p/ = 2/31 p, / 0 *° SKYL(0 sun ) + R H (© s „„)(l-SKYL(0 sun )) d0 su „ (16) 
where 0 SU „ is the solar elevation angle (radians). If we use a mean value of SKYL, SKYL' (unitless): 
SKYL 7 = 2/Jt S? 1 SKYL(0 sun ) d© sun (17) 
we can firstly approximate components of p/ by p 4 where: 
p 4 (0 sun ) = P 2 (SKYL 7 + (l-SKYL 7 ) R H (© sun )) 
If we use the mean of R H , R H 7 : 
R H 7 = 2/71 1? 2 R„(0 S J d0 sun 
(18) 
(19) 
then equation (18) can be approximated by: 
p 5 = p 2 (SKYL 7 + R H 7 ( l-SKYL 7 )) (20) 
which, as a two-parameter model of albedo still allows flexibility over the proportion of diffuse irradiance by 
varying SKYL 7 . 
3 - SIMULATIONS 
3.1. BRDF Data 
A series of experiments are described below, in which various aspects of the equations presented above are 
investigated, along with the implications of any assumptions made. The experiments are performed by simulating 
BRDF data using the model developed by Ahmad and Deering (1992). Values for the model parameters have 
been taken from the same paper for a variety of land cover types, including Konza prairie (June), Konza prairie 
(August), bare soil, desert scrub and an alkali flat. The model is composed of six structural and radiometric 
parameters, namely the single-scattering albedo, two parameters describing the scattering phase function, two 
parameters describing the hot spot, and an optical depth parameter. The values of these parameters were obtained 
by Ahmad and Deering (ibid) by inverting the model against a large dataset of field radiometric measurements 
made using the PARABOLA instrument (Deering and Leone, 1986). The fact that the RMS error on the model 
fits ranged from 0.002 to 0.02, and that the data were acquired over a wide range of solar zenith angles, gives 
us a high level of confidence in using the model parameters to simulate the BRDF over the required range of 
viewing and illumination angles. However, one potential problem with the data set is that a rather ad hoc method 
was used to account for the effects of diffuse irradiance in the inversions, in which it was assumed that the 
exitant radiance could be given by: 
L c (f2,£2 sun ) = E i (Q 5un )[f(Q,Q s J/7i + A + B/(I£LNI + li^NI)] (21) 
where A and B are constant over the day. As will be seen later, this method appears to over-estimate the effects 
of diffuse irradiance on the exitant radiance in most instances, when compared to radiance values computed 
directly from f(Q,i2 sun ) and a model of diffuse sky radiance and direct irradiance. The most probable reason for 
this is that the diffuse reflectance model presented above compensates for some of the inadequacies of the BRDF 
model. However, for the purposes of the simulations performed here, the BRDF, f(Q,£2 sun ), is used directly to 
represent surface reflectance. 
3.2. Sky Radiance 
The sky radiance distribution used in this study is provided by the analytical model described by Zibordi and 
Voss (1988), which also provides the direct irradiance component. The model is driven by parameters describing 
the position of the sun and the composition of the atmosphere. Although the model is formulated for a plane- 
parallel atmosphere, it has been adapted to incorporate atmospheric path length corrections for a spherical 
atmosphere according to the model of Kasten (1966). This permits more accurate simulations at low solar 
elevations. 
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