Horizon brightness revisited: measurements and a model of clear-sky radiances

Horizon brightness revisited:measurements and a model of clear-sky radiances

Raymond L. Lee, Jr.

Clear daytime skies persistently display a subtle local maximum of radiance near the astronomicalhorizon. Spectroradiometry and digital image analysis confirm this maximum's reality, and they showthat its angular width and elevation vary with solar elevation, azimuth relative to the Sun, and aerosoloptical depth. Many existing models of atmospheric scattering do not generate this near-horizonradiance maximum, but a simple second-order scattering model does, and it reproduces many of themaximum's details.

Key words: Horizon brightness, clear-sky radiance, digital image analysis, atmospheric optics,multiple scattering.

Introduction

To the uninitiated, the clear daytime sky seems sucha commonplace that its radiance and brightness'distribution surely must be well known. Research-ers in fields ranging from solar energy engineering2 ,3to atmospheric optics4'5 have repeatedly measuredand modeled the angular distribution of clear-skyradiances, and they have published scores of paperson the subject. What can be left to know?

In fact, a great deal is left to know. In simplemodels of scattering by the clear atmosphere, radi-ance increases monotonically from the zenith to theastronomical (i.e., dead-level) horizon. 6' 7 However, apersistent feature of our cloudless atmosphere is alocal maximum of radiance several degrees above thehorizon, not at it. We have detected this near-horizon radiance maximum in clear daytime skiesranging from midlatitudes to the Antarctic, and frommidcontinent to the open sea. However, no one, tomy knowledge, has written about it. Why?

First, the maximum has rather low contrast, mak-ing it visually subtle. We tend to measure and modelthose sky features that call attention to themselves.Like many phenomena in atmospheric optics, thenear-horizon maximum is obvious only to the initiated.Second, before the advent of narrow field-of-view(FOV) radiometers8 and photographic analysis tech-

The author is with the Department of Oceanography, UnitedStates Naval Academy, Annapolis, Maryland 21402.

Received 27 August 1993; revised manuscript received 4 Novem-ber 1993.

0003-6935/94/214620-09$06.00/0.e 1994 Optical Society of America.

niques, 9- 2 accurate and detailed near-horizon radi-ance measurements were difficult, if not impossible,to make.

As a result, many previous models of clear-skyradiances have been compared with measurementsthat are fundamentally inadequate. This inad-equacy stems from the measurements' limited angu-lar resolution, which is often 5c-20'.13-16 Obviously,any clear-sky features that are angularly smallerthan this will either be eliminated or considerablysmoothed. This imprecision in measurement hasled, in turn, to models that fail to reproduce thenear-horizon radiance maximum.'3 -16 Authors ofmodels that do produce a radiance maximum near thehorizon have not commented on this feature, perhapsbecause they are unaware of its verisimilitude.4 5"17

Thus our goal here is threefold. First, we want toanalyze clear-sky radiances near the horizon, usingboth spectroradiometry and digital image analysis.Second, we want to compare these radiance patternswith those predicted by a simple, but physicallyrigorous, solution of the radiative transfer equationto see if we can account for the near-horizon maxi-mum. Finally, we want to see if our model providesus any insight into a simple physical explanation ofthe phenomenon.

High-Resolution Measurements of Clear-Sky Radiances

We begin by electronically digitizing, at a color resolu-tion of 24 bits per pixel, color slides of clear skies seenat several sites and times of day (see Plates 37-40).Algorithms developed earlier'8 are used to calibratethe images colorimetrically and radiometrically.The digitized color slides yield relative radiance data

4620 APPLIED OPTICS / Vol. 33, No. 21 / 20 July 1994

whose angular resolution is limited only by the film; aresolution of 1/65° is possible. Relative sky radi-ances can then be extracted from the image data anddisplayed as meridional radiance profiles (i.e., plots ofnormalized, azimuthally averaged radiance versusangular elevation). Figure 1 illustrates the majorfeatures of a surface-based observer's clear-sky scat-tering geometry.

At one site, color slides were taken simultaneouslywith narrow FOV spectra19 of the clear sky (see Plate37). Comparisons of these two data sources give agood indication of the photographic technique's poten-tial accuracy (see Fig. 2). For elevation angles 2 0.50in Fig. 2, the root-mean-square (rms) difference be-tween the two normalized meridional profiles is0.00802. The radiometer's maximum radiance oc-curs 1.50 above the astronomical horizon, whereasthe photographic analysis places it at 2.40. Depend-ing on solar elevation, azimuth relative to the Sun,and normal optical depth, the maximum may occur at80 (or higher) and be much broader than that seen inFig. 2. The breadth of the maximum and the closeagreement of the photographic and the radiometerdata suggest that we are not seeing the photographicequivalent of a Mach band (a possibility explored byLynch2 0).

Nevertheless, the photographic and the radiometerdata's disagreement about the maximum's elevationillustrates some of the inherent differences betweenthe two techniques. First, the radiometer data wereacquired over a period of 25 min, meaning that theymay incorporate temporal changes in sky radiancepatterns that the photograph does not. Second,although the photographic data have been smoothedover a 0.50 azimuthal FOV, this digital smoothingmay be somewhat different from that occurring opti-cally in the radiometer. (The error bars of Fig. 3illustrate the radiometric noise typical at this narrowFOV for the photographic technique.) Finally, we

12-

10,

a 8-co

8 6.0

5 4.

2-

O1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8normalized radiance

12-

10-

8-

4-

O-0.

0.9 1 1.1

Fig. 2. Comparison of photographic and spectroradiometric mea-sures of clear-sky radiances at University Park, Pa., at 1605 GMTon 6 October 1992 (see Plate 37). Arunningaverage has smoothedthe detailed photographic data of Fig. 3. The solar zenith angleO = 480, the instruments' azimuth with respect to the sun, Mre, is118°, and the equivalent Lambertian surface reflectance rgfe = 0.25.

must correct the photographic analysis for any expo-sure falloff or vignetting that occurs in the camera.

Our Fig. 2 comparison of photographic and radiom-eter data suggests that, with careful geometric andradiometric calibration of photographic data, we canuse digital image analysis when a radiometer isunavailable. When we do so, the resulting photo-graphic radiance profiles confirm that the near-horizon maximum is a remarkably persistent featureof the clear daytime sky. How can we explain it?

12-

10-

8-

es 6-

. 4

.,q

0

X= T _ 180O

Fig. 1. Clear-sky scattering geometry for an observer at point Xwhose view zenith and azimuth angles are B0 and ),,, respectively.Prey is the difference between the sun's azimuth and 4T. Clear-skyradiances reaching the observer include contributions from di-rectly scattered sunlight, L,, and from multiply scattered surfacelight and skylight, Ldiff. At each elemental scattering volume dValong the observer's line of sight, L. and Ldiff are scattered throughvarious angles ' (Mdir for L, is shown above).

-2 .0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

normalized radiance

12-

10-

4-

0

-211 1.1

Fig. 3. Normalized radiance versus view elevation angle for aclear-sky scene at University Park, Pa., at 1605 GMT on 6 October1992 (see Plate 37). The astronomical horizon corresponds to aview elevation of 0°. These photographically derived data span a0.5 0-wide meridional swath near the scene's center that matchesthe FOV of spectroradiometer data taken simultaneously (see Fig.2). Error bars span two standard deviations an of the azimuthalradiances at selected elevation angles.

20 July 1994 / Vol. 33, No. 21 / APPLIED OPTICS 4621

smoothedphotographic radiances

E radiometer radiances(400-700 mun)

0=48, 0 = 118-, r ,= 0.25

RMS difference = 0.00802 (elev. 2 0.5')

Radiance maximum at 1.5 astronomicalelevation is 27.84 W/(m

2sr).

I/

... ... . . .. . r. . .. . .. .. .. - -I

. . . . . . . . . . . I . .. -... |. . . . . . . . . .

Second-Order Scattering Modelfor Clear-Sky Radiances

As noted above, models of clear-sky radiances are notin short supply.2-5 Indeed, scores of models areavailable, ranging from the purely empirical to thehighly theoretical. Although we can easily rejectmodels that fail to produce the near-horizon maxi-mum, why not use those that do?t 5

Aside from the difficulty of successfully translatingothers' algorithms, their authors themselves raisesome cautions. Dave'7 says of his spherical harmon-ics model that "because of the plane-parallel [atmo-sphere] assumption and nature of the method ofcomputations, it is not possible to evaluate" skyradiance at the astronomical horizon. Furthermore,he describes his algorithm as "somewhat unreliable"at 1 elevation, which is within the range whereradiance maxima can occur. Prasad et al.4 use theirmodel (which incorporates van de Hulst's doublingmethod) to plot horizon-to-zenith profiles of skyradiance for several azimuth angles (see their Fig. 8).Surprisingly, their profiles do not appear to convergeat the zenith where, at a particular time and for aparticular atmosphere, only a single radiance is pos-sible. Thus, despite the fact that these two modelsincorporate higher-order multiple scattering, bothexhibit some shortcomings that are relevant to ourproblem. Given these shortcomings, developing anew model of sky radiance does not seem superfluous.

