Pergamon Solar Energy, Vol. 54, No. 4, pp. 219-226, 1995

Copyright © 1995 Elsevier Science Ltd Printed in the USA. All rights reserved

0038-092X/95 $9.50 + .00

0038-092X(94)00117-0

A o °.

NEW METHOD TO DETERMINE ANGSTROM'S TURBIDITY COEFFICIENT: ITS APPLICATION

FOR VALENCIA

J. M. PINAZO, J. CANADA* and J. V. Bosch** Department of Applied Thermodynamics and the *Department of Applied Physics,

Universidad Politrcnica de Valencia, P.O. Box 22012, 46071 Valencia, Spain

Abstract--Traditionally the .~ngstrrm turbidity coefficient has been derived from either spectral direct solar radiation measurements or broadband direct solar radiation and precipitable water measurements. The new method for calculating the ,~ngstrrm turbidity coefficient presented here is based on the ratio of direct solar radiation to global solar radiation on a horizontal surface and on the " C " model of Iqbal. For this method, it is not necessary to know the precipitable water and the ozone content of the atmosphere. The results of the present model are compared against those of a prior study using experimental data measured at Valencia during 1989-1990. Finally, a graphic method to calculate /3 for any location and standard meteorological conditions is presented.

1. INT RO DUCT I O N

Information on ,~ngs t r rm ' s turbidity coefficient /3, particularly on its variability for a given location, is fundamental because it is needed to characterize not only the direct but also the global and diffuse solar irradiance on any plane on a cloudless day. /3 is an essential parameter in the est imation of the expected max imum performance of a solar installation, in the calculation of the max imum potential in air condition- ing installations, in the level of photosynthetic activity efficacy, etc.

The experimental determinat ion of Angs t r0m ' s tur- bidity coefficient/3 and wavelength exponent c~ is ac- complished by the use of a spectral photometer and by observing the at tenuation of the direct solar irradi- ance at two particular wavelengths where the absorp- tion or scattering of the radiation is only due to aerosol (dust part icles) . The wavelengths h = 0.38 # m and

= 0.5 # m are generally used to get the opt imum optical thickness of the aerosol. Then, /3 and c~ in ,~ngs t r rm ' s Turbidity Formula (Iqbal, 1983) are cal- culated:

kax = /3~- a. (1)

This method can be simplified by assuming a = !.3. Values of a found in the literature range from 0.8 to 1.8, depending on the medium size of the aerosol particles (Leckner, 1978; Gueymard, 1989). These authors determine the absorpt ion and scattering coef- ficient at h = 1 #m, as indicated in the International Geophysical Year (1958) which recommends the use of an RG630 (old RG2) filter.

The method ment ioned above requires data from a spectro-photometer, or at least a phyrhel iometer with

several filters, which is available at few stations only. For this reason, another method is often used to obtain /3 based on all-wave direct solar irradiance (Louche et aL, 1987). Direct solar irradiance can be accurately determined with a pyrhel iometer or almost equally well with two pyranometers for global and diffuse solar irradiance and their subsequent mathematical transformation. This method is based on the model of direct solar irradiance for clear days proposed by Louche et al. ( 1987 ) . The model requires the estima- tion of the ozone content of the atmosphere depending on the t ime of the year, the water vapour content ex- pressed as the thickness of precipitable water which, in turn, is given by temperature, relative humidity, and atmospheric pressure at the ground. The above-stated method of es t imat ing/3 will be referred to as Method I in this article. Using this method, the/3-values were obtained from direct solar radiation data that had been measured at Valencia and presented in a previous pa- per (Cafiada et al., 1993).

The determinat ion of the turbidity coefficient /3 from global solar irradiance only is impossible be- cause the presence of aerosols substantially modifies the direct and diffuse components: but much less their sum, the global solar irradiance, as has been stated by several authors (Iqbal, 1983; Gueymard, 1989).

