Solar Research Institute  National Renewable Energy Laboratory
October 30, 2017  Author: Anonymous  Category: N/A
Short Description
irradiance at each wavelength of interest. A sum (integration) R. Bird and R.L. Hulstrom: Solar Energy ......
Description
SERI/TR33S344 UC CATEGORY: UCS9,61,62,63
DIRECT INSOLATION MODELS
RICHARD BIRD ROLAND L.
JANUARY
HULSTROM
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_ EG77C014042
, '.
Printed in the United States of America Available from: National Technical Information Service U.S. Departm ent of Commerce 5285 Port Royal Road Springfi e1dt VA 22161 Price: Microfiche $3.00 Printed Copy $ 5.25
NOTICE This report was prepared as an account of work sponsored by the United States Govern ment. 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 assumes any legal liability or responsibility for the accuracy, completeness or usefulness of any information, apparatus, product or process dis closed, or represents that its us e would not infringe privately owned rights.
TR3 44
S=�I FOREWORD
This re port documents wo rk pe rfo rmed by the SER I Ene rgy Res ource Ass e s sment Branch for the Di vi s i on of S ola r Ene rgy T echno l o gy of the U . S . De pa rtment of Energy. The re po rt compa re s s e ve ral s im ple d i re c t insolation models wi th a rigorous s o l a r t ransmi s s ion model and des c ribes an improved , s im ple , d i re c t insolation model .
Roland L . Huls t rom , Branch Chief Ene rgy Res ource Asses sment
,
\
Appro ved for: S OLAR ENERGY RESEARCH INST ITUTE
f o r Res ea rc h
iii
TR344
S=�I TABLE OF CONTENTS
1 .0
Int roduc t i on••••••••••••••••••••••••••••••••••••••••••••••••••••••••
1
2.0
Des c ri ption of Models •••••••••••• . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2 . 10
S OL TR.AN' Model••••••••••••••••••••••••••••••••••••••••••••••••• Atwat e r and Ball Model•••••••••••••• . . . . . . . . . . . . . . . . . . . . . . . . . . Hoyt Model •••••••••••••••••••••••••••••••••••••••••••••••••••• L acis and Hansen Model� ••••••••••••••••••••••••••••••••••••••• M acht a Model•••••••••••••••••••••••••••••••••••••••••••••••••• AS RRAE Model •••••••••••••••••••••••••••••••••••••••••••••••••• \V'att Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...... . . . . . M ajumd ar Model •••••••••••••••••••••••••••••••••••••••••••••••• Bi rd Models ••••••••••••••••••••••••••••••••••••••••••••••••••• Addit ional Model s and Othe r Considerat ions ••••••••• ••• •••••••• 2 . 1 0 . 1 W at e r Vapo r••••••••••••••••••••••••••••••••••••••••••• 2 . 10 . 2 O zone ••••••••••••••••••••••••••••••••••••••••••••••••• 2 . 1 0 . 3 Uni formly Mixed G ases ••••••••••••••••••••••••••••••••• 2 • 1 0 . 4 Ai r Mas s •••••••••••••••••••••••••••••••••••••••••••••• 2 . 1 0 . 5 S imple T rans po rt E quat i on•••••••••••••••••••••••••••••
3 4 5 7 7 8 8 10 10 12 12 13 14 15 16
3.0
Model Compa'risons • •••••••••••••••• ••• . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
4.0
S ummary and C onclus ions .
. . . . ••••••••••••••••••••••••••••••••••••••••
39
5.0
Re fe rences ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
43
A ppendix.
T abul ated Model Dat a••••••••••••••••••••••••••••••••••••••••••
v
Al
S=�I IWI � ��
T� R__=3:!.:4� 4 =
________________________ _
LIST OF FIGURES
31
T ransmittance versus Secant o f Solar Zenith Angle f o r Midla t i t ude Summe r Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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T ransmi ttance ve rsus Secant of 'S olar Zeni th Angle f o r Suba rc t i c Winter Model
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Oz one Abs o rptance ve rsus S olar Zenith An gle for 0 . 3 1 em of Oz one ( MLS ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Oz one Abso rptance ve rsus S olar Z enith Angle for 0 . 45 em of O z one (SA��) . . . . . . . . . .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Abso rptance ve rsus S o la r Zenith Angle f o r 2 . 93 cm o f Wat e r Va po r (MLS )
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Abso rptance ve rsus S o lar Zenith Angle f o r 0 . 4 2 cm o f Wate r Va po r ( SAW ) •
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T ransmittance ve rsus Solar Zenith Angle f o r All Molecul a r E f f e c t s E xc e pt H2 0 Abso rpt ion (MLS )
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T ransmittance ve rsus S o lar Zeni th Angle for All Molecul a r Effects Exc e pt H2 0 Abs o rption ( SAW ) . . . . . . . . . . . . . . . .
