yields the 
! approxi- 
(3) 
measure- 
albedo of 
JF values 
nels with 
: exist for 
intensity 
th angles 
pectralon 
changing 
1e BRDF 
(4) 
t of the 
0; v, 0,). 
figures 3 
t and the 
| panel is 
DF’s of 
are nec- 
e Factors 
S: 
sun and 
ght only. 
shadow. 
rlobal ir- 
epending 
yts yields 
on panel 
, Thein- 
alculated 
mined In 
(5) 
  
  
v=45° 
Figure 2: Legend of the 3-dimensional plots 
v = |; — ör| mod 180° (planar symmetry is 
assumed in fig. 3-10), 0 = 0, (angles greater 
than 75° are not shown) 
  
Figure 3: BRDF of the Spectralon panel at 
6; = 25° and À = 850 nm 
3. L, of the sample illuminated by the sun and skylight 
4. L, of the sample illuminated by skylight only 
Here again, the difference between the last two mea- 
surements yields the radiance AL, reflected by the 
sample which can be attributed to direct sunlight only. 
Assuming, that the incoming irradiance E;“™ does not 
change between the two measurements, the value of 
the BRDF of the sample can then be obtained from 
sample Li AL, (0i, V, 07) 
f£ (0i, v,0,) —  EPRT(Ó) (6) 
Of course, the incoming irradiance E;**™ did change 
between the two measurements. The difference (5.4% 
on average) was included into the calculation of the 
error. 
The measurements were performed on the roof of a 20-storied 
building in the month of april at clear skies. 
Aside from azimuthal symmetry, the samples should meet the 
following requirements, if the BRDF of the sample is to be 
measured with the above procedure: 
* The BRDF of the sample must not change between 
the measurements. |n our experiment, several days 
passed between the measurements, this excludes the 
determination of the BRDF of vegetation. 
8,275? 
8,775? 
  
Figure 4: BRDF of the Spectralon panel at 
0; = 75° and. A = 850:nm 
e |f the diffuse component is very weak, the intensity of 
light emitted into a nonspecular direction differs hardly 
between the measurements 3 and 4 described above. 
Thus the difference between these measurements will 
be small and the relative error of AL, will increase 
strongly. E.g., we did not succeed in fitting a BRDF- 
model to the data of a sample of aluminum and a sam- 
ple of black roof paper, both of them have a negligible 
diffuse component. 
3 BRDF-MODEL FITTING 
Several BRDF-models (e.g. polynomials, spherical har- 
monics, and functions proposed by [Oren, 1995] and 
[Minnaert, 1941]) were fitted to the measured data points, 
using a least-squares-algorithm. The compatibility of the 
measured data and the fitted BRDF-models was judged by 
means of a x?-test. The best results were obtained with a 
model proposed by [Walthall, 1985]. This model was modi- 
fied by [Liang, 1994] to be compatible with Helmholtz's the- 
orem of reciprocity. We extended it by an additional Gaussian 
peak accounting for specular reflection: 
fr = a0 +1: (07 +07) as - (82-67) 
+a3 » (0: - 0) - cosy + as : gas (009: . e cil (7) 
where the free parameters ao to aa describe Walthall's model, 
and the last 3 parameters are our extension for specular re- 
flection with r being the angle between the reflected light (di- 
rection ,, v) and the specular angle (0;, v = 180°) (direction 
of the incident radiation: 0;,v — 0?). The term eas (00:07) 
models the increasing intensity of the specular peak with in- 
creasing zenith angles. For 0; and 6, fixed, the term e ^9'* 
produces a Gaussian peak. 
So the total number of free parameters of this model is 7, 
and thus well over-determined by the ca. 35 measured data 
points per sample and wavelength. 
[Liang, 1994] also modeled the 'hot spot' (a peak reflected 
back into the direction of the light source). We did not in- 
clude the 'hot spot' term, because our experimental setup 
prevented to measure at angles where the 'hot spot' occurs. 
495 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B7. Vienna 1996