2 . THE NEW M E T H O D

The method propounded in this paper, referred to as Method II, is based on the est imation of the coeffi- cient K defined as the ratio of the direct solar irradi- ance upon a horizontal surface to the global solar irra- diance that reaches this same surface:

K = & / ~ = ( & - & ) / ~ . ( 2 )

* ISES member. It is easy to determine the coefficient K from measure- ments of the global and diffuse solar irradiance on a

219

220 J. M. PtNAZO, J. C A N A D A ,

0.25 - - e2 = 0°C

{q o.2 _ _ _ 0z = 25oc

oi, I ..... Oz-,ooc ~ . . . . . . . . . . . . . . . - ~ K=o8

. . . . . . . . . . . . . . . . . I o o5~ ~,=o.9 ! 0 ~ 7 ~ i ? ~ i - ~ - ~ ~ ~i?~i~T~?T~?Ti-~??

89.325 91.325 93.325 95.325 97.325 99.325 101.325

Pressure (KPa)

and J. V. BoscA

Fig. I. Variation of/3u with P as a function of 0~ and K (with tJ0 = 0.9, F~ = 0.84, ot = 1.3, and pg = 0.2).

0.25 - - 0Z= O*C

- - - - ~z = 25°C 0 . 2 ~

~ 018 ... . . Oz = 5°°c

m 0.1

" " L'-']L'L "S L'± "2 " - ' - - L "2 - ' L ",~ "

0 L 89.325 91.325 93.325 95.325 97.325 99.325 101.325

Pressure (KPa)

Fig. 2. Variation of ~I with P as a function of 0~ and 1, (with wp = 3 cm, ot = 1.3, and l = 0.3 cm NTP).

horizontal surface with the use of two pyranometers or f rom two consecut ive measurements with the same pyranometer first shaded, then not shaded.

Operat ing with the radiation model proposed by Iqbal (1983) , Gueymard ( 1 9 9 3 ) c o n c l u d e d that Iq- ba l ' s parameterizat ion Model " C " is one of the best currently available. This Model C leads us to the fol- lowing expression for K (see Appendix A ) :

K = (1 - p g p a ) / [ 1 + (0.817-..[0.5(1 - 3-r)

+ Fe(1 - T a / T a a ) l ) / T r T a ( l - - ma "1- ml°Z)] . (3 )

~-, as a funct ion of /3 and a was parameterized by M~ichler ( 1983 ):

T, = 0 .12445~ - 0.0162 + (1.003 - 0 .125~)

× e x p [ - / 3 m , ( 1 . 0 8 9 a + 0 .5123)] . (4)

~-~ = 1 - (1 - w0)(1 - m a + m a l ° 6 ) ( 1 - ~-,). (5)

Pa = 0.0685 + (1 -- F~)(1 - "ra/7-aa ). (6 )

7-~ and m a are given in Model C as:

% = e x p [ - 0 . 0 9 0 3 m ° 8 4 ( 1 + ma - m ] ° l ) ] , (7 )

m, = P / [ 1 0 1 3 2 5 ( c o s 0z + 0 .15(93.885 - 0,)-t253].

(8)

In the standard atmosphere, the relation between the altitude z ( in m) of the location above sea level and the a tmospheric pressure P ( in Pa) is

P = 101325 exp ( - 0 . 0 0 0 1 1 8 4 z ) . (9)

Equations (4 ) through (8 ) relate the above-defined K coefficient with the atmospheric pressure (or the altitude of the location above sea level) , the ground albedo (pg), the solar zenith angle (0z) and the coeffi- c ients /3 and a of Angstr6m, as well as the parameters Fc ( forward scat terance) and ~0 (s ingle scattering al= bedo) .

Using these equations we obtained an expression for /3 depending on five variables or parameters (see Appendix B) :

/3 = f(cx, 0z, pg, K, P ) . (10)

The solar zenith angle 8z is determined by the num- ber of the day in the year, solar hour, and latitude of the location. For the remaining variables and parame- ters, a sensitivity analysis is presented in the fol lowing paragraphs.

