# How solar radiation is calculated

The solar radiation analysis tools calculate insolation across a landscape or for specific locations, based on methods from the hemispherical viewshed algorithm developed by Rich et al. (Rich 1990, Rich et al. 1994) and further developed by Fu and Rich (2000, 2002).

The total amount of radiation calculated for a particular location or area is given as global radiation. The calculation of direct, diffuse, and global insolation are repeated for each feature location or every location on the topographic surface, producing insolation maps for an entire geographic area.

## Solar radiation equations

### Global radiation calculation

Global radiation (Global_{tot}) is calculated as the sum of direct (Dir_{tot}) and diffuse (Dif_{tot}) radiation of all sun map and sky map sectors, respectively.

Global_{tot}= Dir_{tot}+ Dif_{tot}

### Direct solar radiation

Total direct insolation (Dir_{tot}) for a given location is the sum of the direct insolation (Dir_{θ,α}) from all sun map sectors:

Dir_{tot}= Σ Dir_{θ,α}(1)

The direct insolation from the sun map sector (Dir_{θ,α}) with a centroid at zenith angle (θ) and azimuth angle (α) is calculated using the following equation:

Dir_{θ,α}= S_{Const}* β^{m(θ)}* SunDur_{θ,α}* SunGap_{θ,α}* cos(AngIn_{θ,α}) (2)

- where:
- S
_{Const}—The solar flux outside the atmosphere at the mean earth-sun distance, known as solar constant. The solar constant used in the analysis is 1367 W/m^{2}. This is consistent with the World Radiation Center (WRC) solar constant. - β—The transmissivity of the atmosphere (averaged over all wavelengths) for the shortest path (in the direction of the zenith).
- m(θ)—The relative optical path length, measured as a proportion relative to the zenith path length (see equation 3 below).
- SunDur
_{θ,α}—The time duration represented by the sky sector. For most sectors, it is equal to the day interval (for example, a month) multiplied by the hour interval (for example, a half hour). For partial sectors (near the horizon), the duration is calculated using spherical geometry. - SunGap
_{θ,α}—The gap fraction for the sun map sector. - AngIn
_{θ,α}—The angle of incidence between the centroid of the sky sector and the axis normal to the surface (see equation 4 below).

- S

Relative optical length—m(θ)—is determined by the solar zenith angle and elevation above sea level. For zenith angles less than 80°, it can be calculated using the following equation:

m(θ) = EXP(-0.000118 * Elev - 1.638*10^{-9}* Elev^{2}) / cos(θ) (3)

- where:
- θ—The solar zenith angle.
- Elev—The elevation above sea level in meters.

The effect of surface orientation is taken into account by multiplying by the cosine of the angle of incidence. Angle of incidence (AngInSky_{θ,α}) between the intercepting surface and a given sky sector with a centroid at zenith angle and azimuth angle is calculated using the following equation:

AngIn_{θ,α}= acos( Cos(θ) * Cos(G_{z}) + Sin(θ) * Sin(G_{z}) * Cos(α-G_{a}) ) (4)

- where:
- G
_{z}—The surface zenith angle.Note that for zenith angles greater than 80°, refraction is important.

- G
_{a}—The surface azimuth angle.

- G

### Diffuse radiation calculation

For each sky sector, the diffuse radiation at its centroid (Dif) is calculated, integrated over the time interval, and corrected by the gap fraction and angle of incidence using the following equation:

Dif_{θ,α}= R_{glb}* P_{dif}* Dur * SkyGap_{θ,α}* Weight_{θ,α}* cos(AngIn_{θ,α}) (5)

- where:
- R
_{glb}—The global normal radiation (see equation 6 below). - P
_{dif}—The proportion of global normal radiation flux that is diffused. Typically it is approximately 0.2 for very clear sky conditions and 0.7 for very cloudy sky conditions. - Dur—The time interval for analysis.
- SkyGap
_{θ,α}—The gap fraction (proportion of visible sky) for the sky sector. - Weight
_{θ,α}—The proportion of diffuse radiation originating in a given sky sector relative to all sectors (see equations 7 and 8 below). - AngIn
_{θ,α}—The angle of incidence between the centroid of the sky sector and the intercepting surface.

- R

The global normal radiation (R_{glb}) can be calculated by summing the direct radiation from every sector (including obstructed sectors) without correction for angle of incidence, then correcting for proportion of direct radiation, which equals 1-P_{dif}:

R_{glb}= (S_{Const}Σ(β^{m(θ)})) / (1 - P_{dif}) (6)

For the uniform sky diffuse model, Weight_{θ,α} is calculated as follows:

Weight_{θ,α}= (cosθ_{2}- cosθ_{1}) / Div_{azi}(7)

- where:
- θ
_{1}and θ_{2}—The bounding zenith angles of the sky sector. - Div
_{azi}—The number of azimuthal divisions in the sky map.

- θ

For the standard overcast sky model, Weight_{θ,α} is calculated as follows:

Weight_{θ,α}= (2cosθ_{2}+ cos2θ_{2}- 2cosθ_{1}- cos2θ_{1}) / 4 * Div_{azi}(8)

Total diffuse solar radiation for the location (Dif_{tot}) is calculated as the sum of the diffuse solar radiation (Dif) from all the sky map sectors:

Dif_{tot}= Σ Dif_{θ,α}(9)

## References

Fu, P. 2000. A Geometric Solar Radiation Model with Applications in Landscape Ecology. Ph.D. Thesis, Department of Geography, University of Kansas, Lawrence, Kansas, USA.

Fu, P., and P. M. Rich. 2000. The Solar Analyst 1.0 Manual. Helios Environmental Modeling Institute (HEMI), USA.

Fu, P., and P. M. Rich. 2002. "A Geometric Solar Radiation Model with Applications in Agriculture and Forestry." Computers and Electronics in Agriculture 37:25–35.

Rich, P. M., R. Dubayah, W. A. Hetrick, and S. C. Saving. 1994. "Using Viewshed Models to Calculate Intercepted Solar Radiation: Applications in Ecology. American Society for Photogrammetry and Remote Sensing Technical Papers, 524–529.

Rich, P. M., and P. Fu. 2000. "Topoclimatic Habitat Models." Proceedings of the Fourth International Conference on Integrating GIS and Environmental Modeling.