analytical characterization of spectral actinic flux and spectral irradiance in the middle...

Photochemistry and Photobiology Vol. 35, pp. 89 to 101, 1982 Printed in Great Britain. All rights reserved

003 1-8655j82/010089- 13$03.00/0 Copyright 0 1982 Pergamon Press Ltd

ANALYTICAL CHARACTERIZATION OF SPECTRAL ACTINIC FLUX AND SPECTRAL IRRADIANCE

IN THE MIDDLE ULTRAVIOLET

P. F. SCHIPPNICK and A. E. S. GREEN University of Florida. Gainesville, Florida 3261 1, USA

(Received 30 October 1980; uccepted 30 June 1981)

Abstract-We present an analytic characterization of upward and downward diffuse spectral irradiance for the wavelength range 280-380 nm, solar zenith angle range from 0 to 86 , altitude range from 0 to 5 km and for non-zero surface albedo. This work is a modification and extension of the previous work of Green, Cross and Smith based upon the radiative transfer calculations of Braslau, Dave and Halpern. Guided by these irradiance systematics we develop an analytic characterization of diffuse spectral scalar irradiance or actinic flux also broken down into upward and downward components for the above wavelengths, solar zenith angles and altitudes, for non-zero surface albedo utilizing the actinic flux calculations of Peterson.

1. INTRODUCTION

There is at present a considerable need for an inex- pensive and accurate representation of UV radiation in the atmosphere and at the ground occasioned by the present concern over atmospheric pollution and possible climatic effects. Pertaining to problems of tropospheric photodissociation rates (and smog for- mation) the global ‘actinic’ flux (also called scalar irradiance or omnidirectional flux [Smith and Wilson, 19721) is more relevant. The primary goal of the present work was to find an analytic characterization of actinic flux (the term in general use in the photo- chemical community).

Recently Green et a / . (GCS, 1980) presented an analytic characterization of skylight (the diffuse spectral irradiance) in terms of a ratio framework. This approach provides for a significant improvement in accuracy with respect to the earlier work of Green et ( I / . (GSS, 1974b), Shettle and Green (SG, 1974), and Green et u/. (GMM, 1974a). In undertaking to develop formulae for diffuse spectral scalar irradiance or actinic flux, the present authors initially sought to reparametrize the equations of GCS (Green and Schippnick, 1980a) to fit actinic flux calculations for ground level (Peterson, 1976). We then sought to generalize them to the case of the upward irradiance in order to fit the global actinic flux calculations for altitudes ranging to 4.21 km (Peterson, 1977; Peter- son, private communication). During the course of this endeavor we found that by generalizing and modifying the empirical formulae of CiCS we could improve upon the GCS average representation of the irradiance data of Braslau and Dave (BD, 1973a,b,c) and of Dave and Halpern (DH, 1976). These modifi- cations also made it possible to represent the actinic flux calculations of Peterson quite well. The fact that the ratio formalism and our modified empirical func-

tions encompassed both sets of data supports their use for interpolating and extrapolating between the atmospheric models used by BDH and Peterson to different aerosol levels and ozone levels. The results for the downward spectral irradiance at sea level have been presented in an invited conference paper (Green and Schippnick, 1980b).

The organization of this work is as follows. Section 11 describes the prior Florida work which is essentially retained in the present study. Section 111 describes the atmospheric models used by Braslau, Dave and Halpern and presents the modified analytic characterization of empirical functions for spectral irradiance. Section IV describes the atmospheric model used by Peterson and presents the analytic characterization of spectral actinic flux. Section V. the Discussion compares parameter values, while Section VI the Conclusion, describes the advantages of the present modified empirical functions and their possible applications.

11. RELATIONSHIP TO PRIOR FLORIDA WORK

For the sake of continuity we will first repeat the features of GCS and the earlier Florida models from which i t evolved which are retained. Our notation follows that of GCS as closely as possible, with some resystematization of parameter symbols.

We write direct actinic flux and direct irradiance 4

D(A, 0, y ) = H(%) exp - 7;(L)Ni(y) /pi(O) (1) i = l

D(It,O, y) = pD@, 0, y ) (2)

The tilde denotes actinic flux or scalar irradiance, while its lack, where the distinction applies, denotes irradiance. The arguments A, 0 and 1’ stand for wave-

89

90 P. F. SCHIPPNICK and A. E. S. GREEN

length, solar zenith angle and altitude, while

The symbols T ~ , T ~ , T~ and T~ denote the Rayleigh scattering, aerosol scattering, ozone absorption and aerosol absorption optical depths at sea level. The symbols p l , pz, p3 and p4 = pz denote generalized cosine functions of the form

p = coso.

pi(@) = [ (p2 + ti)/(l + (3) where the small value ti allows approximately for the roundness of the earth.

The optical depths at sea level, T ~ , are written as

T i ( 4 = Wi(0) ki(A) (4)

where wi(y) is the total thickness of (scattering or ab- sorbing) medium i above the altitude y with wz(y) = w4(y). The functions N i ( y ) are the total thicknesses above the altitude y, each normalized to unity at sea level (y = 0). These functions in parametrized form along with a parameter table and graphs are given in GCS. They were developed in earlier Florida studies [see Green, 1964; GSS, SG, GMM, and Riewe and Green (RG, 1978)].

Accurate determination of the extra-terrestrial solar flux, H ( A ) has been a persistent problem with reported values varying by as much as 20%. Recent measurements by Heath et d. (1980) with the double mono- chromator aboard the Nimbus 7 satellite launched in October, 1978, provide new promise of accuracy in this spectral region. An analytic representation of this body of measurements provided by Green and Schippnick (GS, 1980b) is given here in Table 1 for the convenience of the user.

Unlike the direct irradiance, the diffuse irradiance can only be obtained by a numerical solution of complex radiative transfer equations. In the earlier Florida models (see GSS, SG and GMM) empirical analytic formulae similar to the direct irradiance for-

Table 1. Analytic representation of the extraterrestrial solar flux

4 Ai ,Ji

279.5 - 0.738 2.96 286.1 - 0.485 1.57 300.4 - 0.243 1.80 333.2 0.192 4.26 358.5 -0.167 2.01 368.0 0.097 2.43

Aiexp - (2 - li)’/2uf

where

K = 0.582 W/m2nm, p = 9.102 and I , = 300 nm.

*The GCS individually tuned value of the parameter Po for model D1 is in error and should be replaced by 1.8174.

mula were developed. These were adjusted to represent diffuse spectra irradiance extracted from the measurements of Bener (1972) and also to the discrete ordinate radiative transfer calculations of Shettle and Green (1974) for various ozone aerosol thicknesses. These expressions worked reasonably well (e.g. to the 25% level of accuracy) but they introduced rapidly varying empirical wavelength dependences not based upon physical parameters. Somewhat later, experi- mental work by Chai and Green (1976) and Garrison et al. (1978) indicated that by taking the ratio of the diffuse to the direct irradiance, one filters out most of the irregular Fraunhofer structure in the extraterrestrial solar spectrum, the irregularities in the ozone absorption coefficient in the Huggins bands (> 309 nm) and the large variations of the irradiances in the Hartley continuum (< 309 nm). Unfortunately, the variation of the diffuse/direct ratio with angle (0) is very great, due largely to the factor p in the direct term (see Eq. 2). Such large variations are difficult to encompass with high accuracy by simple empirical formulae.

In the GCS model the basic strategy was to represent the downward diffuse irradiance as the product of two ratios and the direct irradiance at sea level (y = 0) for an overhead sun (0 = 0’). Following GCS (see their Eqs. 1, 3, 4 and 5), we write

where S, is either upward ((I = u) or downward (a = d ) spectral diffuse irradiance for zero albedo, and

and

MJA Y) = s a ( L O‘, Y ) / D ( A 0-, 0 ) (7)

Two advantages of the GCS ratio form are: (1) the dynamic variation of the M-ratios is limited to less than one decade over the interesting wavelength inter- val, and (2) the @-dependence of the .Y-ratios is similar to that of the analogous direct ratio, D ( i , & y ) /

By taking advantage of these features, GCS found i t possible to represent the diffuse spectral irradiance to a higher degree of accuracy than had GSS, SG or GMM. Essentially they represented the ratios cz and Ma by algebraic expressions in terms of the physical parameters z i (A) and N i ( y ) and ~ ( 0 ) .

Furthermore, GCS used as ‘data’ the results of the precise radiative transfer calculations of Braslau and Dave (1973a,b,c) on the effects of aerosols on U V radiation and the work of Dave and Halpern (1976) which addressed the effects of ozone change (these works will be referred to collectively as BDH).

Parameters for the GCS empirical formulae are given for an ‘average’ or ‘universal’ model (see GCS, Table 4) and for each of the individual BDH models (GCS, Table 5).* The present work insofar as it over- laps GCS has given greater concentration to the average or universal model. It has succeeded here in de-

D ( A 0”. Y ) .

Spectral actinic flux 91

Table 2. Summary of equations and formulae

The following equations apply for both irradiance and actinic flux: ( G and G d i f are Only of interest for the actinic flux)

u =

In the following equations actinic flux quantities are indicated by a tilde:

SU(>., 0, 0, A ) = AGd(A, 0, 0, A )

It then follows that

and

4

Ed(i, y ) = Nl eXp - cidTi(1 - N i ) i = 2

4

E u ( l L , y ) = exp - E ~ ~ T ~ ( ~ - N i ) i = 1

3,,(i, 0, y. A ) = 2SA,(2, 8, y , A ) for a = d or u

p = cos 0

Please refer to GCS (1980) and GS (1980b) for the fitting forms and parametrizations of N i ( y ) , T ~ ( A ) and k 3 ( 2 ) , and the parameters, I ! .

veloping a more accurate universal representation of the BDH models albeit with the use of more elaborate formulae. Table 2 summarizes the new generalized and modified empirical formulae arising from this work. Table 3 gives the parameters for these empirical fitting formulae which we have arrived at primarily by non-linear least square fitting of the 'data' in the irradiance tables of Braslau, Dave and Halpern and the actinic flux tables of Peterson. The reader who is already familiar with the work of GCS and the representations of altitude and wavelength dependent physical functions should, with little additional effort, be able to utilize these empirical data fitting formulae to estimate quantitatively four radiation quantities: the downward diffuse irradiance, &(,I, 0, y, A ) ; the upward diffuse irradiance, &(A, 0, y, A ) ; the downward diffuse actinic flux, s'& 0, y , A ) ; and the upward diffuse actinic flux, &(A, 0, y , A). The altitude dependence of these four quantities was a main cause of prolifer- ation of detail in the present functions. It was found to be necessary to generalize the GCS joint (i,y)- dependence through ~~(1.) Ni(y) to a separate dependence on ri(,i) and Ni(y). Similarly, the w,-dependence cannot be represented simply through rs(i.) = k3(i.)Wj except for the S-ratios.

92 P. F. SCHIPPNICK and A. E. S . GREEN

Table 3. Final parameter values for empirical fitting functions

m = Id l u 2d 12u 2u 4d 4u

0.8041 1.1032 1.437 2.027 0.4373 1.554 6.20 A m 1.261 3.365 2.254 6182 1.334 1.554 6.20 4

A,, retunes at 2.662 and Alu at 10.95 for add = udU = 1.0 (economization option)

U AT,(BD) AfADH) 23. P. P.

d 0.3747 0.3264 0.2959 1.2230 1.2230 It 0.2248 0.1983 1.662 1.1181 1.1181 r

Note: The AT, values marked (DH) are those retuned to the DH data and recommended for implementation. These marked (BD) are the corresponding values optimized to the BD data and

are given for reference.

~~ 1.0132 - 0.3133 0.2797

m = l r 2r 3r 4r

A m 0.4424 0.1000 3.70 am 0.5626 0.88 0.8404 ( 1 .O)

m = Id l u

1.389 1.735 1.389 1.735 0.921 0.921 0.544 0.544 0.5346 0.6440 0.4000 0.3692 0.3475 0.0795 0.3475 0.0795 - 0.4909

0.4909 ~

2 rl

1.12 1.12 0.564 0.564 0.6077 0.4436 0.3445 0.3445 0.4638 0.4638

2u 3d 3u

1.12 0.7555 0.7555 1.12 1.565 0.5505 0.564 1 + 1.7~3 1.0 0.564 1 + 1 . 7 ~ ~ 1.0 0.1020 1.000 1.000 0.1020 1.169 1.169 0.0 2.392 0.0 0.0 2.392 0.0 0.5658 3.502 2.417 0.5658 3.502 2.417

4d

0.88 0.88 0.490 0.490

~

- - ~

0.4638 0.4638

4u

0.88 0.88 0.356 0.356

~

- -

-

0.5658 0.565n

U B" q. q. t"

d 84.37 24.93 0.6776 1.0367 0.0266 0.00933 11 28.80 6.481 1.3% 1.952 0.01 12 0.00933

Readers only concerned with application of these new empirical fitting functions and not their method of development, need not follow the arguments of Sections 111 and IV in detail, but may derive from these sections a working familiarity with the overall componentization in order to make better use of Table 2.

Ill. THE BDH MODEL AND EMPIRICAL FITTING FUNCTIONS

A. Bra.slau-Dave data

The six atmospheric models of Braslau and Dave (BD, 1973a,b) together provide a representative set of physically or theoretically possible atmospheric conditions: (A) pure Rayleigh atmosphere, (B) gaseous absorption included, (C, D) nonabsorptive aerosol included, (C1, D1) absorptive aerosol included. The Dave and Halpern (DH, 1974) model C1* is similar to C1. In models B-D1, total ozone thickness, wj = 0.318 atm-cm. In model C, mean aerosol scattering optical depth, iz = 0.081, and mean aerosol absorptive optical depth i, = 0. In model C1,

5 , = 0.013 while in models D and D1, f 2 and f4 are multiplied by 4.18 from their C and C l values respectively. The DH 'data' consist of downward ground level diffuse and direct irradiance values for w j = 0.2-0.45cm at albedos of 0.0 and 0.3. The BD 'data' consisting of downward and upward diffuse and direct irradiance values are given for various values of albedo and altitude: A = 0.0, 0.1, 0.3, 0.8; y = 0.0, 0.98, 1.06, 3.06 km.

The BDH aerosol distribution functions can be well represented in the parametrized forms used by the Florida group. A table of parameters is given in GCS along with graphs. The two variants of N , ( y ) , namely N,,(y) and N,&) , are the aerosol distributions for the C and D models, respectively. The heavy aerosol distribution drops off faster, representing a greater proportionate burden of low-level component (see Braslau and Dave, 1973b). For the purpose of analytic characterization we have preferred to take the N2,(y) function as a convenient standard aerosol altitude distribution to represent the low-level aerosol altitude- distribution which is essentially independent of the total aerosol burden.


In the BDH inputs, the aerosol absorptive optical depth is around 15% of the total extinction optical depth. We note that the BDH values of z2 and T~ are accurately represented by linear interpolations of the form

ri(i) = ti(i.,)[l + si(l" - io)/i.o] (8)

where i, = 300nm, rzc(&,) = 0.0804, ~ ~ ~ ( 1 ~ ) = 0.0136, s2 = 0.256 and s4 = -0.441 [zZD(J.) = 4.18 zzc(i.), z4,(n) = 4.18 ~~~(i)]. A table of all of the inputs to the BDH calculations is given in GCS as well as in DH. Peterson provides a table of all of the inputs of his calculations in his 1976 paper.

In GS (1980b) a parametrization has been provided for the extinction coefficient ( T ~ + '14 for nominal w 2 , in effect, k 2 + k4) based upon the calculations of Shet- tle and Fenn (1979) for the four atmospheric conditions: maritime, rural, urban and tropospheric. These calculations include the dependence upon the relative humidity, r . For all but urban, T~ is negligible. For urban, their results give ~ 4 / ( ~ 2 + z4) = 3(r36u/, for r = G-707; decreasing to < 1% for r = 997(,:;;.

To represent the ratios corresponding to the BDH data. GCS used certain empirical functions (see GCS Eq. 6, 7, 8, 10 and 12). In the present work we use somewhat more generalized forms. Thus we generalize the GCS function Y'(l., 0) to

.%(%, 0,J') = [ F o + (1 - F,)exp - Y 3 a T 3 N 6 3 J U 4 a ]

'exp - ( y , , ~ , N : ~ a + Y Z ~ T ~ N $ ~ ~ ) ~ ~ (9)

(10)

where

F , = 1/(1 + B a ( T 3 + T4)'")

4" = [(I + t m 2 + LJI+ - 1

and

(11)

The corresponding data and fits to :/((I., 0, y) and .Y;(j., 0, y ) for model Cl are graphed in Figs. 1 and 2. The parameters ; j j u and hi, can be thought of as cor- rective adjustments to the analogous direct form to the extent to which they differ from unity. The parameters, t , are analogous, in effect, to the parameters

The sigmoid gate function Fa provides a slight shape adjustment at mid- and low-range of i.. It has been simplified from its GCS form rendering the w,-dependence of the .+"-ratios more physical in the limit of vanishing ozone thickness. The effect of aerosol absorption on the .V-ratios is slight.

The values of h2" and S 3 . are so small, we have set them equal to zero. Whereas .go., 80", y) increases by -45:), as y ranges from 0 to 3 km for the C1 model, ,K(i, 80 , y) increases by only Sq6, for all of the BD models. This variation is controlled by h l u .

Now we consider the forms Mo(i ,y) . We have generalized the GCS function M ( i ) to

MAi, Y) = (.fdi, Y) + h U ( L Y ) ) Y ~ L Y)Y& Y)

ti of Eq. 3.

(12)

0 30 60

Solor zenith angle

Figure I. Downward diffuse spectral irradiance :I$(?., 11, y) (normalired to I at 0 = 0 ) vs. solar zenith angle, 0. Both data and fit are shown for wavelengths j. = 297.5, 307.5

and 330.0 nm and for altitudes i' = 0 and 3.06 km.

\ Upword - CI -

330.0

0 30 60 90' 10 -

Solar zenith angle

Figure 2. Upward diffuse spectral irradiance : X ( i , 0, y) Innrmal i7rd t n 1 X t /I = 0 i VY w l x r w n i t h angle 0 Rnth data and tit are shown for wavelengths j. = 297.5, 307.5 and 330.0 nm and for altitudes Y = 0.98 and 3.06 km. Fits for these two altitirde lcvcls arc graphically nearly

indistinguishable.


The empirical functions f in and fzn are the Rayleigh and aerosol scattering terms and g3. and gba the ozone and aerosol absorption factors, respectively. One notes that for the model A atmosphere M&, y ) reduces to ft,(A, y ) ; for the model B atmosphere, to flO(i, y) g3&, y), and for the model C and D atmos-

In the following it will be convenient to define the pheres. to (fi& Y ) + fZ& Y ) ) g 3 . ( A Y) .

quantities, hi&, y ) such that

We represent the empirical functions , f i d ( i , y) and ,flu(i, y) by the following analytic forms for which the data and the fits represented by these equations are

A (nm) 3x) 310 290

I I I 1 I /

0.3 0.5 1.0 20 71

Figure 3. Ratios of downward and upward overhead diffuse to ground level direct spectra irradiance, f l d ( i , y) and

y ) for a pure Rayleigh scattering atmosphere vs. Ray- leigh optical depth, t,. Data and fit are shown at altitudes y = 0 and y = 3.06 km for downward radiation, and at

alritudes y = 0.98 and 3.06 k m for upward radiation.

O O W ~ I I I I ' 1 " 1 I 1 X

10-1 loo I o1

73

Figure 4. Ozone 'absorption loss terms', for downward and upward radiation, hAd(j., y) and h3"(j., y) vs. ozone optical depth. 73. Data and fit for altitudes v = 0 and 3.06 km for downward radiation, and data and fit for altitudes,

y = 0.98 and 3.06 km for upward radiation are shown.

The altitude dependence function NI(y:pld) is a generalization of the Rayleigh altitude distribution function, Nl(y) = (a + l ) / (a + exp y/hl) where u = 0.437 is the 'shape parameter', and h l = 6.35 km is the 'height parameter' (see GCS, 1980; RG, 1978). This generalization accomplishes the primary purpose of representing the low-altitude plateau behavior of fld(jL, y) by means of the large value of the parameter 6: we approximately assign D the value ez - 1 where e is the natural logarithm base. The height parameter, h l , roughly delineates the extent of the plateau region. Note that for larger y , N,(y;PId) - N f ~ " ( y ; 1). Although we did not benefit from higher altitude data in this work, simple calculations (single scattering, two-stream) indicate that such a dependence applies. Single-scattering calculations which we have carried out for both the irradiance and the actinic flux atiow us to assign P l d = plu and lald = B l u .

Note that f ld(n , y ) depends on the total overlying column, Nl(y) while f l u ( i , y ) depends in a similar manner on the total underlying column, 1 - N, (y ) . The relaxation indicated in Eq. 16 is apparent from inspection of Fig. 3. Proceeding to the model B data we set

- -

where p3&) = 1 + 1 . 7 ~ ~ and P 3 J i ) = 1. The corresponding data and fits are graphed in Fig. 4. Note that the same power-law dependence on t 3 holds for both h3AA Y ) and h3JL, Y ) .

The altitude dependence of h 3 d ( i r 4 ' ) is sharply dependent on A for 2. < 300. The effect of this altitude dependence on Md(j., y ) is quite marked as inspection of Fig. 8 indicates. We have endeavored to represent this dependence in as physical a way as possible, wherefore the added term 1 in /jJd(j.) and p3,,(i). How- ever, it is impossible to extrapolate the validity of h3d(A, y ) or h3"(2, y) to significantly higher altitudes since 0.97 < N 3 d 1 for y < 3 km, and the altitude

Spectral actigic flux 95

9 Downward

I I I I I I I I I 280 300 320 340 360

A Inm)

Figure 5. Aerosol scattering ‘contributions’ to the downward overhead ratios for the C and D model atmospheres, fZdC(%, y ) and f z d o ( I . y) vs. I for altitudes, y = 0 and 3.06 km. Both data and fits are shown. To within 17;. the

graphs of the fils to the C and D data coincide.

dependence of h3d( jL1 y ) and h3&, y) at higher altitudes must remain to be determined.

If we attempt to incorporate the data of Models C and D into the form of Eq. 12 we must fit the forms fro(;., y j to the data in Figs. 5 and 6. We set

, f2”(ir J’) = A12 t;’”t?’“(l - N I ) ” ~

+ A 2 . tciZu(l - N2)OZu (19)

This form is motivated by the observation that the input channels to backscatter (which is predominantly Rayleighbsee GCS, 1980 and RG, 1978) including aerosol downscatter should result in a non-negligible Rayleigh-aerosol ‘coupling’ term. Equation 19 proved to be the simplest and most apt form to represent this. Note that since t2,JtZc = 4.18 and we can with good accuracy set aad = c12” = 1.12, the D and C data for both upward and downward aerosol scattering terms differ by about a Factor of 5 . This allows us to overlay data for both models on the same graph in Figs. 5 and 6 with conveniently rescaled vertical axes. We set

where

This form was chosen for maximum simplicity and accuracy. The divisor 8 3 d ( i , y ) cancels the factor g &. y ) in Eq. 12, thus allowing M d @ , y) to agree with the paradigm of Eq. 12.

Since the aerosol is concentrated so close to the ground, we may with some confidence interpret Eq. 20 for y = 0 as indicating two channels of input into the aerosol downscatter channel (at y = 0). namely Rayleigh downscatter and the direct beam. The first term (at y = 0) therefore may be said to represent the

280 320 340 360

A tnm)

Figure 6 . Aerosol scattering ‘contributions’ to the upward overhead ratios for the C and D model atmospheres vs. 1. for altitudes, y = 0.98 and 3.06 km. Both data and fits are shown. To within ln,; the graphs of the fits to the C and D

data coincide.

increment to downscatter due to aerosol rescatter of Rayleigh downscatter. This increment is seen to result from the strong anisotropy of the aerosol phase function acting as a by-pass.

At non-zero altitude the mutual feedback of Ray- leigh and aerosol scattering is evidenced by the y-dependence of the first term bearing no difference whatsoever from that of the pure Rayleigh term, and suggests a picture of aerosol channel ‘homogenized’ into the overall Rayleigh multiple scattering process. In fact, the paradigm of Eq. 12 for h f d ( 2 , y) can be rearranged to give

Md(A, Y ) = [ f i d ( A , y) g 3 d ( A * Y)(l + 9 2 d ( A ) )

+ N $ Z d 2 2 d ( A ) 1 g4d(Ar y ) (22) This form explicitly appropriates the ozone factor to the Rayleigh multiple scattering process.

In Fig. 7 we have graphed the data (for y = 0) and the fit to $ 2 d ( 4 . Even the weak I-dependence of the t 2 ( A ) function inputed by BDH is reproduced, thus strikingly confirming the hypothesis of our model.

Lastly, we consider the aerosol absorption factors g40(2, y ) and set

r

. . . . . 0 0 8 O o 9 I ’ . C

I I I I I I 300 320 340 360

(rm)

Figure 7. Downward aerosol scattering ‘response function’, $2d(1) vs. 1 for both C and D model atmosphere ground

level data.

96 P. F. SCHIPPNICK and A. E. S . GREEN

and introduce the notation ‘ O r

Gd( i , 0, y, A ) = S,J (~ , ( I , J’, A ) + D ( i , ( I , 1’) ( 2 5 )

One now defines an air reflectivity function r(i .) such that an upward irradiance, S,. originating at the ground, will be reflected back to thc ground by the entire overhead atmosphere as a downward irradiance at the ground, r (A)Su , One assumes that the reflectivity of the atmosphere is Lambei tian. B r a s h , Dave and Halpern inputed a Lambertian surface albedo into their radiative transfer calculations. Both are borne out by experiment (Doda and Green, 1980, 1981) with deviations only at large zenith angles.

0 3 300 320 340 360

(nrn I

Figure 8. Downward overhead ratio, M d ( L y) vs. i for model C1. Both data and fit are shown for altitudes y = 0,

0.98, 1.96 and 3.06 km.

Now following GSS and SG we set

(26) G d ( & 0, 0, 0) Gd(l, 0, 0, A ) =

1 - r(L)A

The y-dependence of qLu( i r y) is interesting in that it is much like that of 946, with /34d = 0.490 and p4,, = 0.356, while the co-factor Ad,, z 4 A4d.

Note that the B, C and D models provide the mini- mum necessary data base (three points) to determine

and C L ~ ~ , a = d and u, in other words, the w,-dependence off&, y) and gda(A, y) , a = d and u. It is of interest to note the systematic which obtains in these parameters, namely a2“ = 1 + E , a4a = 1 - E ,

where a = d or u and = 0.12 and a,, = 1 - E (see Eq. 29). We conclude that the scattering terms, f2a(l.r y) and the absorption ‘terms’, h4JA, y), which to first order represent absorption losses, are nearly linear in w2. This result is physically reasonable.

Finally, we display the data and corresponding fits for the quantities Md(2, y ) and M&, y ) for model c 1 in Figs. 8 and 9.

Albedo dependence

To account for the effect of surface albedo we employ an approach based on the methodology of Green, Sawada and Shettle (1974) and Shettle and Green (1974). We revise our notation to explicitly exhibit the albedo dependence of the diffuse flux:

S,(L 0, y) - S,(k 0. Y, A ) (24)

and

Equation 26 suffices to extract r ( i ) from the BDH data. Equation 27 actually defines A (see Peterson, 1976). One also finds that r(1.) is independent of 0 and A to within T%,. We represent r ( i ) by the empirical form

(28) 4) = I f 1 r ( 4 g 3 r G ) + f 2 A 4 1 g4rW

where

fl,O.) = A , . ~ 7 ” , f ~ ~ ( 1 . ) = A2,r;%

gd j -1 = 1/(1 + A3rr?r)

and

g4,(4 = 1Al + A4,54) (29)

We define the albedo contribution

S&, 0, y, A ) = &,(A, 0, Y, A ) - S, (L 0, Y, 0) (30)

and its normalized altitude dependence

E,(L y) = S A ~ ( ~ , 0, J’, Ah’SAJj., 0, 0, A ) (31)

We note that E d ( % , y ) and Eu(A,y) are independent of 0 and A to within a few percent for all the 0 and A values of the BD data, and that Ed(;”, y) and Eu(?., y) are well fit by the following forms

4

Ed(2, y) = N1 eXp - 1 EidTi(1 - N i ) (32) i = 2

4

Eu(i,y) = exp - 1 ciur i ( l - Ni) (33) i = 1

where E~~ = € 2 4 and c4,, = E ~ ” .

The data and fits to Ed(;;, y) and Eu(;>, y) are given in Figs. 10 and 11. The N , factor of E d is taken over

300 320 340 360 from the approach of GCS who observed that backscatter is predominantly Rayleigh. As can be seen from Fig. 10, it accounts for over 907; of the altitude variation for E. > 325 nm. The form for E, and the corresponding Pactor in E d represent an attenuation

A Inm)

Figure 9. Upward overhead ratio, Mu(;.. r ) vs. i. for model C1. Both data and fit are shown for altitudes 1’ = 0.98, 1.96

and 3.06 km.


r Y 0.98

I 8

Downward - C I 04V

I I I I I I J 300 320 340 360

X (nm)

Figure 10. Normalized (to 1 at y = 0) altitude dependence factor for the multiply reflected downward albedo radiation vs. /I for model C1. Both data and fit are shown for

= 0.98, 1.96 and 3.06 km.

/ / ?I-

Upward - C I

I I I I I I I 300 3 20 3 40 360

A ( n m l

Figure 1 1 . Normalized (to 1 at I.' = 0) altitude dependence factor for the multiply reflected upward albedo radiation vs. i for model C1. Both data and fit are shown for

v = 0.98, 1.96 and 3.06 km.

effect. Only the coefficients for ozone, c3" and EJd, are greater than unity, and therefore may be said to represent averaged secO values.

We wish to mention an alternate approach to the representation of y-dependcnce of the albedo terms. Equations 26 and 27 can be naturally generalized to

and

One expects A,,(/,, y , A), which is defined by Eq. 35 to be essentially independent of 0 as shown. This approach is important since i t is physically motivated and it in effect forms the basis of the methodology of data reduction employed in the ground reflectivity measurements of Doda and Green (1980). It is in fact the approach indicated by Eq. 13 of GCS.

The form r (A) was built up by a similar process as the M-ratio forms. The simple form of aerosol scattering representation is sufficient to represent the data accurately, sincef2,(i) < fip(i).

B. Daor-Halpern data

This body of data allows the w,-dependence of Md(i, y ) and r( / . ) to be determined. To represent this dependence we must correct the w3-dependence already implicit in the form of h3&, y) given in Eq. 18, by replacing A 3 a ~ Y Y by the expression AT,wg.k;lu, where a = d or r . Due to the C1* model of Dave and Halpern differing somewhat in its ozone-dependence characteristics from those of the C1 model of Braslau and Dave, the value of A:, must be retuned from its effective value optimized for the BD data by a factor of 0.8711 for a = d and 0.8929 for a = r . A retuning factor of 0.8820 = (0.87 11 + 0.8928)/2 and a value of pu = ( p d + p , ) /2 = (1.2230 + 1.0132)/2 = 1.1181 are estimated for the upward case for which data arc not available. Note the near linearity of dependence of h30(j.r y ) on wj, analogous to h&, y ) on w 2 .

We found that the ozone-dependence of .%(j., 0, y) optimized for the BD data is correct as it stands for the DH data. This is not surprising considering the nature of the .z(/L, 0, y ) function: the 0-dependence acts as a spatial sensor whereas the mere spectral dependence of Md(A, y ) for y < 3 km does not. Graphs of Md(i,,O), r(L), and .%(A, 0,O) for w 3 = 0.2, 0.3 and 0.4 are given in GS (1980b). The values of ATd, Pd and ~1~~ given here differ slightly from those in G S (198%) due to minor recalculation.

1V. ANALYTIC CHARACTERIZATION OF ACTINIC FLUX

The fundamental difference between irradiance and actinic flux is illustrated by comparing Eqs. 1 and 2-one notes the absence of the cosine factor in the direct actinic flux. Whereas irradiance represents radiant flux received by a flat horizontal detector (and is therefore also referred to as horizontal flux), actinic flux represents radiant flux received by a spherical detector, and is therefore also referred to as scalar or omnidirectional flux or scalar irradiance (Smith and Wilson, 1972; Peterson, 1976; Spitzer and Wernand, 1979). If one lets I j . (po , p ; y)dp represent incoming flux in local zenith angle element (10 ( p = cosll and p o = cos0, where 0, is solar zenith angle and 0 is


local zenith angle) then

Sd(Ar 80, Y) = lo1 1,1&, P ; Y I P dp

f d ( 4 80, y ) = Jol 12.(&> y)dp

(36)

(37)

with S, and g, taking the integration limits - 1 to 0. Contrary to what one might think, upward and downward actinic flux d o not refer to the radiant flux received by the upper and lower hemispheres of such a detector, but represent flux received by the entire detector impinging from above (local zenith running from C 9 0 ) and below (9C180). Incoming rays at a given incident angle are received by the subtended hemispherical face of the sphere. Ideally, to detect upward or downward actinic flux one would still employ a spherical detector, but shielded from above or below. In all, we see that the essential difference between irradiance and actinic flux is a geometric one.

We originally attempted to retune the GCS analytic form for the downward irradiance to the ground level downward zero albedo actinic flux data of Peterson (1976) in GS (1980a). The success of this effort encour- aged us to extend this plan to the altitude-dependent data as well.

Peterson uses as his values of w 3 , fa, f4 and A the values 0.292 cm, 0.210, 0.032 and 0.05, representative of a heavier-than-average aerosol and an ozone burden and ground albedo, all typical of the vicinity of West Los Angeles. This amount of aerosol loading is intermediate between those of the C1 and Dl models of Braslau and Dave. Since we are fitting actinic flux data for only this one atmospheric condition, application of the fit to other conditions constitutes an extrapolation. This is not a problem as far as the aerosol is concerned, since the form of the fit was developed to accommodate interpolation and extrapolation between atmospheric conditions. However, the ozone burden includes a heavy low-altitude layer, not present in the BDH models. This factor may have introduced a small distortion into our fit-in the behavior at large solar zenith angle, and possibly in the delineation of the y-dependence. This effect should be addressed in a future refinement of the parametrization achieved here, which was not optimized over data broken down into downward and upward components, or into zero and non-zero albedo cases for the altitude dependent data. Nevertheless, an accurate and reliable fit has been obtained with the data presently available.

Due to this limited breakdown of the actinic flux data available, a certain simplified reparametrization had to be devised. Specifically, the data consisted of zero altitude data for both zero and non-zero albedo cases and altitude dependent data for non-zero albedo.

The zero altitude data therefore serve to display f i d ( A , 01, @(a, Q,O) and GA(n, 8,0, 0.05) where

G,(i, f f , y, A ) = (?(A, 0, y, A ) - G(k, 0, y, 0 ) (38)

is the albedo contribution to the global actinic flux,

G(i, 8, Y, A ) = G d ( k 8, Y, A ) + s'.(i, 8, Y , A ) (39)

We have retuned all six parameters of @(A, O,O), and the parameters, Alldl 2 2 d r 2 3 d 3 and E3d of h?d(L90)

while taking over the values of the remaining parameters from the corresponding values for the irradiance. The co-factors, 21, and 22, are fitted over the long wavelength data, imposing the constraint Z l d / A l d = A 2 d / ~ ~ ~ after which the ozone absorption parameters, 23, and 5 3 d are fitted to the short wavelength data. Similarly, the coefficients, F l d and j 2 d and the parameter td are fitted over the long wavelength data, after which j 3 d , & , and qd are fitted over the short wavelength data, as above. The data and the fits to them for @(A, 8,O) and $,,(A, 0) are displayed in Figs. 12 and 13.

We cannot use the actinic analogs of Eqs. 26 and 27 or 34 and 35 to represent the albedo dependence of actinic flux because the input to the sufuce reflection process is the total downward irrudiance at 'ground level (sea level). Instead one must reformulate. Fol- lowing Peterson (1976) we set

gAU(i, Q,O, A ) = 2SAu(jb, O,O, A ) (40)

where Lambertian property of surface reflectivity has been used to obtain the factor of 2. Next, noting the Lambertian property of air reflectivity (in general r(j., y)) we set

(41) gAd(A3 8, y, A ) = 2SAd(i, 0, y , A )

I

I 0'

- 9 %

,cn" a -

10'

10 0 30 60 90'

Solar zenith angle

Figure 12. Downward diffuse ground level spectral actinic flux %(i, 0 , O ) (normalized to 1 at 0 = 0 ) vs. solar zenith angle, 0. Both data and fit are shown for wavelengths

>. = 297.5, 307.5 and 332.5 nm.


T

I I 1 I I I 320 340 36C

h i m )

Figure 13. Ratio of total diffuse spectral actinic flux to ground level direct s p p a l actinic flux for overhead sun for albedo A = 0.05, Gdil(jb, o', y , O.OS)/D(>., 0 ,o) vs. i. for altitudes y =-0, 0.98, 3.46 and 4.21 km. Ground level overhead ratio, Md(i. , 0) is also graphed. Both data and fits are

shown.

The generalization of surface reflectivity to upward reflectivity above altitude y by analytic continuation should also be Lambertian whence

$Au(jb? 0, y, A ) = 2sAu(;9 6, y> A ) (42)

Our resultant form for the albedo component of global actinic flux provides an extremely close and accurate fit as can be seen from Fig. 14.

Proceeding to the altitude-dependent data we first isolate the 0 = 0" portion of this data, 6 d j J (i, 0' , y, 0.05) where we define

G d i , ( b o", y, A) = fd(A9 0'7 y, A ) f gu(J? 8", y, A ) (43)

Although we cannot separate out f l d ( i . 2 y) or $$., y), it is nevertheless instructive to look at

(44)

Our approach in fitting these data was to leave fixed the parameters already determined for the zero altitude data, vary A,,, A,z., A2", PI,, A3". and G3,, and take over the other parameters from the irradiance. The constraints A l u / A l u = A12u/A12u = A L u / A Z u were imposed. Again the ozone-response parameters were determined separately. The constraint, j13u = %3d, in analogy with the irradiance result, was imposed.

One expects the above procedure to return reasonable physical parameter values. We have plotted these data and our fit to them in Fig. 13. The parameter B l u converged near 0.6 in agreement with the single scattering result.

Finally, the remaining parameters of the altitude dependent data are determined, namely, the y-dependence parameters of g(L, 0 , y ) and all of the parameters of $@, 0, y). We take over all of the y-dependence parameters of %(A, 0, y ) and .%(A, 8, y) from the

G A ( i , O", y, 0.05) D(A, o", 0)

+

irradiance case as a simple, reasonable assumption and set i2. = yZur I, f td and y3u = 7 3 d . We then vary i L u over the long wavelength data and then & and Lju over the short wavelength data. The fit to the total diffuse actinic flux G d i J ( i , 0, y, A = 0.05) and the corresponding data are plotted in Fig. 15 for selected values of y at 0 = 0" and 0 = 7 8 ' . The kink in the graphs at i. = 320nm is a result of the irregular dependence of H ( i ) . We have taken over all but 23 of the formal parameters of the irradiance fit and varied these 23 subjects to 7 constaints, in effect utilizing 16 degrees of freedom-3 for fid(j.. y), 4 for fiJi., y), and 6 for $(A, 8, y) and 3 for %(A, 0, y) to differentiate the dependence of the variation of the actinic flux from that of the irradiance. Parameter values for irradiance and actinic flux are given in Table 3.

Although we have actinic flux data for only one value of w3, we may estimate an extrapolatory refinement of dependence on this variable as we did for M,( i ,y ) . As a simple physical assumption we set

- -

;,, = pu and ;d P d .

V. DISCUSSION

It is of interest to compare the results of the return- ing of the parameters to the actinic flux data with their original irradiance values.

t 8.60'

v) - I O I t ,*-

I. 8 = 78.

0.001 I I 1 I I 30 0 3 20 340 36 0

A ( n m )

Figure 14. Ratio of [albedo contribution to ground level total diffuse spectral actinic flux] to ground level direct spectral actinic flux for albedo A = 0.05, G,(i, O,O,O.OS)/D(i, 0 ,O) vs. i. for solar zenith angles, 0 = 0, 60, 78 and 86'. Both data and fit are shown. The $value of the .!/;-parameter, f, = 0.266 is its value determined by the .'d data. The value rd = 0.216 was adjusted to optimize the fit at large zenith angles to the data of this

graph.


Y' 0.0 - 0.98 - 2.35 X-X 4.21

I I I I I I I 300 320 340 360

X (nm) Figure 15. Total diffuse spectral actinic flux for albedo A = 0.05. cdi,(k. 0.?:0.05) vs. *I at solar zenith angles 0 = 0 and 78 for altitudes y = 0. 0.98, 2.35 and 4.21 km.

Both data and fit are shown.

The pair A,,, i2d are a factor of 1.57 greater than the corresponding pair A,,, A Z d . One also notes a near constant value of 1.71 0.03 for the ratio i?& o)/M,(>.. 0) for A. > 315 nm. This is a convenient range for comparison since above 315 nm the ozone absorption is small. This value arises by virtue of the

J2d(A , y) , and depends upon t2. It is of interest to note that it increases from 1.57 towards 1.87 as r 2 increases from zero. This ratio is the most distinguishing feature of the (geometric) difference between actinic flux and irradiance and is a result of the cosine factor in Eq. 36 diminishing contributions from the limbs.

We next turn to the downward ozone response. From Figs. 8 and 13 we obtain Md(310,0)/ Md(300,O) = 1.2 while Gd(310, O)/Gd(300, 0) = 1.8. These graphs apply to atmospheric conditions of w3 = 0.318 cm, t2 = 0.081 and w 3 = 0.292 cm, fz = 0.210, respectively; however, the difference in w 3 is small ( - 10%) and decreased in its effect due to ratioing; S2-dependence of M d and kid for the most part cancels in the ratio.

These values indicate a significantly greater ozone response of the actinic flux. This difference can again be ascribed to the geometric difference between the actinic flux and the irradiance. Radiation from the limb is more attentuated by ozone absorption due to greater slant path. Unmitigated by the cosine factor (Eq. 37), this radiation makes a greater proportionate contribution to the actinic flux than to the irradiance.

products A l d A 2 d and 2 1 d 2 2 d in f ;d ( l ,Y) and

How this physical effect is represented through the parameters of our analytic characterization is scen by comparing the co-factors Jfd and ATd (since i)d = p d ) and the exponents k 3 d and zAd. The co-factors are roughly equal: A'Sd = 0.3747(BD) or 0.3264(DH) while i j d = 0.2959. Thus, the effect is conveyed by the exponents: c 3 d = 2 . 0 7 ~ ~ ~ .

In comparing the empirical functions ,%(>.. 0, J) and $(i,O,y) (see Figs. 1 and 12) we observe the one apparent difference being a faster fall-off with decreas- - ing ,i and increasing 0 due to $ 3 d > yjd and td < r,, respectively. What is most noteworthy about these two quantities is their similarity.

It would be risky to draw detailed conclusions concerning the upward component of the actinic flux since separate data for this quantity were not available. Nevertheless. we can draw attention to the same two basic features concerning A?"()., y) and Mu(;., y) as we observed for G d ( A r y) and Md(>., .I.).

First we note that constraining AlU/Aiu =

A112u/A12u = J2u /A2u , these ratios converged to 3.05. This value is significantly greater than the value of the corresponding downward ratio (which would be between 1.57 and 1.71). This increase in ratio by almost a factor of 2 is striking. The results for A:, and j13, are somewhat puzzling, but due to the lack of separate data for the upward actinic flux. represent the best that can be done at this time as far as physical results are concerned.

Finally, restricting our consideration to the irradiance, we compare .g(jv, 0, y) and .%(L 0, y), and make the interesting observation of depressed aerosol response in .U;. This we tentatively identify with the depressed aerosol backscatter (see GCS 1980; RG, 1978). One also observes depressed aerosol response in M,,(i , y) and r(>.).

VI. CONCLUSION

There is a compelling need in tropospheric photodissociation rate studies for an accurate yet inexpen- sive representation of actinic flux or scalar spectral irradiance in the troposphere. The present work attempts to fulfill this need accurately and economi- cally by analytically representing the dependence of Peterson's calculated actinic flux 'data' on the individual degrees of freedom I., 0, 4; w3, T ~ . T~ and A. Although present techniques for measuring actinic flux are not as advanced as those for irradiance, in the interests of future efforts we have also separated the limited available data into downward and upward components.

The present irradiance work on the other hand utilizes a rather extensive and broken-down body of data and represents the results of a detailed treatment of the dependence on all of the degrees of freedom. The empirical fitting functions are admittedly more complex than those used in GCS; however, this added complexity of detail evolved in response to the additional components of irradiance and the actinic flux


which are considered and in an attempt to achieve a more accurate and physical universal representation.

Following G C S we have characterized the BDH and Peterson data entirely in terms of their physical inputs for H ( A ) , ~ ~ ( l , ) , T ~ ( A ) , T~(A) and ~ ~ ( 3 . ) . Since all of these inputs are subject to change due to new measurements or analyses, it is not unreasonable to retain the present formulae, but t o replace the old physical inputs by new ones (see Green and Schipp- nick, 1980b).

A similar remark might be made with respect t o the low and high altitude components of the aerosol and ozonc distributions. Riewe and Green (1978) have treated the low-level ozone component separately. Such a refinement of approach would be appropriate for da ta such as we were considering here for actinic flux for a polluted urban atmosphere.

In regard to the aerosol distribution, the user is reminded that our analytic characterization utilizes the N,,(v) distribution (see GCS, 1980) as a standard altitude distribution, irrespective of w 2 . This is justi- fied by the fact that the major difference in aerosol thickness between C and D atmospheres of Braslau and Dave lies in the low-level component (the component whose variation is most likely t o be of concern to the typical user) and the normalized altitude distribution of the low-level aerosol burden is essentially independent of the thickness of that burden. Corre- spondingly, the normalized altitude dependence of the empirical aerosol functions, f2,(i, y) are likewise essentially independent of the aerosol loading as can be seen from Figs. 5 and 6 [this is also true of h d . , Y)].

In final summary we should reiterate that the primary purpose of the present work was to broaden the analytic skylight representations of the Florida Group to encompass actinic flux o r scalar spectral irradiance. I t is hoped that these actinic flux representations will be of use in tropospheric photochemistry studies and that the improved irradiance formulae will be helpful in future photobiology studies.

Ackno~lr t l yement s~-The authors would like to thank Dr. J. Peterson for providing additional actinic data for this study, and Olivia Berger. Roxie Wilkerson, and Linda Combs for the production of the manuscript. This work was supported in part by contract EPA-R806373010 from the U. S. Environmental Protection Agency and contract

NAS-5-22909 from the National Aeronautics and Space Administration.

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analytical characterization of spectral actinic flux and spectral irradiance in the middle...

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