638 
L = SE/SS; 
(3) 
j j j 
SE = Z (Es pfl ISf,) (1-f) + Ssff + I (Es pbi 1^) (1-f) + f + I (Eap fl la«) (1-i) + S af f+ 
i=i i=i i=i 
j 
I (Eap bi Iatj) (1-f) +S3,, f + EgflgfU-f) + Sgf f + Eg b Ig b (1-f) + S& f 
i=l 
SS = 1%+ Ssf+ Isj, + Sst, +1%+ Saf + Ia^, + Sa,, + Igf + Sgf + Ig b + Sg b 
where f is the ratio between the radiance of a shaded surface and the radiance of the same illuminated surface 
when the sunbeams are perpendicular to the surface; subscripts f and b refer to the (rf) and (q,) radii of the 
basic view area, respectively. 
The model calculates the radiance for the profile going through the center of the spheroids, 
and then for several next profiles parallel to the first one. The total radiance (Lt) for a given view angle is 
computed as a average values from all the profiles and the flat space between the spheroids, where the L for the 
flat plane equals cosGs. 
The reflectance of the simulated surface is finally expressed by the relative reflectance factor 
(FR) defined as the proportion of the total luminance measured from an off-nadir direction (Lt 0 ) to the 
radiance measured from the nadir (Lt,,): 
FR = Lt 0 /Lt n . 
(4) 
2.2. Observed data 
The model was tested using soil bidirectional reflectance data acquired on a bare field of an alluvial plain of 
the Durance river, named La Crau, located 40 km to the south of Avignon, 25 km south-east from Arles, and 
15 km north of the Mediterranean Sea in southern France. The plain is covered by regularly spread pebbles of 
an average diameter of several centimeters. Medium textured soil, partly overgrown by natural vegetation, 
appears between the stones. This area serves as winter pastures for sheep. 
Soil spectral data were measured by a three-channel (SX1: 0.50-0.59 pm, SX2: 0.61-0.68 
and SX3: 0.79-0.89 pm) field radiometer CIMEL simulating the SPOT (HRV) bands. It collected radiance 
data along the solar principal plane in 13 directions at view zenith angles from 60° towards the sun through 
the nadir to 60° away from the sun at 10° of increments. The duration of the complete sequence is about 4 
minutes. The radiometer observed the soil surface from a distance of 2 m. This instrument with a 12° field of 
view (FOV) integrated reflected energy from a circular area of 0.14 m 2 at a 0° view zenith angle to an elliptical 
area of 0.29 m 2 at a 60° view zenith angle. 
Simultaneously to the radiometric data collecting, photographs of the target, viewed by the 
radiometer, were taken. The photographs were used to measure soil surface roughness parameters. They were 
obtained using photographs taken at the 0° view zenith angle for five selected values of the solar zenith angles. 
In the five photographs, contours of all the pebbles lying within the radiometer FOV (a circle on the ground of 
a radius (r n ) equal to 21.02 cm) were drown. Measuring the total area of the contours (Ap) by a computer 
system for image analysis named VISILOG, the pebble area index (AI) was calculated as the average value for 
all the five photographs as follows: 
AI = Ap/( 7t r n 2 ) (5) 
Then, the average relative distance between the pebbles divided by their average radius (d/a) was computed as: 
d/a=V Tt/AI . (6) 
The parameter describing the shape of the pebbles, i.e., the average proportion between their vertical and 
horizontal radii (b/a), was determined after finishing the reflectance data collection. The b and a sizes of each 
of the pebbles lying within the radiometer nadir FOV was measured directly by a ruler.