The Angström turbidity coefficient β D o g according to Dogniaux is obtained from the empirical formula given by the following equation:

β D o g = T l − [ h + 85 39.5 exp ( − w p ) + 47.4 + 0.1 ] 16 + 0.22 w p (2)

where T l is the Linke turbidity factor, h the Sun elevation angle in degrees and w p the precipitation amount in centimeter. w p is calculated using the following Equation (32):

w p = 0.493 ϕ T exp ( 26.23 − 5416 T ) (3)

where T is the temperature in Kelvin and ϕ the relative humidity in fractions of one.

The expression used to evaluate the Linke turbidity factor T l [8] [11] [12] [13] [14] [15] is:

T l = T l k 1 δ R a ( m a ) 1 δ R k ( m a ) (4)

where T l k , δ R k ( m a ) and δ R a ( m a ) are respectively the Linke factor according to Kasten, the Rayleigh integral optical thickness and the integral optical thickness. The Linke factor T l k is related to the normal incidence solar irradiance expressed by the Equation:

T l k = ( 0.9 + 9.4 sin ( h ) ) ∗ ( 2 ln ( I 0 ( R 0 R ) − ln ( I n ) ) ) (5)

where I n , I 0 , h, R and R 0 are respectively the direct normal solar irradiance in W/m², the solar constant, the Sun’s elevation angle in degrees and the instantaneous and the mean Sun-Earth distances.

δ R k ( m a ) and δ R a ( m a ) are given by the following Equations:

1 δ R a ( m a ) = 6.6296 + 1.7513 m a − 0.1202 m a 2 + 0.0065 m a 3 − 0.00013 m a 4 (6)

1 δ R k ( m a ) = 9.4 + 0.9 m a (7)

m a is the air mass given by [16] :

m a = m r ( P 101325 ) [ sin ( h ) + 0.15 ( 3.885 + h ) − 1.253 ] − 1 (8)

where P is the local pressure in Pascal given by [9] :

P = 101325 exp ( − 0.0001184 z ) (9)

z is the altitude of the location in meter.

Based on Iqbal C model [8] [17] determine the Angström turbidity coefficient β using the solar irradiance data and the aerosol transmittance τ a .

The aerosol transmittance according to Iqbal and Mächler [17] [18] [19] is given by:

τ a = ( 0.12445 α − 0.0162 ) + ( 1.003 − 0.125 α ) exp [ − β m a ( 1.089 α + 0.5123 ) ] (10)

Louche’s et al. [20] expressed the aerosol transmittance for cloudless sky as:

τ a = 1 0.9751 I n E 0 τ g τ 0 τ r τ w (11)

The direct solar irradiance at normal incidence I n in W/m², is directly measured with a pyrheliometer.

The Earth eccentricity correction factor E 0 is given by:

E 0 = ( R R 0 ) 2 (12)

where R and R 0 are the same as defined in Equation (5).

The parameter τ g represents the mixing gases absorption transmittance given by:

τ g = exp ( − 0.0127 m a 0.26 ) (13)

The parameter τ 0 is the ozone absorption transmittance given by:

τ 0 = 1 − [ 0.1611 U 3 ( 1 + 139.48 U 3 ) − 0.3035     − 0.002715 U 3 ( 1 + 0.044 U 3 + 0.0003 U 3 2 ) − 1 ] (14)

where U 3 = m r l . (l is the thickness of the total vertical ozone layer in cm).

The parameter τ r is the Rayleigh scattering transmittance given by:

τ r = exp ( − 0.0903 m a 0.84 ( 1 + m a − m a 1.01 ) ) (15)

The parameter τ w is the water vapor transmittance expressed as follow:

τ w = 1 − 2.4959 U 1 ( ( 1 + 0.79034 U 1 ) 0.6828 + 6.385 U 1 ) − 1 (16)

where U 1 = w p m r and w p is calculated by Equation (3).

The expression of the Angström coefficient denoted β L o u c h in the following, is obtained from a combination of Equations (10) and (11):

β L o u c h = 1 m a D 3 log ( D 2 τ a − D 1 ) (17)

where D 1 = 0.12445 α − 0.0162 , D 2 = 1.003 − 0.125 α and D 3 = 1.089 α + 0.5123 .

