# Direct solar radiation: spectrum and irradiance derived from sun-photometer measurements

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Direct solar radiation: spectrum and irradiance derivedfrom sun-photometer measurements

Wolfram Wobrock and Reiner Eiden

The continuous spectrum of the direct solar radiation from = 330 to 2690 nm, penetrating a cloudlessatmosphere and arriving on the earth surface, is determined by measuring the solar irradiance in ten selecteddiscrete spectral ranges defined by interference filters. Heretofore knowledge of the extraterrestrial solarspectrum has been required as well as of the transmittance functions to describe the spectral opticalproperties of the atmosphere. A set of appropriate and simple functions is given and discussed, which allowscalculation of the molecular, aerosol, oxygen, and ozone optical thicknesses. The influence of atmosphericwater vapor is considered through line by line calculations. The dominant and most fluctuating extinctionparameters are the aerosol optical thickness and the content of precipitable water vapor. These are obtainedby measurements with two sun photometers, developed according to the WMO recommendation. To test thederived solar spectrum at ground level the photometers are also run with nine broadband filters. The valuesobserved differ little from those obtained by integration of the deduced spectral irradiance. Furthermore,the integral value of the resulting entire spectrum agrees reasonably well with the total direct irradiancegained from actinometer measurements.

1. IntroductionVarious transmittance models are available to eval-

uate the direct solar irradiance in a cloudless turbidatmosphere (see, e.g., Bird and Hulstrom,1 Bird et al.,2Bird and Riordan,3 Kneizys et al.,4 and Leckner5).The dominant meteorological parameters for theseschemes are the optical thicknesses or the transmit-tance functions for aerosol particles and water vapor.Only actual measurements of the real atmosphere canyield appropriate information about such highly vari-able parameters. These can be obtained, e.g., with thehelp of high resolution spectral photometers, as usedby the WMO in its "background pollution monitoring"program at a large number of stations.6

Photometer measurements of direct solar irradiancein narrow wavelength intervals of a few nanometershalfwidth yield the magnitude of the instantaneousirradiance and thus, through the optical thickness rx,

Wolfram Wobrock is with Johann Wolfgang Goethe University-Frankfurt, Institute for Meteorology and Geophysics, Frankfurt a.M., Federal Republic of Germany; and Reiner Eiden is with Bay-reuth University, Institute for Geoscience, Bayreuth, Federal Re-public of Germany.

Received 19 August 1987.0003-6935/88/112253-08$02.00/0. 1988 Optical Society of America.

the amount of scattering and absorbing atmosphericconstituents between sun and detector. The relation-ship between observed irradiance EA and the extinc-tion properties of the atmosphere is given by the well-known Lambert-Beer law:

EX = E,OTA, (1)

where Tx is the transmittance function,TA = exp(-rAm), (2)

E,O is the spectral distribution of the known extrater-restrial solar radiation,7 and m is the relative air mass.8Thus the spectral measurements of Ex provide theoptical thickness i-r in the spectral interval X :1 dX, withdX being the halfwidth of the interference filter.

Based on the WMO/PMOD recommendation9 twophotometers were designed which allow determinationof Ex within ten narrow wavelength intervals limitedby interference filters and distributed over the spectralrange from 300 to 1900 nm. If we are able to deducefrom these discrete values of r an actual and continu-ous function for -r over the whole solar range (0.33-2.69 Mim), we are also able to determine the total actualdirect irradiance EB:

2.69EB = I E~dX.

0.33

Therefore, we have to know the spectral transmittancefunctions, i.e., the optical thickness, of all attenuatingconstituents of the atmosphere.

1 June 1988 / Vol. 27, No. 11 / APPLIED OPTICS 2253

To obtain this information we use known transmit-tance functions for ozone absorption, Rayleigh scatter-ing, and water vapor absorption. For oxygen absorp-tion and the aerosol extinction we propose newtransmittance functions. Also, we suggest a simpleanalytical formula to determine the aerosol opticalthickness based on photometer measurements at threedifferent wavelengths and an extension of this formulabeyond 2 Am.

Furthermore, in contrast to methods commonlyused,10 11 we evaluate the total atmospheric water va-por content from measurements with interference fil-ters within H20 absorption bands coupled with de-tailed line-by-line calculations. To check this methodwe compare the results to observations done with aspecially designed water vapor photometer, which wascalibrated using rawinsonde data.

Since in the solar spectrum the narrow 02 absorp-tion bands are prominent, a highly resolved band mod-el following Moskalenko1 2 is used.

