Propagation of Light in the Atmospheric Environment

Abstract

Any daytime photometric measurements or observations outdoors reflect the momentary conditions of sunlight and skylight illuminance at ground level. The natural light sources determine the photic field adjusted by the atmospheric environment. Every constituent of the atmosphere, air molecules, aerosol and dust particles, water vapor droplets, and other gaseous compounds, forms the actual state of the photic field. Basically, the light propagated through the atmosphere undergoes scattering and absorption phenomena with different efficiencies across the visible spectrum. To characterize the daylight behavior it is necessary to understand the underlying physics, especially the radiative transfer of monochromatic radiation.

Appendix 4

4.1.1 Comparison of Trials to Measure and Model the Whole Range of the Indicatrix and Gradation Functions

Owing to the main influences of the momentary sunbeam direction and optical air mass, a homogeneous atmosphere can be characterized by two symmetrical functions determining the scattering and diffusion of skylight:

• 1. Under the clear sky conditions, the momentary air molecules and water vapor as well as aerosol particle content influence the scattering, and transmission properties of the atmospheric layers. Thus, extraterrestrial sunbeams are spread in all directions into space and the resulting luminance distribution can be characterized by quasi-symmetrical luminance solids that are usually determined by their cover curves called indicatrices. Principally three basic types of indicatrices according to the presence of turbidity and cloudiness can occur:

  • Under very clear and clean atmospheres the forward and backward scattering is roughly the same and close to the Rayleigh indicatrix.

  • Under higher turbidities and partly cloudy conditions the forward scattering is prolonged and backward scattering is reduced, forming only a small tail.

  • When dense cloudiness and/or fog scatter the sunbeams, so that the position of their original light source, i.e., the sun’s position in the sky is no longer detectable, the luminance solid has the shape of a sphere, and the indicatrices are uniform and the relative indicatrix function is equal to 1.

• 2. Owing to the thickness of the atmospheric cover of the globe, the second basic influence on the sky luminance distribution is caused by the atmospheric optical air mass through which the sunbeams have to pass. Of course, the smallest thickness is in the direction of the zenith, then gradually rising toward the horizon, but the vertical gradation of sky luminance characterized by the gradation function has several tendencies:

  • Under clear and clean atmospheres when the sky is blue, the denser atmospheric layers close to the horizon owing to their turbidity have usually higher luminance and cause the gradation function increase toward the horizon.

  • In special cases when the dense fog or diffuse cloudiness produces totally uniform sky luminance (Lambert sky), the gradation function becomes unity.

  • Dull overcast skies are extremely dense in the horizontal directions and so the zenith luminance is twice or 3 times as high than at the horizon. Thus, the gradation function has to show this vertical luminance drop prevailing and characteristic for overcast sky conditions. The drop is usually associated with a unity indicatrix.

Because in arbitrary atmospheres both these indicatrix and gradation functions have multiplying influences, their true measurements can be done only when one function is stable so the other can be separated.

The possibility to measure the indicatrix function on the solar almucantar, i.e., the horizontal circle containing the sun position, or using the sky almucantar at different altitude where scan data are available was described by Kittler (1969, 1993) in connection with the CIE trend to standardize the clear sky luminance distribution (Kittler 1967) which was adopted by CIE (1973).

Sky luminance measurements were manually conducted only seldom and even less frequently analyzed, (e.g., clear sky patterns were measured at Bratislava on selected clear mornings in 1961 by Kittler 1962 to find the scattering effects). In 1990–1991 much quicker special indicatrix scans were performed also manually with a portable luminance meter on a tripod enabling the rotation around the solar almucantar. However, neither simultaneous sky illuminances ${D_{\rm{v}}}$ nor zenith luminances ${L_{\rm{vZ}}}$were available; thus, the appropriate sky type after ${L_{\rm{vZ}}}/{D_{\rm{v}}}$ could not be identified or checked. A more sophisticated study deemed wise before sky luminance scans could be evaluated. The first American sky luminance measurements were also made manually, by Kimball and Hand (1921a, b), in Washington and in Chicago, and later measurements were made by Karayel et al. (1984) in San Francisco. From June 1985 to December 1986, data were recorded by a sky luminance scanner produced by Pacific Northwest Laboratories and used by an LBL team and are known as Berkeley scans. From roughly 16,000 all-sky scans, 88 typical sky conditions were selected for indicatrix and gradation analysis by Perez for US-SK project (Kittler et al. 1998). This early American scanner had a special scanning net with 10° vertical steps which was quite well suited for indicatrix and gradation analysis. Australian scans were recorded by a German PRC Krochmann scanner in Sydney (Hayman 1992) and Japanese scans from Tokyo (Igawa 1992) used an EKO scanner with the scanning net proposed by Tregenza (1987) with 12° vertical steps and various azimuth increments. Computer programs were provided for the American and European sky subdivisions and so during the research project Kittler et al. (1998) could analyze not only indicatrix functions, but also gradation functions from sky scans measured in Berkeley, Sydney, and Tokyo.

