A Model of Diffuse Broadband Solar Irradiance for a Cloudless Sky

Warwick Grace   of  Grace Research Network

11 April 2006




An analytic model is presented for the broadband diffuse irradiance received at the earth’s surface through a cloudless homogeneous atmosphere containing  light absorbing agents and isotropic scattering agents. The model is extended to include a contribution from light re-scattered after  ground reflection.





where  S  is the diffuse irradiance,

the constant k represents the concentration of intercepting agents:

ka represents absorbers and ks the isotropic (multiple) scatterers so  k = ka + ks .

a scattering ratio, D, is defined as  ks / k. 

Q represents the top of atmosphere irradiance (being the solar constant, Q ~1367 W/m2 ) and

n  the zenith angle of the sun and set the vertical coordinate as y,  with y = 0 at the ground and y = H at the top of the atmosphere

β . 1.66 and

the albedo is A.

Sometimes the term Transmittance, Tz,  is used instead of kH. The relationship is simply  exp(-kH) = Tz.



How well does it work? – the figure below shows that it works quite well.




Figure   Half-hourly exposures of observed and modelled diffuse  irradiance

 in MJ/m2 (on a horizontal surface) at Adelaide Airport on 17 Oct 2004.

Observed denoted by diamonds, model estimates shown by solid line.


If zenith transmittance Tz  which is defined as exp(-kH)  is assumed known (say ~ 0.75 being a typical value), then rough estimates of the proportion of scattering agents to intercepting agents (say D ~ 0.5) and Albedo (say A ~ 0.2) provide the input parameters for the analytic model (as expressed by Equation 14). The model then provides simple expression to estimate clear sky diffuse irradiance for any desired zenith angle, n.  In this case, the formula reduces to




In summary, the model provides a simple analytic theory and expression to explain or to estimate broadband diffuse irradiance.


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