708 
A wide range of measurement campaigns have shown that, for many land cover types, albedo varies as a function 
of solar elevation (e.g. Ranson et al. (1991), Eck and Deering (1990)). Despite this, many models using albedo 
data to estimate shortwave energy interactions at the earth surface assume that it is invariant with sun angle. Pan 
of the reason for this is because the modellers have previously had no reliable estimates of the spatial, diurnal, 
and longer-term temporal variations in albedo. However, current and especially future remote sensing instruments 
offer the potential to supply such data. Consequently, it is important to consider the sort of data products and 
the levels of accuracy that are required. One of the issues that must therefore be investigated is whether or not 
it is sufficient to use a constant value of albedo. Other related issues are: how should we actually calculate such 
a constant; and, if a constant value of albedo is insufficiently accurate for some applications, how should we 
specify the dependence of albedo on sun angle. 
Retrieval of surface albedo from remote sensing data is dependent on knowledge of the atmospheric optical 
properties, since this conditions both the angular and the spectral distribution of the solar radiation incident on 
the target, as well as the path radiance from the surface to the sensor. It is also apparent that the nature of the 
irradiance field directly affects the surface albedo, as demonstrated in the measurements of surface albedo over 
a forest canopy under ’clear sky’ and overcast conditions (Eck and Deering, 1990). When modelling the effects 
of the sky radiance on surface BRDF, it is common to assume that it is isotropic, i.e. does not vary over the sky 
(e.g. Pinker and Laszlo (1992)) and can be stated as a proportion of the total irradiance. A further simplification 
about the nature of the irradiance field that might then be made is that the ratio of direct-to-diffuse radiance 
remains constant with changes in solar elevation, as this reduces the number of parameters we have to consider 
in modelling BRDF and albedo. 
In this paper, we explore the implications of these assumptions through a series of numerical simulations based 
on separate models of scattering within the atmospheric and at the earth surface — Zibordi and Voss (1989) and 
Ahmad and Deering (1992), respectively. The aim here is to examine the errors in estimates of surface albedo 
that result from assumptions about atmospheric conditions and to study how albedo varies with Sun angle. 
Simulations are performed over a wide range of solar zenith angles and for several different land cover types. 
These are preceded by a section which considers the mathematical derivation of albedo and related measures. 
2 - THEORY 
We start, then, with the assumption that remote sensing can provide adequate samples of TOA hemispherical- 
conical radiance to allow us to derive a surface BRDF, f x (£2,£20 (in sr' 1 ), at an appropriate spatial resolution. 
For the purpose of this study, the BRDF is defined as: 
where X is wavelength. £2 is the viewing vector, £2' is the (directional) illumination vector, dL Xe is the 
irradiance (Wm' : pm' 1 ) (Figure 1). Although the BRDF varies as a function of the wavelength of observation, 
the wavelength subscript, X, is ignored in the remaining equations and all units are expressed in wavelength 
independent terms (e.g. Wm' 2 instead of Wm 2 pm' ; ). It should be noted at this stage that although the BRDF is 
an intrinsic property of the reflecting surface (Nicodemus et al. 1977), it will vary as a function of spatial 
location, spatial scale and the time of measurement, as well as in response to local environmental conditions, such 
as terrain slope and aspect. 
f x (fi.i2'l = dL Xe (£2.£2 / ) / dE-JD.') 
(i) 
incremental spectral radiance exitant from the surface (Wm 2 sr''pm'’) and dE Xi is the incremental incident spectral 
The hemispherical-directional radiance exitant from a surface with unit normal vector N, L e (WrrAsr 1 ), which 
is a function of the viewing vector and the solar illumination vector, is thus: 
where Li is the incident radiance (Wmrsr 1 ) — a function of both £2 / and the sun position Q. un — and where 
the integral is performed over the (upwards) exitant hemisphere of the surface. This can be reformulated to 
separate the direct (L sun (£2 sun )) and diffuse (L sky (i2 sun ,Q / )) components of the incident radiance: