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Bird Clear Sky Model


In our algorithms of the calculation of solar radiation on the surfaces of the static and sun-tracking collectors, we use the "Simplified Clear Sky Model for Direct and Diffuse Insolation on Horizontal Surfaces" developed by Richard E. Bird and published in the report [1].




[1] Bird, R. E., and R. L. Hulstrom, 

"Simplified Clear Sky Model for Direct and Diffuse Insolation on Horizontal Surfaces",

Technical Report No. SERI/TR-642-761, 

Golden, CO: Solar Energy Research Institute, 1981


General notes


In this topic, we discuss the Bird model that calculates different components of solar radiation. In the next topic, we show how the Bird model works in the "Climate & Irradiance" window, and explain how to use the "Climate" window to prepare a climate model of your site for your further work with the "Solar Day" and "Solar Year" windows.


Both topics are closely connected. So do not try to understand all at the first reading. Read through both topics to know in general what they are about. Then get your first experience with the "Bird" window, and return to more detailed reading of this topic. Then repeat the similar procedure with the next topic and the "Climate" window.


Key questions for your orientation


Why we need a solar radiation model for the shading calculations?


The shading of your solar collectors (if it occurs) may substantially vary during a day. The shading is more probable within the morning and evening hours, when the Sun position above the horizon is low. On the other hand, the intensity of the solar radiation varies during a day too. Usually, these factors act contrary one to another (in opposite phases), because the solar radiation is higher just at noon, and is lower in the morning or in the evening. Therefore, we need somehow to evaluate and compare the energy weight of the different time spans during a day to calculate the average daily shading and the corresponding energy losses properly. Just the Bird model helps us to do it.  


Why we can use a unified solar radiation model for the different geographical latitudes?


The extraterrestrial solar radiation (the solar beam radiation out of the atmosphere on the Earth orbit) is a relatively stabile characteristic. The Earth orbit is slightly elliptical, and the Earth-Sun distance regularly varies during a year by +/- 1.5 % from its average value. As a consequence, the extraterrestrial solar radiation also varies by +/- 3% around its average value, the so called extraterrestrial solar constant of 1.367 kW/m2.


Note that the solar radiation out of the atmosphere is a beam radiation: it has only the direct component, and does not have any diffuse component. While passing through the atmosphere, the incoming direct solar radiation is being partly scattered, transforming into the diffuse radiation. Both the direct and diffuse components are also being partly absorbed. The decreasing of the solar radiation intensity by the atmosphere depends exclusively on the atmosphere state and the length of the solar beam path inside the atmosphere (the so called air mass, or the optical length of the beam path) that in its turn depends on the Sun elevation above the horizon.


So the geographical latitude impacts the solar energy balance only indirectly, because it regulates the grades of sunny hours by the Sun elevation but not the solar radiation itself. In all other aspects, the process is identical on any geographical latitude, and can be described by a unified model.


Do users need to learn the Bird model theory before to use Shadow Analyzer practically?


No. You do not need to learn the theory of the Bird model before to use Shadow Analyzer. Just the "Bird Clear Sky Model" window represents the main features of the Bird model for you. It shows main practical results and consequences of the model. Moreover, the window allows you to experiment with the model to get feeling how the different components of the solar radiation depend on the Sun elevation, and how the atmospheric parameters affect these dependencies.


The Bird model is integrated in our software, and works automatically. The "Climate & Irradiance" window includes an array of 12 partial Bird models (one model with the independent settings per each of 12 months of the year). The partial Bird models of the adjacent months are connected by a spline approximation.


OK, we use the Bird model for clear sky, but what to do with cloudy sky?


The "Climate & Irradiance" window contains 12 monthly partial Bird models, 12 monthly probabilities of clear sky, and 12 correction coefficients for the diffuse radiation. Think of the "Climate" window as a climate model that you can fill by input parameters and adjust for the solar radiation statistic of your site.


You set 12 monthly probabilities of clear sky according to the monthly statistic of sunny hours at your site. This values determine the statistical weight of partial Bird models. The rest hours of the total light time of each month are the cloudy hours (or cloudy time intervals), within which only the diffuse radiation reaches the ground surface.


