Tropospheric haze and colors of the cleartwilight skyRAYMOND L. LEE, JR.* AND DUNCAN C. MOLLNER

Mathematics and Science Division, United States Naval Academy, Annapolis, Maryland 21402, USA*Corresponding author: raylee@usna.edu

Received 31 January 2017; revised 21 March 2017; accepted 23 March 2017; posted 24 March 2017 (Doc. ID 285879); published 12 June 2017

At the earth’s surface, clear-sky colors during civil twilights depend on the combined spectral effects of molecularscattering, extinction by tropospheric aerosols, and absorption by ozone. Molecular scattering alone cannot pro-duce the most vivid twilight colors near the solar horizon, for which aerosol scattering and absorption are alsorequired. However, less well known are haze aerosols’ effects on twilight sky colors at larger scattering angles,including near the antisolar horizon. To analyze this range of colors, we compare 3D Monte Carlo simulations ofskylight spectra with hyperspectral measurements of clear twilight skies over a wide range of aerosol opticaldepths. Our combined measurements and simulations indicate that (a) the purest antisolar twilight colors wouldoccur in a purely molecular, multiple-scattering atmosphere, whereas (b) the most vivid solar-sky colors require atleast some turbidity. Taken together, these results suggest that multiple scattering plays an important role indetermining the redness of the antitwilight arch. © 2017 Optical Society of America

OCIS codes: (010.1290) Atmospheric optics; (010.1310) Atmospheric scattering; (010.1690) Color; (010.3920) Meteorology;

(330.1730) Colorimetry.

https://doi.org/10.1364/AO.56.00G179

1. INTRODUCTION

Students of sky color learn early on that sunrise and sunset col-ors become redder as concentrations of small particles increasein the lower troposphere [1–5]. Scattering by solid and liquidparticles that are small compared with the wavelengths λ of vis-ible light (here taken as 400 ≤ λ ≤ 700 nm) greatly increasesthe reddening of direct and forward-scattered sunlight com-pared with that due to molecular scattering alone. Hulburtshowed in 1953 that spectral absorption by ozone’s Chappuisbands helps to explain colors and spectra in sunlit regions of theclear twilight sky, including at the zenith [6]. However, in 1974Adams et al. used a single-scattering model to demonstrate thata purely molecular atmosphere, whether it includes ozone ornot, would yield only yellowish-orange sunset skies rather thanred ones (see [7], Figs. 16 and 17). When they added a normalamount of tropospheric aerosol to their model, this change didindeed produce an orangish-red solar sky near sunset (see [7],Fig. 19).

Yet such single-scattering twilight models [7–9] by defini-tion cannot produce multiple-scattering features (e.g., the darksegment [10,11]), and they may omit important spectral andcolorimetric details in the sunlit atmosphere [12]. In fact,tropospheric haze aerosols’ profound effects on circumsolartwilight colors suggest a corollary question: how do changesin aerosol concentrations (as measured by, say, aerosol spectral

optical depths τaer;λ) affect colors throughout the clear twilightsky? One plausible idea is that redder direct sunlight from largerτaer;λ will produce redder features in the antisolar direction,much as it does for the alpenglow and rainbow [13–15].Similarly, could large changes in molecular surface pressuresand number densities ever yield twilight colors that rival thoseof the real atmosphere?

In Sections 2–4 below, we address these questions by com-paring measured and modeled twilight chromaticities and colormaps. However, we start by considering the effective opticalproperties of real and simulated tropospheric haze particles, aswell as those for atmospheric ozone. In principle, each of ourtwilight hyperspectral scans should be accompanied by detailed,directly sampled soundings of aerosols and ozone throughoutthe local atmosphere. Although this ideal exceeds our equipmentand personnel resources, Fig. 1 illustrates one satisfactory alter-native: normal τaer;λ data acquired by AERONET sun photom-eters [16] near our coastal site in North Beach, Maryland(38.710° N, 76.529° W). Although AERONET data is mea-sured at five discrete wavelengths across the visible, it is wellapproximated by the continuous Angström function [17]

τaer;λ � βλ−α; (1)

with β ∝ spectrally integrated τaer and α specifying aerosols’effective spectral dependence. Thus aerosol optical depth

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increases as β increases, and its spectral selectivity increases asα increases.

Among the twilights measured with our Pika II imagingspectrometer [18], Fig. 1’s three τaer;λ spectra are from datesthat have minimal, intermediate, and high amounts of tropo-spheric aerosols. In 56 clear skies from 2007–2016 for whichLee has measured spectra and colors, local AERONET datayields a sample mean x�α� � 1.5203 and a sample standarddeviation s�α� � 0.3925. Note that molecular scattering hasα ∼ 4.09 so that τmol;λ ∝ λ−4.09 [19]. Table 1 lists the best-fit aerosol α and β parameters for Fig. 1’s three twilights.