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Aeros ol T ransmittance ve rsus S olar Z enith Angle for 5km Vis ib i l i ty Ae rosol
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Solar I rradiance ve rsus S olar Zenith Angle , Vis ibili ty = 5 km (MLS ) •
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S olar I rradiance ve rsus S olar Z enith An gl e , Vis ibility = 2 3 km (MLS ) •
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Ae ro s ol T ransmittance versus S olar Zenith Angle f o r 23km Vis ibility Ae rosol •
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S olar Irradiance versus Solar Zenith Angl e , Vis ibility = 23 km (SAW ) •
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32 34 35 36
S;:: , �I , � {..�
�T R � � �3 4 � 4 � �
____________________________________________________�________
LIST OF TABLES
14
21
S ources o f Data for Cons t ruct ion of Models
22
Comparisons o f Direct Normal Irradiance for Di fferent Forms o f the Trans port E quat i on U s ing Bird Models and S OLTRAN . . . . . . . . . . . . . . . . . .
17
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Com parisons o f Di rect Normal Irradiance for Di fferent Forms o f the Trans po rt E quat i on U s ing Bi rd Models and SOLTRAN . . . . . . . . . . . . . . . . . .
18
AI
Tabulated Data for the Midlat i tude Summer Atmo s phere (V = 23 km) for S everal Models
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Tabulated Data for the Subarctic Winter Atmo s phere (V = 23 km) for S everal Models •
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Tabulated Data for the Midla t i t ude Summer At mo s phere (V = 5 km) for S eve ral Models •
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Tabulated Dat a for t he Subarctic Winter At mos phere (V for S everal Mode ls •
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A3 A4 AS
TR 3 4 4
SECTION 1.0 INTRODUCTION So l a r ene rgy ( insolation) conve rsion systems a re d i f f e rent f rom systems based on o the r sourc es of ene rgy, because the ene rgy source is s ub ject to va rying meteorological cond i t ions . As a res ul t , rel iable insolation data a re requi red at each s ite of int e rest to des ign a solar ene rgy sys tem . His t o rical data have been collected by the Nat ional Weathe r Se rvice (NWS ) on a very l i mi t ed bas is at 26 locations throughout the Uni ted S ta t es , and data a re currently being collected at 38 loca t i ons . Because of the small n umbe r of s ta t i ons in this netwo rk and the va riability of insolation, it is essential to have ac c u rate models to predict insolat ion a t o the r locat ions . The accuracy o f thes e models and e xpe rimental data affects the des ign , pe rfo rmance , and econo mics of s ola r s y s t ems . Nume rous s im ple insolation models have been produced by d iffe rent inve s t iga to rs ove r the pas t half c ent ury . The goal of these models has been to provide an uncompl i cated e s t imat ion of the available insolat i on . Thes e models , by ve ry diffe rent met hods , account for the influence of each a tmos phe ric c on s t i tuent on s o la r radiat ion. Thi s , in t u rn , l eads to confus ion and que s t ions of validity f rom pro s pe c t ive use rs . Thi s s t udy compares s eve ral o f the mo re recent models o f the d i rect component of the insolation for clear sky conditions . The c om pa ri s on includes s even s im ple model s and one rig o rous model that is a bas is f o r dete rmining accu rac y . The res ults o f the compa ri sons a re then used t o f o rmulate two s imple models of d iffe ring c om pl e xi t y . The most useful f o rmalisms of present models have been inco rpo rated into the new mode ls . The c ri t e ria f o r evalua t ing and f o rmul a ting model s a re s implic i t y , accuracy , and the ability to use readily ava i l able met e o ro logical dat a . I n the future , a s imil a r analys is o f model s f o r g loba l and d iffuse ins o lation is planned . S im ple global and d if f use models will be c om pa red with a rigo rous model that use s Monte Ca rlo technique s . Add it iona l compa risons a re planned, with ve ry ca ref ully taken e xpe rimental data for both d i rect and diffuse ins o lation components . As many met e o rological meas u rements as can rea s onably be t aken will be inc l uded with these data . The goal o f this wo rk is to produce a wel ldocumented global insolat ion model that includes the d i rect and diffuse clear sky insolation as well as cloud and g ro und reflected ins olation. Thi s re po rt is the f i rs t s t e p towa rd achieving that goal .
