The soil bidirectional reflectance can be expressed as (Rahman et al., 1993):

rs = r0 M Fhg H (3)

where r0 is an arbitrary parameter characterizing the intensity of the surface cover. The function H is characterised by a large reflection in the illuminating direction. M and FHG are the symmetric and asymmetric angular functions. The variation of this reflectance around the albedo can be greater than 0.1. In the specific case of oceans, the reflectance varies from zero to values greater than cloud reflectance, it also depends on wind speed (Lefevre et al. 2007). Thus, by considering the albedo instead of the bidirectional reflectance, one commits a significant error on irradiance reflected by the ground then backscattered by the atmosphere, thus contributing to the diffuse fraction of the SSI. This omission is very often made for operational reasons because of the lack of data describing the ground.

Very often, for operational reasons, the irradiance calculations are done under the assumption of a flat terrain inside the pixel. However, the cosine of the local incident angle 9 is:

cos9(P, a) = (cosracosScos^i + sinSsin^^cosP

+ cosracosSsin^icosasinP + sinracosSsinasinP — sinScos^icosasinP (4)

where a is the hour angle, S is the solar declination, ф is the latitude of the site, a and /3 correspond to the direction of the local slope, respectively in azimuth and tilt. Thus, the SSI of the pixel should be modified by the ratio:

R = Hpixelcos9(a(p),P(p))dp / cos9(0,0) (5)

2. Conclusion

We have quantified the influence of the different atmospheric properties on the SSI. Clouds (cloud optical thickness and type) are the most important variable for the SSI. They also exhibit high temporal (~ 30 m) and spatial (~ 10 min) variation (Rossow and Schiffer 1999). Aerosol loading and type, water vapour amount and atmospheric profile have a great influence on SSI, particularly in clear skies. Wald and Baleynaud (1999) demonstrate noticeable variation in atmospheric transmittance due to local pollution in cities at scale of 100 m. Ground albedo and its spectral variation have an important influence on diffuse part and spectral distribution of SSI. They exhibit very high spatial variation on a pixel basis and seasonal temporal variations. The influence of ozone amount is large in the ultraviolet range, but remains low on the integrated wavelength SSI.

Data availability of water vapour and aerosol loading is fairly low. We are expecting one value per day for a 50 km pixel. The error induced on the SSI depends on the variation of these parameters within the pixel. The clearer the sky, the greater the error. Comparisons of ground measurements of hourly means of SSI made at sites in Europe less than 50 km apart for clear skies show that the spatial variation in SSI, expressed as the relative root mean square difference, can be greater than 10 %.This result is in agreement with Perez et al. (1997) who stressed the large spatial variability of the SSI.

The influences of vertical position and geometrical thickness of clouds in the atmosphere are negligible. Thus, the solution of the RTM for a cloudy atmosphere is equivalent to the product of the irradiance obtained under a clear sky and the extinction coefficient due to the cloud. In particular, it means that the method Heliosat-4 may be composed of two distinct parts: the clear-sky part and the cloudy-sky part. In addition, given the fact that the cloud parameters may be known every % h and 3 km and the clear-sky parameters every day and 50 km, the adoption of the concept of "a model for the clear sky, another for other types of skies" saves time: the first model, which takes into account all other atmospheric parameters, focuses the bulk of computation resources.

The results of this work form the basis for the establishment of the method, called Heliosat-4, based on the exploitation of a RTM for the operational processing of satellite data to produce assessments of solar surface irradiance every 3 km and % h on Europe and Africa. The necessary inputs to Heliosat-4 have been identified; a gross assessment of the relative importance of their uncertainties on the final assessment has been obtained.

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