Lacking direct samples of North Beach’s local aerosol col-umn, we do not know precisely what its dispersed-phase speciesare for a particular twilight. However, measurements from acoastal Delaware site in 1982–1983 indicate that its aerosolswere predominated by small haze droplets that containedsulfate and organic compounds, plus lesser amounts of fine dustand soil grains [20]. A 2007–2008 study at a coastal New Jerseysite found that most of its aerosol droplets included nitrate,

non-sea-salt sulfates, sodium, ammonium, and chloride, alongwith soluble ferrous species [21]. Consistent with these data,our twilight simulations use the spectral extinction propertiesof what Shettle labels a “tropospheric” mixture of water-solubleand dust-like aerosol species [22]. This broad category isacceptable for our colorimetric and photometric modeling ofNorth Beach twilights because the pre-sunset AERONETτaer;λ data quantifies their effective bulk spectral extinctionproperties.

On ozone changes and twilight colors, Lee previously ana-lyzed [12] whether large seasonal variations in 2005 Antarcticozone concentrations (O3) could affect twilight chromaticities,and he found only weak correlations between the twoquantities. In fact, changes in polar aerosol composition andconcentrations had as much influence on twilight colors asdid changes in either column total O3 or its vertical distribu-tion. In addition, a local Maryland site exhibited weak corre-lations between O3 concentrations and spectrally measuredzenith colors [12]. Finally, note that the 2005 variations inAntarctic O3 concentrations were much larger than thosefor Fig. 1: in the Antarctic, the ozone coefficient of variationCV�O3� � s∕x ∼ 36%, whereas CV�O3� ∼ 8.6% near NorthBeach in October–November 2013 [23]. Given (a) the mooteffects of large ozone variations on measured twilight colors and(b) the North Beach site’s much stabler O3 concentrations,we use a fixed (and realistic) ozone level of 300 Dobson unitsfor its simulated twilights.

2. MEASURED AND MODELED ANTITWILIGHTCOLORS

On 21 clear evenings from November 2012–November 2016,the Pika II was used to measure twilight spectra of clear skiesat two sites: the United States Naval Academy in Annapolis,Maryland and at North Beach. Both sites are on theChesapeake Bay’s western shore. For each scan, the Pika IIcamera’s optical axis panned along an almucantar at a fixedview-elevation angle of h � 15°–16° above the astronomicalhorizon. A calibrated digital inclinometer with a repeatabilityof�0.1° was used to determine h, and each scan’s lateral widthof 196° included both the solar and antisolar skies. Because theNorth Beach site includes a wide, unobstructed view of theChesapeake Bay, we used its astronomical horizon (including

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Fig. 1. Aerosol normal optical depths τaer;λ measured as a functionof wavelength λ by the NASA Goddard Space Flight Center’sAERONET sun photometers at Greenbelt, Maryland (2013 dates)and Edgewater, Maryland (11-27-2016), the two sites closest toour twilight measurements in North Beach, Maryland.

Table 1. Color Gamuts for Measured and Modeled Antitwilights

Figure Measurement Date or Model Parameters Surface h0 (°) h Interval (°) τaer;λ β [Eq. (1)] τaer;λ α [Eq. (1)] gamut g

1, 3a 11-3-2013 −1.56 0–30 0.0061076 1.766 0.0554101, 3a 10-20-2013 −1.77 0–30 0.018211 1.802 0.0415791, 3a 11-27-2016 −1.74 0–30 0.030695 1.741 0.0336444 aerosol, z � 0 km −1.56 0.5–30 0.0061076 1.766 0.0458224 molecular, z � 0 km −1.56 0.5–30 N/A N/A 0.0652065 aerosol, z � 10 km −1.56 −3.2–30 0.0061076 1.766 0.107375 molecular, z � 10 km −1.56 −3.2–30 N/A N/A 0.125476 8-4-2013 −3.03 −2.1–30 N/A N/A 0.0885516 8-9-2013 −2.89 −2.1–30 N/A N/A 0.061952

aFigure 3’s maxima of β and g are in boldface, whereas its minimum values are in italics. Entries labeled N/A indicate “not applicable” for purely molecularatmospheres or “not available” for Fig. 6.

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a small dip-angle correction [24]) to refine inclinometer mea-surements of the camera’s h angle.

Figures 2(a)–2(c) are twilight images calculated from hyper-spectral scans made on three different evenings at North Beach.In Fig. 2, the digital red–green–blue (or RGB) values for eachpixel are determined by its CIE tristimulus values, and these inturn are calculated from the pixel’s radiance spectrum as mea-sured by the Pika II. Lee and Laven [25] describe such spec-trum-to-RGB calculations in more detail. In Figs. 2(a)–2(c),h ∼ 30° at image top, the mean unrefracted sun elevationh0 ∼ −1.6° [26], and measured aerosol optical depths τaer in-crease from Fig. 2(a)–2(c). In particular, τaer�380 nm� �0.033182 for Fig. 2(a), 0.10286 for Fig. 2(b), and 0.16249for Fig. 2(c). See Fig. 1 for the corresponding τaer;λ spectra.