We start by applying the equation of radiativetransfer to a curved-shell atmosphere in which scat-terer number density decreases exponentially withaltitude. Scatterers consist of molecules and spheri-cal haze droplets, and slant optical paths are calcu-lated by the use of Bohren and Fraser's algorithm.2'The scattering model accounts for variations in solarelevation, scatterer optical depth, scale height, phasefunction, and Lambertian surface reflectance. Itproduces clear-sky radiances for a surface-based ob-server as a function of view zenith angle 0, and viewazimuth 4,

In the model, 0 is the zenith angle, with 0 = Tr beingthe nadir and 0 = 0 the zenith. 4) is azimuth angle,and 4)

rel is the relative azimuth (4)rel = 0 is toward theSun, 4)

rel = 1r is in the antisolar direction; see Fig. 1).T(0, 4)) is the scattering angle measured in sphericalcoordinates 0, . For scattering of direct sunlight,tdir is defined by the Sun, a scatterer (aerosol or

molecule), and the observer. Atmospheric absorp-tion is parameterized by the single-scattering albedowo. For an observer at the Earth's surface, thediffuse clear-sky radiance Lsky in the direction 0,, ),, is

Lsky = J(t, 'r)exp(T - Tf )d, (1)

where the source function J(T, 7) = Jdir + Xi=I Jdiffiis the sum of direct single scattering Jdir and allhigher orders of diffuse multiple scattering 1t=1 Jdifalong the total atmospheric slant optical path Tf

coinciding with the observer's viewing direction.Jdir and Jdiffl can easily be expanded as integrals overT, 0, and X), and the integrals can then be evaluatednumerically. For realistic clear atmospheres, weshow that for dir > 200 the multiple-scatteringcontributions to Jff after the first term Jdiff,1 do notaffect L~k. profiles appreciably. In other words, werestrict our model to single plus double scattering,making it a second-order scattering model.

Assuming the solar radiance L, is nearly constantover the Sun's small solid angle o,,

LstXdir(T) Pdir(T)Jdir = Wo exp(-'r) = wnoE exp(-r) 4

(2)

where Es is the Earth's solar constant (= Lsw)Pdir(T) is the scattering phase function for directscattering of sunlight, and T is the integration vari-able in Eq. (1).

For diffuse multiple scattering, Jdiff1 = (/4Tr)

f4w Pdiff(P)Ldiff(T)dwO, or

Jdllffl - 'ITr 3 |_ __ Pdiff(O, -))Ldiff(0, 4))sin(0)dOd,,

(3)

where Pdiff is the scattering phase function for scatter-ing of surface light and skylight from 0, ) into adifferential air volume dV that lies along an observ-er's line of sight in the direction %,, 4),. In general,the angles T(0, 4)) for scattering surface light andskylight into dVwill not be the same as the observer'sTdir as he or she looks toward dV.

The diffuse radiances Ldiff(0, 4)) that illuminate eachdVare calculated by integrating over Tdiff, where Tdiff isthe slant-path optical depth from dV to the top (orbottom, depending on 0) of the atmosphere in thedirection 0, 4). Specifically,

Ldiff(0, 4)) = Lsc exp(-T t f)

+ 1 d E8 exp(-r 4 )P6 ,e exp(r - Tdiff)dT.4rm

(4)

In Eq. (4), L Tsfc is the radiance reflected upwardfrom the surface toward dV. L sfc is attenuatedexponentially along the slant optical path T Tsfc be-tween the Earth's surface and dV. Most of dV's 0, 4)directions do not intersect the Earth's surface, soL T sfc = 0 for them.

We assume that the surface behaves as a Lamber-tian reflector with reflectance rrf0, making L T sfc = E,cos(0o)exp(-t r sf)rsf/. T sfc iS the slant optical pathof direct sunlight (at zenith angle 00, azimuth 4 0)

down to the surface, and is itself a function of 0, 4.


Es cos(OO) is the fraction of the solar irradiance that isprojected normally onto the Lambertian surface.

Returning to Eq. (4), the Tlla are the slant opticalpaths defined by the direction of the sun (00, 40) andpoints along the direction 0, + from dV. Physically,exp(-Te() describes direct sunlight's attenuation be-tween the top of the atmosphere and those pointsalong T as we move toward dV. The phase functionP,, describes the scattering efficiency of direct sun-light toward dV from the direction 0, A. Note thatall the scattering phase functions P('P) used here aref(i) because the ratio of aerosol to molecular scatter-ing changes along the slant T (the ratio changesbecause altitude changes along the slant paths).

In addition, two different aerosol phase functionsare used, and these are based on Mie scatteringcalculations over two polydispersions, the Deirmend-jian haze-L and haze-M size spectra.22 Because weassume that the incident solar radiance L4 is unpolar-ized, the intensity distribution function for aerosolscattering (at the single-scattering level) is given byRef. 22's Eq. (5-9). Our molecular-scattering phasefunction follows the Rayleigh-scattering criterion de-scribed in Eq. (4-37) of Ref. 22. When Tdi = 900,

both multiple scattering and our aerosol size spectrawill considerably reduce skylight's linear polarizationfrom its theoretical upper limit of 100% (see, forexample, McCartney's discussion22 on pp. 195-196and 231-233).

