Zenith luminance and sky luminance distributions for daylighting calculations

Energy and Buildings, 6 (1984) 283 - 291

283

Zenith Luminance and Sky Luminance Distributions for Daylighting Calculations M. KARAYEL, M. NAVVAB, E. NE'EMAN and S. SELKOWITZ Energy Efficient Buildings Program, Lawrence Berkeley Laboratory, University of California, Berkeley, CA 94720 (U.S.A.)

SUMMARY We derive an equation for zenith luminance as a function o f turbidity and solar altitude based on analysis o f large quantities of luminance and illuminance data measured in San Francisco, California, between September 1979 and August 1982. Using only average turbidity values to predict hourly zenith luminance as a function o f solar altitude can produce large errors. We compare the equation derived from our data, which is valid over a wide range o f turbidity and solar altitude, to other published models. We also compare the relationship between horizontal illuminance and zenith luminance from the clear sky and conclude that, when ideal clear days are compared, this relationship is similar to earlier work based on measurements in European climates. Finally, we compare our sky luminance distribution measurements to previous published luminance distributions using the diffusion indicatrix. Our results are intended to help improve daylight availability prediction techniques and define additional requirements for data collection.

INTRODUCTION Daylight availability studies are motivated by the realization that sky conditions differ in different localities and t h a t data are often gathered on only a few solar and climatological variables. As a result, statistical methods are often used to derive key parameters. Extrapolation from limited measured data seems to be a c o m m o n practice. Zenith luminance is integral to m a n y daylight availability calculations. We present results of a study of zenith luminance in which systematic automatic and manual measurements were made for three years in San 0378-7788/84/$3.00

Francisco, California. We compare our results with previous work on this subject and examine the validity of extrapolating results from one locality to others. The collected data are analyzed statistically to determine parameters and functional relationships. Because our data acquisition is automatic, we had to distinguish sky conditions based on criteria derived solely from solar illuminance and irradiance measurements, rather than subjective human judgment. We generally used the criterion that the sky is clear as long as the direct normal irradiance is greater than 200 watts/m 2. This criterion can be met when there are thin layers of high clouds; to exclude these conditions (when solar altitude is higher than 20°), we also require that the ratio of diffuse irradiance to global irradiance be less than 1/3. In this paper we analyze only the clear sky data that satisfy these criteria. Note t h a t the analyzed data may include d a t a from partly cloudy days when the sun was shining.

DATA COLLECTION The data-gathering station is located on the roof of the Pacific Gas and Electric (PG&E) building in d o w n t o w n San Francisco, approximately 140 m (450 ft) above sea level. Measurements of eight irradiance and illuminance variables have been taken there every 15 minutes for more than four years. At the same location, measurements of zenith luminance have been taken every 15 minutes for more than three years. The station instrumentation consists of seven illuminance sensors {photometers) and two irradiance sensors (pyronameters) t h a t measure global and diffuse components. For both diffuse illuminance and diffuse irradiance measurements, the sun is blocked by a shadow band. Four of © Elsevier Sequoia/Printed in The Netherlands

284 the p h o t o m e t e r s measure vertical illuminances for each major compass orientation. In order to eliminate ground reflectance, a circular black h o n e y c o m b plane with a metal ring around its edge is m o u n t e d below the vertical sensors. Zenith luminance is measured by a modified p h o t o m e t e r . A cylindrical steradian shade assembly is m o u n t e d on this sensor, permitting a 15 ° field o f view of the zenith region (see Fig. 1). For clear conditions, the results are corrected to th e standard 1 ° field of view used for zenith luminance measurements. This is done by integrating the sky luminance distribution for a standard CIE clear sky for both o f these angles and comparing the results. The area-corrected error was less than 1% for solar altitudes less than 80 °. Because the m a x i m u m solar altitude at our location is a b o u t 76 °, we believe these results are acceptable.

Fig. 1. Photometric and radiometric instrumentation on the roof of the Pacific Gas and Electric building in downtown San Francisco. Additional manual measurements were made o f the sky luminance distribution with a 1 ° luminance sensor and a camera fitted with a fisheye orthographic lens. These measurements were carried out in two locations: on t o p o f the PG&E building and at t he Environmental Design building on t he University of California campus in Berkeley, across t he San Francisco Bay. The two locations are a b o u t 20 kilometers apart. These manual measurements were made t o obtain typical sky luminance distributions and to test w he t he r the CIE clear sky models are appropriate for sky luminance patterns in this region.

