| 9 | | Exweighted can be calculated by integrating the spectral radiance weighted with a biological action spectrum and the geometry of a human over all azimuth and zenith angles of the upper hemisphere and the exposure time. The complete equation of Exweighted, considering all dependencies, is complex and the calculation can be quite demanding. Therefore, multiple simplifying assumptions have been made, which are described in detail in Seckmeyer et al. 2013. The simpliffed equation of Exweighted, used in this model, is given as: |
| | 9 | Ex,,weighted,, can be calculated by integrating the spectral radiance weighted with a biological action spectrum and the geometry of a human over all azimuth and zenith angles of the upper hemisphere and the exposure time. The complete equation of Ex,,weighted,,, considering all dependencies, is complex and the calculation can be quite demanding. Therefore, multiple simplifying assumptions have been made, which are described in detail in Seckmeyer et al. 2013. The simpliffed equation of Ex,,weighted,,, used in this model, is given as: |
| 18 | | The spectral radiance L is defined as Formel, |
| 19 | | which depends on the wavelength lambda, the time t, the azimuth angle and the incident angle, which is defined as the angle between incident light beam and horizontal plane. In addition, dQ represents the radiant energy, dA the area element and dOmega the solid angle. For a receiver that is not orientated normal to the source, the area element dA must be weighted with the cosine of the angle between the direction of the beam and the normal to the area dA (WMO, 2008; CIE, 2011). The radiance can be a source or a receiver based quantity. However, in this model the radiance is used as a receiver based quantity only. This can be best visualized by a reversed cone with the given solid angle d as its base and the vertex on the area dA. See schematic diagram in Figure 1.1. |
| 20 | | In comparison the spectral irradiance E_lambda is defined as the radiant energy dQ arriving per time interval dt, per wavelength interval d_ and per area dA from any origin incident onto a horizontally oriented area element (Seckmeyer et al., 2010) The quantity irradiance is sometimes also referred to as radiative flux. |
| | 18 | The spectral radiance L is defined as: |
| | 19 | {{{ |
| | 20 | #!Latex |
| | 21 | \begin{equation*} |
| | 22 | L_{\lambda}\;=\; \frac{\text{d}Q}{\text{d}t\cdot \text{d}\lambda \cdot \text{d}A\cdot cos\,\alpha\cdot \text{d}\Omega}, \quad \left[\frac{\text{W}}{\text{m}^{2} \;\text{nm} \;\text{sr}}\right]. |
| | 23 | \end{equation*} |
| | 24 | }}} |
| | 25 | which depends on the wavelength ''λ'', the time t and the incident angle which is defined as the angle between incident light beam and horizontal plane. In addition, dQ represents the radiant energy, dA the area element and d''Ω'' the solid angle. For a receiver that is not orientated normal to the source, the area element dA must be weighted with the cosine of the angle between the direction of the beam and the normal to the area dA (WMO, 2008; CIE, 2011). The radiance can be a source or a receiver based quantity. However, in this model the radiance is used as a receiver based quantity only. This can be best visualized by a reversed cone with the given solid angle d''Ω'' as its base and the vertex on the area dA. See schematic diagram in Figure 1.1. |
| | 26 | In comparison the spectral irradiance E_lambda is defined as the radiant energy dQ arriving per time interval dt, per wavelength ''λ'' and per area dA from any origin incident onto a horizontally oriented area element (Seckmeyer et al., 2010) The quantity irradiance is sometimes also referred to as radiative flux. |
| 32 | | In Figure 2.1, the simulated (diffuse) sky radiance weighted with the vitamin D3 action spectrum is shown for Hannover on 21 March at solar noon. The radiance in Figure 2.1 is visualized as a polar plot, where the zenith is located in the center and the azimuth angles are marked around the plot. It should be noted, that similar to an astronomical map, the directions of east and west are inverted.\\\\ |
| | 38 | In Figure 2.1, the simulated (diffuse) sky radiance weighted with the vitamin D,,3,, action spectrum is shown for Hannover on 21 March at solar noon. The radiance in Figure 2.1 is visualized as a polar plot, where the zenith is located in the center and the azimuth angles are marked around the plot. It should be noted, that similar to an astronomical map, the directions of east and west are inverted.\\\\ |
| 49 | | Figure 2.2: (a) Projection of the 3D voxel model with winter clothing, visualized for incident angles 30_, 60_ and 85_, with the front turned by 30_ in azimuth direction. Only hands and face are exposed to UV radiation, which is shown in light gray color and clothing shown in dark gray. (b)-(d) projection areas of the 3D voxel model oriented towards 180_/south, as a function of incident and azimuth angles. The minimal projection area in each plot is located in the middle of each picture, representing a view from the zenith at an incident angle of 90_. (b) Projection area of a human with winter clothing. The three asterisks mark the projections shown in (a). (c) Projection area of a human without any clothing, which results in nearly identical projection areas between the front and back of the human. (d) Projection area of a human with summer clothing, where face, hands, neck and arms are exposed.\\ |
| | 55 | Figure 2.2: (a) Projection of the 3D voxel model with winter clothing, visualized for incident angles 30°, 60° and 85°, with the front turned by 30° in azimuth direction. Only hands and face are exposed to UV radiation, which is shown in light gray color and clothing shown in dark gray. (b)-(d) projection areas of the 3D voxel model oriented towards 180° (south), as a function of incident and azimuth angles. The minimal projection area in each plot is located in the middle of each picture, representing a view from the zenith at an incident angle of 90°. (b) Projection area of a human with winter clothing. The three asterisks mark the projections shown in (a). (c) Projection area of a human without any clothing, which results in nearly identical projection areas between the front and back of the human. (d) Projection area of a human with summer clothing, where face, hands, neck and arms are exposed.\\ |
| 58 | | To calculate the human exposure the weighted radiances Lweighted("; '; t) must be integrated over all directions of the upper hemisphere and the exposure period t. In order to receive the total human exposure, the biologically-weighted direct normal irradiance multiplied with the projection area of the corresponding sun position is added to Exweighted. |
| 59 | | In order to convert the calculated human exposure from Joule into IU, a conversion factor of 70.97 [IU J^-1] (i.e. IU per Joule of vitamin D3-weighted UV) is used (Seckmeyer et al. 2013) |
| | 64 | To calculate the human exposure the weighted radiances must be integrated over all directions of the upper hemisphere and the exposure period t. In order to receive the total human exposure, the biologically-weighted direct normal irradiance multiplied with the projection area of the corresponding sun position is added to the diffuse exposure Ex,,weighted,,. |
| | 65 | In order to convert the calculated human exposure from Joule into IU, a conversion factor of 70.97 [IU J^-1^] (i.e. IU per Joule of vitamin D,,3,,-weighted UV) is used (Seckmeyer et al. 2013) |