Performance of the Second-Order Model

For a wide range of parameters, the second-orderscattering model does in fact generate a near-horizonradiance maximum. The model's maxima changeboth in angular breadth and elevation with changesin solar elevation, normal optical depth, and relativeazimuth. Bear in mind that this simple model willnot reproduce all the features found in more detailedmodels, such as those by Prasad et al.4 and Dave.'7However, near the horizon and at large scatteringangles from the sun, the second-order model is oftenmore realistic.23 Thus, for our purposes, its strengthsoutweigh its weaknesses. We consider some specificcases.

Unlike Prasad et al.'s model, the second-ordermodel produces azimuth-independent values of thezenith radiance, as shown in Fig. 4. As far as ispossible in Fig. 4, we have matched model parameterswith those used in Prasad et al.'s Fig. 8. Note alsothat, for this solar elevation and aerosol optical depth,the second-order model predicts that, at view eleva-tions of <200, the clear sky will be brighter in thebackward direction (rel = 180°) than at right anglesto the Sun's azimuth ()rel = 90°). Although Dave'smodel behaves similarly near the horizon (see hismodel's behavior in Prasad et al. 's Fig. 8), Prasad etal. predict just the opposite. Which claim is morerealistic?

Although no single set of measurements can beconclusive, Fig. 5 seems to support the second-ordermodel's claims about the azimuthal behavior of clear-

0.01

0 45- x 475 n

Es- 2044

W/(m 2Am)rsfc= . Co= 0.97

-5 0 5 15 25' 35' 45' 55. 65- 75' 85' 90-95view zenith anele 0

- Iv

Fig. 4. Second-order scattering model's meridional radiance pro-files for a combined molecular and aerosol atmosphere. Modelparameters were chosen to closely match those of Fig. 8 in Ref.4. Multiplying the scaled radiances [which include a factor of 1/(Trsr)] by the solar spectral irradiance Es.ie at wavelength X convertsthem into absolute radiances.

sky radiances near the horizon. In Fig. 5, we seehow relative radiances vary azimuthally across Plate38, averaged over view elevation angles of 1°-20°.Now, using the same second-order model parametersthat produce the best fit to the measured meridionalradiance profile, we compare the azimuthal measure-ments with the model's behavior. The agreementbetween measurements and the model is quite goodover the range (4 re = 90°-122° (the curves' rmsdifference is 0.0011). We do not know if the trend ofLsky increasing with (4 rel continues, but at least weknow that L(rei) does not decrease monotonicallybeyond rel = 900. As we look toward higher rel

near the horizon in Plate 38, we are approaching the

90' 95' 100- 105' 110, 115' 120- 125'relative azimuth, Vre

Fig. 5. Comparison of the azimuthal variation of clear-sky radi-ances across Plate 38 with those predicted by the second-ordermodel, with the same model parameters that generate the best-fitmeridional radiance profile of Fig. 6. T is the molecular normaloptical depth, Taer is the aerosol normal optical depth, and Haer isthe aerosol scale height. A constant molecular scale height of 8.4km and a single-scattering albedo so of 0.97 are used throughoutthis paper.


aerosol backscattering maximum. Thus both na-ture and simple physical reasoning suggest that thesecond-order model is correct in saying that, near thehorizon, L(rel = 180) > L(+rel = 900).

The second-order model also produces the solaraureole, as evidenced by the local maximum near O, =450 in Fig. 4's L(4+rei = 00) curve. However, com-pared with aureoles produced by Prasad et al.'s andDave's models, the second-order model appears consis-tently to underestimate the aureole's magnitude.These underestimates are not surprising because wehave considered only two scatterings and thus havedisallowed higher-order scatterings into directionsnear the direct beam. In an atmosphere with apronounced forward-scattering peak, we can expectthis assumption to produce errors.

However, for Tdir > 200, the radiances of the threemodels are nearly equal. For example, at the zenith(Tdir = 450 for O0 = 45), the second-order model yieldsa radiance that is 23% larger than that predicted byDave's 7 model (see his Fig. 12) and - 6% smallerthan the average zenith radiance in Prasad et al. 's 4

Fig. 8. At 0,, = 800 and 4arel = 30 (dir = 43.5°), thesecond-order model differs from the other two modelsby 9%, with the signs of the differences being thesame as at the zenith (all comparisons have beencorrected for spectral variations in E). These andother comparisons of the models are the basis for myabove statement that we can usually ignore higher-order scattering contributions to L~ky profiles if welook at comparatively large scattering angles rdirfrom the Sun.

Comparisons of Measured and ModeledRadiance Profiles

When we compare our measured radiance profileswith those of the model, the agreement is usuallyquite good. By careful (and realistic) choice of modelparameters, we can reduce the rms difference be-tween the measured and the modeled radiance pro-files to <0.04 for radiance profiles normalized bytheir respective maxima.

