**2. Regular daylight measurements and their possible analysis**

Since the CIE (2003) and ISO (2004) fifteen general homogeneous sky luminance patterns were standardised many CIE IDMP (International Daylight Measurement) stations recording regularly long-term daylight parameters try to evaluate the frequency of typical skies in their localities. Because the general CIE IDMP stations without sky luminance scanners sometimes do not record even zenith luminance *Lvz* simultaneously with diffuse skylight illuminance measurements *Dv* there are missing either sky scans or the classifying parameter *L D vz v* / , which could identify the momentary sky type. Thus, usually are only available data of regularly measured illuminance parameters in one minute steps during daytime, i.e.:


Parameterisation of the Four Half-Day Daylight Situations 149

The perpendicular parallel sunbeam illuminance at the ground level can be also calculated

where *LSC* is the luminous solar constant (Darula et al., 2005), which is the normal extraterrestrial illuminance on the outer border of the atmosphere for the average distance between sun and earth, approximately *LSC* 133800 lx., which is corrected for any date by

<sup>360</sup> 1 0.034cos 2

1

*<sup>v</sup> a* - luminous extinction coefficient of a clean and dry (Rayleigh) atmosphere after Clear

0.1 1 0.0045 *<sup>v</sup> <sup>a</sup> <sup>m</sup>*

1 9.9 0.043 *<sup>v</sup> <sup>a</sup>*

*Tv* - luminous turbidity factor, which defines the number of clean and dry atmospheres in the direction of sunbeams that reduces relatively its momentary penetration. In fact, if in eq. (8) *Pv* is measured by a sun tracker or *Pv* is derived from measured *G D v v* data, then

ln / *<sup>v</sup>* ln / *v v*

*v v P LSC E P*

*E LSC v s* sin

Thus, once the momentary illuminance *Pv* or *Pv* is determined the actual sunlight impact on any arbitrary plane can be calculated using the cosine of its incidence angle. However, for the vertical planes oriented either to direct East or West cardinal points this cosine

*a m a m*

 

sin 0.50572 6.07995 *s s*

1.6364

*P LSC a mT <sup>v</sup>* exp *v v* lx, (8)

<sup>365</sup> *<sup>J</sup>* -, (9)



lx. (14)

*<sup>m</sup>* -, (12)


*<sup>z</sup>* is geographical longitude of the time zone in deg.,

the ellipticity factor , which is often approximated by IESNA (1984)

*m* - relative optical air mass approximated by Kasten & Young (1989)

*<sup>L</sup>* - geographical longitude of the location in deg.

*m*

(1982), later published by Navvab et al., (1984)

the actual value *Tv* can be determined as

function is simplified to

*v*

where *PGD vvv* and the extraterrestrial horizontal illuminance *Ev* is

*T*

applying the Bouguer law, i.e.

or

A clock controling system starting every minute count has to be recorded too either in local clock time *LCT* or true solar time *TST* . These regular measurements can serve for the specification of daylight situations during the half-day or to the rough identification of the sky type in any minute, hour or date.

In fact even in absence of the sun tracker the *Pv* illuminance can be derived from *Gv* and *Dv* recordings as

$$P\_{v\perp} = \frac{G\_v - D\_v}{\sin \gamma\_s} = \frac{P\_v}{\sin \gamma\_s} \text{ (lx)}\tag{1}$$

where *P P vv s* sin is the horizontal illuminance caused by only parallel sunbeams in lx, *s* is the momentary solar altitude which can be determined for any station location, date and clock time after:

$$
\sin \gamma\_s = \sin \varphi \sin \delta - \cos \varphi \cos \delta \cos(15^\circ H) \text{ [ ${-}$ ]}.\tag{2}
$$

The station location is given by the geographical latitude in deg., while date is specified by solar declination and hour number *H* during daytime in *TST* .

Solar declination angle can be calculated for any day number within a year *J* (i.e. for 1st January *J* 1 and for 31st December 365 *J* ) using different approximate equations (e.g. Kittler & Mikler, 1986). The simplest is that introduced by Cooper (1969)

$$\delta = 23.45^{\circ} \sin \left[ \frac{360^{\circ}}{365} (284 + f) \right] \text{ [} \text{\textquotedblleft]} \text{.} \tag{3}$$

and a more accurate approximation was recommended by EU after Gruter (1981)

$$\delta = 23.45^\circ \sin \left[ \frac{360^\circ}{365} (J - 80.2^\circ) + 1.92^\circ \sin \left( J - 280^\circ \right) \right] \text{[ $\uparrow$ ]}.\tag{4}$$

Because usually CIE IDMP stations record all measurements in local clock time *LCT* the value *H* in *TST* has to be recalculated without consideration of summer shift time, after

$$dH = LCT + \eta + \frac{\left(\lambda\_z - \lambda\_L\right)}{15^\circ} \text{ [h]},\tag{5}$$

where is the equation of time in hours approximated after a simpler formula by Pierpoint (1982)

$$\eta = 0.17 \sin\left[4\pi \left(f - 80\right) / 373\right] - 0.129 \sin\left[2\pi \left(f - 8\right) / 355\right] \text{ [h]}.\tag{6}$$

or a more accurate formula by Heindl & Koch (1976)

$$\begin{aligned} \eta &= 0.008 \cos \frac{360^{\circ}f}{365} - 0.052 \cos 2 \frac{360^{\circ}f}{365} - 0.001 \cos 3 \frac{360^{\circ}f}{365} - \\ &- 0.122 \sin \frac{360^{\circ}f}{365} - 0.157 \sin 2 \frac{360^{\circ}f}{365} - 0.005 \sin 3 \frac{360^{\circ}f}{365} \end{aligned} \tag{7}$$

*<sup>z</sup>* is geographical longitude of the time zone in deg.,

*<sup>L</sup>* - geographical longitude of the location in deg.