The approach developed by Pinazo et al. [9] is also based on Iqbal C model and on a coefficient K which is defined as the ratio between the direct beam solar irradiance on a horizontal surface and the global solar irradiance received by the same surface. The aerosol transmittance according to Pinazo et al. is expressed as:

τ a = ( 1 − A ) C 1 − A C (18)

with A = ( 1 − w 0 ) ( 1 − m a + m a 1.06 ) and C = C 1 − C 2 .

The parameter w 0 is the single scattering albedo or the ratio between the scattering and the extinction (scattering plus absorption) coefficients of aerosols that are high above the ground.

C 1 and C 2 are given by:

C 1 = ( [ 1 + ( F c B − 1 ) K − ρ g ( 1.0685 − F c ) 2 ρ g ( 1 − F c ) ] 2 + B K [ 0.5 ( 1 − τ r ) + F c ( 1 − F c ) ρ g ] ) 0.5 (19)

C 2 = 1 + ( F c B − 1 ) K − ρ g ( 1.0685 − F c ) 2 ρ g ( 1 − F c ) (20)

where

B = 0.79 0.9751 τ r ( 1 − m a + m a 1.02 ) (21)

F c is the forward scattering parameter defining the radiation fraction scattered in the forward half-space and ρ g is the albedo of the ground.

The Angström coefficient according to this model will be denoted β P i n z and will be calculated using a combination of Equations (10) and (18).

Gueymard and Vignola [10] proposed a method for estimating the Angström coefficient using the relation between the global (or diffuse) and the direct irradiance based on the spectral code SMARTS2 [10] [21]. The Angström coefficient denoted β G y e m is obtained from the following Equation (2):

β G y e m = 0.5 ( [ a 1 2 − 4 ( a 2 − a 3 K a b ) ( a 0 − K a b ) ] 0.5 − a 1 a 2 − a 3 K a b ) (22)

where K a b is the ratio between the diffuse irradiance and the direct beam normal irradiance. It corresponds to a standard value for zero altitude and the total amount of ozone equal to 0.3434 atm-cm. The coefficients a i are function of the zenith Sun angle, the pressure, the perceptible water and the ozone amount. These coefficients and the way they are calculated are detailed in [10].

We take daily mean values of the thickness of the total vertical ozone layer 1 from MODIS satellite data Ichoku 2004 [29] since we have no local measurements for this parameter. Figure 1 plots the temporal variation of the daily ozone layer thickness values for the period 2005-2008 (upper side) and its frequency distribution (bottom side). Annual mean values of the ozone layer thickness are 0.297 cm, 0.296 cm, 0.299 cm and 0.296 cm for 2005, 2006, 2007 and 2008 respectively. Similar 1 values are also obtained with data of the OMI instrument Torres 2002 [30]. We notice a higher value of this parameter and a more pronounced max in 2007. The maximum occurrence value of ozone layer thickness is around 0.285 cm according to Figure 1.

The total precipitable water is defined as the integrated water vapor in a vertical column extending from the surface to the top of the atmosphere. This parameter is important and its influence in calculation should be studied especially that in most cases we have absence of atmospheric sounding or solar spectral measurements [2]. We have used four algorithms in the present study to estimate the precipitable water:

1) Wright’s formula: A linear relationship relates the logarithm of the precipitable water w to the dew point temperature T d [17] :

ln w = a + b T d (23)

Parameters a and b are not universal and have both site and time dependency. The mostly used values of these parameters by several authors are those obtained by [31] for Albany NY: a = − 0.0756 and b = 0.0693 [2]. These values are

suitable for estimating instantaneous precipitable water under cloudless skies [31]. Two sources of error affect calculation of T d . They are associated to local parameters a and b and to the calculation method. The parameter T d is calculated by:

p s ( T d ) = p v ( T ) = Φ p s ( T ) (24)

where T is the temperature, Φ the relative humidity and p s the saturation pressure of water vapor calculated with several algorithms among them the commonly used Magnus and Leckner algorithms. The p s , in mbar, is expressed for each algorithm by Equations (25) and (26):

p s M = 6.107 exp ( 17.38 T 239 + T ) (25)

p s L = 0.01 exp ( 26.23 − 5416 273.15 + T ) (26)

where T is in degrees and Φ in fraction of one. M and L letters associted to p s variable stand for Magnus and Leckner respectively.