In Figs. 6-10 we have plotted the measured meridi-onal radiance profiles of Plates 37-40 and of Plate 41in Ref. 24. With the exception of Fig. 10, each profileis an azimuthal average across the entire scene (inFig. 10, the simulated azimuthal FOV is 0.5°), and atypical standard deviation for these azimuthal radi-ance averages is 0.02. All view elevation and zenithangles are measured with respect to the astronomicalhorizon. The rms differences between modeled andmeasured radiances are limited to astronomical eleva-tion angles above 0.25°-0.75° in order to reduce thehighly variable contributions of surface reflectance(in Figs. 6 and 10, local topography rises 0.5° abovethe astronomical horizon).

Paired with each measured radiance profile is abest-fit profile from the second-order model. Ourbest-fit criterion attempts to minimize rms differ-ences between the modeled and the measured profileswhile simultaneously requiring the model's maxima

20- 20°

18-X 18°

16- measured radiances 16-

14~- Xi modelradiances 1414 12'

0 12' 12'

, 10' 10°

' -. 8°.5 8'

4! 6- RMS difference = 0.0131 (elev. 2 0.75') 60= 63' 0,= 106', rfc=04, 4

2 Tm= 015, Tar = 0035, Haer= 2 km2' 2'

-2- . . -2-0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

normalized radiance

Fig. 6. Comparison of measured and modeled clear-sky radiancesfor the Bald Eagle Mountain scene at University Park, Pa., 1530GMT on 5 February 1987 (see Plate 38).

to occur within 1 of the observations' maxima.The parameters 0 (solar zenith angle), re (meanrelative azimuth in the scene), and rfc (Lambertiansurface reflectance) either are known or can be mea-sured from the digitized images. Because Tol (mo-lecular normal optical depth) and u70 (single-scatter-ing albedo) are taken to be the constants 0.15 and0.97, respectively, the variable parameters are theaerosol normal optical depth Taer and the aerosol scaleheight Haer. Our best-fit values for these two param-eters are quite plausible (see, for example, pp. 224-225 of Ref. 21 and Fig. 3 of Ref. 25).

The best fit occurs in Fig. 10 (rms difference0.00378), and the poorest fit is seen in Fig. 9 (rmsdifference 0.035). The model's performance in Fig.10 is especially reassuring because here we have themost detailed time, elevation, and azimuth data (seePlate 37). These details mean that 0 and 4(r areknown quite accurately, thus reducing uncertaintiesin the fitting algorithm.

12'

10'

00

8-

6'

4-

2'

0.6

- measured radiances

-B-- model radiances

RMS difference = 0.0248 (elev. 2 0.25')0o= 77° re= 100', rC= 0.85,

Tmo= 015, Tr = 0.0725, Haer= 2 kmae.e

_.

0.7 0.8 0.9 1 1.1 1.2 1.3

normalized radiance1.4

12'

10'

8'

4'

0.

1.5

Fig. 7. Comparison of measured and modeled radiances for anAntarctic clear-sky scene (see Plate 39). Although the snow isbrighter than the sky, radiances are still normalized here by a localmaximum occurring above the horizon.


I I . . . I . . . . I . . . . I . . . . I . . I . I . . . I I .

measured radiances|

RMS difference = 0.00378 (elev. 0.75°)

0= 480, 0rel= 118, rHf= 0.25,tm0 0.15, taer~ 0.0925, Haer~ 1 km

_ I 1 ,, ,,,, I, e. . . . . . . . . . . . . . ...

10'

8'

0aO 6'

a 4'it:

0'

-2 c

10-; 10-

18'

6-

8'0a;

6'.

a 4-

2'

0'

a.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

normalized radiance

Fig. 8. Comparison of measured and modeled radiances for aclear-sky scene off Hamilton, Bermuda, at 1530 GMT on 2 June1988 (see Plate 40).

In Fig. 9 we know 00 and krel with nearly the sameaccuracy that we did in Fig. 10, yet Fig. 9 has thelargest rms difference of the five scenes analyzed.Here the measurements and the model disagreechiefly near the horizon. The same error patternoccurs in Figs. 6 and 8, implying that the second-order model's parameterizing of surface contribu-tions to Lky is the source of the problem. Neverthe-less, the model does accurately reproduce the positionand general features of the near-horizon maximum.Because Prasad et al.'s and Dave's models also useLambertian lower boundaries, they may have nobetter success in accurately describing the sky radi-ance distributions seen above real topography.

What general conclusions can we draw about thebehavior of the near-horizon maximum? First, notethat the maximum's angular elevation tends to in-crease with decreasing solar elevation and increasingoptical depth. For example, in Fig. 6 the Sun is wellabove the horizon (00 = 63°), the atmosphere is veryclear (aer = 0.035), and the maximum occurs at 2.10

0'

-2'- 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1

10'

5'

6'

2'

0'

-2'

normalized radiance

Fig. 10. Comparison of measured and modeled radiances for aclear-sky scene at University Park, Pa., at 1605 GMT on 6 October1992 (see Plate 37).

elevation. Conversely, in Fig. 9 the Sun has nearlyset (00 = 860) and, although visibility is still good(Taer = 0.0725), the radiance maximum's elevationhas increased to 8.30. Second, the maximum's angu-lar breadth increases with increasing optical depth.In Fig. 11 we have combined the measured radianceprofiles from Figs. 6, 8, and 10. In order of increas-ing aerosol normal optical depth are the Bald EagleMountain sky of 5 February 1987 (Taer = 0.035, Plate38), the University Park sky of 6 October 1992(Taer = 0.0925, Plate 37), and the Bermuda sky(Taer = 0.245, Plate 40). We can easily see in Fig. 11that as aerosol loading increases, the radiance maxi-mum broadens and becomes more poorly defined.

