Sky Luminance Characteristics

Abstract

Any sunbeam or directionally uniform luminous flux formed by parallel rays reaching the atmospheric border will strike tiny air molecules, aerosol particles, or water vapor droplets, which will cause its absorption, scattering, diffusion, and reflection into space. This phenomenon was studied earlier by Bouguer, but Weber (1885) was probably the first to suggest that the resulting luminance distribution into space should be characterized by an irregular luminance solid formed by directional elemental luminance in different directions relative to the original beam. It was assumed that such solids could be usually rotationally symmetrical around the original beam as an axis and could be specified by their sectional curve, called the scattering indicatrix.

Appendix 5

5.1.1 Comparison of Basic Approaches and Approximations for Defining the Sky Luminance Patterns

Basic for all relevant sky models defining their luminance patterns are the gradation and indicatrix functions, which predetermine their luminance distribution under actual sun positions and the state of the atmosphere. The first luminance models of Fesenkov (1955) indicated the possible separation of both these functions assuming only the first degree of diffusion in the arbitrary volume of the atmosphere. Several refinements were proposed by Sobolev (1943, 1949, 1972) and Kittler (1967, 1985, 1986), but unfortunately the more complex formulae did not result in higher precision.

Some approximations and simplifications were accepted to determine practical sky models for daylighting purposes. Whereas under overcast conditions only the gradation function has to be simulated by different formulae as described in the appendix in Chap. 4, under cloudy and clear skies both functions are important with a multiplying effect. The indicatrix function was applied in different models in its absolute form $s\left( \chi \right)$ after (5.5) only rarely for clear sky models (Nagata 1971, 1983), whereas many others applied the relative indicatrix function normalized to the luminance perpendicular to sunbeams, i.e., $f\left( \chi \right) = {L_{{\rm{\chi }}}}/{L_{{90^\circ }}}$ after (5.6). In this respect many relevant recent sky luminance models (Perez et al. 1993; Kittler et al. 1997, 1998; Igawa et al. 2001, 2004; ISO 2004) have applied the same exponential relations to model the gradation and relative indicatrix functions after (5.21) and (5.23), respectively. The ISO (2004) standard defines the parameters $a\;{\hbox{and}}\;b$ for the gradation functions, $c,\,d$, and $e$ parameters for the relative indicatrix functions for every one of the 15 standard sky types after Kittler et al. (1997, 1998), whereas in both the Perez and Igawa models these parameters are calculated after different indicators, e.g., Perez et al. (1993) specified within the eight ranges of the clearness after four model coefficients for determining the ${a_{\rm{p}}} - {e_{\rm{p}}}$ parameters in the relation containing also the influence of the sky brightness. So the procedure to find the suitable gradation and indicatrix functions is quite complex and tedious.

Sky luminance distribution modeling was followed by Perez et al. (1993a), applying the first measurement results from the multipurpose scanning photometer (Perez et al. 1992) recorded in Berkeley. The data set taken between June 1985 and December 1986 comprised more than 16,000 sky scans. The so-called all-weather model used the ISO/CIE gradation function and a distorted indicatrix function excluding the normalizing member $- \exp \left( {d\pi /2} \right)$ in (5.23). This model is based on the two-parameter description, the parameters of sky clearness ∈ and sky brightness $\Delta$ (Perez et al. 1983), which included the horizontal diffuse irradiance ${D_{\rm{e}}}$, the zenith angle of the sun position ${Z_{\rm{s}}}$, the optical air mass m, the solar constant (${\hbox{SC}}$) and normal parallel beam irradiance ${P_{\rm{en}}}$ as

$$ \in = \frac{{\left( {{D_{{{\rm{e}}\,}}} + {P_{\rm{en}}}} \right)/\,{D_{\rm{e}}} + 1.041\,{Z_{\rm{s}}}^3}}{{1 + 1.041\,{Z_{\rm{s}}}^3}}, $$

(A5.1)

$$ \Delta = \frac{{m{D_{\rm{e}}}}}{\hbox{SC}}. $$

(A5.2)

After these basic two parameters, also Perez’s ${a_{\rm{P}}},\,{b_{\rm{P}}},\, {c_{\rm{P}}},\,{d_{\rm{P}}}$, and ${e_{\rm{P}}}$ subparameters for the gradation and indicatrix functions have to be calculated after separate relations and subcoefficients in eight different ranges of sky clearness.