Although the antitwilight arch (or ATA) and dark segmentoccur at different maximum elevation angles and in slightly dif-ferent compass directions in Figs. 2(a)–2(c), these maxima areall at azimuth relative to the sun ϕrel � 180° or ∼1∕3 of thedistance from each image’s left edge. Sun elevation h0 only dif-fers by ∼0.2° in Figs. 2(a)–2(c), but their twilight colors arestrikingly different: the ATA and dark segment are most vividfor Fig. 2(a)’s minimum τaer, whereas its sunlit blue sky is theleast chromatic of the three evenings. For Fig. 2(c)’s maximumτaer, the ATA is barely distinguishable from the surroundingsky, even though solar-sky colors are vivid near the image’sright edge. This pattern is not coincidental: in all of our hyper-spectral scans, the ATA and dark segment become lesschromatic as τaer increases.

Some of Fig. 2’s colorimetric data is plotted in Fig. 3, whichshows the interior of the CIE 1976 uniform-chromaticity-scale (or UCS) diagram. To good approximation, thisdiagram’s v 0 ordinate is a light source’s proportion of blue toyellow–green (smaller v 0 is bluer), and its u 0 abscissa is the pro-portion of bluish-green to red (larger u 0 is redder). Anothermetric of color, correlated color temperature (or CCT), canbe used as a proxy for a source’s spectral energy distribution.For this reason, Fig. 3 also includes part of the Planckian

locus and its corresponding color-temperature limits. Figure 3’sperceptual scale for color difference is a MacAdam just-noticeable difference (JND) ellipse for this part of the UCSdiagram [27,28].

Plotted in Fig. 3 are u 0v 0�h� chromaticity curves for Fig. 2’sthree antisolar meridians (i.e., at constant ϕrel � 180°), withchromaticity coordinates at each h averaged across an azimuthalwindow of width Δϕrel � 4°. Collectively, these u 0v 0�h� curvesillustrate the wide range of antisolar color gamuts typically seenat the earth’s surface. In particular, note that the normalizedcolorimetric gamut g [29] for any given u 0v 0�h� curve decreasesas τaer increases, which is consistent with the color patterns seenin Fig. 2. Table 1 summarizes the relationship between g and

Fig. 2. Visible-wavelength images calculated from hyperspectral scans made along almucantars at view-elevations h � 15°–16° at North Beach on(a) 11-3-2013 at unrefracted sun elevation h0 � −1.56°, (b) 10-20-2013 at h0 � −1.77°, and (c) 11-27-2016 at h0 � −1.74°. From Fig. 1,(a) has the minimum spectrally integrated τaer and (c) has the maximum τaer.

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Fig. 3. Smoothed meridional chromaticity curves of the antitwilightarch (or ATA) and dark segment (near h � 0°) from hyperspectral scansmade at North Beach on the twilight dates in Figs. 1 and 2. For eachcurve, azimuth relative to the sun ϕrel � 180° and h � 0°–30°.

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surface values of τaer;λ parameters α and β, with the latter valuesrequiring λ in Eq. (1) to be in μm.

To simulate twilight colors, we use a multiple-scatteringMonte Carlo model that (a) includes the spherical atmosphereneeded for twilight, (b) specifies surface and atmospheric scat-tering and absorption either just radially or in all three dimen-sions, and (c) simulates skylight spectral radiances at and farabove the earth’s surface [30,31]. This MYSTIC model (from“Monte carlo code for the phYSically correct Tracing of pho-tons In Cloudy atmospheres”) has been developed over twodecades and used on many atmospheric radiative transfer prob-lems from the ultraviolet to the thermal infrared. For efficiency,

we use MYSTIC’s backward mode in which photons are scat-tered from the observer back toward the sun [32,33]. Wemodel MYSTIC’s τaer;λ using Eq. (1) and for now set itsLambertian surface reflectance r � 0.2 at all λ.

Figure 4 plots the colorimetric consequences for MYSTIC’santisolar twilight of either eliminating aerosols entirely (i.e., apurely molecular atmosphere) or adopting Fig. 1’s minimumaerosol amount. We designate these as MYSTIC atmospheremodels A and B, respectively. In Fig. 4 and later simulations,we set unrefracted h0 � −1.56° to aid comparisons amongfigures [this h0 value is for Fig. 2(a)’s minimum τaer case].As shown by the perceptibly large distance between Fig. 4’stwo u 0v 0�h� curves at their respective minimum CCTs, atthe surface (altitude z � 0 km) the ATA in model A’s molecu-lar atmosphere would have visibly purer reds than those frommodel B (minimum τaer of 11-3-2013). In other words, thesmallest antitwilight CCT in a molecular atmosphere(∼4050 K at h � 5.1° and ϕrel � 180°) would be visibly red-der than the most vivid ATA simulated for North Beach(∼5390 K at h � 6.1°).