1
5=~1
2
!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
SOLTRA N MODEL
The rigorous model, called SOLTRA N, 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 lay ered atmosphere is constructed between sea level and 100km altitude by de fining atmospheric parameters at 33 levels within the atmosphere. The actual sea level layer heights at which atmospheric parameters are defined are: (0.0 km) to 25 km altitude in 1.0km intervals, 25 to 50 km in 5km 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, N 20, CH 4, CO, N2, and 0 2 ) ar! stored in the code at 5cm 1 wavenumber i �tervals with a resoluti9n of 20 cm 1 . The average transmittance over a 20cm 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 theo retical molecular line constants in linebyline 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 para meters from the standard models or by introducing radiosonde data.
3
S=�II*I
4 4:.::!_ R__'3::!.:! :!T�
__ ___ ____________ ____ _
Some of the outputs of the LOWTRAN code include the total transmittance; the transmittance of H20, °3, and the uniformly mixed gases ; and aerosol trans mittance at each wavelength value specified. In the S OLTRAN model these transmittance values are multiplied by Thekaekara I s corresponding extrater restrial 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 spec tral 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: (1) where IO = extraterrestrial solar irradiance, Z
= solar zenith angle,
TM = transmittance for all molecular effects except water vapor ab sorption,
� = absorptance of water vapor, T
A = transmittance of aerosols.
The mathematical expressions for the transmittance, absorptance, and air mass M, are given by: T
0.15[M (949 M = 1.041 
x
106
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
...o:T::..olol R:... 3 "" . 4'"'" 4
________________________
TA = exp (
a
M') ,
2 M = 35/ [(1224 cos Z) M' = P
+
(4)
° 5 1] · ,
(5)
M/I013.
•
where 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, useroriented model. Results from using this model are presented in a later section with a compari sons of other models. 2.3
HOYT MODEL
A model for solar global insolation that includes a model for the direct com ponent is described by Hoyt [11]. The following equation is for the direct z solar insolation on a hori ontal surface: 5 a ) T T I = I (cos Z)(1 AS R O i i l
(6)
�
where a represents the absorptance values for water vapor (i = 1), carbon i dioxide (i = 2), oz one (i = 3), oxygen (i = 4), and aerosols (i 5). The parameter TAS 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' w
+
�
a2 = 0.00235 (U
6.31
+
x
104)
°.3
 0.0121 , ,
0.0129)°·26  7.5 5
x
104 ,
(7)
(8)
S=�II*I
4 4 .:... T.::.: �:... ....; : 3 :: R...:
____________________ _____
(10)
as = O.OS[g(a)]
TAS =[g(a)]
M'
M'
,
,
(ll )
(12)
(13) where u
U
U
w ,
c
t
o
= pressurecorrected* precipitable water in the path (cm), = pressurecorrected 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),
M'
= pressurecorrected air mass,
g(a)
=
a tabulated function bidity coefficient a,
that is related to the angstrom tur
f( M') = a tabulated function of pressurecorrected air mass. See Hoyt [11] for the tabular data. Because the functions g(a) and f(M') are in tabular form rather than in empir ical 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 pressurecorrected 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 2.4
'�I
T�R��3�4�4�
________________ __________________________________________
LACIS A ND 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 where Y path.
=
MD ' w
w
=
2.9 Y [ (1 + 141.5 y)
0.635 + 5.925 y]  1 ,
(14)
with D being the precipitable water vapor (cm) in a vertical w
The expression for ozone absorptivity in the Chappuis band is given by vis a o
=
2 1 0.02118 X (1 + 0.042 X + 0.000323 X ) 
(15)
and for the ultraviolet band by v aU o
=
1.082 X (1 + 138.6 X)
0.805
3 1
+ 0.0658 X [1 + (103.6 X) ]
where X = DoM with Do being the amount (cm) of ozone in a vertical path. total ozone absorptivity is given by the sum a
o
=
vis uv a + a 0
0
(16) The
(17)
Comparisons of the results of these expressions ,.;ith other models is shown in a later section. 2.5
M ACHTA 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 cal culating solar insolation at a given location without the use of mathematical A standard value of direct solar insolation is given expressions. as 887 2 W m , and a standard value of diffuse insolation is given as 142.5 W m2 These standard values are then corrected by the use of graphs and the work sheet. The corrections are made for station altitude, zenith angle, precipi table water, .turbidity, and earthsun 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 A few examples of calculations using this model of Braslau and Dave [14]. will be illustrated in a later section. 7
TR344
S=�I' 2.6
AS HRAE HODEL
The American S o ciety of Heat ing, Refr igera t ion and Air Cond it ioning Engineers , AS HRAE, publ ishes a s imple model [ 1 5 , 1 6] f or es t imat ing s olar ins o la t ion a t l o ca t ions in the Northern Hemis phere . This me thod represents the solar ins o lat ion a t the earth ' s s ur f a ce , under clear sky cond it ions , by us ing IDN
=
A e B s e c Z
(18)
where A
=
the "apparent " extraterres t r ial solar rad ia t ion ,
B :: the "apparent " opt ical a t t enuat ion coef f icient , Z
=
the solar z enith angl e .