For zenith luminance prediction, Perez et al. (1990) proposed an approximation model with parameters ${a_i},\,{ }{c_i},\, { }{c\prime_i},\,{\hbox{ and }}{d_i}$ prescribed for each of the eight sky clearness ∈ ranges in a table to be used in the approximation

$$ {L_{\rm{vZ}}} = {D_{\rm{e}}}\,\left[ {{a_i} + {c_i}\cos Z + {{c\prime}_i}\exp \left( { - 3Z} \right) + \Delta {d_i}} \right]\,({\hbox{kcd}}/{{\hbox{m}}^{{2}}}). $$

(A5.3)

The calculation of ${L_{\rm{vZ}}}$ is based on the pseudo efficacy (kcd/W) to get the zenith luminance expressed in kilocandelas per square meter.

In both the Perez et al. (1993) and Igawa et al. (2004) modeling procedures for momentarily measured irradiance, at least ${G_{\rm{e}}}$ and ${D_{\rm{e}}}$ as well as the simultaneously registered solar zenith angle ${Z_{\rm{s}}}$ have to be available.

In all Igawa’s models, gradation and indicatrix functions and auxiliary relations were determined by regression analysis depending on the momentary solar altitude and the so-called normalized global illuminance ${N_{\rm{evg}}}$ (Igawa et al. 1997).

The normalization was set as a ratio

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

(A5.4)

where $\,{E_{\rm{vgm}}}$ is the so-called relative global illuminance, which is similar to ${G_{\rm{v}}}/{E_{\rm{v}}}$ but is defined as

$$ {E_{\rm{vgm}}} = m\,{G_{\rm{v}}}/{\hbox{LSC}}. $$

(A5.5)

${E_{\rm{vgms}}}\left( {{\gamma_{\rm{s}}}} \right)$ is the relative global illuminance of the CIE standard clear sky under atmospheric transmittance 0.75 estimated in the regression analysis as dependent on the solar altitude γ _s by the equation (Igawa et al. 1997)

$$ {E_{\rm{vgms}}}\left( {{\gamma_{\rm{s}}}} \right) = 0.19 + 2.09{\gamma_{\rm{s}}} - 2.581{\gamma_{\rm{s}}}^2 + 1.486{\gamma_{\rm{s}}}^3 - 0.323{\gamma_{\rm{s}}}^4, $$

(A5.6)

previously determined by Matsuzawa et al. (1997) as

$$ \eqalign{ {E_{\rm{vgms}}}\left( {{\gamma_{\rm{s}}}} \right) = 0.197 + 1.943{\gamma_{\rm{s}}} - 2.376{\gamma_{\rm{s}}}^2 \hfill \\\quad \quad \quad \quad + 1.327{\gamma_{\rm{s}}}^3 - 0.232{\gamma_{\rm{s}}}^4 - 0.031{\gamma_{\rm{s}}}^5. \hfill \\}<!endgathered> $$

(A5.7)

The comparison of the Japanese gradation functions with the ISO/CIE gradation functions is given in Fig. 5.4, and indicatrix functions are compared in Fig. A5.1.

Fig. A5.1

Comparison of ISO/CIE indicatrix functions with those of Igawa et al. (1997)

Later, more measured data were obtained in Japan using a sun tracker (Fig. A5.2) and an EKO scanner (Fig. A5.3).

