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Parameterization of solar irradiation under clear skies Mächler, Meinrad A. 1983

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PARAMETERIZATION OF SOLAR IRRADIATION.UNDER CLEAR SKIES by Meinrad A. Machler Diplom Maschineningenieur, Eidg. Techn. Hochschule Zurich, 1977 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the Department of Mechanical Engineering We accept th i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March 1983 (c) Meinrad Machler, 1983 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of 1 The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date DE-6 (3/81) ABSTRACT This study compares 5 ex i s t i n g insolation models on three levels of complexity. The model by Leckner represents a spectral integration model, the models by Bird & Hulstrom, Davies & Hay and by Hoyt are parameterization models and the model of A.S.H.R.A.E i s a simple seasonal model. The emphasis of the comparison was kept on the attenuation by atmospheric aerosols as well as on the aspect of the scattered radiation. No new model w i l l be proposed; instead, several improvements to increase the accuracy or to make the application easier w i l l be presented, notably for the determination of aerosol attenuation. Si m p l i f i c a t i o n has been achieved by using the co r r e l a t i o n between horizontal ground v i s i b i l i t y and aerosol attenuation, resu l t i n g in easy-to-handle equations. While most models received only minor changes i t was necessary to restructure the A.S.H.R.A.E. algorithm substantially. i i TABLE OF CONTENTS Abstract i L i s t of Tables iv L i s t of Figures v Acknowledgements v i Nomenclature and Glossary v i i I Introduction 1 1.1 Outline of the Atmospheric Attenuation 3 1.2 Rayleigh Scattering 9 1.3 Ozone Absorption 12 1.4 Gaseous Absorption 15 1.5 Water Vapor Absorption 18 1.6 Aerosol Attenuation 20 I. 6.1 The Aspect of V i s i b i l i t y 25 II Treatment of Direct Radiation 28 II . 1 The Spectral Model of Leckner 32 11.2 The Model of Davies and Hay 34 11.3 The Model of Hoyt 38 11.4 The Model of Bird (and Hulstrom) 43 11.5 The A.S.H.R.A.E. Model 46 III Treatment of Diffuse Radiation 58 111.1 General Approach for Single Scattering 60 111.2 Multiple Scattering 61 111.3 Diffuse Radiation in the Models 69 111.3.1 Model of Leckner 69 111.3.2 Model of Davies and Hay 72 111.3.3 Model of Hoyt 74 i i i 111.3.4 Model of Bird and Hulstrom 76 111.3.5 A.S.H.R.A.E. Model 78 IV Concluding Remarks 85 V Further Work 86 VI References 87 VII Appendix 90 iv L i s t of Tables Table I Hoyt's values for Rayleigh scattering Table II Hoyt's values for aerosol scattering Table III Solar spectrum and absorption c o e f f i c i e n t s Table IV A.S.H.R.A.E. c o e f f i c i e n t s V L i s t of Figures F i g . 1 ; The E x t r a t e r r e s t r i a l Solar Spectrum 8 Fi g . 2; The Transmittance due to Rayleigh Scattering 1 1 F i g . 3; The Transmittance due to Ozone Absorption 14 F i g . 4; The Transmittance due to Absorption by 17 Mixed Gases F i g . 5; The Transmittance due to Water Vapor Absorption 19 F i g . 6; Optical Properties of Atmospheric Aerosol 23 F i g . 7; The Transmittance due to Aerosol Attenuation 24 Fi g . 8; Performance of 5 Models, Direct Radiation I 51 Fi g . 9; Performance of 5 Models, Direct Radiation II 52 F i g . 10; Performance of 5 Models, Direct Radiation III 53 F i g . 11; Performance of 5 Models, Direct Radiation IV 54 F i g . 12; Transmittance due to Water Vapor Absorption 55 Fi g . 13; Transmittance due to Aerosol Attenuation 56 Fi g . 14; Performance of the old and new ASHRAE Model 57 Fi g . 15; Multiple Attenuation Pattern 64 Fi g . 16; Multiple Scattering Pattern 66 F i g . 17; Davies and Hay Layer Model 79 F i g . 18; Performance of 5 Models, Diffuse Radiation I 80 Fi g . 19; Performance of 5 Models, Diffuse Radiation II 81 F i g . 20; Performance of 5 Models, Diffuse Radiation III 82 F i g . 21 ; Performance of 5 Models, Diffuse Radiation IV 83 F i g . 22; Performance of ASHRAE Models, Diffuse Radiation 84 ACKNOWLEDGEMENT I would l i k e to express my sincere thanks and appreciation to my supervisor, Dr. Muhammad Iqbal for his help and advice throughout the project. I also would l i k e to thank Douglas T. Brine for many in s p i r i n g discussions. Dr. John Hay's comments on the f i r s t draft of t h i s thesis have been very useful in preparing the f i n a l document. The f i n a n c i a l support of the Natural Sciences and Engineering Research Council of Canada i s g r a t e f u l l y acknowledged. v i i Nomenclature A A.S.H.R.A.E. c o e f f i c i e n t [W/m«m] BA Aerosol forward scattering r a t i o AMS Air mass B A.S.H.R.A.E. c o e f f i c i e n t C A.S.H.R.A.E. c o e f f i c i e n t C to H Factors as defined in text DS Double scattered d i f f u s e radiation [W/m«m] FR Forward scattering r a t i o I unsubscripted: t o t a l irradiance; [W/m«m] subscripted: spectral irradiance [W/m*m«Mm] Idif Diffuse irradiance [W/m»m] Irr Direct irradiance [W/m-m] K Davies' and Hay's t u r b i d i t y parameter; Optical depth M M u l t i p l i c a t o r in A.S.H.R.A.E. algorithm N Molecule density P Pressure [kPa] PR Attenuation pr o b a b i l i t y SC Solar constant [W/m-m] SH Scale height [km] SS Single scattered d i f f u s e radiation [W/m«m] TS T r i p l e scattered d i f f u s e radiation [W/m«m] UG Concentrated atmospheric gas [cm] UO Concentrated atmospheric ozone [cm] UW Precipitable water [cm] v i i i VIS V i s i b i l i t y [km] WO Ratio of (AS)/(AA+AS) f Hoyt's Rayleigh scattering parameter g Hoyt's aerosol scattering parameter k Absorption c o e f f i c i e n t [1/cm] ra Atmospheric albedo rg R e f l e c t i v i t y of the ground (albedo) s Pathlength [km,cm] a Absorptance a Angstrom wavelength exponent 0 Angstrom p a r t i c l e number density X Wavelength [nm,Mm] T Transmittance 6 Zenith angle [deg] Values in Brackets (A) Aerosol (AA) Aerosol absorption (Abs) Absorption (AS) Aerosol scattering (C) Carbon dioxide (G) Gas (0) Ozone (OX) Oxygen (R) Rayleigh scattering (Uv) Ultra v i o l e t part of the spectrum ix (Vi) V i s i b l e part of the spectrum (Sc) Scattering (tot) t o t a l (1).. General o p t i c a l properties ..(4) Subscripts h on horizontal surface n at normal incidence 0 E x t r a t e r r e s t r i a l value X spectral value Glossary Airmass: The r a t i o of the actual pathlength versus the shortest possible pathlength of the radiation through the earth's atmosphere. (See page 6/7 and eq. (1-6). A.S.H.R.A.E: American Society of Heating, Refrigeration and Airconditioning Engineers. Index of r e f r a c t i o n : A complex number; the real part i s a measure for the o p t i c a l d i f r a c t i o n , the imaginary part indicates the d i e l e c t r i c properties. This index i s a function of the wavelength. X Insolation: The t o t a l solar radiation which reaches the ground. Precipitable water: The height of the completely condensed water in a column from ground to the edge of the atmosphere (mostly in [cm]). Scattering phase function: A function in three dimensions which gives the r e l a t i v e d i s t r i b u t i o n of radiation after scattering. Solar constant: The amount of radiation which would reach the earth in the absence of an atmosphere at mean sun-earth distance. The exact amount of the solar constant i s subject of discussions in the s c i e n t i f i c community. Turbidity: 'Turbid' a i r refers to attenuation of radiation in the atmosphere from sources other absorbing gases. Zenith angle: The angle between the normal to the ground and the position of the sun. The zenith angle and the solar a l t i t u d e add up to 90 degrees. 1 I. Introduction The popular interest in solar radiation and i t s research received an appreciable boost in the l a s t decade. Although t h i s has brought the aspects of solar i r r a d i a t i o n to a broader attendance, i t must be said that solar radiation was a subject of research throughout this century; i n f a c t , research on extinction of radiation was done more than a century ago (for example: Lambert [1]). The methods of prediction of solar i r r a d i a t i o n f i n d a wide range of applications in such f i e l d s as: - c a l c u l a t i o n of cooling loads for a i r conditioners - forestry and agriculture - performance of solar c e l l s - heating of buildings - material deterioration under sunlight - thermal power generation The solar radiation reaches the surface of the earth either as direct (beam) radiation or as diffuse radiation after scattering and r e f l e c t i o n . While the beam radiation i s of special interest to such applications as focusing devices, the d i f f u s e radiation contributes a considerable percentage of the insolation and can not be neglected in most cases. While i t i s s a t i s f a c t o r y for many purposes to know the t o t a l amount of i r r a d i a t i o n over the whole spectrum of wavelengths, other applications c a l l for spectral values of 2 diffuse and dire c t i r r a d i a t i o n - notably in the f i e l d of photovoltaics. Often, the knowledge of i r r a d i a t i o n under cloudless conditions i s s u f f i c i e n t because peak loads occur under clear sky conditions. Therefore t h i s study w i l l be limited to i r r a d i a t i o n under clear skies. Five models of di f f e r e n t complexity w i l l be the basis of t h i s study which has the goal to simplify the use and extend the a p p l i c a b i l i t y of various parameterization models. Every model w i l l be described b r i e f l y , followed by suggestions for an improvement of the respective model. Fundamentals of the scattering and absorption of solar radiation through the earth's atmosphere are b r i e f l y discussed in the next chapter. 3 1.1 Outline of the Atmospheric Attenuation The extinction of radiation in the atmosphere i s not constant over the spectrum; t h i s becomes evident through the fact that the d i f f u s e radiation of the clean cloudless sky i s blue and turns red for very high zenith angles, i.e the sun very close to the horizon. Before an attempt can be made to model broadband attenuation, a short description of the spectral (as a function of wavelengths X) attenuation phenomena seems necessary. Solar radiation i s not emitted homogenously over the spectrum. It has a maximum at around 480 nm which i s at the blue end of the v i s i b l e spectrum. F i g . 1 shows the solar radiation a r r i v i n g above the earth's atmosphere, as well as radiation after passing through the atmosphere. Also shown in t h i s figure i s a graph of the radiation which reaches the earth's surface after passing through a t y p i c a l atmosphere. It becomes quite clear that in some bands of the spectrum the attenuation i s very strong and in others i t i s rather weak. The e x t r a t e r r e s t r i a l radiation reaches the outer l i m i t s of the atmosphere (approx. 