background beyond the rain shaft, andfrom the rain shaft. In dark backgrounds the primary and secondary bows first become visible when theoptical thickness of rain shafts τR ≅ 0:0003 and τR ≅ 0:003, respectively. The bows are brightest andmostcolorful for 0:1 ≤ τR ≤ 3, a range that is typical for most showers. The peaks of the scattering phase func-tion for raindrops that correspond to the geometric optics rainbow are so pronounced that rainbows re-main bright and colorful for optically thick rain shafts seen against dark backgrounds, but the bowsappear washed out or vanish as the background brightens or where the rain shaft is shaded by an over-hanging cloud. Rainbows also redden as the Sun approaches the horizon. © 2008 Optical Society ofAmerica

OCIS codes: 010.1290, 010.1615, 010.1690, 010.5620.

1. Introduction

The rainbow, an eternal symbol of beauty that encom-passes our hopes and fears, has attracted the interestof scientists from ancient times. Most attention hasfocused on the scattering properties of individualraindrops. The impact of drop sizes and shapes onrainbow appearance is well understood and has beensimulated beautifully [1–3]. But the impact of the at-mospheric environment on the appearance of therainbows has received less attention. Sunlight thatproduces rainbows must first pass obliquely throughhazy air before it illuminates not only a partiallyshaded rain shaft of finite optical thickness, but thecloud above the rain shaft, the sky, land, or clouds be-yond it, and the sky behind the observer. The vitiatingimpact of the atmospheric environment on the appar-ent brightness and color purity of the bows has longbeen quietly acknowledged as indicated by such re-marks as, “The colors of all rainbows are dilute and

contain much white light.” [4], or “On one occasion Isaw the rainbow standing out clearly against the darkbackground of a wood…” [5]. However, it was only in1991 that Lee analyzed rainbow photographs digi-tally and showed how severely background lightingcan limit the chromatic range of even apparentlybrilliant bows [6].

Here I expand on a series of earlier works in whichrainbows were modeled in their atmospheric envir-onment [7–10]. Here the bows are displayed in skypanoramas that illustrate some of the earlier find-ings, namely, that rainbows (1) can be seen in verylight showers, (2) appear brightest and most colorfulwhen the optical thickness of the sunlit part of therain shaft is of the order of unity and the backgroundis dark, (3) appear brighter near the horizon thanhigher in the sky when the rain shaft is shaded byan overhanging cloud, and (4) are restricted to redand yellow when the Sun is near the horizon.

2. Model

The BOWSKY model used to produce the simula-tions described here is available on the internet

0003-6935/08/34H176-06$15.00/0© 2008 Optical Society of America

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[11]. It displays sky panoramas that show the pri-mary and secondary bows for a range of parameterslisted in Table 1 that must be specified by the user.The treatment of sunlight and atmospheric condi-tions for scattering by molecules, extinction by aero-sols, and absorption by ozone are the same as in thearticle on halos in this issue [12], which contains de-tailed descriptions. Briefly, the Sun is located at ze-nith angle ϕSun and is approximated as a Planckradiator at 5750K with radiance calculated for 61wavelengths from 400 to 700nm.Before reaching the rain shaft, sunlight passes

through an ozone layer represented by a Gaussianprofile at a user selected height in the stratospherewith a 20km width at half-amplitude, and a lowerconstant concentration in the troposphere. Total col-umn ozone content, in Dobson units (DU), is a freeparameter in the model, but all the simulationsshown here use 300 DU. Ozone absorbs visible radia-tion mainly in the Chappuis bands at wavelengthsaround 600nm. Its impact on rainbows and sky coloris relatively small until the Sun is near the horizon.The light passes through an aerosol layer near the

ground whose concentration and effective size arefree parameters. The impact of the Earth’s curvatureon the path length of beams near the horizon is in-corporated through use of the optical air mass[13]. The cloud and rain shaft are illuminated by di-rect sunlight and by scattered skylight from behindthe observer, which is assigned a radiance of onethird that of clear sky because much of the sky isshaded by cloud.The clouds and cloud fields that produce rainbows

have an infinite variety of shapes, so only the prin-cipal elements are modeled here. The geometry ofcloud and rain shaft is the same as in [8] and isshown from both side and plan views in Fig. 1.The model rain shaft is a cuboid or block locatedat distance xR with width ΔxR ¼ xR − xRend, heightzR, and optical thickness τR. It is at least partlyshaded by an overhanging cuboidal cloud at distancexC ≤ xR. The number density of drops within the rainshaft is assumed to be spatially constant even thoughthe heaviest rain is typically embedded deep within

departing storms or sometimes near the leading edgeof approaching squall line thunderstorms.