While we cannot visually experience a purely molecularatmosphere, we can approximate its antitwilight colors froman airplane in the lower stratosphere (at, say, z ∼ 10 km).Combined with Table 1’s g values, Fig. 5 shows just how muchthis higher altitude can improve antitwilight colors: at the sameh0, not only has g for model B nearly doubled from the ob-served North Beach value, but its minimum CCT in the strato-spheric antitwilight arch is now reduced (i.e., further reddened)to ∼3740 K at h � �0.8° [34]. Although an observer atz � 10 km is far above boundary layer aerosols, their opticaleffects are still visible due to backscattering of direct sunlightthat has been reddened after traversing the troposphere at low(or even surface-grazing) altitudes.

Figure 6 shows that the increases in g due to an observer’slarge z can be pronounced: on two different airline flights, Leeused a colorimetrically calibrated RGB camera [12] to measure

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Fig. 4. MYSTIC Monte Carlo simulated meridional u 0v 0�h� curvesfor a purely molecular atmosphere (designated atmosphere model A)and one with minimal τaer (atmospheremodel B based on 11-3-2013;see Fig. 1). The observer is at z � 0 km, ϕrel � 180°, andh0 � −1.56° as in Fig. 2(a).

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Fig. 5. Meridional u 0v 0�h� curves measured for the minimum τaercase on 11-3-2013 (observer at z � 0 km) or simulated byMYSTIC models A and B with the observer at z � 10 km.

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Fig. 6. Smoothed meridional chromaticity curves of the ATA anddark segment calculated from calibrated RGB camera images taken atairplane flight altitudes z ∼ 11 km near Burlington, Wyoming(8-4-2013) and near Sterlin Lake, Yukon (8-9-2013).

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antitwilight colorimetric gamuts that were ∼12%–60% largerthan the largest g from surface observations (see Table 1).

However, one consequence of Fig. 6’s lower h0 is that itsminimum antitwilight CCTs are in fact larger than those simu-lated in Figs. 4 and 5 for h0 � −1.56°. This increase in mini-mum CCT as h0 decreases is similar to that measured forsurface-based twilights [10]. Thus the reddest part of the anti-solar sky (the ATA) becomes steadily bluer as evening twilightprogresses, just as it does in Fig. 6. For naked-eye observers, thepractical import of Figs. 5 and 6 is that to see the best antitwi-light colors (i.e., antitwilights with the largest g), one shouldclimb a tall mountain or, more conveniently, take a high-altitude airplane flight.

3. MODELING COLORS OF THECIRCUMHORIZONTAL TWILIGHT SKY

Our experience to date is that the MYSTIC model tends toshift antitwilight u 0v 0�h� curves along the Planckian locus tocolor temperatures slightly higher than those measured bythe Pika II. However, the simulated chromaticity curves’ shapesand gamuts are quite realistic, meaning that MYSTIC consis-tently produces accurate relative, as distinct from absolute,chromaticities. This is no small accomplishment, given the(a) inherent difficulties of determining τaer;λ spectra exactlyand (b) marked sensitivity of chromaticities to even smalldifferences in skylight spectra. Bearing these caveats firmlyin mind, we now extend the model’s reach across the sky toget a more complete picture of civil twilight’s underlying colorstructure.

As observed earlier [10], the ATA is in fact part of a reddishband of light that encircles the horizon during civil twilight. Fora surface observer, this band’s lowest point is the astronomicalhorizon at the sun’s azimuth where it forms part of whatMinnaert calls the “bright glow” [35]. The band’s highest pointis in the antisolar direction (see Figs. 2 and 3), where its h isseveral degrees above the astronomical horizon at elevationangle hmax. Visualize this reddish band as the intersection ofa tilted plane with the observer’s celestial sphere, withthe plane’s tilt angle being hmax and at any ϕrel its

h � hmax � sin�ϕrel∕2�. Figure 7 illustrates the relationshipsamong this tilted plane, the observer’s horizontal tangent plane,and the vertical principal plane (the latter is defined by the sun,zenith, and observer). In astronomical terms, the reddish bandis a celestial small circle that is tilted slightly with respect to thelocal tangent plane. During civil twilight, hmax increases as h0decreases and as τaer increases at a given h0.

Because this band is the reddest part of the circumhorizontaltwilight sky, using MYSTIC to map its colors is quite instruc-tive. We do so in Fig. 8 for three different atmospheres, eachwith simulated h0 � −1.56° at z � 0 km: (a)model A’s purelymolecular atmosphere, (b) the minimum and (c) maximum τaercases (models B and C, respectively) whose measured antisolarchromaticities are graphed in Fig. 3. The τaer;λ spectra for mod-els B and C are plotted in Fig. 1 and their best-fit α and β valuesare listed in the first and third rows of Table 1. As noted above,each atmosphere model is given the same surface O3 concen-tration of 300 Dobson units. Thus Fig. 8 directly displays thecolor consequences of having fixed O3 and no, minimal, orlarge amounts of tropospheric haze aerosols.