Fig. A5.2

Japanese sun tracker

Fig. A5.3

EKO sky luminance scanner. (Photo by Igawa)

A new version of a model based on radiance distribution was published (Igawa et al. 2004), where differently determined parameters based on global irradiance were given (e.g., ${\hbox{Ce,}}\;{\hbox{Cle,}}\;{\hbox{Seeg,}}\;{\hbox{Si}}$), but the ISO/CIE gradation and indicatrix functions were respected. There are a few weak points in applying the latest Japanese method. The main weakness is the basic dependence on ${G_{\rm{e}}}$ or ${D_{\rm{e}}}$, which have to be obtained either from a meteorological network or an International Daylight Measurement Programme (IDMP) station. Furthermore, the sky luminance distribution is linked more closely to ${D_{\rm{e}}}$ or ${D_{\rm{v}}}$, influenced directly by the luminance patterns, whereas parallel sunbeams in ${D_{\rm{v}}}$ or ${G_{\rm{v}}}$ distort the problematic luminous efficacy of ${D_{\rm{v}}}/{D_{\rm{e}}}$ or ${G_{\rm{v}}}/{G_{\rm{e}}}$ at different solar altitudes. The complex multiple normalizing system introduced by Igawa et al. (2004) is too complicated for practical use because it produces an infinite number of sky patterns linked to momentary or daily changing ${G_{\rm{e}}}$ or ${G_{\rm{v}}}$ and ${D_{\rm{e}}}$ or ${D_{\rm{v}}}$ measured values interrelated by the luminous efficacy.

To address the question of whether luminous efficacy could be a representative parameter for sky type classification, two examples of the monthly occurrence were evaluated comparing prevailingly overcast and clear situations in already documented months, i.e., for August 2001 in Fig. A5.4 and for November 1995 in Fig. A5.5.

Fig. A5.4

Global luminous efficacy spread dependent on turbidity differences

Fig. A5.5

Global luminous efficacy under overcast skies

ISO/CIE clear sky types 11–15 differ in the various turbidities expressed by the luminous turbidity factor ${T_{\rm{v}}}$, and great variance of the luminous efficacy was found, with the highest under ${T_{\rm{v}}} = 2$, but with rising turbidity the variance is suppressed. Probably that it is an effect of the luminous efficacy of sunbeams as well as the diffusivity of the skylight, which is less sensitive to solar altitude changes in the turbid atmosphere.

Data for the global luminous efficacy ${\hbox{Eff}_{\rm g}} = {G_{\rm{v}}}/{G_{\rm{e}}}$ plotted against solar altitude γ _s under the overcast sky type (ISO/CIE sky type 1) show that there is no significant dependence on solar altitude and the values obtained vary by ±10 lm/W around ${\hbox{Eff}_{\rm g}} = 115\;{\hbox{lm/W}}$.

However, the dependence of these models on the inaccurate luminous efficacy is problematic and questionable.

It is true that in reality only on very seldom occasions are repeated sky luminance patterns, either homogeneous or nonhomogeneous, exactly the same, but these can result in a very similar horizontal irradiance or illuminance level measured at ground level. However, creating the sky gradation and indicatrix functions after such scalar values is very doubtful and cannot predict typical distribution patterns.

The magnitude of skylight is the portion of extraterrestrially available sunlight scattered penetrating the atmosphere expressed by the ratio ${D_{\rm{e}}}/{E_{\rm{e}}}$ in irradiance unit or ${D_{\rm{v}}}/{E_{\rm{v}}}$ in illuminance units. Using the ratio ${D_{\rm{e}}}/{E_{\rm{e}}}$ for predicting sky luminance models assumes the application of luminous efficacy recalculation, which can be inaccurate when diffuse skylight is taken as an average efficacy of 120 lm/W, whereas the extraterrestrial ${\hbox{LSC}}/{\hbox{SC}}$ is only 97.6 lm/W (Darula et al. 2005), as

$$ \frac{{{D_{\rm{e}}}}}{{{E_{\rm{e}}}}} = \frac{{{D_{\rm{e}}}}}{{{\hbox{SC}}\,\sin {\gamma_{\rm{s}}}}}, $$