The Sun appears at zenith angle ϕSun and azimuthangle θSun from the normal to the vertical plane of therain shaft. When θSun ≠ 0, one side of the bows isbrighter than the other, but here I only show simula-tions with θSun ¼ 0. The panoramas show the sky upto an angular distance of ψ ≤ 60° from the antisolarpoint, with the appropriate spherical trigonometry toshow variations of light and color of the sky and thebows as a function of zenith and azimuth angles [8].

Figure 1 shows that the most distant point of therainbow beam from the observer illuminated by theSun is xF; zF. If xF < xR for any beam, that part of therain shaft is totally shaded, so no rainbow will ap-pear there. The region beyond the rain shaft, whichcan consist of cloud, clear sky, or landscape, is as-signed radiance of a fraction BB of that of a clearsky. Dark backgrounds below thick thunderstormshave BB as low as approximately 0.01, but brightbackgrounds such as sunlit clouds, which are muchbrighter than skylight, can have a BB ≫ 1.

The simple block model of a cloud and rain shaftand background sky can only indicate the impactof major features on the appearance of rainbowsand cannot begin to reproduce the infinite varietyof lighting patterns produced by the irregular andfragmented cloud fields and landscape of real rain-bow situations. But it does explain geometric fea-tures such as why only the bottoms of somerainbows are seen when the Sun is high enough

Table 1. Table 1. User Selected Parameters in the Model

Parameter Symbol Choice or Range

Surface pressure pSL ≥ 0Solar zenith angle ϕSun 0 → 90°Solar azimuth angle θSun ≥ 0Rain shaft optical depth τ ≥ 0Distance to rain shaft xR ≥ 0Width of rain shaft xR − xRend ≥ 0Height of rain shaft zR ≥ 0Distance to cloud xC ≤ xRBackground brightness BB ≥ 0Atmospheric turbidity β ≥ 1Ångstrom coefficient α -1, 0, 1, 2, 3Ozone layer height H ≥ 12kmTotal ozone content DU ≥ 0 Fig. 1. Side view (top) and plan view (bottom) of the rainbow

cloud and rain shaft model. All the symbols are defined in the text.

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and the cloud’s overhang is so large that all but per-haps the bottom of the rain shaft is shaded.The passage of light through the atmosphere is

governed by Bouguer’s law with two exceptions: (1)that the background lighting is specified and (2) thatmultiply scattered light reaching the observer fromthe rain shaft is given by the albedo equation [14]

AR ¼ 0:13τR1þ 0:13τR

: ð1Þ

Equation (1) is also used to calculate the fraction oflight that penetrates from beyond the rain shaft,which is given by 1 − AR. The angular scatteringphase function PðψÞ, shown in Fig. 2, is based onequations that fit the output of a Monte Carlo geo-metric optics model of light beams that strike largespherical drops at random points and undergo reflec-tion and refraction according to Snell and Fresnellaws [9]. This treatment excludes supernumeraryrainbows and represents the optimal conditions fornear-spectral coloration.

3. Simulations

Sky panoramas of double rainbows are shown for sixdifferent sets of conditions in Fig. 3 and can be com-pared with innumerable photographs of rainbows[15–17]. In all, the primary bow is brighter, more col-orful and narrower than the secondary, which has areverse color sequence. The red fringe of the bows hasthe highest color purity. The dark blue inner fringe ofthe primary bow can be seen when the background isdark (BB ¼ 0:01) and the Sun is well above the hor-izon, but it is overwhelmed by a bright background,as in Fig. 3(e), where BB ¼ 1:0. Blue is removed fromdirect sunlight by scattering before reaching the rainshaft when the Sun is near the horizon (e.g.,ϕSun ¼ 88°) as in Fig 3(f). In that case (i.e., a sunsetrainbow), the bow is typically red and yellow, butfaint green may be seen when there are few aerosolsand surface pressure is much lower than at sea level.Finally, Alexander’s dark band is present in all the

frames but is misrepresented as a bright band inthe sunset bow of Fig. 3(f) because of an artifact inthe model’s translation between radiance and color.For much the same reason the white center in the1931 CIE diagram appears brightest.

