The phrase "daylight availability" refer to the amount of sunlight and skylight which may be present for a specific time and date at a given location.

From instrumented observation over a long period, it has been concluded that the mean values of illuminance from the sun and from the sky on a plane are related to the geometrical position of the sun in the sky, and are also dependent on the condition of the sky.

For any given instant, the shifting clouds in a tropical sky may result in changing illuminance on a particular area on a plane, but the time-average or space average values are not dependent on the cloud pattern. Rather, the values are dependent on the overall amount of cloud cover in the sky.

In some important daylighting applications, the values of illuminance from skylight and from sunlight on a horizontal plane are required to be known. But in more elaborate cases, the luminance values of different patches of sky in different directions in the sky dome are required to be known.

In the following, mathematical models of sunlight and skylight will be presented. These models have been adopted by CIE and IESNA. The values calculated by the models are meant to present average values for the given sky condition. Actual instantaneous values corresponding to the same condition may be twice or half as large as the value from the model.

It is noted that the models have been developed based on observations from high-latitude locations, where the sun do not reach zenith. But the models are implicitly meant to be valid for all geographical locations on earth, including the tropical region. The daylight research station in AIT will soon accumulates sufficient data to verify, modify these models, or introduce new model(s).

3.3.1 Sky Condition

Two physical sky conditions have been adopted by CIE, overcast (or cloudy) and clear. The IESNA has adopted another intermediate sky called partly cloudy sky.

These sky conditions are classified by the amount of cloud cover or alternatively the sky ratio .

Cloud Cover Index

The cloud cover index is graduated in tenths and is used to indicate the amount of cloud in the sky dome, with 0.0 for no cloud to 1.0 for full cloud. This data is obtained from observation and is available as a part of the standard meteorological data.

Sky Ratio

The sky ratio is obtained as the ratio of diffuse radiation to global radiation (sum of sun and sky radiation on horizontal plane).

The sky conditions are classified by the values of the two indices and are as shown in Table 3.3.1.

Table 3.3.1 Classification of sky condition.

The cloud cover index is obtained from observation and is subject to uncertain ly . The sky ratio is based on instrumented measurement and would not be subject to human judgement. However, during sunrise and sunset times the sky ratio approaches one no matter what the actual sky condition is. These two simple indices both possess draw backs.

The present sky classification is more suitable for professional community where practicality in determining sky condition is desirable. However, in the research community, more detailed classifications in terms of more gradation in sky types based on measurement records are often proposed.

3.3.2 Illuminance from Sunlight and Skylight

Sunlight
The illuminance from sunlight just outside the atmosphere of earth (at the mean distance from the sun) is obtained from

where E_SC     =   standard extra-terrestrial illuminance on a plane normal to the sun,
                         also known as solar illumination constant, klux,
           E _fs     =   efficacy of sunlight, lm/W,

           G  (l)   =   power-density spectral function of the radiation of the sun,
           P  (l)    =   standard photopic response function.

The values of E_SC  and E _fs as adopted by CIE are given as 

                   E_SC    =   128 klux, and
                   E _fs    =   94.2 lm/W. 

The orbit of the earth around the sun is elliptic.  For a given Julian day j_d, the corrected illuminance constant is 

E_st  = E_SC  [1+0.034cos(2p(j_d  - 2))/365].                                                            (3.3.1)

 The light flux from the sun is attenuated by the atmosphere as it travels to earth.  The illuminance of sunlight in a plane normal to the sun ray is given as 

E_Dn  =  E_st  exp(-C m ),                                                                                          (3.3.2)

 where  
C     = atmospheric extinction coefficient, the value of which is given in Table 3.3.2,  and      
m     =   optical air mass, the relative distance of travel of the ray through the atmosphere and is given as the inverse of the sine function of the solar altitude angle a_s  i.e. m = 1/sin a_s .

 Table 3.3.2    Values of atmospheric extinction coefficient.

 Note that under the cloudy (or over cast) sky the sun is completely obscured.

 The illuminance of sunlight on a horizontal surface is

E_Dh  =   E_Dn  sina_s .                                                              (3.3.3)

On a vertical plane it is

E_D v  =   E_Dn  cosq_s ,                                                             (3.3.4)

where   q_s    =   the angle between the sun ray and the plane normal.

Skylight
Similar to the case of sunlight, skylight is also a strong function of sina_s. The illuminance from skylight on a horizontal surface is given as

 E_kh  =   A  +B  (sina_s )^C,                                                         (3.3.5)

 where   the values of A , B , and C   are given in Table 3.3.3 for different sky conditions.

Table 3.3.3    Values of A , B , C   for different sky conditions.

Figure 3.3.1 shows plots of the illuminance values due to sunlight and skylight under different sky conditions as a function of the solar altitude angle a_s .  

