# SOLAR SPECTRAL IRRADIANCE IN THE VISIBLE AND INFRARED REGIONS

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Photochemistry and Phorobioiogy Val. 48. NO. 4, pp. 477-486, 1988 Printed in Great Britain. All rights reserved

0031-8655/88 $03.00+0.00 Copyright @ 1988 Pergaman Press plc

SOLAR SPECTRAL IRRADIANCE IN THE VISIBLE AND INFRARED REGIONS

ALEX E. S. GREEN'.* and SHUN-TIE CHAI' 'Department of Mechanical Engineering and *Department of Engineering Sciences, University of

Florida, Gainesville, FL 3261 1, USA

(Received 2 December 1987; accepled 16 May 1988)

Abstract-The analytical formulas previously developed for estimating the spectral irradiance reaching the ground in the ultraviolet are extended into the visible and infrared (35C3000 nm). This approach has two distinct features: (1) all physical inputs for calculating the direct irradiance are given in analytical form, and (2) the diffuse spectral irradiance (skylight) is calculated using dimensionless ratios which relate it to the direct irradiance. In common with other approaches, the global spectral irradiance for arbitrary ground reflectivity is calculated from the sum of the direct and diffuse spectral irradiances and a divisor which depends upon the ground and air spectral reflectivities. The global spectral irradiance on a tilted surface may also be calculated in terms of the above quantities and two angles. As in the case of the ultraviolet. the formulas presented are intended for photobiological applications.

1. INTRODUCTION

Early measurements and calculations of visible and infrared radiation (VIR)? reaching the ground are summarized in a number of specialized studies (Elsasser, 1942; Goody, 1964; Green and Wyatt, 1965; Kondratyev, 1969). Following the launching of Sputnik 1 in 1957, a tremendous increase of effort began which is mainly recorded in specialized reports by aerospace firms and national laboratories and in highly distilled form in the archival literature. The most comprehensive methodology which has emerged is now based upon the Air Force Geophysi- cal Laboratory (AFGL) atmospheric absorption line parameter compilations, (Rothman et al., 1982a,b) a data bank which described several hundred thou- sand molecular transitions. A model and computer

*To whom correspondence should be addressed. ?Abbreviations: D, the downward direct spectral irradiance; G, global irradiance; H, extraterrestrial solar spectral irradiance; I , , the absorption intensities; IR, infrared; K,, aerosol extinction coefficient; K,, ozone absorption coefficient; Kh, mixed gas absorption coef- ficient; M , the ratio of the diffuse irradiance at 0 = 0" to the direct irradiance at 0 = 0"; M,,, sun-earth distance; N . the day of the year (1-365); R , ground reflectivity; r , air reflectivity; S , diffuse sky irradiance; Y, the ratio of the diffuse irradiance to the diffuse; irradiance at 0 = 0"; T , , air scattering transmittance function; T2. aerosol scattering transmittance function; T3, ozone absorption transmittance function; T.,, aerosol absorption transmit- tance function; Ts. water vapor absorption transmittance function; T6. mixed gas absorption transmittance function; t,, small characteristic numbers; UV, ultraviolet; V, vis- ible; VIR, visible and infrared; W , wavenumber; W,, aerosol thickness; W,, the band centers; W,, total ozone thickness; Z, , the aerosol optical depth at 500 nm; z , the angle of incidence of the direct beam on the tilted surface; A , wavelength; p, cos 6; p,, generalized cosine function; T,, Rayleigh scattering optical depth; T ~ , aerosol scattering optical depth; T,. ozone absorption optical depth; T ~ , aero- sol absorption optical depth.

code FASCODE (Fast Atmospheric Signature Code, Smith et al., 1978) performs, with a major computer, line by line calculations calling upon these atlases. Such procedures are useful for the purposes of high resolution detection systems and remote sensing with lasers.

For moderate resolution purposes, a FORTRAN computer code LOWTRAN has been developed through several versions (Kneizyo et al., 1983). These programs and modest data sets (Perluissi and Maragoudakis, 1984), with large computers, provide estimates of atmospheric transmittance and radiance from 350 to 40 000 cm-I (0.25 to 28.5 pm) averaged over 20 cm-' in 5 cm-' steps. Their results are useful for many applications in atmospheric science, climatology and meteorology. For photobiologists, solar radiation and illumination engineers, and other applied scientists lower resolution atmosph- eric spectral transmittance and solar spectral irradiances reaching the ground are more useful (the forest vs the trees problem!). Such needs have been addressed by Leckner (1978), Bird and Rior- dan (1986) and others who have developed simpler numerical programs and smaller data sets, suitable for desk type microprocessors. In the present work we further simplify such applied or engineering type calculations in the 0.35 to 3 pm region by adapting some features of work previously developed (Green, 1966; Green et al., 1974; Shettle and Green, 1974; Green et a l . , 1975) to estimate UV radiation effects (Green et al., 1975; Calkin, 1982). The most recent form of this approach (Green et al. , 1975; Calkin, 1982). The most recent form of this approach (Green et al., 1980; Schippnick and Green, 1982; Green, 1983) has been published as a BASIC program (Bjorn and Murphy, 1985) for use by plant physiologists. The equations have also been used to estimate ultraviolet irradiation of human cornea, lens and retina (Hoover, 1986).