To test the deduced spectrum of Ex (i.e., the trans-mittance function Tx), the two photometers are alsorun with broadband filters of several hundred nanome-ters transmission range. Finally, the integral of theresulting spectrum of Ex is compared to the total directirradiance gained from actinometer observations.11. Theoretical Background

Due to the additivity of the optical thicknesses of theatmospheric constituents Trxj, Lambert-Beer's law forthe transmittance of direct solar irradiance can bewritten as

E = ExO exp -ZA m ). (3)

10o-

t.,a I

10 -

3 4 5 6 7 8 9 103

wavelength (nm)Fig. 1. Wavelength dependence of the aerosol optical thickness T,adetermined by Eq. (7) with measurements at X = 415,576,867, 1023,and 1645 n: - - -,5/15/82,12:33 GMT + 1 h, visibility -12 km; -. - . -,5/13/82,9:03 GMT + 1 h, visibility ;35 km,.....,3/6/82,12:12 GMT+ 1 h, visibility >>35 km; the - curve results from Angstrom'sformula [Eq. (5)] for the observation of 5/13/82, 9:03 GMT + 1 h,using only the measured optical thickness at X = 415 and 867 nm.

in space and time; besides water vapor they are respon-sible for the principal extinction of solar radiation in acloudfree atmosphere. It is, therefore, more reason-able to derive Tx,0 directly from measurements insteadof assuming an aerosol particle size distribution, avertical profile, and a chemical composition of theparticles.

A widely used empirical function for the aerosoloptical thickness is the spectral formula of Ang-strom 6 ' 1 7:

A = ,

The scattering of electromagnetic radiation iscaused mainly by aerosol particles and air molecules(Rayleigh scattering), and the absorption is caused bywater vapor, ozone, oxygen, and again aerosol parti-cles. The scattering and absorption processes by airmolecules and aerosol particles are described by Mie'stheory.1 3"4 The theory allows a continuous wave-length-dependent calculation of the optical thicknessof the Rayleigh atmosphere TX,R and of the aerosoloptical thickness TX,0.

The mixing ratio of the main atmospheric constitu-ents N2 and 02 is practically constant in the lower 100km of the atmosphere, and their molecule numberconcentration can be determined for typical seasonalair pressure profiles. This allows one to calculate theRayleigh-optical thickness TX,R without further as-sumptions using Mie's theory. The result has beenapproximated by Frohlich and Shaw,15 who proposedthe following expression:

TXR = A(-B+cX+D/).For the choice of A, B, C, and D see Frohlich andShaw.15 The quantity A, for example, considers thealtitude of the observation point and the seasonalshape of the pressure profile as a function of latitude.

Aerosol particles on the other side vary considerably

(5)This formula can also be derived from Mie's theory

(see van de Hulst 4) by assuming Junge's power func-tion' 8 for the aerosol particle size distribution:

dN(r)/dr r(,*+I) (6)with aerosol particle radius r and a constant exponentv*. To determine a and Al in Eq. (5) Tx,0 has to bemeasured at least for two different wavelengths. Thusphotometer measurements at three different wave-lengths, e.g., at 415, 576, and 867 nm, allow a threefoldcalculation of a and A3. This procedure, however, gen-erally results in three different values for a and Al, sincethe assumption of a constant exponent v* in the aerosolparticle size distribution is often incorrect.

To remedy this insufficiency we propose a new for-mula that allows a more flexible spectral description of'rea:

TA a = /(aX2 + b + c). (7)In Fig. 1 this function r,a is plotted for three differ-

ent measured turbidities, i.e., visibility conditions. Incontrast to Eq. (5) the formula (7) for Tx,a allows acurvature on a log-log plot, especially the anomaly inthe short wavelength range, as observed by Quenzel19and Eiden.20 The shape of the curves in Fig. 1 can beverified by Mie calculations.2'

2254 APPLIED OPTICS / Vol. 27, No. 11 / 1 June 1988

l. - 1.-

\ - x

2

Water vapor, 02, 03, C0 2 , and N20 are selectiveabsorbing gases. Their absorption spectra or bandsconsist of numerous absorption lines. The Lorentzformula22 describes the shape of a single line by meansof the absorption coefficient KL(v):

KL(v) = Koa /[(V - V0)2 + a'], (8)

where v = 1/X is the wavenumber, vo is the line position,ajL is the halfwidth of the line in wave numbers, and K 0is given by S = 7raLKo, with S being the line intensity.The AFCR Laboratories 2 3 offer a compilation of theseatmospheric absorption line parameters for atmo-spheric absorbers. Out of the huge number of absorp-tion lines listed there, we only use those forming thestrong absorption bands from 705 to 740, 800 to 845,888 to 990, 1070 to 1220, 1370 to 1535, and 1725 to 1900nm.

The optical thickness of water vapor at a singlewavelength X is then given by

TA,H20 = KL(v)WH20 (9)

with WH2 O being the absorbing mass in cm ppw (precip-itable water). Since WH2 O is a temporal and spatialstrongly fluctuating quantity, we determine the pre-cipitable water directly from measurements. The op-tical thicknesses of Rayleigh scattering and aerosolextinction can be calculated using Eqs. (4) and (7)within the spectral range of a water vapor absorptionband. Thus we can calculate iteratively the content ofWH20 from:

J= EAOFxSA

X exp[(-TX,. - rA,R - KLW 2O)m]dX, (10)

where Ji is the photometer signal with an interferencefilter centered in a H20 absorption band, (Xl,X2) is thetransmission range of the interference filter, Fx is thespectral transmittance curve of the interference filter,SA is the spectral sensitivity of the detector system, andwA20 is the precipitable water vapor reduced to airmass m = 1. The KL (V) value has to be calculatednumerically line by line in steps of dX 0.01 nmaccording to Eq. (8). Thus the iterative solution (reg-ula falsi) of Eq. (10) is time-consuming.