To describe our data fully, we must mention San Francisco's unique microclimate. The city is located at 38 ° north latitude and 123 ° longitude on a peninsula between the Pacific Ocean to the west and a large bay to the east. San Francisco is a naturally airconditioned city with cool, pleasant summers and mild winters due to the interplay of westerly air masses and cool ocean currents from the north. Some of the most unusual aspects of this city's microclimate are the sea fog and low clouds in early mornings and the variability of clouds within different parts of the city. Our automatically gathered data include representative samples of all the sky conditions c o m m o n to San Francisco.

RESULTS

Zenith luminance equation Zenith luminance is usually expressed as a function of solar altitude, ~'s, and Linke's turbidity factor, TL. Several authors have developed different analytical expressions for clear sky zenith luminance and for Linke's turbidity factor. We calculated the turbidity factor directly from horizontal irradiance and illuminance measurements. Details of these calculations are presented in the authors' paper on t urbi di t y [1 ]. Because we t o o k measurements over a wide range of turbidity and for higher solar altitudes than have most ot her investigators, their equations proved inadequate to fit our data. Table 1 lists some proposed equations and their limitations. To fit our data, we simplified the general form used by Dogniaux and Liebelt by omitting the linear dependence on turbidity, and refit the parameters: Lz = (1.376TL -- 1.81) tan ~,s + 0.38

(1)

where Lz = zenith luminance (kcd/m2), ~,,-solar altitude (degrees), and TL = L i n k e ' s t urbi di t y factor. The parameters of eqn. (1) were estimated from a set of 1 1 9 0 0 measurements by nonlinear least squares. The standard deviation for this fit was 0.637 kcd/ m 2. This equation is valid for 2°~< ~/~ ~< 75 °. Figure 2 shows the dependence of zenith luminance on turbidity and solar altitude based on eqn. (1). We derived other, more complicated equations to predict zenith luminance. But in general the

285 TABLE 1 Equations for zenith luminance* Author

Equation

Limits TL

Dogniaux Kittler Krochmann Liebelt Gusev LBL

Lz Lz Lz Lz Lz Lz

= = = = = =

( 1 . 2 3 4 T L - - 0 . 2 5 2 ) t a n ~'s + 0 . 1 1 2 T L - - 0 . 0 1 6 9 0.3 + 3.0 t a n ~'s 0.1 + 0 . 0 6 3 7 s + (Ts(Ts - - 3 0 ) / 1 0 0 0 ) e °'°3~(~/s-~8) ( 1 . 3 4 T L - - 3 . 4 6 ) t a n 7s + 0 . 1 T L + 0.9 0.9 + 6.5 t a n ~'s ( 1 . 3 7 6 T L - - 1 . 8 1 ) t a n 7s + 0 . 3 8

7s 7s~< 65 ° ~'s < 65°

2.25 ~< T L ~< 3 . 2 5 T L = 2.75 3 ~< T L < 7.5 5 ~< T L ~< 6.5 1.5 < T L <~ 8.5

7s ~< 65 o 2° < ~,s < 76 °

* I n t h e s e e q u a t i o n s : L z = z e n i t h l u m i n a n c e ( k c d / m Z ) ; 7s = s o l a r a l t i t u d e ( d e g r e e s ) ; a n d T L = L i n k e ' s t u r b i d i t y fact o r . E x c e p t f o r t h e last e q u a t i o n , t h i s T a b l e is b a s e d o n ref. 5.

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5(a) through 5(c) display the actual measurements and predicted zenith luminance from three randomly selected consecutive days in August 1980. Hours with missing data did n o t meet the clear sky criteria. The agreement between our predictions and the measured data is quite good for this limited random sample, We tested how well the equations in Table 1 fit our measured data for a wide range of turbidity. Statistical goodness of fit tests indicated that our equation is an unbiased estimator of the zenith luminance values. On the average, the equation given by Dogniaux overestimated our data by 1.024 k c d / m 2, and that given by Liebelt underestimated our data by 0.679 kcd/m 2 over the range for which each equation is valid. Furthermore, even

Fig. 5(a)(b)(c). Comparison of measured zenith luminance with predicted values using eqn. (1) from three r a n d o m l y selected consecutive s u m m e r days. Hours with missing data did not m e e t clear day criterion. Predicted data +, measured data ©.