100 km above sea level) v i r t u a l l y without any attenuation, simply because the outer space provides an almost complete vacuum. On i t s way from the apparent solar surface to the earth's surface the quantum may be subject to either scattering or absorption (or both) when interference with p a r t i c l e s occurs. The p r o b a b i l i t y of interaction grows with the 4 increasing density of suspended matter. At a given p a r t i c l e density the pr o b a b i l i t y of interaction increases with increasing pathlength. The pr o b a b i l i t y for non-interference i s c a l l e d transmittance and must be a fraction of unity. There i s no interference in vacuum, therefore the transmittance becomes unity. If the medium i s t o t a l l y opaque and thus the mean free pathlength zero, the transmittance becomes zero as well. For most intermediate cases we expect an exponential decay of the amount of d i r e c t l y transmitted radiation at a pa r t i c u l a r wavelength. The main sources of attenuation are gases, which absorb strongly in selected parts of the spectrum ("bands of absorption") and also scatter, and p a r t i c l e s in the a i r , which mainly scatter, but have (weaker) absorptive properties as well. The term "absorption" describes the interaction where the quantum energy i s converted to heat (or some other form of energy) and the quantum ceases to ex i s t . "Scattering" means a change in d i r e c t i o n of the quantum through an interaction with a corpuscle without any loss of the quantum energy. Under the assumption that only one type of attenuator exists in a homogenuous medium, the transmittance for any pa r t i c u l a r wavelength can be written as: r(1) = exp-{K(1) } (1-1) X X The term ' r' stands for transmittance and 'K' for the o p t i c a l depth. The i r r a d i a t i o n can now be written as: 5 Irr = I • exp-{K(l) } (1-2) nX OnX X another attenuators can be added to t h i s atmosphere. The combination of the influences of two (or more) attenuators can be described by way of m u l t i p l i c a t i o n of the transmittances for the various attenuators: r(tot) = T(1) -T(2) -r(3) • (1-3) X X X X or H t o t ) = exp-(Kd) + K(2) + K(3) +...} (1-4) X X X X In atmospheric applications those K's represent the o p t i c a l depth of atmospheric constituents, such as ozone, dry a i r , water vapour, aerosols or mixed gases. This leads to the formulation of a transfer equation which describes the amount of incoming di r e c t radiation: Ir r = I -exp-{K(l) + K(2) + K(3) +...} (1-5) nX OnX X X X Naturally one would l i k e to define exactly the absorption and the scattering of solar radiation at every wavelength. Restrictions in the resolution of the measuring equipment and in the p r a c t i c a b i l i t y of data handling make i t necessary to devide the spectrum in a f i n i t e number of small spectral bands. More than 98% of the solar radiation i s emitted between 290 nm and 4000 nm. For reasons to be explained l a t e r , most of the radiation outside the above l i m i t s does not reach the earth 6 and i s therefore of no in t e r e s t . This study uses the spectral d i v i s i o n of Thekaekara [2] which divides the spectrum into 144 parts within the range of 290 nm to 4000 nm as outlined in Table I I I . The bandwidths are not as small as desirable - especially at the longer end close to 4000 nm - but a spectrum with narrower intervals would not allow the use of many sets of absorption c o e f f i c i e n t s available in the l i t e r a t u r e . Thekaekara's spectrum was adopted by NASA and many sets of absorption c o e f f i c i e n t s are based on i t . Recently, a new spectrum, proposed by the World Radiation Center has been adopted by the World Meteorological Organization [34]. To understand the transfer equation, the mathematical handling of the spectral attenuation for the various atmospheric constitutents has to be understood. In the absence of a complete and comprehensive set of measured data t h i s study uses the o r e t i c a l spectral transmittances for dry a i r scattering, aerosol attenuation, ozone absorption, mixed atmospheric gas absorption and water vapor absorption as c r i t e r i a for any parameterization. These values were taken from Leckner [3]. It i s important to determine the pathlength of radiation. The shortest pathlength possible for the solar radiation i s with the sun in the zenith. Commonly the pathlength i s expressed in non-dimensional form of the 'airmass'. At sea l e v e l , an airmass of unity is the shortest pathlength (zenith angle 6=0); at zenith angles higher then zero the airmass increases. The simplest mathematical formulation to determine the airmass uses the following assumptions: - there i s no curvature of the earth 7 - the index of refraction of the a i r i s equal to unity. Thus the airmass takes on the form: AMS = l/cos(0)=sec(0) (1-6) Other formulations that eliminate the r e s t r i c t i v e assumptions are introduced in later sections whenever they are used. 9 1.2 Transmittance Due To Dry Air Scattering (Rayleigh scattering) Approximately 80% of the atmosphere consists of nitrogen and nitrogen a f f e c t s radiation mainly by scattering. The theory of scattering by p a r t i c l e s much smaller than the wavelength of the radiation - and molecules are just that - i s well founded and goes back to the work of Lord Rayleigh. The next largest constituent of the atmosphere, the biatomic oxygen, i s a molecular scatterer as well. It also absorbs radiation but the absorptive property i s of second order between 290 and 4000 nm. Rayleigh found that the transmittance due to molecular scattering i s wavelength dependent. Under the assumption that the scattering molecules are perfect spheres, the transmission through dry a i r with molecules much smaller than the wavelength of the radiation becomes an exact function of the fourth power of the wavelength and can be put into the following form: r(R) = exp{K(R)-AMS} (1-7) X where the o p t i c a l depth K(R) at unity AMS i s given as (Penndorf [4]): 2 2 4 K(R) =8-7T.(n -1)-N / (3«N -X) (1-8) s o The term 'N' gives the molecule density with 'N subscript o' the density at sea le v e l and 'n subscript s' the index of re f r a c t i o n . 10 Leckner [3] uses an exponent of -4.08 to allow for the departure of molecules from the theory of perfect spheres (based on Rayleigh): -4.08 r(R) = exp{k(R ) - (X )-AMS} (1-9) X with: k(R) = -0.008735 X in [jum] (1-10) Thus the transmittance due to dry a i r scattering as used in t h i s study i s represented by Eq. (1-9) and Eq. (1-10). A plot of T(R , X) against X i s shown in F i g . 2. As we can see from F i g . 2, the transmittance due to scattering by molecules increases with increasing X. Thus Rayleigh scattering has i t s strongest e f f e c t in the short end of the solar spectrum. 11 Figure 2, The Transmittance due to Rayleigh Scattering 12 1.3 Transmittance due to Ozone Absorption Oxygen atoms not only come in pairs but also in t r i p l e t s in form of ozone. Not only does ozone scatter (this i s included in Eq. 1-9), but i t also absorbs radiation. Ozone absorbs radiation in the shorter wavelengths of the solar spectrum and i t i s mainly responsible for the absorption of the u l t r a v i o l e t radiation, excessive amounts of such would harm the l i f e on earth. The transmittance due to ozone can be described by the following equation: r(O) = exp{-k(0) -UO-AMS} (1-11) X X The term 'UO' denotes the amount of ozone in a v e r t i c a l column. Depending on season and lati t u d e the value of 'UO' varies between UO=0.20 [cm] and UO=0.50 [cm] at NTP. Furthermore, the v e r t i c a l p r o f i l e of the ozone concentration shows a d i s t i n c t concentration at very high a l t i t u d e with a peak at around 22 km above sea l e v e l for midlatitudes. The peak elevation decreases s l i g h t l y towards the poles. Ozone absorption c o e f f i c i e n t s k(O) used in t h i s study were taken from Leckner [3] who in turn took the c o e f f i c i e n t s from Vigroux [5]. These c o e f f i c i e n t s are reproduced in Table I I I . The transmittance of ozone as a function of wavelength i s shown in F i g . 3. As we see, the transmittance approaches zero for X-values smaller than 300 nm. Below 290 nm no radiation i s 13 transmitted due to the t o t a l absorption by ozone. This i s the reason why no portion of radiation below 290 nm i s taken into account as mentioned above in Section 1.1. 1 4 1.0 2 .0 Wavelength (micrometers) 3 . 0 4 .0 F i g u r e 3, T r a n s m i t t a n c e due t o Ozone A b s o r p t i o n 15 1.4 Absorption by Atmospheric Mixed Gases Among a l l the dry a i r gases ( i . e . excluding water vapor), ozone has a d i s t i n c t concentration p r o f i l e in the v e r t i c a l d i r e c t i o n and i t i s due to t h i s reason that the absorption by ozone was treated separately in Section 1.3. A l l the remaining gases (such as oxygen and carbon dioxide, etc.) are more or less homogenously dis t r i b u t e d in the atmosphere and their concentration does not vary greatly. In t h i s study they are referred to as "mixed gases". Among these mixed gases, the main absorbers of solar radiation between 290 and 4000 nm are carbon dioxide and oxygen. F i g . 4 shows the monochromatic transmittance due to absorption by atmospheric mixed gases. Most of the absorption occurs outside of the v i s i b l e part of the spectrum with the exception of an oxygen band at 760 nm which i s right at the threshold of human perception. The f i r s t expectation for an equation to formulate the transmittance due to mixed gases would probably be something l i k e the following: T(G) = exp{-K (G)-AMS) (1-12) X X But Fig.4 shows that the attenuation by mixed gases occurs in narrow bands with steep flanks. The set of wavelength intervals i s much too crude for the assumption that the attenuation and the i r r a d i a t i o n within those inte r v a l s are constant. Absorption c o e f f i c i e n t s for mixed gases are therefore averaged over the i n t e r v a l s . There are many approaches to model the absorption by 16 mixed atmospheric gases (Fowle [6], Howard [7]). This study uses the approach of Leckner [3] who in part based his work on publications of Yamamoto [8], McClatchey [9] and Elterman [10]. The shape of absorption bands can be c l a s s i f i e d . Each class of absorption band t h e o r e t i c a l l y c a l l s for a d i f f e r e n t function to describe the extinction, as outlined by Goody [11] and Tiwari [12]. Within the spectral range of 290 to 4000 nm however, i t i s s u f f i c i e n t l y accurate to use one function only; the transmittance due to the absorption by mixed atmospheric gases can then be described by the following equation: / 1.41- K(G) • AMS \ T(G) = exp- (1-13) X I 0.45 / \(1+118.3-K(G) -AMS) / X ' The c o e f f i c i e n t s K(G) used with t h i s equation to produce F i g . 4 where taken from Leckner [3] and are reproduced in Table I I I . 1 7 Figure 4 , Transmittance due to Absorption by Mixed Gases 18 1.5 Absorption by Water The attenuation behaviour of water vapor i s far more complex than the one of ozone and is similar to the one of the mixed atmospheric gases. Water absorbs in certain bands of the spectrum with a very steep r i s e of the absorption at the flanks of these bands which are often close to bands of almost t o t a l transparency. Similar to the mixed gases t h i s study adopted the water absorption treatment of Leckner [3] who obtained the transmittance equation for water vapor as: 1 0.2385- UW • k(W) • AMS \ • (1-14) 0.45 / (1+20.07 • UW «k(W) -AMS) / X 1 Eq.(l-14) i s of the same form as the transmittance equation for mixed gases - Eq.(I-13) - with the extension of an element describing the varying amount of water. The set of water absorption c o e f f i c i e n t s used to produce F i g . 