Readers will of course experience slightly different chroma-ticities than we do when looking at Fig. 8, depending on theirparticular display device or on the illuminant for their printedpage. Those important caveats aside, we created Fig. 8 usingstandard projective geometry techniques that map twilightchromaticities into their RGB equivalents on a computer’s cali-brated color display [36], for which we made white (or maxi-mum gray level R � G � B) correspond to an equal-energy u 0,v 0. Because twilight’s overall luminance dynamic range far ex-ceeds that of most computer displays, we made Fig. 8’s lumi-nances proportional to the square root of MYSTIC modelluminances. Furthermore, some twilight colors may be purerthan a particular display can reproduce [37]. Thus Fig. 8 pro-vides a diligent, but not exact, color reproduction of theMYSTIC twilights.

What qualitative optical insights can Fig. 8 provide? First, asFig. 4 shows graphically, the molecular atmosphere in Fig. 8(a)has the reddest antisolar colors (see the color map’s left side),followed by the far less saturated ones of the minimum τaer case(model B) shown in Figs. 2(a) and 8(b). Note that Fig. 2(a)’s

Fig. 7. Geometrical relationship of the observer’s tangent plane tothe tilted plane of maximum circumhorizontal redness and to the ver-tical principal plane. For clarity, the tilt angle hmax is exaggerated here,and the small rectangle indicates that the tangent and principal planesare perpendicular.

Fig. 8. Simulated twilight colors mapped as f �ϕrel� in a plane thatis tilted up slightly from the solar horizon at ϕrel � 0°, h � 0° forMYSTIC (a) purely molecular model A, (b) minimum τaer modelB, and (c) model C, which has the maximum τaer observed inFigs. 1 and 2(c).

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reddest colors are confined to a very narrow, slightly purplishband just above the horizon near the scan’s left side. Yet whatFig. 8(b)’s minimum τaer atmosphere lacks in antisolar vivid-ness, it gains in its solar sky’s more saturated oranges.

For the maximum τaer atmosphere (model C) in Figs. 2(c)and 8(c), the ATA is marginally visible at best. And althoughFig. 8(c)’s solar sky is now visibly redder than those in Fig. 8’stwo smaller-τaer cases, this red is less saturated. Note too thatFig. 8(c)’s midpoint where ϕrel � 90° is the bluest of modelsA–C. Thus taken as a whole, Fig. 8’s colors suggest that (a)adding even a little of Shettle’s tropospheric aerosol [22] toa molecular atmosphere makes the circumsolar sky redder atthe expense of the antisolar sky and (b) adding more aerosolwill eventually eliminate any visible redness in the antisolarsky while not necessarily making the solar sky a vivid red.These two trends are clearly evident in Fig. 2’s observedtwilights.

Figure 9 translates Fig. 8’s azimuthal trends to the UCS dia-gram. In Fig. 9, the model A molecular atmosphere has thesmallest 0° ≤ ϕrel ≤ 180° circumhorizontal colorimetric gamutwhile model B (minimum τaer) has the largest gamut. Twoimportant color consequences of this larger gamut are that,compared with model A’s molecular atmosphere, model Bhas a less-red ATA (i.e., its CCTs are larger) and a reddersky near the solar horizon. This suggests that, unlike the alpen-glow or rainbow, reddening of direct sunlight does not by itselfexplain the redder ATAs that result from having little or noaerosol. A plausible explanation is that in the molecular atmos-phere, redder indirect sunlight from the solar sky (i.e., multiplescattering) gives model A the reddest ATA in Figs. 4 and 9. ForFig. 9’s model C (maximum τaer), the entire chromaticity curveis shifted toward higher CCTs (i.e., it is bluer at all h) and itssolar half lies farther from the Planckian locus. In fact, thereddest part of this circumhorizontal sky occurs at ϕrel ∼ 16°

rather than at the sun’s azimuth, a displacement that is justdiscernible in Fig. 8(c).

Figure 10 shows the colorimetric results of lower h0 onFig. 9’s tilted-plane azimuthal colors. For both the molecularand minimum τaer atmospheres, the 0° ≤ ϕrel ≤ 180° colori-metric gamut is larger at h0 � −2.78°. This expansion neces-sarily means that the (a) reddest part of each ATA is now bluer(note Fig. 10’s higher CCTs at ϕrel � 180°) and (b) solar skiesare now redder. Thus in either atmosphere, as the sun movesfarther below the horizon the ATA fades into its blue-skysurroundings and circumsolar colors become more vivid.

4. MODELING COLORS OF THE TWILIGHTSKY’S PRINCIPAL PLANE

Figure 11 rotates the plane of simulated twilight colors so that itpasses through the zenith. In other words, this color map showstwilight colors of the clear-sky principal plane, again usingFig. 8’s atmospheres and h0 � −1.56°. As in Fig. 8, we makeFig. 11’s displayed luminances proportional to the square rootof MYSTIC model luminances. The ATA for Fig. 11(a)’s

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min aerosol + O3 (hmax= 11.3°)CIE

197

6 v'

CIE 1976 u'

sunlight aboveatmosphere

φrel= 0°, h = 0°

φrel= 180°, h = hmax

MYSTIC tilted azimuthalpaths through middle

of ATA for h0= -2.78°

(reddest part of ATA)

(solar horizon)model A

model B

Fig. 10. Same as Fig. 9 for models A and B, but here forh0 � −2.78°. As a point of reference, an x marks the chromaticityof sunlight above the atmosphere.

