Explain why you might want to divide the sky up.
There are many ways to divide up the sky into small segments or illuminance/irradiance zones. Early work by Tregenza suggested that the optimum diameter of a sky zone was approximately 0.2 radians (11.5°) (Tregenza, 1987 - algorithm). From this work, the Commission Internationale de l'Eclairage (CIE) recommended the use of a 145 segment equal-area subdivision based on 8 equal altitude bands which, without noticeable error, allows each zone to be considered as approximating a point source (CIE, 1989).
The CIE 145 sky zone method seeks to achieve a roughly equal-area distribution of segments whilst others apply a much simpler latitude/longitude or equal-angle approach. Figure 2 below shows a range of different sky subdivision strategies. To see these in action, use the Overlay options in ECOTECT's Sun-Path Diagram Dialog Box.
To define segments, each altitude band has a different number of azimuth segments apportioned such that they each have a roughly equal area. As this technique requires that there be an integer number of segments in each band, it is not possible for each to have exactly the same area - however the variation across all segments is relatively small. This subdivision method is shown in the top-left in Figure 2.
As can also be seen in Figure 2, it is possible to vary the resolution of this method to accommodate any number of segments. Obviously the variations in area will be greater for some numbers of zones as it is more difficult to evenly apportion them between bands. However the algorithm is very simple so it is possible to test many different numbers in order to minimise this variation if required.
The triangulated subdivision in the centre of the figure is based on methods developed by Song and Kimerling (Song et al. 2002) for the projection of an equal area global grid onto the surface of a sphere by small circle subdivision. In this method the triangular divisions are close to but not exactly equal area, however the greater the number of sub-divisions the lesser the variation.
Figure 2 also shows two equal-angle methods on the right. These simply divide the sky hemisphere into azimuth and altitude bands, much like lines of latitude and longitude around the Earth' s surface. In this approach, these azimuth bands converge to a point at the zenith, resulting in huge variation between the areas of segments at the horizon (largest) compared to those near the zenith (smallest). When using such a system, the relative contribution of each segment is simply weighted by its relative area. To do this, area-weighted averaging is used such that any value in each segment is multiplied by its area and the sum of all such values is then divided by the total summed area, as shown in the following equation:
Implementation in ECOTECT
ECOTECT uses the equal-angle method, based on user-defined azimuth and altitude angles. This obviously biases accuracy towards the zenith of the sky dome when compared to the horizon as each segment is much smaller in that area. However this is exactly the reason this method was used rather than an equal-area method:
- It is much quicker to calculate the exact array index given any solar position (as this is done many tens of thousands of times during an annual calculation, such a consideration is quite important), and
- It is more accurate when considering the effects of shading devices immediately above a vertical surface - which form by far the majority of shading design conditions for windows and walls.
By default, detailed sky calculations in ECOTECT are done using 2° angular segments, however even with 10° angular segments this method is still more accurate at the horizon than the standard 145 zone equal-area method based on 12° bands. Of course where it counts - immediately above the surface - it is much more accurate and able to detect significantly more detail in overhead shading devices.
References
- Commission Internationale de l'Eclairage (CIE)
- 1989, Guide to Recommended Practice of Daylight Measurement. Editor J D Kendrick.
http://www.cie.co.at/publ/abst/108-94.html.
- Song, L. , A.J. Kimerling and K. Sahr.
- Developing an Equal Area Global Grid by Small Circle Subdivision, in M. Goodchild and A. Jon Kimerling (editors), Discrete Global Grids. Santa Barbara, CA, USA: National Center for Geographic Information & Analysis, 2002.
http://www.ncgia.ucsb.edu/globalgrids-book/song-kimmerling-sahr/
- Tregenza, P.
- Subdivision of the Sky Hemisphere for Luminance Measurements, Lighting Research and Technology, Vol 19, 1987, pp13-14.
- Tregenza, P.R., Sharples, S.
- Subtask C2 - New Daylight Algorithms, IEA Task 21, 1995.
http://naturalfrequency.com/Tregenza_Sharples/Daylight_Algorithms/
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