An a tmospher ic cla r it y adjus tment , C N , cal led the cle a rnes s number , is then used to mul t iply the d ire ct normal insolat ion cal culat ed us ing Eq . 1 8 . This clearnes s number correct s for var ia t ions in transmit tance at a part icular Values of A , B, and C N are published by AS HRAE [ 1 5] as well as l o ca t ion. tables of solar ins o la t ion for the Nor thern Hemisphe re [1 6] resul t in g from appl icat ion of these parameter value s . A thorough dis cus s ion o f the or igin o f the AS HRAE model is presented by Hul s t r o m [ 1 7] , and this inf o rma t ion will not be repeated here . It is s uf f i cient to s ay that Eq . 1 8 , commonly called Beer 's Law , is s t r ict ly appl icable only f or mono chroma tic radia t ion . If one takes the na tural l o ga r ithm of Eq . 1 8 , t he res ul t is : I n A  B sec Z
(19)
A plot of th is express ion on a In IDN versus s e c Z axis sys t em resul ts in a s tra ight line , with A the int e r cept of the lo garithmic a xis and B the s l o pe of the l ine . The ver t ical int e r cept A o ccurs for the ext rapolation s e c Z = 0 . 0 , which corresponds to zero air mas s or the e xtraterres trial ins olation . In the model compar is ons given in a later s e ct ion the deviat ions of this model f rom more a ccurate result s are ind icated . Huls t rom [ 1 7] poin t s out that the clearness numbers pub l ished by AS HRAE cor re ct only f o r water vapor var ia t ions . Moreove r , the s e are only average wat e r vapor condit ions , and it is shown here in a later s e ct ion that var iat ions in aerosol a t t enuat ion are normally a much more s ignif icant fact o r . 2.7
WATT MOD EL
A model for global ins olation has been cons t ructed by Wat t [ 1 8] , bas ed par t l y o n the work o f Moon [ 1 9]. The �xp re s s ion f or the d ire ct normal insolat ion is (20) 8
!s::�1
1�1
� 4� 4� �T�R� �3
______________________________________________________________
where the t rans mit t an ce fun ct ions are Twa f or water vapor abs orpt ion, T a for dry air s cat t e r ing, T o for o zone abs orpt ion , Tws for water vapor s ca t t e � ing , T f o r lower level aerosol absorp t ion and s ca t t er ing, and T u f or upper layer L aerosol abs o rp t ion and s ca t t e r ing. These t ransmit tance funct ions are de f ined by =
0 . 9 3  0 . 0 33 log ( Uw M2 )
( 2 1)
(22)
( 2 3)
(24)
(25)
=
TL
=
1 0 Tu M 3 0. 6 ( T
_
(26)
,
0 . 5  0 . 0 1 Uw  0 . 0 3)
,
(27)
( 28 ) The para meter Po is the sea level pres s ure; P is the pre s sure at the surface being cons idered ; U o and Uw are the amount of o zone and water vapor in cm in a ver t ical path; TO . 5 is the at o mo s pher ic turbid ity at 0 . 5]lm wavelength; lobs is unde f ined in Wat t ' s rep o r t ; and 10 is the broadband extraterres tr ial ins o l a t ion . The para met ers Tu and TL are the upper and lower layer broadband t urbid ity or op t ical dep th . TU can be t a ken from p lots in Wat t 's report for The Hi are called path length mod if iers by Wat t , and they serve pas t years . the same purpose as a ir mass in the p revious model s . These path length mod if iers are equal for solar zenith angles ( 70 ° and are e qual to the s e cant of the solar zenith angle ( s ec Z ) . For s o lar zenith angles > 70 °, the pa th len gth mod if ier is def ined different ly for ea ch a t mos phe r ic cons t ituent a ccording to the a l t itudes in the a t mos phere be tween which the cons t ituent is concentrated . A parame ter F z i is cal culated us ing the following e xpres s ion: ( 29 )
9
S=�I
'*1
TR 4 .:.... :.:. ....: :.; ...:::..... 34 ..
__________ ___ ________
6 where r is the earth I s radius (6. 4 x 10 m) and the hi are the atmospheric altitudes (heights) between which the constituent is located. If a constitu ent is concentrated between two altitudes, h I and h 2, an Fzl 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:
M.