The perpendicular parallel sunbeam illuminance at the ground level can be also calculated applying the Bouguer law, i.e.

$$P\_{v\perp} = LSC \in \exp\left(-a\_v \, mT\_v\right) \tag{8}$$

where *LSC* is the luminous solar constant (Darula et al., 2005), which is the normal extraterrestrial illuminance on the outer border of the atmosphere for the average distance between sun and earth, approximately *LSC* 133800 lx., which is corrected for any date by the ellipticity factor , which is often approximated by IESNA (1984)

$$\mathbf{a} \in \mathbf{I} + 0.034 \cos \frac{360^{\circ}}{365} \mathbf{(J-2)} \text{ [-]},\tag{9}$$

*m* - relative optical air mass approximated by Kasten & Young (1989)

$$m = \frac{1}{\sin \gamma\_s + 0.50572 \left(\gamma\_s + 6.07995^{\circ}\right)^{-1.6364}} \text{ [-l]} \tag{10}$$

*<sup>v</sup> a* - luminous extinction coefficient of a clean and dry (Rayleigh) atmosphere after Clear (1982), later published by Navvab et al., (1984)

$$a\_v = \frac{0.1}{1 + 0.0045 \, m} \text{ [-]}\tag{11}$$

or

148 Sustainable Growth and Applications in Renewable Energy Sources

A clock controling system starting every minute count has to be recorded too either in local clock time *LCT* or true solar time *TST* . These regular measurements can serve for the specification of daylight situations during the half-day or to the rough identification of the

In fact even in absence of the sun tracker the *Pv* illuminance can be derived from *Gv* and

sin sin *vv v <sup>v</sup>*

sin sin sin cos cos cos(15 )

Kittler & Mikler, 1986). The simplest is that introduced by Cooper (1969)

365

*GD P <sup>P</sup>* 

*<sup>s</sup>*

The station location is given by the geographical latitude

 

or a more accurate formula by Heindl & Koch (1976)

*s s*

is the horizontal illuminance caused by only parallel sunbeams in lx,

is the momentary solar altitude which can be determined for any station location, date

 

 and hour number *H* during daytime in *TST* . Solar declination angle can be calculated for any day number within a year *J* (i.e. for 1st January *J* 1 and for 31st December 365 *J* ) using different approximate equations (e.g.

> <sup>360</sup> 23.45 sin 284 365

<sup>360</sup> 23.45 sin 80.2 1.92 sin 280

Because usually CIE IDMP stations record all measurements in local clock time *LCT* the value *H* in *TST* has to be recalculated without consideration of summer shift time, after

> 15

is the equation of time in hours approximated after a simpler formula by Pierpoint

*J J*

*z L H LCT*

0.17sin 4 80 /373 0.129sin 2 8 /355 *J J*

<sup>360</sup> <sup>360</sup> <sup>360</sup> 0.008cos 0.052 cos2 0.001cos3

<sup>360</sup> <sup>360</sup> <sup>360</sup> 0.122sin 0.157 sin 2 0.005sin 3

365 365 365

*JJJ*

*JJJ*

365 365 365

and a more accurate approximation was recommended by EU after Gruter (1981)

lx, (1)

*H* (2)

in deg., while date is specified

[°]. (4)

h, (7)

*<sup>J</sup>* °, (3)

h, (5)

[h], (6)

sky type in any minute, hour or date.

*Dv* recordings as

where *P P vv s* sin

and clock time after:

by solar declination

*s* 

where 

(1982)

$$a\_v = \frac{1}{9.9 + 0.043\,\mathrm{m}} \text{ [-]}\tag{12}$$

*Tv* - luminous turbidity factor, which defines the number of clean and dry atmospheres in the direction of sunbeams that reduces relatively its momentary penetration. In fact, if in eq. (8) *Pv* is measured by a sun tracker or *Pv* is derived from measured *G D v v* data, then the actual value *Tv* can be determined as

$$T\_v = \frac{-\ln\left(P\_{v\perp} \mid \left(\text{e}\,\text{LSC}\right)\right)}{a\_v \, m} = \frac{\ln\left(E\_v \mid P\_v\right)}{a\_v \, m} \tag{13}$$

where *PGD vvv* and the extraterrestrial horizontal illuminance *Ev* is

$$E\_v = \in LSC \, \text{sin}\,\gamma\_s \text{ [lx]}.\tag{14}$$

Thus, once the momentary illuminance *Pv* or *Pv* is determined the actual sunlight impact on any arbitrary plane can be calculated using the cosine of its incidence angle. However, for the vertical planes oriented either to direct East or West cardinal points this cosine function is simplified to

Parameterisation of the Four Half-Day Daylight Situations 151

Bratislava, clear days, 1 min data

26th December 2006 Gv Dv

4 6 8 10 12

Clock time

0 10 20 30 40 50 60 70

8th 26 April 2006 th December 2006

Bratislava, clear mornings, 1-minute data

20th July 2006

Fig. 2. *G E v v* / courses under *situation 1* after 1-minute measurements

Solar altitude in deg

22nd September 2007

0

Fig. 1. Illuminance courses during clear morning *situations 1*

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Gv / Ev

20

40

60

Illuminance in klx

80

22nd September 2007 Gv Dv

20th July 2006 Gv Dv

8th April 2006 Gv Dv

100

120

$$P\_{vvE} = P\_{vvW} = P\_{v\perp} \cos \delta \sin(15^\circ |12 - H|) \text{ [lx]}.\tag{15}$$

Note, that the East and West oriented vertical planes or fasades are exposed to the morning and afternoon half-day sunlight and skylight effects as measured by global vertical illuminances *GvvE* or *GvvW* respectively. So, when the direct sunlight using eq. (15) can be subtracted only diffuse skylight components for these orientations can be determined, i.e. *DvvE* or *DvvW* .