The cloudy time intervals (as well as the diffuse radiation itself) are not so interesting for the shading calculations. So we need only to meet the balances and proportions of the direct, diffuse, and total solar radiation. Therefore we simplify the calculation procedure: we use the Bird model for the cloudy time intervals too, but take from it only the calculated diffuse component of solar radiation, and then we apply the correction coefficients for the diffuse radiation using them as the abstract adjustment parameters to meet all the balances that are known from the available site statistic.  


Can one use the "Bird" and "Climate" windows for free for the educational and research purposes?


Yes. You can use the "Bird" and "Climate" windows for free for the educational and research purposes. These options are fully functional and publicly available in the time-unlimited free Trial version of Shadow Analyzer. Download the Trial version of Shadow Analyzer from our Website. Copy the following string into the address bar of your Internet browser:


The "Bird Clear Sky Model" and "Climate & Irradiance" windows are working together with the "Solar Table" window in the Trial version exactly so as they do it in the full version of Shadow Analyzer. So you can learn the behavior of the solar radiation inside the atmosphere, set the climate profile including the monthly probabilities of clear sky for a site on any geographical latitude, and calculate the daily sums of direct and diffuse solar radiation for fully sunny and mean cloudy days over the year. The calculated values are displayed in the "Solar Table" window, and you can copy them into a text document or insert in an Excel table for further numerical analysis.


The "Bird Clear Sky Model" window


The "Bird Clear Sky Model" window represents the Bird model of the solar radiation of the clear sky. It explains main mechanisms and factors of the absorptance and scattering of the solar radiation in the atmosphere. It shows how the solar radiation depends on the Sun elevation and the state of the atmosphere. It calculates the direct and diffuse components of the solar radiation at the ground level. It shows also the extraterrestrial solar radiation (out of the atmosphere).


You can select / change the parameters of the Bird Model using the arrow keys of your keyboard when the window is active. To select a parameter, use the UP / DOWN arrow keys. The selected parameter is marked by a yellow background. You can change the selected parameter using the LEFT / RIGHT arrow keys.


The "Bird Clear Sky Model" window is independent of any other window. However, you can import the Bird model parameters of the "Bird Clear Sky Model" window into the "Climate & Irradiance" window to quickly set the parameters of 12 partial Bird models uniformly for all months of the year. To make such a uniformly setting, activate the "Climate & Irradiance" window, and then press the key "B" on your keyboard.


If you do not have enough information to set all the parameters of the Bird model for your particular site, use the default settings for the unknown parameters.




The window has the following pages:


1. Air mass

2. Transmittance

3. Irradiance


To switch pages, use the keyboard keys "1", "2", "3", or the right mouse click, when the window is active.


The argument of functions of all pages is the same: zenith angle Z (from Zenith to visible Sun), degrees. Note that Z = 90 - E, where E is the Sun elevation above horizon (also called the altitude angle).


To see the numerical values of the calculated function curves, use the "Solar Table" window.


The page 1 shows the dependency of the optical length (or air mass) of the solar beam on the zenith angle Z. The air mass in vertical direction (at Z = 0) is accepted as a unit. The air mass as a function of Z is nearly proportional to 1/cos(Z). However, it does not become infinitely large at low Sun position (near to the horizon Z = 90) because of the spherical form of the Earth surface. If the pressure decreases (for example due to elevation of the site above the sea level), the air mass also decreases. So the actual optical length of the beam is proportional to the pressure-corrected air mass.   


The page 2 shows the partial transmittance coefficients associated with several mechanisms of the scattering and absorptance of the solar radiation inside the atmosphere. The final transmittance of the atmosphere for the direct solar radiation is proportional to the product of multiplication of partial transmittance coefficients. For the diffuse radiation, the Bird model takes into account also the reflection between ground and sky that depends on the ground albedo.


The page 3 shows the resulting picture for the different components of solar radiation as functions of the zenith angle Z.


Restore basic states with keys "8", "9", "0"



Page 1: Air mass


The page shows the relationship between the air mass and the pressure-corrected air mass.