5 with Eq. (1-14) are reproduced in Table III; they were taken from Leckner [3]. The t o t a l amount of the pre c i p i t a b l e water in a v e r t i c a l column can be estimated from a number of observations such as dew point temperature or p a r t i a l pressure of water vapor (see Leckner [3]). 19 J - D 2 .0 3 . 0 Wavelength (micrometers) Figure 5, Transmittance due to Water Absorption 20 1.6 Attenuation by Atmospheric Aerosol The accurate mathematical description of the attenuation by atmospheric aerosols i s extremely d i f f i c u l t . This i s shown by the variety which exists in the most important parameters which influence the determination of aerosol attenuation: - the p a r t i c l e size and i t s d i s t r i b u t i o n - the d i e l e c t r i c properties which determine the absorption - the shape of a p a r t i c l e - the wavelength - the number of aerosol p a r t i c l e s per unit volume - aerosols both scatter and absorb. Probably the most d i f f i c u l t task i s the determination of the p a r t i c l e size d i s t r i b u t i o n and i t s o p t i c a l properties. Over the decades there have been numerous suggestions to determine the number and the size d i s t r i b u t i o n of aerosols in a given volume of the atmosphere - none of them e n t i r e l y s a t i s f a c t o r i l y . One d i f f i c u l t y i s the cut-off point: The smaller the p a r t i c l e the more d i f f i c u l t i t becomes to prove i t s existence. The Rayleigh theory of scattering does not describe the scattering by aerosols because the aerosol p a r t i c l e s (or most of them) are substantially larger than the wavelength of the solar radiation within the spectrum of in t e r e s t . Gustav Mie [13] developed a theory which treated the scattering by p a r t i c l e s larger than the wavelength of the incident radiation but this 21 theory is also r e s t r i c t e d to spherical p a r t i c l e s . Nevertheless i t proved to be a valuable tool for the description of aerosol scattering, though a treatment within the range of engineering applications requires some simplifying assumptions. Under the assumption that the p a r t i c l e diameter (or radius) d i s t r i b u t i o n follows a power law and that absorption i s neg l i g i b l e or i s nonexistent, one can apply Mie's theory in the form of a rather simple r e l a t i o n of wavelength dependence. Negligible absorption means that the complex part of the index of refraction has to be small or zero which in turn means that the aerosols have to have d i e l e c t r i c properties. The fact that water condenses around p a r t i c l e s increases the a p p l i c a b i l i t y of t h i s assumption. A s t i l l widely used approach to implement these assumptions was f i r s t published by Angstrom [14,15]. His formulation for the spectral transmittance due to aersosol attenuation takes on the following form: a r(A) = exp{0 • AMS / (X )} (I-15) X The constants a and 0 are obtained with f i l t e r measurements at two wavelengths. With only two equations to determine a and 0 the result i s a set of two constants. At a given wavelength the value of a i s a function of the p a r t i c l e s i z e . It becomes quite c l e a r , that for every wavelength and for every class of p a r t i c l e size a unique a i s v a l i d . An average constant a i s thus an averaged value over 2 parameters. The p a r t i c l e size d i s t r i b u t i o n of the atmospheric aerosols usually follows a power law quite c l o s e l y . Exceptions from th i s rule have been observed for 22 maritime aerosols and for unusual accumulations of dust from a single source l i k e volcanic eruptions or forest f i r e s of large extension. The average a over the whole spectrum between 290 and 4000 nm has a value between 0.9 and 2.0 with the most frequent observations around a=1.3. Fig.6 shows the relationship of the t u r b i d i t y parameters at p a r t i c u l a r wavelengths and for various r a d i i of the aerosol p a r t i c l e s (from McCartney [16]). Angstrom c a l l e d the constant 0 ' P a r t i c l e Number Density'. This might be misleading because 0 i s not only a function of the p a r t i c l e numbers per unit volume but even more a function of the aerosol mass density. For applications of spectral irradiance values i t might be of interest to increase the number of wavelengths at which f i l t e r measurements are taken to determine 0 not only as a constant but as a function of the wavelength. This study's main concern lays with the parameterization models and the exact value of the spectral transmittance i s not as important as the average transmittance value over the whole spectrum. To determine the l a t t e r i t i s s u f f i c i e n t to obtain 0 as a constant and i t seems that Angstrom's choice of wavelengths was a good one in respect to the accuracy of the broadband aerosol transmittance. F i g . 7 shows the transmittance due to aerosol attenuation as a function of wavelength for average 0=0.1 and an a of unity. 23 TOTAL SCATTERING AND EXTINCTION BY HAZE AEROSOLS Wav*»!ennfh (um) g u r e 6, O p t i c a l P r o p e r t i e s o f A e r o s o l s a s a F u n c t i o n o f W a v e l e n g t h . (From M c C a r t n e y [ 1 6 ] ) 24 Figure 7, Transmittance due to Aerosol Attenuation 25 1.6.1 The Aspect of V i s i b i l i t y The determination of the transmittance due to aerosols from t u r b i d i t y measurements i s quite a formidable task. It would be desirable to link the aerosol transmittance to a more e a s i l y measureable quantity l i k e the horizontal v i s i b i l i t y . The f i r s t problem arises with the d e f i n i t i o n of v i s i b i l i t y . Linke [17] defines the threshold of human perception as a contrast difference of 2%: If the apparent brightness of an ideal black body i s more than 0.98 times the apparent brightness of a diff u s e background, the object i s considered i n v i s i b l e for the human eye, which has a maximum s e n s i t i v i t y at X=550 nm. The li n k i n g of v i s i b i l i t y to a fixed quotient of contrast has the d e f i n i t e advantage that the horizontal o p t i c a l depth follows immediately. Neglecting any amount of molecular scattering: If Rayleigh scattering i s taken into account, the o p t i c a l depth for Rayleigh scattering (0.01162 1/km) [10] has to be subtracted and we obtain: A variety of authors have established relations between the v e r t i c a l aerosol density d i s t r i b u t i o n and the aerosol density k(A,0)= ln(l-0.98)/VIS [1/km] (1-16) k(A,0)= (3.912/VIS)-0.01162 [1/km] (1-17) 26 near the ground. A very convenient way to do so was published by McClatchey [9] who established the 'scale height' of an aerosol d i s t r i b u t i o n . 'Scale height' i s the equivalent pathlength at ground l e v e l aerosol density to obtain the same aerosol attenuation as for the v e r t i c a l path through the atmosphere: SH = K ( v e r t i c a l path)/ Mground, horiz) (1-18) It i s quite c l e a r that the scale height becomes higher with decreasing p o l l u t i o n because the aerosol p o l l u t i o n shows the highest density close to the ground. Under normal circumstances - no aerosol of one single source, i.e volcanic eruption, forest f i r e - i t was found that the scale height SH increases more or less l i n e a r i l y with increasing v i s i b i l i t y . Buckius and King [18], with data from McClatchey [10], found the scale height at a v i s i b i l i t y of 23 km to be SH(23)=1.577 km and at a v i s i b i l i t y of 5 km to be SH(5)=1.132 km. These results f i n d support in data from Zuev [19]. With Eqs. (1-17) and (1-18), 0 can now be written as [18]: a /3.912 X /( 1 .577-1 . 1 32)•(VIS-5) \ 0=(.55) • .01162 • +1.132 (1-19) \VIS J \ 23 - 5 / Although a guess has to be made about the value of the p a r t i c l e size d i s t r i b u t i o n exponent 'a', the equation proves very v a l i d under usual aerosol d i s t r i b u t i o n s where 'a' i s commonly around 1.3. For values of 'a' d i f f e r e n t from 1.3 however, the use of Eq. (1-19) in conjunction with Eq. (1-15) yi e l d s diverging re s u l t s for equal v i s i b i l i t i e s , which can be attributed to the fact that the maximum s e n s i t i v i t y of the eye (at 550 nm) does 27 not coincide with the maximum spectral irradiance (at 480 nm). This study therefore suggests a s l i g h t modification of Eq. (I-19) so that equal transmittances for equal v i s i b i l i t i e s can be obtained: 0=(3.912/VIS-.O1162)•{(16.2385+VIS)•(F-G-a)+H} (1-20) where F= 2.3575E-02 (l-20a) and G= 9.387E-03 (l-20b) and H= 0.278863 (l-20c) These Eqs. (1-19) and (1-20) do not cover v i s i b i l i t i e s in fog where the p a r t i c l e s become very big. The link between ground v i s i b i l i t y and aerosol o p t i c a l depth w i l l be used in the parameterization models as outlined in the next chapter. 28 II Treatment of Direct Radiation There are currently 3 lev e l s of models to estimate d i r e c t (and diffuse) radiation on clear days. Models which use spectral transmittances and irradiances and perform numerical integrations represent the highest l e v e l . This study presents a rather simple model of t h i s kind by Leckner [3] as an example for t h i s approach. The middle l e v e l i s represented by less complex parameterization models which do not perform integrations over the spectrum but s t i l l s p l i t the determination of the extinction into various parts representing atmospheric constituents. This study presents three models of t h i s kind from Davies and Hay [20], Hoyt [21] and from Bird and Hulstrom [22]. The lowest l e v e l of models simulates insolation through just one equation with airmass as a variable and constants v a l i d for entire months not considering any changes of the atmosphere. The most widely known such model i s the A.S.H.R.A.E. algorithm [31] which w i l l be presented in Section II.5 of t h i s study. The basic problem of modeling dir e c t insolation i s the solution of the transfer equation (Eq. 1-5) for a p a r t i c u l a r wavelength and i t s integration over the whole spectrum: 4000 r I r r ( t o t ) = I i • r(tot) • dX (II-1) n OnX X 290 J For a model of the top l e v e l the main problem i s the d e f i n i t i o n of the spectral transmittances for the various attenuators over the whole spectrum. Sections 1.1 through 1.6 have shown one way 29 of solving t h i s task. Other models - not presented here - divide the atmosphere into a number of homogenous layers for which the attenuating properties are defined i n d i v i d u a l l y [24,25]. The transfer equation for a p a r t i c u l a r wavelength or with enough accuracy for a very narrow band i s given below: T = r(R) T ( A ) -T(W) T ( G ) «T(0) (II-2) X X X X X X However, i t i s mathematically not sound to formulate the broad-band transmittance as follows: T = r(R)«T(A)•T(W)-T(G)'T(O) (II-3) Despite t h i s obvious mathematical inconsistency a l l models of the middle l e v e l presented in t h i s study force the determination of the s p e c t r a l l y integrated i r r a d i a t i o n values into the form of Eq.(II-3) or similar expressions. Eq.