### **3. Four typical half-day situations indicated by illuminance courses**

In the previous paper (Darula & Kittler, 2004a) from typical half-day illuminance courses were identified four characteristic daylight situations, which need to be explained in more detail:

*Situation 1*: Absolutely cloudless half-day with relative sunshine duration 0.75 *s* and the parameter of instability 8.4 *U* is almost clear with only few smaller clouds moving over the sky vault. Except these few sunshaded events the clear sky is quite stable and all horizontal illuminance parameters *Pv* , *Gv* and *Dv* follow a fluent increase in level during the morning hours and similar decrease during the afternoon due to solar altitude changes in different seasons.

Examples of selected half-days with *situation 1* are using recorded data from the Bratislava CIE IDMP general station ( = 48°10´N, *<sup>L</sup>* = 17°05´E) with the Central European climatic influences, but these should be taken as instructional and illustrative examples characterising typical cases of *situation 1.* As examples of clear sky mornings in Bratislava, Slovakia, were chosen courses measured during a long summer day on the 20th July 2006 followed by an autumn day on 22nd September 2007, while a short winter day 26th December 2006 represents one of the shortest days and the spring day 8th April 2006 with slight veiling Cirro-Stratus influences is also documented. The measured half-day courses of global horizontal illuminance *Gv* and diffuse sky illuminance *Dv* are documented in Fig. 1. Although the measurement registration is in the local clock time without the summertime shift it is evident that the courses follow the solar altitude changes, i.e. the sin *<sup>s</sup>* tendency of the extraterrestrial horizontal illuminance after eq. (14). Therefore the efficiency parameters *G E v v* / and *P E v v* / should be rather stable and showing a large amount of the extraterrestrially available luminous flux reaching the ground level, therefore these parameters can markedly characterise *situation 1* (Fig. 2). The momentary 1-minute measurements except some slight spreads on the April day show a steady rise with the solar altitude which is even better followed by the hourly averages in Fig. 3 with the stepwise rise of *G E v v* / from 0.45 to 0.75. In consequence, also the luminous turbidity factors *Tv* follow the stable atmospheric conditions without abrupt changes, except when the sun position is shaded by crossing cloud patches and then can reach higher short time peaks as in Fig. 4 on 8th April 2006. However, due to gradual evaporation during morning the turbidity might fluently rise with the formation of Cirrus or Cirrostartus veiling cloudiness as is shown by the trend of rising hourly average *Tv* values in a small range 1.5 to 3 in Fig. 5. Such rising *Tv* effects can be expected especially in equatorial regions with sometimes gradual cloud formation at noontime and in afternoon hours, which no longer belong to *situation 1.* 

Note, that the East and West oriented vertical planes or fasades are exposed to the morning and afternoon half-day sunlight and skylight effects as measured by global vertical illuminances *GvvE* or *GvvW* respectively. So, when the direct sunlight using eq. (15) can be subtracted only diffuse skylight components for these orientations can be determined, i.e. *DvvE* or *DvvW* .

In the previous paper (Darula & Kittler, 2004a) from typical half-day illuminance courses were identified four characteristic daylight situations, which need to be explained in more

Examples of selected half-days with *situation 1* are using recorded data from the Bratislava

influences, but these should be taken as instructional and illustrative examples characterising typical cases of *situation 1.* As examples of clear sky mornings in Bratislava, Slovakia, were chosen courses measured during a long summer day on the 20th July 2006 followed by an autumn day on 22nd September 2007, while a short winter day 26th December 2006 represents one of the shortest days and the spring day 8th April 2006 with slight veiling Cirro-Stratus influences is also documented. The measured half-day courses of global horizontal illuminance *Gv* and diffuse sky illuminance *Dv* are documented in Fig. 1. Although the measurement registration is in the local clock time without the summertime shift it is evident that the courses follow the solar altitude changes, i.e. the

 tendency of the extraterrestrial horizontal illuminance after eq. (14). Therefore the efficiency parameters *G E v v* / and *P E v v* / should be rather stable and showing a large amount of the extraterrestrially available luminous flux reaching the ground level, therefore these parameters can markedly characterise *situation 1* (Fig. 2). The momentary 1-minute measurements except some slight spreads on the April day show a steady rise with the solar altitude which is even better followed by the hourly averages in Fig. 3 with the stepwise rise of *G E v v* / from 0.45 to 0.75. In consequence, also the luminous turbidity factors *Tv* follow the stable atmospheric conditions without abrupt changes, except when the sun position is shaded by crossing cloud patches and then can reach higher short time peaks as in Fig. 4 on 8th April 2006. However, due to gradual evaporation during morning the turbidity might fluently rise with the formation of Cirrus or Cirrostartus veiling cloudiness as is shown by the trend of rising hourly average *Tv* values in a small range 1.5 to 3 in Fig. 5. Such rising *Tv* effects can be expected especially in equatorial regions with sometimes gradual cloud formation at noontime and in afternoon hours, which no

*Situation 1*: Absolutely cloudless half-day with relative sunshine duration 0.75 *s* and the parameter of instability 8.4 *U* is almost clear with only few smaller clouds moving over the sky vault. Except these few sunshaded events the clear sky is quite stable and all horizontal illuminance parameters *Pv* , *Gv* and *Dv* follow a fluent increase in level during the morning hours and similar decrease during the afternoon due to solar

*H* lx. (15)

*<sup>L</sup>* = 17°05´E) with the Central European climatic

*PP P vvE vvW v* cos sin 15 12

**3. Four typical half-day situations indicated by illuminance courses** 

altitude changes in different seasons.