Air mass M corresponds to the normal pressure at sea level (1.013 bars).

The pressure-corrected air mass depends on the pressure parameter (of the considered site). 









Page 2: Transmittance


The page shows several mechanisms of absorptance and scattering of solar radiation in the atmosphere.









Page 3: Irradiance


The page shows some components of solar radiation as functions of zenith angle Z.



Extraterrestrial constant is the extraterrestrial solar irradiance averaged over the year.








At the default values of parameters Pr = 1.01, Oz = 0.34, Wt = 1.4, BT = 0.19, GA = 0.25, the calculated values of the different components of solar radiation are as it is shown in the following table (maximize this help window if the table is deformed)


Z 0 15 30 45 60 75 90

E 90 75 60 45 30 15 0

cos(Z) 1. 0.966 0.866 0.707 0.5 0.259 0

Dir_N 0.854 0.845 0.815 0.756 0.643 0.407 0

Dir_H 0.854 0.816 0.706 0.534 0.322 0.105 0

Dif_H 0.199 0.196 0.189 0.176 0.152 0.105 0

Tot_H 1.053 1.013 0.895 0.710 0.474 0.211 0

Ext_H 1.367 1.320 1.184 0.967 0.684 0.354 0


Use the "Solar Table" window, to generate more detailed table.


Note that we show also the extraterrestrial solar radiation Ext_H in both the "Bird" and "Climate" windows. This variable is useful for some research purposes, because there is a correlation (known from the solar radiation statistics) between two values D = <Dif_H> / <Tot_H> and T = <Tot_H> / <Ext_H>, where brackets < ... > mean the averaging over a daily or monthly interval. The function D = f(T) helps sometimes to verify or correct data from solar databases, or to reconstruct the unavailable data of direct solar radiation.


Basic formulas and further calculations


We use terms "Global" and "Total" solar radiation as synonyms. Also, we use terms "Beam" and "Direct" solar radiation as synonyms (however, some other authors distinguish these terms reserving the word "Direct" for the particular case of the beam solar radiation projected or measured on the horizontal surface).


To specify which a surface we are speaking about, we use suffixes "_N", "_H", and "_S".     


We apply the suffix "_N" to the case of the direct (or beam) solar radiation Dir_N measured on the "normal" surface (or "normal" plane) that is perpendicular to solar beams (like a 2-axes sun-tracking solar collector).


We apply the suffix "_H" to the case of different components of solar radiation measured on the horizontal surface. By the definition, the relationship between Dir_N, Dir_H, Dif_H, and Tot_H are as follows 


Dir_H = Dir_N * cos(Z);


Tot_H = Dir_H + Dif_H = Dir_N * cos(Z) + Dif_H;


where Z is the zenith angle from Zenith to visible Sun.


In general case of an arbitrary oriented surface S, we use the suffix "_S". 


Dir_S = Dir_N * cos(i);


Tot_S = Dir_S + Dif_S = Dir_N * cos(i) + Dif_S;


where "i" means the incidence angle.


The incidence angle "i" is the angle between the normal to the surface S and the direction towards visible Sun. For the horizontal surface i = Z, and cos(i) = cos(Z). For a 2-axes sun-tracking solar collector i = 0, and cos(i) = 1.


The main assumption that we use in further calculations of the "Solar Day" and "Solar Year" windows is that the diffuse radiation on an arbitrary oriented surface S is approximately the same as the diffuse radiation on the horizontal surface. So the value of Dif_S is substituted by the value of Dif_H.


It means that we accept Dif_S = Dif_H as an simplified approximation, and express the total solar radiation on an arbitrary oriented surface as


Tot_S = Dir_N * cos(i) + Dif_H;


Comments. Actually, there is no problem to include a more complex code into the calculation algorithm to calculate the value of Dif_S taking into account two factors: the asymmetry of the distribution of the diffuse radiation coming from different parts of the sky semi-sphere, and the reflectivity of the surrounding objects of the architectural environment of your solar project. The main reason why we apply the simplified approximation Dif_S = Dif_H is that a more rigorous algorithm would require much more complex structure of the input data including actually measured reflectivity coefficients of the surfaces of the surrounding objects that would be difficult for our users to gather for each particular solar project. 