1
=
h
2
 h F l I z z2 h h  I 2
F
(30)
The values of h used for the various constituents are: ozone:
hI
=
20 km, h2
::
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 = Fz2' i 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
=
0.25 1331.0 (O.8644)(MP/I000)(0.8507)(UJM)
(31)
where M is the air mass, P is the surface pressure, and Uw 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 4 =........ . 3 =:.,. ... 4
_____________________
models previously presented. The expressions used were tuned to give the best leastsquares fit to the SOLTRAN 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 SOLTRAN was from 0.3 to 3.0 �m.· The total ifradiance 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 10 = 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 T
0.84 (1.0 + M'  (M,) 1.01) ] R = exp [ 0.0903 (M')
T
,
(33)
0.3035 o = 1.0  0.1611 Xo (1.0 + 139.48 X o) 2 1 0.002715 Xo ( 1.0 + 0.044 Xo + 0.0003 Xo )
Tu = exp  0.0127 (M') 0.26 ,
a = 2.4959 � [ (1.0 + 79.034 �) 0.6828 + 6.385 �J1 w
(34)
(35)
(36) (37)
L A = 0.2758 T A (0.38) + 0.35 T A (0.5)
M
= [cos z + 0.15 (93.885  Z) 1.25J 1
(38)
(39)
where M'= tW/PO' X = UoM, � = UwM, T is the transmittance of the uniformly u o mixed gases, T A is the broadband atmospheric turbidity, and L A (0.38) and T A (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 oz one absorption used by Lacis and Hansen have been dropped in Eq. 34. This expression is still much too complicated when the relative importance of oz one is considered. The second line of Eq. 34 could be dropped without serious effeft� The form of r 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
104 P + 0.051)]0.5
(41)
The expressions for � and TA 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 pro vide results that are nearly as accurate as the more complex model with much less effort. 2.10
ADDITIONAL MODELS AND OTHER CONS IDERATIONS
This study is not a comprehensive comparison of simple direct insolation models. It is a comparison of the more recent models. An excellent compari son 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 H20, °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. [233� and linebyline 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].
1
•
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 absorp tion bands near 0.7 and 0.8�m wavelengths. Because of the poor documenta tion of Fowle's data, McDonald could not claim an absolute accuracy greater than 30%. 12
5=�1 ' Huls trom explicates The ASHRAE model is bas ed on work by Threlkeld. " Threlkeld determined the var iable amount o f water vapor on a monthly bas is by a s emiemp ir ical t e chnique. He used meas urements of the broadband d ir e ct beam ins olat ion [at Blue Hul l , Mas s . ; L incoln , Neb . ; and Mad ison , Wis . J in con j un ct ion with the Moon cal culat ion r out ine , t o derive ind ire ctly the corre spondin g amount of pre cipitable water vapor " [1 7J. Hoyt and also La cis and Hansen used water vapor from Yamamot o [36 ]. La cis and Hansen explain , "Abso rp t ion by maj or wat e r vapor bands has been mea s ured at low spect ral resolution by Howard et ale (1 9 5 6 ) . Yamamoto (1 9 6 2 ) we ighted these abs orp t ivit ies with the s olar flux and summed them , incl ud in g es t imat es for the weak absorp t ion bands near 0 . 7 and 0 . 8 Um which were not measured by Howard et ale, to obtain the total abs orp t ion as a funct ion of wat er vapor amoun t " [1 2 ]. The reference to Howard et a l . is referenced here as [37]. Watt used an expres s ion f o r wat er vapo r that he der ived empirically from data in Chapt er 1 6 of Valley [38 ]. He then compa red this express ion w ith meas ured data f rom severa l l o ca t ions and adj us ted the coef f icients to obt a in the bes t a greement . Ma chta bas ed his model on cal culat ions performed by Bra slau and Dave [1 4 ] with a r igorous rad ia t ive t rans f er code . Bras lau and Dave based their cal culat ions on the database used by LOWTRAN. However , they used a mathemat ical formal ism d if f e rent from L OWTRAN f or band abs o rp t io n. 2 10 . 2 •
Oz one