= 48°10´N,

CIE IDMP general station (

longer belong to *situation 1.* 

detail:

sin *<sup>s</sup>* 

Fig. 1. Illuminance courses during clear morning *situations 1*

Fig. 2. *G E v v* / courses under *situation 1* after 1-minute measurements

Parameterisation of the Four Half-Day Daylight Situations 153

It has to be noted that during sunrise and early morning hours the prevailing daylight is caused by skylight and therefore also on clear days the early *G E v v* / values are equal or quite close to *D E v v* / while under higher solar altitude the *P E v v* / component is rising while *D E v v* / value fluently decreases after Fig. 6 from roughly 0.5 to 0.1. The average hourly decrease is slightly distorting this range showing approximately 0.4 to 0.1

Bratislava, clear mornings, 1-minute data

20th 8 July 2006 th April 2006

26th December 2006

26th December 06

22nd September 2007

Fig. 6. *D E v v* / courses under *situation 1*: after 1-minute measurements

22nd September 2007

Fig. 7. *D E v v* / courses under *situation 1*: after measured hourly averages

Bratislava, clear mornings, hourly data

0 10 20 30 40 50 60 70

Solar altitude in deg

20th 8 July 2006 th April 2006

0 10 20 30 40 50 60 70

Solar altitude in deg

If simultaneous measurements of the zenith luminance is recorded under clear sky conditions the classifying parameters *L D vz v* / , can identify the momentary sky type with the fluent rising tendency dependent on the solar altitude. In Fig. 8 this tendency is shown using 1-minute data while in Fig. 9 the same is documented after hourly mean values. Due to rather constant and fluent trends during *situation 1* besides the momentary one-minute recordings also hourly averages and appropriate parameters are quite satisfactorily reflecting clear half-days which might reduce the number of data considerably (Darula. &

respectively (Fig. 7).

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

> 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Dv / Ev

Kittler, 2005a).

Dv / Ev

Fig. 3. *G E v v* / courses under *situation 1*:after measured hourly averages

Fig. 4. *Tv* courses under *situation 1*: after 1-minute measurements

Fig. 5. *Tv* courses under *situation 1*: after measured hourly averages

0 10 20 30 40 50 60 70

26th

20th

26th

December 2006

December 2006

20th 8 July 2006 th

Solar altitude in deg

8 July 2006 th

0 10 20 30 40 50 60 70

Solar altitude in deg

 20th July 2006

26th

December 2006

0 10 20 30 40 50 60 70

Solar altitude in deg

Bratislava, clear mornings, hourly data

22nd September 2007

April 2006

Fig. 3. *G E v v* / courses under *situation 1*:after measured hourly averages

22nd September 2007

Bratislava, clear mornings, 1-minute data

April 2006

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1

1

10

40

Tv

Fig. 4. *Tv* courses under *situation 1*: after 1-minute measurements

8th

Fig. 5. *Tv* courses under *situation 1*: after measured hourly averages

22nd September 2007

April 2006

Bratislava, clear mornings, hourly data

10

40

Tv

Gv / Ev

It has to be noted that during sunrise and early morning hours the prevailing daylight is caused by skylight and therefore also on clear days the early *G E v v* / values are equal or quite close to *D E v v* / while under higher solar altitude the *P E v v* / component is rising while *D E v v* / value fluently decreases after Fig. 6 from roughly 0.5 to 0.1. The average hourly decrease is slightly distorting this range showing approximately 0.4 to 0.1 respectively (Fig. 7).

Fig. 6. *D E v v* / courses under *situation 1*: after 1-minute measurements

Fig. 7. *D E v v* / courses under *situation 1*: after measured hourly averages

If simultaneous measurements of the zenith luminance is recorded under clear sky conditions the classifying parameters *L D vz v* / , can identify the momentary sky type with the fluent rising tendency dependent on the solar altitude. In Fig. 8 this tendency is shown using 1-minute data while in Fig. 9 the same is documented after hourly mean values. Due to rather constant and fluent trends during *situation 1* besides the momentary one-minute recordings also hourly averages and appropriate parameters are quite satisfactorily reflecting clear half-days which might reduce the number of data considerably (Darula. & Kittler, 2005a).

Parameterisation of the Four Half-Day Daylight Situations 155

 <sup>1</sup> arccos tan tan

This is an normalising amount to calculate relative sunshine duration during the half-

*hd*

*ahd S*

*Situation 2*: Cloudy half-days with possible foggy short periods are characterised by scarce and lower sunlight influences under a range of relative sunshine durations ( 0.03 0.75 *s* and 10 6 *U s* ) and relatively higher diffuse illuminance levels. Such situations are caused by the prevailing area of the sky covered from almost homogeneous presence of clouds layers with different combinations of cloud type, turbidity and cloud cover overlayed in their height positions and movement drifts. Therefore, usually their *Gv* courses are close to *Dv* levels and so are also ratios *G E v v* /

To document cloudy half-days were chosen from the Bratislava data again seasonally typical cases, i.e. a summer day 3rd June 2007, an autumn day on 5th September 2007, a cloudy winter morning on 20th December 2006 and a spring morning on 5th April 2006. The measured half-day courses of global horizontal illuminance *Gv* and diffuse sky illuminance *Dv* are recorded in local clock time again in Fig. 10. In early morning hours under cloudy conditions *G E v v* / and *D E v v* / are almost the same as is not so noticeable from the winter course of illuminances, but evident in Fig. 11 in 1-minute or in Fig. 12 in the hourly alternative compared with Fig. 13 and 14. In this cloudy case the *G E v v* / and *D E v v* / values is very high reaching 0.25 to 0.6 level indicating a very bright but sunless winter half-day which is indicated also by the *Tv* lower values compared with all other cloudy samples (in Fig. 15 and 16) as well as in rather horizontal range of *L D vz v* / parameters in Fig. 17 and especially their averages in Fig. 18 with the data spread within the values 0.2 to 0.38 close to

Due to cloudiness overlays and turbidity changes rather high values of *Tv* factors have to be expected usually dependent on the solar altitude as shown in Fig. 15 or 16. However, within the half-day courses momentary unstable *Pv* can occur, thus there are cases also with higher average relative sunshine durations during the half-day in the range 0.1 to 0.5, but seldom over 0.5 with lower sunlight intensities, which are usually indicated by smaller peaks within the half-day course. These drab sunlight influences are documented by the small differences

*Situation 3*: Overcast half-days are absolutely without any sunlight and are caused by either dense layers of Stratus or Altostratus cloudiness or inversion fog when the sun

between *G E v v* / and *D E v v* / values when comparing Fig. 12 and 14 respectively.

