Atmospheric Parameterization for Clear Sky

Fundaments of Atmospheric Parameterization

The depletion processes include scattering by air molecular (Rayleigh Scattering), scatter-ing and absorption by aerosols, absorption by ozone layer, absorption by uniformly mixed gases, and absorption by water vapor (Iqbal, 1983). The total atmospheric attenuation ef-fect is the result of interaction between atmosphere conditions and solar-earth revolution, because the relative air optical mass changes with the solar ray direction. Different sub-stances scatter or/and absorb solar radiation at certain spectrum range, and the attenuation processes are substance and spectrum independent, therefore can be treated individually as monochromatic attenuation by each substance and then superimposed together. Both monochromatic and the spectrally integrated attenuation can be described by Lambert’s law with 3 mutually convertible parameters: the attenuation coefficientκ_i, the transmittanceτ_i, and the optical depth, also called optical thicknessδ_i:

I˙_i = ˙I_0ie^{−κ si} (2.6)

τ_i =e^{−κ si} (2.7)

δ_i =κ_is_i (2.8)

Hereidenotes a given atmospheric substance, which may or may not exist in throughout the whole atmospheric layer, s_i is the optical path length of a given substance i, I˙_0i and I˙_i is the intensity of spectral radiation before and after passing through the atmosphere.

The attenuation coefficient is an obsolete term, and is nowadays rarely used. Because the processes are independent, fornsubstances, their effects can be superimposed (see Eq.2.10).

For calculating direct radiation, it is also not necessary to differentiate between absorption and scattering, because both effects are the same.

τ_e=τ_rτ_aτ_o τ_g τ_w =

i=n

i=1

τ_i (2.9)

δe=

i=n

i=1

δi (2.10)

with τe : effective transmittance of direct radiation by all atmospheric substances [−]

τ_r : transmittance of Rayleigh (molecule) scattering [−]

τa : transmittance of aerosol scattering and absorption [−]

τ_o : transmittance of ozone absorption [−]

τ_g : transmittance of uniformly mixed gas [−]

τw : transmittance of water vapor [−]

τ_i : spectrally integrated transmittance of a given substances [−]

δ_e : effective optical depth of all substances [−]

δi : spectrally integrated optical depth of a given substances [−]

For diffuse radiation, absorption and scattering have to be treated separately, because the forward scattered radiation will continue traveling to the ground, whereas absorbed radi-ation terminates. Molecule and aerosol have also different scattering effects, namely the molecule scattering is symmetrical in forward and backward direction, while the aerosol has a strong forward scattering. The forward scattered radiation reaching the ground can be expressed with the assistance the atmospheric albedoρ_at, which denotes the diffuse compo-nent reflected back to space:

ρat =τo τg τw(0.5(1−τr)τa+ (1−f_af)fas(1−τa)τr) (2.11) T_d= 1−ρat=τoτg τw (0.5(1−τr)τa+f_affas(1−τa)τr) (2.12) Heref_af denotes the portion of forward scattered radiation. fasis scattered fraction out of the total energy attenuated by aerosol.T_dis the diffuse transmission coefficient.

Regression analysis and inversion methods using radiative transfer model (RTM) are used to derive the attenuation parameters, based on measurable meteorological data, such as tem-perature, solar irradiance, humidity, etc. To be mentioned, these data can be also obtained from satellite observations, i.e. the Heliosat-3 model (Mueller et al., 2004). The objective of RTM is to find out the individual optical depth, the size of the extinction layer, or the equivalent sum.

Besides the aforementioned parameterization for clear-sky direct and diffuse radiation mod-eling, there are many parameterization methods in literature, a good overview of the most popular ones can be found in Ineichen (2006). Although the parameterization methods are so many that it is even confusing to decide for a proper one, they can be generalized into two groups based on whether individual or integrated parameters are used in the radiation model. The individual parameterization applies several attenuation parameters of each ex-tinction layer (substances) together, which is represented by the Page model (Page, 1997), SOLIS model (Mueller et al., 2004), Bird and Hulstrom model (Bird and Huldstrom, 1980), CPCR2 model (Gueymard, 1993). The second group summarize the different processes into Rayleigh optical depthδ_r and the Linke turbidityT_Ln, by which the transmittance of each extinction substance are converted and summed up with reference to the spectrally inte-grated Rayleigh optical depth of the clean and dry atmosphere (Kasten, 1996). It is used by the European Solar Radiation Atlas (ESRA) model (Rigollier et al., 2000) and the Ineichen model (Ineichen and Perez, 2002). Linke Turbidity can be derived by measurements of the beam irradiance using appropriate but expensive equipment. The typical Linke Turbidity values are widely reported in literature for different parts of the world.