when this bias was removed our equation predicted zenith luminance with less deviation*. This is not surprising, since our equation was derived from this data set. We expect to test our predictive equation at other locations to determine if it provides a good fit to data elsewhere. Since turbidity is measured in few locations, it is often necessary to estimate zenith luminance values based on only limited turbidity data. In order to examine the variability in zenith luminance data, we averaged the data over turbidities for each solar altitude. *Standard deviation based on our e q u a t i o n was 0.798 k c d / m 2, whereas the standard deviations according to the equations by Dogniaux and Liebelt were 1.078 and 1.041 k c d / m 2, respectively.

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The standard deviation was also computed. Figure 6 shows this average value plus or minus one and two standard deviations, (OL~ {%)). This Figure shows t h a t for a given solar altitude, zenith luminance can vary considerably from the calculated average value. With low altitudes, the relative difference is fairly high, but the absolute values remain low. This is not the case with altitudes above 60 °, where the relative difference as well as the absolute values are both quite high. In a location for which turbidity data are not available, a calculation of zenith luminance using only the dependence on solar altitude at an average turbidity value can thus introduce a potentially large relative error.

Zenith luminance and diffuse illuminance Several authors have suggested formulae for the relationship between zenith luminance and horizontal illuminance from the sky for clear sky conditions [2]. Measuring these variables simultaneously makes it possible to examine this empirical relationship. Our data indicate that the ratio of horizontal illuminance to zenith luminance for clear sky first rises and then drops with solar altitude, as expected. More significantly, our average data suggest values for the relative zenith luminance, RL (i.e., the ratio of diffuse horizontal illuminance to zenith luminance), of the clear sky that are lower than those derived from both the CIE sky luminance distribution according to Kittler and according to Gusev as reported by Aydinli [2]. We derived the

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following equation for the average value of the relative zenith luminance: RL =

5.42 - - 4 . 1 6 % + 10.87%e-2"2~s

(2)

(% is given in radians). For a comparison of our fitted equation to that of Kittler or Gusev, see Fig. 7. R L c a n also be derived from clear sky luminance distribution data. We measured clear sky luminance distributions for San Francisco for a limited number of very clear days using manual survey techniques (see next section). R L estimated by integrating our measured luminance distributions closely matches that calculated from the CIE distribution for all solar altitudes above 20 °, but is lower for very low solar altitudes (see 'ideal clear sky' curve in Fig. 7). This analysis suggests that the difference between our average equation and others' may be attributed partly to the fact that our cumulative data set represents 'average' rather than 'ideal' clear sky conditions. Aydinli suggests a seven-term power-series expansion for the relative zenith luminance: R L = ~i6=0Ai~fsi [2]. The power-series coefficients (Ai) are given in Table 2. Because power-series equations of this type do not increase one's understanding of the empirical relationships, we prefer to display average measured data with confidence limits (Fig. 8).

288 TABLE 2 Coefficients of relative luminance equations Kittler A0 A1 A2 A3 A4 A5 A6

6.9731 4.2496 --8.5375 --8.6088 1.9848 --1.6222 4.7823

Gusev 7.6752 6.1096 --5.9344 --1.6018 3.8082 --3.3126 1.0343

E-2 E-4 E-5 E-6 E-8 E-11

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This Figure s h o w s t h e r a t i o o f diffuse h o r i z o n tal i l l u m i n a n c e t o zenith l u m i n a n c e as m e a sured, a n d displays t h e s c a t t e r as +1 s t a n d a r d d e v i a t i o n (+ OnL (%)).

S k y luminance distribution T h e s k y l u m i n a n c e d i s t r i b u t i o n , including t h a t o f t h e zenith a n d c i r c u m s o l a r regions, was e x a m i n e d t h r o u g h m a n u a l surveys o f t h e s k y o n clear days. F o r t h e s e m e a s u r e m e n t s a visual e v a l u a t i o n was u s e d as t h e c r i t e r i o n f o r a clear s k y . T h e m e a s u r e m e n t s w e r e t a k e n f o r e v e r y 10 ° a z i m u t h s t a r t i n g at t h e sun a z i m u t h a n d f o r every 10 ° o f altitude. T h e s k y was assumed to be symmetric with respect to the sun a z i m u t h ; t h e r e f o r e , o n l y h a l f o f t h e s k y was s u r v e y e d , so t h e survey was c o m p l e t e d in less t h a n 10 m i n u t e s , avoiding significant changes in sun p o s i t i o n . Figure 9 shows a