(lI-3) i s therefore c a l l e d the 'broadband transfer equation', although there i s an inherent error in t h i s formulation. Because the 'broadband transfer equation' i s widely used despite the obvious mathematical deficiency, i t seems appropriate to assess the error inherent in t h i s formulation. The problem can be s p l i t into two separate questions: 1.) To what extent can a "broadband transmittance", averaged over the whole spectrum, be established for a single attenuator. 30 2.) To what extent can the "broadband transmittances" for attenuators be used m u l t i p l i c a t i v e l y in conjunction with each other and with the solar constant. The averaged transmittance for a single attenuator - an assumed ideal condition where only one kind of attentuation occurs over the whole spectrum - i s now defined by means of the following equation: 4000 r {I • T(1) • dX 290J OnX X T(1) = (II-4) 4000 r I • dX 290 J OnX where: 4000 f Ii • dX = SC (II-5) 290^ OnX It i s possible to integrate t h i s equation and define the respective T as a function of the airmass and the density of the attenuator. This function - which w i l l most l i k e l y not be in a closed form - can then be parameterized with any desired accuracy as a function of the airmass and the density of the attenuating medium. A l l models of the middle l e v e l use t h i s approach. The second question i s not so readily answered: For an easier understanding the example w i l l be r e s t r i c t e d to two attenuators. The effect of three or more attenuators can then be extrapolated. For one attenuator the transmittance i s described by Eq. (II-4) above. For two attenuators the correct equation 31 becomes: 4000 r 11 • T ( 1 ) • T(2) • dX 290 J OnX X X r ( t o t ) = (II-6) SC It i s now quite obvious, that the product of Eq.(II-4) for two d i f f e r e n t attenuators i s not the same as Eq.(lI-6) because the integral of a product of two functions i s not the same as the product of two integrals of the same two functions, unless one (or both) of those functions degenerates to a constant. Despite t h i s mathematical verdict most parameterization models use t h i s approach. The errors involved are rather small because over very large parts of the spectrum only one of the attenuators i s dominant or complete extinction occurs through one attenuator, where another would be quite strong. However, the user of parameterization models is cautioned not to use single elements of the models because the performance of a single transmittance might be less accurate then the product. On the lowest l e v e l no attempt i s made to solve the transfer equation; instead, a simple power law with the airmass as a variable i s adopted. The next sections w i l l outline these models in d e t a i l . 32 I1.1 The Spectral Model by Leckner This study presents the solar insolation model by Leckner [3] as an example for the models of the top l e v e l , where numerical integration methods are used to calculate the solar insolation incident on the earth's surface. The elements of Leckner's model where presented in parts in the sections 1.2 through 1.6. The basic equation of Leckner's model of solar insolation has the following form: 4000 r Irr = I I -T(R) -T-(A) «r(W) -T(G) - T ( O ) -dX (II-7) n 290 J OnX X X X X X The various spectral transmittances are calculated with the respective expressions discussed e a r l i e r in chapter I, sections 2 to 6. The performance of Leckner's model for various amounts of water vapor, ozone and t u r b i d i t y i s shown in Fig's. 8 to 11 (at the end of Chapter I I ) . Also shown are other models as a comparison. The v e r t i c a l axis shows the irradiance in watts per square meter. Mean sun to earth distance was assumed. Turbidity values were obtained with Eq. (1-19). The horizontal axis shows the zenith angle; unless otherwise noted for a p a r t i c u l a r model, Eq. ( 1 - 6 ) was used to obtain the airmass. Leckner's model tends to predict s l i g h t l y higher values then the other models except for conditions of high t u r b i d i t y . While the model of Leckner exceeds the p o s s i b i l i t i e s of a small c a l c u l a t o r , i t is nevertheless easy to use on a computer. 33 Because of the good performance (compared to more elaborate models) combined with a tolerable amount of required computer time the Leckner model has been chosen as a standard for the development of components for the other models presented in t h i s study. If further studies should show some s i g n i f i c a n t systematic deviations of Leckner's model from r e a l i t y , corrections on the elements of the other models should be easy to perform. The next sections present these models and some changes suggested to improve their a p p l i c a b i l i t y and/or p r a c t i c a b i l i t y . 34 II.2 The Model by Davies and Hay Davies and Hay [20] present a model to calculate solar i r r a d i a t i o n on horizontal surfaces. The basis for t h i s model (from now on referred to as Model A) i s the "transfer equation for broadband transmittances" as outlined at the beginning of Chapter II (Eq. II-3) with a s l i g h t modification. Because water vapor and mixed gases absorb in parts of the spectrum where no ozone absorption occurs, the transmittances for water vapor (and gases) and ozone are not mult i p l i e d and the following approach i s used instead: Irr = SC • cos(f5) . [ T (0) • r (R)-a(W) ] • r (A) (II-8) h The absorptance due to ozone and the resulting transmittance are given as follows (Lacis and Hansen [26]). For the ozone absorption band in the u l t r a v i o l e t : 0.02118 • UO a(0,Uv) = :— (II-9a) -4 2 1 + 0.042 • UO + 3.23-10 • (UO) and for the band in the v i s i b l e part of the spectrum: 1.082 • UO 0.0658 • UO a(0,Vi) = + (II-9b) .805 3 (1 + 138.6-UO) 1 + O03.6-UO) and the absorption due to ozone as the sum of the above: a(0) = a(0,Uv) + a(0,Vi) ( l l - 9 c ) and f i n a l l y for the transmittance due to ozone: 35 T(0) = 1 - a(0) (II-9) The combined absorptance by water vapor and mixed atmospheric gases in Model A was taken from Lacis and Hansen [26] as 2.9 • UW a(W) = (11-10) .635 (1+141.5-UW) + 5.925-UW This function approaches asymptotically the value 0.49 for large airmasses. The transmittance r=(1-a) w i l l therefore never be less than 0.5. The transmittance due to scattering by dry a i r (Rayleigh scattering) was presented in tabulated form . The authors also published a polynomial expression to f i t the tabulated transmittance data: 2 3 4 r(R)=.972-.08262AMS+.00933AMS -.00095AMS +.0000437AMS (11-11) The above expression f i t s the table well within reasonable l i m i t s of the airmass. Davies and Hay circumvented the d i f f i c u l t task of establishing a parameterization for the aerosol transmittance. Instead, following a suggestion by Houghton [27], they employed the simple relationship: AMS T(A) = K (11-12) No procedure was given to determine -K- and the user was l e f t with the suggestion to use K = 0.95 "as a global average value". This value would be matched by "clean A t l a n t i c a i r " . For anticylonic days a value of K = 0.88 was suggested; the authors 36 further recommend " l o c a l c a l i b r a t i o n against measurement". It i s obvious that the weak point of the model of Davies and Hay for any application i s the uncertainty involved in the determination of the aerosol transmittance. This study therefore attempts to improve Davies' and Hay's model by replacing their transmittance equation for aerosols. Integrated values were obtained from the model of Leckner [3] in form of the following equation: 4000r -a exp{-/3-X }«dX J OnX 290-r(A) = (11-13) 4000 •dX U r I < OnX J 290 As was shown in chapter II, Eqs. II-4 to II-6, the broadband transmittance for an attenuator with inhomogenous attenuation c h a r a c t e r i s t i c s over the spectrum can never be represented by a power function. As a simplifying step, the f i c t i o n of a power law was upheld to obtain a formulation which i s easy to handle, although t h i s approach necessarily means a loss of accuracy at higher airmasses. Because of the significance of peak radiation values, the constants of the following Eq.(11-14) were chosen to f i t Eq.(11-13) at low airmasses (AMS = 1 and AMS = 2): T(A) = C + D • exp(-E • 0 • AMS) (11-14) where: C = (0.12445 • a - 0.0162) , (ll-14a) 37 D = (1.003 - 0.125-a) (II-14b) and E = (1.089 • a + 0.5123) (II-14c) The Eq.(11-14) makes the assumption that the user has access to measured values of 0. The determination of 0 - as outlined in section 1.6.1 - makes f i l t e r measurement a necessity. To obtain reasonably accurate values of 0 without f i l t e r measurements, i t is suggested that Eq. (1-20) be used in conjunction with Eq. (11-14). The performance of Model A i s shown in Figs. 8 to 11 against other parameterizations. The results of th i s model correspond well with other models except for high t u r b i d i t y . 38 II .3 The Model by Hoyt The model by Hoyt (from now on c a l l e d Model B) uses a derivative of the broadband transfer equation. Hoyt obtains the di r e c t part of the solar radiation as: He defines f i v e d i f f e r e n t absorptances a for water, ozone, carbon dioxide, oxygen and aerosol and two transmittances due to scattering by p a r t i c l e s (aerosol) and by dry a i r (Rayleigh). This Eq. (11-15) is b a s i c a l l y the 'broadband transfer equation' as outlined in chapter II "Treatment of Direct Radiation". This equation has been modified by Hoyt by substituting the transmittances T by r=(1-a) and by then neglecting a l l terms of second or higher order: r(AS) r(R) (11-15) Based on: T = T • T • T 2 3 he substitutes T = (1-a )•(1-a )•(1-a ) 1 2 3 which i s r =1-a -a -a +a »a +a «a +a -a -a «a -a 1 2 3 1 2 1 3 2 3 1 2 3 and i s s i m p l i f i e d to T =1~a -a -a 1 2 (11-16) 3 39 The influence of t h i s substitution and s i m p l i f i c a t i o n i s small for small absorptances but increases with higher airmasses where absorption may become quite large. Hoyt's equation e a s i l y d e l i v e r s results 5% smaller than i f he had used the o r i g i n a l "broadband transfer equation" with the respective absorption values. Hoyt's absorptance values are therefore adjusted to his equation and cannot be used properly with other models at high airmasses. The absorptances for Model B are for water: -4 0.3 a(W) = 0.110-(6.31•10 +UW) -0.0121 (II-17) for ozone: -4 0.38 -4 a(0) = 0.045-(8.34-10 +UO) -3.1-10 (11-18) for carbon dioxide: 0.26 -4 a(C) = 0.00235.(0.0129 + UC) -7.5-10 (11-19) for oxygen: -3 0.875 a(OX)= 7.5-10 - AMS (11-20) for aerosol: AMS a(AA)= (l-wo)-{g(/3)} (11-21 ) Unfortunately, Hoyt did not give closed form formulas for the ca l c u l a t i o n of the two scattering components and the absorption by aerosol. Instead, he used the following rela t i o n s h i p to obtain the transmittances due to scattering: 40 transmittance due to Rayleigh scattering: AMS T ( R ) = {f(AMS)} ( 1 1 - 2 2 ) transmittance due to aerosol scattering: AMS r(AS)= (g(0)} ( 1 1 - 2 3 ) The two functions f(AMS) and g(j3) were given in tabulated form only without a parameterization function (see below). While the AMS f(AMS) 0 g(/3) 0.0 0.0 0.0 1 .000 0.5 0.909 0.02 0.972 1.0 0.917 0.04 0.945 1 .5 0.921 0.06 0.919 2.0 0.925 0.08 0.894 2.5 0.929 0.10 0.870 3.0 0.932 0.12 0.846 3.5 0.935 0.14 0.824 4.0 0.937 0.16 0.802 0.18 0.780 0.20 0.758 0.24 0.714 0.28 0.670 0.32 0.626 Table I Table II table for g(/3) extends to f a i r l y large t u r b i d i t y values, the usefulness of the table for f(AMS) i s r e s t r i c t e d because i t extends only to airmass 4 which i s approximately equivalent to a zenith angle 8= 75 degrees. Even under equatorial conditions the table r e s t r i c t s the use of the model to times of more than one hour before sunset or after sunrise. Although the curve for both functions f(AMS) and g(0) are smooth enough to allow considerable extrapolation, the a p p l i c a b i l i t y of the model i s 41 hampered. This study t r i e s to open t h i s otherwise well performing model (graphical results in comparison with other models are shown in F i g . 