Using the Near-Horizon Maximum as a QuantitativeRemote-Sensing Tool

Clearly the near-horizon radiance maximum dependson many variables, including solar elevation, atmo-spheric and surface absorptivity, and aerosol and

12'

18'

16'

14'

a 12-

0 0a0-

a~ 8'I 6-

.

4- _2-

-2- .0.2

- easured radiance

---i model radiances

RMS difference = 0.035 elev. 2 000= 86', l= 170-, rcS= 0a17

,Cm,= 0.15, r- =00725, Hr 2'

1.2k-75,

! Ia

IJ

., ... .... , I .. .. - - I.. .......

0.3 0.4 0.5 0.6 0.7 0.8

normalized radiance

18'

16'

-14'

12'

-10'

8-

-6-

-4-

-2-

- 0'

. ...9 ..1 . -2-0.9 1 1.1

Fig. 9. Comparison of measured and modeled radiances for aclear-sky scene on the Chesapeake Bay (North Beach, Md.) at 2300GMT on 24 March 1992 (see Plate 41, Ref. 24).

10'

5, 8-0

0

.= 6-

, 4-.S

0'

-2 ' . l . , I 0.2 0.3 0.4 0.5 0.6 0.7 0.8

normalized radiance0.9 1 1.1

12'

10'

8'

6'

4-

0'

-2'

Fig. 11. Comparison of measured clear-sky radiance profiles inatmospheres with different aerosol normal optical depths raer. SeePlate 37 (University Park), Plate 38 (Bald Eagle), and Plate 40(Bermuda) for the original photographs.


-measured radiances

model radiances

RMS difference = 0.0294 (elev. 2: 0.25-)o= 15 t= 50', rfc= 0.23,

rmoI= =O.laer=0.24

,Haer= km

. . . I I . . . . . . . . .

-

_

.. . . . . . . . . . . . . . , . . - - -I*.. . ,- I . . . . .

12- 12- 12-12-

I

. . . I a,.

molecular scale heights and normal optical depths.Can any useful, observable patterns be made toemerge from this welter of details? The answer isyes, provided that we constrain some of our variables.For example, Fig. 12 is a nomogram from the second-order model that shows the maximum's angularelevation as a function of aerosol normal opticaldepth (or turbidity) and solar elevation. (Here weuse the turbidity definition given by McCartney,22 p.206.) The nomogram is strictly valid only for fixedvalues of 4

>rely Haer, and rfc, although in practice thelatter two parameters have a much smaller effect onthe diagram's details than does the relative azimuth.In Fig. 12, as elsewhere, the molecular normal opticaldepth Tmo = 0.15 and the single-scattering albedowo = 0.97.

Figure 12 reiterates some points made above, includ-ing the fact that the near-horizon maximum consis-tently rises with increasing optical depth. The maxi-mum also tends to rise with decreasing solar elevation.To illustrate this trend, trace along a fixed opticaldepth from high to low Sun elevations. Note, how-ever, that at large aerosol optical depths (say,Toer = 0.3), the maximum will set slightly as the Sunmoves downward from very high elevations.

Aside from illustrating the behavior of the second-order model, Fig. 12 serves as a practical observingtool: we can estimate aerosol optical depth based onthe near-horizon maximum's elevation. For ex-ample, if you see a radiance maximum 3 above theantisolar horizon when the Sun's elevation is 210, thenormal aerosol optical depth is - 0.05 (turbidity- 1.33). Naturally such observations can be made

with a radiometer as well as with the naked eye.However, before seizing on Fig. 12 as a panacea for

measuring turbidity, an important caveat needs to beconsidered. The near-horizon maximum often isvisually subtle both because of its low contrast and

because its color and brightness changes are com-mingled (see Plate 38). Not surprisingly, the maxi-mum's contrast is lowest when the optical depth islargest and multiple scattering is most pronounced(see, for example, Plate 40). Thus, although themaximum might exist at, say, 150 elevation, it may beso broad and of such low contrast that a naked-eyeobserver cannot see it or may misidentify its position.Nevertheless, if used judiciously, Fig. 12 gives us away of turning a mere visual curiosity into a practicalremote-sensing tool.

