Luminous Distribution of the Sky

Article 076

Sep. '03

The main source of light in the sky is obviously the Sun itself. However, as a result of atmospheric scattering and reflection off clouds, the entire sky dome also emits light. The overall distribution of light over the sky dome therefore depends on current environmental conditions. Figure 1 below illustrates a range of different sky conditions showing just how variant this lighting distribution can be.

These images are the result of taking photographs using a fish-eye lens. Such images capture the full hemisphere of the sky, with the horizon around the perimeter and the zenith in the centre. An example of such a lens in shown in Figure 2, along with an image taken in a built-up area to clearly illustrate the horizon.

Standardised Sky Models

As clouds form and move through the sky, the distribution of light can change almost minute by minute. This means that we cannot really design for any specific distribution, but must rely on 'average' conditions. The Commission International de l’Eclairage (CIE) has developed a series of mathematical models of ideal luminous distributions under different sky conditions - of which the three most common are clear, uniform and overcast. However, there are many different types of sky and many different mathematical models used to describe them.

Each of these models assume the entire sky dome has some level of luminance, varying with angle from the horizon to the zenith, and with the relative angle from the current Sun position. The sky gets this luminance from sunlight that is scattered by air molecules or suspended particulates and reflected around by moisture vapour and clouds.

As a worst-case, the overcast sky condition is usually used. However in some tropical regions the uniform sky is considered by some researchers to be more appropriate. For the UK the BRE recommend the use of an average sky based on their own mathematical model. For a more detailed analysis of the various sky types and their applicability, see the paper by Roy, Ruck, Reid, Winkelmann & Julian and other references given at the end of this topic.

CIE Overcast Sky

The Overcast Sky distribution model is based on a completely clouded sky where the Sun and its position are not apparent. The passage of radiation through the clouds usually produces close to white light by mixing as moisture droplets are quite large and affect all frequencies of light. However, if the atmosphere is heavily polluted the overcast sky color appearance can be slightly yellow.

The distribution of luminance in such a sky is symmetrical about the zenith and is lower at the horizon than overhead. Given a zenith illuminance (Lz) and its altitude (a), the luminance at any point (L) in the sky is given by:

L   =   Lz	1 + 2 sin(a)
	3

Looking at this formula, you can see that the illuminance at the zenith (a=90) is three times brighter than at the horizon (a=0). This is significant as it means that skylights will be much more effective for daylighting per unit area than side windows as they allow more light in.

CIE Uniform Sky

As the name suggests, this model represents a sky with a constant value of luminance. Thus, no matter where in the sky you look, the model will return a value of 1.0.

CIE Clear Sky

A clear sky assumes that the Sun is visible, resulting in a very non-uniform luminance distribution where the area around the Sun is much brighter than any other area. The CIE Clear Sky model relates the luminance (L) at any point in the sky vault with the zenith luminance (Lz):

L	=	(a + b e-3k + c cos2(k)) (1 - e-d sec(z))
Lz		(a + b e-3zs + c cos2(zs)) (1 - e-d)

where a,b,c,d are adjustable coefficients, k is the angle (in radians) between the point with luminance L and the Sun, z is the zenith angle of the point and zs is the zenith angle of the Sun.

According to CIE Publication 22, Standardization of Luminance Distribution on Clear Skies (Paris, 1973), the CIE clear sky model uses:

Additionally CIE has also standardized a clear sky for polluted atmospheres, using the following coefficients:

The blue color of the clear sky is strongly dependant on the height of the location above sea level and the amount of the atmospheric pollution. Nitrogen dioxide makes the color of the atmosphere slightly brown and this can be seen when looking towards an urban area from the surrounding countryside. High water vapor levels, in the absence of pollution, tend to give the sky a whiter appearance.

UK Average Sky

The BRE Average Sky formula is broadly similar to the clear sky in that it has a circumsolar peak, but treats the effect of clouds differently to better reflect European conditions. This formula differs also in that it does not return a relative luminance value, but an absolute luminance in thousands of candela per metre squared (kcd/m^-2). The luminance (L) at any point in the sky vault is given by:

where a,b are adjustable coefficients, k is the angle (in radians) between the point with luminance L and the Sun and z is the zenith angle of the point.

where zs is the zenith angle of the Sun.

Design Sky Illuminance

The above sky models give only the distribution of luminance over the sky dome, not the total amount of light coming from it, also known as the Sky Illuminance. Whilst two locations may both be experiencing similar overcast skies on the same day that closely match the ideal CIE Overcast Sky model, the total amount of light at each location may be completely different.

The main determinant of overall sky illuminance is the latitude of the location. As landscape artists have long known, light levels near the equator are much brighter and have a completely different quality than those at mid-latitudes and the poles. If we can take cumulative surface UV exposure as a relative indicator of daylight exposure, then Figure 5 below shows just how significant this variation can be over a year (the grey area indicates areas where no data was recorded).

Design Sky Values

Design Sky values are derived from a statistical analysis of outdoor illuminance levels. They represent a horizontal illuminance level that is exceeded 85% of the time between the hours of 9am and 5pm throughout the working year. Thus they also represent a worst-case scenario that you can design to and be sure your building will meet the desired light levels at least 85% of the time.

Design sky values vary from around 12-15,000 lux at the equator down to around 3-4000 lux at a latitude of ±60°, as shown in Figure 6 below.

Obtaining Design Sky Values

Obviously as different calculation methods give slightly different results, the best way to obtain the Design Sky value for any location is from a published source. However, if this is not readily available, then you can calculate it from the average diffuse sky illuminance formula given by Tregenza, as shown below. The value Y° is the current altitude of the Sun in the sky, given in degrees.

Illuminance values from this equation need to be calculated hourly throughout the year to determine a distribution pattern, from which the 15th percentile is derived. As discussed above, this is because the design sky illuminance is taken to be the illuminance level exceeded 85% of the time. Below is a tool that performs this calculation and displays a cumulative frequency graph of average lux levels. Note that the values obtained from the Tregenza formula vary from around 4,000 Lux up to 10,000 Lux at the Equator.

Useful Links

BRE ETSU Daylight Algorithms

http://eande.lbl.gov/Task21/C2/contents.html

The Development of Modelling Strategies for Whole Sky Spectrums
under Real Conditions for International Use

http://eng.murdoch.edu.au/FTPsite/skymap_report.pdf

CBD-17. Daylight Design

http://irc.nrc-cnrc.gc.ca/cbd/cbd017e.html

Satel-Light: Sky Types and Luminances

http://www.satel-light.com/guide/advlum.htm

This information is brought to you by  SQUARE ONE research PTY LTD.

Internet Site: www.squ1.com               Electronic Mail: sales@squ1.com