8 to 11) to wider use by replacing the tables for f(AMS) and g(0) through two parameterization formulas which extend the range of application beyond the present l i m i t s of AMS=4 and 0=0.32. However, as F i g . 11 shows, at very high t u r b i d i t y the results of thi s model are well below the results of other models. This study presents a parameterization not of f(AMS) but of r(ray)=f(AMS)**AMS: T(R)= (.375566) • exp(-.221185•AMS) + .615958 (11-24) The formulation for T(R) (Eq. 11-24) never deviates more than 0.2% from the values given by Hoyt. To estimate the range of reasonable accuracy for extrapolated values beyond AMS=4, the UBC Computer Center Software "*OLSF" was used to compare with Eq. (11-24). The range of confidence (error less than 2%) extends as far as AMS=8. This reduces the limited time range around sunrise/sunset to less than half of i t s previous value. The table for g(0) w i l l be replaced by a parameterization of similar construction as follows: g(0)=1.909267 • exp(-.6670236• 0) - 0.914 (11-25) This equation never y i e l d s a deviation from Hoyt's tables 42 of more than 0.32%. As for the transmittance function an attempt of extrapolation has been made with a powerful software (UBC *OLSF). Against t h i s standard, confidence in Eq. (11-25) can be maintained up to t u r b i d i t y values of 0=0.5. These modifications of Model B open i t to applications of atmospheres of high t u r b i d i t y at small solar a l t i t u d e s . 4 3 II . 4 The Model by Bird and Hulstrom Bird and Hulstrom published a model [ 2 2 ] to calculate d i r e c t i n s o l a t i o n and la t e r extended their model to diff u s e radiation and incorporated some minor changes of the d i r e c t part of the model to improve the performance [ 2 3 ] . This study presents the model of Bird and Hulstrom as published in [ 2 3 ] (from now on c a l l e d "Model C") which i s i t s most recent form. Model C uses the "broadband transfer equation" almost in i t s o r i g i n a l form; only a constant factor i s used: The use of the factor ' 0 . 9 6 6 2 ' stems from the fact that Bird and Hulstrom used a spectrum for their parameterization that only considered the range from 3 0 0 nm to 3 0 0 0 nm. Because there i s some transmission past the above mentioned spectral boundaries the said factor was incorporated. The transmission equations are parameterized as follows: Irr = S O c o s ( t ? ) - 0 . 9 6 6 2 . r ( R ) - T ( A ) «T(W) T ( G ) T ( 0 ) h ( 1 1 - 2 6 ) . 8 4 1 T(R)= exp { - 0 . 0 9 0 3 -(AMS') •(1+AMS'-(AMS') 1 . 0 1 ( 1 1 - 2 7 ) - 0 . 3 0 3 5 r ( 0 ) = 1 - . 1 6 1 1 • X O ( 1 + 1 3 9 . 4 8 • X O ) ( 1 1 - 2 8 ) 2 - 0 . 0 0 2 7 1 5 - X O / ( 1 + 0 . 0 4 4 - X O + 0 . 0 0 0 3 - X O ) 4 4 with X O = U O • AMS (Il-28a) 0.26 r(G)= exp{-0.0127-(AMS')} (11-29) 6 8 2 8 T ( W ) = 1 - 2 . 4 9 5 9 * X W / { ( 1 + 7 9 . 0 3 4 - X W ) + 6 . 3 8 5 - X W } ( I I - 3 0 ) with XW = UW • AMS (Il-30a) .873 .7088 .9108 r(A)= exp{-K(A) •(1+K(A)-K(A) )-AMS } (II~31) with K(A)= 0.2758«K(A) + 0.35-K(A) X=0.38jum X=0.5jum (11-32) Bird and Hulstrom used Kasten's [28] formulation for the airmass as given below in Eq.(11-33): -1.25 AMS = 1/{COS0 + 0.15-(93.885 -0) } (11-33) For Rayleigh scattering and the mixed gas absorption a pressure corrected airmass was used: AMS'= AMS • P/101.3 [P in kPa] (II-33a) Eqs. (11-33) and (ll-33a) are also v a l i d in other parameterization models. As seen in Eqs.(11-31) and (11-32) above, Bird and Hulstrom use atmospheric t u r b i d i t y values at two wavelengths (380 nm and 500 nm); t u r b i d i t y values at these wavelength are measured by the U.S. National Weather Service 45 (Flowers et a l . [29]) for some locations on a routine basis. The use of two weighted spectral t u r b i d i t i e s gives a much more accurate picture of the atmospheric t u r b i d i t y and i t s influence on broadband solar i n s o l a t i o n . But for most locations t h i s procedure c a l l s for either a substantial amount of f i l t e r measurement at two wavelength of the v i s i b l e spectrum or guesswork. This study attempts to make the Model C more accessible and therefore suggests to replace the parameterization for the aerosol transmittance. In a l l cases where detailed t u r b i d i t y data from f i l t e r measurements i s not a v a i l a b l e , the following equation allows the c a l c u l a t i o n of the aerosol transmittance: The Eq. (11-34) was derived by using the link between horizontal v i s i b i l i t y and the t u r b i d i t y as outlined in Section 1.6.1. It may be used to substitute for other aerosol transmittances of models of the same le v e l (Model A and Model B) but not for the model described in the following Section II.5. T(A) = 10.97 - 1 .265 (11-34) 46 II.5 The A.S.H.R.A.E Model This model on the lowest l e v e l uses a very simple algorithm [31]. The extinction over the whole spectrum i s assumed to follow a power law function with the airmass. A l l attenuative influences at an airmass of one as well as the solar constant and i t s v a r i a t i o n over the year are compounded into two sets of 12 constants, one set for every month. Only the var i a t i o n of the water vapour content over the year was taken as variable to determine these constants, which are reproduced in App.IV. The irradiance equation has the following form: Irr = A«cos(0).exp{-B»AMS} (11-35) h On a log-normal plot t h i s function transforms to a straight l i n e with an intercept of 'A' for AMS = 1/cos(t9) = 1. The major advantage of the A.S.H.R.A.E. algorithm i s i t s easy handling and the major drawback i s i t s i n e l a s t i c i t y to any change of concentration of atmospheric constituents. Because the influence of the varying concentration of ozone i s small and the concentration of gaseous absorbers does not change very much (except water vapour), the main targets to increase the e l a s t i c i t y of the model are the absorption by water vapor and the scattering by aerosols. It i s quite interesting to assess the influence of water vapor concentration on the broadband transmittance of solar radiation. This has been done by using Leckner's [3] formulation 47 for transmittance through water vapor in the following form: 4000/" / 0.3* UW • K(W) • AMS I 'exp-OnX . 0.45 \(1+20.07 • UW -K(W) -AMS) X 290 J T(W) = (11-36) The calculations were performed for a multitude of values of pr e c i p i t a b l e water and the results are shown in Fig.12 at the end of Chapter II. From Fig.12, i t i s quite obvious that the f i r s t few millimeters of p r e c i p i t a b l e water have a very strong influence on the transmittance while a difference of one millimeter at higher values of water concentration does not account for very much change in transmittance. The monthly values of 'A' were determined by using average water vapor values for the United States [33],The A.S.H.R.A.E. Model also assumed that 200 dry dust p a r t i c l e s per cubic centimeter were present with no variation over the year. The algorithm to es t a b l i s h the transmitted radiation was taken from Moon [30]. Because of his use of ' p a r t i c l e s per volume' instead of 'aerosol mass concentration' the changing water content in the atmosphere also had an influence on the aerosol transmittance c a l c u l a t i o n due to condensation around the n u c l e i . The other parameter with a great spectrum of variation i s the aerosol content. Similar to Eq. (11-36) above, the influence 48 of t u r b i d i t y on the broadband transmittance was determined by using Leckner's formulation of aerosol o p t i c a l depth in the following form to obtain an (ideal) transmittance through aerosols only (same as ( i i - 1 3 ) : 4000 f -a I I • exp{-/3-X }»dX / OnX 290 J T ( A ) = (11-37) The amount of t u r b i d i t y was varied by varying /3 over a wide range. The results are shown in Fig.13. From t h i s figure i t becomes obvious that the transmittance as a function of t u r b i d i t y i s much more linear than i t is the case for water transmittance. While we can observe a weakened increase of water vapor absorption at higher water vapor concentrations, i t i s quite obvious that the variation of the t u r b i d i t y has an e f f e c t on the transmittance, regardless how much aerosol there i s in the atmosphere. Unfortunately, the values for dust and water concentration, which were used to establish the A . S . H . R . A . E algorithm, represent extremely clear situations which are rarely observed. While t h i s might seem to be of advantage for the c a l c u l a t i o n of peak insolation values, over-estimation for monthly averages resu l t from the use of i t . This fact c a l l s for a change in the A . S . H . R . A . E . algorithm that would make the varying t u r b i d i t y i t s main parameter of influence. This study suggests the use of the ground v i s i b i l i t y 49 as the variable in a modified A.S.H.R.A.E. model. If a further increase of accuracy i s desired, an additional parameter accounting for the var i a t i o n of the humidity can be included. At any rate, as seen from F i g . 13, an error of less than 4% results from a permanent water content of 1.5 cm. With a best f i t method versus other models described in th i s study, the incident radiation can be put in t h i s form: ( °'5) \AMS / Irr = SC • T(A) • M(W) • (0.775) (11-38) n with: / 0.85^ (-0.57) \AMS / r(A)= (1-1.3-VIS ) (11-39) and with: . / 0.27\ \AMS) ) (11-40) M(W) = (1.0223-0.0149-UW) In most cases, M(W) can be set to unity which i s i t s exact value for UW=1.5 cm. The Eq.(11-39) above for the transmittance due to aerosol i s very similar to the one presented in the Bird and Hulstrom model. It has , however, been t a i l o r e d to f i t into the context of Eq. (11-38) together with Eq. (11-40) and should therefore not be used with other parameterization models. If Eq.(11-39) i s replaced by Eq. (11-34), the term (0.775) in ( I I -38) has to be replaced by (0.745). This may create errors of up to 2%. The Eq. (11-38) also does not allow a v a r i a t i o n of the ozone absorption or the rayleigh scattering. An amount of 0.3 cm of concentrated ozone in the a i r was assumed and that the 50 simulation takes place at sea l e v e l . Any adjustment for elevations d i f f e r e n t from sea le v e l has to be made via the AMS r e l a t i o n . In the event of pr e c i p i t a b l e water d i f f e r i n g strongly from UW=1.5 cm, i t i s suggested that corrections are made either by ref e r r i n g to Figure 12 or by using Eq. (11-40) as a m u l t i p l i e r for equation ( 1 1 - 3 8 ) . The suggested changes to the A.S.H.R.A.E algorithm, which extend the range of application to turbid a i r , are d e f i n i t e l y worth the s l i g h t l y increased amount of calculations to use the model. Performance of Eq. ( 1 1 - 3 8 ) is shown in F i g . 