Conclusions

One of our goals is a simple physical explanation ofthe near-horizon radiance maximum. Figures 13and 14 take two different tacks in attempting such anexplanation. In Fig. 13, model radiances for severalviewing directions near the horizon are plotted versusslant optical path length as measured from the ob-server. In other words, at each TsIant along Fig. 13'sordinate, the three curves tell us what clear-skyradiances we would receive from each direction 0 ifthe atmosphere's total optical depth were limited tothe particular T1ant. Obviously in the real atmo-sphere Tslant increases with 0,, so for each 0, in Fig. 13we have labeled the slant optical paths rf(0,) over theentire atmosphere and their corresponding clear-skyradiances Lf(0V).

Notice what happens in Fig. 13 if we limit the totaloptical depth of the atmosphere [i.e., if we examineL(e,) at T

alant < Tf ]. For very shallow atmospheres(say, T slant < 1.6 for O, = 87°-90°), radiance increasesmonotonically as we approach the horizon (0, = 900).For larger and more realistic values of the total slantoptical path (Tlant > 3.6), the integrated clear-skyradiance L(0,) is largest at intermediate elevationangles (0, = 88.5).

Why do we see this change from the shalloweratmosphere? As we thicken the atmosphere, in-

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integrated model radiances (W/(m2sr Ilm))

Fig. 12. Contours of the clear-sky radiance maximum's elevationas a function of solar elevation and aerosol normal optical depth.The observer is looking opposite the Sun (rel = 180), the surfaceLambertian reflectance r = 0.1, and the aerosol scale heightHaer = 1 km.

Fig. 13. Second-order model radiances integrated over slantoptical path Tslat at several view zenith angles 0,. The totaloptical path lengths rf(0,) are measured from the observer to theatmosphere's top and are paired with the corresponding clear-skyradiances Lf (0,).


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70' 72- 76- 80' 84- 88- 90-view zenith angle, 0,

Fig. 14. Second-order model's meridional radiance profiles atkrel = 180° in a nonabsorbing, purely molecular atmosphere withdifferent normal optical depths. In this atmosphere, the near-horizon radiance maximum does not exist for the two smalleroptical depths, but it does for the two larger ones.

creased scattering of solar radiance Ls reduces thedirect-beam source function Jdir at all altitudes.These reductions in Jdir are accompanied by corre-sponding increases in the multiple-scattering sourcefunction Jdiff. However, if we make T'slat large enough,at low altitudes the reductions in Jdir predominateover the cumulative increases in Jdiff. The net resultis that the total source function J(P, r) begins todecrease as we look along atmospheric paths verynear the horizon (i.e., along paths dominated bylow-altitude scattering). All other factors beingequal, the Tlgat contribution to the local maximumwill be most pronounced at the antisolar azimuthbecause TIalnt is largest for Jdir there. Of course, thescattering phase functions P(T) and surface reflec-tance rfc also affect the azimuthal behavior of themeridional radiance profiles.

Figure 14 offers further evidence of the role thatoptical depth plays in creating the near-horizon radi-ance maximum. As does Fig. 4, Fig. 14 shows usmeridional radiance profiles from the second-ordermodel, this time for a nonabsorbing, purely molecularatmosphere. (These curves have not been normal-ized by their respective maxima; multiplying each byEcale yields absolute radiances.) Now, however,rather than keeping the atmosphere fixed and vary-ing 4)rel, we fix Mrel at 1800 and vary the normal opticalthickness of a purely molecular atmosphere. Whenthe atmosphere is thin (Tmo1 = 0.05), radiances doindeed increase monotonically from zenith to horizon.Trebling optical depth to 0.15 still does not produce alocal radiance maximum, although radiances increasevery slowly within 20 of the horizon. However, if weincrease Tmol to 0.25 or 0.4, distinct local maximaappear at 1.750 and 4.50 elevation, respectively.Note too that as T'mol increases, multiple scatteringmakes the zenith progressively brighter, ultimatelyat the expense of the horizon's brightness.

Restricting ourselves to a molecular atmosphereemphasizes the fact that the near-horizon radiancemaximum does not require a highly anisotropic scat-tering medium. Although aerosols are not a prereq-uisite for the maximum's existence, they will usuallychange its details. For example, even in a slightlyhazy atmosphere, a broad, intense solar aureole willdominate the horizon sky in the vicinity of a low sun.Because w = 1.0 in Figure 14, the graph alsoindicates that atmospheric absorption is not neededto produce the near-horizon radiance maximum.Although highly absorbing aerosols will darken thehorizon sky (and the zenith), we do not need to invokethem to account for what we see.

Why does the clear daytime sky often have anear-horizon radiance maximum? A preliminary an-swer is that our atmosphere is just right: it absorbsvery little in the visible, and its optical path lengthincreases monotonically with decreasing view eleva-tion angle (thus increasing total scattering along ourline of sight). At the time, attenuation of directsunlight reduces the source function J(T, T) when welook very near the horizon. For many combinationsof normal optical depth and sun position, a subtle, yetdiscernible, brightness maximum results just abovethe horizon.

