That part of the solar radiation at the Earth's surface which is the direct solar beam, of necessity consists of a parallel flux of radiation at an angle to the vertical equal to the solar zenith angle. Radiance in the direction of the solar disc is very high and falls off sharply just beyond the edge of the disc.
The angular distribution of skylight from a clear sky is complex; it is shown diagrammatically in Fig. 2.4. Because scattering by dust particles, and to a lesser extent by air molecules, is strong in a forward direction, skylight is particularly intense at angles near to the angle of the Sun.928 Radiance from the sky decreases markedly with increasing angular distance from the Sun; it reaches its minimum in that region of the celestial hemisphere approximately 90 ° from the Sun and then rises again towards the horizon. On the basis of more than 3000 scans of clear skies above Calgary, Canada, Harrison and Coombes (1988) arrived at an empirical relationship for the angular distribution of clear sky radiance: this somewhat complex relationship gives the normalized sky radiance, NV(C), as a function of direction
N(C) — (1.63 + 53.7 e"5:94c + 2.04 cos2 C cos 6*)
where 0* is the solar zenith angle, 0 is the zenith angle of any given sky radiance direction, and C is the scattering angle, i.e. the angle between the parallel solar beam (at zenith angle 0*, and azimuth angle f — 0 °) and the sky radiance direction (at 0, f). C is given by cos C — cos 6 cos 6* + sin 6 sin 6* cos f (2.2)
Radiance is the radiant flux in a given direction per unit solid angle per unit area at right angles to the direction of propagation, and has the units watts (or quanta s-1) m-2 steradian-1. To obtain the normalized sky radiance, the actual measured radiance in a given direction, N(0, f), is divided by Eds, the diffuse sky irradiance on a horizontal surface, and then rendered dimensionless by multiplying by p steradians.
Thus, when we want a specific sky radiance in a certain waveband, N(0, f, l), then using an appropriate value however obtained (see later) for the diffuse irradiance in that waveband on a horizontal surface, Eds(l), we rearrange eqn 2.3
p and with the help of eqns 2.1 and 2.2 to provide NV(C), we obtain the desired value.
Contrary to what casual observation might suggest, the radiance distribution beneath a heavily overcast sky is not uniform. In fact, the radiance at the zenith is two to two-and-a-half times that at the horizon. To provide an approximation to the radiance distribution under these conditions, a Standard Overcast Sky has been defined929 under which the radiance as a function of zenith angle, 0, is given by
where B is in the range 1.1 to 1.4. The radiance under a sky with broken cloud must necessarily vary with angle in a highly discontinuous manner, and no useful general description can be given.
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