Further in the "Climate & Irradiance" window, we introduce two more variables Prob and Dcs, and the switch between "100% clear sky" and "mean cloudiness" modes (managed by the keyboard key "V"), where


Prob -- the probability of clear sky in % (or in dimensionless units from 0 to 1),

Dcs -- the correction coefficient for the diffuse solar radiation.


For the "100% clear sky" conditions, we apply the same formulas as shown above:


Tot_H = Dir_N * cos(Z) + Dif_H;

Tot_S = Dir_N * cos(i) + Dif_H.


For the "mean cloudiness" conditions we modify the formulas as follows:


Tot_H = Prob * Dir_N * cos(Z) + Dcs * Dif_H;

Tot_S = Prob * Dir_N * cos(i) + Dcs * Dif_H.


We use these modifications also in "Solar Day" and "Solar Year" windows. See "Comments" in the topic "Solar Day".


How to set the Bird model parameters


Below, we characterize briefly the Bird model parameters by the grades of their impact on the calculated components of solar radiation. You can experiment with the parameter settings of the "Bird Model" window to form your own view on this matter.


The "Bird Model" window allows you to understand how the Bird model parameter setting affects the calculated components of solar radiations. Use the "Bird Model" window together with the "Solar Table" window to see the numerical representation of the calculated curves. This experience is useful for the next step, when you will work with the array of 12 partial Bird models of the "Climate & Irradiance" window.


Then we give some general advices how to set values of the Bird model parameters if you do not have the precise data for your site.


Finally, we add some useful Internet links.


Atmospheric pressure


Although, even a substantial variation of the atmospheric pressure within a diapason of 0.8 - 1.0 bar alters the solar radiation by less than 1.5%, it is the most available data that you can obtain from different sources of information or evaluate by yourself.


Use the following table to evaluate approximately the average atmospheric pressure at your site by its elevation H above the sea level. Use the column "delta P per each next 100 m" if you have the atmospheric pressure data for a site near to your location (but with another elevation H) and need to recalculate it for your site elevation. 


Pressure as a function of the site elevation for U.S. Standard Atmosphere

H, km P, bar delta P per each next 100 m

0.0 1.013 -0.0116

0.5 0.955 -0.0112

1.0 0.899 -0.0106

1.5 0.846 -0.0102

2.0 0.795 -0.0096

2.5 0.747 -0.0092

3.0 0.701  


Read in Wikipedia 




You can keep the default value of the parameter "Oz" (either 0.31 cm or 0.34 cm) in both "Bird Clear Sky Model" and "Climate & Irradiance" windows, because it does not play a noticeable role in our calculations affecting the solar radiation by less than 1.0%.


However, this theme is important by itself for many aspects of our life. Use the following link to read more about the atmospheric ozone. To understand the linked articles, note that the atmospheric ozone is measured either in centimeters (cm) or in the Dobson units (DU). One Dobson unit refers to a layer of ozone that would be 0.01 cm thick under standard temperature and pressure. For example, 300 DU of ozone brought down to the surface of the Earth at 0 C would occupy a layer of 0.3 cm thick.


Read in Wikipedia 




The parameter Wt is the amount of precipitable water in a vertical column from surface expressed in centimeters.


The precipitable water is the total amount of water vapor in a vertical column of air, often expressed as the depth of the layer of water that would be formed if all the water vapor were condensed to liquid water. The Wt parameter substantially depends on the ambient temperature, and as a consequence its average monthly values are varying from month to month.


The variation of Wt parameter within a diapason of 0.5 - 5.0 (around the default setting Wt = 1.4) alters the total solar radiation on a horizontal surface Tot_H at Z = 0 by 6.4%, from +2.6% to -3.8% of its default value. It decreases both the direct Dir_N and diffuse Dif_H components of solar radiation keeping approximately the same proportion between them as in the default case.