3. SENSITIVITY STUDY

A sensitivity study was carried out to observe the relationship between the .&ngstr6m turbidity coeffi- cient/3 and the variables pg (ground a lbedo) , P (a tmo- spheric pressure) , and the parameters a ( ,~ngs t r6m's wavelength exponent ) , w0 (s ingle scattering a lbedo) and Fc ( forward scat terance): all of which were ob- tained by Method II. Further, the dependence of /3 obtained by Method I on the variables P , Wp ( thickness of precipitable water) , l (ozone content of the atmo- sphere) , and the parameter a was analysed.

To observe the variation of fl obtained by Method II with respect to the different variables, condit ions were chosen with three zenith angles (Sz = 0 °, 25 °, and 50 °) and three values of the coefficient K (K = 0.7, 0.8, and 0.9) . When using Method I, the same

0.3

0.25 " ~ / 02 0115

. . . . . i . . . . . . . . . . .

0.05 ~ K=0.9

ol ~ - .............. ?.~?....?.?~?75?.?~7-.~-.:5:~.~--"~ 1,1 1.2 1.3 1.4 1.5 1,6 1,7 1,8 1,9

ot

Fig. 3. Var ia t ion o f / 3 i i w i th <x as a funct ion o f 8, and K (with w0 = 0.9, Fc = 0.84, P = 101325 Pa, and pg = 0.2).

- - e z = o .c

- - - ez = 25oc

.. . . . 0z = 50oc

,~ngstrt~m's turbidity coefficient 221

0,3

o . 2 5 ~ __~0z= ooc

0.2 - - - 0 z = 2 5 ° C

o.~5 . . . . . 0z = 5 ° ° c

01 2 "

0.05

0 1.1 1.2 1.3 1.4 1.5 1,6 1,7 1,8 1.9

o/

Fig. 4. Variation of ~1 with ot as a function of 0z and 1, (with Wp = 3 cm, P = 101325 Pa, and I = 0.3 cm NTP).

0.25 - - 0 z = O° C

0.2 - - . . . . . . . . ~ K=0 .7 - - - e z = 2 5 * c

/ o16 ..... o°c

0.1

0.05 ~ K = 0 . 9

0 .6 0.65 0.7 0.75 018 0.85 0.9 0.95

Fc

Fig. 6. Variation of ~ i with F~ as a function of Oz and K (with w0 = 0.9, a = 1.3, pg = 0.2, and P = 101325 Pa).

angles and the three direct solar irradiances (I , = 700, 800, and 900 W m -2) were chosen.

Figure 1 shows the variation of fl obtained by Method II with the atmospheric pressure. In this Fig- ure, the pressure range corresponds to a standard atmo- sphere between sea level and 1000 m altitude. The fol lowing values were taken as constant: pg = 0.2, w0 = 0.9, Fc = 0.84, and o~ = 1.3. Figure 2 shows the corresponding variation of B by Method I with the atmospheric pressure (wi th wp = 3 cm, a = 1.3, and l = 0.3 cm NTP) , in approximately the same condi- tions. From both figures we can conclude that: • The variation of fl with pressure P is similar in both

methods. • The variation of fl with P is independent of the

value of the coefficient K. • The variation is l inear with respect to P and also

with the locat ion 's altitude above sea level. • fl has a max im um variation of + 10% between the

value at sea level ( P = 101325 Pa) and the value at a level of 1000 meters (pressure of 89325 Pa) independent of the coefficient K and of the solar zenith angle 0~. In Valencia the average atmospheric pressure oscillates between 100900 and 102800 Pa. Figures 3 and 4 show the variat ion of fl with a for

both methods. It can be concluded that both methods present a similar variation. Figure 5 shows the varia- tion in the coefficient fl obtained by Method II in relat ion to the ground albedo pg (for a location at sea level, P = 101325 Pa) . From this figure it can be

observed that: the variation of _+0.1 in pg with respect to the standard value 0.2 produces a max imum devia- tion in fl obtained by Method II of 0.01; and when Method II is used, the variat ion of fl with the ground albedo pg is l inear and independent of the value of the coefficient K and of the zenith angle Oz.