The optical thickness of the rain shaft, τR, and therelative brightness of the background, BB, greatly af-fect the radiance, color, and visual contrast of therainbows. The primary bow first becomes visible un-der dark thunderstorms with BB ¼ 0:01 when τR ≅

0:0003 but is still pale at τR ≅ 0:01. Radiance ofthe bows increases monotonically with increasingτR and at a faster rate than that of the backgrounduntil τR ≅ 1, but for larger τR the background radi-ance increases more rapidly. Therefore, rainbowshave the greatest contrast when τR ¼ Oð1Þ, as Fig. 4suggests, and these bows also tend to have the high-est color purity. Given that optical depths of 1kmwide rain shafts fall into the range of 0:18 ≤ τR ≤

3:3 for rainfall rates of 1 ≤ RR ≤ 100mmh−1 with aMarshall–Palmer drop size distribution [18], vir-tually any sunlit shower with a dark backgroundhas the potential to produce a bright rainbow.

Many rainbows appear brightest just aboveground level and fade or vanish higher up. Shadingof a rain shaft by an overhanging cloud can eitherdull or eliminate rainbows above a certain elevationangle. Figure 3(d) shows bows produced by a broadbut moderate shower with xC ¼ 1300m, xR ¼1700m, ZR ¼ 1000m, ΔXR ¼ 4000m, τR ¼ 4000m,and ϕSun ¼ 75°. This example was chosen to illus-trate the case in which so little of the upper partof the rain shaft is illuminated that the top of the pri-mary bow fades and the top of the secondaryvanishes. Rainbow brightness fades upward fromthe horizon even more rapidly than the model sug-gests when very light rain falls at the outer edgeof the rain shaft and more intense rain occurs deepin the rain shaft, where only the bottom is sunlit.

The radiance of the background light beyond therain shaft can vary by orders of magnitude, from suchobscurity beneath thick thunderstorms that streetlights turn on to an almost blinding brilliance of sun-lit clouds sometimes seen beyond light showers fromsmall storms. As the background brightens, the ap-parent color and visual contrast of rainbows fades.For example, the rainbows of Fig. 3(b) (BB ¼ 0:01) ap-pear much more brilliant and colorful than the rain-bows of Fig. 3(e) (BB ¼ 1:0). When BB ¼ 10 (notshown), as when a sunlit cloud fills the background,the entire sky is so bright that the primary bow ap-pears weak and pale with no blue, and the secondaryis almost invisible.

CIE chromaticity diagrams help quantify howbackground brightness BB constricts the color rangeof the geometric optics primary (heavy curve) andsecondary (thin curve) rainbows. Figure 5 shows thatthe color range and purity of the primary bowdecreases by a factor of approximately 2 when BBincreases from 0.01 to 1.0 and the secondary bowis rendered almost colorless. A similar decrease of

Fig. 2. Angular scattering phase functions for large sphericalraindrops (jagged curve), cloud droplets with a radius of 8 μm (thincurve), and air molecules (thick curve).

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color range and purity according to Mie theory forbows seen against black backgrounds does not occuruntil the drops that produce the bows are decreasedbelow approximately rdrop < 100 μm [3]. The smallcolor range of most primary bows is therefore largelydue to contamination by background lighting. Thebow with the largest range of relative luminanceanalyzed by Lee [6] represents a situation close tothat of the simulation shown here in Fig. 3(e) orthe lower panel of Fig. 5, where BB ¼ 1 althoughthe pale bows that Lee analyzed have BB ¼ Oð10Þ.But even under optimal viewing conditions, the sec-ondary bow has no blue.