Equations (24), (25) and (26) lead to calculate T d with the desired algorithm using the following equations (Equation (27) and (28))

T d M = 239 f ( T , Φ ) 17.38 − f ( T , Φ )     where   f ( T , Φ ) = ln Φ + 17.38 T 239 + T (27)

T d L = 5416 5416 / ( 273.15 + T ) − ln Φ − 273.15 (28)

We have then two precipitable water values w M w and w L w according to Equations (27) and (28) and Wright’s formula 23.

2) Leckner’s formula: This alternative method is often used to calculate the amount of precipitable water w L [32]. It is obtained with the folling Equation:

w L = 49.3 Φ p s L T (29)

3) Gueymard’s formula: Gueymard introduced a new formula in 1994 [33] to estimate the precipitable water w G . It is expressed as follow:

w G = 21.67 H v Φ p s G T (30)

with p s G and H v are given by Equation (31) and (32):

ln p s G = 22.33 − 4914 T − 10922000 T 2 − 0.003902 T (31)

H v = exp ( 13.6897 θ − 14.9188 θ 3 ) + 1.5265 θ + 0.4976 , θ = T / 273 (32)

Annual mean values of the precipitable water w p according to the previous four methods are plotted in Figure 2. We notice that w p obtained with the 4 years of data have the same temporal trend. All methods show a minimum in May and a maximum between July and October. Maximum values are obtained with Leckner model ( w L ) and the minimum with Magnus using Wright’s formula ( w M w ). Gueymards method ( w G ) and Leckner using Wright’s formula ( w L w ) give approximately the same mean values (see Table 1). We will use precipitable water values of each method to estimate the Angström turbidity coefficient with the four broadband models. We notice however, that this parameter obtained from the four methods has not a significant effect on turbidity values for a given broadband model. The difference is about 0.1%.

The wavelength exponent α in Equation (1) is related to size distribution of particles. Low values of α correspond to large particles and vice versa. α = 1.3 ± 0.5 is suggested by many authors for most natural atmospheres [16]. In our case, we will use MODIS satellite data to obtain the values of this parameter since we did not dispose of photometric ground measurements. The variation of its monthly mean values over Ghardaïa city is shown in the upper side of Figure 3. Its yearly mean value is plotted in the lower side of Figure 3 where a slightly increase is observed. The annual mean values are 1.0 ± 0.3 , 1.0 ± 0.3 , 1.0 ± 0.3 , 1.1 ± 0.3 for 2005, 2006, 2007 and 2008 respectively. The mean value of α over the four years is 1.0 ± 0.3 and it is in agreement with values suggested by many authors.

We also used MODIS data to estimate the ground albedo ρ g at Ghardaïa city. Variations of its monthly value are shown in the upper side of Figure 4 ant its

annual mean values plotted in the lower side. The annual mean values are 0.17 ± 0.06 , 0.18 ± 0.05 , 0.17 ± 0.05 , 0.18 ± 0.05 for 2005, 2006, 2007 and 2008 respectively. The ρ g mean value over the four years is 0.17 ± 0.05 . We note also a slightly increase of ρ g between 2005 and 2008 with a litte drop in 2007.

The value of 0.8 for the single scattering albedo w 0 is usually chosen for rural-urban sites as advised by Gueymard [34] [35] while a value of 0.84 is

suggested by [17] for the forward scattering F c . We preferred here to use modeling techniques to find these parameters and their temporal variations rather than a constant value. In a recent study, [36] assessed the intrinsic performance of 18 broadband radiative models using high-quality data sets from five sites in widely different climates. All these models are able to predict direct, diffuse and global irradiance under clear skies from atmospheric data. Intrinsic performances of these models were evaluated by comparison between their predictions and high frequency measurements (1-minute time step in four sites, 3-minute in one site). From the 18 models is the Iqbal C [17] model that requires a relatively large number of atmospheric inputs and showed consistently high scores of statistical indicators. This model will be considered in our present study to estimate the required parameters since it offers a better accuracy than the others more conventional models [36]. In addition, the model inputs are those that we need, namely the Angstrom coefficient β , the average surface albedo ρ g , the wavelength Angstrom exponent α , the forward scatterance F c and the aerosol single scattering albedo w 0 . Only the last two parameters and the Angstrom coefficient β will be considered since the others are obtained from MODIS data (see Sections 3.3 and 3.4). Before proceeding the estimation of the parameters, we recall hereafter the main equations of this model described in detail in [17].