(A5.8)

$$ \frac{{{D_{\rm{v}}}}}{{{E_{\rm{v}}}}} = \frac{{{D_{\rm{v}}}}}{{{\hbox{LSC}}\,\sin {\gamma_{\rm{s}}}}}. $$

(A5.9)

In a slightly alternative form, it was used by Perez et al. (1993) as the sky’s brightness after (A5.2):

$$ \Delta = m\frac{{{D_{\rm{e}}}}}{\hbox{SC}}. $$

(A5.10)

The difference between (A5.8) and (A5.10) is very small except at very low solar altitudes, roughly under 15°, as the optical air mass $m$ deviates from the $\,1/\sin \,{\gamma_{\rm{s}}}$ values. The argument that parallel sunbeams penetrating the layer of Earth’s atmosphere at ${\gamma_{\rm{s}}} = 0^\circ$ produce some normal irradiance reduced by the $m$ value cannot mean that their influence on the horizontal plane is not equal to zero.

Similarly, Igawa and Nakamura (2001) used instead of ${D_{\rm{e}}}/{E_{\rm{e}}}$ the so-called relative global illuminance in the first step of parameterization:

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

(A5.11)

In Igawa’s earlier studies, his normalization parameter was the value of ${N_{\rm{evg}}}$[see (A5.4)], but later he introduced a new normalization factor – the so-called standard global illuminance, which was defined by the best-fit equation

$$ \eqalign{ {S_{\rm{evg}}}\left( {{\gamma_{\rm{s}}}} \right) = - 36.78{\gamma_{\rm{s}}}^5 + 188.79\,{\gamma_{\rm{s}}}^4 - 375.95{\gamma_{\rm{s}}}^3 \hfill \\\quad \quad \quad \quad + 306.2{\gamma_{\rm{s}}}^2 + 15.47{\gamma_{\rm{s}}} + 0.83\,({\hbox{klx}}){.} \hfill \\}<!endgathered> $$

(A5.12)

In Igawa et al. (2004), a similar standard global irradiance assuming a clear sky with the Linke turbidity factor ${T_{\rm{L}}} = 2.5$ was used and defined as

$$ {\hbox{Seeg}} = \frac{{0.84{\text{SC}}}}{m}\exp \left( { - 0.0675m} \right)\,({\hbox{W/}}{{\hbox{m}}^{{2}}}), $$

(A5.13)

as a clear sky index:

$$ {\hbox{Kc}} = {G_{\rm{e}}}/{\hbox{Seeg}}. $$

(A5.14)

Thus, the basic parameter in the Igawa and Nakamura (2001) model is called normalized global illuminance:

$$ {N_{\rm{evg}}} = {G_{\rm{v}}}/{S_{\rm{evg}}}\left( {{\gamma_{\rm{s}}}} \right), $$

(A5.15)

which is used to calculate the gradation and indicatrix coefficients and thus their functions.

However, all these changes in the determination of the normalization did not affect the arguments of Kittler and Darula (2000) that only the ${N_{\rm{evg}}}$ values without further information on the simultaneous solar altitude ${\gamma_{\rm{s}}}$ and turbidity ${T_{\rm{v}}}$ cannot replace fully the classification ratio ${L_{\rm{vZ}}}/{D_{\rm{v}}}$ as well as the approximation owing to luminous efficacy of global horizontal irradiance ${G_{\rm{e}}}$ or ${\hbox{Seeg}}$.