Finally it should be pointed out that the variable pg can be easily obtained with the same measuring equipment because it is the ratio between the global solar irradiance reflected by the ground (upward) to the incident global solar irradiance on the horizontal surface (downward) . Thus, it can be est imated with the supplementary measurement of a pyranometer in a horizontal posit ion facing the ground. A mean value of pg can be used in the whole period of measurements due to its weak variation.

F rom Fig. 5 we can conc lude that ne i the r the a tmospher ic pressure nor the ground a lbedo appreci- ably affect the es t imat ion of the turbidi ty coeff icient f t . There fore fl can be ca lcu la ted for s tandard condi- t ions ( P = 101325 Pa and pg = 0 .2 ) . The error is smal l and falls wi th in the range of errors p roduced by the m e a s u r e m e n t of i r radiances wi th the pyrano- meters .

F igure 6 shows the var ia t ion of fl wi th Fc when Method II is used. We obse rve that fl shows the same t rend wi th respect to Fc as it does wi th respect to a but less s t rongly (see Fig. 4 ) . F igure 7 shows the var ia t ion of r , when measu red by M e t h o d II, wi th ~0. Smal l changes in w0 have a s ignif icant in-

I / 0.25 ~ ~ - - 0 = o °c

/3 o15 . . . . . . . . . . . . . . . Oz =,ooc

° 1 1 ~ . . . . . . . . . . . . . . . . . . . .

o.o5 ~_ _ _ ± _ _ = _ ~ K=O.9 l o ~ , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i : ; . . . . . . . . . i 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.26 0.3

Fig. 5. Variation of flH with pg as a function of /9, and K (with w0 = 0.9, Fc = 0.84, o~ = 1.3 and P = 101325 Pa).

0.15 ~ . . . . . . . . . • •

o.1 t . . . . . . . . . . . . " ~ . . . . . . . . . . . . . . . - -

0.6 0,65 0.7 0.75 0.8 0.55 0.9 0 9 5

60o

- - e z = o ° c

_ _ _ 0 z = 2 5 o c

. . . . . e z = 5 0 o c

Fig. 7. Variation of fin with ~o as a function of 0z and K (with Fc = 0.84, ot = 1.3, pg = 0.2, and P = 101325 Pa).

222 J. M. P1NAZO, J. CAIqADA, and J. V. BOSCA

0.3

0,25 .%

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.2

I n =700 W/rn 2

0.15

0.1 I , =800 W/m 2 i

o.o~ i =90o Wlm 2

- - e z = O°C

- - - e z = 25°C

. . . . . 0 z = 50°C

0.24 0.26 0.28 0.3 0,32 0.34 0.36 0.38

(crn NTP)

Fig. 8. Variation of ~1 with I as a function of 0~ and 1, (with wo = 3 cm, ot = 1.3, and P = 101325 Pa).

f luence on the /~-va lues , as happened with ct in bo th methods . As can be seen f rom Fig. 8, var ia t ions of the ozone layer th ickness do not have a s ignif icant impac t o n / 3 w h e n de te rmined by M e t h o d I, so that it is poss ib le to use s tandard values for this var iab le ( Iqba l , 1983) . Final ly , Fig. 9 shows that fl ob ta ined by M e t h o d I var ies s ignif icant ly with the th ickness o f prec ip i tab le water, s imi lar to its var ia t ion wi th F~, w h e n it is ca lcu la ted by M e t h o d II.

In summary, we conclude that: • Method II has a greater variability than Method I,

especially with w0. • Both methods have a large variability with a . • The thickness of precipitable water, w v , has the

same influence in Method I as F~ has in Method II. • The other variables do not have a significant impact

o n /~.