Fig. 11. MYSTIC simulated twilight colors mapped in the clear-sky principal plane at h0 � −1.56° for (a) models A (purely molecu-lar), (b) B (minimum τaer), and (c) C (maximum τaer).

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purely molecular atmosphere (model A) clearly has the largestvertical extent and most vivid, saturated colors. However, this ispaired on the solar horizon (map’s right edge) with a brightglow that is both narrower and slightly less orange than foreither aerosol atmosphere [Figs. 11(b) and 11(c); models Band C, respectively]. In Fig. 11(c)’s maximum τaer case, thebright glow has its largest vertical extent although its lower edgeis dull-looking and desaturated, and the ATA is barely visibleabove the antisolar horizon. Both color deficiencies stem fromthis turbid atmosphere’s increased multiple scattering.

Figure 12 graphs the colorimetric data that underliesFig. 11’s color map. In Fig. 12, we extend Fig. 4’s antisolarfeatures and also include chromaticities from model C (maxi-mum τaer). Much like its Fig. 9 cousin for circumhorizontalcolors, Fig. 12 indicates that the reddest solar-sky colors willoccur for some unknown minimum τaer (note that both themolecular and maximum τaer cases are less red near the solarhorizon). Exactly what spectrally integrated τaer value yieldsthese minimum CCT colors likely depends on the shape of thecorresponding τaer;λ spectrum.

However, MYSTIC simulations show that in a purelymolecular atmosphere, redder solar-sky colors can also be pro-duced by increasing surface pressure and thus increasingmolecular number density Nmol. This enhanced redness is rea-sonable given that molecular single scattering extinguishes di-rect sunlight at a rate ∝ λ−4.09, whereas our local aerosolstypically do so only at a rate ∝ λ−1.52. Thus from the standpointof spectral extinction by single scattering, aerosols typically areless effective than molecules in reddening sunlight scattered innear-forward directions. (As is often true, though, multiplescattering complicates this simple picture.) The reason thataerosols redden sunlight is not that they are more effectiveat doing so per unit optical depth. Instead, the distinction isbetween the possible and impossible: while τaer can vary widelyin the real atmosphere, its counterpart τmol cannot given thatchanges in surface Nmol are proportionally much smaller.

Equally noteworthy in Fig. 11 are differences among thethree zeniths’ local minima of luminance Lv. The zenith is bestdefined for Fig. 11(b)’s minimum τaer case (model B), since itscombined scattering yields a slightly narrower dark region nearthe zenith than do models A and C. Figure 13 helps to explainthis subtlety. In it, we plot MYSTIC Lv as a function of single-scattering angle ψ , which equals 1.56° on the solar horizon and91.56° at the zenith for h0 � −1.56°. Not only is the near-zenith sky slightly darker for model C (maximum τaer), butits Lv�ψ� curve is also slightly wider here. Taken together, thesedetails describe a darker zenith sky that has lower contrast thanmodel B. The zenith of the model A molecular atmosphere isalso less well-defined because it too has a broader local mini-mum in Lv�ψ�. However, model A has the brightest zenithbecause molecular scattering is symmetric about ψ � 90°.Put another way, single scattering of direct sunlight is strongerfor model A near ψ � 90° than for either model B or C.

Although the minimum CCTs that define the ATA tend notto occur at the same h as their corresponding luminancemaxima [10], Fig. 13 shows that these nearby local maximain Lv�ψ� also behave predictably at constant h0. As aerosolamount increases, the local maxima in antisolar Lv�ψ� widenand move to higher h at a given h0. As a result, the ATA haslower color and luminance contrast as τaer increases (see Figs. 2and 8). Figure 13 has similar Lv�ψ� patterns in the solar sky, butwith one important difference: some unknown minimum τaerproduces the brightest glow. This is because as τaer increases, skybrightness near the sun responds to a changing balance between(a) relatively weak forward scattering by molecules, which iseventually surpassed by (b) stronger forward scattering fromaerosols, which in turn is (c) reduced by increased extinctionfrom ever-larger aerosol slant-path optical thicknesses.

5. CONCLUSIONS

Our analysis of calibrated RGB camera images and ∼80hyperspectral scans made during 21 clear evenings reveals

0.42

0.44

0.46

0.48

0.50

0.52

0.54

0.42

0.44

0.46

0.48

0.50

0.52

0.54

0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32

molecular + O3

min aerosol + O3

max aerosol + O3

CIE

197

6 v'

CIE 1976 u'

solar horizon

MYSTIC principal-plane colors (z = 0 km, h0 = -1.56°)

zenith

antisolarhorizon

ATAmodel A

model C

model B

Fig. 12. MYSTIC simulated meridional u 0v 0�h� curves for Fig. 11’smaps of twilight principal-plane colors that correspond to models A(purely molecular), B (minimum τaer), and C (maximum τaer).

