The Daylight Coefficient Method
The theoretical basis for the calculations involved in the Three-Phase Method, Five-Phase Method, and now F-Matrix Method can be traced back to the concept of Daylight Coefficients introduced by Tregenza and Waters (1983). The core idea behind Daylight Coefficients is that the daylight directly or indirectly incident on a surface inside a room can be accounted for by considering two independent factors: luminance of the sky and the geometry and the optical properties and geometry of the surrounding surfaces. The illuminance at a measurement point in the room from a small patch of sky can be calculated as:
…….…………………………….[1]
where is the luminance of the sky patch and is the angular size of the sky element at an altitude of θ and azimuth of Φ. , the Daylight Coefficient, is a factor that depends on the geometry of the room and the surrounding buildings as well as the reflectances and transmittances of the surfaces that constitute that geometry. Figure 5 shows a schematic diagram for calculating Daylight Coefficients using a terminology similar to the one used in equation [1]. Equation [1] is applicable for illuminance from a single discrete patch of sky. Total illuminance in a room can be calculated by summing up the illuminance contributed by all the patches.
Figure 5. A schematic diagram for the Daylight Coefficient method. The equation for illuminance in the above image accounts for the luminance from a single patch of sky. The total illuminance at a point in the room can be calculated by summing up the illuminance from discrete sky patches. Image Credit: (Reinhart 2001)
The discretized sky model considered for calculating Daylight Coefficients has undergone several modifications over the years. One of the earliest sky models considered for Daylight Coefficient calculations was proposed by Tregenza (1987). This model divides the celestial hemisphere into circular patches and was devised at the time by considering the aperture size of the probe of the scanning luminance meters used for measuring sky luminance. The angular size of the probe considered for the patches in Figure 6 is 11.13o. Tregenza also discussed variants of this model by using different probe sizes. For example, a probe size of 10.15o will lead to a model with 145 sky patches. One major drawback of models with circular patches is that a substantial part of the sky is unaccounted for due to all regions of the sky not being covered by patches. Reinhart (2001) proposed an improved sky model consisting of rectangular sky patches that cover the entire sky hemisphere. Figure 7 shows a Tregenza Sky Model consisting of 145 patches as well as a corresponding continuous division sky model proposed by Reinhart (2001). The center of each of the rectangular patches corresponds to the center of the circular sky patches in Tregenza’s model.
Figure 6. Circular patches proposed by Tregenza for discretizing the hemispherical sky structure. Credit: (Tregenza 1987)
The Reinhart sky model has since been further discretized by equally subdividing the original 145 patches. Discretizing the sky to a higher degree, at the expense of simulation runtime and disc memory, usually leads to more accurate illuminance calculations. This improvement in accuracy can mostly be attributed to the way in which luminance from the sun is considered in Daylight Coefficient based calculation methods. As shown in Figure 8, in discretized sky models the actual position of the sun in the sky at a given time is approximated to 3-4 sky patches. As is also evident from Figure 8, this approximation in position is accompanied by an overestimated assumption of the size of the sun with respect to the sky, especially in the case where a sky with 145 patches is considered.
The luminance values for the skies used in the Daylight Coefficient Method are usually derived from Typical Meteorological Year (TMY) weather data for different geographical locations (National Climatic Center 1981; Reinhart 2006). This data, usually available in the form of Energy Plus Weather (EPW) files, contains hourly values for Direct Normal and Diffuse Horizontal Radiation. The conversion from these irradiance to luminance values is done as per the luminous efficacy and distribution models proposed in (Perez and others 1990) and (Perez and others 1993) respectively.
Figure 7. The image (a) shows the Tregenza sky subdivison scheme with 145 sky patches. Image (b) shows the continuous sky subdivision scheme proposed by Reinhart. Credit: (Bourgeois and others 2007)
Figure 8. Fish-eye projections of a continuous sky model (a) and corresponding discretized versions. Images (b) , (c) and (d) contain 145, 580 and 2305 sky patches respectively. As is apparent in the images above, even with a high degree of discretization, the size of the sun is greatly overestimated in discrete sky models. Credit: (McNeil 2013c)
Calculation of illuminance using Daylight Coefficients has been possible with Radiance for more than a decade. John Mardaljevic’s dissertation contains an extensive discussion on the calculation, and subsequent validation, of Daylight Coefficients with Radiance (Mardaljevic 1999). A detailed account of the implementation of the Daylight Coefficients through Radiance-based programs can also be found in (Reinhart 2001) and (Bourgeois and others 2007). Bourgeois and others (2007) further discuss a more accurate implementation of the Daylight Coefficient Method by proposing a model that allows for 2305 direct solar positions.
The Daylight Coefficient method is implemented in Daysim (2015), an annual Daylighting simulation software. Several other software such as DIVA (Jakubiec and Reinhart 2011), SPOT (Rogers 2006) and Ladybug-Honeybee (Roudsari and Pak 2014) employ Daysim, or one of its derivatives, as the calculation engine for annual daylighting simulations.
A purely Radiance-based implementation of the Daylight Coefficient Method involves the use of matrix calculations. The equation for the Daylight Coefficient Method given in equation [1] can be expressed in terms of matrices as:
…………………………….[2]
where represents the Daylight Coefficient Matrix and represents the sky vector.
For example, assuming a room with 100 illuminance grid-points and a point-in-time Reinhart Sky with 145 patches, the matrix dimensions of and will be [100 x 145] and [145 x 1] respectively. The dimensions of the resultant illuminance matrix will be [100 x 1], where each data point will represent the illuminance at a particular grid-point. Expanding this example to an annual simulation and considering a sky vector for each of the 8760 (365 x 24) hours in a year, the dimensions of , and will be [100 x 145], [145 x 8760] and [100 x 8760] respectively. In this case, the values in E represent time series illuminance at the 100 grid points for 8760 hours a year. The workflow for Daylight Coefficient based simulations using native Radiance programs is discussed in Section 6.2 of Chapter 6.
The standard Daylight Coefficient method is suitable for models involving simple glazing and shading systems that can be modeled as simple glass or translucent materials in Radiance (as “glass” or “trans” primitives respectively). More complex type of glazing and shading systems can also be incorporated into Daylight Coefficient based calculations by modeling these materials using a Bidirectional Scattering Distribution Function (BSDF) primitive and then considering them to be a part of the overall scene2.
Often, the primary objective of daylighting simulations is to parametrically and iteratively evaluate only a certain aspect of the scene. This is especially the case if multiple daylighting simulations are performed to evaluate the performance of different types of glazings or shading systems while keeping everything else in the scene constant. In such instances, using the Daylight Coefficient method, which involves tracing rays from inside the room to the sky in a single step, becomes prohibitively expensive. The Three-Phase Method discussed in the next section is more suited for such simulations.
Considering the overall development timeline of Radiance, the functionality to incorporate BSDFs directly into scenes is a fairly recent development. More details can be found in Greg Ward’s presentation from the 102: th International Radiance Workshop (Ward 2011). ↑