The color of the sky that surrounds the rainbows isalso affected by ϕSun, BB, and τR. Background light inthe simulations is given the same color as clear sky-light, so when τR ≪ 1 the sky around the bows ap-pears blue. As τR increases, less light from beyondthe rain shaft reaches the observer and more sun-light is reflected from the rain shaft. Because rain-drops reflect all colors with equal efficiency, theytend to give optically thick rain shafts the same coloras the Sun and so appear gray or white during theday and orange when the Sun is near the horizon.The purple or lavender color of Alexander’s darkband in the sunset rainbow simulation of Fig. 3(f)

Fig. 3. Simulated sky panoramas of double rainbows for six settings: a solar zenith angle of ϕSun ¼ 75° in all panels but (f), whereϕSun ¼ 88°; optical depth of τR ¼ 1 in all panels but (a), where τR ¼ 10; and (c) where τR ¼ 1. The cloud and rain shaft are 500 and800m from the observer in all panels but (d), where they are 1300 and 1700m, respectively.

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is seen in many sunset rainbow photos, although, asmentioned above, it is darker than the sky outsidethe bows.

4. Summary and Conclusions

The model presented here (BOWSKY) produces skypanoramas that capture many features of the geo-

metric optics rainbows in their atmospheric setting.The model uses a block-shaped rain shaft under anoverhanging block-shaped cloud, and the simula-tions were designed to illustrate how the appearanceof rainbows varies with solar zenith angle, cloud geo-metry and optical depth, and background brightness.Sunlight in the models is absorbed by ozone, scat-tered by air molecules, cloud droplets, and raindrops,and scattered or absorbed by aerosol particles. Therainbows consist of singly scattered sunlight thatis reduced by scattering and absorption as it ap-proaches the observer. Multiply scattered back-ground light is included only in the rain shaft andfrom the background.

The simulations confirm some earlier conclusions.Rainbows with the greatest apparent contrast andhighest color purity are produced by rain shaftswhose sunlit portions have optical depths, τR ≈ 1,and appear against dark backgrounds. Often rain-bows are brightest near the horizon. This can resultfrom shading of the upper parts of the rain shaft bythe overhanging cloud. Rainbows seen at sunrise orsunset are almost entirely red and yellow becausealmost no green or blue light from the Sun reachesthe rain shaft.

At least three important features should be addedto future versions of the model. The first and mostimportant is to include the variation of angular scat-tering phase functions with drop size and shape. Thiswill enable the model to simulate supernumerarybows and the rare displaced twin primary rainbowsproduced by one set of spherical drops and one set offlattened drops. The second feature is to include spa-tial variations of drop concentration in the rain shaft.This will enable the model to better simulate bowsthat appear very bright at the horizon and darkerabove, given that the heavy rain is often deeply em-bedded within the rain shaft. It can also be used tosimulate those relatively unusual situations inwhich the secondary bow appears brighter thanthe primary where, for example, the primary foran observer forms at the edge of a rain shaft. Thethird feature is to make the bows translucent byallowing light from the background scenery topenetrate the bows. If nothing else, this will givethe simulated bows more of the entrancing vibrantappearance that many natural rainbows have.

This research was supported by a NationalOceanic and Atmospheric Administration (NOAA)Cooperative Remote Sensing Science and Technology(CREST) grant and a Professional Staff Congress,City University of New York (PSC CUNY) grant. Iam especially grateful to the academic executivesat all my hosting institutions, including the La-mont–Doherty Earth Observatory of Columbia Uni-versity, Tel Aviv University, the University ofCalifornia at Los Angeles, the University of Granada,and especially the City College of New York, whohave over the years permitted this scientist to per-form research on such almost unfundable but beau-tiful topics as atmospheric optics. Finally, I thank all

Fig. 4. Radiance versus scattering angle ψ measured from theantisolar point for several different values of rain shaft opticaldepth τR when BB ¼ 0:01.

Fig. 5. CIE chromaticity diagrams showing the impact ofbackground brightness (0.01 in the top panel and 1.0 in the bottompanel) on the color range of the primary (heavy curves) and second-ary (thin curves) rainbows for τR ¼ 1 and other conditions as inFig. 3(b).

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my colleagues who have shared my passion andfurthered my knowledge and appreciation of thescience and beauty of nature.

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