The LOWTRAN and t1achta model s used the same original data sour ce s for all the molecular ,ab s orbers . The references given in the previous s e ct ion for water vapor are the same for o zone . A twa t e r and Bal l did not cons ider ozone s e parately but used a general formula that included all mo lecular e f f e ct s except water vapor abs orp t ion. The AS HRAE and Wat t model s are based on the ozone data used by Moon [1 9 ]. Moon , in turn , used data measured by Wulf [ 39 ] in the Chap puis band (0 . 5 t o 0 �7 um ) and data by Lauchli [40 ] in the Hart leyHuggins band below 0 . 35 Urn wavelength. Wat t int e grated the s pe ct ral data given by Moon to obtain broadband ozone absorp t ion. He then modif ied an expres s ion that a greed with these res ul t s t o give the bes t a greement with bro adband t o t a l t ransmit tance data from other s ources . Hoyt used ozone data from Manabe and S t r ickler [4 1 ] to derive an empirical formula . Manabe and S t r ickler based the ir resul ts on expe rimental data from V igroux [42 ] and Inn and Tanaka [4 3]. La cis and Hansen apparently based their emp irical formula for o z one on the same original data s our ces that Hoyt used . They point out that the data for wave lengths greater than 0 . 34 �m were given for 1 8 °C . They us ed the data at 44°C for shorter wavelengths and reduced the longer wavelength data by 2 5 % to compensate for the d if f erence in t emperat ur e . They produced s eparate express ions appropr iate for the ultraviolet and vis ible absorp t ion data , respect ively. 13
55'1
'*'
2. 10. 3
T..::. 4 R..::..:3 ..: 4� =
_______________________
Unif o rmly Mixed Gas es
The unif o rmly mixed gases are CO2 and 02 . LO WTRAN and Machta are based on the s ame o riginal data s ources given in the s e c t ion about water vap o r . Moon did not cons ider The AS HRA E and Wat t model s are bas ed on Moon ' s model . CO2 and 0 3 in insolat ion but migh t have included their effect w ith H2 0 abso rp t ion . Atwa t e r and Ball based their model on an emp ir ical formula that inc l uded all molecular effects except wat er vapor absorp t ion according to Kondratyev [9 ]. This formula was based originally on Fowle ' s data for the cons tant gas es . Hoyt us ed Yamamoto ' s oxy gen abs orp t ion data , which in t urn were taken from Howard e t a l e [44 ]. The carbon d ioxide data were taken f rom Burch et al. [45 J. A summary Table 2  1 .
of
the
data
sources
Table 2 1 .
Atwater and Ball
by
var ious
modelers
Burch et ale (many) Linebyl1ne data
Linebyline data
Burch et al. (many) Linebyline data
Fowle
ICastrov
Fowle
Howard et al. (Yamamoto)
Burch et al.
(many)
(Kastrov)
(1955)
Vigroux
Burch et al.
Inn and Tanaka
Howard et al.
(Manabe & Strickler) Watt
condens ed
SOURC ES OF DAT A FOR CONSTRUCT ION OF MODEL S
(McDonald) Hoyt
is
H2O
}fodel SOLTRAN
used
Valley & Modifications
Wulf
(1960) (1955)
Moon
Lauchli (Moon,) Lacis and Hansen
Howard et al. (Yamamoto)
(1955)
Vigroux Inn and Tanaka (Howard et a1.
ASHRAE
Threlkeld
Wulf
(Best fit 3 locations)
Lauchl1
1961
Handbook) Moon
(Moon) Machta
Burch et al.
(many)
Linebyline data (Dave)
Burch et al.
(many)
Linebyline data (Dave)
14
Burch et al.
(many)
Linebyline data (Dave)
in
S=::!al:.' � ��� 2 . 1 0 .4
34...:. :.,. 4 ....:.. 'T :::..:; R,:... .:::
________________________
Air Mass
The ai r mas s is a coe f f i cient that thro ugh whi ch light rays must pass directly overhead . When the s un is The f o rmal defini t ion of air mass is
acc o unts for the in the atmos phere direct ly overhead , given by Kondratyev
.)(" .)(" �
M
�
increased path length when the s un is no t the air mas s is 1 .0 . [9] as
�
Pd S/
(42 )
pdz
where dz is an increment in the vert ical dire c t ion; ds is an increment along a s lanted pa th; and p the dens ity of a i r , or whatever component of the air tha t is being cons idered . This def ini tion impl ies that differences in alt i t ude be tween d i f ferent s urface locations mus t be acc o unted f or by s o me means other than air mas s . In cal c ulat ing a t mos pheric attenuation over a s lant path , for example , the o ptical de pth f or a vert ical path is mul t i plied by the air mas s t o obtain the total o ptical de pth . The vert ical opt ical depth inc l udes the ef fect of the alt i t ude at which one is working . S o me authors inc l ude the beginning alt i t ude in the air mas s and use the optical depth from sea level , or 1 0 1 3 mb pres s ure . This is called the abs o l ut e , or pres s ur ecorrected , air mas s , given by (43 )
where Po
=
1 0 1 3 mb .
Kondratyev [9 ] s ummarizes the methods of calc ulation for different ranges of For z enith angles < 60 ° , s uffi cient acc uracy can be obtained z enith angle. using M
=
( 44 )
sec Z
where Z is the z enith angle . For 6 0 0� Z � 80 ° , the effect of earth c urva t ure becomes important . Geometric considerat ions give M
=
{ ( r/ H ) 2
cos
2
Z + 2 ( r/H ) + 1
}
1 /2
 ( r/H ) cos Z
(45 )
where r is the earth's rad i us and H the s cale height defined by
(46 )
H
15
S5�1
1*1
T 44 =.;:R::...,;;:.3..:.,. .;..