s a m p l e p l o t o f clear s k y l u m i n a n c e distribut i o n (sun a l t i t u d e = 60 °) derived f r o m analysis o f o u r m e a s u r e d data. In o r d e r to c o m p a r e o u r m e a s u r e m e n t s o f t h e s k y l u m i n a n c e d i s t r i b u t i o n t o o t h e r s ' , we h a v e u s e d t h e c o n c e p t o f a d i f f u s i o n indicatrix developed by Kittler [4]. The diffusion indicatrix m o d e l s t h e d e p e n d e n c e o f s k y luminance distribution on atmospheric s c a t t e r i n g p h e n o m e n a . E q u a t i o n (3) shows t h e d i f f u s i o n indicatrix:

F(O, '7) = P I ( 1 - - e -p" sec(0)) X (1 + P2(e -3~ - - 0 . 0 0 9 ) + P3 cos27) (3) w h e r e P1, P2, P3, a n d P4 are p a r a m e t e r s estim a t e d f r o m m e a s u r e d d a t a , 0 = angle b e t w e e n t h e zenith a n d t h e s k y e l e m e n t P, a n d ~ = s c a t t e r i n g angle b e t w e e n t h e s k y e l e m e n t a n d t h e sun (see Fig. 10). T h e s c a t t e r i n g angle ~/ can be c a l c u l a t e d using t h e e q u a t i o n : = c o s - l ( s i n 0 sin 0 s cos ~ + cos 0 cos 0s)

(4)

w h e r e 0 = angle b e t w e e n t h e s k y e l e m e n t a n d t h e zenith, 0s = angle b e t w e e n t h e sun a n d t h e zenith, a n d ~ = a z i m u t h angle b e t w e e n t h e sun a z i m u t h a n d t h e s k y e l e m e n t a z i m u t h . T h e d a s h e d circular line in Fig. 10 s h o w s t h e locus o f p o i n t s in t h e s k y w h i c h have t h e s a m e ~/value.

289

Z

TABLE 3 Parameters of diffusion indicatrix equations Kittler P1*

1

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0.32

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Fig. 10. Definition of angles for calculation of the scattering angle 3'. The dashed line shows the locus of sky elements having the same 3' angle.

Based on eqn. (3) above, we estimated the parameters P~ through P4 for 172 data points from one clear day for each m o n t h of the year. The multiple correlation coefficients for these fits were high, indicating good fits to data. It was found, however, that the values of the parameters P1, P2, P3, and/)4 derived from analysis of our measured data had considerable scatter around the values given by Kittler and by Gusev in ref. 3. Table 3 compares the values of the P coefficients suggested by Kittler and Gusev to those derived from best fits of our data to the diffusion indicatrix equation. We compared results of our measurements to Kittler's and Gusev's equations using several different approaches. In Figs. l l ( a ) and (b) we plot values of the indicatrix derived from our measured data against Kittler's and Gusev's equations for the indicatrix. All values have been normalized for a scattering angle of ~f = 90 ° by dividing each measured and predicted value by the value of the predicted diffusion indicatrix at 3' = 90 °. The agreement is generally good. If we plot our data with our values for P~ to P4 and normalize to our indicatrix value at 7 = 90 °,

--0.28 (--1.7, 0.73) 0.59

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LBL

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we obtain an excellent fit (Fig. l l ( c ) ) . The fit is unbiased in that the sign of the errors is randomly scattered with respect to scattering angle. This suggests that the form of the equation is suitable for analyzing our data. However, because atmospheric turbidity affects the shape of the diffusion indicatrix, different parameters (PI-P4) may be appropriate for different locations. More detailed measurements and analysis to relate the four coefficients to atmospheric properties will be necessary to understand the basis for the differences between the values for the P parameters derived from our measured data and Kittler's and Gusev's values.