To derive scattering indicatrix courses from either luminance measurements along the sun or any sky almucantar or from a regular sky scan it is important to determine first the position of the normalizing sky element and its normalizing luminance ${L_{90^\circ }}$. For different scan systems the scan step angle in appropriate computer programs has to be respected in order to calculate interpolated data of $f(90^\circ )$ placement on the solar or sky almucantar, which is given generally after (A4.1) or (A4.2),

$$ \chi = \arccos (\sin \,\varepsilon \,\sin \,{\gamma_{\rm{s}}} + \cos \,\varepsilon \,\cos \,{\gamma_{\rm{s}}}\,\cos \,{A})\,(^\circ ), $$

(A4.1)

or when zenith angles are used,

$$ \chi = \arccos (\cos \,{Z_{\rm{S}}}\cos \,Z + \sin \,{Z_{\rm{S}}}\sin \,Z\,\cos \,{A })\,(^\circ ). $$

(A4.2)

Thus, for the normalizing function $f(90^\circ )$ on the solar almucantar where $\varepsilon = {\gamma_{\rm{s}}}$ or ${Z_{\rm{S}}} = Z$ with$\chi = 90^\circ$, i.e., $\cos \,\chi = 0$, the azimuth of this sky element taken from the sun meridian ${A }$ (Fig. A4.1) is

Fig. A4.1

Sun almucantars at different solar altitudes with ${L_{90^\circ }}$ and $f(90^\circ )$ positions

$$ {A_\varepsilon } = \arccos ( - {\tan^2}\,{\gamma_{\rm{s}}}) = \arccos ( - \cot \,a{n^2}{Z_{\rm{s}}})\,(^\circ ). $$

(A4.3a)

Because $\cos \,{A}$ has a minus value, the ${A }$ angle is in the second quadrant; therefore, e.g., the actual ${A } = 180^\circ - \arccos ( - {\tan^2}{\gamma_{\rm{s}}})$, i.e., larger than 90°.

For any sky almucantar with zenith angle $Z$ and scattering angle equal to 90°:

$$ {A } = \arccos \left[ {\frac{{ - \cos \,{Z_{\rm{s}}}}}{{\sin \,{Z_{\rm{s}}}}}\frac{{\cos \,Z}}{{\sin \,Z}}} \right] = \arccos ( - {\rm cotg} \,{Z_{\rm{s}}}\,{\rm cotg} \,Z)\,(^\circ ). $$

(A4.3b)

When sky scans are used for the indicatrix analyses, the regular scan step will usually not suit the needed ${A }$ azimuth distance from the solar meridian; thus, interpolation between nearest luminance readings has to be applied. After the normalizing luminance ${L_{90^\circ }}$ has been obtained, all measured luminances along the sun/sky almucantars can also be normalized as ${L_Z}/{L_{90^\circ }}$ to describe the indicatrix course.

Interpolation is needed also for the gradation function $\varphi (Z)$ as a more spatial distribution of luminances has to be taken into account. The assumption to determine the gradation function has to avoid the influence of indicatrices, i.e., the measured luminance on a section circle of the sky hemisphere which passes the zenith and has the same angular extent from the sun position has to be found. Such section circles according to different sun positions are indicated in Fig. A4.2, which assumes the section is placed through the actual solar meridian. To find the full span of the gradation function, the sky scans with relatively low sun heights have to be selected to satisfy the assumption of a constant value of $f(\chi )$ corresponding to $\chi = {Z_{\rm{S}}}$, and to obtain also the normalizing $\varphi (0^\circ )$function at the zenith corresponding to the normalizing zenith luminance ${L_{\rm{vZ}}}$. Owing to the assumption of the constant scattering influence on the section circle $\varphi ({Z_{\rm{s}}}) = \varphi (0^\circ )$ and thus