_________________________
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. The correct For Z > 800, the effects of refractive index become important. value of air mass in this region can be found in tables (Kondratyev, for A n expression example) or can be calculated with approximate expressions. 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 1
2
3
II
=
=
1
1
0
0 [T
T T T T TA R o u w R
T
o
T  a ] TA u w
1 [T T  a  a ] TA R o 0 w u
=
1
4
=
1
0
[T  a l TA w M
(48) (49) (50) (51)
where T is the transmittance due to Rayleigh scattering, T is the transmit o tance ozone, T is the transmittance of the uniformly mixed gases CO and u 2 0 ' Tw and � are the transmittance and absorptance of water vapor, TA is the 2 t.ransmittance of the aerosol, and T is the transmittance of all molecular M effects except water vapor absorption.
0'
Tables 22 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 23, it appears that Eq. 48 provides the closest agreement with SOLTRA N. 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 estab lished by the authors. However, one assumption implicit in Eq. 48 is that the attenuation by each constituent is independent of every other constituent. I n 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 =..:
 ______ ____
______
C OMPARIS ONS OF DIRECT NORMAL IRRADIANCE FOR DIFFERE NT FORMS OF THE TRANS PORT EQUATION USING BIRD MODELS AND SOLTRAN
Table 22 .
Zenith Angl e
I 2 ( W/m )
12 2 ( W/m )
1 3 2 ( W/m )
14 ( W/m2 )
8 27 . 1
812.5
811.2
816.6
811.0
7 95 . 7
7 94 . 2
8 00 . 1
30 . 0
789 . 0
772 . 8
77 1 . 3
7 77 . 8
40 . 0
754 . 5
736 . 9
7 35 . 2
742 . 8
50.0
7 02 . 1
682 . 3
680 . 4
690. 0
60 . 0
621 . 3
598 . 5
596 . 2
609 . 1
70.0
490 . 2
463. 3
460 . 6
478. 4
75 .0
392 . 3
3 63 . 5
3 6 0. 5
380 . 5
386 . 3
8 0 .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
545 . 8
5 36 . 2
535 . 3
. 5 38 . 9
530 . 6
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
60.0
2 85 . 1
274 . 6
2 73 . 5
279.5
70. 0
1 63 . 8
1 54 . 8
153 . 8
159.8
75 . 0
96 . 2
89. 2
88 . 4
93. 4
97.2
80 . 0
35 . 8
31.9
31 . 4
34 . 0
42 . 4
85.0
3.1
2.5
2.4
2.6
7.0
( de g)
Model Atmosphere MLS
0.0 20. 0
V
2 3 kIn
}ILS
0.0 20 . 0
=
V
=
5 km
I
17
SOLTRAN ( W/m2 ) 833 . 5
756 . 6
617. 7
270.3
Table 23.
Zenith A ngle
COM PARISONS OF DIRECT NORMAL IRRADIANCE FOR DIFFERENT FORMS OF THE TRANSPO RT EQUATION USING BIRD MODELS AND SOLTRAN
II 2 (W/m )
12 2 (W/m )
2 (W/m )
1
4 2 (W/m )
SOLTRAN (W/m2 )
866.0
856.5
855.1
865.5
881.9
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
50.0
737.2
724.0
722.0
737.5
60.0
653.0
637.9
635.6
654.7
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
571.5
565.2
564.3
571.2
568.5
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
50.0
396.2
389.2
388.1
396.4
60.0
29 9.6
292.7
291.6
300.4
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
(deg)
Model Atmosphere SAW
0.0 20.0
0.0 20.0
V
==
23 km
SAW V = 5 km
18
13
804.3
662.5
461.1
297.8
sail
TR34 4
materials . One obvio us s it ua t ion that violates this ass umpt ion is when nea r complete abs o rp t ion in a s ingle cons t it uent occ urs within the spectral band be ing cons idered . If one of the o ther cons t it uents absorbs in the s ame s p e c t r a l locat ion , over attenua t ion will occur in the f inal res ul t s . The form o f E q s . 4 9 and 50 make this s it uat ion even worse .