8 to 11; the performance of Eq. ( 1 1 - 3 8 ) versus the exis t i n g A.S.H.R.A.E. model is shown in Fi g . 14. These graphs show that the suggested changes make the A.S.H.R.A.E. model comparable with other parameterization values. Figure 14 shows that the values of the old A.S.H.R.A.E. model are linked to extremely low t u r b i d i t i e s . 5 1 1 1 1 1 r 30 40 50 "ZENJTH ANGLE Figure 8, Performance of five models. 52 Figure 9, Performance of five models. 53 Figure 1 0 , Performance of five models. 54 CM < —10 v i s i b i l i t y 5 km ozone c o n t e n t 0.31 cm w a t e r c o n t e n t 2 .93 cm o Q O S -ffl 0 ffl Model fl A A A Model B + t t Model C X X X flSHRRE(new) •» * » LecknEr o~1 1 1 i 1 1 r 0 JO 23 33 4 . _ . . _ ? 0 "I 1 1 T -Z E N J ^ H & N GL E 63 70 B3 Figure 11, Performance of five models. 55 F i g u r e 12, T r a n s m i t t a n c e due t o w a t e r v a p o r a b s o r p t i o n . Figure 13, Transmittance due to aerosol 57 58 III The Treatment of Diffuse Radiation A l l the presented models contain ways to calculate the amount of d i f f u s e radiation. Diffuse radiation i s the radiation that reaches the earth's surface not d i r e c t l y but after (possibly: multiple) scattering and/or r e f l e c t i o n . Often a d i s t i n c t i o n i s made between scattered radiation from the sky ("sky radiation") and "multiply r e f l e c t e d radiation" - r e f l e c t e d from the earth back to the sky and then scattered backward to the earth. There i s no way to d i s t i n g u i s h these two components of the d i f f u s e radiation by way of measurement. This d i s t i n c t i o n i s purely for the convenience of the mathematical treatment. Generally the determination of the d i f f u s e radiation under cloudless sky conditions i s more d i f f i c u l t than the treatment of the d i r e c t radiation because: the complexity of the scattering phase function renders the closed formula mathematical treatment of single scattering extremely d i f f i c u l t and that of multiple scattering becomes p r a c t i c a l l y impossible; the inhomogenous nature of the aerosol d i s t r i b u t i o n makes the determination of the scattering versus absorption r a t i o very d i f f i c u l t ; the r e f l e c t i v i t y of the earth's surface (called "albedo") i s variable and i t s determination d i f f i c u l t . Simplifying models are generally necessary to calculate the the diffuse radiation. In the next sections the extent of these s i m p l i f i c a t i o n s w i l l be shown and what results can be expected 59 in return. 60 III.1 General Approach for Single Scattering Most models make use of the simplifying assumption that a l l scattering occurs once only. Secondary or multiple scattering i s neglected. This s i m p l i f i c a t i o n makes the c a l c u l a t i o n of Rayleigh scattering - scattering by p a r t i c l e s much smaller than the wavelength of the radiation - very easy: The phase function for Rayleigh scattering i s symmetrical, provided the f i c t i o n of perfectly spherical p a r t i c l e s i s upheld. This means that 50% of the scattered radiation i s scattered forward and 50% backward. The treatment of Mie (aerosol) scattering i s not as easy as the Rayleigh scattering because, the phase function i s not symmetrical but shows a strong bias in the forward d i r e c t i o n . To determine the amount of diffuse radiation, further assumptions are neccessary, i . e . regarding the sequence of the scattering. They w i l l be introduced with the respective models. 61 III.2 Multiple Scattering None of the models presented in this study approaches the task to calculate the diffuse radiation under consideration of multiple scattering or absorption of scattered radiation. Some models l i k e the one of Lacis and Hansen [26] or the Lowtran Model [25] - not discussed in t h i s study because of the excessive amount of computation time - use algorithms to calculate the dif f u s e i r r a d i a t i o n on the basis of possible multiple scattering. These models divide the inhomogenous atmosphere into many layers (as many as 50) which are homogenous in themselves. These models obtain a high degree of accuracy but demand an excessive computational e f f o r t . One cannot expect a large gain in accuracy for low airmasses and low concentrations of scattering matter in the atmosphere because the amount of multiply scattered radiation a term of second order - becomes very small i f there i s not very much scattered radiation. But i t proves to be worthwile to include the effects of multiple scattering for very turbid a i r at low solar a l t i t u d e . Under these conditions the fra c t i o n of diff u s e i r r a d i a t i o n which i s multiply scattered increases d r a s t i c a l l y . To reduce the errors at large zenith angles and to put the calc u l a t i o n of dif f u s e radiation more in tune with the physical r e a l i t y , t h i s study presents a comparatively easy procedure to include the effects of double and t r i p l e scattering into the 62 spectral Model of Leckner (and into the parameterization models i f so desired). The following assumptions form the basis of t h i s work: The single scattered radiation has a pathlength that i s 50% longer than the d i r e c t radiation. The twice scattered radiation follows a path that i s three times the pathlength of the d i r e c t radiation Radiation which i s scattered more than twice follows a path 5 times as long as the path of the d i r e c t radiation The scattering model of homogenous layers w i l l be replaced by a model in which scattering p r o b a b i l i t i e s are defined according to the function of transmittance. The above assumptions are the result of a semi-empirical approach; it e r a t i o n s were performed to obtain a best f i t against other models which incorporate multiple scattering. At a p a r t i c u l a r wavelength, the d i r e c t radiation i s defined by the transfer equation (1-5). The difference between the e x t r a t e r r e s t r i a l i r r a d i a t i o n and the beam radiation i s the amount that is~depleted in the atmosphere; i t includes the absorbed radiation and the scattered radiation, be i t scattered to the earth or back to the space. It also includes radiation which was scattered and absorbed afterwards. Because the assumption of a layered atmosphere was dropped, the question arises how to divide the depleted radiation into the scattered part and the absorbed part. This i s done by establishing the p r o b a b i l i t y of attenuation for a single quantum. The p r o b a b i l i t y for a quantum to be absorbed or scattered (possibly more than 63 once) i s : PR = (1-T(R) «T(A) «r(W) -T(G) «T(0) ) (II1-1) X X X X X The p r o b a b i l i t y for a quantum to undergo Rayleigh scattering i s defined by the quotient of the o p t i c a l depth for Rayleigh scattering over the sum of the o p t i c a l depths for a l l attenuating processes: PR(R) = PR.log(T(R) ) / lo g ( r ( t o t ) ) (III-2) X X where: r(tot) =T(R) -r(A) -r(W) -T(G) -T(0) (III-3) X X X X X X The p r o b a b i l i t y for a single quantum to undergo only one occurence of Rayleigh scattering (and not more) becomes: PR(R,1) = PR(R).r(tot) (III-4) X Accordingly, the pro b a b i l i t y for a p a r t i c l e to undergo Rayleigh and then one or more further attenuating processes becomes: PR(R,2+)= PR(R)•(1-T(tot) ) (III-5) X which i s : 2 PR(R,2+)= PR • log(r(R) )/ l o g ( T ( t o t ) ) (III-6) X X The Figure 15 i l l u s t r a t e s the process of multiple scattering in a nonlayered atmosphere. 6 4 beam=TaTrTm attenuated: 1-TaTrTm absorbed x(1-TaTrTm) of which-I Rayleigh scat, y(1-TaTrTm) further attenuated: xy(1-TaTrTm) xz(1-TaTrTm) y(1-TaTrTm) ^ I \ of which 2 i" y (1-TaTrTm) yz(1-TaTrTm) further attenuated: aerosol scat, z(1-TaTrTm) scattered and passed: z(1-TaTrTm)TaTrTm y(1-TaTrTm)TaTrTm z(1-TaTrTm) I of which 2 I z (1-TaTrTm) j zy(1-TaTrTm) 2 2 (y+z) (1-TaTrTm) •TaTrTm (y+z) (1-TaTrTm) \ of I which 2 iX 2 x(y+z) (1-TaTrTm) 2 r y(y+z) (1-TaTrTm) z(y+z) (1-TaTrTm) Explanation: PR(Abs)= x= log(Tm)/log(TaTrTm) Ta=r(AS) PR(R) = y= log(Tr)/log(TaTrTm) Tr=r(R) PR(A) = z= log(Ta)/log(TaTrTm) Tm= r(AA)•r(W)• T ( G )•T(O ) For brevity, no multiplicator was used on thi s page Figure 15, Multiple Attenuation Pattern 65 Because 50% of the Rayleigh scattered radiation i s scattered forward but an amount larger than 50% of the Mie scattered radiation i s scattered forward, further assumptions have to be made to simplify the incorporation of multiple scattering: The radiation which i s scattered into the halfsphere facing the earth with a plane p a r a l l e l to the earth's surface i s considered 'forward scattered'. The curvature of earth and atmosphere i s neglected. In the following F i g . 16, the approach to include multiple scattering into the spectral model w i l l be graphically outlined. The forward scattering r a t i o 'FR' of the scattered radiation can be defined as: 0.5'Idif(R)+BA-Idif(A) FR= (III-7) Idif(R)+Idif(A) The term ' I d i f denotes the d i f f u s e radiation from the respective scatterer and BA i s the aerosol forward scattering r a t i o , i . e . the f r a c t i o n of Idif(A) which i s scattered forward [20]. 66 SS* DS* TS* Figure 16, Multiple Scattering Pattern 67 The t o t a l 'sky di f f u s e radiation' then becomes the sum of the single, the double and the t r i p l e scattered radiation: I d i f ( S ) =SS+DS+TS ( I I I - 8 ) X with: SS=I • { ( 1 - r ( S c ) ' ) - r ( A b s ) ' - F R . T ( S C ) " } OhX DS=I . { ( 1 - T ( S C ) ' ) . r ( A b s ) ' * • ( 1 - T ( S C ) " ) • T ( S C ) " ' } • OhX {FR-FR+(1-FR)•(1-FR)} TS=I • { ( l - r ( S c ) ' ) - T ( A b s ) " '•(1-T(SC) " ) • ( 1 - T ( S C ) " ' } • OhX {FR'FR»FR+(1-FR)•(1-FR)'FR.3} where the number of primes corresponds to the class of scattering. As outlined previously in t h i s Section, longer pathlengths of the scattered radiation require higher airmasses. F i n a l l y , there i s a small amount of radiation, which i s ref l e c t e d from the earth back to the sky and scattered back to the earth. This radiation i s : I(MR) =(Irr +SS+DS+TS)•rg•ra (III-9) X hX The term 'rg' stands for the r e f l e c t i v i t y of the ground (albedo) and ra i s the atmospheric albedo. The atmospheric albedo can be defined as: 68 ra=(SS*+DS*+TS*)/lrr (III-10) OhX The t o t a l d i f f u s e radiation can now be defined as: l( t o t ) = I r r +Idif(S)+Idif(MR) (111-11) hX This semi-empirical approach to model multiple scattering was used with the spectral transmittance functions of Leckner. The res u l t s are presented in Figures 18 to 21 as "Leckner, modified". The results are very encouraging and bring the spe c t r a l l y integrated values close to the parameterization models. 69 III.3 Treatment of Diffuse Radiation with the Models III.3.1 The Model of Leckner The part of Leckner's Model to calculate diffuse radiation uses the following basic assumptions: 1. ) The phase function for Mie scattering is assumed to be symmetric. Therefore i t can be treated l i k e Rayleigh scattering. 