Another less fluctuating gaseous absorber is ozone,whose absorption coefficient K0 3 is already knownthrough laboratory measurements.24 The opticalthickness is

TAO 3 = KA,O3w03. (11)

The absorbing ozone mass WO 3 can be determinedthrough photometer measurements in the spectralrange from 550 to 600 nm (see Sec. III) with known TX,Rand Tx,a-

Finally, we also consider 02, which has narrow butstrong absorption bands near 690, 760, and 1260 nm.Assuming a constant amount of 02 (4.45 X 1024 mole-cules/cm2), the transmittance function T, averaged

Table 1. Oxygen-Absorption Coefficients Km as a function of Wavelengthdue to the Method of Moskalenko [Eq.'(12)]

A(nm) K2 X(nm) -02

686 0. 1254 0.0073687 0.3618 1255 0.0123688 0.0974 1256 0.0006689 0.1632 1257 0.0195690 0.1032 1258 0.0048691 0.0727 1259 0.0250692 0.0345 1260 0.0320693 0.0189 1261 0.0465694 0.0036 1262 0.0745

1263 0.0523758 0. 1264 0.1054759 0.2601 1265 0.0597760 1.9001 1266 0.0999761 0.9596 1267 0.0392762 0.4845 1268 0.2245763 0.8640 1269 0.1968764 0.5421 1270 0.0454765 0.3594 1271 0.0417766 0.2154 1272 0.0423767 0.1019 1273 0.0284768 0.0454 1274 0.0484769 0.0360 1275 0.0149770 0.0060 1276 0.0348

1277 0.02271278 0.02101279 0.01291280 0.01321281 0.0240

over dX = 1-nm intervals, can be evaluated by a line-by-line calculation based on the updated AFGL ab-sorption line parameters compilation.2 5 To obtain anabsorption coefficient that is independent of the airmass, we use a transmittance function according to theformula of Moskalenkol2:

TA,2 = exp[-KM0(X)m ), (12)with KO2 being the absorption coefficient resultingfrom the method of Moskalenko. Significant is theintroduction of an exponent m, = 0.57 for the relativeair mass m. Therefore, it is possible to use theseabsorption coefficients for any optical path length.The absorption coefficient KT*(X) is listed in Table I.The oxygen optical thickness TX,02 may now be writtenas

T A,2 = Komm . (13)The transmittance according to Eq. (12) differs lessthan 3% from exact line-by-line calculations for airmasses between m = 1 and 5.

To verify the values for Ta, WH2 0 W0 3, and thetransmittance functions resulting from measurementsand calculations, we used two control methods:

(a) To check the evaluated solar spectrum piecewiseup to X = 1900 nm in 100-400-nm intervals we also runthe photometers with broadband filters.

(b) We measure the integrated value of the directsolar irradiance Ex with a Linke-Feussner actinome-ter.

The signal received by procedure (a) is equivalent to1 June 1988 / Vol. 27, No. 11 / APPLIED OPTICS 2255

1.0

T00.8

(a) (a)

0.0 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

wavelength (jim)(b)

Fig. 2. (a), (b) Spectral shape and position of the interference filtertransmittance TA relative to a direct solar spectrum Ex; the curve, - - -,represents the relative spectral sensitivity S' of the Si photodiodeand, - - , the relative spectral sensitivity S' of the Ge photodiode.

= JX2JB= EA\O S,FA\ exp(-7-xm)dX, (14)

where X1and X2 are cut on and cut off of the broadbandfilters.

The total direct irradiance EB-procedure (b)-isgiven by

p2.69EB = J ESdX. (15)

As the photometer measurements and the appliedtransmittance models only reach up to 1.9 Am, wecalculate the additional incident radiation from 1950to 2690 nm with the spectral transmittance modelLOWTRAN 3,26 using the measured quantity WH2O andan approximation for TX,0 according to Eq. (5) (see Sec.IV).Ill. Experimental Procedure

Two photometers have been developed following therecommendation of WMO/PMOD,9 particularly withrespect to the geometry of the diaphragm and theaperture. As the original WMO/PMOD photometer isequipped with only one immovable interference filter,

1 ' 1w 'v 1e3 1 '4 1 v l g 1 m6 1wavelength ( pm) .8 1.9

(b)Fig. 3. (a), (b) Spectral shape and position of the broadband filter

transmittance TA relative to a direct solar spectrum E.

we changed the construction so that both photometerscan be run with two interchangeable filter wheels.

The main filter wheel contains ten interference fil-ters whose transmittance functions and spectral posi-tions are shown in Figs. 2(a) and 2(b). The filterscentered at X = 415, 576, 867, 1023, and 1645 nm arepositioned outside water vapor absorption bands andare used to determine the optical thickness of aerosolparticles Tx,a. The interference filter at X = 1302 nm,however, has, due to a manufact...

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