Fig. 13. MYSTIC simulated principal-plane luminances Lv�ψ� forFig. 11’s maps of twilight colors that correspond to models A (purelymolecular), B (minimum τaer), and C (maximum τaer).

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several little-known features of civil twilights. First, our dataclearly shows that at a given h0, the antitwilight arch’s verticalextent, colorimetric purity, and redness all increase as τaer de-creases, with redness calculated either by CCT or u 0, v 0 chro-maticities (Figs. 2–4). Second, to make the ATA even morevivid, one can either wait for exceptionally clear, unturbid skiesor ascend above boundary layer aerosols in an airplane or on amountaintop (Figs. 5 and 6). Either approach provides a real-atmosphere approximation to the even more visually strikingATAs of the theoretical molecular atmosphere. Third, for mostturbidities the ATA and dark segment both become visiblybluer as h0 decreases (Figs. 6, 9, and 10).

Fourth, MYSTIC simulations indicate that twilight’s mostvivid solar-sky colors require some, but not too much, turbidity(Figs. 8–12). The additional spectrally selective extinction byaerosols further reddens skylight near the sun, albeit not as ef-ficiently as an equivalent (but unrealistic) increase in molecularscattering would. Fifth, the model A molecular atmospherewould simultaneously produce (a) a redder ATA and (b)less-reddened direct sunlight than our model B minimumτaer case (Figs. 9 and 12). Because the molecular ATA’s greaterredness must come from somewhere other than direct sunlight,the MYSTIC simulations lead us to think that reddish multiplescattering plays an important role in determining the colors ofthe antitwilight arch and the dark segment. Sixth, MYSTIC’sprincipal-plane profiles of twilight Lv (Fig. 13) are realistic andinstructive: not only does the model B minimum τaer case (c)produce the real atmosphere’s brightest, most vivid ATA, but(d) its strong (and minimally attenuated) forward scatteringyields the brightest twilight glow above the solar horizon.Taken together, these measured and modeled results add nu-ance to our understanding of how molecular and aerosol scat-tering affect the coloring of the twilight sky.

Raymond Lee’s research has been generously supported byUnited States National Science Foundation grant AGS-0914535, as well as by the United States Naval Academy’sDepartments of Mathematics and Oceanography. Opinions,findings, and conclusions or recommendations expressed in thispaper are those of the authors and do not necessarily reflect theviews of the National Science Foundation.

Funding. National Science Foundation (NSF) (AGS-0914535).

REFERENCES AND NOTES1. M. G. J. Minnaert, Light and Color in the Outdoors (Springer-Verlag,

1993), p. 326.2. R. Greenler, Rainbows, Halos, and Glories (Cambridge University,

1980), p. 127.3. C. F. Bohren, Clouds in a Glass of Beer: Simple Experiments in

Atmospheric Physics (Wiley, 1987), pp. 92–93.4. D. K. Lynch and W. Livingston, Color and Light in Nature (Cambridge

University, 1995), p. 31.5. C. F. Bohren and E. E. Clothiaux, Fundamentals of Atmospheric

Radiation (Wiley-VCH, 2006), pp. 408–409.6. E. O. Hulburt, “Explanation of the brightness and color of the sky, par-

ticularly the twilight sky,” J. Opt. Soc. Am. 43, 113–118 (1953).7. C. N. Adams, G. N. Plass, and G. W. Kattawar, “The influence of

ozone and aerosols on the brightness and color of the twilight sky,”J. Atmos. Sci. 31, 1662–1674 (1974).

8. G. V. Rozenberg, Twilight: A Study in Atmospheric Optics (Plenum,1966), pp. 245–249.

9. J. V. Dave and C. L. Mateer, “The effect of stratospheric dust on thecolor of the twilight sky,” J. Geophys. Res. 73, 6897–6913 (1968).

10. R. L. Lee, Jr., “Measuring and modeling twilight’s Belt of Venus,” Appl.Opt. 54, B194–B203 (2015).

11. D. K. Lynch, D. S. P. Dearborn, and S. C. Richtsmeier, “Antitwilight I:structure and optics,” Appl. Opt. 56, G156–G168 (2017).

12. R. L. Lee, Jr., W. Meyer, and G. Hoeppe, “Atmospheric ozone andcolors of the Antarctic twilight sky,” Appl. Opt. 50, F162–F171 (2011).

13. J. C. Naylor, Out of the Blue: A 24-hour Skywatcher’s Guide(Cambridge University, 2002), pp. 69–73, 101–102.

14. See Ref. 2, p. 12.15. See Ref. 4, p. 106.16. Figure 1’s τaer;λ data is archived by the AErosol RObotic NETwork

(AERONET) at http://aeronet.gsfc.nasa.gov.17. K.-N. Liou, An Introduction to Atmospheric Radiation (Academic,

1980), p. 239.18. Pika II hyperspectral imaging system from Resonon, Inc., 123

Commercial Drive, Bozeman, Montana 59715 (http://www.resonon.com). The system consists of a digital camera that has an internal dif-fraction grating and is coupled to a rotation stage controlled by a pre-cision stepper motor and laptop computer. In this pushbroom system,the laptop acquires 640 different skylight spectra at each rotation stageposition (i.e., for each line of the resulting hyperspectral datacube).