Fig. A4.2

Section circles for defining gradation courses

$\cos \,{Z_{\rm{s}}} = \cos \,{Z_{\rm{S}}}\,\cos \,Z + \sin \,{Z_{\rm{S}}}\,\sin \,Z\,\cos \,{A}$; thus,

$$ {A} = \arccos \left[ {\frac{{\cos \,{Z_{\rm{s}}}(1 - \cos \,Z)}}{{\sin \,{Z_{\rm{s}}}\,\sin \,Z}}} \right]\,(^\circ ). $$

(A4.4)

So for direct measurements and analysis of the indicatrix and gradation there is a disadvantage – for the high sun positions both functions are reduced and incomplete. But if such analysis has to be done using sky scans some error might also be caused by interpolation of luminances within the recorded scan net. To avoid these discrepancies, an ideal scanning step system could follow the solar almucantar and the “gradation circle” through the zenith and the same $\chi$ angle from the sun would be needed. The computer-regulated tracking along these circles indicated in Figs. A4.1 (in plan projection) and A4.2 (in section) could help the analysis precision.

To draw the indicatrix or gradation course either a polar plot (Fig. A4.3) resembling the luminance body or a semilogarithmic plot (Fig. A4.4), which seems to be more illustrative, can be used.

Fig. A4.3

Clear sky indicatrix in a polar plot with a linear scale

Fig. A4.4

Clear sky indicatrix in a semilogarithmic plot after measured luminance values

As an example, in the latter plot sky almucantar data were used to document the Berkeley clear sky case measured on June 13, 1985 at 5:41 p.m. archived as 164/85 (the isoline luminance map made from scan data is shown in Fig. A4.5). The indicatrix analysis shown in Fig. A4.4 was done for six sky almucantars with zenith angles $\,Z = 30^\circ - 80^\circ$ in 10° steps. It is evident that under homogeneous conditions all indicatrix formations are in a condensed spread area with points close to each other, whether lower- or higher-placed sky almucantars are analyzed. However, the lower the sky almucantar and closer to sun position, the longer the forward and backward indicatrix course as is seen from the comparison of the red and cyan marks.

Fig. A4.5

Example of an isoline interpretation of a luminance scan

For the same scan also relative gradation was analyzed, showing almost a linear course in the polar plot in Fig. A4.6 but the real rather steep increase of luminance toward the horizon is evident in the semilogarithmic diagram in Fig. A4.7.

Fig. A4.6

Clear sky gradation in a polar plot

Fig. A4.7

Clear sky luminance gradation in a semilogarithmic plot

Further examples comparing measured indicatrix and gradation courses in typical cases of other clear and overcast skies derived from Berkeley scans were shown by Kittler et al. (1996). Similar analysis within a US–Slovak project (Kittler et al. 1998) were made from further selected sky scan measurements recorded in Berkeley as well as in Tokyo and Sydney. Further comparisons with the standard indicatrix and gradation functions can be found in the PhD thesis of Markou (2006), who analyzed sky scans measured in Watford, England, whereas Kobav (2006, 2009) utilized his own sky scanning data gathered in Lyon, France.

In addition to several historic measurements and approximations defining the gradation decrease on the overcast sky, e.g., by Schramm (1901) and (Kähler 1908), or Moon and Spencer (1940), few tried to determine the changes of the gradation function in the whole range, i.e., from the overcast to the clear homogeneous sky. A trial was published by Igawa et al. (1997) with the approximation following a sine function (4.18) or cosine function (4.19), i.e.,

$$ \frac{{\varphi (\varepsilon )}}{{\varphi (90^\circ )}} = [1 + {a_i}(1 - {\sin^{0.6}}\varepsilon )]{\hbox{ or }}\frac{{\varphi (Z)}}{{\varphi (0^\circ )}} = [1 + {a_i}(1 - {\cos^{0.6}}Z)], $$

(A4.5)

where ${a_i}$ was defined as the function of the normalized globe illuminance ${N_{\rm{evg}}}$ (see (A4.6), (A4.7)) which roughly follows the corresponding typical range of skies from clear (${N_{\rm{evg}}} = 0.9$) to overcast (${N_{\rm{evg}}} = 0.15$).