Expression (5.20) for the relative luminance distribution is used in a similar form, but instead of the zenith angular distance, elevation angles from the horizon are used; thus,

$$ \frac{{{L_{{{\rm{\chi Z}}}}}}}{{{L_{\rm{vz}}}}} = \frac{{f(\chi )\,\varphi (Z)}}{{f\left( {{Z_{\rm{s}}}} \right)\;\varphi ({0^{ \circ }})}} = \frac{{f\left( \chi \right)\,\varphi \,\left( \gamma \right)}}{{f\left( {\pi /2 - {\gamma_{\rm{s}}}} \right)\,\varphi \,\left( {\pi /2} \right)}}. $$

(A5.16)

The sky radiance distribution was also determined in Igawa et al. (2004), but the gradation and indicatrix functions need an even more complex calculation procedure, as used to determine the gradation and indicatrix coefficients in Igawa and Nakamura (2001) after the sky index ${\hbox{Si}}$

$$ {\hbox{Si}} = {\hbox{Kc}} + {\hbox{Cl}}{{\hbox{e}}^{{0.5}}}, $$

(A5.17)

where ${\hbox{Kc}}$ after (A5.14) is used and the cloudless index first applied in Perradeau’s (1988) sky model is

$$ {\hbox{Cle}} = \frac{{1 - {D_{\rm{e}}}/{G_{\rm{e}}}}}{{1 - {\hbox{C}}{{\hbox{e}}_{\rm{s}}}}}, $$

(A5.18)

where the mean clear sky theoretical cloud ratio value is

$$ \eqalign{ {\hbox{C}}{{\hbox{e}}_{\rm{s}}} = 0.01299 + 0.07698m - 0.003857{m^2} + 0.0001054{m^3} \hfill \\\quad \quad - 0.000001031{m^4}. \hfill \\}<!endgathered> $$

(A5.19)

To determine the gradation and indicatrix coefficients $\,{a_i} - {e_i}$, Igawa et al. (2004) proposed approximation relations dependent on the sky index $\,{\hbox{Si}}$, which is also used to classify five basic sky types in ${\hbox{Si}}$ ranges as follows:

• Overcast sky if ${\hbox{Si}}$ is lower than 0.3.

• Nearly overcast sky when ${\hbox{Si}}$ is between 0.3 and 0.6.

• Intermediate sky if ${\hbox{Si}} = { }0.{6} - {1}.{5}$ _.

• Nearly clear sky if ${\hbox{Si}} = { 1}.{5} - {1}.{7}$ _.

• Clear sky if ${\hbox{Si}} \;{\hbox{is over 1}}.{7}$ _.

Five similar categories based on the cloudless index ${\hbox{Cle}}$ were proposed also by Perradeau (1988), who called the ${\hbox{Cle}}$ value the nebulosity index.

It is evident that all sky models need at least some parameterization base, either the simple expected ${D_{\rm{v}}}/{E_{\rm{v}}}$ ratio or measured ${D_{\rm{e}}},{G_{\rm{e}}},m$, or ${\gamma_{\rm{s}}}$ for a particular situation. Then parameterized cases can be determined for sky radiance or luminance distributions in relative or absolute units. The typology of sky luminance patterns assumes also simple and clear rules to classify sky luminance scans, with certain sky types or standards that can be identified from IDMP regularly measured data. The quasi-homogeneous cases can be selected from all sets using the ${L_{\rm{vZ}}}/{D_{\rm{v}}}$ parameter within the narrow ±2.5% range of the particular ${L_{\rm{vZ}}}/{D_{\rm{v}}}$ ISO standard curves. When diffuse sky illuminance or zenith luminance is not measured or is unavailable, then both the Perez and Igawa sky models, respectively, can be used if regular irradiance data measured by meteorological stations with specified time or solar altitude information can be obtained. When no local data can be found, then for practical tasks sky luminance patterns under a selected sky type can be taken from the set of 15 ISO/CIE standards. The great handicap of the Perez model is the absence of the CIE overcast sky probably owing to only the subtropical sky scans gathered in Berkeley, California. Thus, in the eighth overcast clearness category the indicatrix is not exactly $f\left( {\chi = 1} \right)$ and the gradation ${L_{\rm{vZ}}}/{L_{\rm{vh}}}$ does not follow the 3:1 decreasing tendency. However, owing to manageable and systematically defined relationships, the Perez model was incorporated into the radiance calculation process, where it is especially suited because of its irradiance-oriented base.