19. See Ref. 5, p. 398.20. G. T. Wolff, N. A. Kelly, M. A. Ferman, M. S. Ruthkosky, D. P. Stroup,

and P. E. Korsog, “Measurements of sulfur oxides, nitrogen oxides,haze and fine particles at a rural site on the Atlantic coast,” J. AirPollut. Control Assoc. 36, 585–591 (1986).

21. L. Xia and Y. Gao, “Chemical composition and size distributions ofcoastal aerosols observed on the US East Coast,” Mar. Chem.119, 77–90 (2010).

22. E. P. Shettle, “Models of aerosols, clouds and precipitation for atmos-pheric propagation studies,” in Conference Proceedings No. 454 onAtmospheric Propagation in the UV, Visible, IR and mm-Wave Regionand Related Systems Aspects (Copenhagen, 1989), pp. 1–13.

23. The Meteorological Service of Canada archives surface-based Dobsonspectrophotometer data (www.woudc.org/data_e.html), and its meancolumn total O3 for October–November 2013 from nearby WallopsIsland, Virginia is 296.2 Dobson units. For 11-27-2016, the closestavailable data is at Raleigh, North Carolina, where column-integratedO3 ∼ 290 Dobson units (archived at esrl.noaa.gov/gmd/grad/neubrew).

24. A. T. Young and G. W. Kattawar, “Sunset science. II. A useful dia-gram,” Appl. Opt. 37, 3785–3792 (1998).

25. R. L. Lee, Jr. and P. Laven, “Visibility of natural tertiary rainbows,”Appl. Opt. 50, F152–F161 (2011). See the discussion of itsFig. 11, but note that in our paper Fig. 2’s achromatic stimulus is givenby the equal-energy spectrum.

26. During the 3.5 min needed to acquire a scan at twilight lighting levels,h0 changes by ∼0.65° in Fig. 2. As a result, each of Fig. 2’s verticalscanlines occurs at a slightly different sun position, and this is usedin calculating the scanline’s azimuth relative to the sun ϕrel.

27. CCT and the MacAdam JND are described in G. Wyszecki and W. S.Stiles, Color Science: Concepts and Methods, Quantitative Data andFormulae, 2nd ed. (Wiley, 1982), pp. 224–225, 306–310.

28. J. Hernández-Andrés, R. L. Lee, Jr., and J. Romero, “Calculating cor-related color temperatures across the entire gamut of daylight andskylight chromaticities,” Appl. Opt. 38, 5703–5709 (1999).

29. R. L. Lee, Jr., “Twilight and daytime colors of the clear sky,” Appl. Opt.33, 4629–4638, 4959 (1994). As defined there, g measures thefraction of the CIE diagram spanned by a set of visually contiguouschromaticities: thus g = 0 for a single u’, v’ pair and g = 1 for theentire spectrum locus.

30. B. Mayer, “I3RC phase 1 results from the MYSTIC Monte Carlomodel,” in Extended Abstract for the I3RC Workshop, Tucson,Arizona, 17–19 November 1999, pp. 1–6, MYSTIC may be freelydownloaded from www.libradtran.org.

31. B. Mayer and A. Kylling, “Technical note: the libRadtran softwarepackage for radiative transfer calculations—description and examplesof use,” Atmos. Chem. Phys. 5, 1855–1877 (2005).

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32. B. Mayer, S. W. Hoch, and C. D. Whiteman, “Validating the MYSTICthree-dimensional radiative transfer model with observations from thecomplex topography of Arizona’s Meteor Crater,” Atmos. Chem. Phys.10, 8685–8696 (2010).

33. C. Emde, R. Buras, B. Mayer, and M. Blumthaler, “The impact of aero-sols on polarized sky radiance: model development, validation, andapplications,” Atmos. Chem. Phys. 10, 383–396 (2010).

34. For notational consistency, all h and h0 values in Fig. 5 are given as ifthe observer were still on the z � 0 km tangent plane. However, atz � 10 km one can see the bluish skylight that comes from just

below the local tangent plane, which corresponds to h < 0°in Fig. 5.

35. See Ref. 1, p. 296.36. See Ref. 27, pp. 138–139.37. Figure 8 was developed for a Sony Trinitron display with the

following CIE 1976 primaries: u 0(red) � 0.4289, v 0(red) � 0.5268,u 0(green) � 0.1169, v 0(green) � 0.5594, u 0(blue) � 0.1721, and v 0(blue)� 0.1744; each color channel has a gray-level gamma of 1.8. All ofFig. 8’s colors are well within the triangular u 0, v 0 gamut spannedby this display’s three primaries.

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