CONCLUSIONS

Equations for zenith luminance of the clear sky proposed by Gusev, Kittler, Dogniaux, Liebelt, and Krochmann are all limited to solar altitudes below 65 ° and the range of turbidities given in Table 1. Based on an extensive measured data set, we have derived a new zenith luminance equation that extends zenith luminance predictions to solar altitudes u p to 76 ° and turbidities of 1.5 to 8.5. Given the diversity and quantity of our data, we believe that this equation is valid for a range of climatic conditions. We derived an equation for relative zenith luminance (i.e., ratio of horizontal illuminance to zenith luminance) based on our

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Fig. l l ( a ) . Normalized diffusion indicatrix vs. scattering angle. Measured data (+) vs. Kittler's equation ( ) for solar altitude ( 9 0 - 0s)= 65 °. Parameters for Kittler's equation are: Pl = 1.0, P2 = 10, P3 = 0.45, P4 -- 0.32.

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Fig. 11(c). Normalized diffusion indicatrix vs. scattering angle. Measured data (+) vs. LBL equation ( ) for solar altitude ( 9 0 - 0s) = 65 °. Parameters for LBL equation are: Pl = 0.92, P2 = 12, P3 = 0.60, P4 = 0.29.

Fig. 11(b). Normalized diffusion indicatrix vs. scattering angle. Measured data (+) vs. Gusev's equation ( ) for solar altitude (90 -- 0s) = 65 °. Parameters for Gusev's equation are: PL = 1.0, P2 = 16, P3 = 0.3, P4 = 0.32.

measureddata. Our data suggest lower values than those based on standard CIE distributions. We believe that the differences occur partly because our extensive data set represents 'average' rather than ideal clear sky cond i t i o n s . F u r t h e r w o r k is n e e d e d t o c l a r i f y these differences. We have also analyzed limited sky luminance distribution measurements and evaluated a diffusion indicatrix that describes this d i s t r i b u t i o n as a f u n c t i o n o f t h e s u n - t o - s k y element angle and the zenith-to-sky-element a n g l e . T h e i l l u m i n a n c e l e v e l s o n a g i v e n surface may be predicted by appropriately integrating this equation. The parameters of t h i s e q u a t i o n , as g i v e n b y t h e C I E [ 3 ] , a r e strongly dependent on instantaneous atmospheric conditions. These parameters, as s h o w n in T a b l e 3, v a r y o v e r a f a i r l y w i d e range. However, the estimated average value o f e a c h p a r a m e t e r is in g e n e r a l a g r e e m e n t with the corresponding values proposed by Gusev and Kittler. Additional studies are required to determine direct relationships between turbidity and the parameters of the diffusion indicatrix.

291 ACKNOWLEDGEMENTS This w o r k was s u p p o r t e d b y t h e Assistant Secretary for Conservation and Renewable E n e r g y , Office o f Building E n e r g y R e s e a r c h a n d D e v e l o p m e n t , Building S y s t e m s Division of t h e U.S. D e p a r t m e n t of E n e r g y u n d e r Contract No. DE-AC03-76SF00098. Special t h a n k s are d u e t o R o b e r t Clear o f t h e Lighting S y s t e m s R e s e a r c h G r o u p a t L a w r e n c e B e r k e l e y L a b o r a t o r y f o r his m a n y h e l p f u l suggestions, to M o y a M e l o d y f o r her editorial assistance, a n d t o R u t h Williams f o r word-processing.

REFERENCES 1 M. Navvab, M. Karayel, E. Ne'eman and S. Selkowitz, Measurement and calculation of atmospher-

ic turbidity, Lawrence Berkeley Laboratory Report LBL 17727; Analysis of atmospheric turbidity for daylight calculations, Energy and Buildings, 6 (3) (1984) 293 - 303. 2 S. Aydinli, The Availability o f Solar Radiation and Daylight, 2nd Draft of Daylighting Report for CIE Technical Committee TC-4.2, January 1983. 3 CIE Technical Committee 4.2, Standardization of Luminance Distribution of Clear Skies, CIE Publication No. 22, 1973. 4 R. Kittler and L. Kittlerova, Navrh a Hodnotenie Denneho Osvetlenia, Vydavatel'stvo Technickej a Ekonomickej Literatury, Bratislava, 1968, 1975.

5 CIE Technical Committee 4.2, Circular Letter No. 3/82, CIE, March 1982. 6 P. R. Bevington, Data Reduction and Error Analysis for Physical Sciences, McGraw-Hill, New York, 1969. 7 C. Liebelt, Leuchtdichte und Strahldichtererteilung des Himmels, Dissertation, Lichttechnische Institut der Uni, Karlsruhe, 1978.