Get the average monthly values of Wt from the meteorological statistic of your site, or use any appropriate on-line service in Internet. To see an example of a very convenient database for USA, where you can find the Wt data, use the following link 


Read in Wikipedia 


Broadband turbidity


The parameter BT (broadband turbidity) is the aerosol optical depth from surface in a vertical path. It expresses the clearness of the atmosphere, and is responsible for the proportion between direct and diffuse solar radiation. The greater is BT, the more turbid is the atmosphere, and as a consequence the greater is the part of the initial beam radiation that is scattered on aerosols transforming into diffuse radiation. This process is also accompanied by a slight decreasing of the total solar radiation due to the aerosol absorptance.


The variation of BT parameter within a diapason of 0.1 - 0.4 alters the total solar radiation on a horizontal surface Tot_H at Z = 0 by about 6%, from +2.1% to -4.1% of its default value. Although the changes of the total solar radiation are not so great, the BT parameter plays an important role, because it mainly redistributes the energy from the direct to diffuse component changing dramatically the proportion between the direct and diffuse solar radiation.


The BT variation within a diapason of 0.1 - 0.4 decreases the direct solar radiation Dir_N = Dir_H at Z = 0 by almost 27%, from +8.8% to -17.1% of its default value. Simultaneously, it increases the diffuse solar radiation on a horizontal surface Dif_H at Z = 0 from -17% to +52% of its default value. So the proportion between the direct and diffuse solar radiation at Z = 0 changes from 87/13 at BT = 0.1, to 81/19 at default setting BT = 0.19, and to 70/30 at BT = 0.4.


Unfortunately, it is the least available parameter that is not included in most of the solar databases. Moreover, the aerosol factor is often considered not separately but together with other mechanisms of the general atmospheric turbidity, what make difficult to use the materials of the numerous scientific publication on these theme immediately for the setting of the BT parameter into the Bird model.


Therefore, we propose you to consider the BT parameter as an abstract adjustment parameter of the Bird model, which you can use in the "Climate & Irradiance" window to refine the climate model of your site. Firstly use the default setting of BT, and then, if you see a disagreement of the calculated monthly balances of the direct radiation with the available solar statistic of your particular site, meet the balances adjusting the monthly average values of the BT parameter.


For the advanced users, who want to set the BT parameter using some additional sources of information, we remind its original definition. In the original report [1] mentioned at the beginning of this topic, the broadband turbidity BT denoted as Tau_A is expressed by the following formula Tau_A = 0.2758 * d(0.38) + 0.35 * d(0.50); where d(0.38) and d(0.50) are two values of the aerosol optical depth from surface in a vertical path measured at 0.38 and 0.50 micrometer wavelength. The verification of the Bird model (by the comparison with more complex and rigorous simulation models) reported in [1] was provided with the following parameter values d(0.38) = 0.3538; d(0.50) = 0.2661; Tau_A = 0.191.


Read in Wikipedia 




In general, the albedo is a synonym of the reflectivity.


The parameter GA (ground albedo) means the average reflectivity of a large-scale area (of about one hundred square kilometers) around your site.


The variation of GA parameter within a diapason of 0.1 - 0.9 alters the total solar radiation on a horizontal surface Tot_H at Z = 0 by 8%, from -1.4% to +6.6% of its default value. It affects the diffuse radiation Dif_H only (but not the direct radiation) increasing it within the same diapason by more than 30%. So it works like the BT parameter mostly changing the proportion between the diffuse and direct solar radiation within sunny hours.


The snow cover can play the most significant role in the seasonal variation of GA for some regions. However, its impact on the yearly solar radiation balance is decreased substantially, because this factor usually correlates with the low average Sun position above horizon and low monthly probability of clear sky.


Sample albedoes

Surface  Typical Albedo

Fresh asphalt  0.04

Conifer trees 0.08 - 0.15

Worn asphalt 0.12

Deciduous trees 0.15 - 0.18

Bare soil 0.17

Green grass 0.25

Desert sand 0.40

New concrete 0.55

Snow (old) 0.40 - 0.60

Snow (fresh) 0.80 - 0.90


Read in Wikipedia