Based on the parameterization schemes, the clear sky radiation on horizontal and inclined surface can be expressed as following respectively. Note, the subscriptshandβare denoting horizontalandinclinedsurface respectively, andcandbare used forclearandcloudy(bew ¨olkt in German) sky conditions.

Clear Sky Model on Horizontal Surface

• Direct radiation:

for integrated parameterization:

B_hc= ˙I_SC cosθ_zexp(−0.8662T_Ln m δ_r(m)) (2.13) or for individual parameterization:

B_hc= ˙I_SC cosθ_zτ_rτ_aτ_o τ_g τ_w (2.14)

• Diffuse radiation:

for integrated parameterization:

Dhc = ˙ISC Td(TLn)Fd(θz, TLn) (2.15) or for individual parameterization:

D_hc = ˙ISC cosθz T_d(τi, fas, f_af) (2.16) Hereθzis the solar zenith angle of horizontal surface,is the eccentricity correction factor of solar-earth distance. The diffuse transmission coefficient, for the integrated parameteriza-tion is a funcparameteriza-tion of Linke TurbidityT_d(T_Ln), and for individual parameterization it is given by Eq.2.12. The integrated form also includes a diffuse angular functionF_drelated to zenith angle and Linke Turbidity.

Clear Sky Model on Inclined Surface

For the inclined surface under clear sky, the direct radiation will only be modified by the solar incidence angle (see. Eq.2.17), and the diffuse radiation will be reduced in addition by the reduced skyview fraction ν. Both direct and diffuse radiation on inclined surface can be expressed based on the their counterparts on horizontal surface. For direct radiation on inclined surface, expressed asB_βc, only the incidence angle changed to θinstead ofθz. Diffuse radiation on inclined surface, denotedD_βc, is usually expressed as a portion of the diffuse radiation on horizontal surface at the same locationD_hc:

B_βc =B_hccosθ /cosθ_z (2.17)

D_βc =D_hcψ_β (2.18)

The formulation of coefficient ψ_β depends on how diffuse radiation is treated. The basic component of diffuse radiation is the isotropic part from all direction of the sky, sometimes called dome or sky irradiation. In addition, two anisotropic components may be consid-ered - circumsolar brightening caused by the strong forward aerosol scattering from ap-proximately 5° around the direct solar beam, and the horizon brightening due primarily to

multiple Rayleigh scattering and retro-scattering in clear atmospheres. The diffuse radiation model may consider only the isotropic component as by Liu and Jordan (1961), or in addition the circumsolar component, for example by Hay (1979), or eventually all three components, i.e. by Perez et al. (1990) and Muneer (1990). There are an array of diffuse radiation models which will not be repeated here. These models are applicable for both clear-sky and overcast conditions. For a complete overview of them, the author suggests the article of Evseev and Kudish (2009). Here the Liu-Jordan model and the Muneer model, which will be used in the next section are given briefly.

1. Liu-Jordan model (1961)

ψ_β =cos²(β/2) = (1 +cosβ)/2 (2.19) 2. Muneer model (1990)

ψ_β =T_M(1−F_M) +F_M r_b (2.20)

T_M =cos²(β/2) +U

sinβ−β cosβ−π sin²(β/2)

(2.21) U = 0.00263−0.712F_m−0.688F_m² (2.22)

r_b =max[0,(cosθ/cosθ_z)] (2.23)

F_m =K_hay for sunlit surface and non-overcast sky (2.24)

F_m = 0 for surfaces in shadow (2.25)

K_hay =B_hc/H_0h (2.26)

Here,β is the slope of the incline surface; K_hay is the Hay’s sky-clarity index (Hay, 1979);

denoting the proportion of beam irradiance and extraterrestrial solar irradiance on horizon-tal surface;TM is the Muneer’s tilt factor;Fmis a composite clearness function;r_bandU are some auxiliary variables.

In complex terrain, the topography will exert a major influence on the spatial solar radiation, especially under clear-sky condition where the direct radiation dominates. In Section 2.4 two GIS-based clear-sky radiation models -r.sunand Solar Analyst, will be compared to check the resolution effect and the shading effect under clear sky conditions.