4 . G R A P H I C C A L C U L A T I O N B A S E D O N T H E

S I M P L I F I E D N E W M E T H O D

Appendix B shows the expression that permits cal- culating the value o f /3 as a function of the variables: solar zenith angle (0~), the ratio of n-radiances (coeffi- cient K) , atmospheric pressure ( P ) , ground albedo (pg), and certain parameters. These parameters are: a (Angs t r6m's wavelength exponent depending on the aerosol size distribution, which has a recommended value of 1.3 for natural a tmospheres) , F~ (forward scat- tering ratio which has a recommended value of 0.84),

0.3

025

0.2

0.15

0.1

0.05

_ _ OZ = O°C

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Oz = 25"C

..... 0z = 5o*c

~ ~ - ;,;d ......

1.5 2 2.5 3 3.5 4 4.5

Wp

Fig. 9. Variation of fll with Wp as a function of ez and 1, (with a = 1.3 and 1 = 0.3 cm NTP, and P = 101325 Pa).

0.300 ~ .

0.275 L " - - - - . . . . . " " - ~

0,225 == - _ - " " . " " .

o.loo - . . . . . . . . . . . . . . . . " ' - - . " - . " - . " " - ' . 0.075 L . . . . . . . - - - - . . . . . . . "" - - . _ _ " - . - " . - . . "

0.050 - ' - - ~ " " ~ ~ - ' ~ ' ~ 1

o.o25- - 7 - - _ _ " . . . . . . ----2".-- o o o ,

5 10 15 20 25 30 35 40 45 50 55 60 65 70

Oz Fig. 10. Abacus to obtain/3 graphically as a function of 0z

and K, using Method II.

and ~o (single scattering albedo, which has a recom- mended value of 0.9 for semi-urban sites).

As was discussed before, the model can be simpli- fied considerably by taking standard values ( those rec- ommended by Iqbal [1983]) and taking for the atmo- spheric pressure and the ground albedo the following values:

pg = 0.2, P = 101325 Pa.

This leads to a simplified mathematical expression for

13 =/~(Oz, K). ( 1 1 )

Thus, the only variable that needs accurate measure- ments is the coefficient K. Figure 10 graphically repre- sents the turbidity coefficient/3 versus 0z for different K values. It can be used as an abacus for est imating /3 with acceptable precision. It is important to point out that the stated relationship is valid for any place in the world.

5. V A L I D A T I O N O F T H E P R O P O S E D M O D E L

Upon finishing the validation of the proposed model (Method II) , the pa rame te r /3 was est imated

1000 I n 0 .300 V a l e n c i a 251111990 ~

750 ~ , ~ * i~ . / "~ 0 .225

~; -- ' ° ~ t 0,, ~ ..... . Ig

500 0 .150 ._~

250 0.075

0 0 .000 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

True So la r T ime

Fig. l l. The diurnal variation of 1,, lg, ld, and ~n(Ig , /d) in Valencia.

0.3

0.25

~" 0.2

~ 0.15

~ 0.1

0.05

.A, ngstr6m's turbidity coefficient

. / Y = 0.987 X + 0.0037 = • ~

p '= o 97ol .'3.~.

= " - o

o.o5 oJ o J5 0.2 0.25 0.3

l~= (Direct)

Fig. 12. ~'l(/g, la) versus /~l(/n)-

0.3

0.25

0.2 cI -g __ 0.15

~ 0.1

0.05

0

Y = 0.9293 X + 0.0027

tO2= 0.9510

| l

223

0 0.05 0.1 0.15 0.2 0.25 0.3

1~ t (Direct)

Fig. 14. /~II(l .) versus /~i(/n).

by the method based on the variation of direct solar irradiance (Method I) , and then it was compared with the results obtained from both procedures. The experi- mental data used in the present study were measured on the roof of a building at the Polytechnic University of Valencia outside the city and close to the Mediterra- nean Sea (altitude 20 m, latitude 39.48°N, longitude 0.38°W). The measurements were carried out over approximately 2 years (from April 1989 to December 1990). Measurements of global and diffuse solar irra- diance on a horizontal plane were made with two Ep- pley 8 - 4 8 pyranometers, one of which was equipped with a shadow-band to measure the diffuse irradiance. Measurements of direct solar irradiance were made with an Eppley NIP.