This work was supported by National ScienceFoundation grant number ATM-8917596. AlistairFraser and Craig Bohren of Penn State have providedinvaluable guidance and argument. Michael Churmaof Penn State assembled the indispensable pairing ofsky photographs and radiometer data analyzed in Fig.2. Stephen Mango and colleagues at the U.S. NavalResearch Laboratory's Washington, D.C. Center forAdvanced Space Sensing have provided generoussupport of this project, as has the U.S. Naval Acad-emy Research Council.

References and Notes1. We use radiance and brightness as synonyms in this paper,

while recognizing that they are not linearly related. Forexample, see G. Wyszecki and W. S. Stiles, Color Science:Concepts and Methods, Quantitative Data and Formulae(Wiley, New York, 1982), pp. 259 and 495.

2. R. Perez, J. Michalsky, and R. Seals, "Modeling sky luminanceangular distribution for real sky conditions: experimentalevaluation of existing algorithms," J. Illum. Eng. Soc. 21,84-92 (1992).

3. F. M. F. Siala, M. A. Rosen, and F. C. Hooper, "Models for thedirectional distribution of the diffuse sky radiance," J. Sol.Energy Eng. 112, 102-109 (1990).

4. C. R. Prasad, A. K. Inamdar, and P. Venkatesh, "Computationof diffuse solar radiation," Sol. Energy 39,521-532 (1987).

5. J. V. Dave, "A direct solution of the spherical harmonicsapproximation to the radiative transfer equation for an arbi-trary solar elevation. Part : theory," J. Atmos. Sci. 32,790-798(1975).

6. For basic discussions of optical path length's effect on skyradiance (in a purely molecular atmosphere) see C. F. Bohrenand A. B. Fraser, "Colors of the sky," Phys. Teach. 23,267-272 (1985) and Ref. 7.

7. C. F. Bohren, "Multiple scattering of light and some of its


observable consequences," Am. J. Phys. 55, 524-533 (1987).8. G. Zibordi and K. J. Voss, "Geometrical and spectral distribu-

tion of sky radiance: comparison between simulations andfield measurements," Remote Sensing Environ. 27, 343-358(1989).

9. R. L. Lee, Jr., "What are 'all the colors of the rainbow'?" Appl.Opt. 30, 3401-3407, 3545 (1991).

10. D. K. Lynch and P. Schwartz, "Intensity profile of the 22°halo," J. Opt. Soc. Am. A 2, 584-589 (1985).

11. A. Deepak and R. R. Adams, "Photography and photographic-photometry of the solar aureole," Appl. Opt. 22, 1646-1654(1983).

12. L. J. B. McArthur and J. E. Hay, "A technique for mapping thedistribution of diffuse solar radiation over the sky hemi-sphere," J. Appl. Meteorol. 20, 421-429 (1981).

13. M. A. Rosen, "The angular distribution of diffuse sky radiance:an assessment of the effects of haze," J. Sol. Energy Eng. 113,200-205 (1991).

14. A. W. Harrison, "Directional sky luminance versus cloud coverand solar position," Sol. Energy 46, 13-19 (1991).

15. F. C. Hooper, A. P. Brunger, and C. S. Chan, "A clear skymodel of diffuse sky radiance," J. Sol. Energy Eng. 109, 9-14(1987).

16. C. G. Justus and M. V. Paris, "A model for solar spectral

irradiance and radiance at the bottom and top of a cloudlessatmosphere," J. Cli. Appl. Meteorol. 24, 193-205 (1985).

17. J. V. Dave, "Extensive datasets of the diffuse radiation inrealistic atmospheric models with aerosols and common absorb-ing gases," Sol. Energy 21, 361-369 (1978).

18. R. L. Lee, Jr., "Colorimetric calibration of a video digitizingsystem: algorithm and applications," Col. Res. Appl. 13,180-186 (1988).

19. A Photo Research PR-704 spectroradiometer with a nominal0.50 FOV was used.

20. D. K. Lynch, "Step brightness changes of distant mountainridges and their perception," Appl. Opt. 30,3508-3513 (1991).

21. C. F. Bohren and A. B. Fraser, "At what altitude does thehorizon cease to be visible?" Am. J. Phys. 54, 222-227 (1986).

22. E. J. McCartney, Optics of the Atmosphere: Scattering byMolecules and Particles (Wiley, New York, 1976), pp. 136-138.

23. As in the other models' simulations, all our radiance profilesare monochromatic. The wavelength X used here is 475 nm, atypical dominant wavelength for the clear sky. This wave-length determines both the solar spectral irradiance and theangular scattering phase function for aerosols.

24. R. L. Lee, Jr., "Twilight and daytime colors of the clear sky,"Appl. Opt. 33, 4629-4638 (1994).

25. G. E. Shaw, "Sun photometry," Bull. Am. Meteorol. Soc. 64,4-10 (1983).


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Horizon brightness revisited: measurements and a model of clear-sky radiances