The direct normal irradiance I n (W/m²) is given by:

I n = 0.9751 I s c E 0 τ 0 τ g τ w τ r τ a ( α , β ) (33)

where τ 0 , τ g , τ w , τ r and τ a ( α , β ) are respectively the ozone, gas, water, Rayleigh and aerosol scattering transmittances. I s c and E 0 are respectively the solar constant and the eccentricity correction factor.

The aerosol scattering transmittance, which depends on the Angstrom coefficient β and wavelength Angstrom exponent α , is given by Equation (10).

The global solar irradiance ( I t ) measured with our instruments is the contribution of 2 solar irradiance components given by:

I t = I n h + I d (34)

where ( I n h ) is the normal solar irradiance on an horizontal surface and ( I d ) the horizontal diffuse solar irradiance. The normal solar irradiance I n h (W/m²) is given by:

I n h = I n sin ( h ) (35)

where h is the elevation angle of Sun in degrees.

The horizontal diffuse solar irradiance I d (W/m²) is a combination of three individual components, which are the Rayleigh component, I d r (W/m²), the aerosols scattering component, I d a (W/m²) after the first pass through the atmosphere, and the multiple reflection processes between the ground and sky component, I d m (W/m²):

I d = I d r + I d a + I d m (36)

The I d r component which depends on aerosol single scattering albedo w 0 , is given by:

I d r ( w 0 ) = 0.395 τ 0 τ g τ w ( 1 − τ r ) I s c E 0 sin ( h ) 1 − m a + m a 1.02 τ a a (37)

where τ a a = 1 − ( 1 − w 0 ) ( 1 − m a + m a 1.06 ) ( 1 − τ a ) is the direct radiation transmittance due to aerosol absorptance.

The I d a component is related to the forward scatterance F c :

I d a ( F c ) = 0.79 τ 0 τ g τ w τ a a F c ( 1 − τ a s ) I s c E 0 sin ( h ) 1 − m a + m a 1.02 (38)

The I d m component related to the ground albedo ρ g , is given by:

I d m ( ρ g ) = ( I n h + I d r + I d a ) ρ g ρ a 1 − ρ g ρ a (39)

where ρ a is the albedo of the cloudless sky, which can be computed with:

ρ a = 0.0685 + ( 1 − F c ) ( 1 − τ a τ a a ) (40)

The ( 1 − F c ) term corresponds to the back-scatterance. The second term on the right hand side of Equation (40) represents the albedo of cloudless skies due to the presence of aerosols, whereas the first term is the albedo of clean air.

The global solar irradiance ( I t ) on a horizontal surface is then expressed by:

I t = I n h + I d = ( I n sin ( h ) + I d r + I d a ) 1 1 − ρ g ρ a (41)

We will fit the recorded global solar irradiance ( I t r ) of clear days with Iqbal C model given by Equation (41). The method consists to solve a nonlinear fitting problem in the least-squares sense i.e. we look for the x-vector coefficients ( β , w 0 , F c ) that minimize the following residual function:

‖ I q b a l ( x ) − I t r ‖ 2 = ∑ i ( I q b a l ( x i ) − I t r ) 2 (42)

where I q b a l ( x ) = I t ( β , w 0 , F c ) is the Iqbal C model. Figure 5 plots a recorded global solar irradiance component of a clear day superposed to its fit by Iqbal C model.

We will apply this process to all global solar irradiance of clear days of the recorded data. The clear days are determined using the novel method developed by [37]. Each fit will give us a value of the aerosol single scattering albedo w 0 and a value of the forward scatterance F c . The monthly and the yearly mean values of these two parameters are shown in Figure 6 and Figure 7 respectively. The annual mean values vary between 0.80 ± 0.04 and 0.81 ± 0.04 for the aerosol single scattering albedo w 0 and between 0.82 ± 0.04 and 0.85 ± 0.04 for the forward scatterance F c . We note that the two parameters vary in opposite of phase with each other with particular values during 2007.