Due to the fact that Method I requires data con- taining the measurements of the dry temperature and the relative humidity, it was necessary to gather these variables from the Levante Meteorological Center which is an Official Meteorological Institution. The pressure data were also obtained from the Levante Meteorological Center separated by about 2 km from the Polytechnic University.

The ground albedo was fixed at 0.258 based on 196 measurements of the incident and reflected global solar irradiance in situ and taking a mean value as representative of our measuring station at the Valencia

Polytechnic University. Likewise, the ,~ngstriSm wavelength exponent a was assumed to have the stan- dard value of 1.3 which is usually taken for the aerosol size distribution of a natural atmosphere. With respect to the ozone content (necessary for Method I) , the values proposed by Iqbal (1983) as a function of the latitude and time of the year were accepted.

After sorting out all the collected data, choosing all the cloudless days, it was necessary to make individual graphic representations on a day-by-day basis of the global, diffuse, and direct solar radiation. The coeffi- cient ~ obtained by Method II was represented as well. As an example, Fig. 11 shows the day January the 25th, 1990. In this way, 511 measurements were col- lected and used to validate the model.

6. RESULTS AND DISCUSSION

The two Methods are compared in Figs. 12, 13, and 14. Figure 12 displays the /~-values calculated by Method I from measured global and diffuse solar irradiances versus the /~-values obtained from mea- sured direct solar irradiance. The B-values calculated using Method II are also shown. Figure 13 exhibits the/~-values calculated by Method II from measured global and diffuse solar irradiances versus the/~-val- ues obtained from measured direct solar irradiance.

= o o , , o x + o o o , , o. j . . °25 f , , , , . ~ . [ °'25 r + Ma, hoo .

.~ f p=00517 ,P'..:~ I ~ ~ 02f ~ . . 02 ~ ~° ' ~°15f° " ~ " " I 015r ~%'~ "~ " o . ~

J - ' ; o.1 .... . . . . . ol. ~ * ~ o ~

° . ° s I ~ = ~ n ' . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,05 1959 ~

jl~ I (Direct) Days 182 273 365

Fig. 13. /3,(lg, ld) versus ~(In). Fig. 15. Individual values of/~ at Valencia in 1989-1990.

224 J. M. PINAZO, J. CA/qADA, and J. V. BoscA

0.20

t3

0.15

0.10

0.05

0.00

Method II - = - Method I

/

2 3 4 5 6 7 8 9 10 11

Months

Fig. 16. Monthly average values of/3 at Valencia.

12

The /3-values calculated using Method I are also shown. Finally, Fig. 14 shows the fl-value calculated by Method II f rom measured direct and global solar irradiances versus the fl-values obtained from mea- sured direct solar irradiance. The /3-va lues calculated using Method I are also shown. As can be seen from these figures, both methods give a good fitting, and their deviat ions represent the normal daily variation of the turbidity at this location. In Fig. 15, the values of fl are presented as calculated by each method over the 2 years. This graph displays the variation of the coefficient fl as a function of the day of the year. One can observe a m i n i m u m in winter and a max imum in summer.

Figure 16 presents the average monthly values of ft. Table 1 contains these values with their respective standard deviations which represent the natural vari- ability of the turbidity. It can be seen that both meth- ods give values of fl very close to each other. The typical deviat ion obtained with the proposed Method II is lower than that obtained with Method I. Table 2 gives the average seasonal values of fl for each method.

Table 2. Average seasonal values of ~ll(lg, ld) and ill(1,)

Spring Summer Fall Winter

/311(lg, ld) 0.086 0.154 0.051 0.041 /31(I,) 0.080 0.163 0.052 0.035

ratio of the horizontal direct solar irradiance to the global horizontal solar irradiance.

2. The new calculation procedure established in Method II permits calculation of the value of fl without employing variables such as the thickness of precipitable water and the thickness of the ozone layer. Furthermore, it is not necessary to take mea- surements of the dry temperature and the relative humidity.