477

478 ALEX E. S. GREEN and SHUN-TIE CHAI

One distinct feature of this ultraviolet approach is that all physical inputs for calculating direct irradiance are given in analytical form. This pro- vides unambiguous inputs at all intermediate wave- lengths whereas tabular inputs are defined only at the given wavelengths. A second distinct feature is that the diffuse spectral irradiance (skylight) is calculated using dimensionless ratios which relate it to the direct irradiance. This simplifies estimating diffuse spectral irradiance which in the ultraviolet is usually comparable or greater in magnitude than the direct spectral irradiance. The ratio approach should also be useful in estimating visible and infra- red diffuse spectral irradiances which are usually much smaller than the direct spectral irradiance. In extending the analytic-ratio method to the visible and infrared (VIR) we draw upon the work of Bird and Riordan (1986) and Leckner (1978) and their sources (Neckel and Labs, 1981; Frohlich and Shaw, 1980; Hay and Davies, 1978; Dickinson and Cher- emisinoff, 1980) both for inputs and for validation.

2. DIRECT IRRADIANCE

Extending our previous work (Green, 1983), the downward direct spectral irradiance D(A, 8 ) as a function of wavelength (A) and zenith angle (8) at the ground is placed in the basic form

D(A, 0) = P ff~(A)DseTl T2T3T4TSTb (1) where p. = cos 8, &(A) is the extraterrestrial solar spectral irradiance at one astronomical unit, D,, is a sun-earth distance factor, T I , T2, T3, T4 , Ts and Th denote transmittance functions for, (1) air scat- tering, (2) aerosol scattering, (3) ozone absorption, (4) aerosol absorption, (5) water vapor absorption and (6) mixed gas absorption respectively. For i = 1 to 4 T, = exp -al, the absorbance a,(A,8) = ~,/p., where T, is the corresponding optical depth. For i = 5 and 6 the absorbances obey band model for- mulas (see sections 7 and 8). The symbols pI, p2 (= p4), p3, k5, and k6 which allow approximately for earth roundness denote

p.1 = [(F2 + W ( 1 + G)1"2 (2) where the t,'s are small numbers which depend upon the altitude distribution of the species. (Table 5a)

3. THE EXTRATERRESTRIAL SOLAR SPECTRAL IRRADIANCE

The extraterrestrial solar spectral irradiance H&) used here is essentially the 10 nm resolution version of the Neckel and Labs (1981) solar spec- trum as listed in Bird and Riordan's Table 2-1 (1986) for 122 wavelengths. To fit these data, we extend our U V approach in which we used a black body formula with a series of Gaussian modifiers to simu- late the Fraunhofer structure in the extraterrestrial solar spectral irradiance. Here we use an empirical distribution function which contains the Planck

function as a special case together with a series of Gaussian modifiers

[ 1 + xiAiexp(, - w)] (3) where KO = 2300 W/m2 km, q = 728.5 nm, a g = 4.8, and Po = 2.0. Note that for a 5000 K Planck function ciO = 5, P o = 1, q = c2/T = 28 779 nm where c2 is the second radiation constant and KO depends upon the solar area, emissivity, geometric factors and the first radiation constant. The Gaus- sian.modifier parameters are given in Table 1. Fig- ure 1 illustrates the analytic function and the exper-

Table 1. Gaussian modifiers

- 1. 2. 3. 4. 5 . 6. 7. 8. 9.

10.

386.640 391.138 410.280 456.000 672.941 827.330 837.350 934.950

1377.200 2669.790

Ai

-0.18841 -0.11838

0.22462 0.14190

- 0.02772 - 0.04920 -0.01382 -0.0805 1

-0.34589 0198830

c,

38.9360 2.6710 9.3381

15.4584 20.9260 80.6520 12.0810 19.8210

320.8600 320.1900

2100 r

0 I I I I 1

350 900 1450 ZOO0 2550 3100

WAVELENCTB (nm)

Figure 1. The extraterrestrial spectral irradiance. The curve is the analytic representation and the circles are representative points from the Neckel and Labs compi-

lation.

Solar spectral irradiance in the visible and infrared regions 419

WAVELENGTE (ad

Figure 2. Analytical representations of the optical depths for Rayleigh scattering (7,). aerosol scattering ( T ~ ) and absorption (TJ. and ozone absorption ( T ~ ) for T, = 0.3

and W , = 0.344 atm-cm.

imental data. At wavelengths longward of, 3 pm, thermal atmospheric radiation comes into play, the equations for radiative transfer become much more complex and direct photobiological effects are small.

4. RAYLEIGH OPTICAL DEPTH

To incorporate the Rayleigh scattering optical thickness over this extended range, we use a formula proposed previously (Green, 1983)

n ( X ) = K~(p/p~)(AtJA)'exp[al(ho/X)~l (4) but with parameters K1 = 0.1380, a, = 0.05022, At) = 500 nm and = 1013 mb. Figure 2 illustrates this function which is within the line width of the function used by Bird and Riordan and varies by about 1% from the function of Frohlich and Shaw (1980). The main purpose of this figure is to illus- trate the relative roles of the Rayleigh scattering vis- a-vis aerosol scattering and absorption and ozone absorption at various visible and infrared wave- lengths.

5. AEROSOL OPTICAL DEPTH

The aerosol optical depth is one of the most highly variable inputs needed for VIR calculations and measurement interpretation. Shettle and Fenn (1979) have proposed a series of models for the properties of aerosols of the lower atmosphere and the effects of humidity variations on these optical properties. They consider aerosols which are rep- resentative of those found in rural, urban, and mari- time air masses and in the troposphere at various humidities. Using model size distributions and com- plex indices of refraction they computed extinction coefficients for wavelengths between 0.2 and 40 pm.

For the purpose of this study between 0.35 and 3.1 pm, we concentrate our attention on their rural

model at 50% relative humidity. We take the Shettle-Fenn extinction coefficients computed for the 14 VIR wavelengths as input data and simplify and modify our previous formulas to

ke(h) = ke(An)(l + Pz) '{P2 + e x p [ ( A - W k l ) ( 5 )

(6)

7 2 = (1-r)W,k,(A) (7)

= r W, k,(A) (8)

r(A) = r2(h&)"z

where W , is the aerosol thickness, r = r(A) relates the absorptive to the scattering components, A. = 500 nm, P2 = -0.8174, A2 = 2475 nm, r2 = 0.067, a2 = -0.9369 and k&) = 0.1678.

The values of k,(X,) are fixed using the Shettle-Fenn diagrams for An = 500 nm at 50% relative humidity and 15 000 (particles/cm') particle number density. For different particle number den- sities one should proportion the parameter k,(Ao) accordingly. In this work we refer aerosol par- ameters to the aerosol optical depth at 500 nm using

T , = k, (A") W, (9) Thus using Eqs. 5-8, we may tie .*(A) and T'(X)

to the number T , obtainable with a sun photometer and the assignment of a few parameters P2, A2, r2, and a2 which depend somewhat upon the nature of the aerosols and the relative humidity (Shettle and Fenn, 1979). Aerosol optical depths 'r2(h) and T ~ ( A ) for a rural aerosol at a representative turbidity T , = 0.3 are shown in Fig. 2.

6. THE OZONE OPTICAL THICKNESS

In our UV studies, we consider the Hartley and Huggins bands which are strong ozone absorption bands in the ultraviolet from 180 to 345 nm. In the present work we allow for the Chappuis band which is a weak band from 440 to 740 nm. Here we approximately fit the ozone absorption coefficient data (Leckner, 1978) for the visible wavelengths with the double exponential

where K," = 0.1274 atm-cm, A, = 594.0 nm, A3 = 35.2 nm.

The ozone optical thickness is given by

T g = W&3(A) (111 where W, is the total ozone thickness. In general, W3 is variable averaging 0.38 cm at high latitudes and 0.24 cm towards the equator. The ozone optical depth vs wavelength for a representative ozone thickness is illustrated in Fig. 2.

7. WATER VAPOR ABSORPTION

Many studies have been published on the absorp- tion properties of water vapor and we refer the

480 ALEX E. S. GREEN and SHUN-TIE CHAI

Table 2a. The big bands of water vapor

1. 2. 3. 4. 5 . 6. 7. 8. 9.

3755.9 5331.9 7249.9 8807.0

1061 3.4 12151.3 13830.9 15348.0 16899.0

2662.5 1875.5 1379.3 1135.5 942.2 823.0 723.1 698.3 591.7

241000 .0 27800.0 21000.0

1498.0 877.0 46.0 50.0 16.6 5.0

1 0.52 0.62

1 1 1 1 1

0.40

Table 2b. The secondary bands of water vapor

1. 3657.1 2734.5 16600.0 7.0 2. 5234.9 1910.2 573.0 1 3. 6871.4 1455.3 1896.0 1 4. 9000.1 1111.1 66.0 1 5. 11032.4 906.4 103.0 1

reader to the general works on this topic (Herzberg, 1945; Elsasser, 1942; Goody, 1964; Kondratev, 1969). In the wavelength range of our interest, bands of increasing intensity lie at 600, 700, 800, 900, 1100, 1400, 1900 and 2662 nm. Various band models have been developed for water vapor trans- mittance when the monochromator band width is large compared to rotational line widths and spa- cings, but small compared to vibrational spacings. Here we use the phenomenological IR transmission model of Green [Green and Wyatt, 1965, Eqs. 8-4 (38)] in the simplified form proposed by Leckner (1978)

(12) and x5 = 0.2385, .E5 = 20.07, a5 = 0.45. Rather than use numerical tables for the spectral function k,(A), we incorporate the big band centers and big band intensities of Green and Mann (1987) into a function which represents the water absorption coefficient vs wavelength. Thus we write

where

w = lo7/& w, = 1071h, (14) y = 0.07389 and 6 = 0.005. Here wn denotes the wave numbers of the band centers and I , the absorp- tion intensities as given by the formulas of Green and Mann, Sn denotes the empirical factors which modify some of these band intensities...

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