The original normalization by Matsuzawa et al. (1997) was based on the highest correlation and lowest standard deviation between the sky luminance and global illuminance levels. This correlation was based on the model of the global illuminance under clear sky conditions and the normalization was set as the ratio

$$ {N_{\rm{evg}}} = \frac{{{E_{\rm{vgm}}}}}{{{{E_{\rm{vgms}}}}}({\gamma_{\rm{s}}})}, $$

(A4.6)

where ${E_{\rm{vgms}}}$ as a function of solar altitude is defined in (A5.6) and ${E_{\rm{vgm}}}$ is the so-called relative global illuminance similar to ${G_{\rm{v}}}/{E_{\rm{v}}}$ but defined as

$$ {E_{\rm{vgm}}} = \frac{{m{G_{\rm{v}}}}}{\hbox{LSC}} $$

(A4.7)

and ${a_i} = - 1.334N_{\rm{evg}}^4 + 2.32N_{\rm{evg}}^3 + 4.032N_{\rm{evg}}^2 - 0.591{N_{\rm{evg}}} - 1.$

At the same time, an analysis of various luminance scans by subtracting the gradation function was done by Kittler et al. (1997) and six typical exponential functions with parameters $a$ and $b$ after (A4.9) were proposed for the sky luminance patterns now adopted by the ISO/CIE (2003, 2004) standards shown in Table A4.1. The comparison of three approximations is shown in Fig. A4.8, also with the simple CIE (1955) overcast sky gradation formula.

Table A4.1 Standard a and b parameters for gradation functions $\varphi (Z)$of typical homogeneous cloudiness and turbidity atmospheric conditions for use of (A4.9)

Fig. A4.8

Graphical comparison of approximated gradation functions

Later Igawa and Nakamura (2001) altered the gradation function concept to the exponential fit

$$ \varphi (\varepsilon ) = \frac{1}{{1 + {a_1}/{\hbox{exp(}}{a_2}\,\sin \,\varepsilon {)}}}, $$

(A4.8)

where ${a_1} = 5.5$ and ${a_2} = 1.82N_{\rm{evg}}^2 - 5.82{N_{\rm{evg}}} + 2.26.$.

However, the authors were probably disappointed with both previous gradation functions and therefore applied the ISO/CIE gradation function fully in Igawa et al. (2004).

All Igawa’s studies indicated the link of luminance patterns to global illumination level with connection to gradation and indicatrix parameters, so if no ${L_{\rm{vZ}}}$ data are available to define the classification ratio ${L_{\rm{vZ}}}/{D_{\rm{v}}}$, the ${G_{\rm{v}}}/{G_{\rm{vc}}}$ ratio could serve as a sky luminance classification indicator if ${G_{\rm{vc}}}$ is the horizontal global illuminance under the CIE clear sky standard with ${T_{\rm{v}}} = 2$. Kittler and Darula (2000) documented that only the ${N_{\rm{evg}}}$ or ${G_{\rm{v}}}/{G_{\rm{vc}}}$ values without further information on the simultaneous solar altitude ${\gamma_{\rm{s}}}$ and turbidity ${T_{\rm{v}}}$ cannot replace fully the classification ratio ${L_{\rm{vZ}}}/{D_{\rm{v}}}$.

After studies testing gradations for every sky type by analyzing selected sky scan measurements obtained in Berkeley, in Tokyo, and in Sydney within the US–Slovak project (Kittler et al. 1998), the relative gradation function was chosen for the ISO/CIE (2003, 2004) standards after

$$ \frac{{\varphi (Z)}}{{\varphi (0^\circ )}} = \frac{{{1 } + { }a\,{\text{exp (}}b{\text{/cos}}\,{ }Z{)}}}{{{1 } + { }a\,{ \exp }\,b}}. $$

(A4.9)

The standard relative gradation and indicatrix function is respected in all standard sky distribution models. The only exception is in the Perez exponential indicatrix relation, where the missing component $- { \exp }({\hbox{d}}\pi /2)$ in (4.29) distorts the normalization for the relative indicatrix $f(\chi ) = {L_\chi }/{L_{90^\circ }}$ as well as the relative $f\left( {90^\circ } \right) = 1$ value.