3. The ground albedo, pg, which can be measured with the same equipment used for measuring irradi- ance, has been incorporated in the model. It was found that a standard value pg = 0.2 introduces relatively small errors in fl and can be taken into account by a simple l inear correction.

4. In Method II, Fc, ~v0, and a are taken constant, in accordance with the literature. The fol lowing standard values are suggested for locations similar to Valencia: Fc = 0.84, or0 = 0.9, and a = 1.3.

5. The deviations produced with the new method are smaller than the deviations obtained when Method I is used.

6. For standard condit ions P = 101325 Pa and pg = 0.2, a graphic procedure was generated for ob- taining the turbidity coefficient fl f rom K and solar zenith angle Oz. This graphical method is valid for any location and is independent of latitude, longi- tude, temperature, etc.

7. Using the method proposed in this paper, a monthly distribution and a seasonal distribution of ,~ng- s t r6m's turbidity coefficient fl during the years 1989 and 1990 was obtained for Valencia.

7. CONCLUSIONS

1. A new method has been developed to est imate the turbidity coefficient of ,~ngstr6m, /3, f rom K, the

Eo

N O M E N C L A T U R E

Eccentricity correction factor of the solar orbit (d imens ionless )

Table 1. Mouthy mean values of/3.(lg, ld) and/31(1,) at Valencia with their standard deviations and number of measurements

Standard deviation Standard deviation Number of /3II(/2, /d) X 103 X 103 /31(In) X 103 X 103 measurements

January 39 5 32 7 26 February 43 12 38 15 29 March 0 April 49 10 39 14 35 May 93 23 87 25 67 June 114 17 115 18 28 July 159 35 167 33 102 August 176 53 181 50 52 September 113 41 128 42 40 October 69 38 71 36 30 November 46 15 46 20 102 December 0

F~

Idm

dr

I. In

Isc

kax

K

m a

P l

0.30

Wp

z Ot

h

Pa

P g

7"a

7"aa

7"r

7" 0

Tw

7"g

~z

3i

Angstrrm's turbidity coefficient

Forward scatterance or fraction of the ra- fin diation scattered in the forward half-space i l l ( 1 . )

(dimensionless) Total diffuse solar irradiance on a hori- /~l(Ig, I a )

zontal surface ( W m -2) Diffuse solar irradiance due to scattering 3 n ( I o )

by aerosols ( W m 2) Diffuse irradiance due to multiple rettec- 3i~(lg, ld) tions between the earth surface and the atmosphere ( W m -2) Diffuse solar irradiance due to Rayleigh scattering ( W m -2) Direct normal solar irradiance ( W m 2) Direct solar irradiance on a horizontal sur- face (Wm 2; beam irradiance) Solar constant, 1367 Wm -2 Global solar irradiance on a horizontal surface ( W m -2) Extinction coefficient due to scattering and absorption by aerosols (dimen- sionless ) Ratio of In to lg Relative optical air mass at a given pres- sure (dimensionless) Atmospheric pressure (Pa) Ozone content of the atmosphere (cm NTP) Single scattering albedo or ratio of scat- tering coefficient to extinction (scattering plus absorption) coefficient of the aerosol Thickness of precipitable water (cm) Height above ground (m) Wavelength exponent in ~,ngstrrm's tur- bidity formula (dimensionless) ~,ngstrrm turbidity coefficient (dimen- sionless) Wavelength (/zm) Albedo of the cloudless sky atmosphere (dimensionless; atmospheric backscat- terance) Albedo of the ground (dimensionless) Transmitance of the atmosphere with re- spect to aerosol scattering (dimen- sionless Transmitance of the atmosphere with re- spect to aerosol sionless Transmitance of the spect to molecular sionless

absorption (dimen-

atmosphere with re- scattering (dimen-

Transmitance of the atmosphere with re- spect to absorption by ozone (dimen- sionless) Transmitance of the atmosphere with re- spect to absorption by water vapor (di- mensionless) Transmitance of the atmosphere with re- spect to absorption by uniformly mixed gases (dimensionless) Solar zenith angle (deg)

calculated by Method I

225

fl calculated by Method II 3 obtained by Method I from direct solar irradiance data fl obtained by Method I from global and diffuse solar irradiance data 3 obtained by Method II from direct and global solar irradiance data fl obtained by Method II from global and diffuse solar irradiance data