Figure 3.3.1 Graphs of illuminance values due to sunlight and skylight under different sky conditions

Figure 3.3.2 Daylight illuminance values on horizontal plane under different sky conditions for the 4 reference days.

Figure 3.3.3 shows an annual plot of horizontal illuminance at different times of a day due to skylight under partly cloudy sky condition for a year. Again, the plot shows significant difference only for the months of December and January.

Zenith Luminance
In all models mentioned the luminance of any point in the sky takes on a values relative to that of a reference point. The common reference luminance is the luminance in the zenith direction L_Z.

Figure 3.3.3  Plot of horizontal illuminance due to partly cloudy sky during 5-20 hrs of the 15^th day of each month.

The Zenith luminance is related to the horizontal sky luminance by

  L_Z   =   Z_LE_kh                                                                                             (3.3.6)

 where   Z_L = zenith luminance factor corresponding to a given solar altitude angle a_s.  The value of Z_L is given in Table 3.3.4.

 Table 3.3.4    Values of zenith luminance factor Z_L .

Illuminance on a Horizontal Surface
There are occasions when the illuminance on a horizontal surface needs to be known given the luminance of the sky. Consider the situation in Figure 3.3.4.

Figure 3.3.4 The configuration of the illuminance on a horizontal plane due to an incremental luminous surface.

In the figure, da represents an equivalent incremental surface of the sky dome which possesses a luminance value L (q, f ).  The illuminance on a horizontal plane due to the incremental surface da  is

But da, being part of a spherical surface is related to q, and f as

Substituting this into the above equation gives

                                     (3.3.7a)

                                  (3.3.7b)

Illuminance on a Vertical Surface
Figure 3.3.5 illustrates the situation where the illuminance of a vertical surface is due to the light flux from an incremental area da .

Figure 3.3.5 The configuration of a vertical surface dA receiving flux from an incremental area da .

From the figure , the illuminance on dA  due to the flux from the incremental element da  is

        (3.3.8a)

The illuminance on a vertical surface due to half of the sky in front of the surface is then

                          (3.3.8b)

Illuminance on an Inclined Plane
Consider a plane inclined at an angle b (so that its normal possesses a zenith angle b and an altitude angle (p/2-b ) in Figure 3.3.6.

Figure 3.3.6 An inclined plane with an incremental area dA receiving light flux from an equivalent incremental area da in the sky dome.

Here, the normal vector of the plane at dA  is given as

                                                                  (3.3.9)

The vector to the incremental area da  is given as

                                                                  (3.3.10)

The angle x between the two vectors are obtainable from

                    (3.3.11)

The illuminance dE  on dA  from the luminance of the incremental area da  is then given as

but

Substituting the expression for da  into the relationship for dE  gives

                                                          (3.3.12a)

Integrating over the relevant area of the sky dome gives

                                                       (3.3.12b)

The Relative Position of a Point in the Sky and the Sun
The following figure illustrates the geometric relationship between the sun, the given point, and the cardinal directions.

Figure 3.3.7 The relative geometrical position of a point and the suns.

In all the sky luminance model, the angular variables which determine the relative luminance are:

x     :    the angle between the solar vector and the vector to the given point
f     :    the zenith angle of the given point, and
f_s   :    the zenith angle of the sun.

a) Uniform Sky Model
This model is now used more often to illustrate basic concepts in daylighting calculation, although under overcast sky condition uniform luminance distribution  has also  been observed.  In this model

L (q, f )  =   L_Z   =   L                                                                                   (3.3.13)

The sky luminance for any point in the sky is independent of the position of that point and is identical to that of any other point including that at the zenith.  Applying (3.3.7) with (3.3.13) gives

                                                 (3.3.14)

Similarly, applying (10.4.8) with (10.4.13) gives

                                                                   (3.3.15)

From the result it is noted that for uniform sky
L_Z   =   L   =   E_kh  / p                                                                                        (3.3.16)

and

E_kv  / E_kh    =   1 / 2     =   0.5                                                                          (3.3.17) 

b) Overcast Sky Model
Moon and Spencer proposed a mathematical model for the distribution of overcast sky which has been adopted by CIE since 1955. The original model which is an empirical model is given as

   L (q, f )  =   L_Z  (1 + 2 cosf )/3                                                                      (3.3.18)

An equivalent model which has been derived from a consideration of the luminance distribution of the cloud layer is given as.

                     (3.3.19)

Numerically, the luminance values from the two models are very close. Practically any of the two models can be used. Both models give luminance values which are invariant with respect to the azimuth of the point. The values of the Luminance at zenith and at the horizon are given in Table 3.3.5.

Table 3.3.5 Luminance values at some points from overcast sky models.

