SKYL(Q su „) = L sky (Q sun ) n / E,(Q sun ) 
(10) 
710 
so that, from equations (8) and (10): 
L C (Q,Q S J = E i (Q sun )(SKYL(ii sun ) p,/n + (l-SKYL(Q sun )) mSfJ) 
OD 
If the bihemispherical reflectance p 2 is defined (unitless): 
p2 = f 2K+ p,(£2)/tt IQ.NI dii 
( 12 ) 
then, from equations (6) and (11): 
p 3 (ft sun ) = SKYL(Q sun ) p 2 + (l-SKYL(Q sun )) p,(ii sun ) 
(13) 
through reciprocity with (7). We can consider p 2 as an alternative formulation of albedo, one that it is invariant 
conditions (i.e. no direct component), which might be considered as the ’overcast sky’ albedo. p 3 is also a useful 
approximation of albedo, which is valid as long as we can show that the directional nature of the diffuse 
irradiance field does not affect the albedo too strongly. We can proceed from equation (13) to define an albedo 
anisotropy factor, R H (Q sun ) (unitless): 
It is apparent from equation (13) that, if an isotropic sky is assumed, the albedo is the sum of two component 
parts: one related to the albedo for a pure diffuse sky (the bihemispherical reflectance); and the other, related to 
the albedo for a directional source, the latter of which may vary according to the position of the sun. The 
parameter SKYLIT,,) will vary according to the position of the sun, even for clear blue skies, as demonstrated 
in Figure 2. Note that the behaviour of the curve in Figure 2 depends on atmospheric scattering constituents, 
which are defined for atmospheres 1 and 2 below. It will also vary as a function of altitude, as demonstrated 
by Teillet and Santer (1991). 
If we are able to assume that the sky is isotropic, equation (14b) provides a flexible definition of albedo which 
can account for varying proportions of diffuse irradiance (and cloudiness). In terms of attempting to obtain a 
convenient parameterization of albedo, we need only to define SKYL(Q sun ), as well as p 2 and R H (F> surl ) for each 
cover type. Thus, in providing a useful dataset from remote sensing observations, it would appear that a spatial 
characterization of p 2 and R H (f2 sun ) would generally be sufficient. 
2.2. Angular-Averaged Definitions of Components of Albedo 
If we need to consider supplying data products which give a single estimate of albedo which will be used at all 
Sun angles, we can consider a number of candidate values, the first of which might be p 2 , the ’overcast sky’ 
albedo. We can expect this to be an accurate estimate given the conditions discussed above, but may wish to 
consider alternative definitions more appropriate to clear sky conditions. One way to do this would be to provide 
a diurnally-averaged measure of albedo p 7 through: 
where t is time. This can be simplified by assuming an isotropic sky, but an important consideration with a 
with sun angle. In effect, p 2 is the equivalent of the albedo of the surface under purely isotropic irradiance 
R H (^un) = P,(O sun )/p 2 
(14a) 
so that 
pf^J = P 2 ( SKYL(Q sun ) + (l-SKYL(Q sun )) R„(O sun )) 
(14b) 
(15) 
definition such as this is that since the range of solar elevation angles viewed from any location on the Earth 
varies strongly as a function of latitude and time of year, we would have different values of p' for the same 
land cover type as a function of these variables. It is not immediately obvious what the magnitude of the 
variation in p' might be, but the extra computation required might be prohibitive in an operational context. In 
place of this, we can consider computing angular-averaged albedo values: