SERI/TR-33S-344 

UC CATEGORY: UC-S9,61,62,63 

DIRECT INSOLATION MODELS 

RICHARD BIRD 

ROLAND L. HULSTROM 

JANUARY 1980 

PREPARED UNDER TASK No. 3623.01 

Solar Energy Research Institute 

1536 Cole Boulevard 

Golden. Colorado 80401 

A Division of Midwest Research Institute 

Prepared for the 

U.S_ Department of Energy 

Contract No_ EG-77-C-01-4042 
, 

'. 

Printed in the United States of America 

Available from: 

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U.S. Department of Commerce 

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Price: 

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Printed Copy $ 5.25 

NOTICE 

This report was prepared as an account of work sponsored by the United States Government. Neither the United States nor the United States Department of Energy, nor any of 

their employees, nor any of their contractors, subcontractors, or their employees, makes 

any warranty, express or implied, or asumes any legal liability or responsibility for the 

accuracy, completeness or usefulness of any information, apparatus, product or process 

disclosed, or represents that its use would not infringe privately owned rights. 
S=�I TR-344 

FOREWORD 

This re port documents wo rk performed by the SERI Energy Resource Assessment 

Branch for the Division of S olar Ene rgy Technology of the U.S. Depa rtment of 

Energy. The report compa res seve ral sim ple direct insolation models with a 

rigorous solar t ransmission model and desc ribes an improved , s imple , direct 

insolation model. 

Appro ved for: 

S OLAR ENERGY RESEARCH INSTITUTE 

for Res earch 

iii 

Roland L. Hulstrom, Branch Chief 

Ene rgy Res ource Assessment 

, 

\ 
S=�I TR-344 

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2.0 

3.0 

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TABLE OF CONTENTS 

Int roduction •••••••••••••••••••••••••••••••••••••••••••••••••••••••• 

Descri ption of Models •••••••••••• . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

2.1 

2.2 

2.3 

2.4 

2.5 

2.6 

2.7 

2.8 

2.9 

2 . 10 

SOL TR.AN' Model ••••••••••••••••••••••••••••••••••••••••••••••••• 

Atwater and Ball Model •••••••••••••• ...................... . ... 

Hoyt Model •••••••••••••••••••••••••••••••••••••••••••••••••••• 

L acis and Hansen Model� ••••••••••••••••••••••••••••••••••••••• 

Machta Model •••••••••••••••••••••••••••••••••••••••••••••••••• 

AS RRAE Model •••••••••••••••••••••••••••••••••••••••••••••••••• 

\V'att Model ......................................... . ...... .... . 

Majumdar Model •••••••••••••••••••••••••••••••••••••••••••••••• 

Bird Models ••••••••••••••••••••••••••••••••••••••••••••••••••• 

Addit ional Models and Other Considerations •••••••••••• •••••••• 

2 . 10. 1 W ater Vapor ••••••••••••••••••••••••••••••••••••••••••• 

2.10.2 Ozone ••••••••••••••••••••••••••••••••••••••••••••••••• 

2 . 10.3 Uniformly Mixed G ases ••••••••••••••••••••••••••••••••• 

2 • 10. 4 Ai r Mas s •••••••••••••••••••••••••••••••••••••••••••••• 

2 . 10.5 S imple T rans po rt E quation ••••••••••••••••••••••••••••• 

Model Compa'risons • ••••••••••••••••••• . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Summary and C onclusions ..... •••••••••••••••••••••••••••••••••••••••• 

Re ferences •••••••••••••••••••••••••••••••••••••••••••••••••••••••••• 

Appendix. T abulated Model Data •••••••••••••••••••••••••••••••••••••••••• 

v 

1 

3 

3 

4 

5 

7 

7 

8 

8 

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10 

12 

12 

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14 

15 

16 

21 

39 

43 

Al 
S=�I IWI _________________________ -344 

-� �� 

LIST OF FIGURES 

3-1 T ransmittance versus Secant of Solar Zenith Angle 

for Midlatitude Summer Model ......... .......... .................. 22 

3-2 T ransmittance ve rsus Secant of 'S olar Zenith Angle 

for Subarctic Winter Model •••••••••••••••• •.•.. ••••••••• ••••••••• 23 

3-3 Ozone Abs orptance ve rsus S olar Zenith Angle for 

0.31 em of Oz one ( MLS ) ................ .......... ............ ..... 25 

3-4 Ozone Absorptance versus S olar Zenith Angle for 

0.45 em of Ozone (SA��) . ....... . ................................... 26 

3-5 Abso rptance ve rsus Solar Zenith Angle for 

2.93 cm of Water Vapor (MLS ) ••••••••••••••••••••••••••••••••••••• 27 

3-6 Abso rptance ve rsus Solar Zenith Angle for 

0.42 cm of Wate r Vapor ( SAW) ••••••••••• •••••••••••• •••••••• •••••• 28 

3-7 T ransmittance ve rsus Solar Zenith Angle for 

All Molecular Effects E xcept H2

0 Absorption (MLS ) •••••••••••••••• 29 

3-8 Transmittance versus Solar Zenith Angle for 

All Molecular Effects Except H20 Absorption ( SAW).. ......... .... . 30 

3-9 Aerosol T ransmittance ve rsus Solar Zenith Angle for 

5-km Visibility Aerosol ••••• •••••••••••••• ••• •••••••••••••••••••• 31 

3-10 Aerosol T ransmittance versus S olar Zenith Angle for 

23-km Vis ibility Ae rosol ••••••••••••••• ••••• ••••••••••••••••••••• 32 

3-1 1 Solar I rradiance versus S olar Zenith Angle, 

Vis ibility = 23 km (MLS ) ••••••••••• ••••• •••••••••• •••••••••• ••••• 34 

3-12 Solar I rradiance versus Solar Zenith Angle , 

Visibility = 5 km (MLS ) ••••••••••• ••••••••••••••••••••••••••••••• 35 

3-13 S olar Irradiance versus Solar Zenith Angle, 

Visibility = 23 km (SAW) ••••••••• ••••••••••••••••• •••••.•••••••• • 36 

vii 
S;::�I , ____________________________________________________ � ________ �T�R�-�3�4�4� 

- {.

� 

LIST OF TABLES 

2-1 Sources of Data for Construct ion of Models •••••••••••••••••••••••••• 14 

2-2 Comparisons of Direct Normal Irradiance for Di fferent Forms of the 

Trans port E quation Using Bird Models and SOLT AN............... ... 17 

2-3 Com parisons of Direct Normal Irradiance for Different Forms of the 

Trans port Equation Using Bird Models and SOLT AN.................. 18 

A-I Tabulated Data for the Midlatitude Summer Atmosphere 

(V = 23 km) for S everal Models •••••••• •••••••••••••••••••••••••••• A2 

A-2 Tabulated Data for the Midlatitude Summer At mosphere 

(V = 5 km) for Several Models ••••••••••••••••••••••• ••••••••••• ••• A3 

A-3 Tabulated Data for the Subarctic Winter Atmosphere 

(V = 23 km) for Several Models ••••••••••••••••• ••••••••••••••••••• A4 

A-4 Tabulated Data for t he Subarctic Winter Atmos phere (V = 5 km) 

for S everal Models • • • • •••••••••••••••••••••• •••••••••••••••••••••• AS 

ix 
TR-344 

SECTION 1.0 

INTRODUCTION 

So lar energy ( insolation) conversion systems a re different f rom systems based 

on other sources of energy, because the energy source is s ubject to varying 

meteorological conditions . As a res ul t , rel iable insolation data a re requi red 

at each s ite of interest to design a solar energy sys tem. Historical data 

have been collected by the Nat ional Weathe r Se rvice (NWS ) on a very limited 

bas is at 26 locations throughout the Uni ted S tates , and data a re currently 

being collected at 38 locations . Because of the small number of s tations in 

this netwo rk and the variability of insolation, it is essential to have accurate models to predict insolation at o ther locations . The accuracy of these 

models and e xperimental data affects the design, perfo rmance , and econo mics of 

solar systems . 

Numerous s imple insolation models have been produced by d iffe rent investigato rs over the past half century. The goal of these models has been to provide 

an uncomplicated estimat ion of the available insolation. These models, by 

ve ry different methods , account for the influence of each atmos pheric constituent on solar radiation. This , in turn, leads to confusion and questions 

of validity f rom prospective use rs. 

This study compares several of the more recent models of the direct component 

of the insolation for clear sky conditions . The c ompa rison includes seven 

s imple models and one rigorous model that is a bas is for determining accuracy. The results of the compa ri sons a re then used to formulate two s imple 

models of diffe ring complexity. The most useful formalisms of present models 

have been incorpo rated into the new mode ls . 

The c riteria for evaluating and formulating models a re s implicity , accuracy , 

and the ability to use readily available meteorological data. 

In the future , a s imilar analys is of models for g lobal and diffuse insolation 

is planned . S im ple global and d if fuse models will be com pa red with a rigorous 

model that uses Monte Carlo techniques . Add itional compa risons a re planned, 

with ve ry carefully taken e xpe rimental data for both direct and diffuse insolation components . As many meteorological measurements as can reasonably be 

taken will be included with these data. 

The goal of this wo rk is to produce a well-documented global insolation model 

that includes the direct and diffuse clear sky insolation as well as cloud -and 

g round -reflected insolation. This re port is the first step toward achieving 

that goal . 

1 
2 

5=~1 
!5::�1 !�l __________________________________________________________ T.R�-�3�4�4� 

SECTION 2.0 

DESCRIPTION OF MODELS 

A rigorous atmospheric transmission model has served as a basis for comparing 

the accuracy of simple empirical models. The next few sections present a 

description of the rigorous model and the mathematical expressions that form 

the simplified models. 

2.1 SOLTRAN MODEL 

The rigorous model, called SOLTRAN, was constructed from the LOWTRAN [1-3] 

atmospheric transmission model produced by the Air Force Geophysics Laboratory 

and the extraterrestrial solar spectrum of Thekaekara [4]. 

The LOWTRAN model has evolved through a series of updates and continues to be 

improved with new data and computational capabilities. In this model, a layered atmosphere is constructed between sea level and 100-km altitude by defining atmospheric parameters at 33 levels within the atmosphere. The actual 

layer heights at which atmospheric parameters are defined are: sea level 

(0.0 km) to 25-km altitude in 1.0-km intervals, 25 to 50 km in 5-km intervals, 

and at 70 km and 100 km, respectively. At each of these 33 altitudes the 

following quantities are defined: temperature, pressure, molecular density, 

�vater vapor density, ozone density, and aerosol extinction and absorption 

coefficients. A complete description of the standard model atmospheres 

incorporated in this code is given by McClatchey et al. [5J. 

The absorption coefficients of water vapor, ozone, and the combined effects of 

the uniformlx mixed gases (C02, N20, CH4

, CO, N

2

, and 0 ) ar! stored in the 

code at 5-cm 1 wavenumber i

�

tervals with a resoluti9n of 20 cm 1. The average 

transmittance over a 20-cm- interval as a result of molecular absorption is 

calculated by using a band absorption model. The band absorption model is 

based on recent laboratory measurements complemented by using available theoretical molecular line constants in line-by-line transmittance calculations. 

The effects of earth curvature and atmospheric refraction are included in this 

model. The results of earth curvature become important along paths that are 

at angles greater than 60° from the zenith, and refractive effects then 

dominate at zenith angles greater than 80°. 

The scattering and absorption effects of atmospheric aerosols (dust, haze, and 

other suspended materials) are stored in the code in extinction and absorption 

coefficients as a function of wavelength. These coefficients were produced by 

a MIE code for defined part:i.cle size distributions and complex indices of 

refraction. Four aerosol models are available, representing rural, urban, 

maritime, and tropospheric conditions. 

A user can choose any one of six standard atmospheric models incorporated in 

the code or can construct his own atmosphere by using a combination of parameters from the standard models or by introducing radiosonde data. 

3 
S=�II*I _____________________ TR-34:.:

4 

Some of the outputs of the LOWTRAN code include the total transmittance; the 

transmittance of H2

0, °3

, and the uniformly mixed gases; and aerosol transmittance at each wavelength value specified. In the SOLTRAN model these 

transmittance values are multiplied by Thekaekara I s corresponding extraterrestrial solar irradiance at each wavelength of interest. A sum (integration) 

of the results of these individual multiplications is then performed over the 

spectral interval of interest to produce a value of the broadband terrestrial 

direct beam irradiance. The current version of SOLTRAN is limited to a spectral region between 0.25 and 3.125 ).1m because of a limited extraterrestrial 

solar spectral data file. 

2.2 ATWATER AND BALL HODEL 

A model for the direct solar insolation was published recently by Atwater and 

Ball [6]. This is a modification of an earlier model published by Atwater and 

Brown [7], which also includes a diffuse insolation formalism and the effect 

of clouds, neither of which are discussed here. 

The equation for the direct insolation on a horizontal surface is given by: 

where 

(1) 

IO = extraterrestrial solar irradiance, 

Z = solar zenith angle, 

TM = transmittance for all molecular effects except water vapor absorption, 

� = absorptance of water vapor, 

T

A = transmittance of aerosols. 

The mathematical expressions for the transmittance, absorptance, and air mass 

M, are given by: 

T

M = 1.041 - 0.15[M (949 x 10-6 P + 0.051)]°

·

5 ,* (2 ) 

(3) 

*Atwater and Ball recently published an errata sheet in Solar Energy, Vol. 23, 

p. 275, changing the coefficient, 0.15, in Eq. 2 to 0.16. This change has not 

been incorporated in the results presented here. 

4 
55'11*1 ________________________ ..

T::.R:.-3""4'"-'-4 

where 

T

A = exp ( -a M') , 

M = 35/ [(1224 cos2 Z) + 1]°·

5 

, 

M' = P • M/I013. 

(4) 

(5 ) 

Uw = amount of water vapor in a vertical path through the atmosphere 

(cm) , 

a = total broadband optical depth of the aerosol, 

P = surface pressure (mb). 

A brief discussion of the form of Eq. 1 is given in Paltridge and Platt [8]. 

The form of Eq. 2 is a slight variation ,of an empirical formula derived by 

Kastrov and discussed by Kondratyev [9]. Equation 3 is the form derived by 

McDonald [10], and Eq. 5 is a modification of a formula used by Rodgers for 

ozone and discussed by Paltridge and Platt [8]. 

Equation 4 is discussed in more detail by Atwater and Brown [7]. They used a 

MIE theory calculation to determine the value of a, which is not the approach 

that would be used in a simple, user-oriented model. 

Results from using this model are presented in a later section with a comparisons of other models. 

2.3 HOYT MODEL 

A model for solar global insolation that includes a model for the direct component is described by Hoyt [11]. The following equation is for the direct 

solar insolation on a horizontal surface: 

5 

I = I

O

(cos Z)(1 -

i�l

a

i

) T

AST

R (6 ) 

where a

i represents the absorptance values for water vapor (i = 1), carbon 

dioxide (i = 2), ozone (i = 3), oxygen (i = 4), and aerosols (i == 5). The 

parameter T

AS is the transmittance after aerosol scattering, and TR is the 

transmittance after pure air, or Rayleigh, scattering. The following formulas 

define these parameters: 

0.110 (U' + 6.31 x 10-4)

°.3 

- 0.0121 , , 

w 

a

2 = 0.00235 (U� + 0.0129)°·

26 

- 7.5 x 10-4 

, 

5 

(7) 

(8) 
S=�II*I ________________________ ..

T.:

R-..

3 :.

4 4 

where 

u 

w 

, 

U 

c 

t 

U 

o 

M' 

g(a) 

(10) 

M' 

as = O.OS[g(a)] , (ll ) 

M' TAS =[g(a)] , (12) 

(13) 

= pressure-corrected* precipitable water in the path (cm), 

= pressure-corrected amount of carbon dioxide in the path [cm 

at standard temperature and pressure (STP) , U� = 126 cm for 

air mass 1.0], 

= the amount of ozone in the path (cm at STP), 

= pressure-corrected air mass, 

= a tabulated function that is related to the angstrom turbidity coefficient a, 

f(M') = a tabulated function of pressure-corrected air mass. 

See Hoyt [11] for the tabular data. 

Because the functions g(a) and f(M') are in tabular form rather than in empirical expressions, this model is not as flexible as it could be. The use of 

the tables often requires interpolation between points, and the range of air 

masses and turbidity coefficients listed in the tables is sometimes too 

limited. Results of this model will be presented in a later section. 

*Hoyt calculates the pressure-corrected precipitable water by multiplying the 

total precipitable water from radiosonde data by 0.75 in a recent report. 

This correction has not been made in the analysis performed here. 

6 
S;55�1 '�I __ __________________________________ __________________ T�R�-�3�4�4� 

2.4 LACIS AND HANSEN MODEL 

Lacis and Hansen [12] have described a formalism for total insolation. Since 

they do not separate the direct and diffuse components of the insolation, 

their formalism cannot be considered here. However, they derived useful 

empirical expressions for water vapor and ozone absorption. The water vapor 

absorptance is expressed by 

a

w = 2.9 Y [ (1 + 141.5 y)0.635 + 5.925 y]-1 

, (14) 

where Y = MDw

' with D

w being the precipitable water vapor (cm) in a vertical 

path. 

The expression for ozone absorptivity in the Chappuis band is given by 

vis 

a 

o 

= 0.02118 X (1 + 0.042 X + 0.000323 X

2

)

-1 

and for the ultraviolet band by 

(15) 

aUv = 1.082 X (1 + 138.6 X)-0.805 + 0.0658 X [1 + (103.6 X)3

]

-1 (16) 

o 

where X = Do

M with D

o being the amount (cm) of ozone in a vertical path. The 

total ozone absorptivity is given by the sum 

vis uv 

a = a + a 

o 0 0 

(17) 

Comparisons of the results of these expressions ,.;ith other models is shown in 

a later section. 

2.5 MACHTA HODEL 

A simple model of global insolation has been constructed by Machta [13] in the 

form of graphs and a worksheet. This model is an approximate method for calculating solar insolation at a given location without the use of mathematical 

expressions. A standard value of direct solar insolation is given as 887 

W m

-2, and a standard value of diffuse insolation is given as 142.5 W m

-2 

These standard values are then corrected by the use of graphs and the worksheet. The corrections are made for station altitude, zenith angle, precipitable water, .turbidity, and earth-sun distance. This method has greatest 

accuracy for very clear days and small zenith angles. 

The graphs for making corrections are based on the very rigorous calculations 

of Braslau and Dave [14]. A few examples of calculations using this model 

will be illustrated in a later section. 

7 
S=�I' TR-344 

2.6 AS HRAE HODEL 

The American Society of Heat ing, Refrigeration and Air Conditioning Engineers , 

ASHRAE, publishes a s imple model [ 1 5,1 6] f or es t imat ing solar insolation at 

locations in the Northern Hemisphere . This method represents the solar inso ­

lation at the earth 's s urface, under clear sky conditions , by us ing 

where 

I = A e-B sec Z 

DN 

A = the "apparent " extraterres trial solar radiation , 

B :: the "apparent " opt ical attenuation coef ficient , 

Z = the solar zenith angle. 

(18) 

An atmospheric clarity adjus tment , CN, called the clearness number , is then 

used to multiply the direct normal insolat ion cal culated us ing Eq. 18. This 

clearnes s number corrects for var iations in transmit tance at a part icular 

location. Values of A, B, and CN are published by ASHRAE [15] as well as 

tables of solar insolation for the Nor thern Hemisphe re [1 6] resul t ing from 

applicat ion of these parameter values. 

A thorough discuss ion of the origin of the ASHRAE model is presented by 

Hulstrom [17], and this information will not be repeated here . It is sufficient to say that Eq. 18, commonly called Beer's Law, is strictly applicable 

only f or monochromatic radiation. If one takes the natural logarithm of Eq . 

1 8 , t he res ult is : 

In A - B sec Z (19) 

A plot of this express ion on a In IDN versus sec Z axis system results in a 

s traight line , with A the intercept of the logarithmic axis and B the slope of 

the line . The vertical intercept A occurs for the ext rapolation sec Z = 0.0, 

which corresponds to zero air mass or the e xtraterres trial ins olation . 

In the model compar is ons given in a later sect ion the deviations of this model 

f rom more accurate results are indicated . Hulstrom [17] points out that the 

clearness numbers published by AS HRAE corre ct only for water vapor var ia ­

t ions . Moreover, these are only average water vapor condit ions , and it is 

shown here in a later section that var iations in aerosol attenuat ion are 

normally a much more s ignificant factor. 

2.7 WATT MODEL 

A model for global ins olation has been cons tructed by Watt [18], bas ed partly 

on the work of Moon [ 1 9]. The �xp ression f or the direct normal insolat ion is 

(20) 

8 
!s::�1 1�1 ______________________________________________________________ T�R� -344� 

where the t rans mittance funct ions are Twa for water vapor absorpt ion, T

a for 

dry air s cat tering, T

o for ozone abs orpt ion, Tws 

for water vapor s catte�ing, 

T

L for lower level aerosol absorption and s cattering, and T

u for upper layer 

aerosol absorption and s cattering. These t ransmit tance funct ions are defined 

by 

= 0 . 93 - 0 . 0 33 log (Uw M2

) 

-T M = 10 u 3 _ , 

TL = 0. 6 (T 0 . 5 - 0.0 1 Uw - 0 . 0 3) , 

(21) 

(22) 

(23 ) 

(24) 

(25 ) 

(26) 

(27) 

( 28 ) 

The para meter Po is the sea level press ure; P is the pressure at the surface 

being cons idered; Uo and Uw are the amount of o zone and water vapor in cm in a 

vert ical path; TO.5 is the ato mospheric turbidity at 0 . 5-]lm wavelength; lobs 

is undefined in Watt's report; and 1

0 is the broadband extraterres trial insolation. The parameters T

u and TL are the upper and lower layer broadband 

t urbidity or opt ical dep th. T

U can be taken from p lots in Watt's report for 

pas t years . The Hi are called path length modifiers by Wat t, and they serve 

the same purpose as air mass in the previous models. These path length 

modif iers are equal for solar zenith angles ( 70 ° and are equal to the secant 

of the solar zenith angle ( s ec Z ) . For solar zenith angles > 70 °, the path 

length modifier is defined different ly for ea ch atmospheric cons t ituent 

according to the altitudes in the atmosphere between which the cons t ituent is 

concentrated . A parameter F

zi is cal culated using the following express ion: 

(29) 

9 
S=�I '*1 ____________________ ..

T:.:.

-..

34..

4 

where r is the earth I s radius (6.4 x 106 m) and the hi are the atmospheric 

altitudes (heights) between which the constituent is located. If a constituent is concentrated between two altitudes, hI and h

2

, an F zl and F z2 are 

calculated corresponding to hI and h2' respectively. The values are then used 

in the following expression to obtain the total path length modifier: 

h

2 F

z2 - h

I F 

zl 

M. = (30) 

1 h

2 - h

I 

The values of h used for the various constituents are: 

ozone: hI = 20 km, h

2 :: 40 km 

dry air: hI :: ° km, h2 = 30 km 

upper dust: hI = 15 km, h2 = 25 km 

lower dust 

and water vapor: hI = ° km, h2 3 km . 

When the value of h is equal to zero, the corresponding Fz can be set equal to 

1.0, and M

i = Fz2' 

2.8 MAJUMDAR MODEL 

A model for direct normal insolation has been constructed by Majumdar 

et al. [20]. This model is for clear sky conditions and minimal aerosol 

content, so that the effect of variable turbidity is not considered. A total 

of 161 sets of observations at three locations in India was used to arrive at 

the following regression equation: 

I

DN = 1331.0 (O.8644)(MP/I000)(0.8507)(UJM)

0.25 (31) 

where M is the air mass, P is the surface pressure, and U

w is the amount of 

water vapor in a vertical path. 

Results generated from this model will be presented in a later section. 

2.9 BIRD MODELS 

As a result of comparing the simple models discussed in this section with the 

rigorous model (presented in a later section), the authors formulated two 

additional models. Where possible, these models used formalisms from the 

10 
S=::lI'.1 _____________________ TR ..

 -.. 44 

-� ��� 

models previously presented. The expressions used were tuned to give the best 

least-squares fit to the SOLT AN data. The first model used the following 

expression for direct normal insolation: 

(32 ) 

where Tu is the transmittance of the uniformly mixed gases (C02 and O2), and 

the other parameters are the same as defined earlier. The 0.9662 factor was 

added because the spectral interval considered with SOLT AN was from 0.3 to 

3.0 �m.· The total if

radiance in the extraterrestrial solar spectrum in Zhis 

interval is 1307 W/m , whereas the Thekaekara solar constant of 1353 W/m is 

used in �ost of the other models. The factor of 0.9662 allows one to use 1

0 = 

1353 W/m with the Bird �odels. If a higher value of this solar constant is 

desired (i.e., 1377 W/m ), it can be used without changing Eq. 32. The 

transmittance and absorptance equations are 

TR = exp [- 0.0903 (M') 0.84 (1.0 + M' - (M,)

1.01)] 

T

o = 1.0 - 0.1611 Xo (1.0 + 139.48 X

o

)

-0.3035 

( 2)

-1 -0.002715 Xo 1.0 + 0.044 Xo + 0.0003 X

o 

Tu = exp -0.0127 (M' )

0.26 

, 

a

w = 2.4959 � [ (1.0 + 79.034 �)0.6828 + 6.385 �J-1 

LA = 0.2758 TA

(0.38) + 0.35 TA

(0.5) 

M = [cos z + 0.15 (93.885 - Z)

-1.25J-1 

, (33) 

(34) 

(35) 

(36) 

(37) 

(38) 

(39) 

where M'= tW/PO' Xo = Uo

M, � = UwM, Tu is the transmittance of the uniformly 

mixed gases, TA is the broadband atmospheric turbidity, and LA(0.38) and 

TA (0.5) are the atmospheric turbidity values that are measured on a regular 

basis by NWS at 0.38- and 0.5-�m wavelengths, respectively. If one of the 

turbidity values is not available, its value can be entered as a zero in 

Eq. 38. The values of T (0.38) and T (0.5) are obtained in practice with a 

turbidity meter that measures the total optical depth at each wavelength. The 

optical depth due to molecular scattering is then subtracted from the total 

11 
S=�I, �I _______________________ T=R::.

-.=,.34.!.

4 

optical depth to obtain the turbidity (or aerosol optical depth) at each 

wavelength. Equation 39 is a form derived by Kasten [21]. The forms of 

Eqs. 34 and 36 are patterned after Lacis and Hansen (12], as shown in Eqs. 15 

and 16. Some of the terms in the expression for ozone absorption used by 

Lacis and Hansen have been dropped in Eq. 34. This expression is still much 

too complicated when the relative importance of ozone is considered. The 

second line of Eq. 34 could be dropped without serious effeft�

r The form of 

Eq. 33 can be simplified by removing the (l.0 + M' - (M')· ) term. This 

simplification provides very accurate results for Z ( 70°. 

The second and simplest model is given by 

(40 ) 

TM = 1.041 - 0.15 [M(9.368 'x 10-4 P + 0.051)]0.5 (41) 

The expressions for � and T

A are the same as Eqs. 36 and 37, respectively. 

Equation 40 was used by Atwater and Ball [6], and Eq. 41 is a slight variation 

of Kastrov as presented by Kondratyev [9]. The variable P is the surface 

pressure at the location being considered. This model will be shown to provide results that are nearly as accurate as the more complex model with much 

less effort. 

2.10 ADDITIONAL MODELS AND OTHER CONSIDERATIONS 

This study is not a comprehensive comparison of simple direct insolation 

models. It is a comparison of the more recent models. An excellent comparison of other models is presented by Davies and Hay [22]. 

For the comparison of different models it is useful to know the origin of the 

data used to formulate the models. An attempt will be made to trace the 

origins of the H2

0, °3

, and uniformly mixed gas data used in the models 

described here. In addition, a discussion of air mass and the various forms 

of the transport equation for direct insolation will be presented. 

2.10.1 Water Vapor 

The LOWTRAN model is based on a band absorption model. The parameters in the 

band absorption model are based on comparisons with transmittance data taken 

by Burch et al. [23-3�1 and line-by-line transmittance calculations degraded 

in resolution to 20 cm • The contributors to the line data are too extensive 

to reference here, but are found in a report by McClatchey et al. [34]. 

Atwater and Ball used an empirical expression from HcDonald (10] based on old 

data taken by Fowle [35]. Fowle's data did not account for the weak absorption bands near 0.7- and 0.8-�m wavelengths. Because of the poor documentation of Fowle's data, McDonald could not claim an absolute accuracy greater 

than 30%. 

12 
5=�1 ' 

The ASHRAE model is based on work by Threlkeld. Huls trom explicates 

"Threlkeld determined the var iable amount of water vapor on a monthly bas is by 

a semiempirical technique. He used measurements of the broadband d irect beam 

ins olation [at Blue Hull, Mass. ; L incoln, Neb .; and Madison , Wis. J in conj unct ion with the Moon cal culat ion routine , to derive indirectly the corresponding amount of precipitable water vapor" [1 7J. 

Hoyt and also Lacis and Hansen used water vapor from Yamamoto [36 ]. Lacis and 

Hansen explain, "Absorption by major water vapor bands has been measured at 

low spect ral resolution by Howard et ale (1 956 ). Yamamoto (1 962 ) weighted 

these absorptivit ies with the solar flux and summed them, including est imates 

for the weak absorption bands near 0.7 and 0 . 8 Um which were not measured by 

Howard et ale, to obtain the total abs orption as a funct ion of water vapor 

amount" [1 2 ]. The reference to Howard et al. is referenced here as [37]. 

Watt used an expres s ion for wat er vapor that he derived empirically from data 

in Chapter 16 of Valley [38]. He then compared this expression with measured 

data from several locations and adj us ted the coefficients to obtain the best 

a greement . 

Machta bas ed his model on cal culat ions performed by Braslau and Dave [14] with 

a rigorous rad iative t ransfer code . Bras lau and Dave based their cal culat ions 

on the database used by LOWTRAN. However, they used a mathematical formalism 

different from LOWTRAN for band absorption. 

2 • 10 .2 Oz one 

The LOWTRAN and t1achta models used the same original data sources for all the 

molecular ,absorbers . The references given in the previous sect ion for water 

vapor are the same for o zone . 

A twater and Ball did not cons ider ozone separately but used a general formula 

that included all molecular effects except water vapor absorption. 

The ASHRAE and Watt models are based on the ozone data used by Moon [19 ]. 

Moon, in turn, used data measured by Wulf [39] in the Chap puis band (0 .5 to 

0 �7 um) and data by Lauchli [40] in the Hartley-Huggins band below 0 . 35 Urn 

wavelength. 

Watt inte grated the s pe ct ral data given by Moon to obtain broadband ozone 

absorption. He then modified an express ion that agreed with these res ults to 

give the best agreement with broadband total t ransmit tance data from other 

sources . 

Hoyt used ozone data from Manabe and Strickler [4 1] to derive an empirical 

formula. Manabe and S trickler based their results on experimental data from 

V igroux [42] and Inn and Tanaka [43]. Lacis and Hansen apparently based their 

empirical formula for ozone on the same original data sources that Hoyt 

used . They point out that the data for wavelengths greater than 0 . 34 �m were 

given for 18°C. They used the data at -44°C for shorter wavelengths and 

reduced the longer wavelength data by 25% to compensate for the difference in 

t emperature. They produced s eparate expressions appropr iate for the 

ultraviolet and vis ible absorption data , respect ively. 

13 
55'1 '*' _______________________ T..:

R-..

3::.

44 

2 . 10. 3 Uniformly Mixed Gases 

The uniformly mixed gases are CO2 and 02

. LO WTRAN and Machta are based on the 

same original data s ources given in the section about water vapor. 

The ASHRAE and Watt models 

CO2 and 03 in insolation 

abso rption. 

are based on Moon ' s model . 

but might have included 

Moon did not consider 

their effect w ith H2

0 

Atwater and Ball based their model on an empirical formula that included all 

molecular effects except water vapor absorption according to Kondratyev [9]. 

This formula was based originally on Fowle 's data for the constant gases . 

Hoyt used Yamamoto's oxygen absorption data, 

Howard et ale [44 ]. The carbon dioxide 

et al. [45 J. 

which in turn were taken from 

data were taken from Burch 

A summary of the data sources used by various modelers is condensed in 

Table 2-1 . 

Table 2 -1 . SOURCES OF DAT A FOR CONSTRUCT ION OF MODELS 

}fodel H2

O 

SOLTRAN Burch et ale (many) 

Line-by-l1ne data 

Atwater and Ball Fowle 

(McDonald) 

Hoyt Howard et al. (1955) 

(Yamamoto) 

Watt Valley & Modifications 

Lacis and Hansen Howard et al. (1955) 

(Yamamoto) 

ASHRAE 

Machta 

Threlkeld 

(Best fit 3 locations) 

Burch et al. (many) 

Line-by-line data 

(Dave) 

Burch et al. (many) 

Line-by-line data 

ICastrov 

Vigroux 

Inn and Tanaka 

(Manabe & Strickler) 

Wulf 

Lauchli 

(Moon,) 

Vigroux 

Inn and Tanaka 

(Howard et a1. 1961 Handbook) 

Wulf 

Lauchl1 

(Moon) 

Burch et al. (many) 

Line-by-line data 

(Dave) 

14 

Burch et al. (many) 

Line-by-line data 

Fowle 

(Kastrov) 

Burch et al. (1960) 

Howard et al. (1955) 

Moon 

Moon 

Burch et al. (many) 

Line-by-line data 

(Dave) 
S=::!al:.' ___________________ ____ --'T::

,:.

.:34..4 

-� ��� 

2 . 10.4 Air Mass 

The ai r mass is a coefficient that accounts for the increased path length 

thro ugh which light rays must pass in the atmos phere when the s un is not 

directly overhead . When the s un is directly overhead , the air mass is 1 .0. 

The formal definition of air mass is given by Kondratyev [9] as 

M � .)("

� 

PdS/.)("

� 

pdz (42 ) 

where dz is an increment in the vert ical direction; ds is an increment along a 

s lanted path; and p the dens ity of air, or whatever component of the air that 

is being considered . This defini tion implies that differences in altitude 

between different s urface locations must be accounted f or by some means other 

than air mass. In calculating atmos pheric attenuation over a slant path, for 

example , the optical de pth for a vertical path is multiplied by the air mas s 

to obtain the total optical de pth. The vert ical optical depth incl udes the 

effect of the altitude at which one is working. Some authors include the 

beginning altitude in the air mas s and use the optical depth from sea level , 

or 1013 mb press ure . This is called the absol ute, or pres s ure-corrected , air 

mass , given by 

(43 ) 

where Po = 1013 mb. 

Kondratyev [9] 

z enith angle. 

using 

s ummarizes the methods of calc ulation for different ranges of 

For zenith angles < 60 °, s ufficient acc uracy can be obtained 

M = sec Z ( 44 ) 

where Z is the zenith angle . For 600� Z � 80 °, the effect of earth c urvature 

becomes important . Geometric considerations give 

{ 2 2 } 1 /2 M = ( r/H) cos Z + 2 ( r/H ) + 1 - ( r/H ) cos Z 

where r is the earth's radius and H the s cale height defined by 

H 

15 

(45 ) 

(46 ) 
S5�1 1*1 _________________________ TR::.

-.,;

344.;.

 

The constant Po is the surface level density. For air or the uniformly mixed 

gases, H = PO/ (pog), where g is the acceleration due to gravity and Po is the 

surface level pressure. 

For Z > 800, the effects of refractive index become important. The correct 

value of air mass in this region can be found in tables (Kondratyev, for 

example) or can be calculated with approximate expressions. An expression 

that this author has found to be correct to within 1% for Z < 890 is defined 

by Kasten [21] to be 

M = {cos Z + 0.15 (93.885 - Z)

-1.253} -1 (47) 

� 

This expression was used in the results given in this report (unless otherwise 

noted), and is called the relative air mass by Kasten. 

2.10.5 Simple Transport Equation 

Several forms of the transport equation have been used in the models described 

here. Some possible forms of the equation are: 

1

2 

1

3 

I

I = 1

0 T

R To T 

u 

T

w TA 

= 1

0 [TR 

= 1

0 [TR 

1

4 = 1

0 

T

o 

T

o 

T 

u 

- a ] T 

w A 

- a

w - a ] 

u 

TA 

[TM - a

w

l TA 

(48) 

(49) 

(50 ) 

(51) 

where T is the transmittance due to Rayleigh scattering, T

o is the transmittance 0' ozone, T

u is the transmittance of the uniformly mixed gases CO2 and 

0

2

' Tw and � are the transmittance and absorptance of water vapor, TA is the 

t.ransmittance of the aerosol, and TM is the transmittance of all molecular 

effects except water vapor absorption. 

Tables 2-2 and 2-3 were constructed using all forms of the transport equation 

(Eqs. 48-51) with Bird Models and the results from SOLTRAN. Two standard 

atmospheres that are built into SOLTRAN were used: the midlatitude summer 

(MLS) and the subarctic winter (SAW) models with sea level visibilities of 23 

and 5 km. 

Based on the results in Tables 2-2 and 2-3, it appears that Eq. 48 provides 

the closest agreement with SOLTRAN. The very simple form of Eq. 50 provides 

results that are comparable to Eq. 48. A theoretical basis for selecting any 

one of these forms of the transport equation as the best has not been established by the authors. However, one assumption implicit in Eq. 48 is that the 

attenuation by each constituent is independent of every other constituent. In 

other words, the transmittance measured in pure materials can be combined in 

the form of Eq. 48 to produce the transmittance through a mixture of 

16 
5='1'� ,, ______ ----__________ --=.

TR�- ..:

34::;,:;:.

4 

Table 2-2 . COMPARIS ONS OF DIRECT NORMAL IRRADIANCE FOR DIFFERENT FORMS 

OF THE TRANSPORT EQUATION USING BIRD MODELS AND SOLTRAN 

Zenith 

Angle Model I

I 12 1

3 14 SOLTRAN 

( de g) Atmosphere (W/m2

) (W/m2

) (W/m2

) ( W/m2

) (W/m2

) 

0.0 MLS 8 27.1 812.5 8 1 1.2 816. 6 833.5 

20. 0 V = 23 kIn 811.0 7 95.7 7 94.2 800.1 

30 .0 789 .0 7 7 2.8 77 1.3 7 7 7.8 

40 .0 754 .5 7 36.9 7 35.2 742 .8 756 .6 

50.0 7 02.1 682.3 680 .4 690 .0 

6 0 .0 621 .3 598 .5 596 .2 609 .1 617 .7 

70.0 490 .2 463. 3 460 .6 478 .4 

7 5 .0 392 .3 3 63.5 3 6 0.5 380 .5 386 .3 

80.0 261 . 7 233 .0 229 .9 2 48.7 25 8.9 

85 .0 101 .5 81.8 79.5 84 .3 102 .8 

0.0 }ILS 545 .8 536.2 535 .3 . 5 38.9 530 .6 

20 .0 V = 5 km 522 .4 512.6 5 11 . 7 5 15.4 

30 .0 491 .4 481 .3 480 .3 484 .4 

40 .0 444 .4 434 .0 433 .0 437 .5 

50 .0 377 .4 366 .8 365 .8 370 .9 

6 0 .0 2 85.1 2 7 4.6 2 73.5 279.5 270.3 

7 0 .0 1 63.8 1 54.8 153 .8 159 .8 

7 5 .0 96 .2 89. 2 88 .4 9 3 .4 97.2 

80 .0 35 .8 31.9 3 1 .4 34 .0 42 .4 

85.0 3.1 2.5 2.4 2.6 7.0 

17 
Table 2-3. COMPARISONS OF DIRECT NORMAL IRRADIANCE FOR DIFFERENT FORMS 

OF THE TRANSPO RT EQUATION USING BIRD MODELS AND SOLTRAN 

Zenith 

Angle Model II 1

2 1

3 1

4 SOLTRAN 

(deg) Atmosphere (W/m

2

) (W/m

2

) (W/m2

) (W/m2

) (W/m

2

) 

0.0 SAW 866.0 856.5 855.1 865.5 881.9 

20.0 V == 23 km 849.4 839.5 838.0 848.9 

30.0 826.8 816.3 814.7 826.4 

40.0 791.2 779.7 778.0 791.1 804.3 

50.0 737.2 724.0 722.0 737.5 

60.0 653.0 637.9 635.6 654.7 662.5 

70.0 515.9 498.0 495.1 519.6 

75.0 413.0 393.7 390.5 417.2 423.3 

80.0 275.3 255.9 252.6 277 .3 289.0 

85.0 106.2 92.6 90.2 98.8 120.0 

0.0 SAW 571.5 565.2 564.3 571.2 568.5 

20.0 V = 5 km 547.2 540.8 539.8 546.9 

30.0 514.9 508.3 507.4 514.7 

40.0 466.0 459.2 458.2 465.9 461.1 

50.0 396.2 389.2 388.1 396.4 

60.0 299.6 292.7 291.6 300.4 297.8 

70.0 172.3 166.3 165.4 173.6 

75.0 101.3 9 6.6 9 5.8 102.3 112.0 

80.0 37.6 35.0 34.5 37.9 50.5 

85.0 3.3 2.8 2.8 3.0 9.0 

18 
TR-344 

materials . One obvio us s it uation that violates this ass umpt ion is when nearcomplete absorption in a s ingle constituent occurs within the spectral band 

being cons idered . If one of the o ther cons t it uents absorbs in the same spectral location , over attenuation will occur in the final results . The form of 

Eqs. 49 and 50 make this s it uation even worse. 

19 

sail 
20 
S=�I TR-344 

SECTION 3.0 

MODEL COMPARISONS 

Where possible, each of the models was programmed on a computer to produce 

data for comparison. Parts of the Hoyt model were generated with a programmable hand calculator, and data from the Machta model were generated by 

using the worksheet format presented with the model. 

General transmittance data for the midlatitude summer (MLS) and the subarctic 

winter (SAW) atmospheric models have been generated with the SOLTRAN code. 

These atmospheric models are two of the standard atmospheres defined in 

SOLTRAN. They were chosen primarily because the amounts of ozone and water 

vapor defined in them represent extremes that could be encountered in the 

United States. The Rayleigh scattering due to molecules and the absorption of 

the uniformly mixed gases (C02 and 02) are relatively constant in these 

models. The aerosol conditions can be defined independently of the atmospheric model. As was mentioned previously, the results from the SOLTRAN code 

are for a spectral interval between 0.3 to 3.0�. If the calculations would 

have been made from 0.25 to 10.0 �, the results from SOLTRAN for individual 

atmospheric constituents could change. However, the total transmittance of 

all constituents should not change appreciably « 1%). 

The amounts of ozone and water vapor in the SAW model are 0.45 cm of ozone and 

0.42 cm of water vapor. The amounts in the MLS model are 0.31 cm of ozone and 

2.93 cm of water vapor. Figure 3-1 presents a plot of the broadband (0.3 to 

3. ° �) transmittance versus sec Z for all of the atmospheric components in 

the MLS model. Figure 3-2 presents the same type of results for the SAW 

model atmosphere. Both figures contain the results for a 23-km visibility 

aerosol at sea level. The aerosol used in this paper is the one defined in 

the LOWTRAN 3 version, and is representative of a continental or rural 

aerosol. 

Figures 3-1 and 3-2 show the relative importance of each atmospheric component 

as an attenuater of broadband radiation (0.3-3.0 �). CO2 and 0

2 are the 

least important elements, and they are omitted from some models. The next 

element exhibiting increased attenuation is 0

3

' followed by H20. The flattening of the curve for H

2

0 in Fig. 3-1 with increasing zenith angle suggests 

that the H

2

0 absorption bands are approaching saturation. Molecular scattering (Rayleigh scattering) dominates total molecular absorption at large zenith 

angles and has a greater effect than most individual molecular species at all 

zenith angles. The one exception to this statement appears to be H

2

0 absorption for high concentrations of H2

0 and for air masses < 2. The most significant attenuator at nearly all zenith angles is the aerosol. 

An aerosol that produces a 23-km meteorological range at sea level is considered to produce a relatively clear atmosphere. At higher surface altitudes 

and remote locations, it is not uncommon during winter months to observe 

meteorological ranges greater than 60 lan, which are extremely clear conditions. The data presented in Figs. 3-1 and 3-2 suggest that aerosol attenuation could be the most important attenuator at most locations throughout the 

United States. Unfortunately, it is also the component that is the most 

21 
S=�I �1 ______________________________________________ 

TR �-�3� 44 

1.0 

0.9 

0.8 

0.6 

0.5 

CD 

g 0.4 

-tIS 

-

·S 

� 0.3 

e 

.. 

0.2 

0.11.

--'-_..&.--'-_-'-.. �..I.-__ .. 

o 2 4 6 8 10 12 

sec� 

Figure 3-1. Transmittance versus Secant of Solar Zenith Angle for 

Midlatitude Summer Model 

22 
S=�II 

CD 

U 

c 

-tV 

.--

E 

U) 

c 

12 

I0.5 

0.4 

0.3 

0.2 

�---=====

CO2 + O2 

03 

H20 

0.

1��_��_-=--��� __ ..... 

o 

Figure 3-2. Transmittance versus Secant of Solar Zenith Angle for 

Subarctic Winter Model 

23 

TR-344 
S=�I '*1 __________________ _____ 

TR-_3_4_4 

difficult to measure and hence the least defined . It sho uld be emphasized 

that these conclusions are for broadband ( thermal) d irect insolation , and that 

the situation will change s ignificantly as the bandwidth is f urther restricted 

or global insolation is considered . 

The previo us section noted that the ASHRAE model assumes a plot of the total 

t ransmittance shown in Figs . 3-1 and 3-2 will s cribe a straight line . The 

s lope of this line provides the optical depth. An examination of Figs . 3-1 

and 3-2 demons trates that this ass umption is reasonable o ver a limited range 

of zenith angles . If the plot were of irradiance versus sec Z, the vertical 

intercept at sec Z = 0 would provide the extra te rres trial irradiance , which is 

us ually too low. 

Figures 3-3 and 3-4 present 0

3 absorpt ion data for five of the models described earlier . Figure 3-3 is for 0.31 cm of 03 ( MLS ), and Fig. 3-4 is for 

0 . 45 cm of 0

3 ( SAW) . The models produce s ignificantly different res ults when 

compared in this manner. However, the differences are minor when the effect iveness of 03 is accounted for. The t riangular data points are a res ult of 

performing a least squares fit to the S OLT AN data with the e quations of Lacis 

and Hansen (Bird model ) . The minor deviat ions of the t riangles f rom the 

SOLT k� data are a res ult of attempting to fit a s imple expression to various 

amounts of 0

3

' 

Figures 3-5 and 3-6 present model compar isons of H2

0 absorpt ion for the two 

standard a tmospheres being cons idered. S ince H20 absorption plays a s ignificant role in transmission calcula tions , the differences b etween the models 

sho uld be not iceable in the total transmission. 

Fig ures 3-7 and 3-8 present plots of transmittance vers us solar zenith angle 

for all molecular effects except H20 absorption for the SAW and MLS atmospheric models, respectively . Some of the models did not readily lend themselves to this particular calculation and are not included . The s urprise 

about these data is the accuracy of the s imple expression in the Atwater 

model. 

Fig ures 3-9 and 3-10 illustrate a comparison of aerosol t ransmittance for the 

MLS and SAW a tmospheric models with sea level vis ibilities of 5 and 23 km, 

respectively . In most cases , aerosol attenuation is independent of the atmospheric model; b ut the Watt model for aerosols is d ependent on the amount of 

H20. 

Hoyt 's model provides strong agreement with SOLT AN, b ut its tabular form is 

not as easily used as empirical formulas. 

range of t urbidity coeffic ient s to include 

data are f or aerosol s cattering only , and 

aerosol absorption in th� Hoyt model. 

The table covers an insuf ficient 

a 5 -km visibility aerosol. Hoyt 's 

there is an additional factor for 

The value of the upper layer t urbidity T

u used in Watt 's model was taken from 

historical plots that he produced. The val ue used , T = 0.02, was an approximate average of the his torical data. The rest of t�e parameter values for 

this model were taken directly from SOLT AN. 

24 
S=�·I I.I _______________________ -_344 

-� ��� 

0.09 

0.10 .----------------------, 

0.08 

0.07 

Q) 0.06 

(,) 

c 

ta -

e-0.05 

o 

(I) 

J:2 

< 0.04 

0.03 

0.02 

0.01 

Lacis & Hansen 

ASird 

O __ � ____ � __ � ____ � __ � ____ � __ � ____ � __ � 

o 10 20 30 40 50 60 

Sola� Zenith Angle (degrees) 

70 80 90 

Figure 3-3. Ozone Absorptanee versus Solar Zenith Angle for 0.31 em of 

Ozone (MLS) 

25 
TR-344 

S=�I -� 

:.J �� --

G,) 

u 

0.10 

0.11.- 

0.09 

0.07 A Bird 

lacfs & Hansen lI-

; 0.06 

-

eo 

� 0.05 

« 

0.03'=-

 

1-= 

0.02: _---6-..

 --

--

0.01 

O� __ � __ � ____ � __ � ____ � __ � ____ � __ � __ � 

o 10 20 30 40 50 90 

Solar Zenith Angle (degrees) 

Figure 3-4. Ozone Abso rptanee versus Solar Zenith Angle for 0.45 em of 

Ozone (SAW) 

26 
S=�I 1.11 ________________________ R_-_34

4 _ -� �� 

CD 

u 

c 

-ta 

a. .. 

0 

en 

..c:2 

<C 

0.7 

0.6 

0.5 

0.4 

0.3 

0.2 

. 

0.1 

0 

A Bird 

10 20 30 40 50 60 70 80 90 

Solar Zenith Angle (Degrees) 

Figure 3-5. Absorptance versus Solar Zenith Angle 

for-2:93em of Water Vapor (MlS) 

27 
S=�I TR-344 

0.16 .-

 

CD 

(J 

0.15 

0.14 

0.13 

0.12 

; 0.11 

-Q. ..

 

o . 

.a 0.10 

ct 

0.09 

'0.07 

.. Bird 

0.06 b=:

0.05�--�--------�----------------------� __ 

o 10 20 30 .40 50 60 

Solar Zenith Angle (degrees) 

70 80 90 

Figure 3N6. Absorptanee versus Solar Zenith Angle for 0.42 em of Water 

Vapor (SAW) 

28 
5=�1 TR-344 

Q) 

(,) 

r: 

-ca 

-

E 

en 

r: 

e 

.. 

1.0r-

 

0.9r= 

0.8 ,:

0.7 

0.6 

0.5 

0.4 

0.3 H20 - 2.93cm 

03 - 3.1 mm 

0.2 

0.1 

O� __ � ____ � __ � ____ � __ � ____ � __ � ____ � __ � 

o 10 20 30 40 50 60 70 

Solar Zenith Angle (degrees) 

80 90 

Figure 3-7. Transmittance versus Solar Zenith Ang le for All Molecular 

Effects Except for H20 Absorption (MlS) 

29 
5='. :=}I __________ �--__________ _ 44 

1.0r-

 

CD 

U 

C 

ca 

0.7 

= 

'E 0.5 

(I) 

c 

e 

I- 0.4 

0.3 

0.2 

0.1 

H20- O·4�cm 

03 - 4.5mm 

O� __ � __ � ____ � __ � __ � ____ � __ � __ � __ _J 

o 10 20 30 40 50 60 70 

Solar Zenith Angle (degrees) 

80 90 

Figure 3-8. Transmittance versus Zenith Angle for All Molecular Effects 

Except H20 Absorption (SAW) 

30 
S=�I i*I __ � ________________ TR_ 344 

CD 

u 

c 

-CI:S 

:: 

E 

en 

c 

!l 

I1.0 

, 0.9 

0.8 A Bird 

0.7 

0.6 

0.5' 

0.4 

, 0.3 

0.2 

0.1 

0 10 20 30 40 50 60 70 80 90 

S,?lar Zenith Angle (Degrees) 

Figure 3-9. ��rosol Transmittance versus Solar Zenith 

Angle for S-km Visibility Aerosol 

31 
S=�I 

CD 

(.) 

c 

-CD 

-

e 

CI) 

c 

e 

� 

1 .0 

0.9 

0.8 

0.7 

0.6 

0.5-

A Bird 

0.4 

0.3 Soltran 

0�2 

0.1 

10 20 30 40 50 60 70 80 90 

Solar Zenith Angle ( Degre�s) 

Figu re 3-10. Aerosol Transmittance versus Solar Zeni th 

Angle for 23- km Visibility Aerosol 

. 

32 

TR-344 
S=�I TR-344 

Figures 3-11 thro ugh 3-1 3 illustrate the total terrestrial s olar irradiance as 

a f unction of solar zenith angle for s everal models under different atmospheric conditions . The differences between some of the models are 

s ignificant . However , the differences ill ustrated here are dominated by the 

aerosol attenuation, and the d if ferences d ue to molecular effects are not 

obvio us. If a much clearer atmosphere were modeled or a more res tricted 

bandwidth were used , the difference in molecular effects would be more 

evident . 

In the res ults given for the Atwater model , the broadband o pt ical depth for 

the aerosol was taken from the Bird model. This causes close agreement with 

the SOLTRAN res ults , b ut Figs . 3-7 and 3-8 show that the b ulk of the molecular 

attenuation in this model agrees very well with SOLTRAN. Atwater's H2

0 absorption deviates s omewhat from SOLTRAN res ults . 

The res ults for Bird shown in Figs . 3-1 1, 3-1 2, and 3-l3 used Eq . 49 as the 

transport eq uat ion. 

Because of the diffic ulty of considering aerosol attenuation and Rayleigh 

scattering with Hoyt 's model , it is included only in Fig. 3-12 and gives 

values that are lower than the SOLTRAN res ults over the applicable region. 

Machta 's model agrees with SOLTRAN within ±-5% to 60° zenith angle . It is 

primarily l imited to clear air condit ions and zenith angles less than 60°. 

The values shown at 70° zenith angle are beginning to deviate somewhat. 

The Maj umdar model res ul ts have been g iven for the MLS a tmospheric model with 

V = 23 km ( see Fig. 3-11). The Maj umdar model is for low t urbidity c onditions 

and has no provision for varying the t urbidity . It is evident from Fig. 3-1 1 

that the t urbidity res ul ting from a s ea level visibility of 23 km is too large 

for this model . The model is very s imple , and it could be q uite acc urate if 

provisions were made to vary the aerosol a ttenuation. 

Appendix A presents tabular data for each a ttenua tion element for most of the 

models. Direct normal irradiance for the Bird model is given in Tables 2-2 

and 2-3 for the same atmospheric conditions modeled in Appendix A. 

This comparison and eval uation of exis ting s implified models for calculating 

the direct solar beam energy c onsidered s ix models - Hoyt , Watt, Lacis and 

Hansen, Atwater and Ball , Machta, and Maj umdar . Ideally, s uch models sho uld 

be c ompared according to the attenuation of each atmospheric cons tituent . By 

doing this , differences in calculating the total broadband direct solar energy 

can be specifically associated with differences in how they calculate 

at tenuation arising from each cons tituent . However , this was impossible 

because of the variety of techniques used by the models. For example , some 

models considered all molecular attenuation processes in a single express ion , 

making it imposs ible to distinguish attenuation from specific molecular 

constituents . The s pecific c omparisons performed considered the following : 

• absorptance d ue to 03 ( Lacis and Hansen; Hoyt ; Watt) , 

• absorptance d ue to water vapor (Lacis and Hansen; Hoyt; Watt ; 

Mc Donald) , 

33 
S=�I t: � 

______ _________ __ TR"'"_-�344 

I 

1 000 1&::=======-___ _ 

N -

E 

� -

Q) 

u 

. :; 100 

=a 

f! ... 

Maehta 

.A Bird 

• Maehta 

V= 23 km 

Uw=0.42 em 

Uo·0.45 em 

Majumdar 

�-- Watt 

�-- Soltran 

�-- Atwater 

10

o 

�--�--�--�----�--�--�--�----�� 

10 20 30 40 50 60 70 80 

Solar Zenith Angle (Degrees) 

Figure 3-11. Solar Irradianee versus Solar Zenith Angle, 

Visibility = 23 km (MLS) 

34 

90 
S=�I ' 

COl -

E 

� -

� 

u 

c 

.! 

"C 

ftI ..

 

-..

 

.. 

ftI 

'0 

(J) 

1 000 

1 00 

.& Bird 

V=5 km 

Uw =2.93 em 

Uo= 0.31 em 

1 0

o 

�- 

10 20 30 40 50 60 70 

Solar Zenith Angle (Degrees) 

80 90 

Figure 3-1 2. Solar I rradianee versus Solar Zenith Angle, 

Visibili ty = 5 km (MLS) 

35 

TR-344 
TR-344 

=�1 

I 

--

-� �� 

1 000 

N -

E 

-� 

Q) 

u 

c 1 00 

.! 

-0 

as .. 

-.. 

.. 

as 

'0 

(J) 

o 10 

A Bird 

• Machta 

V= 23 km : 

Uw = O.42 cm 

Uo = O.45cm 

30 40 50 60 70 

Solar Zenith Angle (Deg rees) 

80 

Soltran 

"* Atwater 

90 

Figure 3- 13. Solar Irradiance versus Solar Zenith Angle, Visibility = 23 km (SAW) 

36 
S5�1 '*1 ________________ � ______ __

T__3_44 

• the atmospheric transmittance due to all molecular processes except 

water vapor absorption ( Atwater; Watt) , 

• the atmospheric transmittance due to aerosols ( Hoyt ; Watt) , 

• the direct solar beam irradiance versus s olar zenith angle (Machta; 

Hoyt; Maj umdar; Watt; Atwater ) . 

In general, two sets of atmospheric conditions were considered : midlatitude 

summer ( MLS) and s ubarctic winter ( SAW) . The MLS c onditions are characterized 

by a precipitation water vapor of 2. 93 cm (high) and an ozone . amoun.t of 

0.31 cm ( low) . The SAW conditions are distinguished by a water vapor of only 

0.42 cm ( low) , and an ozone amount of 0.45 cm (high) . Within these 

conditions , two sets of atmospheric aerosol conditions were studied : clear 

( visual range = 23 km at sea level) and turbid ( visual range = 5 km at sea 

level). By using these models and conditions , a reasonable range of 

atmospheric s tates was considered. 

37 
38 
!s::�1 ,�, ______________________________________________________________ TR-344� 

SECTION 4.0 

SUMMARY AND CONCLUSIONS 

The recently developed , rigorous , spectral solar radiation transport model 

( SOLTRAN) was modif ied to have the capability of calculating the broadband 

( thermal solar energy) direct solar beam irradiance. Such energy is utilized 

by solar thermal concentrator devices. The modifications to SOLTRAN allow the 

prediction of the broadband solar irradiance for various atmospheric conditions and for various slant paths ( relative air mass) through the atmosphere. The atmospheric constituents considered are carbon dioxide ( C0

2

), 

oxygen ( 02

)' o zone ( 03

)' water vapor ( H20 ) , aerosols , and the molecular 

s cattering (Raylei'gh) . This modified version of S OLTRAN can be utilized to 

generate , the intensi ty of the direct solar beam versus time of day for various 

locations and atmospheric conditions . Such information is crucial to the des ign and performance analyses of solar concentrator sys tems and central receiver systems . 

After having developed the broadband-thermal version of SOLTRAN, it was then 

used to perform the following investigations and analyses : 

• definition of the relative significance of the various atmospheric 

constituents to the attenuation-transmittance of the direct solar beam 

energy; 

• comparison and evaluation of simplified models/algorithms ; 

• development of an improved , s implified model for solar energy. 

The broadband SOLTRAN calculation delineated the relative significance of the 

various atmospheric constituents in the transmittance of direct solar beam 

energy . Ae rosols appear to dominate the attenuation for a reasonable range of 

atmospheric conditions . Molecular scat tering (Rayleigh) is next in importance , followed by water vapor absorption. These three attenuation processes s-aerosol scattering , molecular s cattering , and water vapor absorptionnearly completely determine the transmittance of the atmosphere to direc t 

solar energy on a clear day. Attenuation caused by CO2, O2, and 0

3 is 

minor. Because aerosols and water vapor are so important in determining the 

available direct solar beam energy, one needs high-quality measurements of 

their geographic and temporal characteris tics. In the absence of actual 

measurements of the direct beam, s uch measurements could be used (with a model 

like SOLTRAN) to assess the availability and character of the direct solar 

beam energy. The SERI Energy Resource Asses sment Branch will determine the 

availability of the aerosol ( turbidity) and the water vapor measurements 

database , their quality, their applicability to assessing direct solar beam 

energy , and whether improved ins tnnnentation and techniques are required to 

meet the needs of solar energy applications . 

The comparisons of the ozone absorptance revealed significant differences 

between the s implified models and S OLTRAN. The magnitude of the differences 

is a function of solar zenith angle and amount of ozone. In general , S OLTRAN 

predicts a lower absorptance due to ozone than the Hoyt , Watt , and Lacis and 

Hansen models . The total range bf the differences is determined by the Hoyt 

39 
S=�I I*I __________________________ ..

3::.

44 

( high) and S OLTRAN ( low) models , being approximately 30% to 50% in absorptan�e. The Watt and the Lacis and Hansen models appear to give q uite s imilar 

res ults , which fall between the Hoyt and S OLTRAN results. Although the 30% to 

50% differences are certainly significant , they are not considered to be a 

s ignificant source of differences in calculating the total broadband direct 

i rradiance because the contribution of o zone to the total broadband attenuat ion is minor . 

For low amounts of water vapor (0. 42 cm) , the comparisons of the s implified 

models with S OLTRAN revealed that the Lacis and Hansen and Hoyt models gave 

results s imilar to S OLTRAN, while the McDonald and Watt models gave s ignificantly lower values for water vapor absorptance . These differences depend on 

solar zenith angle , but they range from 25% to 75%. For high amounts of water 

vapor (2.92 cm) , the Hoyt , McDonald , Lacis and Hansen, and S OLTRAN models 

yield similar resul ts from an approximate solar zenith angle of 0" to 60" . At 

greater angles the models diverge s omewhat. The Watt model resul ts in s ignificantly lower values , by about 50%, than all other models. Again, the 

s ignificance of these differences to the calculation of the broadband direct 

irradiance is determined by the relative s ignificance of water vapor to the 

total attenuation. 

The comparisons of the transmittance due to all molecular effects except water 

vapor revealed close agreement between the Atwater and SOLTRAN results . The 

Watt model gave results that were lower in t ransmittance , depending on solar 

zenith angle . 

The comparison of aerosol t ransmittance versus solar zenith angle for the MLS 

conditions with turbid aerosol condit ions revealed that the Watt model gave 

s ignificantly higher values than the S OLTRAN model. For the SAW with clear 

( visual range of 23 km) c onditions , the Hoyt and SOLTRAN models gave s imilar 

results . The Watt model gave significantly higher t ransmittance values . 

The comparisons of calculated direct beam solar irradiance versus solar zenith 

angle were performed for the MLS condit ions , with a clear atmosphere ( visual 

range = 23 km at sea level) and a turbid atmosphere ( visual range = 5 km at 

sea level ) . The comparisons for the clear atmos phere revealed that the 

Machta, Atwater , and SOLTRAN models agree fairly closely; the Hoyt model 

results in lower values of direct irradiance, but they are within approximately 1 0% of the SOLTRAN/ Atwater/Machta values . However , at large solar 

zenith angles ( greater than 70" ) the models begin to diverge s ignificantly. 

The Watt model agrees favorably with the S OLTRAN/Atwater/Machta values up to a 

solar zenith angle of about 7 0"; at greater angles the Watt model predicts 

s ignificantly higher values. The Maj umdar model gives much higher values of 

direct irradiance versus solar zenith angle than all o ther models , by as much 

as 20%. The comparisons for a turbid atmosphere could only be performed for 

the Watt, Atwater , and SOLTRAN models due to l imitations in the o ther models 

to c lear conditions . It was shown that the Atwater and S OLTRAN model agree 

favorably to a zenith angle of 70" . The Watt model predicts s ignificantly 

higher values , especially past zenith angles of 50 ". Below zenith angles of 

50" , the Watt values are within about 10% of the S OLTRAN/Atwater results . 

Comparisons were also performed for the SAW atmosphere with c lear conditions . This again revealed agreement between the S OLTRAN and Atwater models, 

and characterist ically, that the Watt model tends to predict high values . 

40 
TR-344 

However, for these SAW and clear conditions , the Watt model is in fairly good 

agreement to zenith angles of about 70° . 

Finally, the rigorous SOLTRJU� results were used to develop an improved , 

s implified model for p redicting the direct solar beam energy as a f unct ion of 

a tmospheric c onditions and solar zenith angle . The improvements consist of 

higher accuracy (as compa red to the S OLTRJU� results) and the ability to handle 

readily available , specific inputs to characterize the a tmospheric water vapor 

and aerosols. Mathematical formulations were derived by comparison with 

S OLTRAN res ults for the a tmos pheric transmit tance components of molecular 

( Rayleigh) scattering , ozone absorption, uniformly mixed gas ( C0

2 and 02

) absorption , water vapor absorpt ion, and aerosol scattering and absorption, as 

functions of relative air mass ( solar zenith angle) . The inputs to the model 

( Bird) are s urface pressure , precipitable ozone, precipitable water vapor, and 

aerosol turbidity at 0.38 � and at 0. 500 un. Thus the Bi rd model allows the 

calculation of the direct s olar beam irradiance as a function of available 

atmos pheric parameters , which properly consider the s ignificant water vapor 

and aerosol cons tituents. 

The absolute accuracy of the Bird , S OLTRAN, and o ther models can be determined 

only by comparisons with actual measurements of the direct solar beam irradiance and measurements of the atmospheric inputs. Unfortunately , s uch comparisons and measurements have been lacking in the past in the vis ible 

region [46-50 J ; when done , they will probably resul t in improvements in the 

S OLTRAN and s implified models. The SERI Energy Resource Assessment Branch and 

t he Solar Energy Meteorological Res earch and Training sites ( university research programs s ponsored by DOE to collect insolation and meteorological research dci'ta at eight locations within the United States) will be collecting 

s uch data and performing research that will greatly advance the s tate of 

knowledge concerning the a tmospheric influences on the d irect solar beam irradiance . Consequently, this will lead to improved prediction models for the 

d irect solar beam irradiance and improved design and predict ions of solar 

energy conversion devices . 

41 
42 
S=�I 

r;.1 __________________

___ T;:.:,R:.

-3=.4:.4 

-� �� 

1. 

SECTION 5 . 0 

REFERENCES 

Selby, J. E. A . ; Kneizys, F. X. ; 

Atmospheric Transmittance/Radiance: 

78-0053; 1978. 

Chetwynd, J. H. ; McClatchey, R. A. 

Computer Code LOWTRAN 4. AFGL-TR2. Selby, J. E. A . ; Shettle, E. P. ; McClatchey, R. A. Atmospheric Transmittance from 0.25 to 28 . 5�: Supplement LOWTRAN 3B. AFGL-TR-76-0258; 

1976. 

3. Selby, J. E. A . ; McClatchey, R. A. Atmospheric Transmittance from 0 . 25 

to 28 . 5�: Computer code LOWTRAN 3. AFCRL-TR-75-0255; 1975 . 

4. Thekaekara, M. P. "Extraterrestrial Spectral Irradiance ." The Extraterrestrial Solar Spectrtml, A. J. DrUl

mond and M. P. Thekaekara, 

eds. Mount Prospect: Institute of Environmental Sciences; pp. 71-133 ; 

1973. 

5. McClatchey, R. A . ; Fenn, R. W . ; Selby, J. E. A . ; Volz, F. E. ; 

Garing, J. S. Optical Properties of the Atmosphere (Third Edition). 

AFCRL-72-0497; 1972. 

6. Atwater, M. A.; Ball, J. T. "A Numerical Solar Radiation Model Based on 

Standard Meteorological Observations." Solar Energy. Vol. 21: 

pp. 163-170 ; 1978. 

7. Atwater, M. A.; Brown, P. S. "Numerical Computations of the Latitudinal 

Variation of Solar Radiation for an Atmosphere of Varying Opacity." J. 

Applied Meteorology. Vol. 13: pp. 289-297; 1974. 

--

8. Paltridge, G. W . ; Platt, C. M. R. Radiative Processes in Meteorology 

and Climatology. New York: Elsevier; 1976. 

9. Kondratyev, K. YA. Radiation in the Atmosphere. New York: Academic 

Press; 1969. 

10. 

11. 

Mc Donald, J. E. "Direct Absorption of Solar Radiation by Atmospheric 

Water Vapor ." J. Meteorology. Vol. 17: pp. 319-328 ; 1960. 

Hoyt, D. V. 

Solar Energy. 

"A Model for the Calculation of Solar Global Insolation." 

Vol. 21: pp. 27-35 ; 1978 . 

12. Lacis, A. L. ; Hansen, J. E. "A Parameterization for the Absorption of 

Solar Radiation in the Earth's Atmosphere." J. Atmospheric Science. 

VoL 31: p p • 118-13 3 ; 1 974. 

13. Machta, L. Workbook for Approximate Calibration of Solar Radiation 

Sensors. Silver Spring, MD: National Oceanic and Atmospheric Administration; TM-ERL-ARL-70 j 1978 . 

43 
S=�I i*1 _________ --

 _____________ ..:

TR ..

3..44 

14. Braslau, N. ; Dave , J. V. Effect of Aerosols on the Transfer of Solar 

Energy Through Realistic Model Atmospheres. Palo Al to, CA: I BM Research; RC4114; 1972. 

15. The American Society of Hea ting, Ref rigeration and Air Conditioning 

Engineers . ASHRAE HANDBOOK OF FUNDAMENTALS. 1972 ed. 

16. Jordan, R. C . ; Liu, B. Y. H . , eds . Applications of Solar Energy for 

Heating and Cooling of Buildings . ASHRAE-GRP-170; 1977 . 

17. Hul strom, R. L. Insolation Models) Da ta and Algorithms . Golden, CO: 

Solar Energy Research Institute; SERI/ TR-36-1 10; 1978. 

18. Watt , D. On the Na ture and Distribution of Solar Radiation. U. S. 

Depart�ent of Energy; HCP/T255 2-0 1; 1 978. 

19. Moon, P. "Proposed Standard Solar-Radiation Curves for Engineering 

20. 

21. 

22. 

23. 

24. 

25. 

26. 

27. 

Use ." J. F ranklin Ins titute . Vol . 230 : pp . 583-6 17; 1940. 

Majumdar, N. C. ; Mathur, B. L . ; Kaushik , S. B. "Prediction of Di rect 

Solar Radiation for Low Atmospheric Turbidity." Solar Energy . 

Vol . 1 3 : pp . 383-394 ; 1 97 2. 

Kas ten, F. "A New Table and Approximate Formula for Relative Opt ical 

Air Mass ." Arch. Meteoral . Geophys . Bioklimatal . Series B, Vol. 14 

(No . 2 ) : pp . 206-223 ; 1 966. 

Davie s , A. D . ; Hay , J. E. "Calculation of the So lar Radiation Incident 

on a Horizontal Surface ." Proceedings , First Canadian Solar Radiation 

Da ta Workshop. April 1 7-1 9, 1978. Canadian Atmospheric Environment 

Service ; 1979. 

Burch , D. E. Semi-Annual Technical Report, Inves tigation of the Abs orption of Infrared Radiation by Atmospheric Gases. Philco Ford Co rp .; 

Aeronutronic Report U 4784 ; Contract No . F19628-69-C-0263 ; 1970. 

Burch , D. E.; Gryvnak, D. A. Ab sorption by H2

0 Be tween 5045 and 

14, 485 cm-1 (0.69 to 1.98 un Region) . Aeronutronic Report U-3704; 1 966. 

Burch, D. E . ; Gryvnak, D. A. Strengths , Widths and Shapes of the Oxygen 

Lines Near 7600 Angs troms . Aeronutronic Re port U-40 76; 1967. 

Burch, D. E . ; G��nak, D. A . ; Pat ty, R. R. Absorption by CO! Between 

4500 and 5400 cm (2 � Region) . Aeronutronic Report U-2 955 ; 964. 

Burch, D. E . ; G�1vnak, D. A . ; Patty, R. R. Absorption by CO2 Between 

6600 and 7 125 cm (1.4 � Region) . Aeronutronic Re port U-3127 ; 1 965a . 

i 

28. Burch, D. E . ; Gry�ak, D. A . ; Patty, R. R. Absorption by CO2 Between '" 

8000 and 10, 000 cm (1 to 1.25 � Re gion) . Ae ronutronic Report U-3 200 ; 

1 965b . 

44 
S=�I 

f�.1 ___________ --

_____________ T=R�-..

3..44..;. 

-� ��� 

29. 

30. 

31. 

32. 

Burch, D. E.; Gr1vnak, D. A. ; Patty, R. R. Absorption by C02 Between 

5400 and 6600 cm - (1 .6 \.l

 Region) . Aeronutronic Report U-320 1; 1 965c. 

Burch, D. E . ; Gr1vnak, D. A. ; Patty, R. R. Absorption by H�O Between 

2 800 and 4500 cm- (2. 7 \.l Region) . Aeronutronic Re port U-320 ; 1 965d . 

Burch, D. E . ; Gr1vnak, D. A. ; Patty, R. R. Absorption by CO2 Between 

7125 and 8000 cm- (1.25 to 1.4 \.l

 Region) . Ae ronutronic Report U-3930; 

1967 . 

Burch, D. E . ; Gryvnak , D. A . ; Pa tty, ?-. R. Ab sorption by CO2 Between 

3100 and 4100 cm-1 

��--��������----��--�----��--�----��� 

(2.44 to 3.22 ].

 Region) . Aeronutronic 

Report U-4 132; 1968. 

33. Burch, D. E . ; Gryvnak, D. A. ; Singeleton , E. B . ; France, W. F . ; 

Williams , D. Infrared Absorption by Carbon Dioxide , Water Vapor and 

Minor Atmospheric Constituents. AFCRL ; Re search Contract AF19 (604)-

2 633; Ohio S tate University; 1962. 

34. McClatchey , R. A. ; Benedict , W. S . ; Clough , S. A. ; Burch , D. E . ; 

Calfee , R. F . ; Fox, K. ; Rothman, L. S . ; Garing , J. S. AFCRL Atmospheric 

Line Parameter Compilation. AFCRL-TR-73-0096 ; 1 973. 

35. Fowle, F. 

Vol. 42 : 

E. " The Transparency of Aqueous Vapor ." 

pp . 394-4 1 1; 1 9 15. 

Astrophysics . J. 

36. Yamamoto , G. "Di rec t Ab sorption of Solar Rad iation by Atmospheric Water 

Vapor , Carbon Dioxide and Molecular Oxygen." J. Atmospheric Science . 

Vol. 19: pp . 182-188 ; 1 962. 

37. Howard, J. N. ; Burch, D. E . ; Williams , D. " Infrared Transmission of 

Synthetic Atmospheres , Parts I-V." J. Optical Society of America . 

Vol. 46 : pp . 186-190, 237-2 4 1, 242-245 , 334-338 , 452-455 ; 1 956. 

38. Valley, S. L . , ed . "Solar Electromagnetic Radiation . " Chapter 16 in 

Handbook of Geophys ics and Space Environments. Air Force Cambridge 

Research Laboratories ; 1965. 

39. Wulf , O. R. "The De termination of Ozone by Spectrobolometric Measurements ." Smithsonian Mis cellaneous Collection. Vol. 85 (No . 9): p. 9; 

1 93 1 . 

40. Lauchli, A. "Zur Ab sorption der Ul travioleten St rahlung im Ozon. " Zs . 

F. Phys . p. 92; 1 929. 

41. Manabe , S . ; Strickler, R. F. 

with Convective Adjus tment ." 

pp . 361-385 ; 1 964. 

"Thermal Equilibrium of the Atmosphere 

J. Atmospheric Science . Vol. 2 1 : 

42. Vigroux, E. "Contributions a i ' etude Experimentale de l' absorption de 

l ' o zone." Ann. de Physique . Vol . 8: pp. 709-762; 1953. 

45 
!;::�I I __________________________ �-----------------------------------TR--3 4-4

-� ��� 

43. Inn, E. C. Y . ; Tanaka , Y. "Absorption Coefficient of Ozone in the 

Ultra-Violet and Visible Regions ." J. Optical Society American. 

Vol . 43: pp . 870-873 ; 1 953. 

44 . Howard, J. N. ; Burch , D. E. ; Williams , D. Near-Infrared Transmiss ion 

Through Synthetic Atmospheres. AFCRC-TR-55-213; 1 955 . 

45. Burch, D. E . ; Gryvnak, D. ; Williams , D. The Infrared Ab sorption by 

Carbon Dioxide . Ohio State University Research Foundation; Report on 

Project 778; 1960. 

46. She ttle , E. P. ; Fenn , R. W. "Models of the At"mospheric Aerosols and 

Their Optical Properties ." Proceeding of AGARD Conference No. 183 , 

Optical Propagation in the Atmosphere . pp . 2 . 1 -2 . 16, presented at the 

Electromagnetic Wave Pro'pagation Panel Symposium, Lyngby , Denmark; 27-3 1 

Oc tober 1 975. 

47. Roberts , R. E . ; Selby , J. E. A. ; Biberman, 1. M. " Infrared ContinulIDl 

Absorption by Atmospheric �vater Vapor in the 8-12 ).m Window. " Applied 

Optics. Vol. 14: p. 2085 ; 1 976. 

48. 

49. 

Haught , K. M. ; Corday , D. 

Transmis sion Measurements : 

M. "Long-Pa th High-Re s olution Atmospheric 

Appl ied Optics. Vol . 17: 

Comparison with LOWTRAN 3B Predic tions ." 

pp . 2668-2670 ; 1 978. 

McClatchey, R. A. ; Kneizys , F. X. 

pheric Transmiss ion Mea surements : 

tions ; Comments ." Applied Optics. 

"Long-Path High-Resolution AtmosComparison with LOWTRAN 3B PredicVol . 18: pp . 592-593 ; 1 979. 

50. Haught , K. M. ; Cordray, D. M. "Long-Path High-Re solution Atmospheric 

Transmis s ion Measurements : Comparisons with LOWTRAN 3B Predictions ; 

Reply to Comments ." Applied Optics. Vol . 18: p. 593 ; 1 979. 

46 
5»5'1 � �, __________________ �� ____________ ______ �T=R-�3�4�4 

APPENDIX 

TABULATED MODEL DATA 

A compilation of computer outputs for the Atwater and Ball , l-lat t, Bird , Lacis 

and Hansen, Maj umdar, and the Hoyt models are presented. The a tmospheric 

models used were the Hidlat itude Summer (MLS ) and the Subarctic Winter ( SAW) 

models with 23 km and .5 km sea level vis ibilities for each model. The direct 

normal irradiance (IDN) is not included for the Bird models s ince it was given 

previously in Tables 2-2 and 2-3 . The SOLT AN output is also lis ted in 

Tables 2-2 and 2-3 . The IDN is not given in the Hoyt model, becaus e parts· of 

the calculation that required look-up tables were done on a hand calculator . 

F or the MLS model , 2.93 cm of H20 and 0.31 cm of 03 was used . The SAW used 

0.42 cm of H2

0 and 0.45 em of 0

3

. The turbidity at 0.5 and 0.38 lltn wavelength 

was 0 . 2733 and 0 . 3469 , respect ively , for V = 23 km; and 0 . 9243 and 1 . 1 7 27 , 

respectively, for V = 5 km. In all cases , the incident extraterrestrial irradiance was 1353 W/m2

• 

Al 
Table A-1 . Tab ulated Data for the Midlatitude Summer 

Atmosphere (V = 23 km) for Several Models 

10 13�3. 0000 

Ulol 2.90300 

uc . 3100 

PI! 1 013. 00M T1'ItJ, • . 2733 TMJ3S- . 30,;" T1'ItJ1Ia . 1913 

IiI�T� nrl!CCT I'II'!AI. 

ZENITH IiIIRI'IASS TIi! Al

 TI'I I'DIt 

O. 0000 1 . 0000 . a2159 .1063 .a901 a7'.n"iIa 

20. 0000 1 . 06"1 . 91156 . 1 0S3 . S8!53 8'7.6348 

30. 0000 1 . 1!14!5 .SOIS . 1 110 . 13788 832._1 

40. 0000 1 .30!50 . 7"0 . 1 1'1 . 36e6 n ... I ..

 ' 

150. 0000 1." "8 . 7"27 • til .. . s�a9 73'. 00" 

60. 0000 t.�976 \ liaa4 • lalla . sa77 643. 3732 

70. 0000 2.9148 . '72!5 . 10,;' . 7a33 "';I3.2S2JI

 

7!5. 0000 3.S"19 . "7" . 1!592 . 7"'2 3aO. I_ 

80. 0000 '.68 ..

 .3370 . 17'0 .;;,1

2 aas._ 

8S. 0000 1 0. $068 . li .. 1 .2177 • !5"2'; 15".'''01

 

IoITT DIRECT ffilRI'!AI. 

om. 1'''

 T>lacA 1'>I2I

S Ttl3 TAIR 111

 

0.0 .87,*, . 914;; .';1379 . 97';8 .9016 8'9. 1 987 

30.0 . 9740 . 9137 . 9341 . 9763 . S"74 88".2390 

3 0.0 .S6159 . 9Ia, . 9287 • 97!S7 .S917 863.3708 

40. 0 .S!530 . :HOS .91'" . 9747 . 8826 S3 I. 6"'56 

SO. O . S3a" • "oa3 . '0" • 9724!' ._3 78a. I!!I' 

60. 0 • 79»a . 9047 . an7 . %99 .,04SI 704.3a94 

:'0. 0 . 73� .s"';! • 32,H .%� .8029 !!73.la .. " 

i'!$� 0 . 6aaa . $,",

3 . 7813 . ?'86 . 770l! "77. 0l!26 

80. 0 . !!944 .88% . ii933 .'l'469 . 7144 336. 1S9 

9!5. 0 .4331 . S800 .4$% .n09 . 61a3 143. 9323 

HIlYT nll!eeT MIlRM!. 

ZEl'IITIt ""

 Atlli AIl3 AOi 

O. 0000 .1397 . 007!! • 0215S . 007:1 

a:o. OOI]O .1426 . 0076 . 026" . 0079 

30. 0000 .14&4 .0079 . 0274 • 009!! 

"0. 0000 . 1�23 • alia I . 02ge . OO� 

!O. 0000 . 1612 . ooas . 0310 . OltO 

60. 0000 . 1 747 . 0091 . 0344 . 0137 

70.0000 . 1%9 . 0101 .. 0401 . 0190 

7!I. OOOO • a 1"7 .0109 . 0448 . 0a44 

ao. 0000 . 2442 . 01U • 0�23 . 0338 

85.0000 • 2938 .01 ..

 . 0"9 • Ol!7S 

I. I S >120 FlHII 113 

nN.,.>I AIR_S "" T03 

O. 0000 . 9995 . 13"1 . 9776 

20. 0000 1 . 063" . 1�63 .9769 

30. 0000 1 . 1'36 . 1 392 . 9761 

"0.0000 1 •• 037 .1437 � 9747 

!l0. 0000 1.!5l525 . 1503 . 97<:5 

60. 0000 1 .. 90927 . 1 000 • 968a 

70.0000 2. 139'97 . 17!51 .961a 

7:5. 0000 3.S076 .19';!I • 95!52 

80. 0000 '. !5790 • .!oae . �32 

8!1. 0000 1 0.3163 . 2299 .. 91 ..

 9 

I"!AJU"DAR DIRECT "aRM!. 

ZENI,. III � TT 1'>l21l 11)" 0. 0000 1 1!!0. 6024 .90,,3 �31.a'39a 

ZO. OOOO 1 139.9371 .M67 919. 565� 

30. 0000 I lZS. 0SI5' . S031 903. 5S77 

"0. 0000 1 160.7170 . 7977 979. 0171 

!l0. 0000 1 061.5163 . 7997 93S.a4"!

S 

60. 0000 9!'!5.:"6 . 7778 774. 3037 

70. 0000 972.30S" . ;�as 061 ..-9765 

7!I. 0000 764.20:53 . 7441 :568.6819 

ao. oooo :190. 3:129 . 7aa .. 4a6 ....

 91� 

8:1. 0000 296. 0123 . 6W 202.6061 

BIRD II IREieT "CRI'!AI. 

:EI't TA Ttl3 TCII2 TR TM ""

 

0 .. 0 . 9122 .983" .997 .. . 9137 . 991 0 . 10.19 

20.0 . 9024 . ge46 . 9972 . 9094 . a863 . 123!5 

30. a � 7990 . 9916 _ .913.9 . 9033 . 8799 . 12�7 

<40 .. 0 . 7672 . 97':\9 • 996!! o S�37 . a697 . 1291 

SO. O . 7330 . 5'772 • 99S9 . 871.13 .8S'" . 1340 

6�. 0 .�771 . 9727 . 99"9 .8S31 . 9a93 . 141\ 

70. 0 • !l777 . 96"3 . 9934 �a074 • 78S6 . 1:520 

7!I.0 . 4949 • 9l5�6 • 98a2 � 7ea4 .. 7463 . 1601 

ao. o . 3�94 . 9430 .9$03 . 7078 . �S67 � 1717 

95. a . 174� .�Hl;; . 9770 . 61:17 . S"92 . 1907 

A2 
Table A- 2. Tabulated Data for the Midlatitude Summer 

Atmosphere (V = 5 km) for Several Models. 

10 13,3. 00GO 

uw 2.9300 

ua . 3100 

PR 1 013. 0000 

TAWl - . 9243 

TAU3S- 1 . 1 727 rAUB- . '''

9 

A'NATIR DIRECT J'tCRI!Al. 

ZEl'IInt "U R�ASS TA AW T� IDII 

0 . 0000 1 . 0000 .:5236 .10,;3 . 8901 ,"�. aS9� 

20. 0000 1 . 0641 .SGa4 .loe3 . ae� S2a.I399 

30. 0000 1 . 1:54' ... 738 .111 0 • 87ee .�.2'62 

<-0.0000 1 .30$0 . . 42"9 .11:51 . eee,; 43a.2IS" 

'0. 0000 1." ,,* . 3"7 . 1214 .a'29 3"1 .�9 

60. 0000 1 .9976 . 27'" . 130a . a277 2S8.9480 

70. 0000 2.'11<48 . 1'17 . 1"" .. t$33 131).7:111 

;"S.oooo 3.a41l1 . 0933 - 1'92 . 74:2 66. 0326 

ao. oooo '. 6&46 . 02'3 . 1790 .,,812 17. 1 770 

a,. oooo 1 0.9068 . 0009 .2177 . S4a6 .3790 

>JATT DIRlcr II

I!Al. 

ZEH TA T"aDiI

 TK2DS T03 TA!!

 lim 

0.0 . '�3 . 91'" . 9379 . 37�8 ,, :�Olf; 609."'83 

20. 0 . '8 12 .9137 . 9341 . 9763 . S97" �aa. 03ao 

30. ° . 15622 .91il5 . 9287 .. 9757 .891 7 $60.8678 

4". 0 . :5329 .9108 �9197 . 97017 . ee26 '19.4331 

150. 0 . 4889 . 908S . 90'1 . 9729 . a.as .,9.a331 

60. 0 .4232 .9047 . 8797 . 9693 . S"'5 1 373.41704 

70. 0 . 3210 . 8992 .8291 .963' . eoa9 a'o ... .,.

o 

7:5. 0 ... ,o2 . 39:53 .7813 . 91586 ,, 'lila 1 74. 8369 

80. 0 . 11583 . aa% .6933 . ..

 89 . 7142 99. '032 

as. 0 . osa<- .8aoo . 48"6 . nll9 . 6 123 1 7. ""47 

HOYT DIRECT NCRI!Al. 

UIIIT>! ",.

 ,;ca2 A03 A!l2 

Q. o ooo . 1397 . 0075 . 02sa . 007S 

ao. oooo . 1426 . 007.6 . 0264 . 0079 

:30. 0000 .. 1464 . 0079 . oa74 . 008' 

40. 0000 . 1:'$23 . 00$1 . oalla . 00" 

'0. 0000 . 161a . 00815 . 0310 . 0 ltO 

60. 0000 • 1147 . 0091 . 034 .. . 0)137 

70.0000 . 1969 . 0101 . 0401 . 0190 

7:5. 0000 ... 147 . 01 0' . 0_ . 0l!42 

�o. 0000 .2422 . 012.i . 0'23 . 033a 

a,. 0000 . 2938 . 0144 . 0663 . 0'78 

LAC I S "21l AnD 03 

ZEIIIT>! "'IRIO'ISS AW TIl3 

0.<1000 . 999� . 13<-1 . 3776 

ao. oooo 1 . 0634 . 1363 . j76' 

30.0000 1 . 1S36 . IS92 . 1761 

40. 0000 1.3037 . 1437 . '747 

sO. oooo 1.':'$2' . 1'03 .972' 

60. 0000 1.-j�27 . 1 600 •• 689 

70. 0000 2. 8997 . 1�1 .%IS 

7'.0000 3. 8076 . 19615 . 3"2 

ao. oooo �,,!5790 .2028 . 943;a 

8' . 0000 1 0.3163 .a299 . 9149 

I'IA�U"DAA DI1>£CT l'IOR"fI 

ZEIIITH 10 X TT TK21l tDIi 

0 . 0000 l1S0. 60a. . a093 93 1.;:'298 

aa.Moo 1 t3'.9371 . 9067 9a.,�" 

30. 0000 1 12'. OS�9 .3031 903." 77 

.. 0. 0000 1100.7170 . 7,n 879. 0171 

'0. 0000 1 061.'163 . 79'7 838.2 ..

 " 

60.0000 "S.'666 . 777$ 774. 3037 

70 .. 0000 B72. 30S4 . 7SSS .6 1.8765 

715. 0000 764. 20'3 . 7441 '69.6819 

so. 0000 '<;'0. 3'2a . 7224 <a6.491' 

a'. 0000 2%.01a3 . '34! ua.6061 

III>D DIRECT NaR""'I. 

Z£H TA T03 T�oa T/I TI'\ AW 

0. 0 . '360 . '834 .9�74 .'137 .$91 0 . 1<:19 

20. 0 .�H69 . :ilea6 .. '$0$72 . 90'" . 8$63 . 123' 

�I). 0 . 4�13 . "aI6 . '?90� . 9033 . 37�' . 1257 

.. 0. ° .4l$19 . 97"" . 9'96' . 6<;>37 .36�7 . 1291 

'0. 0 . 3'40 . ')772 . �e" . 8783 . 8'41 . 1340 

00. 0 .3106 . 9727 . >649 . SS31 . sa93 . 1-411 

70. 0 . 19�0 . 9643 . 9834 . S074 . 78:56 . IS20 

7',, 0 . 121" . 9:566 .. 9sea . '(6a4 . 7-483 . 1 601 

ao. o . 050!!

 . 9430 .'9803 . 707$ . 6867 . 1 717 

8'. 0 . 00'<- . '1 16 . 9770 • .0157 . ,l59a . 1907 

A3 
Table A - 3. Tabulated Data for the Subarctic Winter 

Atmosphere (V = 23 km) for Several Models 

III 

Ubi 

\.10 

Pit " 

TAU!5 ,. 

TFIO'3e_ 

TAU):-

13�3. eooo' 

."200 

... �oo 

1013. 0000 

. 2733 

. �""9 

. 1913 

IITWATER DIRECT rlDRM\. 

Z!l1r1 

0. 0 

Zll'tlTK 

0. 0000 

aO. 0090 

30.0000 

40. 0000 

'0. 0000 

� o . 0 0 0 0 

70. 0000 

7�. 0000 

so, 0000 

85. 0000 

m 

. 3498 

ao. o . S4l!9 

30. 0 .933" 

"0. 0 .S181 

50.0 . 7�O 

';0. 0 . 7'''

 

7J>. ,I) . 6,8,,

 

7!5.0 • ..a .... 

SO. O . ,a8, 

95. 0 . 3'91 

ZEKITH 

0. 0000 

20. 0000 

30. 0 000 

40. 0000 

'0. 0000 

60.0000 

70. 0 000 

7�. 0000 

�O. 00.)0 

S�. 0000 

Z!l1r1lTI<I 

0. 0000 

20. 0000 

31) .. 00t)0 

,,,, 01)00 

150 .. 01)01) 

<>0. '1000 

70. (1001} 

7!I. oooo 

ao� 0 0i,10 

3'.0000 

IIIIII'IRSS 

t . OOOO 

1.0641 

1 . 1 '''� 

1 . 30'0 

1." 48 

1.9976 

... 914$ 

:3.a-"19 

lS. 6e4li 

10.90.;8 

TR 

. a2'9 

. 8 1'S 

.. 31llS 

� 77';toO 

• �427 

.6aa .. 

. !l72' 

. .. 1'9s 

. 3370 

. 1241 

WRTT DIRECT rIORlOAL 

TMaOA TIEOS 

. 94<!" . �o<] 

.9415 . �03 

� 941)4 .'9894 

. n96 .�8al 

.. ·�16 1 • 98,a 

� '332!5 . -E'$18 

. 9a71 . 973!1 

. 90031 . '?.o�2 

0 ';'174 .9468 

.9079 .9027 

HOYT DI RECT rlDPI'IAL 

A<j ACC2 

• (\727 0 0(17, 

. 07"3 • 0076 

. 01/54 . 0 073 

.. 0797 . 0081 

. oa"7 • ooa, 

. 09':2 . 0 091 

.10 .. . 1) 1 0 t 

.11 .. 6 . 01 09 

. 1U'!> . 01<lZ 

. 1-5e7 .. 014<4-

!.AClS "' .. 0 ""D 03 

FU�"ASS AIoI 

. 9'�,?� . 07';1 

1 . 0634 .. 0776 

1 . 1'36 . 0796 

1 . $0'3;' . 08;:7 

1 . 5sa� • !)Bra 

lc'�'4:j' • Q .. I 

a .. a�97 . 1 051) 

3. $076 . 1 131i 

,' • .57�O . la�3 

1 0. 3 1';3 . 14S� 

AIoI 

. 0,9« 

. 0';0' 

. 06eo 

s 064-') 

. 0678 

. 0730 

. 0$IS 

. 08S<] 

.1000 

. 1216 

T03 

. 9736 

��no 

. 9>721 

. 9706 

. '�6t.u 

. �6 

.9�44 

. '1474 

. '336 

.90ao 

AC3 

. 0301 

. 1)309 

. 030.:0 

. 0337 

� 0362 

. 041)1 

.0 .. 7 

. 0'21 

. 0';08 

.. 1)77' 

T03 

.. 97:34 

.. �726 

� �71' 

. 969� 

. �";�7 

. 'Mi'i) 

. '!>�2" 

.. ?""3� 

.. 92.76 

. 6907 

�I)l'!_ DIPECT rtORI'IAI. 

ZEKITH III " TT TH2D Illrl 

0. 0000 1 1'0. �O2" . :37aO 1010. 1839 

eo.oooo 1139 . 93 7 1 .. 3762 998. 788 1 

30. 00 00 uas. �lj; . S736 ?>O3.0703 

40 .. OIlOO 1100.7170 . 8701 957. 78� 

�o� 1)001) 1 0ojl .·'163 . 2643 917. 9'8S 

60. 0000 ?9!5 .. '6� .a567 8' ... 91 0' 

70. 0001) $, ... 30, .. .3438 736. 0339 

7S. 0000 7"4.aO!53 . a337 &37 .. 1483 

ao. oooo ,0;.0. 3'2$ . 31S7 4$3 .. 3134 

as. 6000 296 .. 012.:3 . n1' 2.34 .. 41'6 

BI�D D1RECT NCRI'IAI.

 

"1'1 TA T03 TCO .. Tl1 

0 . 0 .8122 . 9783 . 9874 .'7137 

ao. o . :30S:4 . '?-773 . 9972 . Si'O'?04 

30. 0 . 7990 . ·?7'� • "?569 . 9-1):3:3 

40.0 . 76.72 . -:;737 . �a.;, .an7 

.'S O. II .. 73:30 . 9702 . 31$59 . 8783 

01). 0 . 6771 . ::644 . ',7$49 .a�31 

fl). O . '777 . $1'334 . 9'9'34 .S074 

��. 0 , .. 9 .. .,. . ;'4·34 . i'aaa • (,SS4-

J!it) .. (,I . :36'74 . 92'36 . ',803 . 7079 

s� .o . 1 7490 . 9848 . ';'770 •• 157 

A4 

Tl'! 

.8901 

. ae53 

.a7as 

. a�a& 

. aS29 

• san 

. 7933 

. 7"'2 

."$12 

.. S4a6 

TAI� 

. 1'016 

. 3<>74 

• S'3'1 7 

. Ssa6 

. 3�a3 

.8"'1 

.3029 

• 770a 

. 7I4<! 

. 6123 

FtIl2 

. 007S 

, 0079 

. ooas 

� OO'?' 

. 0110 

. 0 137 

. 0 1'9'0 

. 0242 

. 0338 

. 0573 

IDrI 

92a."3�6 

910 .. 4298 

996.1599 

947 . 7640 

788.860' 

696.7169 

�43" "1Z4 

42'. 1"52' 

26:1. 0097 

70.69UI 

IllM 94l! ..... S>7 

9-29. "'26 

?O� •• l 4S3 

a�. 3�76 

S33 .. <i9'O 

7$ 1. (2)3' 

';38. 5'''2 

�"'.3a46 

"IS. 0 .. 5 

221 .;3732 

AW 

. 0759 

. ono 

. 1)7S? 

. 0$13 

. ')851 

. 0907 

. 1).9� 

.11)'a 

. 1 16.) 

. 1326 
Table A-4. Tabulated Data for the Subarctic Winter 

Atmosphere (V= 5 kml for Several Models 

10 U'3 . oaoo 

'JW .4200 

vo . �oo 

PR 1'l'U! 

1 013. 0000 

• . ,a"3 

TAU3S- 1 . 17a7 TAU'- ....

 9 

ilTIoUIT£R D IRIICT NIlRI'IAI. 

Z'EliIT'I FURi'fRSS TFI ""

 TI'I llIti 

0 . 0000 1 . 0000 . �a3' " f./5'S'4 .a,ol ,a9. ,.26 

ao. oooo 1.0 .... 1 _"024 . o,;os .aaS3 �.0. 6i50i5 

SO. OOOO I. IS'" . "739 • tl6ao . a?as sa3.6S00 

40. 0000 1 . 30'0 � 4a,,� . 06"3 . 369. ,..;70< "'t3 

.50. 0000 1.�i5"9 . 3';'7 . 00;79 .asas 3a9 . ..

 $1 

60. (01)0 1.'97� . a7 ..

 . 0730 .�277 oi!aO.4uH 

70. 0000 2.914$ . ISI7 . 0Sla . 7$33 144. 00S. 

7'. 000. 3.3419 .. 0$33 . 09109 . 7�2 73. 7�"9 

eo. oooo s.6a ..

 . oa�3 .1000 . 6Sla 19.8821 

,,15. 0000 1 0.906a . 0009 . lal6 . 15426 .411 1 

!.IATT nI�ECT NIlR� 

ZI!N TA TI

aDA THaos TQ3 TAIIt I D" 

0.0 ."150 .. ioo&a4 .9909 . ·,736 .3016 637.73ao 

zo.o .�fS!l!l: . �"IS . 9903 . 9730 . 3974 Ii 1 7.43.7 

SO. O .S411 .3404 . 9894 . �721 .391 7 S90. 4373 

40 .. 0 .�1I0 . 9386 . 9aSI . �71')6 . aea. �'.a333 

150. 0 . 4663 . 93�1 .��s . %81 .a�3 489 . ..

 37 

60. 0 . 4�OO . 932' .$0$18 . %36 . S�1 40� .. 'O83 

70. 0 .an3 .9271 . 97315 . '�!I"" .soa9 279. 0931 

7'. 0 . a289 . 1'231 .%,a a 9474 . 7702 201 . 31589 

80. 0 . 1407 .'174 • 94a8 . 9336 . 7142 110.1513 • 

a'.o . 043' . 3079 . 9027 .90S0 • • H23 a6.(,999 

HOYT D I�I!CT NOR"'*-

ZEit I TIl oW FlC02 iIQ3 F!02 

0. 0000 . 0727 .007' . 0301 . 007$ 

20. 0000 . 0743 . O07� . 0309 . 0079 

3 •• 0000 . 0764 • 0078 . 03S0 . 008' 

40. 0000 . 0797 . 0081 . 0337 .009' 

50. 0000 . 0847 • 008' . 0362 . 0110 

�G. 0000 . 0322 • 0091 • e"G! • 0137 

70. 0000 . 1 046 • 6101 . 0467 • 0190 

7'5. 0000 . 1 1 ..

 . \) 1 09 . 0521 . 0242 

ao. oooo .1299 . 01.:2 . 0608 . 033& 

S15. 0000 . 115a7 . 01 ..

 . 07715 . 0'78 

I.FlCIS HaD AriD 03 

ZENITH AIRMASS � Ta3 

0 . 0000 . 999' . O(�H .. �734 

20. 0000 1 . 0634 . 0776 .9726 

�O. 0000 1 . 11536 • 079' .97115 

40. 0000 1. 3037 . OS27 .96% 

50. 0000 1."215 . 087a . S667 

J$O.OOM h�*7 . 0341 . 9�la 

70. 0000 2 .. 3S':l7 . 1 0'0 � 9eia4 

715. 0000 3.a076 . 1 136 . 94,6 

$0.0000 ,. 579. . 1263 . �76 

as. 0000 10.3163 . 14815 .9907 

.. JUI'lDRR D I�IICT "�1'IFl

 

ZENITH 10 X TT THaQ IDN 

0. 0000 1 1�O. 6024 . a780 1010. IS3S 

20. 0000 1139.9371 . 8762 998.7a81 

30. 0000 1 12'.0'1$19 • an'S 963. 0703 

40. 0000 1 1 00. 7170 . &701 1'57. 7868 

50. 0000 1 06 1 . 15163 .3'''

8 917. 'r'S' 

60. 0000 ���666 . $:;67 esa.9105 

70. 0000 1372 .. 3 054 .a •• a 735. 0339 

7'. 0000 764.2OS3 . a337 637. 1483 

$0. 0000 1590. 31526 . 3187 483.313" 

$5. 0000 296. 0123 . 7919 23 •• 4198 

lHRD D IREC T NORI'IAI. 

ZErt TFI TOl TCIl1! T� TI'! ""

 

0. 0 • 5J<;0 . 9783 . 9a7. . 91 3 7 .a91 0 • 07159 

an. o .15169 . 9773 .9872 .9034 . 8863 . 0770 

30. 0 . 4913 ,,�7S9 . 9969 .9033 .. $7'9� . 0787 

40. 0 .4:519 . 9737 . 996' . a937 . a697 . 0813 

50. 0 . :3940 .-:'702 . ,a159 . 8783 .S1541 . 08'1 

60 .. 0 .3106 . %44 .0;849 .91531 . 8293 • 0907 

70. 0 . 1930 . 91534 . '3834 • <l074 . 78�6 . O� 

,.�. o . 1 C!1'" • '143 ..

 . %22 . 7684 • 7483 . 1 062 

ao. o . 0'05 . 92�' . �303 � 7079 . �67 .1160 

a15. o . 005. • S9..a . 9770 .6157 . 5'92 . 1326 

AS 
Document Control I

;' SERI Report No. 

Page TR- 335-344 12. NTIS Accession No. 3. Fiecipient's Accession No. 

-

4. Title and Subtitle 5. Publication Date 

Di rect Inso l ati on Mode l s 

6. 

7. Author(s) la. Performing Organization Rept. No. 

Ri cha rd Bi rd; Ro l and L. Hul strom i 

9. Performing OrganizatiOn Name and Address 110. prOkCtlTaSk/Work Unit No. 

Tas #3623.01 

So l a r Energy Re s earch I n sti t ute/ DOE 11. Contract (C) or Grant (G) No. 

1 617 Co le Bo ulev ard (0) 

Go l den , Co l o. 80401 

(G) 

12. Sponsoring Organization Name and Addres� 13. Type of Report & Period Covered 

Techni cal Report 

14. 

15. Supplementa ry Notes 

I 

I 

16. Abstract (Li mit: 200 words ) 

Severa 1 recen tl y publ i s hed model s of the d i rect component of the broadband 

i nsol a t i o n a re compared fo r c l ear s ky condition s . The compa r i son i nc l udes 

seven s impl e mo de l s and one ri gorous mode l t hat is u sed as a bas is fo r de -

termi ning accuracy . Where possi bl e, the compar i son is made between the 

resul ts of each model for each a tmos pheri c consti tuent ( H20, C02, 03 , 02

, 

aerosol and mo l ecul ar scatteri n g ) s epa ra tely as we l l as fo r the combi ned 

effect of a l l of the consti tuents . Two opt i mum simp l e mo de l s of vary i ng 

deg rees of comp l exi ty a re deve l o ped as a resul t of thi s compari son . The 

s tu dy i nd i cates: aerosol s domi nate the attenuati on of the d i rect beam fo r 

rea sona b l e atmo s pheri c condition s ; mo l ecul ar s catteri ng is next in i mportance ; 

water vapo r is an i mportant absorber; and ca rbon dioxide and o xygen a re rel ati vely uni mpo rtant as atten uato rs of the b roa dband sol ar ene rgy . 

17. Document Analysis 

a. Descriptors Inso 1 a ti on ; �1at hema ti ca 1 r�ode 1 s 

Vapor ; Ozone ; Transpo rt Theory ; Gases 

Comparati ve Eval uati ons Wa ter 

b. ldentifiers/Open-Ended Terms SOL TRAN Mod e l ; Atwa ter and Sa 11 r'10de 1 ; H oyt Mod e l ; 

AS H RAE Model ; Wa tt Hod el ; Laci s and Ha nsen Model ; Ma chta Model ; Maj umdar 

Model ; Bird Model 

c. UC Categories 

59 ,61 , 62 , 63 

18. Avaiiabi lity Statement 

NTI S, U.S. Dept . of Comme rce 

5285 P ort Roya l Road 

S pri ngfi eld, Va . 22 1 61 

; 9. No. of Pages 

62 

20. Price 

$5. 25 

Form No. 8200-13 (6-79) 

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