2. ) The absorption by aerosol w i l l be neglected and thus i t i s assumed that the aerosol attenuation i s by means of scattering only. 3. ) The r e f l e c t i v i t y of the earth i s zero. No multiply r e f l e c t e d radiation has to be considered. Leckner's model defines the scattered radiation as the difference between the direc t radiation and a f i c t i o u s beam that has only been subjected to absorption under exclusion of any aerosol absorption. Therefore the formulation" of the diffuse radiation takes on the following form: 4000Y Idif = O.5«cos(0) • 1(1 -T(0) T ( G ) »r(W)-dX)-Irr (111-1 2) 29tV OnX n where the irradiance for normal incidence i s taken from Eq. (II-7). The value for FR (forward scattering ratio) i s assumed to be 0.5 because the phase functions for scattering are taken as symmetric. 70 Eq. ( i l l — 1 2 ) d e l i v e r s f a i r l y good r e s u l t s for low t u r b i d i t y values and low ground r e f l e c t i v i t y : Most assumptions make s a c r i f i c e s on the accuracy of the aerosol scattered radiation; neither aerosol absorption nor phase function have s i g n i f i c a n t influence under conditions of low t u r b i d i t y . The performance of Leckner's approach i s shown in comparison with other models in Figs. 18 to 21: Leckner's d i f f u s e values are very low. This has to be expected because Leckner's approach does not account for secondary d i f f u s e r a d i a t i o n . For the same assumptions as above and with the added p o s s i b i l i t y of multiple s c a t t e r i n g , Berlage [32] published a semi-exact mathematical treatment as early as 1928. Probably because of the language (German), t h i s work seems to have gone unnoticed in the Anglophone world and i t i s well worth presenting his results.'Berlage shows, that the spectral d i f f u s e radiation becomes: 4000 f T {1-T(R) }-dX I d i f = 0.5- /I -cost?-T(A B) • ( 1 1 1 - 1 3) 290J OnX 1 - 1 . 4 - T - l n ( r ( R ) ) where the 'T' i s Linke's Triibungsindex ( t u r b i d i t y index) which i s defined as: T T ( R ) T ( A ) = T ( R ) ( 1 1 1 - 1 4 ) Therefore ( 1 1 1 - 1 3 ) can be written as: 4000r { l - r ( R ) - r ( A ) } - d X Idif=.5' I - C O S 0 - T ( A B ) - ( I I I - 1 5 ) 290 J OnX 1-1 . 4 - l n ( r ( R ) - T - ( A ) ) 71 As shown in section III.2, i t i s possible to change Leckner's d i f f u s e model by implementing the multiple scattering algorithm with Leckner's transmittance functions. 72 III.3.2 The Model by Davies and Hay The basic idea of the model by Davies and Hay i s a d i v i s i o n of the d i f f u s e radiation into three parts: 1. ) The d i f f u s e radiation caused by Rayleigh scattering. 2. ) The d i f f u s e radiation caused by aerosol scattering. 3. ) The d i f f u s e radiation caused by r e f l e c t i o n of a l l radiation from the earth's surface back to the sky and the backscattering from the sky to the earth. An assumption must be made regarding the sequence of the attenuation processes. Davies and Hay use a layered model - the exact construction of which i s shown in Fig.17: no attenuation occurs concurrently. With th i s s i m p l i f i c a t i o n the equations for the primary di f f u s e componeats become: I d i f ( R ) = S C - c o s ( 0 ) • [ T ( 0 ) • ( 1 - T ( R ) ) T ( A ) - 0 . 5 ] (III-17) and Idif(A)=SC-cos{6)•[(T(O)-r(R)-a(W))•(1-T(A))-WO-BA] (III-18) The term 'WO' denotes the r a t i o between scattering and t o t a l attenuation due to aerosol (taken as 0.95) and 'BA' the forward scattering r a t i o . Note that in this model by Davies and Hay the d i f f u s e radiation caused by Rayleigh scattering i s not subject to attenuation by water vapor and the gases! 73 For the multiply re f l e c t e d radiation, Davies and Hay are using the following equation: Idif(MR)=(lrr +Idif(R)+Idif(A)•rg•ra/(1-rg•ra) (III-19) h The values for the ground r e f l e c t i v i t y 'rg' are usually around 0.2 but can go as high as 0.9 for fresh snow. The atmospheric albedo employed by Davies and Hay was taken from Lacis and Hansen [26] and i s given as: ra=0.0685+0.17(1-r(A)')-WO The value of T ( A ) ' i s calculated as r(A) for a zenith angle 0=57 degrees. The t o t a l radiation can now be determined by adding up the three parts of the d i f f u s e radiation and the d i r e c t radiation and i t can be brought into the following form: I(tot)=Irr +Idif(R)+Idif(A)+Idif(MR) (111-20) h The inclusion of the multiple scattering algorithm as outlined in Section III.2 i s possible. With the absence of measured data i t can not be considered b e n e f i c i a l because of the substantially increased complexity without a sure measure of any improvement. An increased data base might change the facts and reverse t h i s recommendation. The performance i s shown in Figs 18 to 21: While high at small zenith angles, the results come in low for zenith angles over 60 degrees. This fact suggests that the chosen function of the zenith angle might not be i d e a l . 74 III.3.3 The Model by Hoyt The model by Hoyt b a s i c a l l y uses the elements of the d i r e c t model as outlined in II.3. Hoyt also s p l i t s the d i f f u s e radiation into three parts as did Davies and Hay. The main difference to the previously outlined Model A i s the assumption that the scattered radiation from Rayleigh scattering and from Mie scattering is subject to the same absorptive influences as the d i r e c t radiation. Thereby the equations to determine the diffuse radiation become: The Rayleigh forward scattering r a t i o was taken as 0.5 while the aerosol forward scattering r a t i o was not given as a function but as a constant BA=0.75. Hoyt's approach to determine the multiply r e f l e c t e d d i f f u s e (MR) radiation resembles somewhat the one by Davies and Hay: (111-21) and [ . 7 5 . ( I - T ( A S ) ] (111-22) / 5 Idif(MR)=(Irr +Idif(R+A))•rg• 1 - [ a h V 1 (111-23) [.5.(1-T(R)')+(1-T(AS)')..25] But i t has one important difference; It i s obviously assumed that the multiply r e f l e c t e d d i f f u s e radiation i s subject to 75 further attenuation. Physically and mathematically, t h i s treatment i s more accurate then the one outlined in the previous section. The t o t a l radiation i s the sum of the dire c t part, the sky diffuse part and the multiply r e f l e c t e d radiation: I(tot) = Irr +Idif(R)+Idif(A)+Idif(MR) (111-24) h As for the inclusion of the multiple scattering algorithm, the comments to Davies' and Hay's model are v a l i d for th i s model too. The performance i s shown in Figs 18 to 21: The results of th i s model as well are reasonably close to the results of Model C. 76 III.3.4 The Model by Bird and Hulstrom Like the models we previously dealt with, Bird and Hulstrom's model divides the diffuse radiation: One part describes the radiation which reaches the earth after scattering; the other part i s made up of di f f u s e radiation that was r e f l e c t e d from the earth and backscattered to the earth's surface. The equation for the 'sky d i f f u s e ' part of the irradiance i s given as: Idif =1 - c o s ( 0 ) • ( 0 . 7 9 ) - r ( O ) T ( W ) - T ( G ) « T ( A ) • s On (111-25) 1 .02 {.5*(1-r(R))+BA-(1-T(AS)}/{1-AMS+(AMS) } where: BA =0.82 and T(AS)= r ( A ) / r ( A A ) (111-26) and: 1 .06 r(AA)=1-KS-(1+AMS - A M S ) • ( 1 - T ( A ) ) (111-27) The t o t a l radiation, including the multiply r e f l e c t e d radiation becomes thus: I(tot)= (Irr + I d i f ( S ) ) / ( l - r g - r s ) (111-28) h where 77 rs = 0 . 0 6 8 5 + ( 1 - B A ) • ( 1 - T ( A S ) ) ( 1 1 1 - 2 9 ) The factor KS i s dependent on the aerosol size d i s t r i b u t i o n ; Bird and Hulstrom used a value of K S=0 .0933 for a l l their c a l c u l a t i o n s . The diffuse part of Model C can be changed to the 'multiple scattering' pattern as developed e a r l i e r in t h i s study; the difference in performance, however, i s so small that i t cannot be considered an improvement. This study suggests therefore to use the d i f f u s e part of Model C unchanged. The performance of the model i s shown in Fig's 18 to 2 1 : Model C was developed with the aid of more elaborate models and in the absence of better standards can be considered "primary standard" for the d i f f u s e part of t h i s study. 78 III.3.5 Diffuse Radiation with the A.S.H.R.A.E. Model The A.S.H.R.A.E algorithm [31] uses a simple factor 'C to multiply the d i r e c t radiation: Idif = C • I r r (111-30) n where: Irr = A • exp-(B-AMS) (11-31) n While i t i s accurate within reasonable l i m i t s to determine the diffuse radiation as a constant fracti o n 'C of the d i r e c t radiation, the constants proposed by A.S.H.R.A.E. are very low and represent fixed t u r b i d i t y values of unusually clear days. Therefore an underestimation occurs for a l l but the most clear atmospheric circumstances. Because the proposed change of the A.S.H.R.A.E formula (Eqs.II-38/39) allows variable v i s i b i l i t i e s , i t i s necessary to adapt the cal c u l a t i o n of the di f f u s e radiation as well. Comparison with other models, notably the Model of Bird and Hulstrom, lead to the following simple parameterization: Idif = I r r -(3/VIS+0.1) (111-32) n The performance of t h i s equation i s shown in Figs. 18 to 22: The resu l t s are in very good agreement with the res u l t s of Model C 79 Downward scattered componend E x t r a t e r r e s t r i a l Irradiance Absorption by Ozone Rayleigh Scattering Absorption by Water vapor Attenuation by aerosol Downward scattered component Attenuation by aerosol Diffuse radiation (aerosol) Direct beam Diffuse radiation (Rayleigh) Figure 17; Davies and Hay Layer Model; Two Stream Approximation of Diffuse Radiation g u r e 18; P e r f o r m a n c e o f 5 M o d e l s , D i f f u s e R a d i a t i o n 8 1 Figure 19; Performance of 5 Models, Diffuse Radiation 82 gure 20; Performance of 5 Models, Diffuse Radiation 83 Figure 21; Performance of 5 Models, Diffuse Radiation 84 Figure 22; The ASHRAE Models, Diffuse Radiation 85 IV Concluding Remarks It was the goal of t h i s study to put together some of the wealth of data and achievements in the f i e l d of clear sky solar radiation to the user with an engineering background. The use of the horizontal meteorological range - the v i s i b i l i t y - as the main parameter for t u r b i d i t y calculations has to be seen in l i g h t of t h i s goal: Local conditions have a strong influence on the c o r r e l a t i o n between v i s i b i l i t y and t u r b i d i t y . This deficiency i s outweighted, however, by the s i m p l i c i t y and a v a i l a b i l i t y of t h i s method which promises much better results with only s l i g h t l y increased complexity. The t u r b i d i t y i s a main contributor to di f f u s e radiation. The proposed new A.S.H.R.A.E algorithm for' d i f f u s e radiation incorporates t h i s for the f i r s t time. The proposed treatment of multiple scattering also has the applied sciences in mind: A tool i s offered to assess the influence of atmospheric components on multiple scattering without o u t l i n i n g a mathematical approach that disallowed general applications. 