In the half-day system relative sunshine duration during the morning half-day is *s s hd m* while its afternoon relative duration is *hd a s s* either in absolute values or % respectively. If regular minute recordings are measured, then *Shd* can be calculated as the sum of all data after the WMO (1983) and CIE 108 (1994) when the direct irradiance *Pe* 120 W/m2 taken

*hd*

*s*

 [h]. (18)

*<sup>S</sup>* -. (19)

15 *Sahd*

day *hd s* if the true measured sunshine duration in hours *Shd* is available:

in hours or their decimals.

and *D E v v* / typical for s*ituation 2*.

overcast sky (Darula & Kittler, 2004b).

Fig. 8. *L D vz v* / courses under *situation 1*: after 1-minute measurements

Fig. 9. *L D vz v* / courses under *situation 1*: after measured hourly averages

It is evident that the time period close to sunrise is untrustworthy due to an interval when solar altitude is zero and average *G E v v* / ratios are also reduced due to close to horizon mist or high turbidities. The minute courses are intersected by the hourly level in the point of hourly average solar altitude after Kittler & Mikler (1986)

where *H*1, *H*2 are consecutive hours

$$\gamma\_{sH} = \frac{180^{\circ}}{\pi} \arcsin\left[\sin\varphi\sin\delta + \frac{12}{\pi} \left(\sin\frac{\pi H\_1}{12} - \sin\frac{\pi H\_2}{12}\right)\right] \cos\varphi\cos\delta \quad \text{(16)}$$

Sunrise hour *Hsr* when 0 *<sup>s</sup>* is for any location and date defined by

$$H\_{sr} = \frac{1}{15^\circ} \arccos\left(\tan\varphi \tan\delta\right) \text{ [h]}\tag{17}$$

and due to symmetry around noon the hour of sunset *Hss* = 24 - *Hsr* and the astronomically possible sunshine duration *Sahd* during a half-day is

22nd September 2007

Fig. 8. *L D vz v* / courses under *situation 1*: after 1-minute measurements

Bratislava, clear mornings, hourly data

22nd September 2007

Fig. 9. *L D vz v* / courses under *situation 1*: after measured hourly averages

12 12 *sH*

15 *Hsr*

 

of hourly average solar altitude after Kittler & Mikler (1986)

December 06

 8th April 2006 20th July 2006

26th

 December 2006 Bratislava, clear mornings, 1-minute data

 20th July 2006

 8th April 2006

26th

0 10 20 30 40 50 60 70

Solar altitude in deg

0 10 20 30 40 50 60 70

Solar altitude in deg

It is evident that the time period close to sunrise is untrustworthy due to an interval when solar altitude is zero and average *G E v v* / ratios are also reduced due to close to horizon mist or high turbidities. The minute courses are intersected by the hourly level in the point

1 2 <sup>180</sup> <sup>12</sup> arcsin sin sin sin sin cos cos

 

is for any location and date defined by

 <sup>1</sup> arccos tan tan

and due to symmetry around noon the hour of sunset *Hss* = 24 - *Hsr* and the astronomically

 *H H*

[h], (17)

  rad, (16)

0.1

0.1

0.2

0.3

Lvz / Dv

where *H*1, *H*2 are consecutive hours

possible sunshine duration *Sahd* during a half-day is

Sunrise hour *Hsr* when 0 *<sup>s</sup>*

0.4

0.5

0.6

0.2

0.3

Lvz / Dv

0.4

0.5

0.6

$$S\_{\rm dhd} = \frac{1}{15^{\circ}} \arccos\left(-\tan\varphi \tan\delta\right) \text{ [h]}.\tag{18}$$

This is an normalising amount to calculate relative sunshine duration during the halfday *hd s* if the true measured sunshine duration in hours *Shd* is available:

$$s\_{hd} = \frac{S\_{hd}}{S\_{old}} \tag{19}$$

In the half-day system relative sunshine duration during the morning half-day is *s s hd m* while its afternoon relative duration is *hd a s s* either in absolute values or % respectively.

If regular minute recordings are measured, then *Shd* can be calculated as the sum of all data after the WMO (1983) and CIE 108 (1994) when the direct irradiance *Pe* 120 W/m2 taken in hours or their decimals.

*Situation 2*: Cloudy half-days with possible foggy short periods are characterised by scarce and lower sunlight influences under a range of relative sunshine durations ( 0.03 0.75 *s* and 10 6 *U s* ) and relatively higher diffuse illuminance levels. Such situations are caused by the prevailing area of the sky covered from almost homogeneous presence of clouds layers with different combinations of cloud type, turbidity and cloud cover overlayed in their height positions and movement drifts. Therefore, usually their *Gv* courses are close to *Dv* levels and so are also ratios *G E v v* / and *D E v v* / typical for s*ituation 2*.

To document cloudy half-days were chosen from the Bratislava data again seasonally typical cases, i.e. a summer day 3rd June 2007, an autumn day on 5th September 2007, a cloudy winter morning on 20th December 2006 and a spring morning on 5th April 2006. The measured half-day courses of global horizontal illuminance *Gv* and diffuse sky illuminance *Dv* are recorded in local clock time again in Fig. 10. In early morning hours under cloudy conditions *G E v v* / and *D E v v* / are almost the same as is not so noticeable from the winter course of illuminances, but evident in Fig. 11 in 1-minute or in Fig. 12 in the hourly alternative compared with Fig. 13 and 14. In this cloudy case the *G E v v* / and *D E v v* / values is very high reaching 0.25 to 0.6 level indicating a very bright but sunless winter half-day which is indicated also by the *Tv* lower values compared with all other cloudy samples (in Fig. 15 and 16) as well as in rather horizontal range of *L D vz v* / parameters in Fig. 17 and especially their averages in Fig. 18 with the data spread within the values 0.2 to 0.38 close to overcast sky (Darula & Kittler, 2004b).