19
20
TR344
S=�I 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 pro grammable 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 atmos pheric 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 31 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 32 presents the same type of results for the SAW model atmosphere. Both figures contain the results for a 23km 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 31 and 32 show the relative importance of each atmospheric component as an attenuater of broadband radiation (0.33.0 �). CO 2 and 02 are the least important elements, and they are omitted from some models. The next element exhibiting increased attenuation is 0 3' followed by H 20. The flat tening of the curve for H20 in Fig. 31 with increasing zenith angle suggests that the H 20 absorption bands are approaching saturation. Molecular scatter ing (Rayleigh scattering) dominates total molecular absorption at large zenith angles and has a greater effect than most individual molecular species at all z enith angles. The one exception to this statement appears to be H 0 absorp 2 tion for high concentrations of H20 and for air masses < 2. The most signif icant attenuator at nearly all zenith angles is the aerosol. An aerosol that produces a 23km meteorological range at sea level is consid ered 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 condi tions. The data presented in Figs. 31 and 32 suggest that aerosol attenua tion could be the most important attenuator at most locations throughout the United States. Unfortunately, it is also the component that is the most
21
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Figure 31. Transmittance versus Secant of Solar Zenith Angle for Midlatitude Summer Model
22
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Figure 32. Transmittance versus Secant of Solar Zenith Angle for Subarctic Winter Model
23
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_ _ _ _ _ _ _ _ _ _ ______________
diffi c ult t o mea s ure and hence the least defined . It sho uld be emphas i ze d t ha t the s e conc l us i ons are f o r b roadband ( thermal ) d irec t insolation , and tha t the s i t ua t ion will change s igni f icantly a s the bandwidth is f ur ther res t ricted o r global insolation i s considered . The previo us s ec tion noted tha t the ASHRAE model a s s ume s a plot o f the total t ransmittance shown in Figs . 3  1 and 32 will s cribe a s t raight l ine . The s lope of this l ine provides the opt ical depth . An examinat ion of Figs . 3  1 and 3  2 demons trates that this ass umption i s reasonable o ver a l imited range If the plot were o f i rrad iance ver s us sec Z , the vert ical of zeni th angles . int e rcept at sec Z = 0 would provide the ext ra te rres t rial irradiance , which i s us ually too low. Figures 33 and 34 present 03 absorpt ion data for five of the models de Figure 3 3 is for 0 . 3 1 cm of 0 3 ( MLS ) , and Fig . 34 is for s c ribed earlier . 0 . 45 cm of 0 3 ( SAW) . The models produce s ignif icantly different res ul t s when compared in this manner . However , the d i f f e rences are minor when the ef fec t ivenes s of 03 is acco unted f o r . The t riangul a r d a t a p o int s a r e a res ul t o f performing a l ea s t sq uares f it t o the S OLTRAN data with the e qua t i ons of Lac i s and Hansen ( B i r d model ) . The minor deviat ions of the t riangles f rom the S OLTRk� data are a res ult of at tempt ing to fit a s imple expre s sion to various amo unt s o f 0 3 ' Fig ures 35 and 36 present model c ompar isons o f H 2 0 abso rpt ion for the two st andard a tmos pheres being cons idered. S ince H 2 0 absorption plays a s igni f i cant role in transmiss ion cal c ula t ions , the differences b etween the models sho uld be not iceable in the to tal t ransmis sion . Fig ures 37 and 38 present plots o f t r a nsmi t tance vers us s olar zenith angle for all molec ular effects except H 2 0 absorption for the SAW and MLS a tmos pheric models , respectively . Some of the mod e ls did no t readily lend them s elves to this part i c ular cal c ulation and a re not inc l uded . The s ur prise abo ut the s e data is the acc uracy of the s imple expres s ion in the Atwa t e r model . Fig ures 39 and 3  1 0 ill us t rate a c omparison o f aerosol t ransmit tance f o r the MLS and SAW a tmospheric m odels with s ea level vis ib ilities of 5 and 23 km , respect ively . In most cases , aero sol a t t en ua t io n is independent of the a t mo s pheric model ; b ut the Wat t model for aero s o ls is d ependent on the amo unt o f H2 0 . Ho yt ' s model provi d es s trong agreement wi th S OLTRAN, b ut i t s tabular form is The table covers an ins uf ficient not as eas ily used as empirical form ulas . Hoyt ' s range o f t urbidity coef f ic ient s t o inc l ude a 5 km visibility aero s o l . data are f or aero sol s cat tering only , and there is an addit ional fac t o r for aerosol absorption in th � Hoy t model. T he val ue o f the upper layer t urbidity T u used in Wat t ' s model was taken f r o m h isto r ical plots that h e p ro d uced . The val ue used , T = 0 . 0 2 , was a n approx ima t e average of the his torical dat a . The r e s t o f t�e parameter val ue s for this model were taken d i re c t ly from S O LTRAN.
24
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Figure 33. Ozone Absorptanee versus Solar Zenith Angle for 0.31 em of Ozone (MLS)
25
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Ozone Ab so rp tanee versus Solar Zenith Ang le for 0.45 em of
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