R E F E R E N C E S

Cafiada, J., Pinazo, J. M., and BoscL J. V., Determination of Angstrrm's turbidity coefficient at Valencia, R e n e w a b l e E n e r g y 3, 621-626 (1993).

Gueymard, C., A two-band model for the calculation of clear sky solar irradiance, illuminance, and photosynthetically active radiation at the earth's surface, S o l a r E n e r g y 43, 253-265 (1989).

Gueymard, C., Critical analysis and performance assessment of clear sky solar irradiance models using theoretical and measured data, S o l a r E n e r g y 51, 121-138 (1993).

I n t e r n a t i o n a l g e o p h y s i c a l y e a r i n s t r u c t i o n m a n u a l , part VI, Radiation Measurements and Instruments, Pergamon Press, Oxford (1958).

Iqbal, M., A n i n t r o d u c t i o n to s o l a r rad ia t ion , Academic Press, Toronto (1983).

Leckner, B., The spectral distribution of solar radiation at the earth's surface: Elements of a model, S o l a r E n e r g y 20, 143-150 (1978).

Louche, A., Maurel, M~, Simonot, G., Peri, O., and Iqbal, M., Determination of Angstrrm's turbidity coefficient from direct total irradiance measurements, S o l a r E n e r g y 38, 89-96 (1987).

M~ichler, M. A., Parameterization of solar irradiation under clean skies, M.A.Sc Thesis, University of British Colum- bia, Vancouver, Canada (1983).

A P P E N D I X A

In the parameterization Model C of Iqbal (1983) the global solar irradiance on a horizontal surface is given by:

lg = lnCOS 0z + ld = InCOS 0z + ldr + Iaa + /am,

and the direct solar irradiance is given by:

1. = 0.9751Eol~7"orgT-wrar r.

The three components of diffuse solar irradiance are

Ida = 0.79FoE0/~,ccos 0zToT-g7"aaTw(l - - T a / r a a ) /

(1 - ma + m~°2),

lar = 0 . 3 9 5 E o l ~ c o s O~rorgrwr .~ ( 1 - r ~ ) / ( 1 - m~ + m~°2),

/am = p g p a ( l , c o s 0z + /dr + laa)/(l -- PgPa).

With the help of these expressions, eqn (2) leads to eqn (3)

K = lb / lg = (lg - l d ) / lg = (Ig -- la~ -- Ida -- ldm)/Ig.

2 2 6

A P P E N D I X B

T h e e x p l i c i t e x p r e s s i o n f o r / 3 is g i v e n by :

/3 = - ( l n [ D - 0 . 1 2 4 4 5 a + 0 . 0 1 6 2 ) /

( 1 . 0 0 3 - 0 . 1 2 5 a ) ] ) / ( m , ( 1 . 0 8 9 a + 0 . 5 1 2 3 ) )

w h e r e

D = ( T J T ~ a ) ( I -- (1 -- W0)(1 - - m~ + mJa'06)) /

( ! - - ( Y , / V , , ) ( l - - W 0 ) ( I -- m., + m'~°6 ) ) ,

T . I T ~ = E + F ,

J. M . P1NAZO, J . C A N A D A , a n d J. V. B o s c h

a n d

E = - ( l - p g ( 1 . 0 6 8 5 - Fo)

+ K ( F c B - 1 ) ) / ( 2 ( 1 - F c ) p g ) ,

F = [ E 2 + K B ( 0 . 5 ( I - r , ) + F ~ ) / ( ( I - F ¢ ) p g ) ] '+2,

B = 0 . 8 1 / ( % ( 1 - m a + mJam) ) .