Due to cloudiness overlays and turbidity changes rather high values of *Tv* factors have to be expected usually dependent on the solar altitude as shown in Fig. 15 or 16. However, within the half-day courses momentary unstable *Pv* can occur, thus there are cases also with higher average relative sunshine durations during the half-day in the range 0.1 to 0.5, but seldom over 0.5 with lower sunlight intensities, which are usually indicated by smaller peaks within the half-day course. These drab sunlight influences are documented by the small differences between *G E v v* / and *D E v v* / values when comparing Fig. 12 and 14 respectively.

*Situation 3*: Overcast half-days are absolutely without any sunlight and are caused by either dense layers of Stratus or Altostratus cloudiness or inversion fog when the sun

Parameterisation of the Four Half-Day Daylight Situations 157

5th September 2007

20th December 2006

 3rd June 2007 5th April 2006 Bratislava, cloudy mornings, hourly data

0 10 20 30 40 50 60 70

Solar altitude in deg

20th December 2006

0 10 20 30 40 50 60 70

Solar altitude in deg

3rd 5 June 2007 th April 2006

20th December 2006

0 10 20 30 40 50 60 70

Solar altitude in deg

Fig. 12. *G E v v* / courses under *situation 2*: after measured hourly averages

5th September 2007

Fig. 13. *D E v v* / courses under *situation 2*: after 1-minute measurements

5th September 2007

Fig. 14. *D E v v* / courses under *situation 2*: after measured hourly averages

Bratislava, cloudy mornings, hourly data

Bratislava, cloudy mornings, 1-minute data

3rd 5 June 2007 th April 2006

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

> 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Dv / Ev

Dv / Ev

Gv / Ev

position is uncertain as it cannot be seen or guessed behind the overall dense clouds. Under such conditions *Gv* = *Dv* , 0 *Pv* and average relative sunshine duration during the half-day 0.03 *s* . While the *Dv* illuminance levels and the ratio *D E v v* / are quite low, usually in the range 0.02 - 0.25, the ratios *L D vz v* / are over 0.3 and stable during the half-day, i.e. without any dependence on the solar altitude (Darula & Kittler, 2004c). Under overcast sky conditions when sunbeam influences are absent the sky luminance patterns in all azimuth directions are uniform, so only gradation luminance distribution can cause the *Dv* illuminance rise from sunrise to noon.

Fig. 10. Illuminance courses during cloudy morning *situations 2* 

Fig. 11. *G E v v* / courses under *situation 2*: after 1-minute measurements

can cause the *Dv* illuminance rise from sunrise to noon.

Fig. 10. Illuminance courses during cloudy morning *situations 2* 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Gv / Ev

0 10 20 30 40 50 60 70

April 2006

Solar altitude in deg

Fig. 11. *G E v v* / courses under *situation 2*: after 1-minute measurements

Bratislava, cloudy mornings, 1-minute data 5th

 3rd 5 June 2007 th

20th

September 2007

December 2006

position is uncertain as it cannot be seen or guessed behind the overall dense clouds. Under such conditions *Gv* = *Dv* , 0 *Pv* and average relative sunshine duration during the half-day 0.03 *s* . While the *Dv* illuminance levels and the ratio *D E v v* / are quite low, usually in the range 0.02 - 0.25, the ratios *L D vz v* / are over 0.3 and stable during the half-day, i.e. without any dependence on the solar altitude (Darula & Kittler, 2004c). Under overcast sky conditions when sunbeam influences are absent the sky luminance patterns in all azimuth directions are uniform, so only gradation luminance distribution

Fig. 12. *G E v v* / courses under *situation 2*: after measured hourly averages

Fig. 13. *D E v v* / courses under *situation 2*: after 1-minute measurements

Fig. 14. *D E v v* / courses under *situation 2*: after measured hourly averages

Parameterisation of the Four Half-Day Daylight Situations 159

Bratislava, cloudy mornings, hourly data

3rd June 2007

Fig. 18. *L D vz v* / courses under *situation 2*:after measured hourly averages

shows also the *L D vz v* / courses in Fig. 22 and 23 within the average range 0.3-0.4.

Fig. 19. Illuminance courses during overcast morning *situations 3*

5th 20 April 2006 th December 2006

5th September 2007

0 10 20 30 40 50 60 70

Solar altitude in deg

To document overcast half-days by Bratislava recordings again four seasonal examples were chosen, i.e. a winter morning on the 23rd January 2001 and a spring case on 3rd March 2001, an exceptional summer half day on 4th June 2001 and an autumn case on 6th September 2007. The half-day courses of measured global and diffuse illuminances in 1-minute intervals are in Fig. 19 with the *GE DE vv vv* / / analysis in Fig. 20 in 1-minute and in Fig. 21 in hourly alternatives. All four cases document the low and stable efficiency of penetration in the range 0.05-0.2 without any dependence on the solar altitude. The same stable and independent trend

0.1

0.2

0.3

Lvz / Dv

0.4

0.5

0.6

Fig. 15. *Tv* courses under *situation 2*: after 1-minute measurements

Fig. 16. *Tv* courses under *situation 2*: after measured hourly averages

Fig. 17. *L D vz v* / courses under *situation 2*: after 1-minute measurements

0 10 20 30 40 50 60 70

0 10 20 30 40 50 60 70

Solar altitude in deg

5th

20th

3rd

5th

Bratislava, cloudy mornings, hourly data

September 2007

December 2006

June 2007

April 2006

Solar altitude in deg

5 June 2007 th

Bratislava, cloudy mornings, 1-minute data

3rd

20th

December 2006

September 2007

April 2006

5th

Fig. 15. *Tv* courses under *situation 2*: after 1-minute measurements

Fig. 16. *Tv* courses under *situation 2*: after measured hourly averages

Fig. 17. *L D vz v* / courses under *situation 2*: after 1-minute measurements

1

1

10

40

T

v

10

40

Tv

Fig. 18. *L D vz v* / courses under *situation 2*:after measured hourly averages

To document overcast half-days by Bratislava recordings again four seasonal examples were chosen, i.e. a winter morning on the 23rd January 2001 and a spring case on 3rd March 2001, an exceptional summer half day on 4th June 2001 and an autumn case on 6th September 2007. The half-day courses of measured global and diffuse illuminances in 1-minute intervals are in Fig. 19 with the *GE DE vv vv* / / analysis in Fig. 20 in 1-minute and in Fig. 21 in hourly alternatives. All four cases document the low and stable efficiency of penetration in the range 0.05-0.2 without any dependence on the solar altitude. The same stable and independent trend shows also the *L D vz v* / courses in Fig. 22 and 23 within the average range 0.3-0.4.