Reference(s) presents further classification of overcast sky into several models. One of the submodel is the uniform sky. Applying (3.3.18) to (3.3.7) gives

E_kh  =   (7p/9)L_Z                                                                                      (3.3.20)

 Similarly for (3.3.8), it gives

 E_kv   =   (p/6+4/9)L_Z                                                                               (3.3.21)

 The ratio E_kv  /E_kh  gives              

E_kv  /E_kh     =   0.397                                                                                (3.3.22)

In overcast sky, the luminance is highest at zenith and declines towards the horizon to one third of that at zenith.

Figure 3.3.8 Zenith luminance as a function of altitude angle.

c) Clear Sky Model
Kittler proposed a clear sky model which was adopted by CIE in 1973.  The form of the model is

                      (3.3.23)

Referring to Figure 3.3.7, the angle x is related to the other angles by the following relationship which is obtained as a product of the solar vector and the vector of point P.

cosx  =   sina_s cosf  + cosa_s sinf cos(q - g_s )                                                 (3.3.24)

 In order to obtain the illuminance due to clear sky on a horizontal plane or on a vertical plane, (3.3.23) can be applied to (3.3.7) and (3.3.8).  However the integrals cannot be evaluated analytically and numerical integration must be used.  It is noted also that the integrals contain g _s and a _s as parameters.  The illuminance on a horizontal plane due to clear sky is a function of the solar altitude angle a _s .  The illuminance on a vertical plane due to clear sky is a function of both the solar altitude angle and the solar azimuth angle.  Figure 3.3.9 illustrates the pattern of the illuminance as a function of altitude angle.  In the figure, the line labeled Azm = 60 corresponds to the luminance values on a plane which sees the sun at 60º, or that the vector from the plane to the sun makes (90º - 60º) = 30º with the normal of the plane.  The line labeled Azm = -60 is applicable to the case where the sun is at the back of the plane.  The line from the plane to the sun makes an angle 30º to the normal of the back of the plane.  The figure illustrates clearly that the area of the sky close to the sun is much more luminous than those away from it.

Figure 3.3.9 Illuminance on vertical surfaces from clear sky.

Clear sky occurs naturally during summer in temperate climate. The duration is relatively long and the occurrence is frequent in such climate in summer.

In tropical climate, the sky is very rarely very clear. There is always a certain amount of clouds in the sky. However, relatively clear sky with a small amount of clouds can occur during any month of the year.

d) Partly Cloudy Sky Model
Pierpoint v also proposed a model for the luminance of a partly cloudy sky. The model is similar in form to that of clear sky and is given as

                         (3.3.25)

It is noted that this model also includes the variable x, and the luminance of a patch of the sky is also strongly related to the angular distance from the sun.  Figure 3.3.10 illustrates the pattern of the illuminance on a horizontal plane as a function of the solar altitude angle and the azimuth angles of the sun and that of the plane.  The pattern is similar to that in Figure 3.3.9 but the values of illuminance from skylight here for partly cloudy sky are larger.

Figure 3.3.11 illustrates, for the luminance of a given point in the partly cloudy sky, its variation with respect to the angular distance from the sun, and the altitude angle of the point.  Three figures, corresponding to that for solar altitude angle of 22.5º to 67.5º respectively also illustrate the variation with respect to the position of the sun in the sky.

Figure 3.3.10 Illuminance from vertical surfaces from partly cloudy sky.

a) Solar altitude angle 67.5º.

Figure 3.3.11 Features of the luminance from the partly-cloudy-sky model.

b) Solar altitude angle 45º.

c) Solar altitude angle 22.5º.

Figure 3.3.11  (Continued)

The common features of the figures are:
•  the higher the solar altitude angle, the higher the luminance of any point,
•  the luminance values of those points near the sun are higher, but tapering off to a constant value an angular distance of 90º and continue at the same value toward 180º ,
•  the luminance values for those points at the same angular distance from the sun but at lower altitude angles (closer to the horizon) are higher than those at higher altitude angle, the ratio of horizontal luminance to that near zenith being around 2:1.

Clear sky luminance also exhibits similar features, the relative values at different angular positions are even higher. The variational features of the luminance from the partly-cloudy-sky model is some where between those of clear sky and uniform sky.

Sky Luminance Models Used in the Lumen Method
The Lumen method adopted by IESNA for calculation of daylight illuminance in building interior from sidelighting and toplighting assumes the use of the following models shown in Table 3.3.6.

Table 3.3.6 Methods of sky luminance distribution used in the Lumen method.

In these models, the relative luminance of any point in the sky varies only with the altitude angle of that point, and invariant with respect to the azimuth angle. Models a) and b) represent overcast and uniform sky distributions. The other three models represent clear and partly-cloudy-sky distributions without the influence of the circumsolar illuminance, i.e. the phenomenon of high sky luminance near the sun.