86 V Further Work At present, extrapolations from low and medium range v i s i b i l i t i e s to high v i s i b i l i t i e s are used in t h i s study. It would be most desirable to have an extensive data base on the c o r r e l a t i o n between v i s i b i l i t y and t u r b i d i t y for various locations, conditions and seasons. Studies in t h i s f i e l d are comparatively simple to conduct, although the a v a i l a b i l i t y of advanced data processing capacities i s considered e s s e n t i a l . It would be in the best interest of the solar radiation research to contact simultaneous measurements of the effects of a l l attenuators, notably water, ozone, aerosols and various gases as well as simultaneous measurements of quantities with possible high c o r r e l a t i o n such as the meteorological range. Such studies would allow to v e r i f y many exis t i n g models and other work that has been based on extensive simulation, including much of t h i s study. Further work i s also recommended in the f i e l d of d i f f u s e radiation, both broadband and spectral d i f f u s e radiation. The exi s t i n g models s t i l l show wide discrepancies and measured data i s sketchy at best. t 87 References [I] Lambert, J.H.; "Photometrie", (1760). [2] Thekaekara, M.P.; "Solar energy outside the earth's atmosphere", Solar Energy 14, 109-127 (1973). [3] Leckner, B.; "The spectral d i s t r i b u t i o n of solar radiation at the earth's surface - elements of a model", Solar Energy 20, 143-150, (1978). [4] Penndorf, R.; "Tables of the r e f r a c t i v e index for standard a i r and the Rayleigh scattering c o e f f i c i e n t " , J . Opt. Soc. of Am. 47, 176, (1957) [5] Vigroux, E.; "Contribution a 1'etude experimentale de l'absorption de l'ozone", Annals de Physique 8, 709-762 (1953). [6] Fowle, F.E.; "The transparency of aqueous vapor", Astrophysics Journal Vol. 42, (1915) [7] Howard, J.N. et a l . ; "Infrared transmission of synthetic atmospheres", J . Optical Soc. of Am., Vol.46, (1956) [8] Yamamoto, G.; "Direct absorption of solar radiation by atmospheric water vapor,.carbon dioxide and oxygen", J . Atmos. Soc. 19, 182, (1962) [9] McClatchey, R.A. et a l t . ; "Optical properties of the /atmosphere", AFCRL-72-0497, (1972) [10] Elterman, L.; "Relationship between v e r t i c a l attenuation and surface meteorological range", J. Appl. Optics, Vol.9, I804ff, (1970) [II] Goody, R.M.; "Atmospheric Radiation I", Oxford: Clarendon Press, (1964). [12] Tiwari, S.N.; "Models for infrared atmospheric radiation", Advances in Geophysics, Vol. 20, 1-85, (1977) [13] Mie, G.; "Beitrage zur Optik truber Medien", Annalen der Physik, Vol.25, 377ff, (1908) [14] Angstrom, A.; "On the transmission of sun radiation and on dust in the atmosphere", Geografisk Annaler 2, 156-166), (1929) 88 [15].Angstrom, A.; "On the atmospheric transmission of sun radiation I I " , Geografisk Annaler 3, 130-159, (1930) [16] McCartney; "Optics of the atmosphere", (1976) [17] Linke, F; "Handbuch der Geophysik", Vol. 9, 621ff, Berl i n (1941) [18] Buckius, R.O. and King, R.; "Direct solar transmittance for a clear sky", Solar Energy Vol.22, 297-301, (1979) [19] Zuev, V.E.; "Atmospheric transparency in the v i s i b l e and the IR", (1970) [20] Davies, J.A. and Hay, J.E.; "Calculation of the solar radiation incident on a horizontal surface", Proceedings of the F i r s t Canadian Solar Radiation Data Workshop, Ed. T.Won and J.E.Hay, (1980) [21] Hoyt, D.V.; "A model for the ca l c u l a t i o n of solar global i n s o l a t i o n " , Solar Energy, Vol.21, 27-35, (1978) [22] Bird, R.E. and Hulstrom, R.L.; "Direct insolation models", Solar Energy Research Inst i t u t e TR-335-344, (1980) [23] Bird, R.E. and Hulstrom, R.L.; "A s i m p l i f i e d clear sky model for direc t and di f f u s e insolation on horizontal surfaces", Solar Energy Research Inst i t u t e TR-642-761, (1981) [24] Selby, J.E. et a l . ; "Atmospheric transmittance from 0.25/xm to 28.5 Mm: Computer code LOWTRAN 3", AFCRL-75-0255; (1975) [25] Selby, J.E. et a l . ; "Atmospheric transmittance / radiance: Computer code LOWTRAN 4", AFGL-78-0053; (1978) [26] Lacis, A.L. and Hansen, J.E.; "A parameterization for the absorption of solar radiation in the earth's atmosphere", J. Atmospheric Science, Vol.31, 118-133, (1974) [27] Houghton, H.G.; "On the heat balance of the northern hemisphere", J. Meteorology, Vol.11, 1-9, (1954) [28] Kasten, F.; "A new table and approximation formula for the re l a t i v e o p t i c a l a i r mass", Archiv fur Meteorologie, Geophysik und Bioklimatologie, Vol. 12, 206-233 (1966). [29] Flowers, E.C. et a l . ; "Atmospheric t u r b i d i t y over the United States 1961-1966", J. Appl. Meteor., Vol.8, 955, (1969) [30] Moon, P.; "Proposed standard solar radiation curves for engineering use", Journal of the Franklin Institute 230, 583-617, (1940) 89 [31] A.S.H.R.A.E.; Handbook of Fundamentals, (1972) [32] Berlage, H.P.; "Zur Theorie der Beleuchtung einer horizontalen Flache durch Tageslicht", Meteorologische Z e i t s c h r i f t 45-5, 174-180, (1928) [33] Threlkeld and Jordan; "Direct Solar Radiation Available on Clear Days", A.S.H.R.A.E. Trans., Vol. 14, 45-56, (1958) [34] Eighth Session of The Commission for Instruments and Methods of Observation; WMO (1981) 90 Table III; spectrum, absorption c o e f f i c i e n t s Col 1: Interval number Col 2: AX in [nm] Col 3: Center of wavelength i n t e r v a l , [jum] Col 4: Ozone absorption c o e f f i c i e n t s for Eq.(I-1l) Col 5: Water absorption c o e f f i c i e n t s for Eq.(I-14) Col 6: Mixed gases absorption c o e f f i c i e n t s for Eq.(l-13) Col 7: Fraction of Solar Constant within i n t e r v a l , [W/(m-m-nm)] 1 2 3 4 5 6 7 1 5.0 0.290 38.00 0.0 0.0 482.0 2 5.0 0.295 20.00 0.0 0.0 584.0 3 5.0 0.300 10.00 0.0 0.0 514.0 4 5.0 0.305 4.800 0.0 0.0 605.0 5 5.0 0.310 2.700 0.0 0.0 689.0 6 5.0 0.315 1 .350 0.0 0.0 764.0 7 5.0 0.320 0.8 0.0 0.0 830.0 8 5.0 0.325 0.38 0.0 0.0 975.0 9 5.0 0.330 0. 160 0.0 0.0 1059.0 10 5.0 0.335 0.075 0.0 0.0 1081 .0 1 1 5.0 0.340 0.04 0.0 0.0 1074.0 12 5.0 0.345 0.019 0.0 0.0 1069.0 13 5.0 0.350 0.007 0.0 0.0 1093.0 14 5.0 0.355 0.0 0.0 0.0 1083.0 15 5.0 0.360 0.0 0.0 0.0 1068.0 16 5.0 0.365 0.0 0.0 0.0 1132.0 17 5.0 0.370 0.0 0.0 0.0 1181.0 18 5.0 0.375 0.0 0.0 0.0 1157.0 19 5.0 0.380 0.0 0.0 0.0 1120.0 20 5.0 0.385 0.0 0.0 0.0 1098.0 21 5.0 0.390 0.0 0i0 0.0 1098.0 22 5.0 0.395 0.0 0.0 0.0 1189.0 23 5.0 0.400 0.0 0.0 0.0 1429.0 24 5.0 0.405 0.0 0.0 0.0 1644.0 25 5.0 0.410 0.0 0.0 0.0 1751 .0 26 5.0 0.415 0.0 0.0 0.0 1774.0 27 5.0 0.420 0.0 0.0 0.0 1747.0 28 5.0 0.425 0.0 0.0 0.0 1693.0 29 5.0 0.430 0.0 0.0 0.0 1639.0 30 5.0 0.435 0.0 0.0 0.0 1663.0 31 5.0 0.440 0.0 0.0 0.0 1810.0 32 5.0 0.445 0.003 0.0 0.0 1922.0 33 5.0 0.450 0.003 0.0 0.0 2006.0 34 5.0 0.455 0.004 0.0 0.0 2057.0 35 5.0 0.460 0.006 0.0 0.0 2066.0 36 5.0 0.465 0.008 0.0 0.0 2048.0 37 5.0 0.470 0.009 0.0 0.0 2033.0 38 5.0 0.475 0.012 0.0 0.0 2044.0 39 5.0 0.480 0.014 0.0 0.0 2074.0 40 5.0 0.485 0.017 0.0 0.0 1976.0 41 5.0 0.490 0.021 0.0 0.0 1950.0 42 5.0 0.495 0.025 0.0 0.0 1960.0 43 5.0 0.500 0.03 0.0 0.0 1942.0 91 Table I I I , cont. 1 2 3 4 5 6 7 44 5.0 0.505 0.035 0.0 0.0 1920.0 45 5.0 0.510 0.04 0.0 0.0 1882.0 46 5.0 0.515 0.045 0.0 0.0 1833.0 47 5.0 0.520 0.048 0.0 0.0 1833.0 48 5.0 0.525 0.057 0.0 0.0 1852.0 49 5.0 0.530 0.063 0.0 0.0 1842.0 50 5.0 0.535 0.07 0.0 0.0 1818.0 51 5.0 0.540 0.075 0.0 0.0 1783.0 52 5.0 0.545 0.08 0.0 0.0 1754.0 53 5.0 0.550 0.085 0.0 0.0 1725.0 54 5.0 0.555 0.095 0.0 0.0 1720.0 55 5.0 0.560 0.103 0.0 0.0 1695.0 56 5.0 0.565 0.11 0.0 0.0 1705.0 57 5.0 0.570 0.12 0.0 0.0 1712.0 58 5.0 0.575 0. 122 0.0 0.0 1719.0 59 5.0 0.580 0.12 0.0 0.0 1715.0 60 5.0 0.585 0.118 0.0 0.0 1712.0 61 5.0 0.590 0.115 0.0 0.0 1700.0 62 5.0 0.595 0.12 0.0 0.0 1682.0 63 5.0 0.600 0. 125 0.0 0.0 1666.0 64 5.0 0.605 0.13 0.0 0.0 1647.0 65 7.5 0.61 0.12 0.0 0.0 1625.0 66 10.0 0.62 0. 105 0.0 0.0 1602.0 67 10.0 0.63 0.090 0.0 0.0 1570.0 68 10.0 0.64 0.079 0.0 0.0 1544.0 69 10.0 0.65 0.067 0.0 0.0 1511.0 70 10.0 0.66 0.057 0.0 0.0 1486.0 7.1 10.0 0.67 0.048 0.0 0.0 1456.0 72 10.0 0.68 0.036 0.0 0.0 1427.0 73 10.0 0.69 0.028 0.160E-01 0.0 1402.0 74 10.0 0.70 0.023 0.240E-01 0.0 1369.0 75 10.0 0.71 0.018 0.125E-01 0.0 1344.0 76 10.0 0.72 0.014 0.100E+01 0.0 1314.0 77 10.0 0.73 0.011 0.870E+00 0.0 1290.0 78 10.0 0.74 0.01 0.610E-01 0.0 1260.0 79 10.0 0.75 0.009 0.100E-02 0.0 1235.0 80 10.0 0.76 0.007 0.100E-04 0.300E+01 1211.0 81 10.0 0.77 0.004 0.100E-04 0.210E+00 1185.0 82 10.0 0.78 0.0 0.600E-03 0.0 1159.0 83 10.0 0.79 0.0 0.175E-01 0.0 1134.0 84 10.0 0.80 0.0 0.360E-01 0.0 1109.0 85 10.0 0.81 0.0 0.330E+00 0.0 1085.0 86 10.0 0.82 0.0 0.153E+01 0.0 1060.0 87 10.0 0.83 0.0 0.660E+00 0.0 1036.0 88 10.0 0.84 0.0 0.155E+00 0.0 1013.0 89 10.0 0.85 0.0 0.300E-02 0.0 990.0 90 10.0 0.86 0.0 0.100E-04 0.0 968.0 91 10.0 0.87 0.0 0.100E-04 0.0 947.0 92 10.0 0.88 0.0 0.280E-02 0.0 926.0 93 10.0 0.89 0.0 0.630E-01 0.0 908.0 94 10.0 0.90 0.0 0.210E+01 0.0 891 .0 92 Table I I I , cont. 1 2 3 4 5 6 7 95 10. 0 0.91 0 .0 0. 160E+01 0. 0 880. 0 96 10. 0 0.92 0 .0 0. 125E+01 0. 0 869. 0 97 10. 0 0.93 0 .0 0. 270E+02 0. 0 858. 0 98 10. 0 0.94 0 .0 0. 380E+02 0. 0 847. 0 99 10. 0 0.95 0 .0 0. 41OE+02 0. 0 837. 0 00 10. 0 0.96 0 .0 0. 260E+02 0. 0 820. 0 01 10. 0 0.97 0 .0 0. 310E+01 0. 0 803. 0 02 10. 0 0.98 0 .0 0. 148E+01 0. 0 785. 0 03 10. 0 0.99 0 .0 0. 125E+00 0. 0 767. 0 04 30. 0 1 .00 0 .0 0. 250E-02 0. 0 748. 0 05 50. 0 1 .05 0 .0 0. 1OOE-04 0. 0 668. 0 06 50. 0 1 .10 0 .0 0. 320E+01 0. 0 593. 0 07 50. 0 1.15 0 .0 0. 230E+02 0. 0 535. 0 08 50. 0 1 .20 0 .0 0. 160E-01 0. 0 485. 0 09 50. 0 1 .25 0 .0 0. 180E-03 0. 730E-02 438. 0 10 50. 0 1 .30 0 .0 0. 290E+01 0. 400E-03 397. 0 1 1 50. 0 1 .35 0 .0 0. 200E+03 0. 11OE-03 358. 0 12 50. 0 1 .40 0 .0 0. 110E+04 0. 100E-04 337. 0 13 50. 0 1 .45 0 .0 0. 150E+03 0. 640E-01 312. 0 14 50. 0 1 .50 0 .0 0. 150E+02 0. 630E-03 288. 0 15 50. 0 1 .55 0 .0 0. 170E-02 0. 100E-01 267. 0 16 50. 0 1 .60 0 .0 0. 100E-04 0. 640E-01 245. 0 17 50. 0 1 .65 0 .0 . 0. 100E-01 0. 145E-02 223. 0 18 50. 0 1 .70 0 .0 0. 510E+00 0. 100E-04 202. 0 19 50. 0 1 .75 0 .0 0. 400E+01 0. 100E-04 180. 0 20 50. 0 1 .80 0 .0 0. 130E+03 0. 100E-04 159. 0 21 50. 0 1 .85 0 .0 0. 220E+04 0. 145E-03 1 42. 022 50. 0 1 .90 0 .0 0. 140E+04 0. 710E-02 126. 0 23 50. 0 1 .95 0 .0 0. 160E+03 0. 200E+01 1 14. 0 24 75. 0 2.00 0 .0 0. 290E+01 0. 300E+01 103. 0 25 100. 0 2.1 0 .0 0. 220E+00 0. 240E+00 90. 0 26 100. 0 2.2 0 .0 0. 330E+00 0. 380E-03 79. 0 27 100. 0 2.3 0 .0 0. 590E+00 0. 110E-02 69. 0 28 100. 0 2.4 0 .0 0. 203E+02 0. 170E-03 62. 0 29 100. 0 2.5 0 .0 0. 31OE+03 0. 140E-03 55. 0 30 100. 0 2.6 0 .0 0. 150E+05 0. 660E-03 48. 0 31 100. 0 2.7 0 .0 0. 220E+05 0. 100E+03 43. 0 32 100. 0 2.8 0 .0 0. 800E+04 0. 150E+03 39. 0 33 100. 0 2.9 0 .0 0. 650E+03 0. 130E+00 35. 0 34 100. 0 3.0 0 .0 0. 240E+03 0. 950E-02 31. 0 35 100. 0 3.1 0 .0 0. 230E+03 0. 100E-02 26. 0 36 100. 0 3.2 0 .0 0. 100E+03 0. 800E+00 22. 6 37 100. 0 3.3 0 .0 0. 120E+03 0. 190E+01 19. 2 38 100. 0 3.4 0 .0 0. 195E+02 0. 130E+01 16. 6 39 100. 0 3.5 0 .0 0. 360E+01 0. 750E-01 14. 6 40 100. 0 3.6 0 .0 0. 310E+01 0. 100E-01 13. 5 41 100. 0 3.7 0 .0 0. 250E+01 0. 195E-02 12. 3 42 100. 0 3.8 0 .0 0. 140E+01 0. 400E-02 1 1 . 143 100. 0 3.9 0 .0 0. 170E+00 0. 290E+00 10. 3 44 100. 0 4.0 0 .0 0. 450E-02 0. 250E-01 9. 5 93 Table IV Coef f i c i e n t s A,B,C for A.S.H.R.A.E algorithm [31] A B C January: 1230 .0 0. 142 0. 058 February 1215 .0 0. 144 0. 060 March 1 186 .0 0. 156 0. 071 A p r i l 1 136 .0 0. 180 0. 097. May 1 104 .0 0. 196 0. 121 June 1088 .0 0. 205 0. 134 July 1085 .0 0. 207 0. 136 August 1 107 .0 0. 201 0. 122 September 1151 .0 0. 177 0. 092 October 1 192 .0 0. 160 0. 073 November 1221 .0 0. 149 0. 063 December 1233 .0 0. 1 42 0. 057 

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