Fig. 19. Illuminance courses during overcast morning *situations 3*

Parameterisation of the Four Half-Day Daylight Situations 161

0 10 20 30 40 50 60 70

3rd 23 March 2001 rd January 2001

Solar altitude in deg

*Situation 4*: The dynamic courses in horizontal illuminance levels happen during those half-days when the clear sky is covered by smaller cloud patches passing the sun position and shade direct sunlight in many short-term intervals or moments. Thus the overall *Gv* course trends can be usually kept but with many drops of temporary loss or reduction of *Pv* components, which mean dynamic variations between *Gv* and *Dv* levels. Because *Dv* levels are not affected by the *Pv* changes, *L D vz v* / ratios indicate the sky patterns when the zenith luminance is not influenced by passing clouds significantly. However, dynamic changes are reproduced also in *G E v v* / and *P E v v* / courses. In case of dynamic situations it is problematic to use hourly averages which are levelling the momentarily occurring peaks and drops replacing them by an even horizontal line. Thus is also distorted the wide range of *Tv* values that have to be

Due to the irregularity and occasional movement of the shading cloud patches there is a multiple number of different cases, so the selection of characteristic courses is very problematic. However, from Bratislava data were selected also four seasonal representatives, i.e. for winter the morning on 12th January 2007, for spring 14th March 2001, for the summer example the course on 29th June 2007 and for the autumn example was chosen the dynamic morning on 26th November 2007. The actual global and diffuse illuminance courses in Fig. 24 document the dynamic changes during the chosen half-days. The same dynamic variations of *G E v v* / parameters in minute representation are in Fig. 25 while hourly means erase the highest peaks and drops (Fig. 26) considerably. The *D E v v* / courses are relatively more stable and document the low borderline (Fig. 27 and 28) from which additional sunlight influences the peaks. Similarly to *Gv*/*Ev* also *Lvz*/*Dv* courses are very distorted in hourly averages in Fig. 30 in comparison to 1-minute fluctuating values in Fig. 29, but the former indicate a tendency of the background spring and summer clear skies. However, these background scene is also influenced by gradually increasing turbidity, which is low with lower solar altitude and considerably rising when the sunheight is over 35

6th September 2007

Bratislava, overcast mornings, hourly data

4th June 2001

Fig. 23. *L D vz v* / courses under *situation 3*: after measured hourly averages

0.1

0.2

0.3

Lvz / Dv

expected in s*ituation 4.*

degrees (Fig. 31 and 32).

0.4

0.5

0.6

Fig. 20. *G E v v* / courses under *situation 3*: after 1-minute measurements

Fig. 21. *G E v v* / courses under *situation 3*:after measured hourly averages

Fig. 22. *L D vz v* / courses under *situation 3*: after 1-minute measurements

Bratislava, overcast mornings, 1-min data

January 2001

 4th June 2001

23rd

Fig. 20. *G E v v* / courses under *situation 3*: after 1-minute measurements

 4th June 2001

23rd

Fig. 21. *G E v v* / courses under *situation 3*:after measured hourly averages

0 10 20 30 40 50 60 70

Solar altitude in deg

Bratislava, overcast mornings, hourly data

January 2001

6th

3rd

September 2007

March 2001

0 10 20 30 40 50 60 70

Solar altitude in deg

0 10 20 30 40 50 60 70

Solar altitude in deg

6th

September 2007

Bratislava, overcast mornings, 1-minute data

January 2001

Fig. 22. *L D vz v* / courses under *situation 3*: after 1-minute measurements

3rd 23 March 2001 rd

6th

3rd

September 2007

March 2001

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0

0.1

 4th June 2001

0.2

0.3

Lvz / Dv

0.4

0.5

0.6

0.2

0.4

0.6

Gv / Ev

0.8

1.0

Gv / Ev

Fig. 23. *L D vz v* / courses under *situation 3*: after measured hourly averages

*Situation 4*: The dynamic courses in horizontal illuminance levels happen during those half-days when the clear sky is covered by smaller cloud patches passing the sun position and shade direct sunlight in many short-term intervals or moments. Thus the overall *Gv* course trends can be usually kept but with many drops of temporary loss or reduction of *Pv* components, which mean dynamic variations between *Gv* and *Dv* levels. Because *Dv* levels are not affected by the *Pv* changes, *L D vz v* / ratios indicate the sky patterns when the zenith luminance is not influenced by passing clouds significantly. However, dynamic changes are reproduced also in *G E v v* / and *P E v v* / courses. In case of dynamic situations it is problematic to use hourly averages which are levelling the momentarily occurring peaks and drops replacing them by an even horizontal line. Thus is also distorted the wide range of *Tv* values that have to be expected in s*ituation 4.*

Due to the irregularity and occasional movement of the shading cloud patches there is a multiple number of different cases, so the selection of characteristic courses is very problematic. However, from Bratislava data were selected also four seasonal representatives, i.e. for winter the morning on 12th January 2007, for spring 14th March 2001, for the summer example the course on 29th June 2007 and for the autumn example was chosen the dynamic morning on 26th November 2007. The actual global and diffuse illuminance courses in Fig. 24 document the dynamic changes during the chosen half-days. The same dynamic variations of *G E v v* / parameters in minute representation are in Fig. 25 while hourly means erase the highest peaks and drops (Fig. 26) considerably. The *D E v v* / courses are relatively more stable and document the low borderline (Fig. 27 and 28) from which additional sunlight influences the peaks. Similarly to *Gv*/*Ev* also *Lvz*/*Dv* courses are very distorted in hourly averages in Fig. 30 in comparison to 1-minute fluctuating values in Fig. 29, but the former indicate a tendency of the background spring and summer clear skies. However, these background scene is also influenced by gradually increasing turbidity, which is low with lower solar altitude and considerably rising when the sunheight is over 35 degrees (Fig. 31 and 32).

Parameterisation of the Four Half-Day Daylight Situations 163

Bratislava, dynamic mornings, hourly data 12th January 2007

26th November 2007

 29th June 2007 14th March 2001

26th November 2007

 29th June 2007 14th March 2001 12th January 2007

26th November 2007

0 10 20 30 40 50 60 70

Solar altitude in deg

Bratislava, dynamic mornings, 1-minute data

0 10 20 30 40 50 60 70

Solar altitude in deg

14th 12 March 2001 th January 2007

0 10 20 30 40 50 60 70

Solar altitude in deg

Fig. 26. *G E v v* / courses under *situation 4*: after measured hourly averages

Fig. 27. *D E v v* / courses under *situation 4*: after 1-minute measurements

29th June 2007

Fig. 28. *D E v v* / courses under *situation 4*:after measured hourly averages

Bratislava, dynamic mornings, hourly data

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

> 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Dv / Ev

Dv / Ev

Gv / Ev

Fig. 24. Illuminance courses during overcast morning *situations 4* 

Fig. 25. *G E v v* / courses under *situation 4*: after 1-minute measurements

14th

 March 2001 Gv Dv

 Bratislava, dynamics day 1 min data

> January 2007 Gv Dv

 June 2007 Gv Dv

 November 2007 Gv Dv

4 6 8 10 12

Clock time

0 10 20 30 40 50 60 70

March 2001

Solar altitude in deg

Fig. 25. *G E v v* / courses under *situation 4*: after 1-minute measurements

Bratislava, dynamic mornings, 1-minute data

26th

12th

29th 14 June 2007 th

November 2007

January 2007

0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Gv / Ev

Fig. 24. Illuminance courses during overcast morning *situations 4* 

20

40

60

26th

12th

29th

Illuminance in klx

80

100

120

Fig. 26. *G E v v* / courses under *situation 4*: after measured hourly averages

Fig. 27. *D E v v* / courses under *situation 4*: after 1-minute measurements

Fig. 28. *D E v v* / courses under *situation 4*:after measured hourly averages

Parameterisation of the Four Half-Day Daylight Situations 165

0 10 20 30 40 50 60 70

Solar altitude in deg

In the paper by Kittler & Darula (2002) a P-D-G diagram was published to show Bratislava 5-minute data covering the whole July 1996. From 5315 cases were 3113 with sunshine while 2202 measured cases were without sunshine according to the WMO (1983) classification. The monthly relative sunshine duration after 1-minute recordings was in July 1996 on the average *s* 0.52 with daily changes within the range 0.022 – 0.946 which indicates the possibility of half-day situations in all four categories. Due to the averaging distortion it would seem that the prevailing sunny 5-minute intervals 3113/5315 indicate the sunshine

The review of daily measured illuminance courses representing July 1996 by 62 half- days

It is evident that neither the number of sunshine or sunless cases within a month in a P-D-G diagram nor *L D vz v* / and *G E v v* / time-averaged ratios are capable to differentiate the halfday situations when data are summarised during a day, a week or month in these mixed groups. Therefore the first step to identify, select or classify the half-day situations is to check the overall courses of *Gv* and *Dv* illuminance trends and levels and their relative efficiencies compared to the momentary extraterrestrial availability levels expressed in *G E v v* / and *D E v v* / ratios. Of course the stable or discontinuous sunshine duration follows the changes in *G E v v* / and the momentary presence of *P E v v* / ratios indicating the penetration of available extraterrestrial sunshine intensity. These half-day courses roughly characterise also the range of prevailing sky luminance patterns that can be expected and principally belong to the particular half-day situation. While situation 1 and 3 and sometimes even 2 are approximately homogeneous with evenly distributed turbidities and cloudiness cover over the whole sky vault, the situation 4 is characteristic for its unstable dynamic illuminance changes caused by complex layers of different cloud types and



**4. Approximate dependence of the four daylight situations on relative** 

26th November 2007

Bratislava, dynamic mornings

 29th Jun 2007 14th March 2001 12th January 2007

hourly data

1

**sunshine duration** 

duration roughly 0.586.

can be classified into:

Fig. 32. *Tv* courses under *situation 4*:after measured hourly averages

10

40

Tv

Fig. 29. *L D vz v* / courses under *situation 4*: after 1-minute measurements

Fig. 30. *L D vz v* / courses under *situation 4*: after measured hourly averages

Fig. 31. *Tv* courses under *situation 4*: after 1-minute measurements

29th 14 June 2007 th

12th

January 2007

Bratislava, dynamic mornings, 1-minute data

November 2007

March 2001

0 10 20 30 40 50 60 70

Solar altitude in deg

0 10 20 30 40 50 60 70

June 2007

January 2007

Solar altitude in deg

26th

14th 12 March 2001 th

November 2007

0.1

1

10

40

Tv

0.2

0.3

Lvz / Dv

0.4

0.5

26th

Fig. 29. *L D vz v* / courses under *situation 4*: after 1-minute measurements

Fig. 30. *L D vz v* / courses under *situation 4*: after measured hourly averages

Bratislava, dynamic mornings

40

Fig. 31. *Tv* courses under *situation 4*: after 1-minute measurements

29th

1-minute data

0.6

Fig. 32. *Tv* courses under *situation 4*:after measured hourly averages
